Millimeter-Wave Radar Targetsand Clutter

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Millimeter-Wave Radar Targetsand Clutter

Gennadiy P. Kulemin

Technical EditorDavid K. Barton

Artech HouseBoston • London

www.artechhouse.com

Library of Congress Cataloging-in-Publication DataKulemin, G. P. (Gennadii Petrovich)

Millimeter-wave radar targets and clutter / Gennadiy P. Kulemin.p. cm. — (Artech House radar library)

Includes bibliographical references and index.ISBN 1-58053-540-2 (alk. paper)1. Radar targets. 2. Radar—Interference. 3. Millimeter waves. I. Title.II. Series.

TK6580.K68 2003621.3848—dc22 2003060064

British Library Cataloguing in Publication DataKulemin, Gennadiy P.

Millimeter-wave radar targets and clutter. — (Artech House radar library)1. Radar—Interference 2. Backscattering 3. Radar targets 4. Millimeter wavedevices I. Title621.3’848

ISBN 1-58053-540-2

Cover design by Yekaterina Ratner

2003 ARTECH HOUSE, INC.685 Canton StreetNorwood, MA 02062

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International Standard Book Number: 1-58053-540-2Library of Congress Catalog Card Number: 2003060064

10 9 8 7 6 5 4 3 2 1

Contents

Preface ix

Acknowledgments xi

CHAPTER 1Radar Characteristics of Targets 11.1 Introduction 11.2 Target RCS 3

1.2.1 RCS Models 31.2.2 RCSs of Real Targets 7

1.3 Radar Reflections from Explosions and Gas Wakes of OperatingEngines 181.3.1 Analysis of Radar Reflection Mechanisms 181.3.2 Spatial-Temporal Characteristics of Explosion and Fuel

Combustion 231.3.3 Radar Reflections from Explosion and Gas Wake 281.3.4 Centimeter Wave and MMW Attenuation in Explosions 341.3.5 Radar Backscattering from Sonic Perturbations Caused by

Aerodynamic Object Flight 411.4 Statistical Characteristics of Targets 55

1.4.1 Target Statistical Models 551.4.2 Real Target Statistical Characteristics 581.4.3 Echo Power Spectra 62

1.5 Surface Influence on the Statistical Characteristics of Radar Targets 721.5.1 Diffuse Scattering Surface Influence on the Statistical

Characteristics 721.5.2 Multiple Surface Reflection Influence 78References 84

CHAPTER 2Land Backscattering 892.1 Classification and Physical Characteristics of Land 89

v

vi Contents

2.2 State of the Theory 952.2.1 RCS Models 952.2.2 Power Spectrum Model 101

2.3 Normalized RCS 1082.3.1 Normalized RCS of a Quasi-Smooth Surface 1082.3.2 Normalized RCS for Rough Surfaces Without Vegetation 1092.3.3 Backscattering from Snow 1142.3.4 Backscattering from Vegetation 1182.3.5 Normalized RCS Models 120

2.4 Depolarization of Scattered Signals 1232.5 Statistical Characteristics of the Scattered Signals 1262.6 Power Spectra of Scattered Signals 128

References 133

CHAPTER 3Estimation of Land Parameters by Multichannel Radar Methods 137

3.1 Estimation of Soil Parameters 1373.1.1 Introduction 1373.1.2 Soil Backscattering Modeling 1383.1.3 Efficiency of Multichannel Methods 145

3.2 Soil Erosion Experimental Determination 1503.2.1 Set and Technique of Measurement 1503.2.2 Statistical and Agrophysical Characteristics of Fields 1513.2.3 On-Land Radar Measurement Results 1553.2.4 Aircraft Remote Sensing 157

3.3 Methods of Multichannel Radar Image Processing 1593.3.1 Image Superimposing 1593.3.2 Methods of Multichannel Radar Image Filtering 163

3.4 Soil Erosion Determination from Ratio Images: Experimental Results 166References 168

CHAPTER 4Sea Backscattering at Low Grazing Angles 171

4.1 Sea Roughness Features for Small Grazing Angles 1714.1.1 Sea Roughness Characteristics 1714.1.2 Shadowing and Peaks in Heavy Sea 184

4.2 Sea Backscattering Models 1894.3 Sea Normalized RCS 1934.4 Depolarization of Scattered Signals 1974.5 Sea Clutter RCS Model 2024.6 Sea Clutter Statistics 206

Contents vii

4.7 Radar Spike Characteristics of Sea Backscattering 2094.8 Backscattering Spectra 213

References 222

CHAPTER 5Microwave and MMW Backscattering by Precipitation and OtherMeteorological Formations 227

5.1 Structure of Meteorological Formations 2275.2 Atmospheric Attenuation 2335.3 Backscattering Theory 2365.4 Experimental Results Review 239

5.4.1 Precipitation Backscattering 2395.4.2 Cloud Backscattering 242

5.5 The Statistical Characteristics of Scattered Signals 2425.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo) 250

5.6.1 Point Reflections 2505.6.2 Backscattering from the Turbulent Atmosphere 254References 256

CHAPTER 6Sea and Land Radar Clutter Modeling 259

6.1 Land Clutter Modeling 2596.1.1 Initial Data 2596.1.2 Peculiarities of Land Clutter Simulation 262

6.2 Sea Clutter Modeling 2676.2.1 Peculiarities of Sea Clutter Simulation 2676.2.2 Algorithm of Sea Clutter Simulation 268

6.3 Clutter Map Development 2766.3.1 Initial Data for Modeling 2766.3.2 Software Input and Processing Components 2776.3.3 Raster Image Processing Module 2786.3.4 Automatic Highlighting of Contours on the Raster 2806.3.5 Steady Algorithm of Surface Recovery from Contours 2826.3.6 Simulation of the Absolute Reflectivity 283References 284

CHAPTER 7Clutter Rejection in MMW Radar 287

7.1 Influence of Propagation Effects on MMW Radar Operation 2877.1.1 Introduction 2877.1.2 Multipath Attenuation 288

viii Contents

7.2 Influence of Rain and Multipath Attenuation on Radar Range 2907.3 Influence of Land and Rain Clutter on Radar Detection Range 2927.4 Land and Rain Clutter Rejection in Millimeter Band Radar 297

7.4.1 General Notes 2977.4.2 Land and Sea Clutter Rejection 2987.4.3 Rain Clutter Rejection 305References 311

About the Author 313

Index 315

Preface

For the last 40 to 50 years, the intensive development of the millimeter band ofradiowaves has taken place to address communication, radar, remote sensing, andmany other problems. The interest in this band is due to a number of millimeterwave (MMW) advantages in comparison to longer wave bands.

With this interest comes the possibility of developing super wide bandpasscommunication paths between on-land points. The possible development of narrowbeam formations for acceptable antenna sizes would enable better tracking, detec-tion, and surveillance in modern radar. In addition, the reserve and stability tocountermeasures would be higher.

The successful solving of problems for low-altitude, on-land, and maritimetarget detection and tracking has determined the propagation effects of MMWsnear land and sea surfaces and in the troposphere. Among them, we can note themultipath propagation attenuation and the attenuation in precipitation (e.g., rain,fog, and snow) limiting the maximal range of detection. The small influence ofmultipath attenuation in comparison with radar of the centimeter or longer waveband is the essential advantage of MMW radar. The precipitation influence doesnot show itself in microwaves, and it is necessary to take this limiting factor intoconsideration in the millimeter band at ranges of more than few kilometers.

The second problem limiting the application of MMW radar is the land andsea clutter conditioned by backscattering from distributed scatterers and the volumeclutter from such scatterers as precipitations; the latter role increases in the milli-meter band and results in limitations in radar frequency.

The investigations in propagation of MMWs have been carried out in theInstitute for Radiophysics and Electronics of the National Academy of Science ofthe Ukraine for more than 50 years, and great experimental data were collectedduring this time. Part of these results obtained by the author or with his participationhas been included in works presented to the reader. Millimeter Wave Scattering byEarth’s Surface at Small Grazing Angles by G. P. Kulemin and V. B. Razskazovskypublished in 1987 in Russian was the first monograph in the former Soviet Unionin which the problems of forward scattering and backscattering of MMWs by landand sea surfaces were discussed and the theoretical and experimental results were

ix

x Preface

presented, including the statistical characteristics of arrival angles due to multipathpropagation over the surface. This book remained unknown for a wide circle ofreaders. Since then, many new results have been obtained, and no other new bookswere published.

Acknowledgments

The preparation and publishing of this work was made possible by the enthusiasticsupport of David K. Barton. He made significant technical and scientific contribu-tions, as well as providing valuable editorial suggestions.

It has been a pleasure to work with personnel of Artech House Publishers.Special thanks are due to Tiina Ruonamaa for support and attention.

xi

C H A P T E R 1

Radar Characteristics of Targets

1.1 Introduction

The knowledge of radar statistical characteristics for targets to be detected isthe usual starting point for radar system designers. The approach to statisticalcharacterization differs significantly depending upon the radar system functions tobe investigated. For instance, the tasks of target recognition, target identificationin noise and clutter, and simple target detection require different amounts ofinformation on the target scattering properties. In the first case, the signal mustbe presented as a multidimensional random vector in sine space; in the last situation,it is enough to know the average signal power or energy. The knowledge of thelatter characteristic is the starting point for any radar system design and analysisof predetermining feasibility and nature of all further radar signal-processing tech-niques.

Therefore, the first and the most important target characteristic is the radarcross section (RCS). It is also necessary to know the probability density function(pdf) of RCS fluctuation for given conditions of target observation for derivationof radar energy requirements; the minimally needed characteristic is the averageRCS value, which is contained in the expression for the target echo power.

A sufficient number of theoretical and experimental papers are devoted to theinvestigations of the RCS for different targets, but this material is mainly presentedin the periodical references and requires analysis and generalization. Besides, experi-mental data on target statistical characteristics in the shortwave part of microwaveand MMW bands are limited, and this obstacle stimulated the author’s interest intarget characteristics at these wave bands. Discussion of the results of target model-ing using simple shapes and experimental investigations of radar characteristicsfor cone-cylinder bodies in the resonance area (ka ∼ 1, where k = 2p /l is the wavenumber, l is the signal wavelength, and a is the object diameter) seem to be mostinteresting.

Chemical explosions and gas exhausts of operating engines are rather complexmedia involving some mechanisms for microwave backscattering [1, 2]. Thesemechanisms are competing for different stages of the explosions, and for long-lifereflections the effects of microwave interaction with turbulent media possessing

1

2 Radar Characteristics of Targets

time-variable parameters are of great interest for physics. Moreover, there arepossibilities of applications dealing with radar observation of explosions underbattlefield conditions and with the detection of air targets with small RCS whenbackscattering is observed from gas wakes.

The theoretical analysis of the radio wave scattering by a slightly ionizedturbulent wake was performed in [3], and the radar method of measurements forthe turbulent wake was considered in [4]. While considering the problem of amicrowave reflection from an explosion area, attention is focused on the primarystage when the reflection from a shock wave ionized front (SWIF) is observed, andthe reflection coefficient has the value approximately equal to unity (i.e., it ispossible to consider the reflection from an ideally conducting body) [5].

Chemical explosions are the main object of investigations because the mecha-nism of backscattering at later stages of an explosion and that produced by a gaswake is the same. For this reason, the results concerning radar reflection from anexplosion are significant.

The operation of some types of radar and communication systems must bereliable when chemical explosions occur on propagation paths as found, forinstance, in quarries or operation under battlefield conditions. In such situations,the spatial volume important for wave propagation becomes fully or partiallyblocked by the cloud formed by the explosion products and by accompanyingparticles of soil. This results in wave absorption. Further, the space surroundingthe charge of explosives (where the explosion products—both solid and gas—andparticles of soil are present) will be called simply the explosion volume. The mainfocus of investigators in studies of radio wave propagation through the explosionvolume (as can be seen, for example, from survey [6]) was on the influence of thesand-dust cloud formed by soil particles and the influence of artificial smoke (e.g.,phosphorus, hexachlorine, ethane, and oil fog). At the same time, obviously, thephysical processes of microwave attenuation in the explosion volume of chemicalexplosives when there is no involvement of other substances are also interestingbecause even in this case rather large values of attenuation are observed experimen-tally.

Detection of low-RCS airborne vehicles using the secondary effects observedduring their flight through the atmosphere is also a significant issue [7]. Such effectsmay involve the following phenomena: the forming of an atmosphere shock wavedue to object flight at ultra- and supersonic speeds; the presence of strong soundperturbation resulting in modulation of the atmosphere parameters; and the varia-tion of turbulent troposphere parameters caused by intensive sound fields. Micro-wave scattering from atmosphere inhomogenities, which arise due to sound andshock wave propagation, can increase the total air-target cross section and improveradar efficiency.

The statistical models of radar targets are considered using the commonapproach. It is shown that for derivation of detection characteristics, the standard

1.2 Target RCS 3

Swerling models can be extended to include chi-square distributions with smallnumbers of degrees of freedom. Special attention is paid to the results of experimen-tal investigations of the echo power spectra for different classes of targets.

The statistical characteristics of the echo from low-altitude targets are changedbecause of two circumstances [8]. First, the electromagnetic field from the targetin the presence of multipath propagation is the sum of the direct wave and onereflected by the rough surface (sea or land). As it is known [9, 10], the statisticalcharacteristics of a point nonfluctuating target placed over the surface are describedby the Rician distribution. This is due to the influence of the statistical propagationfactor, which introduces a diffuse component of electromagnetic field for the longpaths with many random scatterers. In the shortwave part of the centimeter bandand, particularly, in the millimeter band, the diffuse component of the electromag-netic field scattered by the surface increases along with the destruction of thespecular reflection. In this situation, the spatial correlation radii of the field diffusecomponents over the target are often greater than the geometric dimensions of thetarget [10]. Then the received signal is a product of two terms: the first describesthe target signal fluctuations in free space, and the second describes the fluctuationsof the propagation factor.

In addition, the statistical characteristics of the target echoes are changedbecause of the signal reflection from the target to the radar via the surface (multiplereflections). This effect, conditioned by the multiple reflections, can be significantfor comparatively small ranges from the target to the surface when it is possibleto neglect the propagation losses. Such interaction of the target and surface wasfirst considered in [11] for a plane plate placed at an angle of 45° to the surfaceand the possibility of the RCS growth was shown.

The analysis results of these mechanisms of target and surface interaction arepresented and their influence on the radar target statistical characteristics is shown.

1.2 Target RCS

1.2.1 RCS Models

The choice of a radar target mathematical model is a rather complicated problemin the majority of situations, and it does not yield to exact analysis.

While deriving diffraction from complex objects, we often use the resultsobtained for simple cases. The complex surface of the object is divided into severalsimple areas for which the reflection can be easily determined and described, andthen the summing of partial contributions is performed according to techniquesproposed in [12, 13]. Usually, the following geometrical shapes are used for approxi-mation of the real targets and their parts: segments of spheres, ellipsoids and ogiveobjects, cone segments, cylinders or wires, plane plates, and dihedrals. Only theilluminated parts of such bodies must be taken into account. For derivation of

4 Radar Characteristics of Targets

reflection from these typical surfaces, Kirchhoff’s technique is the most widelyapplied because of its simplicity. The main difficulty for this approach lies indetermining of the angle sector (target aspects) for which the derived expressionsare valid and in the formula transformation for aspect changes. According to thisapproach, one supposes the independence of the bright points that contributemainly into the total reflected signal (i.e., the effects of multiple reflection are nottaken into consideration). This can result in significant errors in the determinationof the object RCS because shapes such as the corner reflector are not taken intoaccount.

A separate problem is the technique of combining the bright point reflections.The calculation of the phase for every component makes sense only when there isan accurate description of the surface and knowledge of the operational frequency,such that the errors of relative phase predictions between the different target partsdo not exceed a fraction of one wavelength. In this case, the technique permits usto estimate sufficiently accurately the complex target scattering pattern with theerrors less than 1–2 dB [14, 15]. Only rough estimation of RCS is possible bymeans of signal noncoherent addition from all bright points if the accuracy ofsurface and frequency description is insufficiently high. It was proposed in [16] touse random relative phase with uniform distribution in the interval of [0, 2p ] forRCS estimation. This approach is valid if the number of bright points is ratherlarge and the linear distances between them exceed the wavelength. Commonly,the use of the random phase model provides RCS estimation and permits us todetermine its most important statistical characteristics.

The technique of scattering pattern calculation for complex objects based onthe geometrical optics approximation was proposed in [17, 18]. For example,the results of the scattering pattern derivation for a Convair-990 aircraft and acomparison with experimental data at a frequency of 10.0 GHz from these papersare presented in Figure 1.1; the experimental RCS values are integrated in thesector of aspect angles of 10°.

The techniques of RCS derivation for comparatively simple objects are elabo-rated carefully for two cases:

• The wavelength is significantly greater than the target dimensions (Rayleighscattering);

• The wavelength is significantly less than the target dimensions, correspondingto surface and edge scattering.

For the intermediate resonance region (wavelength comparable with the targetdimensions), the establishment of some RCS relationships is a rather complicatedproblem, but there exist some techniques for approximation presented, in particu-lar, in [16].

1.2 Target RCS 5

Figure 1.1 The scattering pattern of the Convair-990 aircraft at X-band: (a) horizontal plane and(b) vertical plane. (After: [17].)

Let us consider the techniques of the RCS evaluation for rather simple aerody-namic objects, which can be considered as combinations of round cones and roundcylinders, both having limited dimensions.

Much attention has been paid to derivation for the round finite cone [19, 20].The scattering from its tip is given by

st =l2

16? p tan4 g (1.1)

where g is the half angle at the cone tip and l is the wavelength. This value isquite small in comparison with the scattering from the edge (cone base).

For a cone with dimensions significantly greater than l the RCS along thesymmetry axis (w = 0) is determined as [20]

s

l2 =1p Ska sin p /n

n D2 Scospn

− cos3pn D (1.2)

where n = 3/2 − g /2, and a is the cone base radius.

6 Radar Characteristics of Targets

Equation (1.2) shows that the RCS of a cone for the forward aspects does notdepend on the wavelength and is determined by the base diameter strongly. Thisdependence is approximately s ∼ a2.

For other angles, the RCS depends upon the electromagnetic wave polarization.In particular, the RCS for the vertical polarization is determined as [20]

s

l2 =ka

4p2 Ssin p /nn D2 1

sin w |expF−iS2ka sin w −p4 DG

× FScospn

− 1D−1

− Scospn

− cos3p − 2w

n D−1G (1.3)

+ expFiS2ka sin w −p4 DG × FScos

pn

− 1D−1

− Scospn

− cos3p − 2w

n D−1G |2for 0 < w < g

and

s

l2 =ka

4p2 Ssin p /nn D2 1

sin w× FScos

pn

− 1D−1

− Scospn

− cos3p − 2w

n D−1G2

for g < w <p2

Derivation of the cone RCS using (1.2) and (1.3), as a rule, gives estimatesthat are greater than those ones obtained experimentally. For example, for a finiteround cone (g = 15°, 2a = 150 mm, w = 0° ) the RCS derivation gives the value ofs = −15.5 dB(m2), while the experimental RCS is about −(20 − 25) dB(m2). Abetter match to the experimental results for a cone-cylindrical body, for the axialdirection (w = 0° ), is defined by expression [20]

s

pa2 =4p2 sin2 p2

p + g

(p + g )2 Fcosp2

p + g− cos

2p2

p + g G(1.4)

RCS estimation for the same cone (g = 15°, 2a = 150 mm) according (1.4) givess = −18 dB(m2), closer to experimental results.

For the second simple body—the cylinder of rather small length—the RCS canbe determined as

1.2 Target RCS 7

s

l2 =3

8p?

1

(p /2)2 + {ln [cka (1 − q2)1/2/2]}2 ?(1 − q2) sin 2qx

4q2 (1.5)

where q = cos w , x = kl , l is the half-length of the cylinder, and c = 0.5772 isEuler’s constant. The analysis of (1.5) shows that the maximal RCS for cylindertakes place at angles of incidence close to normal with respect to the cylinder axis.Reducing the wavelength reduces the maximal RCS.

The application of (1.1)–(1.5) for targets having dimensions comparable withthe wavelength (i.e., in the resonance region) provides estimates only of the orderof expected RCS in the most favorable situations. In particular, we performed acomparison of the predicted and experimental RCS for cone-cylinder objects using(1.2)–(1.5). The experiments were carried out at wavelengths of 3.0 and 0.8 cm,and the results are presented in Figure 1.2. It is worth noting that the RCS is higherby 10–15 dB for objects with diameters of 15.0 mm in comparison with diametersof 7.5 mm; this is significantly greater than that predicted by the geometrical opticsapproximation. The experimental RCS decreases by 14–20 dB for the smallerwavelength, while according to the derivations this should be only 4–12 dB.

Thus, the modeling of the real radar target RCS, including objects with rathersimple geometrical shapes, provides only the expected order of the RCS. Moredetailed data on the RCS and the scattering patterns can be obtained by naturalexperiments, especially for X- and Ka-bands, where the small-dimension construc-tive and technological target peculiarities are important.

1.2.2 RCSs of Real Targets

The RCSs for different classes of targets including marine, land, and air objectshave been thoroughly investigated at X-band and longer wave bands, but lessaccurate and complete data are available for millimeter bands.

RCS dependence on the wavelength, as a rule, is not evident or is totally absentfor most radar targets having dimensions many times greater than the wavelength.It can be observed only for the objects for which the reflection is mainly causedby the corner reflectors on their surface. Such constructive elements are especiallytypical of ships and other marine vessels. In connection with this phenomenon,their RCS significantly exceeds their projected area in the plane perpendicular tothe illumination direction. The RCS of objects usually approximately equals suchprojected area if the scattering is mainly caused by quasi-flat or curved surfaceelements. These effects allow us with some carefulness to use the quantitative dataobtained in X-band for RCS estimation in millimeter bands.

Let us more thoroughly consider the results of RCS measurements for targetsof different classes.

The RCSs of marine vessels are rather high, and their mean values are presentedin Table 1.1 [21].

8 Radar Characteristics of Targets

Figure 1.2 The scattering patterns of cone-cylinder bodies at X- and Ka-bands: (a) diameter7.5 mm and (b) diameter 15 mm.

Table 1.1 RCS of Large Marine Vessels

Type Mean RCS (m2)Ships with over 104 tons displacement > 2 ? 104

Middle-class vessels with 103–3 ? 103 tons displacement 3 ? 103–104

Small vessels with 60–200 tons displacement 50–250Submarine in above water state 35–140Submarine periscope (height is 0.5m over water surface) 0.3–0.4Source: [22].

For practically all microwave bands (1–10 GHz), the median value of RCSfrom side aspects can be determined using the empirical expression from [22]

s0.5 = 52 ? f 1/2D3/2 (1.6)

1.2 Target RCS 9

where f is the operational frequency in GHz, and D is the ship displacement inkilotons. The mean RCS of these objects decreases with increasing range, as aresult of the ship’s structure falling into the shadowing zone, and this dependenceis presented in Figure 1.3 for ships of three classes [23].

For MMW bands, the mean RCS of large ships increases with frequency morequickly than suggested by (1.6). For ships with displacements less than 200 tons,this increase is 3–5 dB, and for ships such as motor vessels, it is 15–20 dB. Thisconfirms the assumption that in millimeter bands, the corner reflector shapes ofship superstructures influence the mean RCS. As an illustration, the mean RCSdependence on range for three types of ships is presented in Figure 1.4.

The RCS of the small marine targets presented in Table 1.2 are significantlyless [24]. Such objects as marine buoys have a special place among small marinetargets because they are characterized by rolling motions with height oscillationsdue to rough sea and the presence of an anchor. Their mean RCS decreases withincreasing sea states due to the shadowing effect by sea waves. For instance, suchchange of RCS for a small marine buoy is about 7 dB for sea state changing from1 to 5, while for the same change of sea state, the RCS change is 18 dB for a buoyof medium size and 9 dB for a large one. RCS values for various marine objectsreported in [25] are presented in Table 1.3.

The RCS of land objects also varies within rather wide limits depending onthe object type. The mean forward aspect RCS for some land targets obtainedwhile moving along a dirt road are presented in Table 1.4 [26]. The measurementsare carried out at a 3-cm wavelength.

Figure 1.3 RCS dependence on the range for (1) trawler, (2) dry cargo ship, and (3, 4) tankers.

10 Radar Characteristics of Targets

Figure 1.4 RCS dependences on range at wavelengths of 3.0 and 0.8 cm for (1) patrol boat,(2) tanker, and (3) motor ship.

The RCS of air targets at microwave have also been investigated. As shownin [27], the mean RCS for the piston-engine B-26 aircraft at forward aspects inthe ±10° sector is 20–25 dB(m2), and a similar mean RCS is typical for the C-54at the 3-cm wavelength. As was shown in [28], the mean RCS values are 8–15dB(m2) for large jet aircraft, about 1 m2 for light aircraft of the L-200 type, andabout −(0.9–3.3) dB(m2) for a Russian Mi-4 helicopter [23].

One of the main trends in modern military airplane construction is the designof low-observable vehicles, decreasing their detection probability by air defense

1.2 Target RCS 11

Table 1.2 RCS of the Small Marine Targets

Mean RCS (m2)Object l = 3 cm l = 8 mmYacht, sailboat 10–20 12–14Scull boat 2–4 0.8–5.0Gum boat 1.0–2.0 1.2–2.5Large marine buoy with radar reflector 20–20 —Medium marine buoy with radar reflector 7–10 —Small marine buoy 10 —Channel cone buoy 10 —Man on windsurf 2.5–3.0 2.5–3.5Source: [24].

Table 1.3 RCS of Some Small Marine Objects

Source: [25].

Table 1.4 Mean RCS of Land Targets for Forward Aspects

Object Mean RCS (m2)Tank 6.0–9.0Armored car 8.9–30.0Heavy artillery tractor 15.0–20.0Light artillery tractor 10.0–15.0Truck 6.0–10.0Source: [26].

radar systems. The efforts of the airplane designers led to RCS reduction over thepast decades as illustrated in Table 1.5.

The contributions of reflections from the different elements of the aircraftstructure to the total RCS are determined by the aspect relative to the radar. Forside aspects, reflections from the fuselage and vertical stabilizer are predominant,along with reflections from the leading edges of the wing and stabilizer. Thecontributions to the total RCS from different aircraft structures are illustrated inFigure 1.5.

12 Radar Characteristics of Targets

Table 1.5 Airplane RCS (m2) Decreasing over Past Decades

Airplane Type 1970s 1980s 1990sBomber 50–100 5–10 0.5–1.0Fighter 5–15 1–3 0.1–0.3

Figure 1.5 Contributions of different structures to total aircraft RCS.

The main directions and trends of aircraft design with decreased RCS arepresented in Table 1.6.

Taking into consideration these trends, it is possible to predict that one canexpect light and medium-weight aircraft with RCS of order 10−2–10−3 or less. Thissignificantly decreases their detection range. Hence, it is necessary to find newcharacteristics of targets to ensure their reliable detection at great ranges.

Data on the scattering properties of biological objects have significant interestfor short-range radar designers. In some cases they are the desired targets, and inother cases they are false targets.

The backscattering from a human body is determined by its mass and the radaroperational frequency. The connection between frequency and RCS of a man isseen from data presented in Table 1.7.

1.2 Target RCS 13

Table 1.6 Main Methods of Decreasing RCS

Direction Technical Realization PossibilitiesFlying apparatus aerodynamic shape and The removal of the sharp selvages, corner forms,structural element improvements gaps in aerodynamic surfaces.

Decreasing the vertical stabilizer area.Use of the aerodynamic shape of the ‘‘flyingwing’’ type.Integration of the glider-engine and glider-armament systems.

Radio transparent and radio-absorbing Use of composite material.material applications Application of radio-absorbing coatings.

Application of conducting material for gapremoval.

Decreasing visibility of on-board antenna Antenna scattering in directions other thansystems specular reflection to the radar.

Decreasing numbers of antennas.Ionized absorbing cloud (IAC) creation Electronic gun use.

Application of coatings using radioactiveisotopes.

Table 1.7 Man RCS Dependence on the Frequency

Frequency (GHz) Mean RCS (m2)0.4 0.033–2.331.1 0.1–1.02.9 0.14–1.054.8 0.37–1.889.4 0.5–1.22

Source: [28].

It is seen that RCS of a man does not practically depend on the operationalfrequency in microwave bands. The scattering pattern presented in Figure 1.6 [29]shows that the RCS at 3-cm wavelength is maximal for frontal aspect and minimalfor side aspect.

The detailed investigation of backscattering from birds and insects is a ratherhard problem for several reasons. The ratio of object dimension to wavelength canchange over several orders, while the difference of the shape from spherical (evenwithout considering the wings) leads to strong dependence of RCS on the observa-tion aspect and the polarization of radiation.

Besides, the target is not rigid, and its shape periodically changes with the wingflaps and respiration. The problem is more complex for determination of temporalRCS dependence because this is determined by target behavior (i.e., migration,local food extraction). The RCS dependences on the aspect for three bird speciesat 3-cm wavelength are presented in Figure 1.7.

The RCS dependence on bird mass obtained in [30] is presented in Figure 1.8.The predicted values of the water sphere RCS at wavelengths of 3 and 0.8 cm are

14 Radar Characteristics of Targets

Figure 1.6 Scattering pattern of a man. (From: [29]. 1984 Radio and Communication.)

Figure 1.7 The scattering pattern for three bird species at X-band: (1) pigeon, (2) starling, and(3) crow.

shown by the lines, the experimental data at the 3-cm wavelength are presentedby the points. The simplest model for RCS estimation of biological objects is theequivalent water sphere for which mass is equal to the object mass. However thelength-to-diameter ratio for bird body parts, containing water, equals to 2:1 or3:1 [30].

1.2 Target RCS 15

Figure 1.8 RCS dependences on the bird mass at X-band (solid line) and Ka-band (dotted line).Points are the experimental data. (After: [30].)

The nonspherical shape of scatterers leads to the appearance of a cross-polarizedcomponent of the echo. This component value equals to −12 to −13 dB in compari-son to the main one for the objects with RCS greater than 5 ? 10−3 cm2.

The dependences on wavelength of bird and insect RCS are presented in Figure1.9. At wavelengths of more than 10 cm, the bird RCS can be approximated bythe relation s ∼ l−4 where l is the radar wavelength (i.e., Rayleigh scattering takesplace). The maximal RCS value of birds at this band equals to 0.1–20 cm2, thisvalue decreasing by 10 dB at a 3-cm wavelength and by 15 dB at a 30-cm wavelength.The insect RCS is 10−1–10−4 cm2 up to the wavelength of 8 mm. RCS values forsome bird and insect species are presented in Tables 1.8 and 1.9 [30, 31].

In the period from spring to autumn, the majority of birds are above most ofthe land surface at heights of 1 to 2 km. The migrating birds of some speciesregularly fly at heights more than 4 km and appear at distances from the nearestland of more than 1,000 km. At the heights from 0 to 2 km, the volume densityof the bird distribution is often from 10−7 to 10−6 m−3, and in regions of flockaccumulations, densities of order 10−5 can be found for durations of days. If theradar resolution cell volume equals to 106–107 m3 (as typical of many radars atranges less than 20 km), many such cells will contain at least one bird.

Table 1.10 shows the order of bird density in the some regions of flocking (thebird number is summed in height); the data is averaged over the considerablegeographical area [32]. It is necessary to take into consideration that factors ofsocial behavior can raise the local density of the flocks, especially in approachesto places of night rest, bird colonies, or flock nutrition. In the last column of Table

16 Radar Characteristics of Targets

Figure 1.9 Bird and insect RCS dependences on the wavelength.

Table 1.8 RCS of Birds at 10-cm Wavelength

RCS (m2)Type From Side From Front From BehindPigeon 1.0 × 10−2 1.1 × 10−4 1.0 × 10−4

Starling 2.5 × 10−3 1.8 × 10−4 1.3 × 10−4

Sparrow 7.0 × 10−4 2.5 × 10−5 1.8 × 10−4

Seagull 1.5 × 10−2 2.0 × 10−3 —Source: [30].

Table 1.9 RCS of Insects at 10-cm Wavelength

Type Wing Span (cm) RCS (cm2)Butterfly 10.0 1.0Butterfly 3.0 5.0–10−3

Bee 1.0 2–10−3

Dragonfly — 10−3

Source: [30].

1.10, the estimates of the top averaged RCS are given. RCS bounds are widebecause the target aspect, radiation polarization, and wavelength are not takeninto consideration. These estimates are acceptable for wavelengths from X- toS-bands. The bird-averaged distribution by altitudes is presented in Table 1.11[33].

1.2 Target RCS 17

Table 1.10 Bird Density in Flocking Places

Accumulation Type Area (km2) Bird Number Density (m−3) RCS (cm2)Winter refuges for crows,seagulls, geese, ducks in thelittoral waters <103 104–106 10−9–10−6 10–500Stormy petrel migration byCalifornia coast <103 >106 10−5 50–500Coastal and sea birds in thereproduction period <105 109 10−7 50–500Source: [32].

Table 1.11 The Bird Distribution by Altitude

Percentage of Birds at HeightsHeight (m) Lower Than Shown

250 45500 65

1,000 801,500 902,000 100

Source: [33].

The distributions of insect density in the air often can be greater by manyorders than for birds. Some data from the works of Rainy and Johnson [30] arepresented in Table 1.12; the RCS are given for X-band, and they represent themost probable values and are close to the real data.

It appears that the insect distribution density often can exceed 10−5–10−4 m−3,and densities of order 10−3–10−2 apparently are found regularly for cases whenthe converging winds concentrate the insects. The greatest densities have mostprobably local behavior, and the typical concentration area for these cases doesnot exceed 100 km2.

Thus, considerable data permit us to estimate the RCS of objects that can beeither targets or interference to radar systems designed for the detection of smallRCS objects.

Table 1.12 Distribution Density for Some Insects

DistributionObject Height (m) Density (m−3) RCS (cm2)

Middle butterfly during intensive migration Near surface 10−2 10−2–100

Night butterfly Near surface 10−2 10−1–100

All insects for 1 hour Near surface 101 10−3–100

Source: [30].

18 Radar Characteristics of Targets

1.3 Radar Reflections from Explosions and Gas Wakes of OperatingEngines

1.3.1 Analysis of Radar Reflection Mechanisms

Let us first consider the possible mechanisms of microwave backscattering froman explosion volume. This point is the most complex for the analysis, and it includesthe models of backscattering from a turbulent gas wake as a particular case.

During the primary stage of an explosion, a dominant factor is the reflectionfrom the SWIF. Depending on the ratio between the pressure po of an undisturbedgas and the SWIF pressure p = po + Dp , it is possible to classify three typical stagesof the shock wave:

• Strong shock waves (p >> po );• Shock waves of a middle range intensity (Dp ∼ po );• Weak shock waves (Dp << po ).

For strong and weak shock waves, it is possible to derive the analytical solutionfor the shock wave propagation equation. Its accuracy is determined only by theaccuracy of some primary assumptions. The problem of the strong point-sourceexplosion is solved in [34]. The author of the paper ignored the contra pressureof the medium. Taking contra pressure into account, this problem was solved onthe basis of numerical integration in [35]. For a weak shock wave, the asymptoticexpressions were also obtained (primarily in [36] and those in generalized form in[37]).

For a strong shock wave, the gas density decreases sharply from the SWIF tothe center; practically all of the gas mass is concentrated in the thin layer near thefront surface. With increasing distance from the SWIF to the center, the pressurereduces by two to three times and remains practically stable for the entire sphere.The temperature increases from the front to the center—especially very quickly inthe area of constant pressure, which is determined by the presence of particlesheated by the shock wave, which in turn have high entropy and are located nearthe center.

With the passage of time, the shock wave amplitude decreases and the frontpressure diminishes asymptotically to the atmospheric level. Correspondingly, theSWIF gas compression decreases, and the shock wave propagation speed asymptoti-cally diminishes approaching the sound velocity. The SWIF propagation law trans-forms from r f ∼ t2/5 to r f ∼ ao t where r f is the SWIF radius, t denotes time, andao is the sound speed (i.e., the shock wave transforms gradually to a sound wave).The zone with low pressure follows the compression area in such a wave, afterwhich the air resumes its original state.

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 19

The high temperature of the SWIF, reaching 105K for the initial explosionstage, causes ionization of the air and creation of a plasma layer. In this case, theelectromagnetic field reflection coefficient is described [38] in the following way:

|G |2 = F1 − (1 − ne /ne*)1/2

1 + (1 − ne /ne*)1/2G2

(1.7)

where ne is the plasma electron density in the SWIF; ne* = (pm /e2) f 2 is the electrondensity corresponding to the plasma resonance frequency, f ; and m and e are themass and the charge of the electron, respectively.

According to [39], the critical electron concentration corresponding to theplasma resonance frequency in the SWIF can be reached in the air at temperaturesof 3,000K for the 3-cm wavelength and 4,000K for the 8-mm wavelength. Thepresence of admixtures can decrease the temperature at which the critical concentra-tion appears. Such temperatures in the SWIF exist for 80–100 ms after the explosionfor the chemical explosives (like trotyl) that have a weight of 1 kg. In later stages,the ionization becomes lower than critical, resulting in a rapid decrease of therefractive index. Table 1.13 presents the RCS and the SWIF radius values derivedfrom (1.7) and taking data [35] into consideration. The results are obtained forthe explosion of a 1-kg trotyl charge at 3-cm wavelength; it was assumed that|G | was constant for the first Fresnel zone on the spherical SWIF.

It is worth mentioning that for t > 200 ms (corresponding to r f ≈ 1m), therapid decreasing of RCS begins, and for t > 1 ms, the contribution of the shockwave ionization to a total reflected signal becomes insignificant.

Microwave reflection from the discontinuity in concentration at the shock wavefront is the second possible mechanism. For this case, the reflection coefficient canbe determined as

|G |2 = Sn − 12gpo

D2 ? Dp2 (1.8)

for an infinitely thin layer on the shock wave front and

Table 1.13 The Temporal Dependence of the SWIF Radius and RCS of an Explosion at 3-cmWavelength

Time from theExplosion Start (ms) 4.7 6.2 20 50 84 140 180SWIF radius (m) 0.15 0.164 0.264 0.383 0.47 0.64 0.7RCS (m2) 0.071 0.085 0.226 0.46 0.69 0.13 0.031Source: [1].

20 Radar Characteristics of Targets

|G |2 = SDn2 D2 ?

1

1 + 16p2D2/l2 (1.9)

for a layer of finite effective width D. In these expressions (n − 1) = 320 ? 10−6 fora standard atmosphere when the altitude is 500m, g is the adiabatic constant equalto 1.4 for air, Dp is the pressure discontinuity in SWIF, l is the radar wavelength,and po is the pressure of an undisturbed atmosphere.

Taking into consideration the data on the SWIF pressure discontinuity derivedfrom [17], we obtain | G |2 ≈ 8 ? 10−8 for the explosion of a 3-kg trotyl charge at0.5s after the explosion start and | G |2 ≈ 1.5 ? 10−8 for 30 ms after the start of theexplosion (i.e., the reflection coefficient is not large and it decreases rather slowly).

The SWIF expansion, which results from a finite viscosity and the presence ofturbulent pulsations of temperature, pressure, and speed, gives a considerablereflection coefficient decrease in comparison to values derived from (1.8).

The estimates of the explosion RCS stipulated by this scattering mechanismhave values comparable to the background reflection from the troposphere, andthey are much less than the experimental data. Moreover, both mechanisms cancause reflections existing only for short intervals of time for which the SWIF doesnot exceed the limits of the radar resolution volume. But the reflections exist fora rather great time interval for area volume of relatively small dimensions. Theanalogous estimations of microwave reflections from the Mach cone taking place foran air vehicle flight with supersonic and ultrasonic speeds show that the ionization ofa shock wave front becomes considerable for speeds greater than 2.5 km/s. Forother speeds, the shock wave RCS is approximately equal to 10 (i.e., it is comparableto the troposphere reflective ability). Therefore, the reflections from the front ofan explosion or from the Mach cone can be the important mechanism for thedetection of these objects only for short time periods after the explosion or duringthe air vehicle flight at supersonic speeds.

One more mechanism of reflected radar signal formation is conditioned by theperturbations arising in the explosion area after passage of the shock wave front.The refractive index pulsation intensity increases due to medium turbulence as wellas to the chemical content changing in this area. For aerodynamic object detection,the reflections from the turbulent gas wake of an operating engine can be used.

For real explosions, a volume occupied by explosion products is formed. Theircompositions can be easily specified. The explosion products composition for 1 kgof trotyl C6H2(NO2)3CH3 is presented in Table 1.14.

It is evident that the main components of explosion products are CO, CO2,H2O, and N2 gases and amorphous carbon (soot); the contribution of the otheringredients is negligible. During the later explosion stages, the partial combustionof CO and C, with some additional carbon oxide formation, takes place. Thenoncombusted particles of carbon with dimensions of about 10−6–10−7 mm [40]

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 21

Table 1.14 The Composition of 1-kg Trotyl Explosion Products

WeightProducts Mole Gram Dipole Moment (Debye) Polarizationability

CO2 1.92 84.5 0.1 ± 0.05 2.6 ? 10−24

CO 11.64 326 0.112 ± 0.15 2.02 ? 10−24

H2O 10.96 192 1.65 ± 0.25 1.5 ? 10−24

C 17.32 208 — —N2 6.6 185 — 1.84 ? 10−24

NH3 0.04 0.6 — —Source: [1].

according to these derivations cannot explain the RCS values observed for theexplosion volume.

For the total refractive index estimation in the explosion products volume, letus use the ratio presented in [41]:

N = (n − 1) ? 106 = 2pAorM Sao +

m2

3kT D ? 106 (1.10)

where n is the refractive index; Ao is the Avogadro number; r denotes the density;M is the gas molecular weight; ao and a are the polarization ability and the dipolemoment of the gas molecule, respectively; k is the Boltzmann’s constant; and T isthe temperature in K. The contribution of different components and the totalreflection coefficient of the formed mixture are presented in Table 1.15.

It is seen that the maximum contribution to the refractivity is made by waterin a molecular phase, while among the other components the influence of thecarbon oxide is the most significant. It is worth noting that at high temperatures(typical for explosion products) the contribution caused by the molecule polariza-tion can increase by several orders.

Analogous phenomena occur for fuel combustion in turbojet and turbopropengines. In Table 1.16, we present the data on gas volumes for 1-kg kerosenecombustion (kerosene is now the main type of fuel for modern aircraft) exhausting

Table 1.15 Gas-Like Explosion Products Refractivity for a 1-kg Trotyl Explosion

Percentage Contribution in Mixture N, N-UnitsGas Type in Mixture Due to a0, t = 20°C Due to mCO2 4.0 26 0.82CO 24 140 5.2N2 13.6 66 —H2O 22 90 358.0SN 322 404.0Source: [1].

22 Radar Characteristics of Targets

Table 1.16 The Chemical Composition and Refractivity of Combustion Products for 1 kg ofKerosene

PercentageGas Type Volume m3 ? kg−1 N for Pure Gas of Content RefractivityCO2 1.6 700 12.8 90.0H2O 1.8 1760 14.5 255.0N2 9.1 410 72.7 300.0Total refractivity 645.0Source: [1].

to the atmosphere under normal pressure. The results of refractivity derivationfrom (1.10) are also presented.

According to the obtained estimates in the volume occupied by explosionproducts and fuel combustion products, the mixing of gas-like products with thesurrounding air takes place under the influence of atmosphere turbulence andgravity. The sharp margins of areas with different refractive indices remain intactbecause of turbulent diffusion, the speed of which is greater than the speed of themolecular gas diffusion. Later, blurring of the turbulent product wake marginsoccurs, resulting from intermolecular diffusion.

The dimensions of the volume occupied by the explosion products are limited.For an air explosion, the shock wave front moves more rapidly than the explosionproducts, so from the very beginning of the expansion process the pressure decreasesin the area occupied by the explosion products. A short time later, after the explo-sion, its products will occupy the maximum volume V∞ , which is described for aspherical charge by radius r∞ [42]

r∞ = (20 − 30)ro = (20 − 30)b √3 C (1.11)

where C is the charge weight in kg; b is the coefficient depending on the explosionsubstance density (for pressed trotyl it is equal to 0.053), and ro is the sphericalcharge radius in meters. The dimensions of the gas wake of an operating enginein the cross direction are also limited; the wake diameter is four to six times greaterthan that of the nozzle.

Using the simplifying assumption on a turbulent isotropy for the area occupiedby the explosion products and combustion products, it is possible to estimate aspecific volume RCS of this area using [43]

h =p8

? ⟨Dn2 ⟩ ? k2 ? Fn (k ) (1.12)

where k = 2p /l is the wave number; ⟨Dn2 ⟩ denotes the refractive index fluctuationvariance; and Fn (k ) is the one-dimensional spatial spectrum of refractive indexfluctuations.

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 23

Consequently, for the estimation of RCS of the explosion and engine fuelcombustion volumes, it is necessary to know two local turbulence characteristics:the variance and the spatial spectrum of the refractive index.

It is possible to expect that the normalized one-dimensional spectrum of therefractive index pulsations can be described as a spectrum of a random telegraphsignal with a Poisson distribution of refractive index steps

Fn (k ) = [1 + (kL )2]−1 (1.13)

where L is some typical effective turbulence scale. For kL << 1, we have Fn (k ) ≈ 1and the volumetric normalized RCS h ≈ l−2; for kL >> 1, the specific RCS has avery insignificant dependence on radar wavelength.

Therefore, from the point of view of radar detection, the latter mechanismseems to be the most important among those considered because it provides thegreatest duration of the reflected signal. The next section is devoted to the estima-tions of the variance and of the spatial spectrum of the refractive index fluctuationsfor a disturbed volume.

1.3.2 Spatial-Temporal Characteristics of Explosion and Fuel Combustion

Experimental investigations into the spatial and temporal characteristics of therefractive index of the explosion volume were carried out using a refractometerand a thermoanemometer. The first instrument directly obtained the temporalfluctuations of refractive index differences at two different points of space; thesecond determined the speed pulsations in the air flow.

The necessity of the use of two instruments was conditioned by considerabledimensions of the microwave refractometer open resonators, which prevented theestimation of the spectrum with wave numbers more than 1 cm (i.e., with lineardimensions less than 10–20 cm). The use of a thermoanemometer with a timeconstant about 0.01 second permitted us to investigate the inhomogeneous mediawith scales about 1 to 2 mm. Taking into consideration the similarity of the spatialspectra for a velocity field and the refraction index [43]. It was possible to combinethe data obtained by these instruments.

The data were recorded by a high-speed photoelectric recorder. For data pro-cessing, a sample with a duration of about 10 seconds was divided into segments,each having a duration of about 1 second. Two segments preceded the beginningof the explosion. For each segment, statistical and spectral processing was carriedout. For the spatial spectra determination from the temporal spectra, the hypothesisof frozen turbulence was used [43], according to which the whole spatial stochasticfield moves with a mean velocity of an air flow. This allowed us to obtain thespatial spectra of the refraction index fluctuations for spatial dimensions from 2to 100 cm.

24 Radar Characteristics of Targets

The experimental investigations were conducted on an open flat surface. Thetrotyl charges with weights of 1–3 kg were placed at a height of about 1.5m abovethe surface. The refractometer and the thermoanemometer sensors were placed ata 10-m range from the explosion center. About 40 explosions were carried out fordifferent wind speeds.

Refractometrical investigations into the turbulence local characteristics for theexplosion volume showed the following:

1. For 2–4 seconds at 10m from the center of the explosion of a 3-kg trotylexplosive, the root mean square (rms) value of pulsations exceeded (2–3)N -units, compared to (0.1–0.5) N -units for the undisturbed atmosphere.This phenomenon was observed both for calm weather and for a winddirection toward sensors. For the cross-wind direction, there was a 3–5 dBincrease in the refractive index pulsation in comparison to the undisturbedatmosphere.

The time interval when the effective value of fluctuations was morethan 1 N -unit was equal to 3–5 seconds. The illustration in Figure 1.10presents the refractive index pulsation values as the temporal functionsfor wind absence—Figure 1.10(a)—and for an explosion product movingtoward the sensors—Figure 1.10(b).

2. The refractive index temporal fluctuation spectra retained their shape.Besides, for the frequency band from 5 to 30 Hz, the slope of the disturbedarea spectra did not change in comparison to the undisturbed atmospherespectra corresponding closely to the phenomena predicted theoretically fora homogeneous turbulent atmosphere. Figure 1.11 presents the refractiveindex temporal fluctuation spectra for the different moments of time afterthe explosion for the same experiment. Using the hypothesis of frozenturbulence [16], the transformation to spatial fluctuation spectrum wascarried out (the lower horizontal axis). For the frequency region F < 5 Hz,a modification of the spectrum slope was observed for several experiments,probably resulting from the finite dimensions of the refractometer baseline(0.7m) acting as a spatial lowpass filter. Its influence also resulted in thestructural functions that had a tendency to saturation for the baseline dimen-sions of 0.5–1m.

3. For approximately 70% of the experiments carried out in conditions ofan explosion product movement toward the sensors, the difference of thecorrelation intervals of the flow velocity fluctuations was observed beforeand after the explosion. The decrease of the correlation interval at 2–5seconds after the explosion start was typical (Figure 1.12) in comparisonto the correlation interval for the undisturbed atmosphere. It was the evi-dence that the typical dimensions of the explosion product turbulence dimin-ished for the case of movement toward the sensors in comparison to theundisturbed atmosphere.

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 25

Figure 1.10 Temporal dependence of the refractive index fluctuation rms values in explosion(a) without wind and (b) with a longitudinal wind. (From: [1]. 1997 IEEE. Reprintedwith permission.)

26 Radar Characteristics of Targets

Figure 1.11 Instantaneous power spectra of the refractive index fluctuations in explosion. (From:[1]. 1997 IEEE. Reprinted with permission.)

4. For a distance between the sensors and the explosion center equal to 17m,the fluctuation intensity before and after the explosion remained almost thesame, excluding the cases of movement toward the sensors of the expandingvolume occupied by the explosion product.

Analogous results were obtained when the experimental study of local spatial-temporal characteristics was carried out for a gas wake of an operating jet engine.The investigations were made both for a jet engine model with fuel expenditure2 g/s and for a MIG-21 aircraft engine.

For the model experiments when the distance from the nozzle was of about2–3m, the refractive index fluctuations were 10–20 dB more than ones for theundisturbed atmosphere reaching ⟨Dn2 ⟩ = 100 (N -units)2. Reduction of the fluctua-

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 27

Figure 1.12 Temporal dependence of an air velocity decorrelation time after a 1-kg trotyl explosion.(From: [1]. 1997 IEEE. Reprinted with permission.)

tion intensity occurred with increase of the distance from the nozzle; when thedistance exceeded 8–9m, they decreased to the level of the undisturbed atmosphere.

The shape of the spectrum of fluctuations for the gas wake was practicallyidentical to the spectrum of undisturbed atmosphere; for their description thefunctional dependence ∼F −5/3 could be used. The spatial spectra obtained usingthe freezing approach with taking the local speed of the wake into account werecharacterized by the shift to the area of the large spatial wavelengths L for anincrease of the distance from the nozzle. This resulted from the greater degree ofgenerating small perturbations.

The refractive index fluctuation intensity for a gas wake of the MIG-21operating engine for the different distances from the nozzle and the regimes ofoperation is presented in Table 1.17.

It is seen that with fuel expenditure increase, the same fluctuation intensity isobserved for the greater distances. The frequency and spatial fluctuation spectraof the refractive index are analogous to those typical for the model of a gas wake,and they can be described using ∼F −5/3 and ∼ (1/D)−5/3 dependencies.

Therefore, the refractive index fluctuations for the area occupied by the explo-sion products and for the operating engine gas wake possess the following features:

28 Radar Characteristics of Targets

Table 1.17 The rms Values of Refractivity in the Gas Wake of an MIG-21 Aircraft

Engine Operating Regime Distance Along the Axis (m) from Nozzle XDN 2 C1/2? N-units

Minimal 20 5.525 4.5

Nominal (normal) 50 4.4Maximal 65 3.7Source: [1].

• In the disturbed volume, the refractive index fluctuation intensity increasesgreatly in comparison to that of the undisturbed atmosphere, proving theapplicability of the model proposed earlier for this region;

• The refractive index fluctuation spatial-temporal spectra shape of the dis-turbed areas is similar to the undisturbed atmosphere spectra;

• The dimensions of the disturbed volume are limited by the volume of explo-sion products propagation and by the nozzle gas flow.

1.3.3 Radar Reflections from Explosion and Gas Wake

The experimental study of the radar characteristics for the explosion area of atrotyl explosive was carried out at wavelengths from 10 cm to 4.1 mm. Theparameters of the pulsed and continuous-wave (CW) radars used for experimentalinvestigations are presented in Table 1.18.

The data from the pulsed radars were recorded by a high-speed photoelectricrecorder and by a 10-channel spectral analyzer of parallel type covering the fre-quency band from 10 to 500 Hz. With use of the spectral analyzer of this type, it

Table 1.18 The Parameters of the Pulsed and CW Radars

Parameters 1 2 3 4 5Type Pulsed Pulsed CW CW CWCentral frequency(GHz) 3.0 10.0 10.0 10.0 74.0Transmitter power:–Pulsed (kW) 250 250 — — —–Average (W) — — 10 4 0.6Polarization VV, HH VV, HH VV VV VVPulse duration (ms) 0.5 0.7 — — —Repetition frequency(Hz) 1,750 1,100 — — —Antenna pattern width:–Azimuthal 2° 0.5° 2° 1.5° 0.6°–Elevation 2.3° 0.75° 2° 1.5° 1.0°Threshold sensitivity(Wt) 10−12 0.5 ? 10−12 2 ? 10−18 10−17 5 ? 10−17

Frequency band ofanalysis (Hz) 500 500 0.10–40,000 0.10–40,000 0.10–40,000

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 29

was possible to obtain the instantaneous power spectra of the reflected signals.Moreover, the multichannel gate unit was used, which permitted us to obtain theexplosion volume spatial dimensions.

The data from the CW radars were recorded by the same recorder and by a10-channel spectral analyzer of parallel type covering the frequency band of analysisfrom 0.01 to 40 kHz.

When the position of the explosions was chosen, great attention was paid tothe selection of the surface area with a clutter minimum level.

The explosions of the trotyl charges with weights of 1 and 3 kg were carriedout at a range of about 2 km from the pulsed radars and over 50m from the CWradars. Calibration of the radars was carried out by a set of the corner reflectors.The rms error of the RCS estimation was equal to 2 dB.

We should like to note the following peculiarities of experiments:

1. The long-life reflections from the explosion products volume were the sub-jects of the study, but not the reflections from the short-time high-tempera-ture nucleus;

2. Coherent processing techniques and Doppler frequency filtering in the fre-quency domain F < 10 Hz (in some experiments, 5 Hz) were used forremoval of the obstructing reflections from environment. The RCS of theexplosion in this case was determined as

s = EFu

FL

G (F ) dF

where G (F ) is the power spectrum of the reflected signal, and FL and FUare the low and the upper bandpass filter frequencies. Obviously, for strongdependencies of spectral density on frequency (which according to the resultspresented later did take place for reflections from the explosion), the RCSvalue depended significantly upon FL and FU for this method of dataprocessing.

The experiments showed that the maximum reflection level is observed for aradar antenna beam directed to the explosion center, and for wavelengths of 10cm and 3 cm, the total RCS was equal to several square meters, reaching 10m insome cases. Besides, the RCS did not depend on the wavelength and polarizationof the signal; only a dependence on wind direction was observed.

The RCS for the crosswind was less than that for radial wind direction becauseof an explosion product drift from the explosion area by a cross-direction wind(i.e., the linear azimuth resolution of the radar was better than the radial one).

30 Radar Characteristics of Targets

The mean RCS values obtained in the frequency band from 10 to 500 Hz and0.6s after the explosion are presented in Table 1.19. Table 1.20 presents the averageRCS values measured for wavelengths 3 cm, 0.8 cm, and 0.4 cm in the band ofanalysis 5–200 kHz. It is seen, in particular, that a significant decrease in totalRCS occurs for 3.2-cm wavelength due to the growth of the lower bound offrequency analysis. At the same time, the RCS for the surface explosion of 1-kgtrotyl reached 0.2–0.3 m2 for a 3-cm wavelength and the same frequency band.

The measurements of the dimensions of the explosion volume that formed theecho were carried out by means of the estimation of the azimuthal cross sectionsof this volume using narrow-beam antennas with main lobe widths less than 1°.They showed that the volume diameter was about 6–7m for the explosion of thetrotyl charge (its weight was 1 kg) without the envelope and reached 8–10m forthe explosion of charge with a metal envelope. Demonstrating this effect, Figure1.13 presents the dependencies of the echo power when the antenna axis rotatesby some angle with respect to the explosion center (curve 1 corresponds to thecharge without the envelope, curve 2 corresponds to an enveloped charge). If wetake into account that, according to (1.11), the limit diameter of the area occupiedby the explosion product of a 1-kg trotyl explosion is equal to 2.2–3.2m, it ispossible to suppose that the reflected signal is partially formed by a turbulentatmosphere created by the passing of the shock wave front.

It is worth noting that the dimensions of the reflecting volume are determinedin sufficient degree by the band of analysis of the echo and increase with decreasinglow-bound frequency. For instance, for the wavelength of 3 cm, the reflectingvolume effective dimensions are equal to 2.5–6m for a frequency band of 30–60kHz and to 8.5–11m for a frequency band of 8–27 kHz. This phenomenon, in ouropinion, is explained by the fact that the power spectrum of the reflected signalbecomes poor in the high frequency area as the explosion products volume expands.

Table 1.19 The RCS of the Explosion for Band from 10 Hz to 500 Hz

Wavelength (cm) sDF ? m2

3.2 0.0170.8 0.020.4 0.0035

Source: [1].

Table 1.20 The RCS of the 1-kg Trotyl Explosion for the Frequency Band of 5–200 kHz

RCS (m2)Charge Mass (kg) Wavelength (cm) Cross Wind Longitude WindTrotyl, 3 kg 3.0 16.3 1.75Trotyl, 3 kg 10.0 4.2 2.3Source: [1].

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 31

Figure 1.13 The angular dependencies of the echo at wavelengths of (a) 3 cm and (b) 8 mm:(1) trotyl explosion without metallic envelope, and (2) trotyl explosion with metallicenvelope. (From: [1]. 1997 IEEE. Reprinted with permission.)

The band of analysis essentially determines the duration of the signal reflectedfrom the explosion. This is illustrated by the dependence of the signal duration onthe lower bound frequency of the analysis band presented in Figure 1.14. If forFL = 10 Hz, the total duration was equal to 1–3 seconds; it decreased to 0.5–0.7second when FL = 350 Hz.

The power spectrum analysis of echoes has shown that in the frequency bandof 10–500 Hz, the spectra are described by the relationship G (F ) ∼ F −5/3 (see

32 Radar Characteristics of Targets

Figure 1.14 Echo duration as a function of filter low-band frequency. (From: [1]. 1997 IEEE.Reprinted with permission.)

Figure 1.15). As the explosion evolves, the spectrum becomes poorer in the high-frequency region. When the wind had the direction from the explosion centertoward the radar, the dependence could have the shape G (F ) ∼ F −1 − F 0 atfrequencies less than 30 Hz due to the Doppler shift.

The shape of the echo power spectrum does not remain the same in the highfrequency region. The rapid decrease of the spectral intensity occurs during theexplosion products volume expansion at F > 5 kHz. From the analysis of theinstantaneous power spectra presented in Figure 1.16, which were obtained at a3-cm wavelength for the 1-kg trotyl explosion, it is seen that during 6 ms after theexplosion, the spectrum shape looks like G (F ) ∼ F −1; later, with the processevolution, it approaches ∼F −4 − F −5.

Finally, it is worth mentioning that the use of circular polarization does notresult in a change in the radar characteristics of the signals scattered by the explo-sion, in particular, reduction in the RCS. This obstacle excludes the use of scatteringby ground particles as a possible model because in this case circular polarizationwill attenuate the intensity of the reflected signal.

The results of radar observation of gas wake of MIG-21 and AN-24 aircraftare quite similar to those described earlier. These investigations were carried outwith pulsed Doppler radar at a 10-cm wavelength. The experiments showed thatthese aircraft, moving both with sonic and ultrasonic speeds, have tails in the echodetectable up to distances 1000m. The existence or absence of the inverse opticallyvisible track did not influence the intensity of radar tail essentially. These facts

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 33

Figure 1.15 Echo power spectra at the different moments of time after the explosion start obtainedfor the radar wavelength of 10 cm. (From: [1]. 1997 IEEE. Reprinted with permission.)

Figure 1.16 Instantaneous echo power spectra from the explosion at a 3-cm wavelength. (From:[1]. 1997 IEEE. Reprinted with permission.)

34 Radar Characteristics of Targets

permitted us to conclude that presence of this phenomenon was caused by scatteringfrom the track created by the gas-like fuel combustion products. The estimatesobtained showed that the RCS of the gas wake was about 10−4–10−2 m2, reachingvalues of 0.1 m2 in some cases, and the track length reached 500–1,000m.

These RCS values corresponded well to ones predicted using the model consid-ered in the first section of this chapter. If we suppose that the gas wake can beapproximated by a cylinder with the diameter four to six times that of the enginenozzle and the refractive index fluctuation variance does not change along andacross the axis, from (1.12) one can obtain the specific RCS s = 3 ? 10−7 m2/m(RCS for 1m of the track). We suppose here that DN 2 = 25 (N -units)2 (see Table1.5) and F (k ) = 0.1 for the nozzle diameter 1m at a 10-cm wavelength. Then thetotal RCS is equal to 10−4–10−3 m2, conforming enough well with the RCS of thetrack obtained experimentally.

Experimental investigations into the microwave radar reflections from chemicalexplosions enable us to conclude that the most important mechanisms of scatteringare the following:

• The reflection from the SWIF for the initial explosion stage;• The reflection from the gas-like explosion products for further evolution of

the explosion.

The refractive index fluctuation intensity of the explosion volume is significantlygreater than the fluctuation intensity in the undisturbed atmosphere. The spatial-temporal fluctuation spectra of the refractive index do not differ practically fromthe spectra of the turbulent atmosphere. The RCS of the explosion volume forchemical explosives with the weight about several kilograms reaches 10 m2, andthe intensive reflections exist during intervals less than 1 second or several seconds.These characteristics do not practically depend on radar wavelength in the frequencyband of 10–75 Hz.

The reflected signal spectrum is rather wide, especially during the initial stageof the explosion products cloud formation. These peculiarities permit us to realizeeffective radar detection with clutter filtering in some situations.

The backscattering from the turbulent gas wake of jet engines results in anincrease in the RCS of aerodynamic objects. This phenomenon can be used fortheir detection, especially in the case of a premeditated RCS decrease of the objectitself.

1.3.4 Centimeter Wave and MMW Attenuation in Explosions

The first important characteristic of the explosion volume that influences the totalattenuation is the volume of explosion products flying away at the final stage. Itis determined in [42] by

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 35

V∞ =43

p ? r3∞ = (1.5 − 6.5) ? 103 ? p ? b3 ? C (1.14)

where C is the charge weight (kg) and b is the factor depending on explosionsubstance density (e.g., its value for compressed trotyl is equal to 0.053). Thecontent of gas-like and solid explosion products can be easily determined for allknown explosives and is presented in Table 1.14. It is seen that the main part ofthe explosion products is formed by gases CO2, CO, N2, H2O, and amorphouscarbon C; the contribution of other substances is negligible. During the final stageof explosion evolution, partial burning of CO and C takes place forming carbondioxide; this leaves part of the carbon sediments as a dust.

Among gas-like products, only water vapor and oxygen, concentrated underhigh pressure at the shock wave front of the explosion, possess comparatively largedipole moments and attenuation spectra in the microwave band. The attenuationin carbon dioxide is significant only in the wavelength band 12.9–17.1m [44] (i.e.,far from the microwave band). In the microwave band, there exist weak absorptionlines (frequencies) of CO and NO [45], but their dipole moments have values about0.1 Debye (approximately 20 times less than the dipole moment of water vapor)and the CO concentrations only 1.5 times higher than that of H2O. Therefore,these gases cannot play important roles in microwave absorption.

For nonpolar molecules (N2), the dipole moment can appear as the result ofcollisions, but for usual conditions the absorption factor g resulting from thisphenomenon is much less than that for water vapor (gN2

/gH2O = 10−6).The derivation of the oxygen absorption factor for a pressure of 10 atmospheres

(this value corresponds to the shock wave front pressure at 0.5 ms after a 1-kgtrotyl explosion) has shown that for wavelengths from 0.4 to 3 cm, g had thevalues 1.27 ? 10−3–2.3 ? 10−2 m−1. Thus, the attenuation caused by this phenomenonis very small (taking into account that the width of its layer following the shockwave front equals several centimeters).

Estimation of the water vapor absorption factor has shown that for the explo-sion of 1-kg trotyl, this factor does not exceed 4 ? 10−3–6 ? 10−2 m−1 in the samewaveband (i.e., it has the same level as the absorption in oxygen).

The second cause of microwave attenuation is the temperature ionization ofair at the shock wave front and its heating due the burning of nonreacted remainders,which generates temperatures of about (2–3) ? 103K. This effect can result inthe longtime existence of plasma in the explosion products volume. The electronconcentration in this situation is much less than critical, so in the microwave bandthe condition v >> n is satisfied and the absorption factor can be expressed by thefollowing expression [46]

g = 0.1n ? ne

v2 + n2 S1 − 0.3ne

v2 + n2D−1/2

(1.15)

36 Radar Characteristics of Targets

where n is the number of efficient collisions of electrons with molecules, v is thefrequency of the radar, and ne denotes the electron concentration.

For typical plasma parameters of burning and for atmospheric pressure we usen = 1011 s−1, ne = 108 cm−3. For f = 10 GHz, the absorption factor derived from(1.13) is equal to g ≈ 7.5 ? 105 cm−1 (i.e., the attenuation is very small). Furthermore,the absorption factor in plasma should decrease with decreasing wavelength,although experiments showed its growth.

Finally, the third cause of microwave attenuation in explosions is absorptionby solid explosion products. As is seen from Table 1.12, a large amount of carbonis given off during the detonation process. Carbon particle dimensions have themost probable radius m0 = 0.05–0.15 mm [47] (i.e., usually they are much lessthan the wavelength). For derivation of absorption in such particles, the theory ofMie [48] can be used. For particles having m << l , the attenuation cross sectionis determined as

sa ≈4p2m2

l?

6e″(e ′ + 2)2 + (e″ )2 (1.16)

where e ′ and e″ are the real and imaginary components of the dielectric constant,respectively.

Taking into consideration that according to [49], the particle dimension distri-bution can be approximated by an exponential law with sufficient accuracy

N (m ) = No ?m

m3o

? expS−m2

2m2oD (1.17)

where No is the total number of particles in a volume equal to 1 m3, the attenuationfactor (specific attenuation per 1m) is

g = EN (m )sa (m , l ) dm ≈450No mo

l F 6e″(e ′ + 2)2 + (e″ )2G (1.18)

It is seen from (1.18) that the attenuation factor grows inverse proportionallyto the wavelength l . However, it is necessary to take into account the frequencydependence of the dielectric constant, which can results in a weaker dependenceof g with wavelength. The results of the carbon complex dielectric constant deriva-tion for burning temperatures using the data of [50] are presented in Table 1.21.It is seen from Table 1.21 that the real component of dielectric constant doesnot depend on wavelength, but the imaginary component quickly reduces withdecreasing l , especially in the millimeter band. Consequently, for frequencies higher

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 37

Table 1.21 Complex Dielectric Constant of Carbon for Burning Temperatures

Wavelength (mm) e ′ e″30 2.02 10.88.0 2.02 2.74.0 2.02 1.35

Source: [2].

than 30–40 GHz, the influence of this effect causes some decrease of the attenuationfactor wavelength dependence in comparison with the g ∼ l−1 dependence.

The derivation according to (1.18) for an explosion of 1-kg trotyl, which wasdone under the assumption that the probable radius of the particles was m = 0.1mm and the total number of particles per specific volume occupied by the explosionproducts was equal to N ≈ 4 ? 1019 m3, gave the results presented in Table 1.22.

Let us estimate the total microwave attenuation for propagation through theexplosion products volume. The length of the propagation path can be easilydetermined from the simplest geometry analysis (see Figure 1.17).

Table 1.22 Dependence of Attenuation Factor in Carbon Derived According to (1.16)

Wavelength (mm) 30.0 8.0 4.0g , m−1 0.29 1.68 1.8Source: [2].

Figure 1.17 The geometry for derivation of microwave total attenuation in explosion area (T = thetransmitter position, Rec. = the receiver position, O = the explosion center).

38 Radar Characteristics of Targets

Assuming r1 >> l , R >> l , r2 >> l , and supposing r1 = r2 = r for simplicity,it is easy to calculate the path length of the transmitter-receiver line for angle uwith respect to the explosion center,

l = 2r√q2 − sin2 u , tan u << q (1.19)

where q = R /r.If we assume that the particle distribution along this path is described by the

expression

N =NoD

expS−R ? tan u

D D (1.20)

where D is the effective radius of the volume occupied by the explosion products(i.e., that the particle density exponentially decreases with increasing distance fromarea center), we get

S = g ? l =450No m3

ol ? D F 6e″

(e ′ + 2)2 + (e″ )2G 2r√q2 − sin2 u ? expS−RD

tan uD(1.21)

Refractive focusing effects can exert influence on the signal intensity when thesignal propagates through the explosion volume. Using the results of [43], it iseasily shown that the relative growth of signal intensity Dp /p beyond the sphericalarea is determined as

Dp /p ≈ 2rR

(n − no ) (1.22)

where r is the distance from the area center to the reception point, and R is theradius of dielectric sphere with refractive index n (usually n − no << 1, where nois the refractive index of the atmosphere). Substituting the experimental valuesr = 25m, R = 2.5 m, and n − no = 300 ? 10−6 (this value corresponds to maximalpredicted ones) into (1.22), we obtain Dp /p ≈ 6 ? 10−3 (i.e., the influence ofrefraction is negligible). These estimates of the influence of diffraction on thedielectric sphere show that the inhomogeneous sphere with dimensions R << r ,where r is the distance from the explosion center to reception point (see Figure1.17), does not create significant focusing or defocusing effects [43] and, conse-quently, cannot change the intensity of the received signal significantly.

As mentioned earlier, the experimental investigation into microwave attenua-tion in the explosion area was carried out at wavelengths of 3 cm, 0.8 cm, and

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 39

0.4 cm. The propagation path was of length 50m; the transmitter and receiverwere placed at the ends and the explosions took place in the middle of the path.Trotyl explosives were used; their weights were 1 kg. To eliminate the influenceof ground debris, the charges were placed at heights 1–2m over the land surface(on slabs or poles). Magnetron oscillators were used as transmitters, operating inthe continuous regime with output power 1–10W. For measuring system calibra-tion, polarization attenuators were used. They provided calibration accuracy of0.2–0.5 dB; the worst value corresponded to the system with wavelength 4 mm.

A typical temporal diagram of total explosion signal attenuation is presentedin Figure 1.18. At the moment of charge explosion, rapid drop in the receivedsignal intensity is observed. Maximal attenuation occurs for 30–50 ms for allinvestigated operating frequencies, increasing slightly (up to 10%) with increasingfrequency. The sharp splash in the curve labeled ‘‘4’’ is the time of explosion. Therecovery of the signal level takes place more slowly than its decay at the start ofthe explosion. The total duration of significant attenuation is about 80–100 ms.

The results of the entire radar signal attenuation study for charge detonationon the transmitter-receiver line of sight are presented in Table 1.23, and the resultsof the attenuation factor measurements are given in Table 1.24.

The significant differences between results obtained for the different experi-ments can be explained by the nonidentity of charges and conditions of theirexplosion (technical trotyl charges were used for the experiments).

A comparison of Tables 1.23 and 1.24 shows that for 3-cm wavelength, satisfac-tory agreement of predicted and experimental attenuation coefficient values isobserved, but for shorter wavelengths the difference increases.

Figure 1.18 Samples of temporal diagram for received power at wavelengths of (4) 3 cm and(3) 8 mm. (From: [2]. 1997 IEEE. Reprinted with permission.)

40 Radar Characteristics of Targets

Table 1.23 Maximal Total Attenuation (dB) for 1-kg Trotyl Explosion

Wavelength (mm) Attenuation (dB) Average Attenuation (dB)30 3.2; 4.3; 6.5; 7.5; 11.0; 7.2; 7.1; 4.3; 3.2 6.03 ± 2.48.0 13.8; 14.6; 11.7; 14.2; 11.3; 14.3 13.3 ± 1.714.0 11.8; 8.7; 11.0; 9.0; 12.0 10.5 ± 1.94

Source: [2].

Table 1.24 Experimental Attenuation Factors for Explosions of a 1-kg Trotyl Explosive

Wavelength (mm) g , m−1 g , m−1

30.0 0.15; 0.3; 0.2; 0.35 0.25 ± 0.0758.0 3.3; 3.6; 2.8 3.2 ± 0.194.0 3.8; 2.9; 2.8; 3.6 3.3 ± 0.22

Source: [2].

When the explosion center is shifted by some angle with respect to the transmit-ter-receiver line of sight, less signal attenuation occurs. The observed experimentaldependence S = f (u ) and the results of derivations performed according to (1.22)are presented in Figure 1.19.

Satisfactory agreement of prediction and experimental data is observed. For adeviation angle u ≈ 6° from the explosion center (for experiment geometry, this

Figure 1.19 The angular dependencies of total attenuation for wavelengths of 3 cm, 0.8 cm, and0.4 cm. The solid line is the result of derivation from (1.21). (From: [2]. 1997 IEEE.Reprinted with permission.)

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 41

corresponds to 5–6-m shifting), attenuation is practically absent. This deviationcorresponds to a distance between the line of transmitter-receiver and the chargeposition equal to 40ro (the lower axis of abscissas in Figure 1.19), where ro isdetermined by (1.12) and characterizes the dimensions of area occupied by theexplosion products.

The lack of knowledge of dissociation processes occurring in the explosionmakes it difficult to explain the attenuation temporal dependence. One of theprobable mechanisms is connected with the fact that after primary explosion prod-ucts dispersal, burning of the small dispersed carbon particle takes the place dueto the presence of a high temperature that is sufficient for burning (higher than2,500K). In these conditions, the carbon particle concentration quickly decreases,resulting in recovery of the received signal level. The probability of this hypothesisis supported by data presented in [1], the author of which gives the followingresults: using brisant explosive (the weight was 36 kg), the dust was present for5–10s after start of the explosion, and the attenuation for frequencies 35–140 GHzwas not higher than 0.4 dB. Dust particle concentration had a value of about108 m−3 (i.e., it was less than the primary carbon particle concentration thatappeared as the result of detonation).

In the explosion area occupied by trotyl explosive explosion products, signifi-cant microwave attenuation occurs, and it increases with increased radar frequency.The most probable mechanism for explaining the experimental data is the absorp-tion by solid products of transformation, in particular, by amorphous carbon.

1.3.5 Radar Backscattering from Sonic Perturbations Caused byAerodynamic Object Flight

1.3.5.1 Reflection from the Shock Wave Front Caused by Supersonic Flight

For straight flight of an object when its speed is V, a shock wave is formed in thehomogeneous atmosphere. The shock wave possesses axial symmetry, and its shapeis close to conical (Mach cone) with the object at its apex. For a cone angle w0 ,the following expression is valid

w0 = arcsin (a0 /V ) = arcsin M −1 (1.23)

where a0 is the sound speed and M = V /a0 is the Mach number. This wave spreadingto the sides of the trajectory includes increasing air mass, simultaneously withwhich its intensity diminishes, and transformation into a sound wave is observed.The pressure jump Dp at the shock wave front (SWF) for flight of a body ofrevolution is determined as [51]

Dpp0

=(M 2 − 1)1/8

(H /L )3/4 ? kS ?DL

(1.24)

42 Radar Characteristics of Targets

where L is the body length, H is the normal distance to the body flight axis, D isthe equivalent maximal body diameter, p0 is the pressure of surrounding gas, andkS is the body shape coefficient equal to

kS =1.19

(g + 1)3/4 ? 1Eto

0

F (t ) ? L ? dL2 ?LD

(1.25)

Here g denotes the adiabatic constant, which for air is 1.4; F (t ) is the effectivearea distribution function [7, 52]

F (t ) = √ L2p

? Et

0

Aeff (t )

(t − t )1/2 dt

in which Aeff (t ) is the body cross section.As a rule, when the body of revolution configuration is simple the value Dp

can be calculated analytically, but for more complicated bodies (like aircraft)reliable data relating p to the SWF are usually obtained experimentally. Accordingto [53], the SWF pressure difference is 50–200 newton m−2 (the larger valuecorresponding to a bomber) when aircraft speed equals (M1.5–M2) for distances6–8 km from the aircraft trajectory. In a homogeneous atmosphere, Dp decreaseswith the increase of distance to object according to the law Dp ∼ r−3/4 and the rateof pressure variation in the interval between jumps is inversely proportional to r[54]. The pressure impulse, being the important effect of the shock wave on themedium, is expressed as

I = Et1

to

Dp (t ) dt ≈12

Dp ? T (1.26)

where T = t1 − to and the parameter T is calculated on basis of the followingexpression for rate of pressure variation as a function of time

d (Dp )dt

= 0.2 ?M ? r−1

√M 2 − 1(1.27)

The considered dependences permit us to extrapolate the experimental datagiven in references into the area of comparatively small distances from the object.

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 43

Figure 1.20 presents the SWF parameter values for distances less than 1 kmfrom the airplane (see shaded areas; the lower bound corresponds to a fighter andthe upper one corresponds to a transport aircraft). Point 1 in Figure 1.20 is obtainedfor an F-4C moving at an altitude of 30m [55]. Satisfactory agreement is observedbetween predicted data presented in Figure 1.20 and the experimental data givenin [1, 51, 55] and extrapolated to small distances. In Figure 1.20, the dependencesDp (r ) and I (r ) are also presented for flight with the speed of M2 of a cone-cylinderbody with diameter of 120 mm (curves 2). While pressure jump values for aircraftand cone-cylinder body have almost the same level, the pressure impulse for thesecond is one or two orders less.

From the point of view of radar observation of the SWF and the interactionof the SWF with a turbulent atmosphere, the most interesting items are the spectraof pressure jump (see Figure 1.21). The duration of the N -profile, depending onthe object shape and its speed, can vary within the limits 0.01–0.5 second. Thetheoretical spectrum of such an impulse decreases with increased frequency at therate of 6 dB/octave (line 1).

The experimental spectra (curves 2 and 3) are obtained in [51] for differentatmospheric conditions. With increased frequency, the experimental spectradecrease with the slope close to that predicted. The spectra maxima are observedat frequencies of 10–20 Hz. Expansion of the front, due to atmosphere turbulence,results in reduction of the spectrum high frequency components.

Electromagnetic field reflection from an infinitely thin SWF is determined bythe refractivity jump, the reflection coefficient G given by

|G |2 = Sn1 − non1 + no

D2 ≈Dn2

4(1.28)

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

Figure 1.20 (a) Pressure jump and (b) pressure impulse as functions of distance from an aircraft and acone-cylinder body. (From: [7]. 1997 SPIE. Reprinted with permission.)

44 Radar Characteristics of Targets

Figure 1.21 Power spectra of the pressure jump. (From: [51]. 1966 Acoustical Society of America.)

where n1 and no are the refractivity coefficients for the peak and atmosphere,respectively.

Taking into account the adiabatic link between density and pressure for weakshock waves and the proportionality of refractivity coefficient to density, we get

D(n − 1)n − 1

=1g

?Dpp

(1.29)

and

|G |2 = Sn − 12gp D2 ? Dp2 (1.30)

where (n − 1) = 320 ? 10−6 for a standard atmosphere at height about 500m abovesea level.

Because the SWF expands as it propagates in the atmosphere, the reflectioncoefficient is less than its value determined by (1.30) for an infinitely thin jump.Let us assume that the refractivity coefficient varies

n0 = Hn0 for x < 0

n0 + Dn [1 − exp (−x /D)] for x ≥ 0(1.31)

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 45

where D is the effective front width and x denotes the direction normal to thefront. It is easy to show that for this case

|G |2 = SDn2 D2 ?

1

1 +16p2D2

l2

(1.32)

Two obstacles cause SWF expansion in the turbulent atmosphere. The firstis that the presence of turbulent temperature pulsations results in sound speedfluctuations. The second is that the sound waves are carried by moving air, andtherefore the presence of turbulent motions, described by turbulent pulsations ofspeed U ′, contributes to additional random distortions of front shape. In this case,the effective SWF width can be determined by the [56]

D = 16g

g + 1? e2(m + c )2Lo

poDp

(1.33)

where Lo is the outer scale of the atmosphere turbulence and e , m , and c are theparameters characterizing the atmospheric state.

The value of Lo increases linearly with altitude, changing from 1–2m near theland surface to 300–400m at the upper boundary of atmosphere. According todata presented in [57], for altitudes 100–200m, the typical value is Lo ≅ 10m.

The value of e2 lies within the limits 10−7–10−5 and decreases with the altitude.The coefficient m characterizes the turbulent pulsations of the sound speed aT ,according to m = aT /eao , and is usually close to unity. The coefficient c character-izes the turbulent component of the flow speed Ux′, which is parallel to the SWFnormal and is equal to c = Ux′ /Uo , where Uo is the mean speed of the flow. Whenthe wave propagates in the vertical direction the typical value of c is 0.1–0.2.

The results of deriving the effective width of the SWF for several parametersof turbulence e2(m + c )2—carried out using (1.33)—are represented in Figure 1.22by the solid lines. In the same figure, the experimental data [58] are shown for shockwave resulting from supersonic airplane flights. The dashed curve corresponds todependence D = f (Dp ) for dry air when the SWF width is determined by intermolecu-lar bonds and viscosity only.

It is seen from Figure 1.22 that the real SWF width is two orders greater thanthe theoretical value for dry air, leading to reduction of the reflection coefficient.The dependences of the reflection coefficient upon the distance to the front derivedaccording to (1.30) and (1.32) are presented in Figure 1.22(a). For distances about10m from the object, the reflection coefficient is approximately equal to 10−10—almost the same in comparison with cases considered in Figure 1.22(b) and diminish-ing for greater distances.

46 Radar Characteristics of Targets

Figure 1.22 (a) The effective front width and (b) the reflection coefficient as functions of distance to pressurepeak e2<(m + c )2> equals to (1) 3 ? 10−3, (2) 3 ? 10−4, and (3) 3 ? 10−5. (From: [7]. 1997 SPIE.Reprinted with permission.)

Taking into consideration that in the turbulent medium, besides SWF expan-sion, the SWF refractivity varies irregularly and reflection is possible in directionsother than normal to the front. In this case, backscattering occurs due to the frontroughness. This permits observation of reflections from the shock wave front inan angular region determined by the scattering pattern. For preliminary estimation,it is possible to use the assumption of fully isotropic scattering (meaning thatthe front distortions have amplitude less than the wavelength and a rather smallcorrelation radius).

This assumption results in higher estimates in comparison to the real data overmuch of space excepting the narrow region of angles near the direction of specularreflection. The effective radar cross section s in this case is equal to

s = ESil

G2(r ) dS

where the integration must be carried out over an all-illuminated area.The estimates show that for 10-cm wavelength, when the antenna pattern

width in both planes is about 0.01 radian and the distance to the object is 10 km,the value of the RCS of the SWF front at the distances from the object less than10m is about 10−5–10−6 m2. This value is comparable with background reflections

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 47

from the turbulent troposphere. Taking into account that the spectrum maximumof the SWF pressure is observed at the frequencies 10–20 Hz (see Figure 1.21), theRCS of the SWF front can be greater at decimeter and meter wavelength in compari-son with the estimates obtained for the centimeter waveband.

1.3.5.2 Microwave Scattering from Sound Perturbations in the Atmosphere

The airplane flights with near-sonic and supersonic speeds result in the appearanceof a powerful sound wave field. For instance, the sound radiation power of ‘‘Phan-tom’’ airplane is about 10 millimeter waves. The sound wave propagation is fol-lowed by a dielectric constant variation of the medium and, as result, byelectromagnetic wave scattering from these dielectric constant perturbations. Letus evaluate the scattered field intensity for this case.

The dielectric constant of the medium in the sound wave field can be presentedas

e ( ›r , t ) = eo + e1( ›r , t ) (1.34)

where eo = const denotes the dielectric constant of undisturbed medium,e1( ›r , t ) is the sound perturbation of the dielectric constant, usually e1( ›r , t ) << 1.The value of e1 can be expressed from the Lorentz-Lorenz relationship [59] by thefollowing

e1( ›r , t ) ≈eo − 1

r? dr = 2

no − 1r

? dr (1.35)

where dr is the air density variation caused by the pressure changing due to thesound or shock wave propagation. It is expressed as

drr

=1g

?dpp

where r is the air density.Taking into account that the sound speed in an undisturbed medium is equal

to

ao = √g ? poro

after simple derivations we get from (1.13)

48 Radar Characteristics of Targets

e1( ›r , t ) ≈ 2no − 1

r?

Dp

a 2o

(1.36)

The pressure variation in atmosphere during sound wave propagation can bepresented as [60]

Dp ( ›r , t ) =Ro √2rao Io Ga exp [−a (r − Ro )]

r? expF−iSVt − ka r + ka

›Vao

DG(1.37)

where Ga is the normalized sound source directivity pattern, V denotes the fre-quency of the sound wave, ka = 2p /la is the acoustic wavenumber, V is a windspeed (later it is supposed that it is constant in the scattered area), Io denotes thesound field intensity measured for some sample range Ro from source alongthe axis of radiation pattern, a is an atmosphere absorption coefficient, and r isthe current range from the sound source (see Figure 1.23).

Substituting (1.37) into (1.36), we get

e1( ›r , t ) =e1r

Ga cosSVt − ka r + ka

›VaoD (1.38)

where

e1 = 2√2r−1/2a −3/2o I 1/2

o Ro (no − 1) exp [−a (r − Ro )] (1.39)

aaaaaaaaaaaaaaaaaaaa

Figure 1.23 Geometry for derivation.

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 49

The electromagnetic field reflected from the sound wave package can bedescribed in the Born approach as

›E 1 (R1 , t ) =

k2 exp (−iv t )4peo

E E(V )

E e ik |R—›

1 − r–› |

| ›R1 − ›r |

e1 ( ›r , t )[ ›n (eo›n )] dV (1.40)

Here›

R 1 is the radius vector in the point of electromagnetic wave receptionand v denotes the frequency of electromagnetic field.

Omitting the rather complex intermediate derivations, let us write the ratio ofreceived-to-transmitted electromagnetic field power as

PrPt

= 2Io SRoR1

D2 (no − 1)2

ra 3o

e −2a (R1 −Ro ) k2a 23

D4R41 |C3 − C2 /2R1 |2

(1.41)

× exp3−4Vx

D2a 2o

− DSC1 + C2

VxaoD2

2SC3 −C2

2R1D 4 ?

sinpa3

l F2 7l

loS1 −

VzaoDG

pa3l F2 7

llo

S1 −VzaoDG

Here a3 is the radial length of the sound package, k = 2p /l is the wavenumber,D is the radar directivity pattern width, Vx and Vz are the speed vector projectionsonto the coordinate axes,

C2 = ikF2 7l

laS1 −

VzaoDG: C3 = ik

2 7 l /lo2R1

−1

R 21S 1

D2a

+1

D2Dand Da denotes the width of the acoustic source directivity pattern.

It follows from (1.41) that the scattering of the electromagnetic field has aresonant character; its maximum occurring for l /la determined by the expression

2 −l

laS1 −

VxaoD = 0 (1.42)

The resonant maximum value is proportional to the number of wavelengthsin the sound package (i.e., to the sound package length). Besides, the resonantlength of the radiowave depends upon longitudinal component of wind speed Vzbecause the sound wave length is functionally connected with Vz .

The wind speed cross component Vx leads to two effects, decreasing the levelof reflected signal. The first phenomenon consists of sound wave package expanding

50 Radar Characteristics of Targets

out of the volume of the radar resolution cell; it is taken into consideration by thefactor exp (−4Vx

2 /D2ao2). The second effect deals with the distortion of the spherical

shape of the sound wave phase front. In natural conditions, one more factor appearsdiminishing the backscattering efficiency. It is connected with turbulent pulsationsof the atmosphere refractive index, which result in additional distortions of thesound wave phase front.

The results of numerical derivation of the reflection coefficient for resonantscattering without wind, which were obtained for two radiowave bands (S- andX-bands), are presented in Figure 1.24. The derivations are carried out forthe following values of parameters included in (1.39): aS = 200la , Ro = 10m,Io = 10 Wm−2, D2 = 0.37 ? 10−2, Da

2 = 0.03, T = 20°C, no − 1 = 2.68 ? 10−4,ao = 340 m/sec, r = 1.29 kg/m3, a = 2.5 ? 10−2.

It is seen that for the most favorable geometry and for distances betweenthe radar and the sound wave source less than 100m, the ratio Pr /Pt is about10−8–10−12, and it decreases with further distance (to −200 dB when the distanceis about 500m). This means that the observation of radar reflections from soundwaves is practically impossible at ranges typical for radar operation, especially fordirections that do not coincide with normal to sound wave front.

It is also worth noting that the main energy component in the jet propulsionengine sound radiation spectrum is concentrated in a frequency interval having a

Figure 1.24 Reflected signal ratio as a function of distance between the radar and sound wavesource. (From: [7]. 1997 SPIE. Reprinted with permission.)

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 51

width less than 10 Hz [61]; the maximum of the weak shock wave spectrum alsolies in the low frequency area (see Figure 1.21). This means that for radar detectionof signals scattered by sound waves, it is more expedient to use the UHF or L-bandsas in the case of detection of SWF reflections for supersonic flights.

1.3.5.3 Microwave Backscattering from Atmosphere Turbulent PerturbationsCaused by the Interaction with Sound Field

Microwave signal backscattering from atmosphere turbulent perturbations is theresult of their refractive index variations caused by temperature gradients or localhumidity variations. The volume RCS in this case is determined as [62]

h =p8

⟨Dn2 ⟩ ? k2 ? Fn (k ) (1.43)

where k = 2p /l is the wave number, Fn (k ) denotes one-dimension refractive indexspectrum, and ⟨Dn2 ⟩ is the variance of refractive index fluctuations. It is known[43] that the RCS is connected with the speed of energy dissipation e by thedependence h ∼ e2/3 and, consequently, e increase caused by any reasons leads toincreased RCS.

Shock wave or sound wave propagation in the atmosphere is accompanied byenergy losses; part of the power is transformed into the power of turbulent motion,which can result in variation of energy dissipation speed and, as a result, in changingof the intensity and fluctuation spectrum of the refractive index.

The sound intensity of the source of power PS at some range r is expressed as

I (r ) =PS

4pr2 ? exp (−2ar ) (1.44)

It is easy to show that the sound energy loss of the sound wave is

b (r ) =I (r )

r? m ; m = 2a (1.45)

or

b (r ) =mPS

4prr2 exp (−mr ) (1.46)

The sound absorption coefficient in a homogeneous atmosphere depends onhumidity and temperature, and for a standard atmosphere, it is equal to m = 4 ?

10−5 at a frequency of 10 kHz. The presence of turbulent viscosity of nT, which

52 Radar Characteristics of Targets

exceeds the cinematic viscosity of n as nT /n ∼ Re [62], results in significant increasein the absorption coefficient; here Re is the Reynolds number. Taking into accountthat in the near-ground layer of the atmosphere Re ≥ 30–60 × 109, the soundenergy loss rate is 10 times greater than that determined by (1.46). For example,for a sound source power PS = 5 MW and for range from the source equal to 10mthe value of b , derived using (1.46), is b ≈ 0.01 m2sec−3, but taking into consider-ation the turbulent viscosity, it increases to b ≈ 0.2 m2sec−3. At the same time, theturbulent energy dissipation rate in lower atmosphere layers does not exceed0.01–0.1 m2sec−3 [57] (i.e., it is approximately 10 times less than the additionalenergy obtained by atmosphere and due to the sound absorption).

In several papers devoted to the concept of turbulence, it is shown that in someconditions the sound oscillations, even having relatively low power, can result insignificant change of the turbulence characteristics. Omitting the theory, let usconsider some conclusions and experimental results of such interaction investiga-tions on the basis of papers [63–66].

1. For rather great Reynolds numbers, there exist two-dimension speed fluctua-tions of a sinusoidal type (in atmosphere, their appearance is stipulatedby Helmholtz instability). Under the influence of sound wave, having thefrequency equal to the frequency of autooscillations of sinusoidal fluctua-tions, a sharp increase of fluctuations occurs. Sometimes several maximaat several frequencies occur, but always the main frequency can be distin-guished. According to [65], for initial turbulence intensity approximatelyequal to 16%, the acoustic influence leads to the increasing of turbulentenergy by 97%.

2. The spectra of the speed pulsations in the flow after acoustic influence havea resonant form. From the point of view of instability theory, the generaltendency of resonant frequency behavior seems to be interesting. First ofall, the resonant frequency is a continuous function of the flow speed, andfor speeds less than 20 m/s it lies in the interval from 10 Hz to 1,000 Hz.

3. If the sound oscillations affect turbulent flow rather strongly, increase inthe turbulence intensity occurs at the sonic frequency while its decrease takesthe place at other frequencies [i.e., a transformation of F (k ) is observed].

4. The influence of acoustic waves shows itself even when their intensity isnot high. According to [67], the sound field amplitude must have the valueabout 0.01–0.1% of intensity of the main flow, and the sound field intensitydoes not exceed 0.001–1 W/m2. The forecasting estimates show that sucha level of intensities can be expected at ranges less than some tens of metersfrom the object (e.g., airplane or missile), which produces this acoustic field.

Experimental investigations were carried out, taking into account the complex-ity of the theoretical estimates of such sound wave interaction with atmospheric

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 53

turbulence and their influence on characteristics of the microwave backscatteredfield. An operating jet airplane engine was used as the source of intensive soundfield. The parameters of the turbulent atmosphere were analyzed by two techniques.

In the first, the measurements were made of flow speed turbulent pulsationsfor an engine operating in different regimes by means of thermoanemometer. Themeasured parameter in this case was the variance of the flow speed differences atseveral spatially separated points. The distance between sensors was 5 cm, and thisprovided a decrease in the influence of the direct sound field on the results ofmeasurements. Because the air oscillations for frequencies of hundreds of Hertz,where the main part of the engine noise power is concentrated, were practicallyidentical for both sensors, the signals received by them were mutually canceled.The experiment scheme is presented in Figure 1.25(a).

The directions between the sensors and the axis of jet wake formed the angle40°–60°, which excluded the entry of the combustion products into the displacementzone of the sensors. A sample of the process for a shutdown engine and for oneoperating in its nominal regime is demonstrated in Figure 1.25(b). It is seen thatthe speed fluctuation difference variance for the operating engine is larger thanthat with an undisturbed atmosphere. A forced engine operation regime did notlead to significant change in fluctuation variance in comparison with the nominalregime.

The cumulative distributions of different variance are represented in Figure1.26 (abscissa axis has the normalized values). It is seen that the influence of thesound field results in an increase in the speed difference variance in comparison

Figure 1.25 (a) The scheme of experiment and (b) the sample of flow speed turbulent pulsations for muffledand operating engine. (From: [7]. 1997 SPIE. Reprinted with permission.)

54 Radar Characteristics of Targets

Figure 1.26 Cumulative distributions of flow speed different variance. (From: [7]. 1997 SPIE.Reprinted with permission.)

with the experiment when the engine is shut down. For the 50-m range, thisdifference is less evident and shows itself in the area of large variances only.Consequently, in the area of spatial scales of some centimeters, the sound influenceon the turbulent atmosphere is observed only for ranges less than 10–20m.

The second direction of investigations consisted of direct estimation of themicrowave backscattering electromagnetic field level in the area of an operatingengine. These measurements were performed with X-band continuous-wave radar.The illuminated spot was raised to the height of 100m over the aircraft to removethe reflections from its vibrating surface and from the area behind the engine. Theminimal detectable RCS of this radar in the area of supposed backscattering wasabout 10−4–10−5 m2 in the filter bandwidth of 10 Hz.

The observations have shown that the reflections from the atmosphere zonesurrounding the aircraft did not exceed the minimal detectable level of RCS forall operation regimes of the jet engine. Therefore, further radar measurements areexpedient using more sensitive radars.

The results of investigation show that among three considered mechanismsof possible radar observation of aerodynamic objects using atmospheric soundperturbations caused by their flight, the most promising is the backscattering from

1.4 Statistical Characteristics of Targets 55

the shock wave front occurring during supersonic flight. The microwave backscat-tering from perturbations of the dielectric constant of the air, resulting from soundwave influence, turns out to be comparable with backscattering from naturalturbulence of the atmosphere. The backscattering from the turbulent atmospheredue to variations in its spatial and temporal characteristics, caused by sound energytransformation into the power of turbulence, seems to be promising for radardetection and requires the further investigations with more sensitive radars.

The general requirement is the use of L- or UHF-band radars for which thesemechanisms appear to work more efficiently.

1.4 Statistical Characteristics of Targets

1.4.1 Target Statistical Models

The statistical model of a target, as in general the description of its properties,must be developed with clear orientation to the scope of the problems beingconsidered. Here the model analysis is chosen for applicability to the task of thedetection and radar range estimation. The target model for this case has to correctlyreflect only the fluctuation statistics of the signal amplitude or power (proportionalto RCS) at a given moment of time. These properties are fully determined by theamplitude and phase relations of the elementary electromagnetic waves created bythe target scattering elements in the radar receiver.

For rather general assumptions about the characteristics of the bright pointsforming the radar target, it is possible to form the statistical description of theecho. The diffraction field (or signal amplitude) can be presented in the followingway [68]

S (u ) = ∑j

A j ejwj = X + iY (1.47)

where Aj is the echo amplitude of j th bright point. The phase w j of every ele-mentary signal depends on the range dj from the j th point to the plane of signalregistration and the character of this point.

w j = 2kdj + w jo (1.48)

The coordinate of dj can depend on the target aspect according to dj = djo cos uor in a more complicated manner for continuous change of the surface radius ofcurvature. For an unknown type of target motion and bright point distribution,the values dj can be considered as the random values with some distribution lawp (d ). The pdf of phase does not practically depend on the distribution p (d ) andapproaches a uniform distribution in the interval [−p , p ] if d /l >> 2p .

56 Radar Characteristics of Targets

For slow change of amplitude Aj in comparison with phase variation, the signalcomponents in X and Y have asymptotically Gaussian distributions with equalvariances (i.e., the echo is a narrowband Gaussian process). Generally, for symmetri-cal pdf p (w ), the mean value of the sine-component is equal to zero; the mean of thecosine-component is a ; and the variances of each component s1 , s2 are equal to

a = X = nA 2 Ep (w ) cos w dw

s1 = nA 2 Ep (w ) cos2 w dw −a2

(1.49)

s2 = nA 2 Ep (w ) sin2 w dw

The pdf of amplitude A is expressed as

p (A ) =A

√s1s2expF−

d 2

2s1−

(s1 + s2)2

4s1s2G ? ∑

m(−1)mImSs1 − s2

4s1s2A2DI2mS a

s1D

(1.50)

For uniform phase distribution, we have a = 0, s1 = s2 and the amplitude pdfis Rayleigh:

p (A ) =2An

expS−A2

n D (1.51)

Such targets are called the Rayleigh targets.If among the elementary scatterers there exists a bright point nonfluctuating

both in amplitude and phase, then the signal amplitude is described by the Ricianpdf (a = c , s1 = s2)

p (A ) =2An

expS−A2 + c2

n D ? I0S2cn D (1.52)

For more generalized case with s1 ≠ s2 one can find

p (A ) =k2 + 1

kA exp3−

k2 + 12 SB 2 +

k2 + 1

2k2 A2D (1.53)

× ∑m

(−1)mImSk2 − 1

4k2 DA2I2m (k2 + 1)BA4where k = √s1 /s2 ≠ 1, B = c2 /(s1 + s2) ≠ 0.

1.4 Statistical Characteristics of Targets 57

The set of parameters B = 0, k = 1 corresponds to the Rayleigh distribution,and B ≠ 0, k = 1 corresponds to the Rician distribution.

For the objects with reflecting area dimensions many times larger than thewavelength, the parameter k differs from unity very little because the phase fluctua-tions in this case exceed p . It permits wide use of the Rayleigh target model. It isworth noting that it describes well the scattering characteristics of complex targetsif accuracy is not required in the pdf tails.

To improve agreement between the model and experimental pdf, some authorsmade attempts to use more complicated concepts of real target reflection models.For example, there was the model in which the reflector forming the Rician distribu-tion was augmented by high directivity reflectors with large RCS (comparable withtotal contribution from the other elements). Other models used strong specularreflections appearing only for narrow angle regions, resulting in RCS spikes forspecific target aspects. However, these models did not receive the wide applicationin practice because of complexity or impossibility of multiple parameter estimationsfrom the real data of these distributions.

In the practice of statistical theory of radar, priority was given to the chi-squaredistribution, which for appropriate selection of parameters coincides rather wellwith the Rayleigh or produces enough accurate approximations of most empiricalpdfs. The Swerling models [69] are widely used in radar theory as variants of themodels with RCS variation according to chi-square distributions. The chi-squarepdf takes the form [69]

p (s, s ) =1

G(n /2)?

n2s

? Sns2s D

(v −2)/2

expS−ns2s D (1.54)

=n

s 2v /2G(n /2)(x2)(v −2)/2 expS−

x2

2 Dwhere x2 = ns /s, G(n /2) is the gamma-function, s is the mean RCS value, and nis the number of degrees of freedom determining the ratio of the square of themean RCS to RCS variance. Widely applied in radar, Swerling’s models are shownin Table 1.25.

It is supposed in the first and third Swerling models that the received signals arenonfluctuating ones for one scan (slowly fluctuating pulse train) and uncorrelatedbetween two scans. There are the varying signal fluctuations within one scanduration for the second and fourth models (independently fluctuating pulse train).For the Rayleigh case (1 and 2), the RCS pdf is described by the exponential law

p (s, s ) =1s

expS−ss D (1.55)

58 Radar Characteristics of Targets

Table 1.25 The Swerling Models

Number of FreedomType of Model RCS pdf Echo Fluctuations DegreesModel 0(Marcum’s model) Stable target No ∞Model 1 Rayleigh Slow 2Model 2 Rayleigh Fast 2Model 3 Four degrees of freedom Slow 4Model 4 Four degrees of freedom Fast 4

For the two other models, the RCS pdf has the form

p (s, s ) =4ss

expS−2ss D (1.56)

They are similar to the RCS pdf of a target having one dominant stable scatterertogether with many randomly distributed scatterers. In this sense, the expressions(1.55) and (1.56) give close results in the region of large probabilities.

For the majority of complex targets, the Swerling models 1 and 2 are applicable,especially for observations over the angle range of about 360°. The application ofthe standard Swerling models, as a rule, leads to decreased estimations of RCSprobabilities for the area of small values of probability, precluding reliable estimatesof detection probability for a single signal at the area of correct detection probabili-ties of about PD ∼ 1.

Besides, the Swerling models are poorly applicable to rather simple targets withsmall numbers of bright points. These difficulties were overcome by Weinstock[70, 71]. He extended the area of application of the chi-square distribution todegrees of freedom less than two. The number of degrees of freedom for somebodies of simple shape evaluated by Weinstock is presented in Table 1.26.

This approach permits us to use the chi-square distribution for a wide class ofapproximations from Rayleigh and Rician ones to the popular log-normal distribu-tion by adjusting of the numbers of degrees of freedom.

1.4.2 Real Target Statistical Characteristics

The most extensive experimental data are obtained for air target RCS distributions[72–75]. These results permit us to confirm that for air targets, the amplitude

Table 1.26 The Number of Degrees of Freedom for Simple Bodies

Type Freedom Degrees NumberRandomly oriented cylinder 0.6–1.4Stabilized cylinder 4.0Randomly oriented bodies of spheroid shape 4.0Source: [70].

1.4 Statistical Characteristics of Targets 59

distributions can be approximated rather well by the Rayleigh law up to amplitudevalues six times greater than the mean. As illustration, the cumulative RCS distribu-tions for piston-engine and jet airplanes are presented in Figure 1.27 obtained in[72] at the 3-cm wavelength; the exponential distribution is showed by the dashedline.

It is seen that the experimental results agree with the Rayleigh model in thearea of most probable RCS values, although the RCS mean value is somewhatgreater than in this model (within limits from 1.1 to 1.5) (i.e., the distribution doesnot predict the largest observed signal intensities). For jet airplanes, the agreementof the experimental results with the Rayleigh model is more satisfactory than forpiston-engine aircraft. These results are obtained for long-period observation oftargets or for averaging of signals over a wide aspect sector. If the observationtime is small relative to the echo fluctuation period, a different chi-square lawdescribes the experimental data. For long observation time and large airplane aspectchange, the data approach the Rayleigh statistics (n = 2), and for small time ofobservation the degree of freedom number increases up to n = 10. The results of

Figure 1.27 Cumulative RCS distributions of (a) piston-engine and (b) jet airplanes at the X-band.(After: [72].)

60 Radar Characteristics of Targets

freedom degree number determination for three types of airplanes and for a helicop-ter obtained at the wavelength of 10 cm are presented in Figure 1.28 [73]. It isseen that for a spectrum width of 20 Hz, good agreement between the experimentaldata and the Rayleigh model is observed for observation time more than 5 seconds.

For air target observation over a sector of 10°, the freedom degree numberdecreases to values from 0.9 to 2.4.

The RCS distributions for large land and marine targets, as a rule, coincidewell with the Rayleigh model in the area of the most probable RCS values [23,26]. The distributions tails, characterizing the probability of large RCS values, areusually higher than for the model. However, chi-square distributions with freedomdegree number n ≥ 2 can approximate the experimental results. As an illustration,the distributions of the instantaneous values of signal from the output of a phasedetector are presented in the Figure 1.29(a) for onshore and inflatable boats. Thesedata were obtained at the 3-cm wavelength and are plotted on a scale linearizing theGaussian distribution. The amplitude distributions for an inflatable are presented inFigure 1.29(b) on a scale linearizing the Rayleigh law.

Similar results are obtained for small marine targets. The amplitude distribu-tions for marine buoys in X-band are closer to the Rician distribution, while forthe majority of targets like motor boats, rowboats, spheres, and corner reflectorsplaced on the sea surface, the amplitude distributions are approximated by Rayleighand Rician laws in a satisfactory way. As an illustration, the RCS cumulativedistribution for an anchored sphere obtained at a 3-cm wavelength for sea stateabout of two is presented in Figure 1.30 [27]. The solid curves correspond to chi-square distributions with different number of degrees of freedom. It is seen that

Figure 1.28 Freedom degree number for airplanes and helicopter at S-band. (After: [73].)

1.4 Statistical Characteristics of Targets 61

Figure 1.29 (a) The cumulative instantaneous value distributions for inshore and inflatable boatsand (b) amplitude distribution for inflatable boats at X-band.

the experimental data satisfactory coincide with the chi-square distribution forn = 5–6.

Some RCS distribution peculiarities are observed for simple cone-cylinderbodies. In the majority of situations during experimental investigations, their differ-ence from Rayleigh model was noticed; in particular, the ratio of the power of thefluctuation component to the total power was equal to 0.03–0.19, which is lessthan the corresponding value for a Rayleigh target [23, 74]. RCS cumulativedistributions for a cone-cylinder body with 150-mm diameter and about 500-mmlength obtained at X-band are presented in Figure 1.31 [23, 26] (solid lines arethe chi-square distributions with different number of freedom degrees). The compar-ison of the experimental data with the model permits us to assert that with reason-able confidence, the approximation of these data is possible by chi-squaredistributions with 0.8 ≤ n ≤ 4.

62 Radar Characteristics of Targets

Figure 1.29 (Continued.)

In spite of comparatively poor data for the statistical characteristics of biologicobjects, apparently, it is possible to establish as a working hypothesis that the RCSdistribution for man, large animals, and birds is lognormal with a standard deviationof 8–10 dB [29, 30, 75] and that the amplitude distribution of insect backscatteringis Rayleigh.

Consequently, the analysis of RCS experimental distributions for practicallyall types of targets confirms the validity of their approximation by chi-squaredistribution with different number of degrees of freedom for solving detectionproblems. For targets of complex shape, it is possible the use of the standardSwerling models for practically all cases.

1.4.3 Echo Power Spectra

The radar signals scattered from moving objects have Doppler frequencies deter-mined by target velocity and aspect, and these frequencies can reach rather highvalues. Besides a linear motion, the motion of object can have angular and linear

1.4 Statistical Characteristics of Targets 63

Figure 1.30 Cumulative RCS distributions for anchored sphere at X-band and comparison withchi-square distribution.

oscillations in three planes. As a result of object vibration and oscillatory motion,the individual parts of the object can have different radial velocities with respectto the radar. This phenomenon creates the echo spectrum. For a complex objectmotion, including linear motion with some velocity V and an oscillating componentmV cos (V + w ), where m is the modulation coefficient, the echo is frequencymodulated with the spectral components that are equally spaced with respectto the Doppler frequency determined by target velocity and spatial orientation.Acceleration of the object results in spreading of the spectrum.

As D. K. Barton showed [75, p. 82], the power spectrum of the echo from atarget of complex shape can be presented as the spectrum of a Markov’s processwith intensity reduced at frequencies above some defined DF

64 Radar Characteristics of Targets

Figure 1.31 Cumulative RCS distributions for cone-cylinder targets with diameter of 150 mm atX-band.

G (F ) = G0F1 + SF − F0DF D2G−1

(1.57)

where G0 is the spectral density at frequency F0 determined as F0 = 2V /l , V isthe object radial velocity, and DF is the half-width of the spectrum at the half-power level. Then the spectral density decreases at the rate of 6 dB per octave for|F − F0 | >> DF.

The signal power spectra at the X- and Ka-bands for all air targets includingairplanes and helicopters are characterized by comparatively small spectrum widthand large value of F0 due to linear motion. So, for the L-200 aircraft, the effective

1.4 Statistical Characteristics of Targets 65

spectral width at X-band is 20–30 Hz at the −6 dB level, almost independent ofthe polarization This corresponds to the measured spectra results for piston-engineand jet airplanes of [72], the author of which expressed the normalized autocorrela-tion function as

r (t ) = exp (−t /t0) (1.58)

where t0 is the correlation interval having a value close to 0.05-second 3-cmwavelength.

For all piston-engine and jet airplanes for forward aspects, pronounced peaksin the spectra are observed that result from propeller or compressor vane rotation,as illustrated in Figure 1.32. For lower levels (less than −30 dB), one can expectspectral components with frequencies corresponding to fuselage vibrations. As isshown in [76], the fuselage vibrations for piston-engine and jet airplanes aredescribed by the dependence G (F ) ∼ F −2 and have frequency components of 500–600 Hz. Those components with vibration amplitude of Imax ≥ 2.5 are found inthe frequency band less than 40 Hz. In this situation at a wavelength of 8 mm,the phase modulation index is approximately equal to two, which is why thespectral components reach 150–200 Hz (i.e., noticeable spectrum spreading isobserved).

Figure 1.32 Power spectrum for L-200 airplane at X-band. (From: [26]. 1995 SPIE. Reprintedwith permission.)

66 Radar Characteristics of Targets

The echo spectra of helicopters have evident propeller modulation. To demon-strate this phenomenon, the power spectra of echoes from the Russian helicopterMi-4 is presented in Figure 1.33, obtained at a wavelength of 3 cm for differentaspects. For flight directions toward the radar, the power spectra are quite narrow;their width at the −10-dB level does not exceed 300–400 Hz. For increasing rangebetween helicopter and radar, the spectrum spreads significantly. This can be causedby modulation of the echo by the rotating rear propeller. For a hovering helicopter,the spectrum central frequency F0 shifts into the zero frequency area, and significantspectrum spreading at the level of (1–2) ? 10−6 m2/Hz is observed up to frequenciesof 10 kHz. Besides, there are the peaks of propeller modulation that are at6–10 dB higher than the surrounding average level. The propeller modulation peakswere not observed at a wavelength of 8 mm, and the spectra were still wider,reaching 1 kHz at the −10-dB level.

The echo power spectra for land objects are also comparatively narrowband,their width at the −3-dB level lying within the limits of DF /F0 = 0.06–0.23. Thespectrum width values for some land objects at the 3-cm wavelength are presentedin Table 1.27.

Figure 1.33 Power spectra of helicopter Mi-4 at X-band. (From: [26]. 1995 SPIE. Reprinted withpermission.)

1.4 Statistical Characteristics of Targets 67

Table 1.27 Land Target Spectrum Width at Wavelength of 3 cm

Spectrum Width (Hz) at LevelObject −10 dB −20 dBTank 50 200Armored car 50 190Heavy artillery tractor 115 400–600Light artillery tractor 100–300 300–550Truck 50–135 200–450

Examples of power spectra for tank and truck at the wavelength of 8 mm arepresented in Figures 1.34 and 1.35.

It is seen from Table 1.27 that at low levels of intensity the spectrum widthfor targets of the first two types (caterpillar objects) is significantly less than forobjects having a large number of independently moving parts (wheeled objects).The central frequency F0 and spectral width DF increase in inverse proportionalto the wavelength.

Figure 1.34 Power spectra for tank at X-band.

68 Radar Characteristics of Targets

Figure 1.35 Power spectra of GAZ-63 truck at X-band.

The power spectra for large marine targets caused by pitch, roll, and heave inrough sea are characterized by comparatively small width, not exceeding a fewhertz in X-band, and rather large central frequencies. Let us determine the powerspectra shape, taking into account that the spectrum of echo G (v ) is connectedwith the spectrum of target heave Gz (v ) by the following relation [77]

G (v ) = v2 ? Gz (v ) (1.59)

The spectrum of heave is determined by the spectrum of the waves Gm (v )through the transfer function K (v ) of the linear dynamic system.

It is known that a dynamic system like a ship has a rather narrow passband,and its transfer function can be presented as [77]

1.4 Statistical Characteristics of Targets 69

|K (v ) |Z =kZ

√(1 − v2/v 2Z )2 + 4mZ v2/v 2

Z

(1.60)

where vZ is the natural frequency of ship oscillations, mZ is the heave decrement,and kZ is the reduction coefficient.

Then one can get the following expression for the power spectra of heave

GZ (v ) = |K (v ) |2Z ? Gm (v ) =k2

Z Gm (v )

(1 − v2/v 2Z )2 + 4mZ v2/v 2

Z

(1.61)

For the majority of ships and other marine objects, especially small ones, thenatural frequencies of oscillations lie in the area vZ >> v, then

GZ (v ) ≈ k2Z Gm (v ) (1.62)

(i.e., the spectrum of heave is proportional to the wave power spectrum). Takinginto consideration that the spectrum of fully developed waves in the gravitationaldomain can be approximated as

Gm (v ) = bg2v−5 expF−0.74S gvU D2G (1.63)

where U is the wind velocity, g is the gravitational constant, and b is the parameter,which weakly depends on the frequency varying from 2 ? 10−3 in the low-frequencyregion to 10−2 in the high frequency region of the gravitational spectrum.

Then it is possible to predict from (1.60) and (1.61) that the echo spectrumin the high-frequency region is described by the dependence G (v ) ∼ v−3. Takingthis into account, we derive the spectrum approximations for marine objects as[23, 26]

G (F ) = G0F1 + | F − F0DF |nG

−1

(1.64)

where G0 is the spectral density at frequency F0 determined as F0 = 2V /l , DF isthe spectrum half-width at the −3 dB level, and V is the radial component ofthe target velocity in respect to the radar. The spectrum width and power indexexperimentally obtained for some marine targets at the 3-cm wavelength are pre-sented in Table 1.28 and the spectrum width at wavelengths of 8 mm and 4 mmare shown in Table 1.29 (data were obtained for sea state 1–2). It can be noticed

70 Radar Characteristics of Targets

Table 1.28 Spectrum Width and Power Index of Marine Targets at Wavelength of 3 cm

Spectrum Width (Hz)at Level (dB)

Object −3 −10 Mean Index nOnshore motor boat–Anchored 2.0 4.6 2.6–Moving 5.0 10.0 3.2Message cutter 7.0 15.0 2.75Cruiser yacht 4.0 10.0 2.6Sport yacht 6.0 12.0 3.7Motor boat 3.0 11.0 1.9Rowboat 2.0 5.5 2.6Inflatable boat 8.5 20.0 2.5Windsurf 7.0 19.0 2.4Anchored barrel 5.0 11.5 3.0

Table 1.29 Spectrum Width of Marine Targets at Millimeter Band

Object Wavelength (mm) Spectrum Width (Hz) at Level −3 dBMotor boat 8 13.0Boat 8 8.5Inflatable boat 8 8.0Navigation buoy 4 5.5Spherical buoy 8 0.6–0.8Spherical buoy 4 0.65

that the power degree indices in the expression for target power spectrum usuallydecrease with decreasing object size or increasing sea state.

The power spectra of biological objects have some rather typical peculiarities.The spectra of human echoes are narrowband, their width for the −3-dB level equalto 20–30 Hz and mean Doppler frequency F0 for the motion is 80–100 Hz at the3-cm wavelength. The echo amplitude and frequency modulation take a formcontrolled by motion of arms and legs and is expressed in the appearance of stepmodulation peaks. As an illustration, the current spectra of moving man andswimming man obtained at a 3-cm wavelength are presented in Figures 1.36 and1.37.

The signals scattered from birds are, as a rule, amplitude modulated ones. Thefrequency of modulation is inversely proportional to bird size. In [63], it wasexperimentally determined that the modulation frequency Fm (Hz) is connectedwith the wing length l (mm) as

Fm l 0.897 = 572 (1.65)

The frequency of wing strokes for the white heron equals to 2–4 Hz, and isabout 10 Hz for the swallow, 3–4 Hz for the seagull, and 6–7 Hz for ducks. The wing

1.4 Statistical Characteristics of Targets 71

Figure 1.36 Power spectrum of moving man.

oscillations are nonsinusoidal, and there are a number of harmonic components inthe echo. The velocity of birds can be about 15 m/s and more for migration andlocal flights, corresponding to Doppler frequencies of about 1 kHz at a 3 cmwavelength.

The Doppler spectrum width can be determined as [63]

DF = 10.9Ad 0.21l−1 (1.66)

where A is the angular motion amplitude of wing for stroke and d is the distancefrom body to forearm end (m) excluding the length of flapping feathers. It is seenfrom (1.66) that a bird size change by 10 times leads to only 60% spectrumwidening. As a rule, these spectra are rather narrowband; their width being notgreater than 10–20 Hz at a 3-cm wavelength. The spectrum of the echo from aseagull obtained at X-band is presented in Figure 1.38.

Thus, we have presented the experimentally obtained power spectra at micro-wave and millimeter bands for practically all types of targets and a convenientapproximation of spectrum shape by expression of (1.64).

72 Radar Characteristics of Targets

Figure 1.37 Power spectra of swimming man at X-band.

1.5 Surface Influence on the Statistical Characteristics of RadarTargets

1.5.1 Diffuse Scattering Surface Influence on the Statistical Characteristics

In the shortwave part of the centimeter band and, particularly, in the millimeterband, the diffuse component of the electromagnetic field scattered by the surfacegrows significantly together with the destruction of the specular reflection. In thissituation the spatial correlation radii of the reflected diffuse component over thetarget is larger in most cases than the geometric dimensions of the target [78].Then the received signal is a product of two terms: the first describes the radartarget signal fluctuations in free space and second one describes the fluctuationsof the propagation factor. This division was used in some papers [79, 80].

Later we will consider the point target placed over the rough surface that isstatistically equivalent (in its RCS distribution functions and power spectra) to thereal target. In this case, the surface influence reduces to modulation of the targetecho. The signal amplitude and RCS in the point of reception can be presented as

1.5 Surface Influence on the Statistical Characteristics of Radar Targets 73

Figure 1.38 Power spectrum of seagull at X-band.

At* = At ? F 2 (1.67)

s t* = s t ? F 4 (1.68)

where At , s t are the amplitude and RCS of the target in free space, respectively,and F is the surface propagation factor.

Using the relations for the pdf the product of two random values [80] andassuming that the RCS probability function in free space is described by the standardSwerling models [69, 81], we obtain the expression for the probability function ofthe normalized RCS as

pI (j ) = A0 E∞

0

expF− Sjx

+ b ? xDG ? I0(g ? x1/2)dxx

(1.69a)

74 Radar Characteristics of Targets

for Swerling models 1 and 2, and

pII (j ) = 4jA0 E∞

0

expF− S2jx

+ b ? xDG ? I0(g ? x1/2)dx

x2 (1.69b)

for Swerling models 3 and 4.Here

j =s i*

s i*is the target RCS normalized to mean value.

A0 =1

2rdexpS−

f 20

2r2dD.

b =1

2r2d

; g =f 2

0

2r2d

.

f0 = √1 + r20 − 2r0 cos u is the specular reflection coefficient.

r0 , rd are the specular and diffuse reflection factors depending on surfaceroughness, radar wavelength and determined, for example, in [11].

I0 is the Bessel function of zero order.

Calculating (1.69) we obtain finally

pI (j ) = A0 ∑∞

n =0

g2n

22n(n !)2 ? S jb D

n /2

Kn X2√jb C (1.70a)

pII (j ) = 8jA03∑∞

n −1

g2n

22n(n !)2 ? S jb D

(n −1)/2

Kn −1 X2√jb C + S jb D

1/2

? K1 X2√jb C4(1.70b)

where K (?) is the modified Bessel function.Equations (1.70a) and (1.70b) permit us to obtain the moments for the RCS

probability functions in the first and second Swerling models as

ml (I ) = A0 l ! E∞

0

exp (−b ? x ) ? I0(gx1/2)xl ? dx (1.71a)

1.5 Surface Influence on the Statistical Characteristics of Radar Targets 75

ml (II ) = 2−lA0(l + 1)! E∞

0

exp (−b ? x ) ? I0(g ? x1/2)xl ? dx (1.71b)

Comparing (1.71a) and (1.71b), it is easily to obtain the dependence connectingthe moments for two models in the form

ml (II ) = 2−l ? (l + 1) ? ml (I ) (1.72)

Equations (1.71a), (171b), and (1.72) permit us to obtain all l-moments of thenormalized target RCS including the mean value, rms, and the skewness andasymmetry for most models.

Let us consider some particular cases of the rough surface effect on the radartarget statistical characteristics.

For a weakly rough surface in the maximum of the interference lobe, whenthe condition f0 /rd > 1, is fulfilled, and for a surface with considerable roughnessat great ranges, when the diffuse scattering coefficient rd decreases more quicklythan the specular reflection coefficient r0 , the propagation factor density functionis close to the Gaussian distribution [78]

p (F ) =1

√2p ? rdexpF−

(F − f0)2

2r2d

G (1.73)

and for f0 /rd >> 1

p (F ) = d (F − f0) (1.74)

where d (?) is the Dirac function. Then the RCS distribution of the target can bepresented as

pI (s t*) =1

s t*? expS−

2s t*

s t*D (1.75a)

pII (s t*) =4s t*

(s t*)2 expS−2s t*

s t*D (1.75b)

for the first and second Swerling models, respectively, where s t* = s t /f 20 .

The comparison of (1.75) with the RCS distributions in the standard Swerlingmodel shows that for a weak diffuse surface scattering, the common shape of theRCS distributions is retained and a scale transformation takes the place.

76 Radar Characteristics of Targets

For small values of f0 /rd << 1, typical for pure diffuse scattering and also inthe interference minima of the electromagnetic field (f0 → 0), the density functionof the surface propagation factor has the form

p (F ) =F

r2d

expS−F 2

2r2dD (1.76)

and the RCS distributions equal

pI (j ) = 2B0K0 X2√bj C (1.77a)

pII (j ) = 8jB0K0 X2√bj C (1.77b)

The comparison of (1.75) and (1.76) shows the significant difference of theRCS distributions from the Swerling’s models at the last case.

For great values of j (i.e., in the band of the intensive spikes of the echo), usingthe asymptotic form [82]

Kv (z ) = √ p2z

? e −z (1.78)

z → ∞

one can show that the rate of decrease of the distribution for large values of targetRCS in (1.77) is determined by the term exp (−2b1/2j1/2) and is less than in theSwerling models for all values of b ≠ 0.

An examination of these models was carried out using experimental data onthe statistical characteristics of the RCS for some types of the cone-cylindricaltargets with a diameter of 100 mm moving over a land surface with a velocityof about 180 m/s. The cumulative functions of the RCS distribution obtainedexperimentally (curve 1) and derived with use of (1.77) for two values of the diffusescattering coefficient (curve 2 for rd = 0.2 and curve 3 for rd = 0.4), and the RCSdistribution for the first Swerling model (line 4) are presented in Figure 1.39; line4 describes in the best way the target RCS distribution in free space [83, 84]. It isseen that except in the regions of high confidence, the data coincide with the derivedmodel. The difference between the derived and experimental results in the initialpart of the cumulative function can be explained by the use of the approximatemethod of derivation based on the inclusion of only diffuse echo (i.e., the applicationof the Gaussian distribution of the diffuse scattering coefficient instead of the Riciandistribution) and the large errors of the experimental results in this part of thecurve.

The RCS distributions for targets over a land surface smoother decrease forthe large values of the RCS in comparison with the standard Swerling distributions.

1.5 Surface Influence on the Statistical Characteristics of Radar Targets 77

Figure 1.39 The RCS cumulative functions for cone-cylindrical targets. (From: [8]. 2001 SPIE.Reprinted with permission.)

Besides, the increasing of the RCS fluctuations is seen at small ranges when thespecular component of the electromagnetic field is absent because of the increasinggrazing angle and, as consequence, the increase of the roughness parameter. Atlarge ranges, the specular component of the field increases, and, as result, the RCSdistribution is near that for free space.

The results of derivation of the RCS distribution quantiles for experimentaldata and distributions of (1.75) for two values of the diffuse scattering coefficientsare presented in Table 1.30.

It is seen from Table 1.30 that the surface effect leads to an increase in theintensive spikes of the RCS. Increasing the diffuse scattering coefficient by a factor

78 Radar Characteristics of Targets

Table 1.30 The Comparison of the Experimental and Derived Quantiles of RCS Distributions

Distribution Quantiles, dB (m2)for Confidence Level

Conditions 0.1 0.5 0.9Experimental data for free space −13 −15 −24.5Derivation for rd = 0.2 −12 −18 −26Derivation for rd = 0.4 −8 −16 −26Source: [8].

of two leads to RCS growth by 4 dB for confidence level of 0.1. At the same time,the RCS values practically coincide for large confidence levels.

Therefore, a diffuse scattering surface leads to RCS distribution transformationof radar targets, especially for regions of large RCS.

1.5.2 Multiple Surface Reflection Influence

Let us estimate the RCS change for low-altitude targets with very small altitudes.The geometry of this problem is presented in Figure 1.40. For echo power reflectedby the target in the direction of the radar (the solid lines 1), the effect of the roughsurface is absent. For this case, the target is characterized by RCS s t ; for most ofthe targets, these characteristics are known rather well.

Power is also reflected to the surface and back to the radar (dotted lines 2),increasing the target total RCS

s S = s t + s t* (1.79)

where s t* is the component created by the interaction of the target with the surface(bistatic scattering).

One can estimate the power reflection from the target to surface direction usingthe bistatic RCS of the target in the direction of the surface. The power density atthe surface can be determined as

Figure 1.40 The geometry of the problem: (a) the scattering directions and (b) the lighted area.

1.5 Surface Influence on the Statistical Characteristics of Radar Targets 79

P2 =Pt Gt sb

(4p )2r2h2 (1.80)

where

Pt , Gt are the transmitter power and antenna gain, respectively.

r is the range from the radar to the target.

h is the target altitude.

sb is the target bistatic RCS.

According to Crispin’s theorem [84, 85], the mean value of the bistatic RCSfor all target aspects can be determined through the monostatic RCS. The powerdensity at the target equals

P3 =P2s 0Seff

4ph2 (1.81)

where s 0 = s 0(g ) is the surface normalized RCS depending on the incidence angleg and Sef

−f is the area of the illuminated surface that effectively reflects in the target

direction.The normalized RCS of the sea surface for small incidence angles g strongly

depends on the value of this angle. The normalized RCS for incidence angles from0° to 15°, obtained in [86] at the wavelengths of 3.2 cm and 0.86 cm for differentwind velocities, are presented in Table 1.31.

The value of the effective incidence angle g eff , determining the incidence angleinterval for which the surface effectively reflects the field energy to the targetdirection can be determined as

Table 1.31 Dependence of the Sea Normalized RCS on the Incidence Angle

Wind s 0 (dB) for Incidence Angles (degree)Wavelength (cm) Velocity (ms −1) 0 2.5 5.0 7.5 10.0 12.5 15.0

0.86 2.5–5.0 16 11 7 2.5 −5 −13 —5.0–7.5 14 13 12 11 9 6 27.5–10.0 13 12.5 12 11 9.5 8 6.5

10.0–12.0 12 11.5 10.5 9.7 8 7 6.03.2 2.5–5.0 5 3 0.5 −2.0 −5.0 −7.5 −10.5

5.0–7.5 3 2.5 1.5 0.5 −0.5 −2.5 −4.07.5–10.0 −1 −1.5 −3.0 −4.0 −5.0 −6.5 −8.0

Source: [8].

80 Radar Characteristics of Targets

g eff =

Ep /2

0

s 0(g ) ? dg

s 0(0)(1.82)

The derivation results of g eff from the data of Table 1.2 are presented inTable 1.32.

It is seen from Table 1.32 that for increasing wind velocity, a broadening ofthe angle cone takes place, determining the effective reflection of the signal fromthe surface to the target direction. This area equals to

Seff = p (g eff ? h )2 (1.83)

The reflected energy density at the radar is also determined by the target bistaticRCS

P4 =P3sb

4p ? r2 (1.84)

Then, taking into consideration (1.80)–(1.84), the received power can be pre-sented as

Pr =Pt G 2l2

(4p )3r4 ?s 0Seff s

2b

(4ph2)2 (1.85)

This expression is a radar equation for which the mean RCS for reflectionsfrom the surface equals to

s t* =s 0s

2b Seff

(4ph2)2 =s 0s

2b g

2eff

16ph2 (1.86)

It is seen from (1.86) that s t* is proportional to the square of the target bistaticRCS. It is inversely proportional to the square of target altitude, and it increases

Table 1.32 Dependence of geff on Wind Velocity

geff (degree), for Wind Velocity (ms−1)Wavelength (cm) 2.5–5.0 5.0–7.5 7.5–10.0 10.0–12.5

0.86 3.75 8.75 11.1 11.13.2 6.0 10.5 10.5 —

Source: [8].

1.5 Surface Influence on the Statistical Characteristics of Radar Targets 81

for increasing wind velocity because of the increasing effective angle. The derivationresults of s t* for some objects at the altitude h = 1m are presented in Table 1.33(g eff values are taken from the Table 1.32).

It is seen from Table 1.33 that significant growth of s t* is observed for targetshaving a bistatic RCS considerably exceeding, in the surface direction, the mono-static RCS.

For a nonfluctuating target moving over the surface, as shown in Figure 1.40,the reflected signal becomes a fluctuating one because of the surface effect. Let usconsider this phenomenon for the movement over the sea.

For known dependencies of s 0(g ), one can determine the probability functionof s 0 and, consequently, s t*, taking in consideration the amplitude modulation ofthe normalized RCS of the sea, because of the slope angle change of the scatteringarea caused by large waves. As is known [87], the probability function of the slopeangles u of the sea surface can be expressed as

p (u ) =1

√2psuexpS−

u 2

2s2uD (1.87)

where su is the rms value of the surface slope angles, which do not exceedsu ≤ 0.1 for sea [87]. Then we obtain using the results of [80]

p (st*) = p [u1(st*)] ? | du1

dst* | + p [u2(st*)] ? | du2

dst* | (1.88)

Taking into consideration that the s 0(u ) dependence at the angular band ofinterest can be approximated with sufficient accuracy by

s 0(u ) = s0 exp (−au ), u ≥ 0

we obtain

Table 1.33 Values of s t* for Different Objects

s t* (dB) forWind Velocity (ms−1)

Type of Target Wavelength (cm) sb /sb (dB) 2.5–5.0 7.5–10.0Bullet of caliber 22 3.2 1.25 −29.2 −30.3

0.86 7.5 −9.6 −4.3Sphere-cone-sphere with tophalf-angle 15° 3.2 6.0 −12.6 −7.3Cone with top half-angle 15° 3.2 −15.0 −54.6 −49.3Source: [8].

82 Radar Characteristics of Targets

u = −12

lnFs 0(u )s0

G (1.89)

Using (1.86) for s t* determination, we present it as

s t* = s ? s 0(u ) = Hs ? s0 exp (−au1), u1 ≥ 0

s ? s0 exp (au2), u2 < 0(1.90)

where

s =s

2b ? g

2eff

16ph2

Then

u1 = −12

lnS s t*

s ? s0D (1.91a)

u2 =12

lnS s t*

s ? s0D (1.91b)

and

| du1

ds t* | = | du2

ds t* | =1

as t*(1.92)

Taking in consideration (1.91), (1.92), (1.88) can be transformed to the form

p (s t*) =2

√2p ? asu s t*exp3−

ln2S−s t*

s ? s0D

2su a2 4 for 0 ≤ s t* ≤ s ? s0

(1.93)

It is seen from (1.93) that the probability density of s t* conforms to a truncatedlogarithmic Gaussian law.

Introducing y = s t*/ss0 and b = asu , we have finally

p (y ) =2

√2pbyexpS−

ln2 y

2b2 D, 0 ≤ y ≤ 1 (1.94)

1.5 Surface Influence on the Statistical Characteristics of Radar Targets 83

and the cumulative function

F (y ) = 52FS ln yb D, y ≥ 0

0, y < 0

(1.95)

where F(?) is the Laplace function.The obtained distribution of s t* has higher level of tails in comparison with

the standard Swerling models. The probabilities of exceeding the median value byan amount y at wavelengths of 3.2 cm and 0.86 cm are presented in Figure 1.41.

The analysis of these results shows that the increase of sea echo fluctuationsin the millimeter band in comparison with the centimeter band leads to increasingthe probability of the s t* value exceeding the median value. For a simple shapetarget at small altitude the over the surface of a rough sea, the increase in the RCScaused by the reflection from the surface exceeds the monostatic RCS of thesetargets with 10% confidence.

Figure 1.41 Probability of median value exceeding by y. (From: [8]. 2001 SPIE. Reprinted withpermission.)

84 Radar Characteristics of Targets

Thus, it is possible to arrive at the following conclusions:

• The flight of low-altitude targets over a diffuse scattering surface leads toa change in their distribution laws in the region of large values of RCS, incomparison with Swerling’s models.

• An analogous result gives the electromagnetic field reflection for the pathtarget to surface. The RCS increase and higher level of tails in the RCSdistributions are most visible for targets when the bistatic RCS in the surfacedirection exceeds the monostatic RCS. The effect of RCS increase is observedmost strongly for target altitudes of meters over the surface.

• The differences of the RCS distributions from the Swerling models increasewith the radar wavelength reduction.

It is necessary to note that these conclusions can be similarly applied to theland surface.

References

[1] Kulemin, G. P., and V. B. Razskazovsky, ‘‘Radar Reflections from Explosion and GasWake of Operating Engine,’’ IEEE Trans. on Antennas and Propagation, Vol. 45,No. 4, 1997, pp. 731–739.

[2] Kulemin, G. P., and V. B. Razskazovsky, ‘‘Centimeter and Millimeter Radio Wave SignalAttenuation in Explosion,’’ IEEE Trans. on Antennas and Propagation, Vol. 45, No. 4,1997, pp. 740–743.

[3] Neuringer, J. L., ‘‘Derivation of the Spectral Density Function of the Energy Scatteringfrom Underdense Turbulent Wake,’’ AIAA Journal, Vol. 7, No. 4, 1969, pp. 728–730.

[4] Fox, J., ‘‘Space Correlation Measurements in the Fluctuating Turbulent Wakes BehindProjectiles,’’ AIAA Journal, Vol. 5, No. 2, 1969, pp. 362–368.

[5] Velmin, V. A., Yu. A. Medvedev, and B. M. Stepanov, ‘‘Radar Echoes from ExplosionArea,’’ Journal of Experimental Techniques and Physics, Vol. 7, December 1968,pp. 455–457 (in Russian).

[6] Boynton, F. P., C. B. Ludwig, and A. Thomson, ‘‘Spectral Emissivity of Carbon ParticlesClouds in Rocket Exhausts,’’ AIAA Journal, Vol. 6, No. 5, 1968, pp. 116–124.

[7] Kulemin, G. P. and V. B. Razskazovsky, ‘‘Radar Backscattering from Sonic PerturbationsCaused by Aerodynamic Object Flight,’’ Proc. SPIE Radar Sensor Technology II, Vol. 3,No. 066, June 1997, pp. 194–202.

[8] Kulemin, G. P., ‘‘Sea and Land Surface Influence on the Statistical Characteristics of theLow-Altitude Radar Targets,’’ SPIE Proc., Vol. 4, No. 374, 2001, pp. 156–164.

[9] Shtager, E. A., Radiowave Scattering on the Complex Objects, Moscow, Russia: Radioi svyaz, 1986 (in Russian).

[10] Pavelyev, A. G., ‘‘About Scattering of Electromagnetic Waves by the Rough Surface andFrequency Spectrum of the Scattered Signal,’’ Radiotechnics and Electronics, Vol. 14,No. 11, 1969, pp. 1923–1931 (in Russian).

References 85

[11] Kulemin, G. P., and V. B. Razskazovsky, Millimeter Radiowave Scattering by Earth’sSurface at Small Angles, Kiev, Russia: Naukova Dumka, 1987 (in Russian).

[12] Blore, W. E., ‘‘The Radar Cross-Section of Ogives, Double-Backed Cones, Double-Rounded Cones and Cone-Sphere,’’ IEEE Trans. on Antennas and Propagation,Vol. AP-12, No. 5, 1964, pp. 582–590.

[13] Ruck, G. T., Radar Cross-Section Handbook, Vol. 1, New York: Plenum Press, 1970.

[14] Crispin, J. W., and A. L. Maffett, ‘‘Radar Cross Section Estimation of Complex Shapes,’’Proc. IEEE, Vol. 53, No. 8, 1965, pp. 972–981.

[15] Beckmann, P., and A. Spizzichino, The Scattering of Electromagnetic Waves from RoughSurfaces, London, England: Pergamon Press, 1963; Norwood, MA: Artech House, 1987.

[16] Crispin, J. W., and K. M. Siegel, Methods of Radar Cross-Section Analysis, New York:Academic Press, 1968.

[17] Weiss, M. R., ‘‘Numerical Evolution of Geometrical Optics Radar Cross Section,’’ IEEETrans. on Antennas and Propagation, Vol. AP-17, No. 2, 1969, pp. 229–231.

[18] Howell, N. A., ‘‘Computerized Ray Optics Method of Calculating Average Value of RadarCross Section,’’ IEEE Trans. on Antennas and Propagation, Vol. AP-16, No. 5, 1968,pp. 569–574.

[19] Crispin, J. W., and A.L. Maffett, ‘‘Radar Cross-Section Estimation of Simple Shapes,’’Proc. IEEE, Vol. 53, No. 8, 1965, pp. 972–981.

[20] Keller, J. B., ‘‘Backscattering from a Finite Cone,’’ IRE Trans. on Antennas and Propaga-tion, Vol. AP-8, No. 2, 1960, pp. 175–182.

[21] Dimova, A. I., M. E. Albats, and A. M. Bonch-Bruevich, Radiotechnical Systems, Moscow,Russia: Soviet Radio, 1975, p. 438 (in Russian).

[22] Skolnik, M. I., ‘‘An Empirical Formula for the Radar Cross Section of Ships at GrazingAngles,’’ IEEE Trans. on Acoustics, Speech, and Signal Processing, Vol. AES-10, No. 2,1974, p. 292.

[23] Kulemin, G. P., and V. B. Razskazovsky, The Statistical Characteristics of Radar Targets,Inst. Radiophysics Electr., Kharkov, Ukraine, report ‘‘Shore,’’ 1993, p. 185 (in Russian).

[24] Kulemin, G. P., et al., ‘‘About Selection Small Marine Targets from Sea Backscattering byCoherent Radar,’’ Signal Processing in Radiotechnic Systems, Aviation Institute, Kharkov,Ukraine, 1988, pp. 88–98 (in Russian).

[25] Williams, P. D. L., H. D. Cramp, and K. Curtis, ‘‘Experimental Study of the Radar Cross-Section of Maritime Targets,’’ Adv. Radar Techn., 1985, pp. 69–83.

[26] Kulemin, G. P., and V. B. Razskazovsky, ‘‘The Radar Characteristics of Targets atX- and Ka-Bands,’’ Proc. SPIE, Vol. 2,469, 1995, pp. 132–141.

[27] Kulemin, G. P. and V. B. Razskazovsky, ‘‘The Statistical Characteristics of Radar Targets,’’Preprint IRE NASU, No. 92-2, Kharkov, Ukraine, 1992, p. 32.

[28] Skolnik, M. I., Radar Handbook, New York: McGraw-Hill, 1970, pp. 13–27.

[29] Nebabin, V. G., and V. V. Sergeev, Methods and Technique of Radar Recognition, Mos-cow: Radio and Communication, 1984, p. 152 (in Russian); trans., Norwood, MA: ArtechHouse, 1995.

[30] Von, T. P., ‘‘Birds and Insects As Radar Targets: A Review,’’ Proc. IEEE, Vol. 75,No. 2, 1985, pp. 35–63.

[31] Mueller, E. A., ‘‘Differential Reflectivity of Birds and Insects,’’ Proc. 21st Conf. RadarMeteor., 1983, pp. 465–466.

86 Radar Characteristics of Targets

[32] Haykin, S. S., C. R. Carter, and M. V. Patriarche, ‘‘Identification of Areas of Bird Clutterand Weather Clutter Using Air Traffic Control Radar,’’ IEE Conf. and Exhib., 1975,pp. 172–173.

[33] Edgar, A. K., E. J. Dodsworth, and W. P. Warden, ‘‘The Design of a Modern SurveillanceRadar,’’ Int. Conf. Radar: Present and Future, London, England, October 1973,pp. 8–13.

[34] Sedov, L. I., Methods of Similarity and Dimensionality in Mechanics, Moscow, Russia:Ed. of Phys.-Math. Lit., 1967 (in Russian).

[35] Ohocimsky, D. Y., ‘‘The Derivation of Point Explosion Taking Contrapressure intoAccount,’’ Proc. of Math. Institute (named after Steklov), Vol. 87, April 1966 (in Russian).

[36] Landau, L. D., ‘‘On Shock Waves,’’ USSR Ac. of Sc. Proc., Physics Ser., Vol. 6, January–February 1942, pp. 64–69 (in Russian).

[37] Korotkov, P. F., ‘‘On Nonlinear Geometrical Acoustics: Weak Short Waves,’’ Journal ofAppl. Math. and Phys., No. 5, 1964, pp. 30–35 (in Russian).

[38] Lin, S. C., and J. D. Teare, ‘‘Rate of Ionization Behind Shock Waves in Air: TheoreticalInterpretations,’’ Phys. Fluids, Vol. 6, 1963, pp. 355–375.

[39] Krinberg, I. A., ‘‘Air Electric Conductivity for Admixtures Presence,’’ Journal of Appl.Math. and Phys., No. 1, 1965, pp. 78–87 (in Russian).

[40] Baum, F. A., K. P. Stanyukovich, and B. I. Shehter, Explosion Physics, 1959 (in Russian).

[41] Nasilov, D. N., Radiometeorology, Moscow, Russia: Ed. Nauka, 1966 (in Russian).

[42] Adushkin, V. V., ‘‘On Shock Wave Forming and Explosion Products Flying Away,’’J. Appl. Match. and Phys., No. 5, 1963, pp. 107–120 (in Russian).

[43] Tatarski, V. I., Wave Propagation in a Turbulent Medium, New York: McGraw-Hill,1961.

[44] Kondratiev, K. L., Ray Thermal Exchange in Atmosphere, Leningrad: GidrometeoizdatEd., 1956 (in Russian).

[45] Britt, C. O., C. W. Tolbert, and A. W. Straiton, ‘‘Radio Wave Absorption of SeveralGases in the 100 to 117 kMc/s Frequency Range,’’ J. Res. NBS., Vol. 6-D, January 1961,pp. 15–18.

[46] Bazhenova, T. V., Shock Waves in Real Gases, Moscow, Russia: Nauka Ed., 1968 (inRussian).

[47] Boynton, F. P., C. B. Ludwig, and A. Thomson, ‘‘Spectral Emissivity of Carbon ParticlesClouds in Rocket Exhausts,’’ AIAA Journal, Vol. 6, No. 5, 1968, pp. 116–124.

[48] Van de Hulst, H. C., Light Scattering by Small Particles, New York: John Wiley, 1957.

[49] Aleksandrov, Ye. N., et al., ‘‘Solid Explosion Substances Crushing in Shock Wave,’’ Physicsof Burning and Explosion, No. 3, 1968, pp. 84–89 (in Russian).

[50] Still, V. R. and G. N. Gloss, ‘‘Emissivity of Dispersed Carbon Particles,’’ J. Opt. Soc.,Vol. 50, February 1960, pp. 186–191.

[51] Von Gierre, H. E., ‘‘Effects of Sonic Boom on People: Review and Outlook,’’ J. AcousticSoc. Amer., Part 2, Vol. 39, No. 5, 1966, pp. S43–S50.

[52] Rizer, Y. P., ‘‘Sonic-Boom Propagation in Inhomogeneous Atmosphere to Side of DensityDecreasing,’’ J. Applied Mechanics and Technical Physics, No. 4, 1964, pp. 269–274 (inRussian).

[53] Hubbard, H. H., ‘‘Nature of the Sonic-Boom Problem,’’ J. Acoustic Soc. Amer., Part 2,Vol. 39, 1966, pp. S1–S9.

References 87

[54] Witham, G. B., ‘‘On the Propagation of Weak Shock Waves,’’ J. Fluid. Mech., Vol. 1,No. 9, 1952, pp. 290–318.

[55] Kane, E. J., ‘‘Some Effects on the Nonuniform Atmosphere on the Propagation of SonicBooms,’’ J. Acoust. Soc. Amer., Part 2, Vol. 39, No. 5, 1966, pp. S26–S30.

[56] Plotkin, K. J., and A. R. George, ‘‘Propagation of Weak Shock Waves Through Turbu-lence,’’ J. Fluid Mech., Part 3, Vol. 54, 1972, pp. 449–467.

[57] Lamley, J., and H. Panofsky, The Structure of Atmospheric Turbulence, New York:Interscience Publishers, 1964.

[58] Maglieri, D. J., ‘‘Some Effects of Airplane Operations and the Atmosphere on the Sonic-Boom Signatures,’’ 2nd Conf. on Sonic Boom Res., NASA, 1968, pp. 19–23.

[59] Landau, L. D., and Ye. M. Livshits, Mechanics of Continious Media, Moscow, Russia:Gostehizdat, 1954 (in Russian).

[60] Blohintsev, D. N., Acoustics of Inhomogeneous Moving Medium, Moscow, Russia: Fizmat-giz Ed., 1981 (in Russian).

[61] Melnikov, B. P., ‘‘Noise Caused by Tu-124 Airplane Take-Off and Landing,’’ AcousticJ., Vol. 1, No. 2, 1965, pp. 138–146 (in Russian).

[62] Lane, J. A., ‘‘Radar Echoes from Tropospheric Layers by Incoherent Backscatter,’’Electronic Letters, Vol. 3, No. 4, 1967, pp. 173–174.

[63] Shlihting, G., The Origin of Turbulence, Moscow, Russia: Foreign Literature Edit., 1962(in Russian).

[64] Sato, H., ‘‘The Stability and Transition of a Two-Dimensional Jet,’’ J. Fluid Mech.,Part 3, Vol. 7, 1960, pp. 321–329.

[65] Isatajev, S. I., and S. B. Tarasov, ‘‘On Acoustic Field Influence on Wake Directed AlongStream Axis,’’ USSR Acad. Sci. Proc., Fluid and Gas Mechanics, No. 2, 1971, pp. 164–167(in Russian).

[66] Crow, S. C., and F. H. Champagne, ‘‘Orderly Structure in Jet Turbulence,’’ J. Fluid Mech.,Part 3, Vol. 48, 1971, pp. 547–591.

[67] Shade, H., ‘‘Contribution to the Nonlinear Stability Theory of Inviscid Shear Layers,’’Phys. Fluids, Vol. 7, No. 5, 1964, pp. 623–628.

[68] Delano, R. H., ‘‘Theory of Target Glint or Angular Scintillation in Radar Tracking,’’Proc. IRE, Vol. 41, No. 3, 1953, pp. 61–67.

[69] Swerling, P., ‘‘Probability of Detection for Fluctuating Targets,’’ IEEE Trans. Inf. Theory,IT-6, No. 4, 1960, pp. 269–308.

[70] Mayer, H. A., and D. P. Meyer, ‘‘Chi-Square Target Models of Low Degrees of Freedom,’’IEEE Trans. Aerosp. Electr. Syst., Vol. AES-11, No. 5, 1975, pp. 694–707.

[71] Mayer, H. A. and D. P. Meyer, Radar Target Detection, New York: Academic Press,1973.

[72] Moll, P. L., ‘‘On the Radar Echo from Aircraft,’’ IEEE Trans. Aerosp. Electr. Syst.,Vol. AES-3, No. 3, 1967, pp. 574–577.

[73] Muchmore, R. B., ‘‘Aircraft Scintillation Spectra,’’ IEEE Trans. Antennas and Propaga-tion, AP-8, No. 2, 1960, pp. 201–212.

[74] Wilson, I. D., ‘‘Probability of Detecting Aircraft Targets,’’ IEEE Trans. Aerosp. Electr.Syst., AES-8, No. 6, 1972, pp. 757–762.

[75] Barton, D. K., Radar System Analysis, Dedham, MA: Artech House, 1979.[76] Krendell, C., (Translat. Ed.), Random Oscillations, Moscow, Russia: Mir, 1967, p. 361

(in Russian).

88 Radar Characteristics of Targets

[77] Nogid, L. M., Ship Stability and Its Behaviour at Rough Sea, Leningrad, Russia: Sudostroe-nie, 1967, p. 240 (in Russian).

[78] Barton, D. K. and H. R. Ward, Handbook of Radar Measurement, Englewood Cliffs, NJ:Prentice Hall, 1969; Dedham, MA: Artech House, 1984.

[79] Eaves, J. L., and E. K. Reedy, Principles of Modern Radar, New York: Van NostrandReinhold, 1987, p. 712.

[80] Levin, B. R., Theoretical Foundations of the Statistic Radioelectronics, Moscow, Russia:Soviet Radio, 1969 (in Russian).

[81] Berkowitz, R. S., (ed.), Modern Radar, New York: John Wiley, 1968.[82] Gradshtein, I. S., and I. M. Ridzik, Tables of Integrals, Sums, and Products, Moscow:

Fizmatgiz, 1963 (in Russian); trans. New York: Academic Press, 1965.[83] Kulemin, G. P. and V. B. Razskazovsky, ‘‘Complex Effects of Clutter, Weather and

Battlefield Conditions on the Target Detection in Millimeter-Wave Radars,’’ Proc. SPIE,Vol. 2,222, 1994, pp. 862–871.

[84] Kelly, R. E., ‘‘On the Derivation of Bistatic RCS from Monostatic Measurements,’’ Proc.IEEE, Vol. 53, No. 8, 1965, pp. 983–988.

[85] Chernyak, V. S., Fundamentals of Multisite Radar Systems, Amsterdam, the Netherlands:Gordon and Breach, 1998.

[86] Skolnik, M. I., Radar Handbook, New York: McGraw-Hill, 1970.[87] Cox, C., and W. Munk, ‘‘Statistics of the Sea Surface Derived from Sun Glitter,’’ J. Mar.

Res., Vol. 13, 1954, pp. 198–227.

C H A P T E R 2

Land Backscattering

2.1 Classification and Physical Characteristics of Land

Classification of land for backscattering prediction must take into account thepresence or absence of vegetation; for a surface without vegetation, the roughnessis also an essential additional parameter, depending on the wavelength and grazingangle. According the Rayleigh criterion, the surface can be considered rough if therms height deviation sh meets the criterion sh > (l /8) sin c [1], where l is thewavelength and c is the grazing angle. Among surfaces without vegetation, quasi-smooth surfaces (e.g., a road with concrete or asphalt paving, rocks, salt-marshes,or dry salt lakes) are distinguished, to which a simple mathematical model can beapplied. For estimating scattering properties of such surfaces, the essential parame-ters are the statistical characteristics of the roughness and the complex dielectricconstant. The real and imaginary components e1 and e2 of the dielectric constantfor concrete and asphalt in the MMW band are shown in Table 2.1.

The statistical characteristics (rms height, correlation functions, spatial correla-tion radii, and angle slope deviation) of various surfaces are shown in Table 2.2.

Surfaces without vegetation are in a separate group among rough surfaces.Their scattering properties are also determined by the complex dielectric constantand height deviation. The complex dielectric constant data for various surface typesin the MMW band are shown in Table 2.3.

The numerous experimental data sets on the dielectric constant of agriculturalfields obtained in [4, 5] support the following conclusions.

The frequency behavior of the dielectric constant for wet soils has the followingtendencies: e ′ decreases and e″ increases with increasing frequency from 4 GHz to

Table 2.1 Dielectric Constants of Concrete and Asphalt

Concrete AsphaltWavelength (mm) e ′ e″ e ′ e″

30.0 6.5 1.5 4.3 0.18.6 5.5 0.5 2.5 0.62.2 5.55 0.36 2.25 0.18

Source: [2, 3].

89

90 Land Backscattering

Table 2.2 The Statistical Characteristics of Various Surfaces

Cover Type sh , cm r (r ) r0 , cm √g 2

Concrete 1.6 ? 10−2 exp (−7r ) 0.14 0.14Asphalt 0.4 ? 10−1 (1 + 2r2)−3/2 0.22 0.3Sand 0.2–0.6 — 0.6–2.5 0.1–0.4Snow 0.1–0.3 — 0.2–200 <0.2Soil harrowed 2.0 — 10.0–80.0 0.5Source: [2–4].

Table 2.3 Complex Dielectric Constant for Some Covers at Wavelengths of 8 and 2 mm

Wavelength of 8 mm Wavelength of 2 mmCover e ′ e″ e ′ e″Soil 3.0 0.4 — —Soil (clay) 7.0 3.5 2.5 9.4 ? 10−2

Sand 3.0 0.56 2.5 6.2 ? 10−2

Sand 7.0 4.4 — —Snow 2.0 0.4 ? 10−2 1.4 10−2

Ice 3.1 0.7 ? 10−2 — —Brick (red) — — 3.2 1.1 ? 10−1

Brick — — 3.3 1.4 ? 10−1

Pine board — — 2.0 8 ? 10−2

Source: [3].

18 GHz. These dependences for a loam field, obtained for soil moisture from0.02 g/m3 to 0.37 g/m3, are shown in Figure 2.1.

The dependence of the dielectric constant on the volumetric moisture, measuredat 10 GHz and 18 GHz, show that e ′ and e″ depend very weakly on the soil type.As an illustration, Figure 2.2 shows the soil moisture dependence for five fieldswith different contents of sand, clay, and silt. It is seen that the soil content hasinsignificant influence.

The roughness parameters of agricultural fields are determined by the meansof tillage and change over wide limits. As an illustration, the summary of roughnessparameters is presented in Table 2.4 for four fields with different tillage.

The dielectric properties of snow cover play a significant role in backscatteringmodel development and explanation of results. Much experimental data havebeen gathered for snow dielectric characteristics covering the frequency band of0.8–37 GHz [6–9]. Models have been developed in which snow is represented asa two-component or three-component mixture [10, 11]. The model replaces theinhomogeneous medium by a homogeneous one with an effective dielectric constante. It is possible to determine the dielectric constant of the mixture with rather simplemodels if the inclusions in the heterogeneous medium are small in comparison withthe wavelength (permitting consideration only of the absorption while neglectingthe scattering losses) and their shape is known.

2.1 Classification and Physical Characteristics of Land 91

Figure 2.1 Soil dielectric constant dependence on frequency. (From: [4]. 1985 IEEE. Reprintedwith permission.)

Figure 2.2 Soil dielectric constant dependence on soil moisture for five fields at frequencies of10.0 GHz and 18.0 GHz. (From: [4]. 1985 IEEE. Reprinted with permission.)

92 Land Backscattering

Table 2.4 Summary of Roughness Parameters

Type rms Height (cm) Correlation Length (cm) rms SlopeField 1 0.40 8.4 0.048Field 2 0.32 9.9 0.032Field 3 1.12 8.4 0.133Field 4 3.02 8.8 0.485Source: [5].

For dry snow (i.e., a mixture of air and ice) the real part of the dielectricconstant in the microwave band does not depend on temperature or frequency butis a function of snow density. This dependence can be represented as [11]

e ′ds = H1 + 1.9r s for r s ≤ 0.5 g/cm3

0.51 + 2.88r s for r s > 0.5 g/cm3 (2.1)

where r s is the snow density, which under natural conditions rarely exceeds0.5 g/cm3. The experimental results obtained in the frequency band 0.8–37 GHzare described by (2.1) very well, as is shown in Figure 2.3.

The imaginary part of the snow dielectric constant depends on frequency andtemperature in a rather complex way. The dependence of the loss tangent onfrequency is shown in Figure 2.4 [12] for snow density of 0.45 g/cm3 [9].

The dielectric constant of wet snow increases sharply because of the influenceof water in its liquid phase. The content of liquid water in snow is usually describedby a volumetric moisture mv that is the ratio of water volume to total snow volume.There are several models of wet snow dielectric properties. For the millimeter band,the modified Debye model gives the highest accuracy for the dielectric constant [13].Numerous experiments for wet snow [12, 14] support the following conclusions.

The real part e ′ of the dielectric constant increases with an increase in thevolumetric moisture from 1.4 for dry snow to 3.3 for mv = 12% at a frequencyof 2 GHz. It decreases with an increase in frequency to 1.35 for the millimeterband. With increasing frequency the real part of dielectric constant decreases. Thisis illustrated by Figure 2.5(a) obtained at a frequency of 10.0 GHz for moisturechange from zero to 12%. The imaginary part e″ for wet snow increases withincreasing volumetric moisture—Figure 2.5(b). The maximal values are found atapproximately the relaxation frequency, beyond which it decreases with increasingfrequency.

Land territories with vegetation form the third group. The complex dielectricconstants of vegetation media are determined by their biometrical indices (e.g.,vegetation volumetric density and square cover degree) and by the relative moisturecontent in the cellular tissue.

The experimental data on the dielectric constant for corn leaves in the band1.5–8.0 GHz, for volumetric moisture from zero to 0.8 g/cm3, are shown in Figure

2.1 Classification and Physical Characteristics of Land 93

Figure 2.3 Dry snow dielectric constant versus snow density. (From: [12]. 1986 Artech House,Inc. Reprinted with permission.)

2.6 [12]. The dielectric constant for wheat grains as a function of the weightedmoisture is shown in Figure 2.7.

The vegetation biometrical indices change continuously as a function of vegeta-tion growth. The vegetation moisture can change as a function of season andweather conditions. The biometrical indices of some crops and grass for differentperiods of growth are shown in Table 2.5 [15].

In [3], the notion of the effective complex dielectric constant of a medium isintroduced for investigations of propagation peculiarities in vegetation

eeff = eeff (1 − i tan d eff ) (2.2)

94 Land Backscattering

Figure 2.4 Dry snow loss tangent versus frequency. (From: [9]. 1986 IEEE. Reprinted with permis-sion.)

where eeff and tan d eff are the dielectric constant and loss tangent, respectively.The use of this parameter permits derivation of the reflection coefficients from theparameters of homogeneous dielectric media [16].

In [17], a simple expression for effective complex dielectric constant is obtainedfor homogeneous vegetation covers

eeff = 1 + Bdew M (2.3)

where d is the weight of scattering elements in a unit volume, B is an empiricalparameter equal to 0.3 for coniferous forest and to 0.6 for deciduous forest, ew isthe dielectric constant of water at the radar frequency, and M is the relative weightedmoisture content in the vegetation.

It has been established that huge tracts of forest are homogeneous scatterers,just as are grass, cereals, and some types of bushes—all having an inhomogeneityof different degree. The averaged parameter values for coniferous and deciduousforests are shown in Table 2.6 [18].

Data on dielectric properties of vegetation are insufficient for general modeldevelopment of land surfaces with vegetation.

2.2 State of the Theory 95

Figure 2.5 Frequency dependence (a) real and (b) imaginary parts of dielectric constant of wetsnow for different water contents. (From: [12]. 1986 Artech House, Inc. Reprintedwith permission.)

2.2 State of the Theory

2.2.1 RCS Models

Radars for detection of on-land and low-altitude targets are subject to land back-scattering, characteristics of which depend on the land surface type, the vegetationstate, the season, the radar operating frequency, and the wind velocity.

96 Land Backscattering

Figure 2.6 Dielectric constant versus volumetric water content for corn leaves. (From: [12]. 1986Artech House, Inc. Reprinted with permission.)

There are significant difficulties in the development of theoretical models forland backscattering because of the variety of land types. The major theoreticalworks can be divided into two groups. In the first, the land surface is consideredto consist of elementary scatterers (e.g., spheres and cylinders), for which thescattering properties are known and for which parameters are chosen to securesatisfactory agreement with experimental results. These models usually predictquite well the angular dependence of the normalized RCS, which is a measure ofthe scattered signal intensity, and the scattered signal statistical characteristics.

The second group of works [1–3, 19–21] requires the choice of individual,exactly formulated backscattering parameters; however, these choices give largeerrors in derivations of scattered signals because of the idealization of the surface.

The simplified models of first group are shown in Table 2.7 [22–24]. Here thegrazing angle, related to the incidence angle u by c = p /2 − u, is used as a parameter.

In the model proposed in [2] for grassy terrain, it is assumed that the scatteringcomes from long thin cylinders with considerable loss, permitting multiple scattering

2.2 State of the Theory 97

Figure 2.7 Dielectric constant versus water content for grain: (a) real and (b) imaginary parts.(From: [12]. 1986 Artech House, Inc. Reprinted with permission.)

Table 2.5 Biometrical Indices of Crops

Cover Steam SteamWeight Height Degree Moisture Amount Section

Type (g/cm2) (cm) (%) (%) (cm−2) (cm2)Grass 0.12 10 100 86 1.5 0.008Grass 0.16 17 95 85 0.6 0.016Alfalfa 0.077 18 90 87 0.64 —Sugar beet — 55 95 — 0.02 —Wheat 0.25 110 25 36 0.043 0.096Source: [15].

98 Land Backscattering

Table 2.6 Experimental Values of Model Parameters for Forests

ParametersForest Type d MConiferous forest, thick 10−4 0.4Coniferous forest, thin 5 ? 10−3 0.6Deciduous forest, thick 10−4 0.6Deciduous forest, thin 5 ? 10−3 0.8Source: [18].

Table 2.7 Simplified Models of Land Backscattering

Angular Dependence RCS DependenceModel of Normalized RCS on Wavelength ReferenceDiffuse scattering sin2 c — —Diffuse scattering (Lambert law) sin c — —Ament’s model C l−2 —Facet model exp [−cot2 (c /2s2)] l−2, l−6 [22]Model of half-cylinders c4, c2 l−2, l0, l−3, l−1 [23]Model of half-spheres c4, c2 l−2, l0, l−4, l−1 [24]Vegetation model as cylinders sin2 c — [2]

effects to be neglected. For small incidence angles close to normal incidence (grazingangle c → 90° ), models have been developed to satisfactorily explain the experimen-tal data. For example, in [25] a cover cloud model is proposed in which a cloudof identical particles with volumetric homogeneous distribution represents thevegetation. For simplicity in the cover cloud model, the approach of single scatteringis used and the contributions of all particles are summed, taking into considerationthe double attenuation in the cover layer between its surface and the particle. Thismodel satisfactorily describes the experimental results in the band 8.6–17.0 GHzfor incidence angles less than 70°.

In [26], a dielectric layer model is proposed, in which the vegetation is repre-sented by a homogeneous dielectric medium. This model is satisfactory at wave-lengths from 1.65–3.75 cm for incidence angles close to nadir.

A great number of discs with different spatial distribution is used as a modelof forest foliage in [27].

Attempts have been made to formulate models of the second group via exactsolutions to the electrodynamic backscattering problem, using measured surfaceparameters as initial data for derivation of the scattered signal. These modelsassume that all land surfaces can be divided into two classes: those with and thosewithout vegetation. The surface without vegetation has a significant additionaldependence on the degree of roughness.

According to the Rayleigh criterion, the surface is rough if the rms deviationfrom the average surface is sh > (l /8) sin c . Otherwise it can be described asquasi-smooth.

2.2 State of the Theory 99

Theoretical models are based on methods of solving the problem of scatteringby a statistically rough surface. Rather accurate models of backscattering basedon Kirchhoff’s or small perturbation methods are strictly applicable only for quasi-smooth surfaces without vegetation (e.g., concrete or asphalt). For these surfacemodels, it is sufficient for estimation of RCS to know the surface electrophysicalcharacteristics such as the dielectric constant and surface roughness.

For Kirchhoff’s method, it is assumed that for sufficiently gently sloping surfaceswith the radii of curvature considerably greater than the wavelength, limitingconditions can be represented as those of the plane facets of which this surfaceconsists. The problem is solved by introduction of local Fresnel coefficients followedby integration along the surface currents. The vector form of Green’s theorem isused to obtain the full scattering matrix [28].

For a strongly rough surface, the normalized RCS can be represented as in[19]

s 0 =R 2

f0

sin4 c? cos2 b0 ? expS−

cot2 c

tan2 b0D (2.4)

where tan b0 = 2sh l and l is the spatial correlation radius of surface roughness.Consequently, the value of b0 can be interpreted as the mean slope of the roughsurface. The value of Rf0 is the Fresnel reflection coefficient of the plane surface,a function of its electrophysical characteristics. The use of reflection coefficientsfor the plane surface does not permit us to take fully into account the polarizationeffects on the scattered signal.

The determination of the scattered electromagnetic field in the small perturba-tion method is based on the following assumptions, applicable to quasi-smoothsurfaces. Let the surface height deviation be set as

z = f (x , y ) (2.5)

Let us choose the plane z = 0 such that (2.5) describes the deviation from meanheight z = 0. Then the small perturbation method is applied for small gently slopingroughness

|ksh sin c | << 1 (2.6a)

| df (x , y )dx | << 1; | df (x , y )

dy | << 1 (2.6b)

The normalized RCS of a rough surface with complex dielectric constant e canbe expressed as in [28]

100 Land Backscattering

s 0HH = 4pk4 sin4 c |aHH |2S (k0) (2.7)

s 0VV = 4pk4 sin4 c |aVV |2S (k0)

where k = 2p /l is the wavenumber; and S(k0) is the spectral density of surfaceroughness with a wavenumber

k0 = S4pl D cos c (2.8)

The physical sense of (2.8) is that the backscattering takes place for the spectralcomponent with a period one-half the radar wavelength. The Fresnel coefficientsin (2.7) are determined by the surface dielectric properties and have the form

aHH =e − 1

Xsin c + √e − cos2 c C2 (2.9a)

aVV =(e − 1)[e (1 + cos2 c ) − cos2 c ]

Xe sin c + √e − cos2 c C2 (2.9b)

aHV = aVH = 0 (2.9c)

The absence of scattered signal depolarization is inherent in estimation of thenormalized RCS in the small perturbation method characterized by (2.9c). Thiscontradicts experimental results even for quasi-smooth surfaces.

The coefficients of aHH , aVV depend very weakly on wavelength for smallgrazing angles, their change being not greater than 1–3 dB in the band 10–140GHz.

From (2.7) we see a strong dependence of normalized RCS on wavelength(inversely proportional to the fourth power of wavelength) and on grazing angle(s 0 ∼ sin4 c ). The normalized RCS for vertical polarization is greater than thatfor horizontal by 8–10 dB.

A comparison of derived and experimental results for quasi-smooth concreteor asphalt surfaces shows that the model (2.7) does not apply for these surfacesin the shortwave part of millimeter band because the surface becomes stronglyrough.

A comparative analysis of the two latter models leads to the conclusion thatnormalized RCS estimations using Kirchhoff’s method can be used for rather smallincidence angles u near nadir, while estimations using the small perturbationmethod can be used for incidence angles u > 30–40°, but neither applies at extremelysmall grazing angles when the influence of shadowing is significant.

2.2 State of the Theory 101

This is why these models are not applicable for such surfaces as rough oneswithout vegetation or those with vegetation. The main way to derive general modelsin these cases is the development of empirical models based on experimental results.

2.2.2 Power Spectrum Model

The backscattering power spectra of land surfaces with vegetation are determinedby the wind-induced motion of scatterers as the radar scans its volume and/or byradar platform motion.

For identical radar transmitting and receiving antennas or for a single combinedantenna, the signal at the receiver input from the j th scatterer in the far zone canbe represented as

uj (t ) ~√Pt s j

r2j

? G (uj ) ? exp [i (v0 t − kr j − w j )] (2.10)

G (u ) = E∞

−∞

f (x ) ? exp (ikx sin u ) dx

where Pt is the transmitter power, s j is the RCS of the j th scatterer, w j is the phaseof the signal from the j th scatterer, v0 is the radar frequency, k = 2p /l is thewavenumber, r j is the range to j th scatterer, G (u ) is the antenna voltage pattern,and f (x ) is the electromagnetic field distribution at antenna aperture.

The variables in (2.10) can be divided into two groups. The first group consistsof the values r j and u j , which change with radar platform motion and antennascanning, and which can be functionally expressed through the current antennaposition and time. The variables of the second group, s j and w j , characterizingthe scatterer state at the present moment of time, are random functions of time inmost cases and depend on the scatterer number. If the extent of each scatterer issmall in comparison to its range, (2.10) can be written as the product of twofactors, each containing the variables of one of these groups. Consequently, thescattered signal spectrum is the convolution of two spectra: The first describes thesignal fluctuations resulting from scatterer motions and the second results fromantenna scan and radar platform motion.

The conclusions for a single scatterer can be generalized for the sum of scat-terers. Assuming that all scatterers are in the far zone, the signal at the receiverinput is

U (t ) ~ ∑∞

j =1

1

r2j

G (u j , t )√s j (t ) ? exp {i [v0 t − w j (t )]} (2.11)

102 Land Backscattering

Then the autocorrelation function can be found as

⟨U (t )U*(t ) ⟩ ~ ∑j

∑j ′

⟨ 1

r2j r2

j ′√s j (t )s j ′ (t ′ ) ? G (u j , t )G (u j ′ , t ′ ) (2.12)

× exp {i [v0(t − t ′ ) − w j (t ) + w j ′*(t ′ )]}⟩Here the sign * denotes the complex conjugate value.

Assuming that the scatterers are statistically independent and the phases ofscattering are distributed uniformly in [0, 2p ], (2.12) can be written in form

⟨U (t )U*(t ′ ) ⟩ ~ ∑∞

j =1

1

r4j√s j (t )s j (t ′ ) (2.13)

? G (u j , t )G (u j , t ′ ) exp {i [v0(t − t ′ ) − w j (t ) + w j (t ′ )]}

Taking into consideration the independence of scatterer fluctuations andantenna scanning, this expression can be transformed to

⟨U (t )U*(t ′ ) ⟩ ~ ∑∞

j =1⟨Aj (t )Aj (t ′ ) exp {−[w j (t ) − w j (t ′ )]} exp [iv0(t − t″ )]⟩ (2.14)

× ⟨G (u j , t )G (u j , t ′ ) ⟩

where Aj (t ) ~ √s j (t ) ⁄ r2j is the amplitude of the j th scatterer signal.

Let us note that

Rsj (t ) = ⟨Aj (t )Aj (t ′ ) exp {i [v0(t − t ′ ) − w j (t ) + w j (t ′ )]} ⟩ (2.15)

RGj (t ) = ⟨G (u j , t )G (u j , t ′ ) ⟩ (2.16)

The expression (2.15) represents the autocorrelation function of the signal fromthe j th scatterer and (2.16) characterizes the antenna scanning law. Substituting(2.14) and (2.15) into (2.14), we obtain

⟨U (t )U*(t ′ ) ⟩ ~ ∑j

Rsj (t )RGj (t ) (2.17)

The power spectrum is the Fourier transformation of the autocorrelation func-tion, and hence for our case it is the convolution of the scatterer fluctuationspectrum and antenna-scanning spectrum. This leads to the conclusion that for

2.2 State of the Theory 103

determination of the spectrum of land backscattering caused by antenna scanningand radar platform motion, it is sufficient to find separately the spectra of scattererfluctuation and of antenna motion (or the electromagnetic field modulation acrossits aperture).

Let us consider the spectrum model formed only by the motions of scatterersunder the influence of wind or other reasons.

The scattering surface can be represented as a linear system with a transientresponse

K (t ) = K0 + DK (t ) (2.18)

where K0 characterizes the reflectors that are stable in time (e.g., buildings, baresoil, or rocks) and the second term characterizes the fluctuating scatterers (e.g.,grass, branches, or leaves of trees). Then the autocorrelation function of this systemcan be represented as

g (t ) = K 20 + r (t ) (2.19)

If the continuous signal is u (t ) = A cos v0 t at the system input, the autocorrela-tion function at the system output (the scattered signal) will have the form

Rout (t ) = Rin (t ) ? g (t ) =A2

2[K 2

0 + r (t )] cos v0t (2.20)

The power spectrum is the Fourier-transform of autocorrelation function andcan be written as

G1(v ) = E∞

−∞

Rout (t ) ? e −jvt dt = a2d (v − v0) + G (v ) (2.21)

where d (v − v0) is the Dirac function, and a2 is the ratio of stable to fluctuatingcomponents of scattered power (RCS).

It is seen that the total power spectrum of the scattered signal consists ofone component that is produced by scattering from stable reflectors and anotherproduced by moving scatterers. The first term can considerably exceed the secondfor terrains without vegetation.

Let us consider the power spectra of the surface model, for which the scatterersof branch, leaf, and grass types move under the influence of wind. The basis ofthis model is the derivation technique developed by G. S. Gorelik [29, 30] forradiowave scattering problems with moving inhomogeneities. Assume that the

104 Land Backscattering

scatterer velocity autocorrelation function reproduces with some scale correctionthat of the wind velocity.

Assuming for simplicity that the signal amplitude from the j th scatterer is unity,one can obtain from (2.9) the expression for signal phase of j th scatterer

w j = 2kr j (t ) + w0 (2.22)

where r j (t ) is the range from the radar to the j th scatterer, and w0 is the initialphase.

From Figure 2.8 we have

rj = r + xj sin u (2.23)

where xj is the scatterer displacement relative to its initial position.Without violating generality, one can choose the initial phase w0 such that

w j = 2kxj (t ) sin u (2.24)

Assume that the velocities of the scatterers are a stationary random process,mutually independent, and have identical statistical properties and zero meanvalues.

For a displacement velocity of the j th scatterer uj (t ), the displacement duringtime t is

(Dxj )t = xj (t + t ) − xj (t ) = Et +t

t

u j (s ) ds (2.25)

Figure 2.8 Geometry of derivation.

2.2 State of the Theory 105

This value is stationary by the previous assumptions, and the phase differencehas the analogous properties.

The autocorrelation function is

R (t ) = ⟨u (t ) ? u (t + t ) ⟩ = ∑j

∑j ′

⟨cos [v0 t − w j (t )] cos [v0 t ′ − w j ′ (t ′ )] ⟩

(2.26)

where t ′ = t + t .Because of statistical independence of w j (t ) and w j ′ (t ′ ) for j ≠ j ′ (2.26) can be

written as

R (t ) = ∑j

⟨cos [v0 t − w j (t )] ⟩ + ∑j

∑j ′

⟨cos [v0 t − w j (t )] cos [v0 t − w j ′ (t ′ )] ⟩

(2.27)

where terms having j = j ′ in the second term are omitted.Taking into consideration the independence of the statistical properties of

separate scatterers, (2.27) can be written in the form

R (t ) = n ⟨cos[v0 t − w (t )] cos[v0 t ′ − w (t ′ )] ⟩ (2.28)

+ (n2 − n ) ⟨cos[v0 t − w (t )] cos[v0 t ′ − w (t ′ )] ⟩

Under the assumption that the mean scatterer velocities are zero,

⟨sin Dw ⟩ = ⟨sin w ⟩ = ⟨sin w ′ ⟩ = sin (w + w ′ ) = 0 (2.29)

and for a uniform phase distribution in [0, 2p ] one can approximately assume[31]

⟨cos w ⟩ = ⟨cos w ′ ⟩ = ⟨cos (w + w ′ ) ⟩ = 0 (2.30)

Then (2.28) has the form

R (t ) =n2

cos v0t ? ⟨cos Dw ⟩ (2.31)

The envelope of the autocorrelation function

r (t ) = ⟨cos Dw ⟩ (2.32)

106 Land Backscattering

is determined by the distribution of scatterer displacement probabilities becausefrom (2.24) and (2.25) we have

Dw = 2kDx (t ) sin u (2.33)

Letting m = 2k sin u, we obtain

r (t ) = ⟨cos mDx (t ) ⟩ = E∞

−∞

cos mDx ? p (Dx ) ? d (Dx ) (2.34)

If the scatterer displacements are described by a Gaussian distribution

p (Dx ) =1

√2p (Dx )2expF−

(Dx )2

2(Dx )2G (2.35)

we obtain [32]

r (t ) = expF−m2(Dx )2

2 G (2.36)

Consequently, the envelope of the autocorrelation function is determined bythe mean square value of scatterer displacement, which is easily expressed throughthe autocorrelation function of scatterer velocities [30]

Ru (t ) = ⟨u (t ) ? u (t ′ ) ⟩ = (Du )2 ? r (t ) (2.37a)

where (Du )2 is the scatterer velocity variance.Actually [30],

(Dx )2 = 2Et

0

(t − j )Ru (j ) dj

and

R (t ) =n2

cos (v0t ) ? exp3−m2(Du )2 Et

0

(t − j )ru (j ) dj4 (2.37b)

2.2 State of the Theory 107

The spectral density is found as Fourier-transform of the autocorrelationfunction

G (v ) =2p E

∞

0

R (t ) cos (vt ) dt (2.38)

=np E

∞

0

cos (v − v0)t ? exp3−m2(Du )2 Et

0

(t − j )ru (j )4 dt

Taking into account that the autocorrelation function of the horizontal compo-nent of wind velocity at small heights above the surface is determined as [33, 34]

ru (t ) = e −a | t | cos bt (2.39)

(2.38) can be written in the form

G (v ) =np E

∞

0

cos (v − v0)t ? exp3−m2(Du )2 Et

0

(t − j )e a | t | cos bt4 dt

(2.40)

The solution of this integral has the form [35]

G (v ) =np

expSm2(Du )2

k2 D ∑∞

m =0

(−1)m

m !? Sm2(Du )2

k2 Dm (2.41)

?1m2(Du )2

k2 + km

FSm2(Du )2

k2 + kmD2 + (v − v0)2G2where k = (a2 + b2)/2a .

108 Land Backscattering

Letting m2(Du )2 = s2f , we obtain

G (v ) =np

expSs2f

k2D ∑∞

m =0

(−1m )m !

? Ss2f

k2Dm

?

s2f

k2 + km

Ss2f

k2 + kmD2

+ (v − v0)2

(2.42)

The analysis of this expression shows that the power spectrum of land scatteringhas different forms as a function of s

2f /k2. For s

2f /k2 >> 1, the expression for

spectral density has the form [35]

G (v ) ≅np

(2ps2f )1/2 expF−

(v − v0)2

2s2f

G (2.43)

Thus, for slow scatterer oscillations (small Doppler frequencies less than the−3 dB spectrum bandwidth), the power spectrum is practically Gaussian. Thespectrum width is

DF =s f

p√2? √(Du )2 =

2√2l √(Du )2 (2.44)

From the analysis of (2.44), it can be seen that the land backscattering spectrumwidth is inversely proportional to wavelength and proportional to the rms windvelocity fluctuations.

In the region of higher frequencies when s2f /k2 << 1, the power spectrum has

the form [35]

G (v ) =np F s

2f /2k

(s 2f /2k )2 + (v − v0)

−s

2f /2k

4k2 + (v − v0)2 + . . .G (2.45)

where only the first terms are significant. Thus, for the frequency region greaterthan the spectrum width, the change of spectral density is described better by apower function with an exponent approaching two.

2.3 Normalized RCS

2.3.1 Normalized RCS of a Quasi-Smooth Surface

The normalized RCS is one of the most important and universal parameters charac-terizing the land backscattering, and it depends on such variables as the surface

2.3 Normalized RCS 109

type, wavelength (or frequency), grazing angle, transmitting and receiving polariza-tions, the season of year, and weather conditions. Although the normalized RCSconcept is strictly applied only to homogeneous rough surfaces or flat land areas,it is also used in practice for a description of backscattering from nonuniformterrains (e.g., for towns or land areas with sparse trees). In these cases, it is necessaryto use the normalized RCS concept with caution. The influence of the factorsmentioned earlier has been investigated in the wide band from meter to shortmillimeter wavelengths and was generalized in [20]. The main conclusions forquasi-smooth surfaces are the following.

Smooth surfaces are characterized by weak scattering (except at angles nearnadir), and the angular dependence of normalized RCS for these surfaces is s 0 ∼sin3 c , where c is the grazing angle. As was noted in Section 2.2.1, theoreticalmodels have been developed most completely for quasi-smooth surfaces, and thederivations of normalized RCS within the framework of these models are satisfacto-rily adjusted to experimental results. The rms error does not exceed 2–3 dB, andmaximal error is 10 dB [12].

The dependence of s 0 on wavelength can be represented as s 0 ∼ l−4. Forgrazing angles less than 30°, the RCS is from 6 dB to 10 dB higher for verticalpolarization than for horizontal. As an example, the angular dependences of nor-malized RCS for concrete at wavelengths of 3.2 cm and 8.6 mm for vertical andhorizontal polarizations of transmitting and receiving are shown in Figure 2.9, andthe dependence of s 0 on wavelength is shown in Figure 2.10 for concrete, asphalt,and gravel. The dependence of experimental normalized RCS on frequency fordifferent types of road surfaces for horizontal polarization is presented in Table2.8, confirming the conclusions on strong frequency dependence of the normalizedRCS.

The presence of a water film on the surface decreases the normalized RCS by10 dB for small grazing angles, as shown in Figure 2.11; this is explained by thesmoothing of surface roughness by the water. This same smoothing increases thenormalized RCS for nadir radiation.

For nadir radiation, the rms roughness height strongly influences the normalizedRCS, leading to its decrease with increasing sh . The values of normalized RCSfor nadir operation in the frequency band 40–135 GHz are shown in Table 2.9.

2.3.2 Normalized RCS for Rough Surfaces Without Vegetation

Rough surfaces without vegetation have greater values of normalized RCS (by15–25) dB and other angular and frequency characteristics in comparison withquasi-smooth surfaces In Figure 2.12, the angular dependences of the normalizedRCS are shown for rough surfaces without vegetation at wavelengths of 1–3 cm.

The normalized RCS of a rough surface without vegetation is a function ofthe surface roughness sh and volumetric soil moisture mv [i.e., s 0 = f (sh , mv )].

110 Land Backscattering

Figure 2.9 The normalized RCS of concrete versus grazing angle for (a) vertical and (b) horizontalpolarizations. (After: [36].)

The growth of surface roughness changes the angular dependence of normalizedRCS to s 0 ∼ sin c for highly rough terrains and the dependence of s 0 on wavelengthto s 0 ∼ l−1. As an illustration, the angular dependences of the normalized RCSfor ploughed fields with different types of cultivation (degrees of roughness) areshown in Figure 2.13. It is seen that the angular dependence of the normalizedRCS for ksh ≥ 1.2 can be approximated as s 0 ∼ sin c . However, right up to thesevalues of ksh , the experimental angular dependences are satisfactorily adjusted toderivations according Kirhhoff’s method. For greater values of ksh the surface ispractically diffuse, as shown in Figure 2.13. All of these surfaces scatter diffusely

2.3 Normalized RCS 111

Figure 2.10 Dependence of normalized RCS for quasi-smooth surfaces on radar wavelength. (After:[36].)

Table 2.8 Normalized RCS (dB) Frequency Dependence of Different Surfaces at the GrazingAngle of 10°

Frequency (GHz) Concrete Asphalt Asphalt-Gravel Slag-Gravel10.0 −(30–54) −(26–46) −(25–41) −(25–44)15.5 −(29–45) −(25–39) −(20–33) −(18–34)35.0 −(20–43) −(18–33) −(15–29) −(18–28)

Source: [36].

in the millimeter bands, and the angular dependence of the normalized RCS canbe approximated as s 0 ∼ sin c .

The second factor determining the normalized RCS is the dielectric constant,which depends on frequency, soil type, and its volumetric moisture. Increasing thevolumetric soil moisture leads to increasing s 0 and changes the frequency depen-dence of s 0 because of increasing soil dielectric constant. However, the normalizedRCS change of bare soil is less than 10 dB over the entire range of field moisturechange; this is considerably less than the change caused by surface roughness.

112 Land Backscattering

Figure 2.11 Angular dependences of normalized RCS for dry and wet asphalt at wavelength of8 mm and vertical polarization (VV). (After: [36].)

Table 2.9 Average Values of Normalized RCS for Nadir Radiation

Normalized RCS (dB) at Frequencies,Surface Type 10.0 GHz 40–90 GHz 70.0 GHz 135.0 GHzLake (smooth surface) 11.4 20.0 15.2 —Asphalt — 16.0 — —Concrete — 15.2 11.5 —Sand, gravel 6.5 −7.4 −1.2 —Brick — — — 4.0Veneer, thickness 5 mm — — — −10.0Source: [36–39].

As was shown in Section 2.1, the influence of soil type on its dielectric constantis rather small. The Fresnel coefficients in (2.7) depend rather weakly on thefrequency. In this case, the dependence of normalized RCS on relative moisturecan be represented by following empirical expression [11]

s 0 = 0.148mf − 15.96 (dB) (2.46)

where mf = (mv /Cv ) ? 100%, mv is the volumetric soil moisture in grams percubed centimeter, and Cv is the field moisture capacity. The dependences of the

2.3 Normalized RCS 113

Figure 2.12 Angular dependences of the normalized RCS for three types of land.

normalized RCS on volumetric soil moisture are shown in Figure 2.14. It is seenthat a volumetric moisture change from 0.05 g/m3 to 0.3 g/m3 leads to an increasein normalized RCS of 7–8 dB.

Vegetation masks the soil surface, and in this case the total normalized RCSis determined as [11]

s 0S (c , l ) = s 0

veg (c , l ) +s 0

soil (c , l )

L2(c )(2.47)

where s 0veg and s 0

soil are the normalized RCS of vegetation and soil, respectively,and L (c ) is the attenuation coefficient in the vegetation layer.

At frequencies above 8 GHz and for grazing angles less than 60°, the contribu-tion of the first term in (2.47) is predominant, because one can neglect the soilinfluence, and the contribution of the second term is significant at lower frequenciesand for c ≥ 60°. Figure 2.15 shows the attenuation in a vegetation layer for differenttypes of plants as a function of the penetration depth at the 3-cm wavelength(X-band).

Under conditions of incomplete surface screening, the normalized RCS depen-dence on the soil moisture takes the form [11]

114 Land Backscattering

Figure 2.13 Angular dependence of normalized RCS for different sh . (From: [40].)

s 0 = 0.113mf − 13.84 (dB) (2.48)

Thus, it is not as steep as for soil without vegetation, for which the normalizedRCS is given by (2.46). The s 0 dependences on soil moisture for bare soil and forsoil with vegetation are shown in Figure 2.16.

2.3.3 Backscattering from Snow

Microwave backscattering from snow involves surface scattering by the air-snowand snow-soil boundaries and volumetric scattering by ice crystals within the snowlayer. Multiple scattering also takes place, conditioned by the upper and lowerboundaries. The normalized RCS of snow is determined by several factors: theradar frequency, the transmitted and received polarizations, and the grazing angle,as well as the electrophysical and geometrical properties of the snowpack [41].

The first snow parameter influencing the normalized RCS is its water equivalent

2.3 Normalized RCS 115

Figure 2.14 Soil normalized RCS dependences on moisture at 3–4.5 GHz. (After: [12].)

Figure 2.15 Attenuation dependences on the penetration depth for three types of vegetation.

116 Land Backscattering

Figure 2.16 Normalized RCS dependences on soil moisture for bare and vegetated surfaces. (From:[12]. 1986 Artech House, Inc. Reprinted with permission.)

W = r s h (2.49)

where r s is the snow density and h is the snowpack height.The second parameter is the volumetric water content mv . The surface

roughness parameters and ice crystal size distribution are also significant character-istics of snow.

In most cases, the normalized snow RCS (if one does not take into considerationmultiple scattering) can be expressed as [42]

s 0 = s 0ss (l , u ) + s 0

s (u ′ ) +g

2sa (u ′ )

L2(u ′ )? s 0

soil (u ′ ) (2.50)

where s 0ss is the normalized surface scattering RCS of the air-snow boundary,

s 0s (u ′ ) is the normalized volumetric scattering RCS of the snow, g

2sa (u ′ ) is the

power displacement factor of the air-snow boundary, and s 0soil is the normalized

RCS of the underlying soil. The angle u ′ is related to the incidence angle u by

2.3 Normalized RCS 117

sin u = √es ? sin u ′ (2.51)

where es is the dielectric constant of snow.For dry snow and small grazing angles (i.e., s 0

s ≈ 0), one can neglect thescattering from the air-snow boundary. In this case, only volumetric scattering andreflection from the snow-soil boundary determine the normalized RCS in (2.50),and the expression for the normalized RCS becomes

s 0(W ) = A0 − (A0 − s 0soil ) ? exp (−C0W sec u ′ ) (2.52)

where A0 and C0 are coefficients depending on radar frequency, incidence angleand, polarization [43]. The dependences of the normalized RCS on the waterequivalent of snow are shown in Figure 2.17. It is seen that the saturation of thenormalized RCS is observed for snowpack height and its water equivalent growth,assuming the absence of soil influence on the scattering signal intensity.

For wet snow, one cannot neglect the surface scattering of the air-snow bound-ary because of the large value of the snow dielectric constant resulting from water

Figure 2.17 The normalized RCS of snow versus depth of snow cover. (From: [43].)

118 Land Backscattering

in the liquid phase. In some cases, this surface scattering predominates in formingthe scattered signal. A rapid decrease of the normalized RCS takes place withincrease in liquid water content, as is shown in Figure 2.18. For mv ≥ 4–5% therate of change of s 0 decreases as a result of the considerable attenuation of thefree water. The result is a predominant influence of the surface scattering.

The frequency dependence of the normalized RCS for dry and wet snow isshown in Figure 2.19 [41]. First of all, one can see the interesting dependence ofs 0 on the liquid water content. A change of mv from 0% to 1.25% gives a decreasein the normalized RCS of about 1 dB in the decimeter band; the difference growsfor increasing frequency, and it reaches the maximal value of 10–12 dB at35.0 GHz. This difference decreases with further increase in frequency and is2–3 dB at 140.0 GHz.

2.3.4 Backscattering from Vegetation

Vegetation leads to changes in RCS angular and frequency dependences. The small-est value of the normalized RCS is observed for thin vegetation covers. This valuevaries within limits of −(20 to 25) dB for grazing angles less than 10°. The back-

Figure 2.18 The normalized RCS of snow versus the volumetric moisture. (From: [43].)

2.3 Normalized RCS 119

Figure 2.19 The frequency dependence of snow normalized RCS. (After: [12].)

scattering is essentially diffuse, with an angular dependence of form s 0 ∼ sin cand very weak frequency dependence for all grazing angles less than 10°.

Dense forests have the largest RCS in spring and summer as a result of thecorrelation of the scattering intensity with vegetation biomass. A high correlationbetween the normalized RCS and biomass is observed in the shortwave part ofcentimeter and in millimeter bands for grazing angles less than 60°–70°, where thesoil influence is practically negligible. Thus, at 11 GHz for c ≤ 60°, the correlationbetween normalized RCS and biomass exceeds 0.9 for both copolarization andcross-polarization [41]. An increase in biomass to 0.4 kg/m2 leads to increasingthe normalized RCS by 3 dB (i.e., RCS change at a rate of 7.5 dB/kg/m2).

There is considerable seasonal dependence of normalized RCS for vegetationcovers. In the spring–summer period, the scattering intensity increases by 10–20 dBin comparison with the autumn–winter period, and the largest values are observed inJune–July. This corresponds to an increase in biomass and vegetable water contentduring this time period. As an illustration, the seasonal changes of the normalizedRCS and leaf cover coefficient are shown in Figure 2.20(a), and the seasonal changesof the normalized RCS and the vegetable water content are shown in Figure 2.20(b).

There is no great difference in the normalized RCS for deciduous and coniferousforests in summer and autumn [44]. As noted in [20], no differences in the normal-ized RCS for coniferous forest in summer and winter are observed, while fordeciduous forest this difference is 20–22 dB.

There is a very significant influence of weather conditions on the normalizedRCS. After a rain, the backscattering of grass increases by about 3 dB for all

120 Land Backscattering

Figure 2.20 (a) Cover coefficient and normalized RCS and (b) water content and normalized RCS depen-dence on season. (After: [40].)

grazing angles, and for forest this increase is 5 dB in comparison with dry weather[42, 43].

The dependence of normalized RCS for vegetation surfaces on the wavelengthis very weak in centimeter and millimeter bands, having the form s 0 ∼ f n where0 ≤ n ≤ 1. The best approximation for frequency dependence for both forest andfields covered by the grass and agricultural plants has the form s 0 ∼ f 0.6, givingsatisfactory results from 3 GHz to 100 GHz. For grazing angles less than 1°, theexperiments do not disclose even this much frequency dependence, as illustratedby Table 2.10 [20, 23, 38, 45].

2.3.5 Normalized RCS Models

The features of land backscattering derived earlier permit us to develop severalempirical models for normalized RCS of different terrain types.

Table 2.10 The Vegetable Surfaces Normalized RCS for c < 1°

Wavelength (mm)Vegetation Type32 12.5 8.6 8.15 4.1

Meadow (flat surface) −27 −30 −15 −25.5 −30Steppe (roughness heights up to 0.5m) −23 −23 −23 −24 −2Dry meadow −20 −20 −27 −21 −22.5Sparse mixed forest, bush −22 −20 −23 −21 −21.5Dense foliage forest −12 −14.5 −11 −8 −9.5

2.3 Normalized RCS 121

For the land surfaces with vegetation, the angular and frequency dependencesfor incidence angles from 0° to 60° (grazing angles greater than 30° ) and in theband 1–18 GHz can be represented by the empirical expression [46]

s 0 (dB) = a0 + a1e a2u + (a3 + a4e −a5u ) ? exp [−(a6 + a7u ) f ] (2.53)

where u = 90° − c is the incidence angle and a1 − a7 are coefficients determinedby the transmitted and received polarizations; their values are shown in Table 2.11.

The frequency dependence of the normalized RCS given by (2.53) is very weakat frequencies above 4 GHz, and at frequencies less than 4 GHz, s 0 decreasesvery rapidly with decreasing frequency (i.e., frequency dependence appears in thedecimeter band). At the same time, the angular dependence is that of diffusebackscattering.

A simpler empirical expression (with fewer number of coefficients) for thenormalized RCS of these surfaces was proposed in [47]

s 0 = D + 10a ? log f + 8.6b ? f a − Mu (dB) (2.54)

where the coefficient values are as shown in Table 2.12.Another model applies to the band 3–100 GHz, for grazing angles less than

30°, and is suitable for RCS description of various surfaces including quasi-smooth,rough with and without vegetation, snow, and city and country areas. The normal-ized RCS in this model has the form [21, 48]

s 0 (dB) = A1 + A2 log c /20 + A3 log f /10 (2.55)

Table 2.11 Coefficient Values for Use in (2.53)

Polarization a0 a1 a2 a3 a4 a5 a6 a7Horizontalpolarization(HH) 2.69 −5.35 0.014 −23.4 33.14 0.048 0.053 5.1 ? 10−3

VV 3.49 −5.35 0.014 −14.8 23.69 0.066 0.048 2.8 ? 10−3

Crosspolarization(HV) 3.91 −5.35 0.013 −25.5 14.65 0.098 0.258 2.1 ? 10−3

Source: [46].

Table 2.12 Coefficient Values for Use in (2.54)

a b D M0.8 0.04 −15.5 0.1

Source: [47].

122 Land Backscattering

Here f is the frequency in gigahertz, and c is the grazing angle in degrees. Thecoefficients A1 − A3 for various types of terrain are shown in Table 2.13.

The significant feature of this model lies in the replacing of the real terrainvariety by a limited number of land surface types, which include all of the mainterrain types. This is sufficient for land clutter estimations in typical conditions ofradar operation.

As an illustration, the angular and frequency dependences of the normalizedRCS for deciduous forest are shown in Figures 2.21 and 2.22, obtained for agrazing angle of 10° by various authors, where the solid and dotted lines correspondto derivations according to (2.55). The variance of experimental data amounts to10–15 dB, and this is associated with differences both in various land types andin the methods of experimental data processing.

Table 2.13 The Coefficients A1–A3 in Land Clutter Model

Terrain Type A1 A2 A3Concrete −49 32 20Arable land −37 18 15Snow −34 25 15Deciduous and coniferous forests, summer −20 10 6Deciduous forest, winter −40 10 6Grass with height more than 0.5m −21 10 6Grass with height less than 0.5m −(25−30) 10 6Urban territories (town and country buildings) −8.5 5 3Source: [48].

Figure 2.21 Normalized RCS s 0 versus grazing angle for forest at wavelengths of 8 mm (curve 1)and 3 mm (curve 2).

2.4 Depolarization of Scattered Signals 123

Figure 2.22 Normalized RCS versus frequency for forest at c = 10°.

Consequently, we have normalized RCS estimations for various types of landfor practically all microwave and millimeter-wave bands.

2.4 Depolarization of Scattered Signals

The polarization differences of backscattered signals are determined, to a majordegree, by the ratio of surface roughness to wavelength.

For quasi-smooth terrain (concrete, asphalt), the normalized RCS at grazingangles less than 30° is greater by 8–10 dB for VV than for HH, and HV componentsare practically absent [20]. This is in good agreement with the land surface modelobtained by the small perturbation method for which the depolarization compo-nents are zero (s 0

HV = s 0VH = 0).

For bare rough surfaces, the difference between RCS values for HH and VVpolarizations disappears as the rms roughness sh increases, and the difference doesnot exceed 2–3 dB for grazing angles less than 60°. The normalized RCS ratio forco- and cross-polarizations for these surfaces is 7–15 dB [1] decreasing with reduced

124 Land Backscattering

wavelength and grazing angle and with increasing sh . For example, in Figure 2.23,the dependence of s 0 on incidence angle is shown for three soils with different shmeasured at the 3-cm wavelength, and in Figure 2.24 the copolarization to cross-polarization ratios are shown for wavelengths of 3 cm and 8 mm for soils withdifferent erosion state [49].

The polarization ratios are small for land surfaces with vegetation. The ratios 0

VV /s 0HH decreases with vegetation biomass growth. For agricultural plantings

(e.g., potatoes, alfalfa, or sugar beets), this ratio is 2–3 dB at 10 GHz [50] and3–4 dB between 10 and 100 GHz [51].

The copolarization to cross-polarization ratio varies over wide limits. As shownin [51], this ratio is 7–12 dB at the 3-cm wavelength, decreasing with grazing angledecrease, and is 10–12 dB at 9-mm wavelength (Figure 2.25). Thus, for variousland surfaces, the polarization ratios at a 3-cm wavelength are shown in Table2.14.

The polarization features of signals scattered by snow are not pronounced.There is little difference between RCS for VV and HH polarizations for snow atfrequencies above 10 GHz, and copolarization to cross-polarization ratio is about10 dB for dry snow and 10–15 dB for wet snow. This ratio decreases with furtherincrease in frequency, to 5 dB at 95 GHz and 3 dB at 140 GHz [41].

For sea ice, the ratio of s 0VV /s 0

HH is practically unity, and the copolarizationto cross-polarization ratios are 3–10 dB in the band 10–40 GHz [52], dependingon surface roughness.

These data permit us to use the following hypotheses for description of landbackscattering:

• The ratio s 0VV /s 0

HH is 10 dB for quasi-smooth surfaces at 10 GHz, and itsdependence on frequency is rather weak.

Figure 2.23 (a–c) The normalized RCS versus incidence angle for three types of soil. (After: [12].)

2.4 Depolarization of Scattered Signals 125

Figure 2.24 Ratios s 0cop /s 0

cross versus incidence angle at wavelengths of (a) 3 cm and (b) 8 mmfor soils with different erosion states (curves 1–3). (From: [49]. 1995 SPIE. Reprintedwith permission.)

• This ratio is zero for all types of rough surfaces.• The copolarization to cross-polarization ratio is 10 dB with the exception

of quasi-smooth surfaces for which the cross-polarization components ofthe backscattered signal are very small.

Thus, the following expressions can be used for normalized RCS of land clutter

126 Land Backscattering

Figure 2.25 (a–c) Polarization ratios for vegetation at a 9-mm wavelength for different crops.

Table 2.14 Ratios of s 0VV /s 0

HH and s 0copal /s 0

cross , in Decibels, for 3-cm Wavelength

Surface Type s 0VV /s 0

HH s 0copal /s 0

cross

Quasi-smooth 8.0–10.0 —Rough bare soil 2.0–3.0 7.0–15.0Rough surface with vegetation 2.0–3.0 7.0–12.0Snow 0 10.0–15.0Ice 1.0–2.0 3.0–10.0Town and country areas 0 2.0–3.0

s 0VV ≅ 5s 0

HH + 10S f10D

−1/2

for quasi-smooth surfaces

s 0HH for other surfaces

(2.56a)

Here s 0HH is the value of s 0 from (2.55), and f is the frequency in gigahertz.

The cross-polarized component of the normalized RCS is

s 0cross = s 0

HV = s 0VH ≈ s 0

HH − 10 (dB) (2.56b)

for all types of surfaces other than quasi-smooth. For quasi-smooth surfaces, thecross-polarized component of the normalized RCS in model is zero (−∞ in decibels).

2.5 Statistical Characteristics of the Scattered Signals

The fluctuations of backscattered signals from land are associated with the motionof scatterers within the resolution cell and the shift of the resolution cells due to

2.5 Statistical Characteristics of the Scattered Signals 127

radar platform motion and antenna scanning. This determines the normalized RCStemporal and spatial probability distributions

p (s 0) = E∞

−∞

p Xs 0 |m C ? p (m ) dm (2.57)

Here p (m ) is the pdf that characterizes the spatial distribution of the fluctua-tions, p Xs 0 |m C is the conditional pdf that describes the temporal fluctuations ina single resolution cell, and m is the mean value of the normalized RCS.

The pdf of RCS can depend in various ways on the distribution of mean ormedian values p (m ) of s 0 within the scan volume. This is because the RCS distribu-tion in a single resolution cell is transformed, and the resulting distribution haslarger tails (i.e., greater probability of occurrence of large RCS values).

The temporal statistics of the scattered signal follows the RCS distributionwithin the limits of a single resolution cell p Xs 0 |m C. Typical approximations ofexperimental distributions of normalized RCS are the Rayleigh and Rician [1].For these cases, the probability function of the in-phase (I ) and quadrature (Q )components of the scattered signal are Gaussian with variance equal to the fluctuat-ing component power and with nonzero mean value (the mathematical expectation)equal to the amplitude of the stable component.

The stable component of the scattered signal is formed by reflectors that arestable in time (e.g., rocks, buildings, or bare surfaces). In some cases, its value isconsiderably larger than the fluctuating component. As result, the normalized RCScan be represented as a sum of two terms

s 0 = s 0st + s 0

fl = s 0fl (1 + a2) (2.58)

where s 0st and s 0

fl are the normalized RCS of the stable and fluctuating components,respectively, and a2 is the ratio of stable-to-fluctuating RCS.

For various land types, the values of a2 are determined by the mean windvelocity and the wavelength and are

a2 = 530.7l2 ? V −2.9 for forests

63l2 ? V −3 for short vegetation

100l2 ? V −3 for bare land

(2.59)

For town and country areas, the a2 values are approximately 104–105. In theseexpressions, l is the wavelength in centimeters and V is the wind velocity in metersper second.

128 Land Backscattering

Two models are applied rather widely for modeling the probability distributionfunctions of land backscattering. In the first (the Rayleigh model), an ensemble ofindependent reflectors with random amplitude and phase produces the scatteredsignal. This is a typical model for land surfaces with dense vegetation. The pdf ofthe normalized RCS in this model is

p (s 0 /m ) =1m

expS−s 0

m D (2.60)

For terrains with the stable scatterers, a model using the Rician pdf is applied:

p Xs 0 |m C =1 + a2

m? expF−a2 − (1 + a2)

s 0

m G ? I0S2a√(1 + a2)s 0

m D(2.61)

where I0 is the Bessel function of the first kind and zero order.The normalized RCS differs from the Gaussian model only for small sizes of

the radar resolution cell, and this function can be represented as

p (s ) =bsb −1

aexpS−sb

a D (2.62)

This is the Weibull distribution of RCS. Here a is the shape parameter and bis the slope parameter, b = 1 for the Rayleigh model (2.10). The experimentalresults of [53] showed that the b parameter for forest and grass in summer and inwinter and for the wind velocities of 5–10 m/s is 10–15 (i.e., the Weibull distributionis rather similar to the Gaussian). Examples of these distributions for scatteringfrom grass at 3.2-cm wavelength are shown in Figure 2.26.

Therefore, it is possible to use the Gaussian pdf to describe the statistics oftwo quadrature components of the scattered signals from the land surfaces.

2.6 Power Spectra of Scattered Signals

The power spectra of land backscattering are determined, as a rule, by fluctuationsof scatterers that move under wind conditions and by volume scanning and radarplatform motion. The influence of volume scan for various scanning methods thatlead to power spectrum broadening is analyzed in detail in [10, 20], and radarplatform motion is considered in [11].

The scatterer motions induced by wind fluctuations in the troposphere controlthe power spectrum of backscattering from terrain with vegetation. In the frame-

2.6 Power Spectra of Scattered Signals 129

Figure 2.26 Histograms of relative amplitude distributions for backscattering from grass at a 3-cmwavelength and for wind velocity (a) 5 m/s and (b) 7 m/s. (From: [53].)

work of this model, the power spectra are described as functions of the fractaltype in which the spectral width and the power exponent are the functions of thewind velocity. The power spectrum can be represented as [20, 54]

G ′ (F ) = a2d (F ) + G (F ) (2.63)

where d (F ) is Dirac’s function, which characterizes the scattering from the stablereflectors and G (F ) is the power spectrum of the fluctuating component. Theexperimental investigations of land clutter power spectra carried out by the authorin 1966–1969 and later ones [20, 54, 55] showed that the power spectrum of thefluctuating component can be represented by the following empirical expressions

G (F ) = G0F1 + S FDF D

nG−1

(2.64)

n =2(U + 2)

U + 1? S100

f D0.2

(2.65)

where F is the Doppler frequency; DF is the −3 dB width of the power spectrum,which is the function of wind velocity, the operating frequency, and the land surfacetype; and n is the power exponent, which depends on wind velocity and vegetationtype. For formal description of these values, one can use the following expressionsfor surfaces with vegetation

130 Land Backscattering

DF = 1.23 ? S3.2l D ? U 1.3 (2.66)

where l is the wavelength in centimeters, f is the frequency in gigahertz, and U isthe average wind velocity in meters per second. The latter expression does not takeinto consideration the seasonal change and variations for various vegetation covers.

As an example, the dependence of DF on wind velocity is shown in Figure2.27, where the dots are the experimental results and the solid line is from (2.66).

The presence of vegetation and increase in biomass leads to an increase in thescattering intensity [i.e., increase of 10–15 dB in the spectral components for allfrequencies in summer as compared with autumn and winter, with some decreasein the power exponent of (2.65)]. As an example, the backscattering power spectrafor a swamp with grass and bushes are presented in Figure 2.28 for the 3-cmwavelength (the straight line 1 for summer, the line 2 for winter).

The power spectra of backscattering from forest and grass at 3 cm in summerand winter are shown in Figure 2.29 [20, 53]. It is seen that the power exponentin (2.65) decreases in winter for forest and does not change for grass.

It has been determined experimentally [20] that the power spectra of backscat-tering from land with vegetation extend over Doppler frequency bands in the order

Figure 2.27 The −3-dB spectrum bandwidth versus wind velocity for a 3-cm wavelength.

2.6 Power Spectra of Scattered Signals 131

Figure 2.28 Doppler spectra of backscattering from swamp with bushes at the 3-cm wavelengthin summer (1) and in winter (2).

of 10–20 kHz. Examples of such spectra obtained at wavelengths of 3 cm, 8 mm,and 4 mm for forest and grass are shown in Figure 2.30.

The power spectra of intensity (the amplitude spectra) have practically thesame properties as the power spectra of coherent signals. First of all, square-lawdetection of signals with power spectra of the form G (F ) ∼ F −n does not broadenthe amplitude spectrum in comparison with the power spectrum if the signal-to-noise ratio is chosen correctly. The amplitude spectral shape can be described by(2.64) and the spectral width by (2.66). For a light wind (about 1 m/s), the correla-tion interval decreases with reduced wavelength more rapidly than for moderatewind. This can be explained by the greater efficiency of the phase modulation ofthe scattered signal because the influence of small radial movements of the scatterersis more visible. There may be flatter spectra at low-frequency F for further reductionin wavelength.

132 Land Backscattering

Figure 2.29 Doppler spectra of backscattering from forest and grass at the 3-cm wavelength in(a) summer and (b) winter. (From: [53].)

Figure 2.30 The power spectra of backscattering from forest and grass at wavelengths of 3 cm,8 mm, and 4 mm. (From: [48]. 1994 SPIE. Reprinted with permission.)

References 133

References

[1] Melnik, Y. A., Radar Techniques of Earth Investigation, Moscow, Russia: Soviet Radio,1980 (in Russian).

[2] Peake, W. H., ‘‘Theory of Radar Return from Terrain,’’ IRE Nat. Conv. Rec., Part 7,1957, pp. 34–43.

[3] Andreev, G. A., and V. A. Golunov, ‘‘Scattering and Radiation of Millimeter Waves byNatural Formations,’’ Results of Science and Technique, Vol. 20, 1980, pp. 3–106 (inRussian).

[4] Hallikainen, M., et al., ‘‘Microwave Dielectric Behavior of Wet Soil—Part I: EmpiricalModels and Experimental Observations,’’ IEEE Trans. Geosc. Remote Sens., Vol. GE-23,1985, pp. 25–34.

[5] Dobson, M. C., F. Kouyate, and F. T. Ulaby, ‘‘A Reexamination of Soil Textural Effectson Microwave Emission and Backscattering,’’ IEEE Trans. Geosc. Remote Sens.,Vol. GE-22, No. 6, 1984, pp. 530–536.

[6] Cumming, W., ‘‘The Dielectric Properties of Ice and Snow at 3.2 cm,’’ J. Appl. Phys.,Vol. 23, 1952, pp. 768–773.

[7] Sweeney, B. D., and S. C. Colbeck, ‘‘Measurement of the Dielectric Properties of Wet SnowUsing a Microwave Technique,’’ Cold Region Res. and Eng. Lab. (CRREL), Hanover, NH,1974, p. 84.

[8] Hallikainen, M. F., F. T. Ulaby, and M. Abdelrazik, ‘‘The Dielectric Behavior of Snowin the 3 to 37 GHz Range,’’ 1984 IEEE Int. Symp. (IGARSS-84) Digest, San Francisco,CA, August 1984, pp. 169–176.

[9] Hallikainen, M. F., F. T. Ulaby, and M. Abdelrazik, ‘‘The Dielectric Properties of Snowin the 3 to 37 GHz Range,’’ IEEE Trans. Antennas and Propagation, Vol. AP-34,No. 5, 1986, pp. 1329–1340.

[10] Polder, D., and J. H. Van Santen, ‘‘The Effective Permeability of Mixtures of Solids,’’Physica, Vol. 12, 1946, pp. 257–269.

[11] Ulaby, F. T., A. Aslam, and M. C. Dobson, ‘‘Effects of Vegetation Cover on the RadarSensitivity to Soil Moisture,’’ IEEE Trans. Geosc. Remote Sens., Vol. GE-20, No. 4, 1982,pp. 476–481.

[12] Ulaby, F. T., R. K. Moore, and A. K. Fung, Microwave Remote Sensing, Vol. 3: Activeand Passive, Norwood, MA: Artech House, 1986, p. 1098.

[13] Boyarsky, D., et al., ‘‘Electric Models of Dry Snow Cover for Remote Sensing: Some Resultsof Theory and Experiment,’’ Scient.-Techniq. Conf. Statistical Methods and Systems ofRemote Sensing Data Processing of Environment, Minsk, USSR, November 1989,pp. 82–83 (in Russian).

[14] Linlor, W. I., ‘‘Permittivity and Attenuation of Wet Snow Between 4 and 12 GHz,’’J. Appl. Phys., Vol. 51, 1980, pp. 2811–2816.

[15] Andreev, G. A., and A. A. Potapov, ‘‘Millimeter Waves in Radar,’’ Foreign Radioelectron-ics, No. 11, 1984, pp. 28–62.

[16] Armand, N. A., et al., ‘‘Radiophysical Techniques of Environment Investigation,’’ inProblems of Modern Radiotechnics and Electronics, Moscow, AN USSR Pub., 1978 (inRussian).

[17] Zubkovich, S. G., Statistical Characteristics of Radio Signals Scattered from the EarthSurface, Moscow, Russia: Soviet Radio, 1968 (in Russian).

134 Land Backscattering

[18] Redkin, B. A., and V. V. Klochko, ‘‘Derivation of Averaged Tensor of Vegetation ComplexDielectric Constant,’’ Radiotechnics and Electronics, No. 8, 1975, pp. 1596–1603 (inRussian).

[19] Beckmann, P., and A. Spizzichino, The Scattering of Electromagnetic Waves from RoughSurfaces, Pergamon Press, 1963; Norwood, MA: Artech House, 1987, p. 303.

[20] Kulemin, G. P., and V. B. Razskazovsky, Scattering of Millimeter Radiowaves by theEarth’s Surface for Small Grazing Angles, Kiev, Ukraine: Naukova Dumka, 1987 (inRussian).

[21] Ishimaru, A., Wave Propagation and Scattering in Random Media, New York: AcademicPress, 1978.

[22] Ament, W. A., F. C. MacDonald, and R. D. Shewbridge, ‘‘Radar Terrain Reflections forSeveral Polarizations and Frequencies,’’ Proc. Symp. Radar Return, Arlington, NY,October 1959, pp. 346–349.

[23] Spetner, L. J., and I. Katz, ‘‘Two Statistical Models for Radar Terrain Return,’’ IRE Trans.Antennas and Propagation, Vol. AP-8, No. 5, 1960, pp. 242–246.

[24] Twersky, V., ‘‘On Scattering and Reflection of Electromagnetic Waves by Round Surfaces,’’IRE Trans. Antennas and Propagation, Vol. AP-8, No. 1, 1960, pp. 81–89.

[25] Attema, E., and F. T. Ulaby, ‘‘A Radar Backscatter for Vegetation Targets,’’ Proc. OpenInt. Symp. EM Propag. Nonionized Media, La Baule, France, May 1977, pp. 579–584.

[26] Bush, T., and F. Ulaby, ‘‘Radar Return from a Continuous Vegetation Canopy,’’ IEEETrans. Antennas and Propagation, Vol. AP-24, No. 3, 1976, pp. 269–276.

[27] Zamaraev, B. D., Y. F. Vasilyev, and V. G. Kolesnikov, ‘‘Foliage Forest Normalized RCSEstimation at Millimeter Band of Radiowaves,’’ Spatial-Temporal Signal Processing inRadiotechnical Systems, Aviation Institute, Kharkov, Ukraine, 1985, pp. 50–53 (inRussian).

[28] Bass, F. G., and I. M. Fuks, Wave Scattering from Statistical Rough Surfaces, Moscow,Russia: Soviet Radio, 1972, p. 424 (in Russian).

[29] Gorelik, G. S., ‘‘On Scatterer Velocity Correlation Influence on the Statistical Propertiesof Scattered Radiation,’’ Radiotechnics and Electronics, Vol. 2, No. 10, 1957,pp. 1227–1233 (in Russian).

[30] Gorelik, G. S., ‘‘To Radiowaves Scattering Theory on the Wondering Inhomogeneities,’’Radiotechnics and Electronics, Vol. 1, No. 6, 1956, pp. 695–703 (in Russian).

[31] Tikhonov, V. I., Statistical Radiotechnics, Moscow, Russia: Soviet Radio, 1966, p. 678(in Russian).

[32] Gradshtein, I. S., and I. M. Rizhik, Tables of Integrals, Sums, Series and Products, Fizmat-giz, Moscow, 1963, p. 1100 (in Russian).

[33] Pinus, N. Z., and S. M. Shmeter, Aerology, Moscow, Russia: Gidrometeoizdat, 1965,p. 351 (in Russian).

[34] Zubkovsky, S. L., ‘‘Fluctuation Spectra of Wind Velocity Horizontal Component at Heightof 4m,’’ Izv. AS USSR, Physics of Atmosphere and Ocean, No. 10, 1962, pp. 1425–1428.

[35] Malakhov, A. N., ‘‘About Generator Spectral Line Shape for its Frequency Fluctuations,’’Journal of Experimentental and Theoretical Physics, Vol. 30, No. 5, 1956, pp. 884–889.

[36] Taylor, R., ‘‘Terrain Return Measurements at X-, Ku, and Ka Bands,’’ IRE Nat. Conv.Rec., Part 1, 1959, pp. 19–30.

[37] Long, M. W., Radar Reflectivity of Land and Sea, 3rd ed., Norwood, MA: Artech House,2001.

References 135

[38] Trebits, R. N., R. D. Hayes, and L. C. Bomar, ‘‘MM-Wave Reflectivity of Land and Sea,’’Microwave J., Vol. 21, No. 8, 1978, pp. 49–53.

[39] King, H. E., et al., ‘‘Terrain Backscatter Measurements at 40 to 90 GHz,’’ IEEE Trans.Antennas and Propagation, Vol. AP-18, No. 6, 1970, pp. 780–784.

[40] LeToan, T., ‘‘Active Microwave Signatures of Soil and Crops,’’ Proc. IGARSS’82, Digest,Vol. 1, 1982, pp. Tp 2.3/1–Tp 2.3/5.

[41] Kulemin, G. P., T. N. Kharchenko, and S.E. Yatsevich, ‘‘Snow Remote Sensing by the RadarTechniques,’’ Preprint of IRE NASU, No. 92-8, Kharkov, Ukraine, 1992 (in Russian).

[42] Ulaby, F. T., W. H. Stiles, and M. Abdelrazik, ‘‘Snowcover Influence on Backscatteringfrom Terrain,’’ IEEE Trans. Geosc. Remote Sens., Vol. GE-22, No. 2, 1984, pp. 126–132.

[43] Ulaby, F. T., and W. H. Stiles, ‘‘The Active and Passive Microwave Response to SnowParameters: Part II—Water Equivalent of Dry Snow,’’ J. Geophys. Res., Vol. 85, No. C2,1980, pp. 116–122.

[44] Kim, Y. S., et al., ‘‘Surfaces-Based Radar Scatterometer Study of Kansas Rangeland,’’Remote Sens. Environ., Vol. 11, 1980, pp. 253–265.

[45] Grant, C. R., and B. S. Yaplee, ‘‘Backscattering from Water and Land of Centimeter andMillimeter Wavelengths,’’ Proc. IRE, No. 6, 1957, pp. 976–982.

[46] Ulaby, F. T., ‘‘Vegetation Clutter Model,’’ IEEE Trans. Antennas and Propagation,Vol. AP-28, 1980, pp. 538–545.

[47] Vasilyev, Y. F., B. D. Zamaraev, and G. P. Kulemin, ‘‘The Angular and Season Backscatter-ing Dependences of Millimeter Radiowaves by Vegetation Cover,’’ Preprint of IRE NASUNo 91-3, Kharkov, Ukraine, 1991, p. 26, (in Russian).

[48] Kulemin, G. P., and V. B. Razskazovsky, ‘‘Complex Effects of Clutter, Weather andBattlefield Conditions on the Target Detection in Millimeter-Wave Radars,’’ Proc. SPIE,Vol. 2,222, 1994, pp. 862–871.

[49] Kulemin, G. P., et al., ‘‘Soil Moisture and Erosion Degree Estimation from MultichannelMicrowave Remote Sensing Data,’’ Proc. Europ. Symp. SPIE on Satellite Remote Sensing,Paris, Vol. 2,585, September 1995, pp. 144–155.

[50] Moore, R. K., K. A. Soofy, and S. M. Purduski, ‘‘A Radar Clutter Model: AverageScattering Coefficients of Land, Snow and Ice,’’ IEEE Trans Aerosp. Electr. Syst.,Vol. AES-16, 1980, pp. 783–799.

[51] Skolnik, M. I., (ed.), Radar Handbook, New York: McGraw-Hill, 1970.[52] Kulemin, G. P., ‘‘Growler Detection Method in Sea Clutter by Coherent Radar,’’ Proc.

Sixth Int. Conf. Remote Sensing for Marine and Coastal Environment, May 2000,Charleston, SC; Veridan ERIM Int., Vol. 2, pp. II.47–II.54.

[53] Savchenko, A. K., S. J. Haimov, and G. P. Kulemin, ‘‘On the Experimental Study of RadarBackscattering from Land,’’ XVII Europ. Microwave Conf., Stockholm, October 1988,pp. 705–709.

[54] Kulemin, G. P. and V. B. Razkazovsky, ‘‘Land Backscattering Spectra of Centimeter andMillimeter Radiowaves for Small Grazing Angles,’’ Preprint of IRE NASU, No. 195,Kharkov, Ukraine, 1982, p. 39 (in Russian).

[55] Brukhovetskiy, A. S., and A. A. Puzenko, ‘‘About Signal Spectrum for Transversal Move-ment of Shadowing Reflectors,’’ Radiotechnics and Electronics, Vol. 15, No. 12, 1970,pp. 2533–2538 (in Russian).

C H A P T E R 3

Estimation of Land Parameters byMultichannel Radar Methods

3.1 Estimation of Soil Parameters

3.1.1 Introduction

Efficient use of agricultural fields requires application of modern remote sensingtechniques for soil characteristic determination because the traditional methods ofin situ measurements do not provide data detailed enough for practice and are toolabor consuming.

During recent decades, besides optical and infrared methods, radar methodsof soil characteristic study have been intensively developed—in particular, near-surface soil moisture estimation, determination of humus content, and degree ofsoil erosion. The high resolution and the weather independence of radars permittheir practical application to remote sensing of large terrain areas through the useof airborne and spaceborne radars—synthetic aperture radar (SAR) or side-lookingradar (SLAR). In radar remote sensing, the electromagnetic field scattered by theobjects serves as a source of information about the physical and chemical propertiesof the surface. For this reason, the processes of microwave scattering from baresoil have been the subject of theoretical and experimental investigations for manyyears. This research activity was directed at finding a correlation between theparameters of the scattered electromagnetic field and the statistical and agrophysicalcharacteristics of soil.

The capability of active microwave techniques to sense near-surface soil mois-ture has a considerable research interest. The basis for microwave remote sensingof soil moisture is the dependence of the soil’s dielectric properties on its moisturecontent due to a large contrast between the dielectric constant of water and thatof dry soil. Boundary conditions affecting soil characteristics include the small-scale random surface roughness generated by cultivation, soil erosion processes,azimuthally dependent ridge/furrow patterns, and the slope of a terrain elementaffecting the local angle of incidence.

As is well known, the scattered signal intensity is determined by the statisticalcharacteristics of the surface and the dielectric constant of the medium. However,

137

138 Estimation of Land Parameters by Multichannel Radar Methods

the range of the soil specific RCS variations due to moisture does not exceed8–10 dB, while the roughness variations can lead to 15–25 dB of RCS variations.Consequently, the effects of surface roughness create obstacles to correct estimationof soil moisture.

The use of the dual-polarization ratio of normalized RCS for obtaining unbiasedestimation of the real part of the complex dielectric constant was first proposedin [1], and this ratio was also used in [2]. The application of multifrequency radartechnique to determination of surface roughness was proposed in [3], where thisapproach was used for small incidence angles, and joint processing was not applied.In papers [4–8], the application of multichannel techniques to separate determina-tion of near-surface soil moisture and surface roughness was proposed, and jointprocessing of data was used. Taking into account that roughness parameters dependmainly on mechanical and aggregate soil structure, the successful retrieval of thisinformation permits determining soil areas with different degrees of erosion andanalyzing the dynamics of their evolution. This chapter presents the results oftheoretical and experimental investigations into multichannel remote sensing tech-niques applied to estimation of soil parameters in the microwave band.

3.1.2 Soil Backscattering Modeling

In modeling wave scattering from natural terrains, one generally expects a combina-tion of surface and volume scattering, especially when a dry soil medium is inhomo-geneous. The soil medium can be treated as a volume consisting of variable fractionsof soil solids, aqueous fluids, and air. Soil solid material is a mixture of sand, silt,and clay, and it is characterized by a distribution of particle sizes (texture) andmineralogy of their consistent particles (particularly, a clay fraction). The particlediameter of clay is less than 2 mm, the particle diameter of silt is between 2 mmand 50 mm, and the particle diameter of sand is between 50 mm and 200 mm.The water in soil is classified as free or bound water.

Modeling of scattering from bare natural and agricultural soils starts from acharacterization of the surface roughness and dielectric behavior of materials. Thelatter depends on several parameters, including the bulk density, the particle sizedistribution, the mineralogical composition, the content of organic matter, thecontent of bound and free water, the soil salinity, and the temperature.

The problem of electromagnetic wave scattering from rough surfaces has beenstudied extensively for many years. However, because of the complexity of theproblem, satisfactory solutions are available only when very stringent restrictionsare imposed on the rough surface profile, the electromagnetic parameters of theirregular boundary, or the frequency. Models can be divided into two groups:field-approach models and intensity-approach models [9]. The disadvantage of theexisting field-approach models is that they cannot in practice include multipleincoherent scattering beyond the second order. On the other hand, the intensity-

3.1 Estimation of Soil Parameters 139

approach models can include more multiple scattering terms and the interactionbetween surface and volume scattering, but it assumes a far field interaction betweenscatterers. We restrict ourselves to rough surface scattering models only and neglectvolume scattering, although in practice these two effects are sometimes difficult toseparate. For bare soil, the surface scattering is dominant only if the terrain canbe considered homogeneous.

Two rough surface scattering models are widely applied because of their simplic-ity: Kirchhoff’s model and the first-order small perturbation model [10]. In mostbackscattering applications, Kirchhoff’s model is used over the incidence angularregion 0 < u < 20°. This model is restricted to high frequencies and indicatesthat for perfectly conducting surfaces the backscattered fields do not depend onpolarization.

The small perturbation model is used over the angular range 20° < u < 60°.Larger incidence angles are not considered because the scattering mechanismsfor grazing incidence are likely to be different from a purely surface scatteringphenomenon. Using a perturbational approach derived for surfaces with smallgradients (ksh < 0.3, where k = 2p /l is the wavenumber and sh is the rmsroughness height), the backscattering field is shown to be strongly dependent onthe polarization of the incident and scattered waves.

Other scattering models are, perhaps, potentially more powerful than the previ-ous ones, but they have not yet been extensively applied to the interpretation ofexperimental measurements. Without going into detail, we mention three methodsthat seem to be the most promising: the full-wave method of Bahar [11] bridgingthe wide gap existing between the perturbational solutions for rough surfaces withsmall slopes and the quasi-optics solutions, the diagram method [12], and thestochastic Fourier method [10]. These methods are not yet fully developed, butthe results obtained appear to be very encouraging. In all of the models of roughsurface scattering, the normalized RCS of surface characterizing the intensity ofthe scattered field is a product of two functions

s 0pp ( f , ui ) = Dpp [es ( f ), ui ] ? S ( f , ui ) (3.1)

The first function D (?), or so-called dielectric function, characterizes the dielec-tric properties of the scattering medium and depends upon the polarizationpp = HH or VV, the angle of incidence ui , and the dielectric constant es . For thesemodels, the dielectric functions are equal or proportional to the Fresnel reflectivity(i.e., they present equally the RCS dependence as a function of dielectric constantof the medium). Their main limitation is an inadequate estimation of the scatteringcoefficients for cross-polarized components of the scattered signals.

The second function S (?) takes into account the surface roughness influenceon normalized RCS, where f denotes the radar frequency.

140 Estimation of Land Parameters by Multichannel Radar Methods

For analysis of the capabilities of the multichannel method, we use the simplesurface scattering models, particularly the small perturbation model described inthe previous chapter. There are several reasons for this. First of all, the conditionksh > 1 is usually fulfilled for agricultural soils with different methods of cultivationin the microwave band. Then, the small perturbation model can be used only forapproximated estimation of the normalized RCS. However, the model derivationsof normalized RCS dependences as functions of incidence angles, rms surfaceroughness, and soil moisture coincide rather accurately with the experimentalresults up to and beyond ksh > 2.5–3.0 [12]. The most significant differencesbetween the model and experimental results are observed for the cross-polarizedcomponents of the scattered signals. Besides, for the considered multichannelmethods, we have used the ratios of the normalized RCS at different frequenciesand polarizations, where the absolute error of RCS estimation from the perturba-tional model do not greatly influence the ratio values.

For the case of backscattering we obtain

Dpp [es ( f ), ui ] = |app (ui ) |2 (3.2)

S ( f , ui ) = 8(k cos ui )4 ? s

2h ? W (2k cos ui ) (3.3)

aVV =(es − 1)[sin2 ui − es (1 + sin2 ui )]

(es cos ui + √es − sin2 ui )2(3.4)

aHH =es − 1

(cos ui + √es − sin2 ui )2(3.5)

where W (?) is the surface roughness spectrum.The moisture content determination is based on the correlation between the

dielectric function (3.2) and the soil dielectric constant, as well as on the dielectricfunction dependence on frequency and soil moisture. The results of simulation [5–7]have shown that the dielectric functions had a weak dependence upon frequency inthe microwave band, while the soil dielectric constant differed significantly fordifferent frequencies. This is illustrated in Figure 3.1.

Analysis shows that the maximal differences of dielectric functions do notexceed 1 dB if the ratio of two frequencies satisfies the condition

1 < f2 / f1 < 2 − 3 (3.6)

(i.e., for the microwave band, the dielectric functions are practically frequencyindependent). The most obvious dependence of D on f takes place for moisture

3.1 Estimation of Soil Parameters 141

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bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

Figure 3.1 The dielectric functions D versus soil moisture for horizontal polarization, frequencies1–18 GHz, and incidence angle 60°. (From: [5].)

0.05–0.2 g ? cm−3. The lower bound of this interval deals with the appearance offree water molecules in the soil, and when the moisture content exceeds 0.2–0.25g ? cm−3, saturation of dielectric functions occurs. This phenomenon leads to twoimportant conclusions:

• The potential moisture content sensitivity is equal for all frequencies in themicrowave band.

• The weak frequency dependence of the dielectric functions permits soil mois-ture estimation for the upper soil layer and measurement of other statisticalcharacteristics on the basis of joint analysis of multichannel remote sensingdata.

As shown in [5], in the framework of the small perturbation approach, it istheoretically possible to separate and to accurately estimate the roughness and themoisture parameters on the basis of multichannel measurements without a prioriknowledge of surface statistical characteristics. In the most simple case of two-frequency remote sensing, it is possible to select the frequencies f1 , f2 and the

142 Estimation of Land Parameters by Multichannel Radar Methods

angles of incidence u1 , u2 in such a way that the ratio of corresponding functionsof roughness does not depend on surface statistical characteristics. The conditionof such independence is described by the equation

k1 sin u1 = k2 sin u2 (3.7)

Then the ratio of the RCS for two polarizations is

s 0pp ,1

s 0qq ,2

=s 0

pp ,1 ( f1 , u1)

s 0qq ,2 ( f2 , u2)

=Dpp (e1 , u1)

Dqq (e2 , u2)? Sk1 cos u1

k2 cos u2D4 (3.8)

Here pp , qq = HH or VV, e1 = e ( f1), and e2 = e ( f2). The ratio (3.8) is afunction of dielectric constants, wavelengths, and angles of incidence, and it doesnot depend on surface statistical characteristics. Remote moisture determinationcan be performed with (3.8), taking into account that the dielectric constants arefunctions of the volume moisture content of the soil upper layer. Here we canconsider five polarization ratios

R1 =s 0

HH,1

s 0HH,2

; R2 =s 0

HH,1

s 0VV,2

; R3 =s 0

VV,1

s 0HH,2

; R4 =s 0

VV,1

s 0VV,2

; R5 =s 0

HH

s 0VV

(3.9)

The polarization ratio R5 is the particular case when f1 = f2 = f , pp = HH,and qq = VV. It was proposed for the first time in [1], where it was shown thatthis ratio served for obtaining unbiased estimation of the real part of the complexdielectric constant; this ratio is also used in [5]. Moisture dependence R5 forf = 10 GHz and for different angles of incidence is shown in Figure 3.2. Thenonlinear behavior of this dependence is evident. The variation range of R5 increaseswith an increase in the angle of incidence and for u = 50°–60°, it is approximatelyequal to 6 dB. The maximal moisture sensitivity is observed for volume moisturecontent less than 0.25 g/cm−3 and is approximately equal to 0.24 dB/0.01 g/cm−3.For wetter soil, saturation is observed, and the sensitivity decreases by a factormore than 3.

Analysis of ratios R1–R4 as functions of the moisture is done in [5], where itis shown that forming of these estimates provides a maximum moisture contentsensitivity greater than 0.1 dB/0.01 g/cm−3 when frequency ratio f1 /f2 = 1.2–2.5;this is practically available. Multichannel techniques for estimation of statisticalcharacteristics for some surfaces in the framework of the small perturbation methodis based on the use of the following relationship [4]

SRi =s 0

pp (u i , f1)

s 0pp (u i , f2)

=Dpp [u i , es ( f1)] ? S (u i , f1)

Dpp [u i , es ( f2)] ? S (u i , f2)(3.10)

3.1 Estimation of Soil Parameters 143

Figure 3.2 The R5 ratio versus soil moisture at 10 GHz for incidence angles 20°–60° (From: [5].)

The weak frequency dependence of the dielectric function in the microwaveband permits us to assume that

Dpp [u i , es ( f1)] ≅ Dpp [u i , es ( f2)] (3.11)

This assumption is permissible taking into account the fact that the instrumentalaccuracy of normalized RCS estimation is 1.5 dB, especially if the operation fre-quencies f1 and f2 are comparable [i.e., when ( f2 − f1) << ( f2 + f1)/2 andf2 / f1 < 1.5]. In such conditions (3.10) is a function of surface parameters onlyand does not depend on es ( f ). That is,

SRi = Sk1k2D4 ?

W (2k1 sin u i , 0)W (2k2 sin u i , 0)

(3.12)

where, as noted earlier, W is the two-dimensional surface roughness spectrum.It is possible to assume for simplicity that

SKi = SRi ? Sk2k1D4; L1, i = 2k1 ? sin u i ; L2, i = 2k2 ? sin u i (3.13)

144 Estimation of Land Parameters by Multichannel Radar Methods

The model of surface roughness spatial spectrum determines the further estima-tion of the statistical characteristics.

Most often, surfaces with a Gaussian surface height distribution and any surfaceautocorrelation function are used. For surfaces with this height distribution and aGaussian autocorrelation function, analytical expressions are available for crosschecking. In addition, many surface scattering theories have been reported in theliterature using the assumption of Gaussian height distribution, although theroughness spectrum of naturally occurring surfaces generally has more high fre-quency components than the Gaussian. For a Gaussian surface, the autocorrelationfunction is

r (j ) = s2h ? exp (−j 2 ⁄ l 2) (3.14)

and the spatial power spectrum of roughness is

W (L) = ps2h l 2 expF−SLl

2 D2G (3.15)

Here l is the spatial correlation radius, L is the spatial wavenumber.The exponential autocorrelation function of the surface is also used

r (j ) = s2h expS−

|j |l D (3.16)

with a spatial power spectrum

W (L) = 4s2h l 2 ⁄ (1 + L2l 2) (3.17)

The rapidly increasing number of applications of fractal models and fractalgeometry in physics deserves close attention in studies of various areas, and particu-larly in remote sensing of land. It was first demonstrated by M. Berry [13] thatfor Gaussian statistics the surfaces with a simple power spectrum of the type

W1(L) = C1L−a, 1 < a < 3 (3.18)

are fractal rough surfaces. We note that the spectrum W is defined for any real L,and therefore, the corresponding surface has no characteristic scales and the rmsheight is not defined (sh → ∞). For the values of the spectral exponent a given by(3.18), the Fourier transform of W is a divergent integral; thus, the correlationfunction does not exist.

3.1 Estimation of Soil Parameters 145

The second surface type has the same expression for the roughness spectra butwith an outer (or large scale) cutoff L0

W2(L) = HCL−a for L ≥ L0

0 for L ≤ L0(3.19)

In view of the experimental data, type II surfaces appear to be of particularimportance in the practice of remote sensing (here the usual case is a = 2). Weshould like to note that for large L the spectra (3.17) and (3.18) are identical, buttype II spectra have finite rms height sh and infinite rms slope.

For a Gaussian surface autocorrelation function, we find the correlation radiusestimate as

l g = √4 ln (SKi ) ⁄ XL22, i − L

21, i C (3.20)

For surfaces with an exponential correlation function, the estimate of thecorrelation radius can be easily obtained as

l e = F(SKi − 1) ⁄ XL22, i − SKi L

21, i CG1/2

(3.21)

The estimate of the fractal spectrum exponent is

a = ln (SKi ) ⁄ ln(L2, i /L1, i ) (3.22)

The constant C is linked with sh by the expression

sh = X√C ? L1−a /2o C ⁄ [a ? sin (p /a )] (3.23)

The value of L0 is not greater than 2 when a = (1–3L); for a = 2, the valueof C = sh .

Therefore, it is obvious that in the multichannel approach, we can obtain onlyone surface characteristic from the radar measurements: its correlation radius orthe fractal spectrum exponent.

3.1.3 Efficiency of Multichannel Methods

The efficiency of considered multichannel approaches was investigated on the basisof the experimental results obtained in [3, 14]. As an example, the first surface

146 Estimation of Land Parameters by Multichannel Radar Methods

type is asphalt, for which the small perturbation method can be applied withoutlimitations. The angular dependences of the normalized RCS for asphalt areobtained in [14] at frequencies of 8.6 GHz, 17.0 GHz, and 35.6 GHz for twopolarizations (horizontal and vertical) and for the incidence angle region of 20–80°.

Before analysis of asphalt dielectric and statistical characteristics, it is worthnoting that the corresponding values are presented in some references (e.g., in [15,16]). At frequencies of 10 GHz and 35 GHz, the averaged dielectric constants ofasphalt are ea = 4.3 ± 0.5 and 2.5 ± 0.3, respectively. The surface roughnesscorrelation radius and rms height are l = 0.22 cm and sh = 0.04 cm.

The derivation of the dielectric function D for the small perturbation modelshows that, for asphalt, its frequency dependence should be taken into consider-ation. The difference between dielectric functions for frequencies of 10 GHz and35 GHz is approximately 2–4 dB for incidence angle variation from 20° to 60°.That is why the use of (3.10) for asphalt statistical characteristics estimation atfrequencies of 17 GHz and 35.6 GHz is impossible, as (3.11) is not valid for thesefrequencies.

The dielectric constant of asphalt was estimated using the polarization ratioR5 that is a monotonic function of ea . The averaged values obtained for all threefrequencies are given in Table 3.1.

These results match well with other reference data, and they confirm thepractical applicability of polarization ratio R5 for dielectric constant estimation.Using (3.10), estimation of asphalt surface roughness correlation radius and fractalspectrum exponent was performed. In this case (3.10) is a function only of l anda . The data for every frequency and polarization were considered statisticallyindependent. The estimates of surface parameters obtained for different channelsand polarizations as well as their averages are presented in Table 3.2.

It is seen from Table 3.2 that from the point of view of statistical reliability,the averaged estimate of the fractal spectra exponent is the best in comparison tosimilar estimates of correlation radius. Its confidence interval is approximately

Table 3.1 Averaged Values of ea

f (GHz) 8.6 17 35.6ea 3.56 ± 1.22 3.34 ± 0.38 1.77 ± 0.51

Table 3.2 Estimates of Asphalt Surface Roughness Correlation Radius, Height rms Values, andFractal Spectra Exponent

f (GHz) lg (cm) lex (cm) shg (cm) shex (cm) a sC (cm)8.6 0.71 0.69 0.12 0.14 1.34 0.107

17.0 0.41 0.50 0.23 0.25 1.66 0.11135.6 0.20 0.26 0.20 0.21 1.61 0.093

Averagedvalues 0.44 ± 0.2 0.48 ± 0.24 0.20 ± 0.05 0.20 ± 0.05 1.39 ± 0.16 0.103 ± 0.008

3.1 Estimation of Soil Parameters 147

20% in respect to a , while the range of respective confidence intervals for estimatesof l is in the limit 50%.

Estimates of the last unknown parameter sh (or C ) were formed using (3.1)by means of minimizing the difference between experimental data of s 0

pp andtheoretically derived approximate values of s 0

pp . Because the amplitude of fluctua-tions depends on (a − C ) value, it is easier to analyze not C but the value sC ,taking into account that sC = C /M 4−a [m] (i.e., expressed in meters as well assh ). The estimates obtained for different frequencies are shown in Table 3.2.

As seen from Table 3.2, the estimate of sC is the best from the point of viewof its statistical reliability. Its confidence interval is less than 5% with respect toits average value, while the analogous intervals for sh lie in the limits 20%–30%of the estimates for Gaussian and exponential surfaces.

As illustration of fractal approximation possibilities, Table 3.3 shows thederived values for the surface roughness spectra W (Li ) obtained using C estimatesat different frequencies (here Li = k1 , k2 , k3 and u i = 30°).

These generalized data for all frequencies with application to average valuesof asphalt dielectric and statistical characteristics show that the Gaussian approxi-mation is not appropriate. It is illustrated well by Figure 3.3, where the experimen-tally obtained data at 35.6 GHz for horizontal and vertical polarizations and thetheoretical curves for Gaussian surface are compared.

Figure 3.4 represents the angular dependence of s 0 of asphalt at 35.6 GHz andmodel representations for exponential and fractal surfaces. The absolute averagediscrepancies of these curves from obtained values are 1.15 and 1.3 dB for exponen-tial and fractal surfaces, respectively. In practice, the two approximations areequally valid.

Analogous estimates of asphalt dielectric constant and its surface statisticalcharacteristics were obtained for 8.6 GHz and 17 GHz. They coincide well withl and a estimates obtained earlier, on the basis of which the dielectric constantestimates for asphalt were derived. Because for practical purposes we were interestedin applicability of fractal spectra for description of surface roughness, this approxi-mation is a subject of further consideration. The asphalt dielectric constant valuesobtained at 8.6 GHz and 17 GHz (a = 1.38) were equal to 3 ± 1.9 and 4.5 ± 2.0,respectively. The confidence intervals of these estimates were rather wide butthey overlapped. This could result from the accuracy of measurements and thevalidity of assumption (3.10) for asphalt at these frequencies. It is worth noting

Table 3.3 The Spectral Density of Asphalt Roughness (in Decibels) at Frequencies f1 = 8.6,f2 = 17.0, and f3 = 35.6 GHz for u i = 30°

f (GHz) W (k1) W (k2) W (k3)8.6 −97.8 −101.9 −106.4

17.0 −97.4 −101.5 −106.035.6 −99.4 −101.0 −105.5

148 Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.3 Angular dependence of s 0 for VV and HH polarizations and their comparison withGaussian models. (Solid and dotted curves are derived from models; signs are theexperimental data.)

the very close values obtained for the latter model parameter C (sC ). We gotsC = 0.146 cm and 0.148 cm for 8.6 GHz and 17 GHz, respectively.

The second type surface is a bare field. The experimental angular dependenceof the normalized RCS for this field obtained at 4.7 GHz for two polarizations ispresented in [3]. The soil is characterized by its high clay content (about 40%).The measurements have been carried out for dry soil (gravimetric moisture 4.3%)and for wet soil with gravimetric moisture (about 30.2%). The high clay contentresults in a temporally invariant rms surface roughness of about 2.5 cm.

The estimates of exponent a were formed from data of s 0 using (3.22) andthe regression analysis method, where the second frequency was 8.6 GHz. The|e | and a data for dry and wet soils are shown in Table 3.4.

The a values for dry soil are practically identical for both polarizations, andthey differ significantly for wet soil. Using C estimations, we derived the values ofsurface roughness spectral density, also presented in Table 3.4. The comparisonof W (k ) for soil with W (k ) for asphalt at 17.0 GHz shows that the rough-ness influence on the normalized RCS of a field is greater than that for asphalt by15–20 dB (see data in Table 3.3).

The application of Gaussian and exponential models for the autocorrelationfunction permits us to estimate the correlation radius of field roughness. The surfaceroughness for this field does not satisfy the small perturbation model. In theseconditions, the derived values of l are approximately 1 cm and do not correspond

3.1 Estimation of Soil Parameters 149

Figure 3.4 Angular dependences of s 0 for VV and HH polarizations and their comparison with(a) exponential and (b) fractal models.

150 Estimation of Land Parameters by Multichannel Radar Methods

Table 3.4 The Values of |e |, a , and W (k1) for Dry and Wet Soil

Polarization HH HH VV VVGravimetric volume, % 4.3 30.2 4.3 30.2|e | 3 35 3 35a 0.8–1.1 0.8–1.1 1.0–1.3 2.3–2.5W (k1) (dB) −(76.7–78.5) −(76.7–78.5) −(79.8–80.5) −(87.4–88.1)

to real values of l for agricultural fields. At the same time, the fractal approximationof surface spectra seems to be more suitable because the spectral exponent is in agood agreement with the value range (a = 1–3).

Thus, the application of multichannel methods permits us to obtain the soildielectric constant and, consequently, the soil moisture and the soil statisticalcharacteristics using the fractal model for the surface spectra—even for roughsurfaces when the limitations of the small perturbation model are inapplicable.

3.2 Soil Erosion Experimental Determination

3.2.1 Set and Technique of Measurement

One of the purposes of multichannel methods of remote determination of soilparameters is estimation of the degree of erosion, because this determines the soilfertility. The erosion influences the radar characteristics of the soil, and, conse-quently, one can find the correlation between the scattered signal and the soilstatistical characteristics.

An experimental study of the multichannel method of soil characteristics estima-tion has been carried out [17]. The agricultural field was located on a hill, and itsdifferent areas had average slopes between 2° and 8°. The cultivation performed3 weeks earlier across the slope included plowing, disking, and harrowing. The soiltype was chernozem (black soil) with four stages of soil erosion. Visual inspectionindicated that the periodic row structure was almost destroyed by many rainfalls.

A dual-frequency CW radar was used for experimental investigation withcharacteristics as shown in Table 3.5. The radar system included two radars withantennas placed on a common platform, the antennas receiving backscatteredsignals from the same area of surface. A peculiarity of this radar was the possibilityof signal radiation with any linear polarization and simultaneous reception of twoorthogonal components of the scattered signal.

Radar measurements were followed by simultaneous in situ measurements ofsoil characteristics, including obtaining the surface profiles along and across thedirection of cultivation, the moisture of the near-surface soil layer (from 2 cm to5 cm in depth), and the agrophysical characteristics of the soil.

Determination of the normalized RCS was made using the method of compari-son with the RCS of a corner reflector for calibration. The latter was installed at

3.2 Soil Erosion Experimental Determination 151

Table 3.5 The Main Characteristics of the Radar

Wavelength (cm) 3.2 0.8Modulation CW CWTransmitted power (mW) 50.0 50.0Sensitivity (dBW) −165 −155Antenna pattern width (degree) 10.0 10.0Polarization—Radiation H or V H or V—Reception Two orthogonal components Two orthogonal componentsPolarization isolation (dB) 35 30Maximal bandwidth of receiver (kHz) 2.0 2.0

a height of 3m over the land, providing a negligible influence of background surfaceclutter on the calibration accuracy.

The main radar characteristics were the normalized RCS dependence on theincidence angles for two polarizations (horizontal and vertical) and for two wave-lengths (3 cm and 0.8 cm), (i.e., four angular dependences—for copolarized andcross-polarized components of the scattered signal—for every polarization of theradiated signal). The measurements were carried out for an incidence angularinterval 35°–70° with step of 5°. For the normalized RCS estimates, the averagingof angular dependence for several azimuthally spaced angles was used. This resultedin decreasing of the normalized RCS fluctuations to 1.0–1.5 dB.

3.2.2 Statistical and Agrophysical Characteristics of Fields

The erosion state of the soil areas was determined by the average-weighted diameterof water-stable aggregates and by the aggregation coefficient Ka [18]. This dependson the quantity and quality of humus, granulometer, and mineral content of thesoil and characterizes the genetic particularity of the soil. The moisture and aggre-gate content characteristics are shown in Table 3.6.

As seen from Table 3.6, the agrophysical characteristics of the soil in areas1–3 are rather similar, and the most different properties are observed for area 4.

The moisture measurements at the reference points were carried out by thethermostat-weight technique and showed that the moisture for the near-surfacelayer of the soil was rather homogeneous for all areas and the gravimetric soilmoisture was near the field capacity. This was caused by weak soil drying because

Table 3.6 Weighted Soil Moisture and Aggregate Content Characteristics

Terrain area 1 2 3 4Erosion degree Noneroded Weakly eroded Middle eroded Heavy erodedSoil moisture (%) 16.2 12.9 17.0 11.8d (mm) 0.85 0.46 0.38 0.25Ka 0.57 0.39 0.37 0.23

152 Estimation of Land Parameters by Multichannel Radar Methods

of the low air temperature (it was about 5°–6° ). Only the moisture in area 3 washigher than for the other areas because it was obtained for the day after a rain.For this reason, it was possible to neglect the moisture influence on the normalizedRCS for different areas of soil.

The surface roughness was measured with a profile meter with density of 1measurement per 10 mm, which ensured errors not greater than 5 mm. The commonlength of every surface profile was 6–7.5m. The profile processing included theobtaining of autocorrelation functions and spatial spectra along and across thedirections of cultivation.

The method of cultivation can result in a significantly different degree ofroughness for the same agricultural region. It is worth noting that for fields witha periodic structure, the rms surface height depends greatly on the profile orientationin respect to the plowing furrows. However, for fresh-plowed fields, the periodicityis often significantly distorted by presence of clods. The small and rather largeclods appearing for breast plowing are oriented randomly. Their dimensions dependon the moisture and the soil type. In general, the surface normalized autocorrelationfunction can be approximated by

r (j ) = exp (−g ? j ) ? cos (2pj /b ) (3.24)

where j is the spatial coordinate. No statistical dependence between parameters gand b was detected. A more general approximation can be used for dual-scaleroughness (the typical cases are the plowed and harrowed fields), taking intoaccount the peculiarities of the cultivation. For this case

r (j ) = (1 − A ) ? exp (−gj ) ? cos (2pj /Lp ) + A ? exp (−hj ) ? cos (2pj /Lh )(3.25)

where the coefficient A is determined by the ratio of harrowing to plowing depths,Lp denotes the distance between the furrows of plowing, and Lh defines the distancebetween the furrows of harrowing.

The observations show that the dynamics of change of field roughness (if thefields are not subjected to additional tillage) are fully determined by the atmospherehumidity conditions—the aggregate and mechanical soil contents. Initially, theroughness smoothing caused by rainfall can result in an increase of periodicitybecause the process of furrow destruction is more prolonged in comparison to theprocess of clod destruction. Table 3.7 shows the approximate range of sh variationsfor different types of cultivation (the measurements were performed for along andacross directions).

It is worth noting that the percent content of clay in the soil is a basic factordetermining the degree of field roughness after natural furrow and clod destruction.The mechanical and aggregate soil contents are subjected to slow permanent change

3.2 Soil Erosion Experimental Determination 153

Table 3.7 Roughness rms Values sh (cm) for Some Fields

Plowed and HarrowedMeasurements Fresh Plowed Washed Off Harrowed and RolledAlong direction of cultivation >4–5 3–5 1.5–3 <1.5Across direction of cultivation 3–4 2–3.5 1–2 <1.5

and, as the result, to erosion. The field slope of about 5°–10° is the reason for theformation of heavily and medium-eroded areas. The mechanical and hydrophysicalproperties of such soils differ from those of noneroded soils. Therefore, we expectthat in the borders of the same terrain, different degrees of erosion can be foundas the result of the same cultivation. For such fields, successful separation ofroughness and moisture effects and their accurate estimation offers the ability toretrieve information about the erosion process and its evolution.

Figure 3.5 presents the correlation functions of roughness across the directionof cultivation just after plowing (curve 1) and 20 days later (curve 2) when thefield became smoothed and rolled. The process of furrow destruction manifestsitself in the removal of the periodic structure of the autocorrelation function. Theautocorrelation functions of surface roughness along the direction of cultivationhardly differ from the autocorrelation functions of isotropic surfaces.

Examples of spatial spectra for investigated land areas are shown in Figure3.6. The spectra were obtained using the standard fast Fourier transform (FFT)and the maximum entropy methods (Berg’s algorithm). It is seen from Figure 3.6(a)

Figure 3.5 The spatial autocorrelation functions of plowed field (1) right after plowing and (2) 20days after plowing.

154 Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.6 The spatial spectra of investigated land lot obtained by (a) FFT and (b) maximal entropymethod.

that harrowing leads to a second maximum appearing in the soil spatial spectrum,while the maximal entropy method smoothes this maximum. The rms values ofsurface roughness sh for the cross direction are greater than for along furrowdirection. In the spatial spectra, the second maximum appeared rather often; it is

3.2 Soil Erosion Experimental Determination 155

connected with the cultivation type. However, in our measurements, it is notobserved due to washing off of the soil.

For determination of roughness characteristics, the fractal approximation ofspatial spectra was used. The average values of sh and fractal exponent a areshown in Table 3.8. The averaged exponent in the experimental spectra was equalto a = 1.61 ± 0.89. For profiles with a strongly marked periodic row structure,the average value of a of the fractal spectra does not change significantly; its valuesare a = 1.64 ± 1.02.

The data analysis showed that the value of sh along the direction of cultivationdid not practically depend on the degree of soil erosion, but we observed a decreaseof sh with an increase of erosion degree across the direction of cultivation. At thesame time, a decrease of a was observed along the direction of cultivation whenthe erosion degree increased. This shows an increase of the spatial correlationradius for noneroded soils in comparison to eroded ones.

3.2.3 On-Land Radar Measurement Results

The results of the angular normalized RCS dependence study have shown thefollowing [6, 19, 20]:

• The normalized RCS for an 8-mm wavelength is greater by 3–6 dB than fora 3-cm wavelength for all selected surface areas and for both polarizationswith reception of copolarized components.

• The normalized RCS values for vertical polarization exceed those for hori-zontal polarization by 0.2–8 dB, depending upon the degree of surfaceerosion.

• The speed of the angular RCS variations depends on the degree of erosion.

The longitudinal furrow structure of the field is a dominating factor in formingthe scattered signal for horizontal polarization. The washing off of roughness forheavily eroded lots results in decreasing RCS in comparison with noneroded regions.At the same time, for vertical polarization, the RCS values for strongly eroded lotsare greater than for noneroded ones. This is probably explained by the differentdegree of cross-furrow microstructure destruction, which is larger for stronglyeroded soil. For example, Figure 3.7 shows the angular dependence of normalized

Table 3.8 Roughness Characteristics of Investigated Lots

Erosion Degree Noneroded Weakly Eroded Middle Eroded Heavy ErodedAlong furrows sh (cm) 1.7 1.54 1.46 1.28

a 1.8 1.6 1.35 1.3Across furrows sh (cm) 2.1 2.45 1.9 1.8

a 1.9 1.75 1.85 1.7

156 Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.7 Specific RCS vs incidence angle for areas with different erosion degree at 0.8 cm for(a) vertical and (b) horizontal polarizations; curve 1 is obtained for noneroded areas,the curve 2 for weakly and middle-eroded areas, the curve 3 for heavily eroded areas.

3.2 Soil Erosion Experimental Determination 157

RCS for areas 3 (middle eroded) and 4 (heavily eroded) obtained at 3 cm and0.8 cm for horizontal polarization. The mean rate of angular variation of thenormalized RCS is approximately 0.4–0.5 dB/degree for area 3 and about 0.2–0.3dB/degree for area 4. This angular dependence can be understood because theexponent of the surface roughness fractal spectrum decreases with increasing degreeof erosion, as seen from Table 3.8.

Dual-channel polarization processing (using ratio R5) has permitted us toemphasize its sensitivity to variations of soil agrophysical parameters. This is seenfrom Table 3.9 where the polarization ratios s 0

HH /s 0VV are shown averaged for

all regions of incidence angles. The maximal value of this ratio at 3 cm is −6.6 dBfor heavy eroded areas and 7.7 dB for non-eroded areas. Good correlation betweenthe ratio R5 and the agrophysical parameters of soil are observed.

For incidence angles u = 35°–45°, the maximum sensitivity of R5 to agrophysicalparameter variations is observed at 3 cm, but at 0.8 cm the maximum sensitivityis observed for the incidence angles greater than 60°.

The angular dependence of the normalized RCS ratios for cross-polarizedreception seems to have no correlation with soil erosion degree. The depolarizationincreases with wavelength decrease from −(10–13) dB for 3 cm to −(5–8) dB for8 mm, and correlation of these values with erosion degree is not observed. InFigure 3.8, this dependency is shown for 3 cm and 0.8 cm and for the horizontalpolarization.

3.2.4 Aircraft Remote Sensing

Sections 3.2 and 3.3 give the radiophysical model an experimental basis for determi-nation of bare soil erosion from multichannel remote sensing data. According tothis approach, one needs to have available a few radar images of the same terrainarea (bare soil region) formed for VV and HH polarizations of transmitted andreceived sensing signals in the microwave band. But the availability of such imagesis not the only prerequisite for successfully measuring soil erosion using multichan-nel radar remote sensing data. There are also some normalized requirements on theimaging systems used, along with the algorithms of multichannel data processing.

First, radar remote sensing systems should provide appropriate spatial resolu-tion in order to be sensitive to local variations of normalized radar cross sectionsin the terrain region of interest. The desirable resolution is about 20 × 20 m2 orbetter, and this can be provided by either SLAR or SAR systems.

Table 3.9 The Ratio R5 of Investigated Areas at 0.8 cm and 3 cm

Erosion Degree Noneroded Weakly Eroded Middle Eroded Heavy Eroded

11.4 6.0 3.4 −4.5s 0HH

s 0VV

(dB)0.8 cm3 cm 7.7 0.2 −3.1 −6.6

158 Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.8 The ratios s 0HV /s 0

HH versus the incidence angle at (a) 3 cm and (b) 0.8 cm (curves 1–3are labeled as in Figure 3.7).

Second, RCS estimates for each pixel obtained from the observed radar imagesshould be sufficiently accurate. This means that the imaging systems have to beproperly calibrated [21], and it is also desirable to use either images with low levelsof noise or to properly smooth them. At the same time, the presence of multiplicativenoise is inherent in radar images, and this noise, also referred as speckle, is especiallyintense in images formed by SARs. Therefore, multichannel images are to be

3.3 Methods of Multichannel Radar Image Processing 159

processed in such a manner that the multiplicative noise is effectively suppressed,while the edges and fine details corresponding to local variations of RCS (and,respectively, different degree of soil erosion) are well preserved.

Finally, the component images of multichannel remote sensing data should beproperly registered (i.e., represented in a common spatial coordinate system withcorrespondence of the multichannel image cell to the element of the sensed terrain).In practice, the resolutions of different radars used for remote sensing are various,the linear resolution depending on range for all radars. Besides, the sensed terrainstrips are sometimes obtained by radar during different flights of the radar platform,and, certainly, the images do not fully coincide. For example, we obtained 8-mmimages from two approximately opposite directions of platform trajectory for radaroperation with different polarizations while we were conducting the experiment.

Terrain relief and Earth’s surface curvature are additional sources of geometricdistortions of the radar image in multichannel systems for remote sensing.

The images were obtained simultaneously by two radar systems at 10 GHzand 35 GHz installed onboard an IL-18D (Russian) aircraft [22]. The spatialresolution of the Ka-band SLAR was about 20 × 20 m2; for the X-band SLAR,it was approximately 30 × 30 m2; the average altitude was about 7,000m; andthe agricultural field under investigation was observed at an incidence angleabout 50°. The field dimensions were 1.2 × 1.2 km2, and the field occupied about50 × 50 pixels.

3.3 Methods of Multichannel Radar Image Processing

3.3.1 Image Superimposing

The radar images are to be superimposed, using both linear and nonlinear tech-niques. For a case when we deal with analyzed terrain areas having relatively smallsizes, it is possible to use the linear aphine transformation; otherwise, it becomesreasonable to apply a preliminary geometrical correction of images and then morecomplicated methods of nonlinear transformations of data arrays.

In practice, several factors obstruct getting the image-to-topology map andimage-to-image registration without errors. These factors are the fluctuations ofthe imaging system platform trajectory (for airborne systems), the curvature of thesensed terrain relief, and the nonlinear relationship between slant and groundrange. So the task of accurate registration, transformation, and interpolation ofmultichannel remote sensing data is also important because these registration errorscan result in errors in further interpretation for soil erosion determination, especiallyin the neighborhood of edges and details [23].

The procedure for image-to-image and image-to-topology map superimposinginclude the following stages:

160 Estimation of Land Parameters by Multichannel Radar Methods

1. Selection of common reference points in all images. This operation is per-formed by qualified experts using the corners of agricultural fields, or inmore general cases any bright quasi-point objects (e.g., bridges, buildings,or highway crossroads) easily recognizable in all images can be selected asreference points and found interactively. Contour (edge) features of imagescan be used for reference point selection as well; sometimes image prelimi-nary filtering is expedient.

2. Derivation of coefficients describing the aphine transform of the superim-posed image to the coordinate system of the reference image or topologymap. Being represented in a matrix form this aphine transform is [8]

F = A ? H + D (3.26)

where operator D = HDx

DyJ denotes the transformation of superimposed

image H and matrix A = HA11A12

A21A22J defines the scaling and rotation opera-

tions to be done over the same image H. Denoting the reference point

coordinates as Fxyi = HFxi

FyiJ and Gxyi = HGxi

GyiJ, i = 1, 2, . . . , l for the

superimposed and reference images (maps), respectively, the superimposingintegral errors are determined by horizontal and vertical components dhand dv

dh = ∑l

i =1(Fxi − A11Gxi − A12Gyi − Dx )2 (3.27)

dv = ∑l

i =1(Fyi − A21Gxi − A22Gyi − Dy )2

Equation (3.27) produces an equation system that can be uniquely solvedwith respect to A and D parameters for l = 3. But for l ≥ 4 (the greater lis, the higher the accuracy of superimposing), some optimization procedureshould be applied; obviously, the requirement of total square mean errorminimization is typical and reasonable. This leads to a solution of theequation system obtained after calculation of partial derivatives on matrixtransform coefficients. These operations are rather easy; at the first stage,the matrix D parameters are estimated, and then the matrix A coefficientshave to be derived. We also analyze the problem of how many referencepoints it is reasonable to select and what their optimal spatial locations are.

3.3 Methods of Multichannel Radar Image Processing 161

Usually the selection of 7–12 reference points forming an equilateral polygonis applicable to many practical situations. They should be placed as sparselyas possible in the sense of maximal total distance between them. Thisconclusion is quite trivial [24] and intuitively understandable.

3. Interpolation of superimposed image H to the coordinate grid nodes of thereference image G. The necessity of this operation is explained by the factthat both images are sampled, and after spatial transformation the majorityof grid nodes for images do not coincide. Several methods can be used fortransformed image interpolation after superimposition: nearest neighbor,bilinear, and bicubic [25]. The first method happened to be more expedientbecause it resulted in minimal dynamic errors in the neighborhood of edgesand fine details. Besides, this method is very simple and requires minimalcomputational efforts. Bilinear and bicubic interpolation methods providebetter visual perception but are more complicated. Because of these advan-tages, the first technique was applied in data processing. It is based onsubstitution by the nearest sample value or on linear (or median) approxima-tion taking into account the four nearest neighbor pixel values, and itprovides appropriate interpolation accuracy. The errors and distortionscaused by interpolation are usually smaller than multiplicative noise existingin real images. Completing the discussion on image superimposing, it isworth mentioning that the errors of this operation can exceed one or twopixels; therefore, it is desirable to decrease them if possible during furtherstages of data processing.

Let us demonstrate these effects for real multichannel radar remote sensingdata. The airborne SLAR image of an agricultural region in the Ukraine is shownin Figure 3.9 (l = 8 mm, HH polarization). The multiplicative noise is not veryintensive, its relative variance s2

m ≈ 0.005.The SLAR image of the same region for VV polarization is shown in Figure

3.10.The multiplicative noise in this image has the same relative variance s2

m ≈ 0.005.The general appearance of this image is in a sense similar to Figure 3.9 (becausethe same terrain region is sensed) and in another sense different, as Figure 3.10 isformed for another polarization. Just this simultaneous similarity and differencebetween component images of multichannel remote sensing data are exploited forretrieval of useful information from them and separation of factors affecting theRCS for different wavelengths and polarizations. Figure 3.11 shows the airborneSLAR image formed by another radar (l = 3 cm, HH polarization). The multiplica-tive noise in this image is more intensive (s2

m ≈ 0.011), although it is still Gaussianwith a unity mean, as with both 8-mm images considered earlier. This image isalso similar to those ones shown in Figures 3.9 and 3.10, but a small differenceof resolution cell size between the images is seen.

162 Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.9 Radar image of agricultural region of Ukraine at 8 mm for HH polarization.

Due to similarity of component images of multichannel remote sensing data,it is quite easy to select the set of ground control points (GCPs) in all images and,if needed, the topology map of the sensed region. An example of such set selectionis shown in Figures 3.9 and 3.11 (10 GCPs are indicated by flags). This approachto multicomponent image joint registration is rather traditional (see [26] for details)and, in our opinion, does not require too many additional explanations. Theonly thing worth mentioning is the following: For joint registration of small-sizecomponent images (or when the component images have a common fragmentof size tens to tens pixels) even linear (affined) transforms provide appropriateaccuracy—the residual errors are usually about one pixel. In case of larger size ofthe fragment common for all component images, the nonlinear spatial transformsare worth applying [26] to minimize the residual superimposing errors. The applica-tion of nonlinear transforms at the image registration stage permits us to keepthese errors at the level about one resolution element.

An example of a three-channel jointly registered radar image obtained forcomponent images in Figures 3.9–3.11 and represented in the monochrome modeis given in Figure 3.12.

The image in Figure 3.9 was chosen as a strong point, providing the greatestsizes of surface fragments common for three images. Only the fragment commonfor all component images is shown in Figure 3.12, which is why this image doesnot have a rectangular shape. Noise in this image is observed, and the edges area little bit smoothed (not sharp) due to residual registration errors.

3.3 Methods of Multichannel Radar Image Processing 163

Figure 3.10 Radar image of agricultural region of Ukraine at 8 mm for VV polarization.

The total accuracy of Ka-band image-to-image superimposition was character-ized by an rms value of 1.97; for superimposing the X-band image to Ka-bandHH, it was 3.06.

Thus, the tasks of multichannel radar preprocessing resulting from the previousanalysis are the following: The noise in these images should be suppressed whilethe edges and small size objects are to be preserved. Besides, if the residual imageregistration errors are rather large, it is desirable to sharpen these edges to avoidpossible data interpretation errors in their neighborhoods.

3.3.2 Methods of Multichannel Radar Image Filtering

Noise in multichannel radar images or in ratio images obtained as the ratios (3.9)calculated for each pixel can be suppressed in several different ways. First, therequired ratio image can be calculated as the pixel-by-pixel ratio of the correspond-ing jointly registered component images without prefiltering. Then, the resulting

164 Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.11 Radar image of agricultural region of Ukraine at 3 cm for HH polarization.

ratio image can be postprocessed by a filter [27, 28]. However, according to recentexperience, this is not the best method. The reasons are the following:

1. The relative variance of noise in the ratio image is s2mrat ≥ s

2m1 + s

2m2 , where

s2m1 and s

2m2 are the relative variances in component radar images used for

obtaining the ratio image; thus, efficient filtering of the ratio image can beproblematic because of rather intensive noise.

2. The residual image-to-image registration errors cannot be removed due topostprocessing of the ratio image.

Another way is to perform multichannel radar image filtering before gettingthe required ratio image(s). In this case, there are two methods:

1. To perform component processing of multichannel data using filters wellsuited for each component image (the proper filter selection depends onthe properties of noise in the corresponding component image [27]);

2. To use vector filtering of multichannel radar remote sensing data.

The comparison of these two approaches shows that the latter one (i.e., thevector filtering of multichannel remote sensing data) is more useful [29]. This is

3.3 Methods of Multichannel Radar Image Processing 165

Figure 3.12 The joint radar image in monochrome regime.

because the vector filtering methods take into account the correlation that practi-cally always exists between the component images of multichannel data.

Among vector filtering methods, we designed two techniques to best suit thetasks that should be solved for the situation at hand. The first method is theuse of adaptive nonlinear vector filter [28]. This filter exploits the advantages ofcomponent and vector filtering. Due to application of the vector median for severalsubapertures at the final stage of data processing, it provides sharpening of thesmeared edges and the fine details arising because of residual registration errors.This is clearly seen in Figure 3.13, where the output multichannel image appearsin monochrome representation.

Comparing the images in Figures 3.12 and 3.13, it is seen that the adaptivenonlinear vector filter [16] also effectively suppressed noise in homogeneous regionsof the image. The benefits provided considerably improve the interpretation ofmultichannel data (see the results for test data in our paper [26]).

Another vector filter that performs appropriately well in the situation consid-ered is the modified vector sigma filter [27]. This filter is unable to remove the

166 Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.13 Multichannel radar image after adaptive nonlinear vector filtering.

residual registration errors, which is why its application is recommended for caseswhen these errors are negligibly small. The noise-suppressing efficiency of themodified vector sigma filter is approximately the same as for the adaptive nonlinearvector filter. The obvious advantage of the modified vector sigma filter is that itis able to preserve the low-contrast edges in component images. For the applicationconsidered, this is a very important property, as the contrasts between the areaswith different erosion degree are usually not too large.

3.4 Soil Erosion Determination from Ratio Images: ExperimentalResults

The use of vector filtering of multichannel radar remote sensing data, as shownearlier, leads to preservation of valuable information and simultaneous removal ofnoise and errors. These results offer ratio images with appropriately high quality.

Keep in mind that the ratio images are needed only for estimation of soil-erosion degree from multichannel radar remote sensing data, according to theradiophysical model approach (Chapter 2). For other types of surfaces, like ice,water surface, forest, and agricultural fields with vegetation, other models of back-scattered signals and RCS can be valid; therefore, ratio image forming can be of

3.4 Soil Erosion Determination from Ratio Images: Experimental Results 167

no use. In other words, this means that one should know a priori what pixels ofjointly registered multichannel remote sensing data and ratio images correspondto bare soil lots. In our case, it was an agricultural field of size approximately1 × 1 km2. In all images presented in Figures 3.9–3.13, this field is placed approxi-mately in the center (up and to the left from flags 5 and 7 in Figures 3.9 and 3.11).

Two ratio images 10 lg XI HHij /I VV

ij C that characterize the RCS ratio have beenobtained using the images presented in Figures 3.9 and 3.10. Before getting theseratio images, the component 8-mm images with different polarizations have beenjointly registered using the nonlinear transform and nearest neighbor interpolation.Then they have been processed by the adaptive nonlinear vector filter (the firstratio image) and by the modified vector sigma filter (the second ratio image). Thedifference between the RCS ratios estimated from the two ratio images was almostalways less than 0.3 dB, and thus the considered methods of vector filtering producesimilar results.

Erosion state classification has been performed by means of setting the thresholdvalues 8 dB, 4.5 dB, and 0 dB. Then, if the ratio image value is larger than 8 dB,the corresponding pixel is considered a noneroded area and indicated by whitecolor in the erosion classification map given in Figure 3.14. The values in the ratioimage that are within the limits 4.5–8 dB are indicated by light gray color in thisfigure (weakly eroded areas). The pixels with values within the limits 0–4.5 dB

Figure 3.14 Erosion state classification.

168 Estimation of Land Parameters by Multichannel Radar Methods

correspond to middle-erosion degree (dark gray color), and the pixels with valuesbelow 0 dB correspond to heavily eroded areas of soil (black color).

In situ measurements of erosion state have been performed for several controlpoints—locations of five of them are shown in Figure 3.14. Due to this, it hasbecome possible to compare the remote sensing classification results with dataobtained from in situ measurements. This comparison has demonstrated the fol-lowing:

1. For all five control points, the remote sensing classification results agreewith conclusions based on in situ measurements.

2. The estimates of the ratios I HHij /I VV

ij differ from the ratios s 0HH /s 0

VVdefined for the centers of the classes (6 dB for weakly eroded lots and2.3 dB for middle-eroded ones) in the points of in situ measurement by lessthan 1.3 dB.

Such coincidence and accuracy of the obtained remote sensing and in situmeasurement results, to our mind, can be considered appropriate for practicalapplications.

Thus, it is shown theoretically that the multichannel methods permit us toseparate and to estimate the roughness parameters and the dielectric constant ofthe near-surface layer in the soil. The obtained ratios for multichannel approachescan be used when the small perturbation model conditions are not fully satisfied.The fractal spectra are the best approximations of the surface roughness spectra.

The correlation between the radar and agrophysical characteristics of soil wasproved experimentally. In particular, the use of dual-polarization processing permitsus to increase the sensitivity of radar remote sensing techniques to the soil erosionchange.

References

[1] Yakovlev, V. P., ‘‘On Radiometer and Radar Abilities for Surface Remote Sensing,’’ Proc.of State Scientific Center of Natural Resources Study, No. 26, Gidrometeoizdat, Leningrad,1986, pp. 27–31 (in Russian).

[2] Shi, I., et al., ‘‘SAR-Derived Soil Moisture Measurements for Bare Fields,’’ Proc. of IGARSS91, 1991, pp. 393–396.

[3] Mo, T., J. R. Wang, and T. J. Schmuggle, ‘‘Estimation of Surface Roughness Parametersfrom Dual Frequency Measurements of Radar Backscattering Coefficients,’’ IEEE Trans.Geoscience and Remote Sensing, Vol. GE-26, 1988, pp. 574–579.

[4] Zerdev, N. G., and G. P. Kulemin, ‘‘Surface Statistical Characteristics Determination byMultichannel Radar Methods,’’ Proc. Scientific Apparatus Design for MM and subMMRadiowave Bands, Institute of Radiophysics and Electronics of Ukrainian Academy ofScience, Kharkov, Ukraine, 1992, pp. 90–98 (in Russian).

References 169

[5] Zerdev, N. G., and G. P. Kulemin, ‘‘Soil Moisture Determination by Multichannel RadarTechniques,’’ Soviet Journal of Remote Sensing, 1993, Vol. 1, No. 11, pp. 139–152.

[6] Kulemin, G. P., et al., ‘‘Soil Moisture and Erosion Degree Estimation from MultichannelMicrowave Remote Sensing Data,’’ Proc. Europ. Symp. SPIE on Satellite Remote Sensing,Paris, France, September 1995, Vol. 2585, pp. 144–155.

[7] Kulemin, G. P., et al., ‘‘Soil Erosion Characteristics Estimation Techniques Using Multi-polarization MM-Band Remote Sensing Radar Systems,’’ Proc. URSI Open Symp.,Ahmedabad, India, November 1995, pp. 185–188.

[8] Kulemin, G. P., et al., ‘‘Radar Dual-Polarization Remote Sensing of Soil Erosion,’’ Proc.European Symp. Aerospace Remote Sensing, London, England, September 1997,Vol. 3222, pp. 89–100.

[9] Ishimaru, A., Wave Propagation and Scattering in Random Media, New York: AcademicPress, 1978.

[10] Bahar, E., ‘‘Full Wave Solutions for the Scattered Radiation Fields from Rough Surfaceswith Arbitrary Slopes and Frequency,’’ IEEE Trans. Antennas and Propagation,Vol. AP-28, 1980, pp. 11–21.

[11] Zipfel, C. C., and J. A. DeSanto, ‘‘Scattering of a Scalar Wave from a Random RoughSurface: A Diagrammatic Approach,’’ J. Math. Phys., Vol. 13, 1972, pp. 1903–1911.

[12] Brown, G. S., ‘‘A Stochastic Fourier Transform Approach to Scattering from PerfectlyConducting Randomly Rough Surfaces,’’ IEEE Trans. Antennas and Propagation,Vol. AP-30, 1982, pp. 1135–1144.

[13] Feder, J., Fractals, New York: John Wiley, 1988.

[14] Ulaby, F. T., R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active andPassive, Vol. III: From Theory to Applications, Norwood, MA: Artech House, 1986.

[15] Ulaby, F., ‘‘Radar Measurement of Soil Moisture Content,’’ IEEE Trans. Antennas andPropagation, Vol. AP-22, 1974, pp. 257–265.

[16] Kulemin, G. P., and V. B. Razskazovsky, The Scattering of Millimeter Radiowaves by theEarth Surface at Low Angles, Kiev: Naukova Dumka, 1987 (in Russian).

[17] Zerdev, N. G., and G. P. Kulemin, ‘‘Soil Erosion Effects in Microwave Backscatteringfrom Bare Fields,’’ 24th Europ. Microwave Conf. Proc., Vol. 1, Cannes, France, September5–8, 1994, pp. 431–436.

[18] Bulygin, S. J., and F. N. Lisitsky, ‘‘Microaggregation as the Indicator of Anti-Erosion SoilStability,’’ Pochvovedenie, No. 12, 1991, pp. 98–104 (in Russian).

[19] Kulemin, G. P., ‘‘Soil Erosion Estimation by Dual-Polarization Radar Remote Sensing,’’IGARSS’99 Proc., Vol. 2, 1999, pp. 846–848.

[20] Kulemin, G. P., ‘‘Radar Estimation of Soil Parameters by Multichannel Methods,’’ Physicsand Engineering of Millimeter and Submillimeter Waves, MSMW Third Int. KharkovSymp., Vol. 1, 1998, pp. 53–59.

[21] Zelensky, A. A., et al., ‘‘Locally Adaptive Robust Algorithm of Image Processing,’’ Inst.Radiophysics and Electr. NANU, Preprint 93-8, Vol. 39, Kharkov, Ukraine, 1993(in Russian).

[22] Kalmikov, A. I., et al., ‘‘Multipurpose Aircraft Radar System for Earth’s Investigations,’’IRE NANU, Preprint 90-21, Kharkov, Ukraine, 1990 (in Russian).

[23] Kulemin, G. P., et al., ‘‘MM-Wave Multichannel Remote Sensing Systems and Algorithmsof Image Processing,’’ Conf. Digest of Int. Conf. on MM-waves and Infrared Science andTechnology, Guanzhou, China, August 1994, pp. 359–362.

170 Estimation of Land Parameters by Multichannel Radar Methods

[24] Richard, J. A., Remote Sensing Digital Image Analysis: An Introduction, Berlin, Germany:Springer-Verlag, 1986.

[25] Achmetyanov, V. P., and A. Y. Pasmurov, ‘‘Processing of Radar Images in Tasks of Earth’sRemote Sensing,’’ Foreign Electronics, No. 1, 1987, pp. 70–81.

[26] Kulemin, G. P., et al., ‘‘Soil Erosion State Interpretation Using Pre- and Postprocessingof Multichannel Radar Images,’’ Proc. Europ. Symp. SPIE, Vol. 3499, 1998, pp. 142–151.

[27] Kurekin, A. A., et al., ‘‘Processing Multichannel Radar Images by Modified Vector SigmaFilter for Soil Erosion Degree Determination,’’ Proc. Europ. Symp. SPIE, Vol. 3868, 1999,pp. 162–173.

[28] Lukin, V. V., et al., ‘‘Data Fusion and Processing for Airborne Multichannel System ofRadar Remote Sensing: Methodology, Stages, and Algorithms,’’ Proc. SPIE, Vol. 4051,2000, pp. 188–197.

[29] Lukin V. V., et al., ‘‘Digital Adaptive Robust Algorithms for Radar Image Filtering,’’Electronic Imaging, Vol. 5, No. 3, 1996, pp. 410–421.

C H A P T E R 4

Sea Backscattering at Low GrazingAngles

4.1 Sea Roughness Features for Small Grazing Angles

4.1.1 Sea Roughness Characteristics

Sea roughness is determined by different atmospheric factors, the most significantof which is the wind. The roughness state can be determined by the significantwave height: the mean (peak-to-trough) height of the highest one-third of the waves.There are two numerical scales for roughness description. One of them is theDouglas scale for description of surface roughness and swell, shown in Table 4.1.

The second is the Beaufort scale, characterizing the surface wind velocity,shown in Table 4.2.

Measurement of the probability distribution of wave heights shows that 45%of oceanic waves have heights less than 1.2m, and 80% have heights less than3.6m.

The sea surface is a complex natural formation on which wind speed, sharpcrests, breaking waves, water spray, and foam are all observed to affect the scatteredsignal and, along with radar frequency and grazing angle, to determine the radarclutter level.

Table 4.1 Douglas Scale

Roughness and Swell State Significant Height (m) State Characteristics0 0 Calm1 0.3 Smooth2 0.3–0.9 Slight3 0.9–1.5 Moderate4 1.5–2.4 Rough5 2.4–3.6 Very rough6 3.6–6.0 High7 6.0–12.0 Very high8 12.0 Precipitous9 — Confused

171

172 Sea Backscattering at Low Grazing Angles

Table 4.2 Beaufort Scale

Wind State Wind Velocity (m/s) Wind Characteristics0 0.5 Swell1 0.5–1.5 Light air2 2.0–3.0 Light breeze3 3.5–5.0 Gentle breeze4 5.5–8.0 Moderate breeze5 8.5–10.0 Fresh breeze6 11.0–13.5 Strong breeze7 14.0–16.5 Moderate gale8 17.0–20.0 Fresh gale9 20.5–23.5 Strong gale

10 24.0–27.5 Whole gale11 28.0–31.5 Storm12 32.0 —

The origin of these phenomena on the surface is the near-water wind, whoseduration T is related to the fetch XS by

XS = 1/2 ? c (w ) ? T (4.1)

where c (w ) is the phase velocity of sea waves. For different wind velocities, thefetch changes within bounds of one to a few hundred kilometers (i.e., the normalizedfetch for fully developed roughness xS = XS g /u2

* equals 107 to 108, where g is theacceleration constant and u* is the friction velocity usually introduced for wavestructure analysis of the near-water atmospheric layer and independent of the heightof wind velocity measurement. Usually, this value is related to the mean windvelocity V at height of z above the surface by

V (z ) = u*ϖ lnS gz

0.035u2*D (4.2)

where ϖ is Karman’s constant, equal to 0.4 according to results of many measure-ments.

The approximate friction velocity dependence on the mean wind velocity canbe presented as [1]

u* ≈ 0.05V (4.3)

As seen from the experimental data obtained in wave tanks, the dependenceof XS , the wave rms height sw and its slope variation s=j on the wind velocityV for fixed fetch has the form shown in Figure 4.1. At first, the rms height andwave slope angles increase slowly for increasing wind velocity, and after a certainvelocity is reached, depending on the fetch, a more rapid growth is observed. The

4.1 Sea Roughness Features for Small Grazing Angles 173

Figure 4.1 Sea wave height dependence on wind velocity for different fetches. (After: [2].)

wind velocity for which this change takes the place is called as the critical velocityVc .

The dependence sw = f (V ) is described rather well by the function [2]

sW = ASV − VcVc

D2 (4.4)

and the slope angle variation by

s2=j = g2 ≅ FlnSra nu*

a DG2

(4.5)

where ra is the air density, a is the surface tension coefficient, and n is the waterviscosity.

The critical wind velocity varies inversely with the fetch.The wave height and slope angle increase for further increase in wind velocity

right up to loss of stability when the wave top angle decreases. As shown by Stokes[3], the angular stream with top angle w is stable if w = p /n (n = 3/2). However,for n ≠ 3/2, the system is unsteady and boundary deformation takes place. It is

174 Sea Backscattering at Low Grazing Angles

shown that this deformation occurs with small vertical acceleration, as the waterjet is ejected from the top. This eruption of a water jet from the wave top is calledwave breaking. According to the accepted classification [4], waves break by spilling,plunging, or surging, depending on the wave steepness.

Spilling occurs when the wave crest becomes unstable at the top and the crestflows down the front face on the wave, producing an irregular, foamy water surfacethat eventually takes the aspect of a bore. Plunging occurs when the wave crestcurls over the front face and falls into the base of the wave, resulting in a highsplash and the development of a bore-like wave front. Surging occurs when thewave crest remains unbroken while the base of the front face of the wave, withminor breaking, advances up a beach. It is necessary to note that breakings of theplunging type are 40%–45% of the total.

Such a description of sea waves applies for any arbitrary small but final valuesof vertical acceleration. Wave top breaking occurs when the local inertial forcesexceed the gravitational ones.

From Stoke’s theoretical papers, the breaking criterion was obtained and intro-duced to practice in the form [5]

HC = 0.027gT 2 (4.6)

which connects the sea wave maximal height HC and period T. Experimentalinvestigations confirm this relationship, giving coefficient values between 0.02 and0.022.

In the period preceding breaking, the wave parameters are changed; first, thewavelength L decreases, leading to increasing steepness g = H /L. Then, horizontaland vertical asymmetry appears, determined by the coefficients of horizontal mand vertical l asymmetry, which are (see Figure 4.2)

Figure 4.2 Asymmetry of sea wave.

4.1 Sea Roughness Features for Small Grazing Angles 175

m = h /H ; l = F2 /F1 (4.7)

Initially, the coefficient of horizontal asymmetry is close to 0.5, quickly increas-ing to values close to 0.9, after which vertical asymmetry appears. The experimentaldata of m and l obtained in [5] are shown in Table 4.3.

The breaking process can be divided into three stages: the start of overturning,the appearance of one or a few splashes, and a chaotic movement. As the frontslope steepness increases to the stability threshold, a water jet is ejected in thedirection of wave movement. When the jet falls onto the undisturbed water surface,the splash appears, after which the process becomes chaotic (i.e., turbulent move-ment of the air-water mixture occurs).

After splashing, the inertial forces quickly decrease and the water particlesmove in parabolic trajectories.

Consider the characteristics of foam formations, because these permit us toindirectly judge the breaking process.

The important quantitative characteristics of foam activity are the dependencesof isolated regions for different type formations (crests and striped structures)and their relative areas on the surface wind velocity and the sea surface state.Investigations of these formations and their division into two types were carriedout in [6]: the crests (dynamic foam with typical lifetime of some seconds corre-sponding to wave crest breaking) and striped structures (static foam with lifetimeof minutes).

We first analyze the wind dependence of the crest areas and relative areas ofsurface cover (and wind at a height of 20m) in a spatial window of 100 × 100 m2

for wind velocities of 5–10.5 m/s.The experimental pdf of relative areas of foam structures [6, 7] differs strongly

from the Gaussian model, having positive values of skewness and kurtosis, andthe gamma distribution is most suitable for their description

p (S ) =l (lS )h −1 exp (−lS )

G(h )(4.8)

where l = (h /s2)1/2; h = 6/g1 = 4/g2 ; G(h ) is the gamma-function; g1 is thecoefficient of skewness; and g2 is the kurtosis coefficient. The parameters of foamcharacteristics are shown in Table 4.4.

Table 4.3 The Asymmetry Coefficients of Plunging

Asymmetrical CoefficientsAsymmetry Coefficients Symmetrical Wave Minimal Maximal Average

m 0.5 0.63 0.93 0.76l 1.0 0.9 2.70 1.85

Source: [5].

176 Sea Backscattering at Low Grazing Angles

Table 4.4 Distribution Parameters of Foam Characteristics

Wind Velocity (m/s) Mean Value (m) s /m g1 g25.7 0.0146 1.24 2.31 6.979.5 0.04 0.71 1.67 3.22

10.5 0.0695 1.08 4.36 16.6Source: [6].

The probability of a match of the experimental histograms to the gammadistribution was estimated according Pierson’s goodness-of-fit test and was foundto be greater than 0.5–0.6. The mean value m and variance s2 have a tendencyto grow with increasing wind velocity.

The relationships between mean values of foam cover and single formationareas and wind velocity show that the threshold wind velocities for foam structureformation are as noted earlier in papers [7–9]. For the Black Sea, this thresholdvelocity is 4.5–5.5 m/s. An analogous threshold appeared for mean values of relativearea of foam cover, approximated by a step function of the form [6]

SS = H0, V < 5 m/s

0.015[1 + 2.2 ? 10−2(V − 5)3], V ≥ 5 m/s(4.9)

The dependence of SS = f (V ) is shown in Figure 4.3(a).The wind velocity dependence of single foam formations is approximated by

S (m2) = H0, V < 5 m/s

0.4 + 0.0384(V − 5)2, V ≥ 5 m/s(4.10)

Dependence S = f (V ) for single foam regions are shown in Figure 4.3(b).

Figure 4.3 Dependences of (a) foam cover–relative sea area and (b) single foam formations on wind velocity.(After: [6].)

4.1 Sea Roughness Features for Small Grazing Angles 177

One of the conclusions of power spectral theory for developed sea roughness[7] is the relationship between the dissipation energy of the breaking wave and thecrest foam cover area. From this, there follows a dependence of the dissipationenergy threshold for breaking on the cube of wind velocity. The SS /S ratio permitsus to evaluate the threshold density of dissipation energy regions, equal to about300 per km2 [6], above which it varies with the cube of velocity to about 580 perkm2 for V = 10.5 m/s.

From [10], the mean number of crest foam formations depends on the windvelocity; this dependence has a very sharp slope from a few units for wind velocitiesless 6 m/s to 10–15 for wind velocities greater than 7 m/s over a sea surface areaof 104 m2. For further wind velocity increases up to 16 m/s, the mean number ofsuch formations remains practically constant.

During breaking and boiling surf formation, an intensive drop-spray phasearises, with a duration of a few seconds. The foam formation occurs after the largedrops enter the water. The foam spot achieves its maximal size in 4–5 secondsafter breaking, and after that a collapse takes place. The foam spot size is 10–15m;the foam structure has a velocity of about 1.0–1.5 m/s at birth, and then it stopsmoving.

The spray component originates with foam crest formation as a result of theaerodynamic jet gap at the water surface, accompanied by steepening and destruc-tion of the wave profile. The water particle horizontal velocities are 0.7–0.9 timesthe wind velocity; the average diameter of drops is 10−2 cm, and their maximaldiameter is about 0.2 cm. The lifetime of the spray approximately equals the surflifetime up to transition of the foam structure to striped foam.

The foam bubble structure is the source of the drop component. The burstbubble ejects water drops into air. Their size is rather small, as a rule, becausethey form from film material. The drop diameter in surf zone foam is determinedby the film thickness h and can be up to 1.5 mm [11]. The large drops arise, as arule, at the end of their injection and mostly do not remain long in the air, withthe exception of the turbulent regions of the surf zone. The experimental diametersof drops from [11] are shown in Table 4.5.

Some of these drops move almost horizontally along the sea surface; verticallymoving drops appear for bubble diameters greater than 1 mm. The vertical velocities

Table 4.5 The Drop Diameters Formed for Bubble Bursting

Drop Number Drop Diameter2 8.4h4 13.6h6 19.4h8 24.3h

10 29.5hSource: [11].

178 Sea Backscattering at Low Grazing Angles

of so-called reactive drops can reach 10–50 m/s. The height of drop ejection alsodepends on the bubble diameter d . From data of Figure 4.4 [12], the maximalejection heights are 15–20 cm for foam bubble diameters of about 1.0–1.5 mm.It is noted that with the exception of the first few seconds, the drop ejection heightis practically constant and is 4–5 cm over the sea surface.

The Phillips-Miles model for sea wave generation [13] is the most developedand is accepted by many authors as a working hypothesis for wave evolution withlimited wind duration and large fetch. This evolution is observed at low frequencies,and spectral component saturation takes place in the high frequency band, wherethe slope can be approximated by a dependence proportional to v−n where n = 4.The Phillips mechanism explains the rise and temporal linear growth of wavespectral components only at the earliest stages of their development. For laterstages, the spectral component energy grows exponentially according to the Milesmechanism of instability. The main factors limiting the wave growth are theirbreaking and the nonlinear interaction of spectral components. The quantitativeestimations of energy transfer rate for real spectra at the expense of instability aregiven in [14, 15]. From these papers, one can reach the conclusion that for thefrequency band lower than the frequency of the sea wave spectral maximum, theenergy transfer from wind to waves contributes the main increase in the waveenergy.

As the main parameter for sea surface description in oceanography, the wavespectrum is used, containing general information about the sea surface state and

Figure 4.4 Dependences of drop heights on the bubble diameter (curves A–F: the first drop, curveG: the second drop, curve H: the third drop, curve I: the fourth drop; temperature A:4°; temperature B: 16°; and temperatures C–I: 22–26°C). (From: [12].)

4.1 Sea Roughness Features for Small Grazing Angles 179

not taking into consideration the earlier noted features. It is also the main parameterused in Bragg’s hypothesis for microwave scattering from sea surface.

The cause of sea roughness is the wind, but this does not mean that the localwind fully determines the sea roughness structure. When the wind influences thesea surface over a rather long time and for large fetches, we obtain the fullydeveloped roughness regime. For such a regime, the main expressions are obtaineddescribing the sea roughness spectrum. The frequency spectrum of sea roughnessfor the simplest conditions of wave formation, when one wave system is observed,can be presented as

S (v ) = A ? v−S exp (−Bv−n ) (4.11)

The Pierson-Moskovitch spectrum is often used for fully developed rough-ness described by (4.11), for which the constants are equal to A = 8.1 ? 10−3g,B = 0.74 ? (g/V)4, S = 5, n = 4, where V is the mean wind velocity at a height of10m, g is the gravitational constant. This spectrum has its maximum at frequency

v0 ≈ 0.9(g /V ) (4.12)

and decreases with increasing frequency v ≥ v0 . It is seen from (4.12) that thespectral maximum shifts to a lower frequency band with increased wind velocity.The Pierson-Moscovitch spectrum width at the half power level is

DFeff ≈ 0.133 ?gV

(4.13)

The roughness power spectrum can distinguish the energy transfer intervalwith its maximum at the frequency v0 in which the energy transfer from the airflow to the sea waves takes place. The energy decreases quickly at lower frequenciesbecause the wave energy transfer at large temporal scales from nonlinear effectsis negligible.

Phillips [16] advanced a hypothesis about the equilibrium region at frequenciesabove the frequency of the spectral maximum. He proceeded from the assumptionthat the wave dissipation process (crest breaking to achieve limited stable configura-tion and the transfer of wave energy to turbulence, capillary wave formation atface fronts of the primary waves, and secondarily nonlinear interaction) determinesthe spectral shape in this region. In this spectral interval, there is a consistentmechanism of energy transfer from larger to smaller turbulence, with dissipationof mechanical energy to thermal. Some 70%–80% of the energy is in the energytransfer interval and 20%–30% is in the equilibrium interval. Neglecting the surfacetension and molecular viscosity in the equilibrium interval, he obtained from similarconsiderations the expression for the frequency spectrum

180 Sea Backscattering at Low Grazing Angles

S (v ) = ap g2v−5, v0 ≤ v ≤ vm (4.14)

where ap is the universal dimensionless constant defined in (4.16). The upperboundary of the equilibrium region is determined by the frequency for which theinfluence of surface tension becomes significant:

vm = Srg3

a D1/4

(4.15)

where r is the water density.The constant of ap varies with the fetch function XS [17]

ap = 0.076SgXS

u 2*D−0.22

(4.16)

(i.e., the value of ap increases rather slowly with increasing fetch). Often thisconstant is equal to (6.9 ± 2.4)10−3 for developed roughness and to (12.9 ± 2.5)10−3

for limited fetches.The equilibrium interval ends with a sharp cut off at the frequency vm , above

which the spectral density decreases more rapidly than for Phillips’s spectrum. Thephysical mechanism of this cut off is the mechanical energy dissipation to thermalin the frequency band v > vm for breaking waves.

From (4.11), it is easy to obtain the approximate expression for wave heightvariance

s2W = E

∞

0

S (v ) dv = const ? g2 ? v−40 (4.17)

This result was first obtained by Hicks [18], with the constant equal to about0.04.

Besides the frequency spectrum, the wave spectrum or the wavenumber spec-trum is often used. The frequency and the wavenumber for gravitational wavesare connected by the dispersion relation

k =2pL

=v2

g(4.18)

where L is the sea wavelength. The wave and frequency spectra are related by

4.1 Sea Roughness Features for Small Grazing Angles 181

S (k ) = S (v ) ?dvdk

(4.19)

Transforming the frequency spectrum to an isotropic spectrum of wavenum-bers, Phillips obtained the spectrum in the form

S (k ) = H0.005/k4, k > g /V 4

0, k < g /V 4 (4.20)

where k determines by (4.18) and equal to k0 ≤ k ≤ km .Here k0 is the wavenumber corresponding to the wavelength of the spectral

maximum and km is the wavenumber corresponding to the upper limit of theequilibrium interval determined as km = (g /a )1/2. According to analytical estimatesin the spectral model JONSWAP [19], the microscale value Lm = L0 /km can berepresented as Lm = L0 /65 and is approximately 0.5m. This result agrees withdata of [20], according to which the transfer from the equilibrium interval to theinterval of gravitational-capillary waves takes place for wavelengths of about 0.2m.

The period and wavelength are related to the wind velocity by the simpleexpressions

T = 2p /v = 0.64V (4.21a)

L = 2p /k (4.21b)

where V is the wind velocity.The sea surface statistical description [21] permits us to confirm that the wave

height distribution is rather close to a Gaussian distribution, with a variance thatcan be obtained from (4.17). The rms wave height dependence on the wind velocitycan be approximately represented as

sw = 0.005V 2 [m] (4.22)

It should be noted that the sea roughness autocorrelation function can beapproximated with good accuracy by [13]

R (t ) = s2W ? e −bt cos v0t (4.23)

where b is related with the spectral width by

DFeff = b /4 (4.24)

182 Sea Backscattering at Low Grazing Angles

In conclusion, note that the sea wave slope variance can be determined fromthe wave spectrum. For Phillips’s spectrum (4.20), one can obtain

s2=j = E

km

k0

k2S (k ) dk = B ? ln (km /k0) (4.25)

where B ≈ 0.005.Using for s

2=j a value determined from (4.6) with k0 from (4.18) and for

km = 2p /Lm , the minimal wavelength of equilibrium interval Lm ≈ 2 cm, we obtainthe rms slope value for wind velocity V = 10 m/s

s =j = F0.005 lnS 2p0.02

0.819.8

102DG1/2

≈ 0.203 (4.26)

The experimental data of Cox and Munk [10] show that the slope variancelies in the interval 0.02 to 0.07—that is, the derivations from (4.26) give resultscoinciding with experimental data.

The sea roughness characteristics described earlier reflect the main features ofroughness from its birth through the wave breaking process.

Beginning with 1958, Phillips’s spectrum for the equilibrium interval of thesea roughness spectrum was used as the universal law for a description of developedroughness. However, the spectral density decrease with increasing k depends onthe environment, particularly on wave development [22–24]. This led to a general-ization of Phillips’s law [22, 25] and to presentation of the wavenumber spectrumin fractal form.

The main wind wave feature for open sea, unlike in lakes, shallows, and wavetanks, is the large width of the wave spectrum. This means that there is greatnumber of secondary waves in the roughness spectrum, besides those determinedby the frequency of the spectral maximum. This leads to validity of the assumptionabout Gaussian statistics of sea roughness as the traditional first approach. Thenthe sea roughness spectrum for the equilibrium interval in terms of wavenumberscan be presented as [25]

S (k , c ) ≈ b (x ) SV 2

g D2m

k−(4−2mm)Y (c ) (4.27)

where Y (c ) describes the angular distribution of wave energy, and b and m aredimensionless functions of normalized fetch xs .

4.1 Sea Roughness Features for Small Grazing Angles 183

The exponent (4 − 2 mm) of k , characterizing the equilibrium interval, is amonotonic function of the degree of roughness development, as observed in[26, 27]. The fractal surface scale for spectrum (4.27) is D = 2 + m , and thereforem is the fractal surface codimension. The particular consequence of the conditionm > 0 is that it represents the large number of secondary waves forming the wavepacket on the sea surface between two main waves with wavelength of L = 2p /k0 .This leads to increasing numbers of brilliant points and single facet curvature.

For a rather wide equilibrium interval or well-developed roughness, the spec-trum falls with increasing k . So, for a relative fetch xs > 105, the maximal valueis m = 1/3 in the low frequency part of the equilibrium interval, decreasing tom = 1/4 in the high frequency part.

The value of m is zero in the dissipative region and for small fetch 103 < xs <104. Because of the rather small change in the fractal codimension m = m (x ), it canbe considered as independent of k over the entire equilibrium interval, conforming tothe conclusion of similarity theory [22].

The coefficient b is a nondimensional parameter called Phillips’s generalizedconstant. It depends slightly on the fetch and is [17]

b ≈ 0.0331x−0.2 (4.28)

It is emphasized that rapid decrease of spectral density in the dissipative interval,proportional to k−4, is a property inherent in developed roughness. The upperboundary of the equilibrium interval is conditioned by the hydrodynamic instabilityof high gravitational waves that have the tendency to break for rms wave angularslopes g reaching some threshold value G ≈ 0.4. As shown in [28], for microscalekm , the variance of slope angles can be determined as

g2 = s2=j ≈

b2 FG(m ) −

d 2m

m G ? S V 2

gLmD2m

(4.29)

where d = k0Lm .For well developed roughness when m ≥ 0.2 and d << 1, the wave slope is

practically independent of L0 . Particularly, for m = 1/4 (the Zacharov-Philonenko-Toba spectrum), we have

g2 ≅bG(1/4)

2?

V

√gLm(4.30)

For fully developed roughness (m → 0), the influence of the main wavelengthL0 is significant and

184 Sea Backscattering at Low Grazing Angles

g2 ≈ b lnScL0h D (4.31)

where c = 0.1193. It is seen from these expressions that the wind dependence ofslope angle variance is significantly changed as a function of the roughness regime.

4.1.2 Shadowing and Peaks in Heavy Sea

Microwave backscattering from the sea surface for small and extremely smallgrazing angles has some peculiarities. First of all, the larger part of surface is inthe shadowing zone for low grazing angles, and the scattered signal is formed onlyby areas that rise above the shadowing zone. Consequently, the signal changesfrom being spatially continuous to discrete. In these conditions, the term normalizedradar cross section loses its sense because this term assumes a homogeneous surfacewith uniform illumination, while in our case the scattered signal is formed by localareas of the sea surface.

Besides, spikes of the scattered signal with specific statistical features areobserved even for large grazing angles in addition to the range continuous signal.A strict physical model of the spikes is unavailable, preventing us from explainingtheir statistics. It is noted only that the spatial statistics of spikes are associatedwith breaking sea waves and presence of foam on the sea surface.

In these situations, the rough sea is not a stationary random process, and onecan identify two phases in the backscattered signal: spikes for which the meanintensity is much greater than the mean level, and gaps in which the scatteredsignal level is considerably less than the mean level or in which the signal ispractically absent for extremely small grazing angles because of the surface shadow-ing. Recently, the interest in investigations of sea backscattering for small grazingangles has increased considerably. This deals with attempts to explain the physicalnature of the scattered signal spikes because of the necessity to take spikes intoconsideration in sea radar clutter models [29–39].

The total RCS of sea clutter for a single resolution cell is determined by thefollowing. In the illuminated area, the total RCS is

s S = ssea + sspray (4.32)

Here ssea is the RCS of the sea surface, as determined by the general laws ofsea backscattering without shadowing and spikes. The second term in the right-hand part, sspray , takes into account the spatial statistics of spikes. The spatialstatistics are determined by the statistics of random sea surface peaks above somethreshold.

The sea surface is a complex natural formation that includes wind waves,spray, and foam. Each component takes part in forming the backscattered signal

4.1 Sea Roughness Features for Small Grazing Angles 185

as a function of the radar frequency and the grazing angle and determines theradar sea clutter level. The main reason for these phenomena on the sea surfaceis the surface wind.

First of all, let us consider the shadowing function of the sea surface. Thesurface can be presented as a stationary random process j (x , y ) with a Gaussianpdf of surface heights, zero mean and variance s 2

W

p (j ) =1

sW √2pexpS−

j 2

2s2WD (4.33)

The incident field vector lies in the plane XZ, the incidence angle is denotedby u, and the grazing angle is c = p /2 − u (see Figure 4.5).

The simplest technique to describe surface shadowing is to introduce a shadow-ing function S (x , y ) that is unity for the illuminated part of surface and zero forthe surface in the shadowing zone. Then, a mean shadowing functionS (u ) = ⟨S (x , y , z )⟩ is determined. The mean power of the backscattered signal isproportional to S (u ). In our case S (x , y , z ) is a function only of the coordinatex .

Using the results of [40, 41], the shadowing function can be represented as

S (u ) =F(m /s=j )L(m ) + 1

(4.34)

where F(?) is the probability integral, m = tan c , and

Figure 4.5 Geometry for derivation.

186 Sea Backscattering at Low Grazing Angles

L(m ) = (2p )−1/2 s =j

mexpS−

m2

2s =jD − erfcS m

s =j √2D (4.35)

The variance of the surface slopes s2=j is determined by the autocorrelation

function of the surface because s2=j = |−r ″(0) | and r is the normalized autocorrela-

tion function. For the two most common spatial surface autocorrelation functions(Gaussian and exponential), one can obtain

s2=j = H2s

2w /T 2 for Gaussian function

s2w /T 2 for exponential function

(4.36)

where s2w is the wind velocity variance and T = 1/4DFeff ; DFeff is the effective

spectral width of the surface determined by (4.13). The variance of the sea surfaceslope is

s2=j ≅ 0.533S g

V D2 (4.37)

Introducing the normalized height z = j /sw and slope h = m /s=j , one canfinally obtain

S (z , h ) = [F(z )]L; (4.38)

L(h ) =1

√2p?

1h

? e h2/2 + F(h ) − 1

For small h << 1

L(h ) ≈1

√2p F 1

h (1 + h2/2)− √2(1 − h )G (4.39)

The dependence of the shadowing function on the grazing angle c for z = 1 isshown in Figure 4.6 (for rms values of slope equal to 0.14 and 0.26, whichcharacterize the boundary values of the sea surface slope variance [24]). The transi-tion from the illuminated zone to the strongly shadowing zone takes place in thenarrow range of the grazing angles from 0.5° to 5° and the sea surface is illuminatedalmost fully for grazing angles larger than 5°.

Let us consider the problems of sea wave peaks (i.e., the case in which a seawave rises above some boundary), using the first approximation of rough sea as

4.1 Sea Roughness Features for Small Grazing Angles 187

Figure 4.6 The shadowing function versus grazing angle. (After: [24].)

a stationary random process with finite second spectral moment and continuousspectrum, at all stages up to wave breaking. For such a random process, there isa sufficiently developed theory of peaks [42, 43]. In the framework of this theoryand for the condition of high threshold when the number of crossings by thestationary random process is small, the peak distribution can be approximatelydescribed by Poisson’s law

P (k , T ) =[N (j0)]k

k !? exp [−N (k , T )], k = 0, 1, 2, . . . (4.40)

where N (j0) is the mean number of peaks.For a Gaussian random process, the mean number of peaks is determined by

Rice’s formula [42]

N (j0) =√−r″ (0)

2p? expS−

j02 D (4.41)

where r″ (0) is the second derivative of the autocorrelation function of the processfor t = 0, and j0 is the threshold normalized to the rms wave height.

According to evaluation of N (j0) from (4.41), the mean number of peaksdecreases with increasing wind velocity. This contradicts the experimental dataand is the result of the use in the derivation of the Pierson-Moscovitch spectrum

188 Sea Backscattering at Low Grazing Angles

of sea waves. This spectrum is not applicable to a heavy sea because it does nottake into consideration the breaking of large numbers of waves.

For the fractal model of the sea wave spectrum, the mean number of peaks isa cubic function of the wind velocity or friction velocity [16]

N (j0) ~ g−1 ? km ? u 3* (4.42)

The dependence of the peak frequency on friction velocity experimentallyobtained in [36] for a unimodal sea wave spectrum is shown in Figure 4.7, andthe analogous results are given in [35].

The dependence of the mean number of peaks on range for two wind velocitiesis shown in Figure 4.8. One can see a decreasing mean number of peaks withincreasing range.

The distribution of the mean peak duration for the normal random processcan be approximately represented as

Pt (t = kt0) = [1 − F(j0)]k −1 ? F(j0) (4.43)

where t0 is a discretization interval that can be determined from the sample theoremas t0 ≅ 1/DFeff ; k = 0, 1, 2, . . . ; DFeff is the effective Pierson’s sea wave spectralwidth. From this expression, it is seen that for increasing threshold j0 the probabilityof short-time spikes increases and the probability of long-time spikes decreases.

The mean spike and gap duration can be determined as

Figure 4.7 The mean number of peaks versus the friction velocity for (a) all conditions of searoughness and (b) sea wave with unimodal spectrum. (From: [36].)

4.2 Sea Backscattering Models 189

Figure 4.8 The spike mean number dependence on range for wind velocities (1) 7 m/s and(2) 10 m/s.

t s (j0) = 2p [1 − F(j0)] ⁄ F√r″ (0) ? expS−j02 DG (4.44)

tp = 1 ⁄ F√−r″ (0) ? expS−j02 DG (4.45)

The mean duration of the spikes and gaps as functions of the threshold fortwo wind velocities are shown in Figure 4.9.

Thus, the results presented permit us to estimate the main characteristics ofshadowing and peaks of the sea surface above some boundary under the assumptionthat the sea surface can be described by a differential random process (i.e., wavebreaking is absent). If breaking is taken into account, the results must inevitablychange.

4.2 Sea Backscattering Models

Theoretical and experimental investigations of backscattering from the sea surfacehave provided a basic understanding of the phenomenon. However, the develop-ment of theoretical models of backscattering has proven to be a challenging task.

190 Sea Backscattering at Low Grazing Angles

Figure 4.9 The mean duration of spikes and gaps as the functions of relative threshold.

Among the sea backscattering models, two models are best known. The first isthe facet model [44, 45], in which geometrical optics techniques are applied forbackscattering description, and the surface is represented by the totality of small,flat plates. It is supposed that the radar reflection is formed by the facets that arenormal to the direction of radiation. If the facet slope distribution is known, onecan determine the fraction that is perpendicular to a given radiated beam and,consequently, the scattered signal intensity. The facet model proved to be usefulfor qualitative analysis of clutter phenomena and was modified [46] on the basisof the actual scattering of finite-length radiowaves by facets of finite sizes, includingthe influence of wavelength on the effective number of facets contributing to thescattered signal.

In those cases when the facets extend in height to at least one interference lobewidth, the radiation conditions are equivalent to a homogeneous field, and thenormalized RCS does not depend on the incidence angle. If the facets occupy onlypart of the interference lobe, the radiation intensity decreases very quickly withincreasing range, and the normalized RCS varies as r −4 where r is the range fromthe radar.

For small grazing angles, this model leads to the following conclusions:

• Facets whose circumference is near l /2 scatter most intensively;

4.2 Sea Backscattering Models 191

• The scattering varies with the square of crest slope because the facets at thewave crest scatter most intensively;

• The normalized RCS dependence on the frequency is related to the facetsize distribution, leading to its varying inversely with wavelength.

Surface shadowing for small grazing angles and heavy sea and scattering fromspray and white caps formed for breaking waves are not considered in the facetmodel. In addition, the variation of normalized RCS dependence with wavelengthdoes not conform with experimental investigation results; this dependence is consid-erably stronger in the model than in experiments.

A more detailed description of the observed effects [47, 48] is given in the two-scale scattering model, in which the sea surface is represented as a superpositionof irregularities with pronounced differences in scale length. The model providesa relatively good explanation of the experimental results, especially at decimeterwavelengths and the longer wavelength part of the centimeter band for grazingangles larger than several degrees.

The two-scale model is more suitable for theoretical interpretation of the phe-nomenon. In the framework of this model, the normalized RCS is [47]

s 0VV = 16pk4 |e |2 sin4 c ? f (e , c ) ? S (ϖ0) (4.46)

s 0HH = 16pk4 sin4 c ? S (ϖ0)

where

e is the dielectric constant of sea water;

f (e , c ) = [(1 + h1 sin c )2 + h2 sin2 c ]−2;

h1 + h2 = √e ;

ϖ0 = 2p /L0 =4pl

cos c ;

S (ϖ ) is the roughness spectral density for sea wavenumber ϖ ;

L is the sea wavelength;

c is the grazing angle.

The sea roughness spectrum for grazing angles 20° to 60° can be representedby the expression [49]

S (ϖ ) ≅Bp

ϖ−4; B = (2 − 6) ? 10−3 ? [. . .]

192 Sea Backscattering at Low Grazing Angles

Analysis of these expressions shows that the normalized RCS does not practi-cally depend on the frequency in the entire UHF band, and the normalized RCSis smaller for horizontal polarization than for vertical.

For small grazing angles, the two-scale model fails to account for a numberof features in the backscattered signal. An attempt was made [50] to make use ofthe model to gain an insight into such experimentally noted phenomena as thelarge normalized RCS and the increase in the central frequency in the spectrumfor horizontal polarization as compared with vertical. Yet it was not possible togive an adequate explanation for all of the features observed. One effect is thelower rate of change (with respect to that predicted by the model) of the normalizedRCS as a function of grazing angle for small c .

The problem of signal characteristics observed for breaking waves and subse-quent generation of sprays has not been addressed so far. Some authors [51, 52]describe experiments that were performed in an attempt to distinguish betweentwo model backscattering mechanisms: facet reflection and two-scale scattering. Itwas pointed out that in X-band, the measured normalized RCS exceeded thatpredicted (according to the two-scale model) by as much as 12 dB. Besides, thenormalized RCS was equal for horizontal and vertical polarizations, and a moresignificant central frequency shift was detected. Estimates showed that, along withBragg scattering, reflection from facets with sufficiently great radii of curvature rcould occur (it should be noted that for wind speed greater than 4 m/s, the contribu-tion from the facets for 10 ≤ kr ≤ 100 proves to be significant). As a result, withwind speed from 0 to 7 m ? s−1, the contribution from the facets to the totalscattered signal is 30%–40%, while the lifetime of these reflections is 20% of thetotal observation period. Moreover, as the wave crests steepen prior to breaking,it becomes necessary to take into account wedge diffraction that (as reported in[50]) can result in an increase of the normalized RCS for horizontally polarizedradio waves in comparison with the vertical by 10–20 dB. Thus, s 0 can be equalfor two polarizations and can even give a result that is inconsistent with the standardtheory, for which s 0

H ≤ s 0V .

As far as the shorter wavelength part of X- and Ka-bands is concerned, thecontribution to the scattered signals due to spray blown from the wave crestsduring wave breaking is of great importance. As a consequence, wave breakingcaused by the wind is accompanied by the appearance of a variable density layerat the air-water interface, which is a mixture of finite volumes of these components.The contribution of the spray to the total scattered signal can be calculated as anintegral over the volume V [32],

d spray = EV

h (h, u ) dV (4.47)

4.3 Sea Normalized RCS 193

where h (h, u ) is the specific volume RCS of spray, which is a the function of theheight h above the sea surface and the wind speed u . Assuming the specific volumeRCS to be uniform in the cross-sectional plane h = const, (4.47) can be simplifiedand if a pulsed signal is used it becomes

d spray =ctu2

ru0E∞

0

h (h, u ) dV (4.48)

where tu is the transmitted pulse duration, u0 is the beamwidth in the azimuthalplane, and r is the range.

The volume normalized RCS of spray in (4.48) can be defined as [32]

h (h, u ) = E∞

0

s (D, l )N (D, h, u ) dD (4.49)

where s (D, l ) is the RCS of a water drop of radius D and N (D, h, u ) is thenumber of drops with radii from D to D + dD in a unit volume.

The RCS of an individual particle can be calculated using the well-known Mietheory, which is valid up to the shorter wavelength part of the millimeter band.

The size distribution of water drops can be described by the Marshall-Palmerformula [53]

N (D ) = N0 exp (−LD ) (4.50)

where N0 = 8 × 104 m−3, L = 26.7v−0.24 [32], and v is the water content in g/m3.Analysis of experimental data for water content at small altitudes above the

sea surface permits derivation of the following empirical formula

v = A0 expS0.83u −hh0D (4.51)

where A0 = 2.4 × 10−11 g ? cm−3 and h0 = 11 cm.Taking the effect of spray into account, it becomes possible to explain both

the angular dependence of the normalized RCS in the region of small grazing angles(which cannot be done with the two-scale model) and some other features of thepower spectra and polarization characteristics of the scattered signals.

4.3 Sea Normalized RCS

The experimental investigations of the normalized RCS variations that have beencarried out for a variety of radar system parameters and external factors over the

194 Sea Backscattering at Low Grazing Angles

frequency range from 3 GHz to 100 GHz at small grazing angles [32, 54–57]reveal several phenomena inherent in behavior of s 0 at millimeter wavelengths.These involve the effect of sprays generated during wave breaking on the angulardependence and magnitude of s 0 and the quicker saturation of the normalizedRCS with increasing wind speed in the millimeter band as compared with theX-band.

Three regions can be distinguished in the angular dependence of the normalizedRCS of the sea surface: the quasi-specular region, the plateau, and the interferenceregion—the latter being of greatest interest for our applications. The magnitudeof the normalized RCS at grazing angles smaller than few degrees shows a suddendrop as the angle decreases, s 0 varying with c . The transition from the plateauregion to the interference region takes the place at the critical angle c cr , whosevalue depends on the wavelength and the sea surface state:

c cr =l

5H≈

l

0.015u2.5 (4.52)

where u is the mean wind speed in m ? s−1 and H is the effective sea wave height,which is lower than the peak-to-trough height by a factor about three.

In the decimeter and the longer-wavelength part of the centimeter band, therelationship s 0 ∼ c4 applies for c < c cr . However, concerning the shorter wave-length part of the centimeter and the MMW band, the boundary between theplateau and interference regions should be set with care. As was noted in [55], thes 0 ∼ c4 dependence holds in the X-band under standard refraction conditions,whereas superrefraction at grazing angles less than 0.1° leads to a s 0 ∼ c relation-ship. The results of measurements performed over the frequency range 10 GHz to75 GHz [32, 54] identified some additional features in the angular dependence ofthe normalized RCS. Between the plateau and the interference region, there existsa transition zone. In this transition zone, the normalized RCS is proportional to∼c2, which corresponds to a range dependence Prange ∼ r −5 for received power. Atshorter wavelengths, the transition zone shifts towards smaller grazing angles.

To explain the presence of this region in the angular dependence, two mecha-nisms might be conceived [32]. One is the presence of increased refractivity withsteep gradients in the near-surface layer of troposphere. Under such circumstances,the real grazing angle is larger than that determined from geometric considerations;this may result in increased normalized RCS.

For the interference region we can use the relationship

s 0 = kc4 (4.53)

where

4.3 Sea Normalized RCS 195

k = const;

c = cg + c r ;

cg ≈ h /r is the geometric grazing angle;

h is the height of the radar antenna;

c r = 1/2 ? r ? grad n is the grazing angle due to refraction;

grad n is the refractivity index gradient.

Then, assuming a constant grad n in the near-surface layer, (4.53) transformsto

s 0 = kFc r +h

2c rgrad nG4

(4.54)

(i.e., the normalized RCS magnitude at small grazing angles is seen to be gradientdependent).

The meteorological measurements that were performed together with the radarmeasurements showed that the gradient, which was highly variable throughout theday, was equal to 2–2.5 N ? m−1, and the value was normally larger in the layer0m to 5m than 0.157 N ? m−1. In other words, there existed super-refraction thatcould even give rise to atmospheric ducting. Within the layer from 5m to 10m, suchconditions existed during 50% of the observation period. However, the measuredgradients on the average appeared to be lower than those required to account for thenormalized RCS angular dependence (i.e., the superrefraction mechanism cannotprovide a complete explanation of the results obtained).

Under wave-breaking conditions, a variable density layer appears at the air-water boundary, representing an air-water mixture. Accounting for the microwavescattering caused by spray blown from the wave crests and generated as the wavesbreak, the total RCS can be written as (4.32) where s spray is given by (4.47)–(4.51).

Then, as follows from [32],

s 0 = s 0M +

E∞

0

h (h, u ) dh

s 0M

(4.55)

The analysis of [32] shows that the rate at which s 0 decreases with decreasinggrazing angle slows down at higher wind speeds (i.e., in MMW band at smallgrazing angles the normalized RCS increases), mainly due to the contribution ofreflections from spray. The decrease in the normalized RCS at still lower grazing

196 Sea Backscattering at Low Grazing Angles

angles can be attributed to the shadowing effect. As a major part of the surface islocated in the shadow zone, and the contribution to scattering is only made bypartial areas (crests of the larger waves), the very concept of a normalized RCSbecomes meaningless. Owing to quick changes in the fine surface structure producedby the wind, the normalized RCS of the sea in the shorter wavelength part of thecentimeter band and at millimeter wavelengths becomes dependent on windspeed. The intensity of reflections starts to increase at wind speeds of more than2.5–3 m ? s−1 and reaches its maximum values at 3–10 m ? s−1. With further increaseof the wind speed, the rate of growth of the RCS drops down and amounts to0.4–0.6 dB/ms−1 [58]. At millimeter wavelengths, the normalized RCS begins tobe saturated more quickly than in the shorter wavelength part of the centimeterband [32] starting at wind speeds of 5–7 m ? s−1.

Many authors make special reference to the dependence of the normalized RCSupon the wind direction with respect to the beam. Under moderate sea conditionsand at small grazing angles, the normalized RCS normally is maximum in theupwind direction and minimum crosswind. According to the data of [59, 60] theratio of upwind to downwind measured at wavelengths of 1–3 cm for HH polar-ization is 8 dB at 10° grazing angle and varies with angle. For VV polarization,it did not exceed 3–4 dB. The upwind-to-crosswind ratio at l = 3 cm reaches5–6 dB. At millimeter wavelengths the upwind-to-crosswind normalized RCS ratiois 5 dB or 6 dB, rising to 10–15 dB for a calm sea [32].

The normalized RCS can be greatly influenced by the transmitting and receivingpolarizations. In the X-band, the normalized RCS for HH polarization exceedsthat of VV by 8–12 dB for grazing angles less than 1–2° [60]. The normalized RCSfor HH polarization is 1–2 dB higher than for VV both in the X-band and forMMWs, over high sea and for grazing angles less than 0.5° [32]. When the seawas calm, the normalized RCS for VV polarization was greater than for HH by5–7 dB.

Analysis of the cross-correlation function for scattered signals at the outputsof synchronous and amplitude detectors and of the sliding-average amplitudesshows the following:

• The correlation of cross-polarized signal components from the outputs ofsynchronous and amplitude detectors is relatively small (the correlationfactors never exceed 0.5).

• The sliding-average amplitudes of two orthogonally polarized signals arebetter correlated; the correlation factor increases with the volume of thepulse packet integrated and with the wind speed. It varies from 0.45 at lowsea states up to 0.95 for a stormy sea with pronounced periodic structureof roughness.

The correlation factors for l = 8 mm and l = 4 mm are shown in Table 4.6.

4.4 Depolarization of Scattered Signals 197

Table 4.6 Correlation Factors for Cross-Polarized Components

Correlation Factorsfor Cross-Polarized

ComponentsWavelength Wind Speed Sliding Number of

(mm) Polarization and Sea State Channels Amplitudes Average Experiments8.0 Linear, at 45° uw < 3 m ? s−1, −0.15 0.13 0.49 6

calm8.0 Linear, 45° uw < 16 m ? s−1, 0.04 0.22 0.6 4

Sea state 54.0 Linear, 45° uw < 11 m ? s−1, 0.19 0.42 — 7

Sea state 44.0 — uw < 11 m ? s−1, 0.19 0.21 0.71 12

Sea state 4

4.4 Depolarization of Scattered Signals

Measurements at frequencies of 10–75 GHz and wind speed up to 14–15 m ? s−1

show strong depolarization of the signals, with depolarization factors of −3 dB to−6 dB.

Investigations of the signal depolarization at 10 GHz and 35 GHz, in whichtwo states of the scattering process could be clearly identified (i.e., spikes causedby reflections from breaking waves and gaps in the absence of wave breaking)were reported in [61–63].

The most widely used criterion of depolarization degree is the depolarizationcoefficient

Di , j = 10 log (Ii , j /Ii , i ) (4.56)

where Di , j is the depolarization coefficient (in decibels), i and j are the polarizationsof transmitted and received signal, respectively, and I is the corresponding signalcomponent intensity.

It is easy to recognize intervals of comparatively high and low intensity (spikesand gaps) in temporal dependences of the copolarized component intensity andthe scattered signal depolarization coefficient for both wavelengths and both polar-izations. In the future, the term spike means the intervals for which signal intensityis 3 rms values greater than its average value. It is seen from the analysis of temporaldependences that for vertical polarization, the scattered signal depolarization levelincreases in spikes for both wavelengths. Meanwhile, for horizontal polarizationand wavelength of 3 cm, it is practically impossible to determine the differencebetween the depolarization coefficients in spikes and gaps, corresponding closelywith results of [60]. Averaged depolarization coefficients in spikes and gaps fordifferent beam directions with respect to wave direction are given in Table 4.7.The analysis of that data shows the following:

198 Sea Backscattering at Low Grazing Angles

Table 4.7 The Mean Depolarization Coefficients in Spikes and Gaps

Direction of Radiation Wavelength (cm) Polarization Di , j spike (dB) Di , j gap (dB)Upwind 3.0 H −8.0 −9.0

3.0 V −15.1 −17.90.8 H −5.3 −8.10.8 V −11.7 −15.0

Crosswind 3.0 H −8.0 −8.03.0 V −15.5 −16.50.8 H −8.0 −8.00.8 V −10.0 −15.5

Surf zone 3.0 H −9.0 −9.03.0 V −16.0 −16.50.8 H −3.5 −8.50.8 V −11.5 −17.0

Source: [34].

• Depolarization coefficients do not practically depend on the beam and waveangles.

• Depolarization coefficient values for horizontal polarization are usuallyhigher by 2–8 dB in spikes and 7–9 dB in gaps for both wavelengths thancorresponding values for vertical polarization.

• In the millimeter band, for both polarizations, the difference of depolariza-tion coefficients in spikes and gaps does not depend upon beam directionand is 3–5 dB.

• For a 3-cm wavelength, there is no difference in depolarization coefficientsin spikes and gaps for horizontal polarization.

• For vertical polarization, the difference varies from 0 dB for surf illuminationto 2.5 dB or 3 dB for upwind illumination.

• Increasing depolarization coefficient is observed in approximately 40%(X-band) and 60% (millimeter band) of cases when spikes appear in sam-plings having a duration of 3 minutes.

• In gaps, comparatively weak depolarization change is observed with radarfrequency change (approximately Di , j ∼ f 0 − f 0.2) for both polarizations,while in spikes more rapid depolarization coefficient increasing occurs(Di , j ∼ f 0.5 − f 0.9).

Experimental investigations conducted by author [63] at 6.3 GHz showed asimilar difference between depolarization coefficients for horizontal and verticalpolarizations. Supposing that s 0

HV = s 0VH , where in HV and VH, the first letter

denotes the transmitted polarization and the second denotes the received one (thisassumption results from the reciprocity theorem [62]), the author explained thisphenomena by the supposition that s 0

VV > s 0HH . But experiments at X- and

Ka-bands and generalized in [32, 33] showed that specific RCS values for horizontaland vertical polarizations are practically equal (i.e., s 0

VV ≈ s 0HH , excluding the

4.4 Depolarization of Scattered Signals 199

mechanism presented in [63] as an explanation of sea clutter polarization differ-ences).

Consequently, we must search for other mechanisms dealing with the presenceof small-structure elements on the sea surface (created by wind and resulting inpartial depolarization of the scattered signal). One of these elements is the contribu-tion to scattered field by the sharp crest of a breaking wave. In particular, thelarger average specific RCS for horizontal polarization over vertical was explainedby this fact in [63]. Taking [64] into account, let us estimate the ratio s 0

HH /s 0VV

for an ideally conducting wedge having the geometry shown in Figure 4.10. Thederivation uses

s 0HH

s 0VV

= | 1 + cosFpa Sp +

c2 DG

2 cosp2

a+ cosFp

a+ Sp +

c2 DG − 1 |2 (4.57)

where c is the grazing angle and a is the angle of the wedge top.The results of this derivation are shown in Table 4.8. They show that only for

the angle range from 100° to 120°, exactly corresponding to wave crest breaking,the scattered signal for horizontal polarization can be greater than for vertical,resulting in increase in the depolarization component s 0

HV and Di , j . But thismechanism cannot explain difference in gaps where sea wave breaking is absent.

Figure 4.10 Geometry of wedge.

200 Sea Backscattering at Low Grazing Angles

Table 4.8 The Ratio of sHH /sVV , dB for Different Angles in Wedge Top

Grazing Angle 100° 110° 120° 130° 140° 150°1 2.05 14.8 0.23 −9.9 −17.4 −24.82 2.04 14.5 0.45 −9.7 −17.2 −24.53 2.03 14.16 0.68 −9.5 −17.0 −24.24 2.02 13.86 0.9 −9.3 −16.7 −23.95 2.0 13.57 1.14 −9.1 −16.5 −23.76 1.99 13.3 1.36 −8.9 −16.3 −23.4

Source: [34].

The second mechanism results from signal scattering by spray formed duringwave breaking. According to [65] data, during crest breaking and boiling, anintensive drop-spray phase appears, lasting for about 1s, after which foam forms.During crest breaking, drops of large size fall into the water and stimulate theforming of small spray, the lifetime of which can exceed several seconds [11]. Thesedrops are moved by the wind, forming a layer saturated by water in differentphases over the sea surface. Greater depolarization for horizontal polarization incomparison with vertical can be explained by the flattening of the spheroids (drops)in the vertical plane. The higher intensity for horizontal polarization results fromthe horizontal axis a being larger than the vertical one b. According to [16] data,when b /a = 0.2 the ratio sHH /sVV ≈ 8.3 dB for a /l ≈ 0.1 to 0.15 (i.e., when thedrop diameter is significantly less than the wavelength).

A spheroid of this type can be approximately considered as a dipole randomlyoriented in space. If the dipole is oriented along axis OA and the plane wave ispropagating along the axis OZ (Figure 4.11), the dipole cross section sxx (u , w )for a polarization plane a coinciding with axis OX is [63]

sXX (u , w ) = smax ? sin4 u ? cos4 w (4.58)

where smax = sxx (p /2, 0) is the cross section of a dipole oriented along axis OX.Correspondingly, the cross section sXY of this dipole when received on the

polarization coinciding with OY is

sXY (u , w ) = smax ? sin4 u ? sin2 w ? cos2 w (4.59)

For the ensemble of randomly oriented dipoles

s XXsXY

=

E2p

0

dw Ep

0

sXX (u , w ) ? sin u ? du

E2p

0

dw Ep

0

sXY (u , w ) ? sin u ? du

(4.60)

(i.e., significant depolarization occurs).

4.4 Depolarization of Scattered Signals 201

Figure 4.11 Geometry of the spheroid.

Depending on the phase of breaking and the wavelength, the contribution ofdrops to the total scattered signal differs for copolarizations and cross-polarizations,explaining both the increase in depolarization coefficient for horizontal polarizationin comparison with vertical and the increasing depolarization in spikes relative togaps.

Using the model of normalized RCS proposed in [32, 62] and assuming signifi-cant reflection from spray during the spikes, we may write:

s 0spike = s 0

sea + s 0drop

s 0spike cross = s 0

sea cross + s 0drop cross (4.61)

Di , j spike =s 0

sea cross + s 0drop cross

s 0sea + s 0

drop

= Di , j gap ?11 +Ddrop

Di , j gap?

s 0drop

s 0sea

1 +s 0

drop

s 0sea

2where

202 Sea Backscattering at Low Grazing Angles

Di , j spike and Di , j gap denote depolarization coefficients for spikes and gaps,respectively.

s 0sea , s 0

sea cross , s 0drop , and s 0

drop cross are the specific sea and spray RCS forcopolarized and cross polarized components, respectively.

Ddrop = s 0drop cross /s 0

drop is the spray depolarization coefficient.

Derivation of d = Di , j spike /Di , j gap is carried out under the assumption thatthe maximal spray depolarization coefficient is Ddrop = 4.8 dB, resulting from(4.59), and that the ratio

s 0drop

s 0sea

= H−(10 to 15) dB for l = 3 cm

−(3 to 5) dB for l = 8 mm

is determined using data presented in [66]. The results of this derivation are shownin Table 4.9, which also contains the experimental values of difference betweendepolarization coefficients in spikes and gaps.

Analysis of data in Table 4.9 shows that spray-drop fraction depolarizationmechanism (the fraction formed due to sea wave collapsing) permits us to explainthe experimental results for the excess depolarization coefficients in spikes com-pared to gaps.

4.5 Sea Clutter RCS Model

The investigations that have been carried out permitted us to develop an empiricalmodel of backscattering at centimeter and millimeter wavelengths and low grazing

Table 4.9 The Derived and Experimental Values of d

The Direction Polarizationof Radiation Wavelength (cm) of Radiation dderived (dB) dexperim. (dB)Upwind 3.0 H 0.26 1.0

3.0 V 4.1 2.80.8 H 1.5 2.80.8 V 6.2 3.3

Crosswind 3.0 H 0.32 03.0 V 3.5 1.00.8 H 1.34 00.8 V 6.3 5.5

Surf zone 3.0 H 0.6 03.0 V 3.5 0.50.8 H 1.6 5.00.8 V 7.9 5.5

Source: [62].

4.5 Sea Clutter RCS Model 203

angles [66]. The presence of super-refraction in the near-surface troposphere istaken into account for this model in that the real grazing angle appears to begreater than one determined from geometric considerations, and is

c =hr

+12

r |grad n | (4.62)

The transition from the plateau region to the interference region occurs at acritical angle

cc ≈3.6 ? 103

p f u 2.5(4.63)

where

u is the average wind speed in m ? s−1.

f is the operating frequency in gigahertz.

l is the wavelength in meters.

The existence of a transition zone near c ≅ c c , where the angular dependenceof the normalized RCS changes in character, is described by the coefficient

Ac =S c

ccD4

1 + S cccD4

(4.64)

The factor taking into consideration the wind-dependent saturation of thenormalized RCS can be written as

Au =S u

u0Dn

1 + S uu0Dn

(4.65)

where u0 is the critical wind speed equal to u0 = 7 m ? s−1 and n the power indexdepending upon the operating frequency and defined as

n = 1.25f 0.5 (4.66)

204 Sea Backscattering at Low Grazing Angles

The coefficient permitting the variations of the normalized RCS with a varyingangle between the beam and the general sea wave run can be taken in the formsuggested in [59],

Aa = exp [0.375 cos c (1 − 2.8a ) f 0.33] (4.67)

The operating frequency dependence of the normalized RCS in the saturationregion and at grazing angles greater than the critical value is relatively weak andcan be approximated as s 0 ∼ f 0.4.

The contribution due to sea spray can be taken into account through (4.48).Making use of the relations of [32] and assuming the volume RCS to be constantwithin the plane h = const, it is then not difficult to obtain the value

s 0spray ≈ 1.36 ? 10−18 f 4 exp (1.4u ) (4.68)

Taking into consideration that the maximum normalized RCS of the sea surfaceat 10 GHz is of the order of 35 dB to 37 dB in the plateau region, its dependencieson these parameters can be written as

s 0 = 7 ? 10−4 S f10D

0.4S c10D

0.5

Ac Au Aa + 136 ? 10−18 f 4 exp (1.4u ) (4.69)

The comparison of the experimental results with the normalized RCS calculatedaccording to (4.69) have shown good agreement. This is illustrated in Figure 4.12(a),representing the angular dependences of the normalized RCS at l = 3 cm: 8 mmand 4 mm for mean wind velocities of 8–10 m ? s−1. The similar dependence ofthe normalized RCS on wind speed for various grazing angles is given in Figure4.12(b). The lines represent computations according to (4.69), while the signs showthe measured results.

Figure 4.12(c) shows the normalized RCS as a function of the operatingfrequency for grazing angles c > c cr , where it can be seen to agree with thefrequency dependence of (4.69), and for grazing angles 0.3° and 0.15°. It is seenthat the RCS dependence on the operating frequency is stronger at small grazingangles (c < c cr ). The effect is due to the decrease in c cr , with increased operatingfrequency as seen from (4.55), and to the growth of s 0 for the same frequencies[see (4.64)]. Once again, the experimental results are in satisfactory agreementwith the data calculated from (4.69).

At frequencies less than 10 GHz, the normalized RCS depends on the polariza-tion of transmission and reception. This dependence can be presented as

s 0VV

s 0HH

≅ e 0.25(10− f )

4.5 Sea Clutter RCS Model 205

Figure 4.12 Dependence of the normalized RCS: (a) upon the grazing angle; (b) upon the wind speed forl = 3 cm (1), l = 8 mm (2), and l = 4 mm (3); (c) upon the operating frequency for thegrazing angle c > ccr (1), c = 0.3° (2) and c = 0.15 (3); here the experimental data show bycircles at 3 cm, by crosses at 8 mm, and by squares at 4 mm.

or

(s 0VV − s 0

HH ) dB ≈ H1.08(10 − f ) for f < 10 GHz

0 for f > 10 GHz

Thus, the normalized RCS dependence on the sea wave parameters in thefrequency band 1–100 GHz can be represented by (4.69). In the frequency band

206 Sea Backscattering at Low Grazing Angles

1–10.0 GHz, the polarization dependence of the normalized RCS is taken intoconsideration in the form

s 0HH ≅ 7 ? 10−4 S f

10D0.5

Ac AV Aa (4.70a)

s 0VV (dB) = s 0

HH + 1.08(10 − f ) (4.70b)

The depolarization of the scattered signals can be taken into account by

s 0cross (dB) = s 0

VH = s 0HV ≅ s 0

HH − 10 (4.71)

Thus, the empirical model estimates the normalized RCS of the sea surfacewith both superrefraction and scattering from spray in a frequency range 10 GHzto 100 GHz at grazing angles c < 30° and wind speeds u < 15 m ? s−1. It givessatisfactory agreement with experimental results and can be used to evaluate thecontribution of sea clutter during operation of radar detection systems.

4.6 Sea Clutter Statistics

The signals returned from the air-water boundary are fluctuating due to scatterermotions within a single resolution cell (a surface area limited by the pulse lengthand the antenna azimuth beamwidth), as well as to shift of the surface areas viewedby a moving radar. Therefore, the pdf of the normalized RCS is a function of spaceand time. Measurements at X-band at small grazing angles (1° to 5° ) and withsmall illuminated cells [67, 68] show that the best approximation to the amplitudeprobability density is provided by the log-normal distribution

p (s ) =1

sss √2pexp3−

lnS ssm

D2s

2s4 (4.72)

where sm is the median RCS and ss the rms deviation of ln (s ). This distributionis characterized by longer tails in comparison with the Rayleigh distribution, whichtakes into account the higher probabilities of larger signal amplitudes than inRayleigh statistics. The experimental distributions are markedly different from theRayleigh for horizontal polarization, although some differences can be noticed forvertical polarization as well. These differences become sharper for shorter pulses,longer ranges, and greater wind speeds. One can also notice that the distributionof instantaneous signal voltages obeys the composite Gaussian law,

4.6 Sea Clutter Statistics 207

p (x ) = (1 − g )1

s√2pexpS−

x2

2s2D +g

ks√2pexpS−

x2

2k2s2D (4.73)

where g is the weight coefficient and k is the ratio of variances of two Gaussianprobability densities.

It should be noted that these parameters depend only weakly on the anglebetween the wind direction and the beam. Similar results were obtained in [32,69] for the shorter wavelength part of the centimeter band and for MMWs. Thebest approximations of the amplitude probability functions were Johnson’s SB andlognormal distributions for horizontal polarization and the Rayleigh distributionfor vertical. At shorter wavelengths and higher sea waves, the standard deviationin (4.72) decreases, as illustrated by the data of Table 4.10 [70].

The differences between the measured distributions and the Rayleigh modelare less significant for VV polarization. Therefore, the amplitude distributions areoften represented for VV polarization by the Weibull distribution [30]

P (A ) =BC

AB −1 expS−AB

C D (4.74)

where C is the shape parameter and B the slope parameter.It should be emphasized that for a distribution characterized by two indepen-

dent parameters, it is relatively easy to select them in such a manner that they fitthe experimental distributions. With B = 2 (4.74) becomes the Rayleigh amplitudedistribution, while with B = 1 it becomes an exponential. The values of B and Cfor (4.74) that were derived at l = 3 cm for HH and VV polarizations are listedin Table 4.11 [30].

The measurements in the shorter wavelength part of the centimeter band andfor MMWs [32, 69] have shown that instantaneous signal strengths (outputs ofa synchronous detector) differed from the standard Gaussian model, the majordistinctions being observed, as in [68], for HH polarization. In the case of verticalor circular polarization, the measured data can be approximated reasonably wellby the Gaussian distribution. However, the probabilities of signal large values are

Table 4.10 sS Dependences on Frequency and Wave Height

sS (dB)Frequency (GHz) Wave Height (m) Wind Speed (m ? s−1) HH VV

10.0 0.24 0 9.0 5.20.48 6.1 7.7 4.41.10 5.6 5.7 5.4

35.0 0.24 0 7.2 5.20.48 6.1 5.7 4.3

Source: [70].

208 Sea Backscattering at Low Grazing Angles

Table 4.11 Dependences of B and C on Sea State

VV HHSea State B C B CSea state 2 0.622 0.065 0.833 0.006Sea state 5 0.495 0.228 0.625 0.034Source: [30].

seen to be higher than predicted by this distribution. Figure 4.13 presents thecumulative probability functions of the signal instantaneous values at l = 3 cmand l = 8 cm using a scale linearly representing the Gaussian law for a range of0.6 km and wind speed of 6 ms−1.

The best approximation of these distributions for HH polarization is providedup to the probability level of 10−4 to 10−5 by the composite Gaussian law, with aratio of component variances k = 10 to 20 and the weight coefficient g < 0.1. Atsmall grazing angles, the signal returned by the sea surface shows a set of specific

Figure 4.13 Cumulative probability functions of signal instantaneous values.

4.7 Radar Spike Characteristics of Sea Backscattering 209

features. Along with the continuous noise-like signal, spikes at 10 to 15 dB abovethe average level are observed at short ranges (1–1.5 km). As the grazing angledecreases (range increases), the returned signal acquires a pulsed character, whichcan be explained by the shadowing of a large part of the surface by such largewaves that only the crests of the larger waves extend above the shadow zone.Under these circumstances, the normalized RCS is no longer a characteristic thatwould give a complete description of the intensity of echoes from the sea surface.To estimate the radar immunity against clutter, the statistical characteristics of thesignal spikes become of particular importance.

4.7 Radar Spike Characteristics of Sea Backscattering

As can be seen, the sea normalized RCS is a sum of two components for practicallyall models (the idea of discrete scatterers is discussed in the introduction). Herewe will discuss the radar characteristics of scattered signal spikes using the resultsobtained in [71–73]. For spikes [16]

s 0spike ≅ F2(c , a ) ? [u 2

* /g ]3/2 (4.75)

where F2(c , a ) is a function of the grazing angle c and the angle a between thewind and beam directions. This function characterizes the scattering of a singlespike.

Let us consider the ratio between the components in (4.69). In the normalizedRCS model [66], the second term is determined by scattering from spray, giving alarge contribution to the total signal in X-band and more especially in the millimeterband. In the framework of this model, the ratio is

s 0spike

s 0 = [1 + 0.65 ? 1014Ac ? AV ? Aa ? c0.5 ? f −3.6 ? exp (−1.4V )]−1

(4.76)

where

f is the frequency in gigahertz.

Ac , AV , and Aa are the coefficients determining the dependence of the normal-ized RCS on the grazing angle, the wind velocity, and the angle between thewind and radiation directions.

V is the wind velocity.

Strong dependencies of this ratio on the wind velocity and the frequency areobserved. For decreasing grazing angle, the contribution of the first term in the total

210 Sea Backscattering at Low Grazing Angles

scattered signal decreases because of shadowing, and the backscattering assumes amore discrete shape.

The experimental data permit us to confirm that the probability of spikesobtained in a single resolution cell at a certain time can be approximated by thePoisson distribution

P (n ) =Nn

n !? exp (−N ), N > 0, n > 0 (4.77)

where N is the mean number of spikes. This probability depends on the grazingangle, the threshold, and the sea state. From our data [32], the mean intensityof spikes at 3 cm is 10–12 min−1 for a grazing angle of 0.4° and decreases to6–8 min−1 for grazing angles of 0.1°–0.15°. This agrees rather well with the dataon the mean number of crossings of some boundary by the Gaussian randomprocess shown in Figure 4.7 for wind velocities of 7–10 m/s. Our measurementscarried out at 140 GHz for wind velocities less than 5 m/s and zero thresholdshowed that for grazing angles less than 3.5°, the mean number of spikes was 20to 24 per minute.

The dependence of the mean number of spikes on the friction velocity obtainedin [30] at 15 GHz for two polarizations shows that it can be approximated by

N = C1 ? u a1* (4.78)

where log C1 = 1.1 and a1 = 2.9 ± 0.6.This agrees with Philips’ model [16]. The ratio s 0

sp /N = const, and with increas-ing wind velocity, the mean RCS of the spikes increases. As the result, the spikenormalized RCS distributions are independent of the friction velocity for velocitiesfrom 25–45 cm/s. The spike probability decreases with increased shadowing.

The spike statistics at the output of the phase detector of a coherent radarmeasured at 3.0 cm and 0.8 cm for the grazing angles less than 0.3° is shown inTable 4.12.

Here the polarization dependence of spike probability is not significant, butits great dependence on the threshold is seen rather clearly. Besides, the spike

Table 4.12 The Spike Probabilities at 3.0 and 0.8 cm for Different Polarizations

Polarization CoincidenceRelative of Radiation Spike Probability Spike Probability ProbabilityThreshold and Reception at 3.0 cm at 0.8 cm of Spikes

1.5 HH 0.27 0.26 0.061.5 VV 0.22 0.26 0.052.0 HH 0.09 0.10 0.022.0 VV 0.09 0.19 0.01

4.7 Radar Spike Characteristics of Sea Backscattering 211

probability as a function of the threshold coincides with conclusions for sea wavepeaks over some boundary. The weak coincidence of spikes for the instantaneousvalues of backscattering at two frequencies can be expected from the differentmechanisms of the scattered signal formation at these frequencies.

The spike and gap probabilities are considerably greater for the sliding meanamplitudes of the scattered signals at 3 cm and 0.4 cm. The joint probability datafor orthogonally polarized scattered signals and for horizontal, vertical, and tiltedpolarizations are shown in Table 4.13 (here the symbols 0 and 1 refer to gaps andspikes, respectively; the first argument corresponds to vertical polarization and thesecond to horizontal received polarization).

It is seen that for the sliding mean amplitudes, coincidence of orthogonallypolarized components for spikes and gaps is observed 30%–40% of the total timeand the absence of coincidence only 10%–20%. At the same time, the spikes ofthe instantaneous signal are observed 10% of time at both polarizations.

In [32, 51, 71], the results for maximal RCS distributions of spikes for athreshold level of 6 dB above the mean RCS at frequencies of 10 GHz and 37 GHzand for horizontal, vertical, and circular polarizations of radiation are presented.The following conclusions are made:

• For identical weather conditions, the maximal spike RCS increases withdecreasing wavelength, and at 8 mm it is larger than at 3 cm by 3–7 dB.

• The maximal spike RCS increases for decreasing grazing angle, explainedby the presence of more intense scatterers in the shadowing zone.

• Polarization dependence of maximal spike RCS is not seen clearly. The prob-ability of the spike RCS larger than 0.1 m2 is higher for horizontal polariza-tion than for vertical polarization, while the probability of spikes with largeRCS at 8 mm is larger for vertical polarization. The spike RCS decreasesfor circular polarization in comparison with linear by about 10 dB.

• The most probable duration of spikes is 0.4–0.6 second, and the maximalduration is 4–5 seconds. These results agree with the sea surface peak dura-tion shown in Figure 4.9.

Table 4.13 Statistics of Spike Amplitudes at the Wavelengths of 3.0 cm and 0.8 cm

SpikeWavelength Transmitted Probabilities Joint Probabilities(cm) Sea State Polarization PV PH P(0,0) P(0,1) P(1,0) P(1,1)

3.0 Sea state V 0.53 0.50 0.34 0.13 0.17 0.373.0 5, wind H 0.51 0.50 0.32 0.17 0.19 0.323.0 velocity Tilted 0.51 0.46 0.46 0.40 0.10 0.140.4 15 m/s V 0.45 0.15 0.49 0.05 0.36 0.100.4 V 0.42 0.49 0.37 0.21 0.14 0.28

212 Sea Backscattering at Low Grazing Angles

• The mean duration of gaps increases for increasing threshold, especially forsea state 2, where the results of the peak theory are clearly applicable. Thesedata also agree with the results in Figure 4.9.

• In our investigations, the mean duration of spikes and gaps depends on thewind velocity. The mean spike duration is 0.1–0.3 second for the windvelocities of 2–4 m/s and 0.2–0.5 second for the wind velocities greater than6 m/s. The mean gap duration is 0.5–5 seconds for the wind velocities of4–8 m/s.

• The mean duration of spikes increases and the gap duration decreases withdecreased wavelength.

As an example, the histograms of spike duration distribution at 3.0 cm and0.8 cm for vertical and horizontal polarization are shown in Figure 4.14, and thedistributions of spike and gap duration at 2.0 mm are shown in Figure 4.15.

As shown in [62], a rather intense depolarization in spikes is observed forvertical polarization, while the difference of depolarized components for spikesand gaps is small for horizontal polarization. The depolarization coefficients forgaps depend on frequency rather weakly, and the spike depolarization coefficientsincrease with increasing frequency.

The total power spectra of the scattered signals are determined by the scatteringfrom sea and spray. There is an increase in Doppler frequency for spikes (by two

Figure 4.14 Spike duration distribution at 3 cm and 8 mm for (a) horizontal and (b) verticalpolarizations.

4.8 Backscattering Spectra 213

Figure 4.15 (a) Spike and (b) gap duration distributions at 2 mm.

to three times) in comparison with gaps. In 15%–20% of the spectra, a secondmaximum appears that is caused, in our opinion, by the scattering from sprayblown by the wind.

4.8 Backscattering Spectra

The power spectra of the signals backscattered from the sea are generally determinedby fluctuations of scatterers driven by the wind as well as by antenna scanningand radar platform motion. The effects of various scanning techniques and platformmotion on the spectra are not discussed here. The following discussion addressesthe power spectra of scattered signals with a fixed radar antenna.

214 Sea Backscattering at Low Grazing Angles

The power spectra of X-band and MMW signals scattered by the sea surfaceshow a marked dependence on the wind speed and antenna polarizations. As thewind speed increases, the central frequency F0 and the spectral width DF tend toincrease. In this case, the spectra for vertical and circular polarizations have lowermagnitudes at the central frequency F0 , as compared with horizontal polarization,while all of the spectral widths are the same. The difference in the central frequencydecreases with decreasing grazing angle. For instance, for l = 3 cm the ratioF0HH /F0VV was 1.2 to 1.5 with a grazing angle about 1°, whereas the differenceof frequency shifts vanished at grazing angles less than 0.3°.

To approximate the scattered signal spectra over the wide band of radiowaves(from 1 GHz to 140 GHz) and small grazing angles, the authors suggest therelationship [32, 74]

G (F ) = G0S1 + | F0 − FDF |nD

−1

(4.79)

where G0 is the maximum value of the spectral density at F0 ; DF is the spectralhalf-width at the −3-dB level, and F is the current frequency. The rate of spectraldensity decrease characterized by the power exponent n is dependent on the windspeed and sea surface state; decreasing with an increase in the wind speed. Figure4.16 shows, as an illustration, the spectra measured at l = 3 cm with horizontalpolarization for the frequency range F < F0 .

Triangles show measurements at wind speeds less than 4 ms−1, and circles andsquares represent the experimental data obtained at wind speeds of 5–8 ms−1. The

Figure 4.16 (a) Power spectra of radar returns from the sea surface at l = 3 cm for horizontalpolarization: curve 1: n = 2; curve 2: n = 3; curve 3: n = 4 and curve 4: n = 5. Thedots correspond to measured results. (b) Power spectra of intensity at l = 3 cm withthe wind speed of 10–12 ms−1; 1: exponential function; 2: power law with n = 2;3: power law with n = 3; the dots correspond to measured results.

4.8 Backscattering Spectra 215

power exponent is seen to decrease with increasing wind speed. Similar magnitudesof the exponent were obtained in [32] for the high-frequency spectral region(F > F0), which leads to the conclusion that the spectrum is symmetrical withrespect to the central frequency shift F0 .

In order to describe the spectral density dependence on the radar operatingfrequency (wavelength) and wind speed, the following empirical relationship canbe used

n = 8.9l0.1V −0.5 (4.80)

where l is the wavelength in centimeters and V is the mean wind speed (ms−1)(see [6]).

The central frequency and the spectral width vary proportionally to theoperating frequency. The measured values happen to be larger than predicted ones,apparently due to the contributions from wave crests and from breaking wavesthat move at considerably greater speeds.

Let us consider the change of central frequency F0 because of the Bragg’sscattering and the phase speed of sea waves in the framework of this mechanism,the orbital movement of sea waves, and the wind drift. Within the framework ofthe two-scale model, the account of these factors permits the reception of thefollowing expression for determination of the spectrum central frequency

F0 = √g ? cos wpl

+ 16pm

rl3 +2V0

lcos a +

2VWDl

cos a (4.81)

where a is the angle between directions of radiation and sea wave movement; V0is the orbital speed of large gravitational sea waves; VWD is the speed of a winddrift; m is the factor of the surface tension; r is the density of water; and l is thewavelength of radiowave.

The first item in the right part (4.81) is determined by the phase speed ofmovement of the great sea waves and defined from the dispersive equation [50].The expression under the root square caused by the surface tension is small incomparison with item of g /pl in practically all range of frequencies except for ashortwave part of a millimeter range.

Besides, the point on sea surface moves in close to a circular orbit. For a caseof simple harmonious movement, orbital speed is determined as

V0 =pHT

(4.82)

where H and T are the height and the period of sea wave accordingly.

216 Sea Backscattering at Low Grazing Angles

For small grazing angles when scattering elements are near to sea wave crests(i.e., the other part of a wave is in a zone of shadowing), the orbital speed maybe submitted as

V0 =pHT S1 + 2p2 H

gT 2D (4.83)

The last expression is the sum of the classical orbital speed (4.82) and the contribu-tion of Stocks drift. The second item, usually rather small, nevertheless brings thecontribution of a spectrum of the reflected signal to the formation.

For rough seas, the height and the period of a sea wave are unequivocallyconnected to speed of wind by the dependences

H = 7.2 ? 10−3 ? V 2.5 (4.84a)

T = 0.556 ? V (4.84b)

Therefore, (4.82) and (4.83) may be presented as

V0 ≅ 1.3 ? 10−2 ? pV 1.5

V0 = 1.3 ? 10−2 ? V 2.5 S1 + 4.66 ? 10−2 p2

gV 0.5D (4.85)

= 1.3 ? 10−2 ? V 2.5 (1 + 4.69 ? 10−2 ? V 0.5)

Here V is the mean speed of a wind at height of 10m above the sea surface.Due to the interaction of a wind and sea surface, one more movement of a sea

surface named a wind drift is observed. Available experimental results specify thatspeed of a wind drift does not exceed 3% from wind speed at height of 10m[11, 12] and often the following ratio is used:

VWD ≈ 0.02V (4.86)

For the estimation of a role considered above three factors, forming the centralfrequency in a spectrum of scattered signals, we shall consider dependence of F0on wavelength presented in Figure 4.17. The experimental data of different authorsobtained in the frequency band of 1–140 GHz is marked by the different signs;the straight lines 1–4 are some approximations of F0 dependence, presented underthe figure. A better agreement of the dependence of F0 on the wavelength with theexperiment is provided by the following empirical expressions [74]

4.8 Backscattering Spectra 217

Figure 4.17 The spectrum central frequency dependences on the wavelength 1:

F0 = √ gpl

+2VWD

l+

2V0l

; 2: F0 = √ gpl

; 3: F0 = 44.4Vl

; 4: F0 = √ gpl

+2VWD

l;

signs show the experimental data.

F0 = 44.4Vl

(4.87)

As marked earlier, the F0 value is determined by the polarization of radiationand reception. Its value is smaller for the vertical polarization in comparison withhorizontal, and this difference decreases for the grazing angle decreasing.

The experimental data is marked as in Figure 4.17.The DF = f (l ) dependence is presented in Figure 4.18. Here the different signs

are the same as in the Figure 4.17. The experimental results are satisfactorydescribed by

DF = 30.7V 0.75

l(4.88)

These are valid for wind speeds of 2–15 m ? s−1 (local wind velocities achieve20 m ? s−1 and more) and were checked at frequencies of 10–140 GHz and grazing

218 Sea Backscattering at Low Grazing Angles

Figure 4.18 The spectral width dependence on the wavelength.

angles c < 5°. As the grazing angle is changed, the power spectra of the backscatteredsignals undergo changes that are especially noticeable for vertical and circularpolarizations. As has been noted, at lower grazing angles F0VV → F0HH , whereasF0HH is virtually independent of the grazing angle. Moreover, as c decreases, thespectra become somewhat broader, as described by the empirical expression [74]:

DF

DF= 0.63 + 0.064

hcc

(4.89)

where h is the radar height. This expression holds for ranges r ∈ (0.3 to 3.0) km.Varying the angle a between the illumination direction and the general wave run(which coincides with the wind direction in the case of developed roughness) leadsto variations of the central frequency in the power spectrum for all polarizations.The central frequency is maximal if the surface is illuminated normally to the wavefront and is practically equal to zero for illumination along the wave. The azimuthaldependence can be approximated by

F0F0max

= cos2.5 a , |a | ≤ 30° (4.90)

4.8 Backscattering Spectra 219

Neither the spectral width nor the exponent of power depends on the anglea . Besides, the spectral width does not depend on the polarization of radiationand reception. Finally, the spectral parameters can be represented in the form

F0HH = F0HV = 44.4Vl

(4.91)

F0VV = F0VH = HF0HH (1 − 0.4c ) for c ≤ 2°0.5F0HH for c > 2°

(4.92)

DFVV = DFHH = DFVH = DFHV ≅ 30.7V 0.75

l(4.93)

Basically, the power spectra of the scattered signal intensity (i.e., the spectraat the amplitude detector output) have properties similar to those of the powerspectra of the signals themselves. Figure 4.16(b) presents the measured powerspectra of intensity and the curves corresponding to an exponential spectrum (curve1) and to the power law

G (F ) = G0 F1 + S FDF D

nG−1

(4.94)

The comparison of Figure 4.16(a, b) shows the spectrum of (4.94) to retainthe general form of those obtained from (4.79), but with a lower power index. Inother words, the rate of the drop off in spectral density slows down in comparisonwith the power spectrum of the signal.

As shown in [75], the intensity spectra (amplitude spectra) are characterizedby a small width in their energy-carrying part, not exceeding 1.5–2 Hz at the−20-dB level. With the antenna pattern oriented parallel to the sea wave, theirspectral peaks lie in the frequency range 0.1–0.15 Hz, coinciding with the maximumof the sea spectrum. The spectra are characterized by the absence of peaks forbackscattering from the surf zone. At frequencies above 0.5–0.15 Hz, the spectrumcan be approximated by an expression similar to (4.88), G (F ) ∼ F −n. The probablevalues of the exponent n lie between 1.9 and 2.6 for l = 3 cm and 2.6 to 3.4 forl = 8 mm. Regression analysis shows that in the X-band, the spectra in their energy-carrying part are, on the average, 1.5 to 1.8 times wider than at l = 8 mm. Nostable differences have been observed in the spectra of cross-polarized components.Here, the analysis revealed a high correlation of the orthogonal scattered compo-nents on both operating frequencies. The correlation factors were 0.7 to 0.95,the lower values being observed for illumination perpendicular to the sea waveand the higher noted in the surf zone. This tendency is even more pronounced in

220 Sea Backscattering at Low Grazing Angles

cross-correlation factors of the signals at 10 GHz and 35 GHz. The lowest valuesare 0.3–0.4 for the calm sea; they tend to increase with the appearance of wavebreaking and reach 0.8 when the breaking is intense.

Because the sea state is far from being steady under the wave-breaking condi-tions (which result in spikes and pauses in the scattered signals), the current powerspectra reveal some distinctive features never shown by the average power spectra.These effects have been studied experimentally in the 3-cm, 8-mm, and 2-mm bands[33, 76] and can be summarized as follows.

In breaking conditions the current spectrum of sea backscattering can be pre-sented as

G (F ) = Gsea (F ) + Gsp (F ) (4.95)

where Gsea (F ) and Gsp (F ) are the spectral densities of sea and spray backscattering,respectively. The first component is described by (4.79), and dependences of itsparameters on the sea state and wind velocity have been discussed earlier. Somefeatures of second component were considered in [33].

The spectral width characteristic of the spike period of the returned signal isconsiderably greater (up to a factor of two) than that one shown in gaps. Thecentral frequency also increases. Table 4.14 presents the averaged data on thespectral widths and central frequency at two wavelengths, with different transmittedand received polarizations.

The results in Table 4.14 show that the central frequencies during the spikeand gaps are higher for horizontal polarization than for vertical polarization. Thisis in agreement with the conclusions drawn from the averaged spectra. The formsof the power spectra for the cross-polarization components are similar to thoseobtained in the copolar channels. A relatively high correlation between the Dopplershift and the intensity of the return signal is observed under the heavy sea, for allpolarizations, whereas similar correlation is practically absent for the calm sea.Instantaneous power spectra of the scattered signal spikes often show, along with

Table 4.14 The Central Frequency and Spectral Width in Spikes and Gaps

Spikes GapsFrequency (GHz) Polarization F0 (Hz) DF (Hz) F0 (Hz) DF (Hz)

9.6 HH 256 ± 56 56 ± 20 141 ± 25 38 ± 18HV 255 ± 66 65 ± 22 141 ± 30 34 ± 22VV 222 ± 82 78 ± 26 62 ± 31 45 ± 23VH 233 ± 83 84 ± 19 63 ± 32 59 ± 21

35.0 HH 890 ± 145 575 ± 153 521 ± 85 219 ± 44HV 969 ± 190 629 ± 52 471 ± 11 271 ± 53VV 606 ± 230 356 ± 118 222 ± 63 157 ± 52VH 720 ± 200 301 ± 70 260 ± 96 224 ± 50

Source: [33].

4.8 Backscattering Spectra 221

the effect of power concentration at higher frequencies than in the gaps, a secondmaximum in their high-frequency part (around 220–360 Hz or 900–1,300 Hz forthe 3-cm and 8-mm wavebands, respectively). The second maximum frequencycan be derived from the empirical relationship

Fsp ≅2U0

l S hh0D0.25

cos a (4.96)

where U0 is the mean wind velocity at height of h0 and h is the height of wavebreaking.

The experimental results of [34] show that the second spectral width can befrom 50 Hz to 2.0 kHz. The mean spectral width is presented in Table 4.15.

The second peak was observed in 15% and 21% of the processed spectra atl = 3 cm and l = 8 mm, respectively. The relative frequency of appearance of thesecond peak reported in [32] was lower and was 1.5% to 6%. Similar multimodespectra from breaking waves were observed at l = 8 mm in [33]. The presence ofthe second peak can be explained in two ways.

First, at the moment of wave spiking, just before breaking, its orbital velocityincreases. Second, the sprays generated after wave breaking are carried away bythe wind, their drift velocity reaching 60% of the wind speed. The predominanteffect of the second mechanism is confirmed by the increased depolarization factor,which can be attributed to the emergence of fast-moving nonspherical droplets.

The degree of correlation between the copolarized and cross-polarized compo-nents of the scattered signal can be found from the coherence function. Analysisof its behavior indicates that in the X-band, the major contribution to the scatteredsignal during the gaps was made by scatterers having relatively low speeds (Dopplerfrequencies below 100 Hz). At the moments when spikes occur, additional high-speed scatterers appear (the corresponding Doppler frequencies are 200–400 Hz),which are characterized by a low degree of coherence between the orthogonallypolarized components. At l = 8 mm the coherence of cross-polarized componentsis low over the entire range of Doppler frequencies, both in the spikes and in thegaps.

Table 4.15 Mean Spectral Width (kHz)

Wavelength (mm) 30 8.0 4.0 2.0DFsp (kHz) 0.05–0.08 0.36–0.58 0.62–0.9 1.25–2.0

222 Sea Backscattering at Low Grazing Angles

References

[1] Kraus, E. B., Atmosphere-Ocean Interaction, Oxford, England: Clarendon Press, 1972.[2] Fabrikant, A. P., ‘‘About Nonlinear Development of Wind Roughness for Light Breeze,’’

Proc. of Academy of Sciences of USSR, Physics of Atmosphere and Ocean, 1980,Vol. 16, No. 9, pp. 985–988.

[3] Price, R. K., ‘‘The Breaking of Water Waves,’’ J. Geophys. Res., Vol. 76, No. 6, 1971,pp. 1576–1581.

[4] Ramberg, S. E., and O. M. Griffin, ‘‘A Laboratory Study of Steep and Breaking DeepWater Waves,’’ Proc. ASCE, J. Waterways, Port, Coastal Ocean Div., Vol. 113, 1986,pp. 493–506.

[5] Boumarin, P., ‘‘Geometric Properties of Deep-Water Breaking Waves,’’ J. Fluid Mech.,Vol. 209, 1989, pp. 405–433.

[6] Bondur, V. G., and E. A. Sharkov, ‘‘Statistical Characteristics of Foam Formations onSea Rough Surface,’’ Oceanology, Vol. 22, No. 3, 1982, pp. 372–379.

[7] Ross, D. B., and V. Cardonne, ‘‘Observation of Oceanic Whitecaps and Their Relationto Remote Measurements of Surface Wind Speed,’’ J. Geophys. Res., Vol. 79, No. 3,1974, pp. 444–452.

[8] Monahan, E. C., ‘‘Oceanic Whitecaps,’’ J. Phys. Oceanogr., No. 1, 1971, p. 139–144.[9] Bortkovsky R. S., and M. A. Kuznetsov, ‘‘Some Results of Sea Surface Investigations,’’

Typhoon-75, Vol. 1, Gidrometeoizdat, Leningrad, 1977, pp. 90–105.[10] Cox, C., and W. Munk, ‘‘Statistics of the Sea Surface Derived from Sun Glitter,’’ J. Mar.

Res., Vol. 13, 1954, pp. 198–227.[11] MacIntire, F., ‘‘Flow Pattern in Breaking Bubbles,’’ J. Geophys. Res., Vol. 77, No. 27,

1972, pp. 5211–5288.[12] Blanchard, D. C., ‘‘The Electrification of the Atmosphere by Particles from Bubbles in

the Sea,’’ Prog. Oceanogr., No. 1, 1963, pp. 71–202.[13] Phillips, O. M., The Dynamics of the Upper Ocean, New York: Cambridge University

Press, 1977.[14] Phillips, O. M., ‘‘On the Generation Waves by Turbulent Wind,’’ J. Fluid Mech., Vol. 2,

1957, pp. 417–495.[15] Davidan, I. N., L. I. Lopatuchin, and V. A. Rozkov, Wind Roughness: Probabilistic

Hydrodynamic Process, Leningrad: Gidrometeoizdat, 1978.[16] Phillips, O. M., ‘‘Spectral and Statistical Properties of the Equilibrium Range in Wind-

Generated Gravity Waves,’’ J. Fluid Mech., Vol. 156, 1985, pp. 505–531.[17] Hasselmann, K., et al., ‘‘Measurements of Wind-Wave Growth and Swell Decay During

the Joint North Sea Wave Project (JONSWAP),’’ Erg. Dtsch.Hydrogr. Inst. Humburg,No. 12, 1973, pp. 95–116.

[18] Hicks, B. L., and E. A. Huber, The Generation of Small Water Waves by Wind, CSLReport M-87, Univ. of Illinois, 1960.

[19] Hasselmann, K., ‘‘On the Non-Linear Energy Transfer in a Gravity-Wave Spectrum:’’General Theory, J. Fluid Mech., Vol. 1, Vol. 12, 1962, pp. 481–500.

[20] Kononkova, G. E., and K. V. Pokazeev, Sea Wave Dynamics, Moscow, Russia: MoscowState University Publishers, 1985.

[21] Phillips, O. M., and M. L. Banner, ‘‘Wave Breaking in the Presence of Wind Draft andSwell,’’ J. Fluid Mech., Vol. 66, 1974, pp. 625–640.

References 223

[22] Barenblatt, G. I., and I. A. Leykin, ‘‘On the Self-Similar Spectra of Wind Waves in theHigh-Frequency Range,’’ Izv. Atmosp. Oceanic Phys., Vol. 17, No. 1, 1981, pp. 35–41.

[23] Kitaigorodskii, S. A., ‘‘On the Theory of the Equilibrium Range in the Spectrum of Wind-Generated Gravity Waves,’’ J. Phys. Oceanogr., Vol. 13, 1983, pp. 816–827.

[24] Glazman, R. E., and P. Weichman, ‘‘Statistical Geometry of a Small Surface Patch in aDeveloped Sea,’’ J. Geophys. Res., Vol. 94, No. C4, 1989, pp. 4998–5010.

[25] Glazman, R. E., G. G. Pihos, and J. Ip, ‘‘Scatterometer Wind-Speed Dias Induced by theLarge-Scale Component of the Wave Field,’’ J. Geophys. Res., Vol. 93, 1988,pp. 1317–1328.

[26] Donelan, M. A., J. Hamilton, and W. H. Hui, ‘‘Directional Spectra of Wind GeneratedWaves,’’ Philos. Trans. R. Soc. London, Ser. A, Vol. 315, 1985, pp. 509–562.

[27] Glazman, R. E., and S. H. Pilorz, ‘‘Effects of Sea Maturity on Satellite Altimeter Measure-ments,’’ J. Geophys. Res., Vol. 95, No. C3, 1990, pp. 2857–2870.

[28] Glazman, R. E., ‘‘Statistical Problems of Wind-Generated Gravity Waves Arising in Micro-wave Remote Sensing of Surface Winds,’’ IEEE Trans. on Geosci. Rem. Sens, Vol. 29,No. 1, 1991, pp. 135–142.

[29] Wetzel, L. B., ‘‘A Model for Sea Backscatter Intermittency at Extreme Grazing Angles,’’Radio Science, Vol. 12, No. 5, 1977, pp. 749–756.

[30] Olin, J. D., ‘‘Amplitude and Temporal Statistics of Sea Spike Clutter,’’ Adv. Radar Techn.,London, 1985, p. 212–216.

[31] Olin, J. D., ‘‘Characterization of Spiky Sea Clutter for Target Detection,’’ Proc. 1984IEEE Nat. Radar Conf., New York, 1984, pp. 27–31.

[32] Kulemin, G. P., and V. B. Razskazovsky, Scattering of the Millimeter Radiowaves by theEarth’s Surface for Small Grazing Angles, Kiev: Naukova Dumka, 1987 (in Russian).

[33] Atanasov, V. B., et al., ‘‘Experimental Study of Non-Stationarity X- and Q-Band RadarBackscattering from Sea Surface,’’ IEE Proc., Part F, Vol. 137, No. 2, 1990, pp. 118–124.

[34] Kulemin, G. P., M. G. Balan, and Y. A. Pedenko, ‘‘Polarization Characteristics of Micro-wave Backscattering by Sea Wave Non-Stationarities,’’ Proc. Int. Conf. Radar97, IEEPublication No. 449, October 14–16, 1997, pp. 90–94.

[35] Jessup, A. T., W. C. Keller, and W. K. Melville, ‘‘Measurements of Sea Spikes in Micro-wave Backscatter at Moderate Incidence,’’ J. Geophys. Res., Vol. 95, No. C-6, 1990,pp. 9679–9688.

[36] Jessup, A. T., W. K. Melville, and W. C. Keller, ‘‘Breaking Waves Affecting MicrowaveBackscatter,’’ J. Geophys. Res., Vol. 96, No. C-11, 1991, pp. 20561–20568.

[37] Loewen, M. R., and W. K. Melville, ‘‘Microwave Backscatter and Acoustic Radiationfrom Breaking Waves,’’ J. Fluid Mech., Vol. 224, 1991, pp. 601–623.

[38] Trizna, D. B., and J. P. Hansen, ‘‘Laboratory Studies of Radar Spikes at Low GrazingAngles,’’ J. Geophys. Res., Vol. 96, No. C-7, 1991, pp. 12529–12537.

[39] Phillips, O. M., ‘‘Radar Returns from the Sea Surface,’’ J. Phys. Oceanogr, Vol. 18, 1988,pp. 1065–1074.

[40] Beckmann, P., ‘‘Shadowing of Random Rough Surfaces,’’ IEEE Trans. Antennas andPropagation, Vol. AP-13, No. 5, 1965, pp. 384–388.

[41] Kulemin, G. P., ‘‘Microwave Sea Backscattering Features at Very Small Grazing Angles,’’Foreign Radioelectronics, No. 12, 1998, pp. 17–47 (in Russian).

[42] Tichonov, I. I., The Surges of Random Processes, Moscow, Russia: Nauka, 1970 (inRussian).

224 Sea Backscattering at Low Grazing Angles

[43] Fomin, A. A., The Surges Theory of Random Processes, Moscow, Russia: Svjaz, 1980 (inRussian).

[44] Katz, I., and L. M. Spetner, ‘‘Polarization and Depression-Angle Dependence of RadarTerrain Return,’’ J. Res. NBS, Vol. D, No. 5, 1960, pp. 483–485.

[45] Spetner, L. M., and I. Katz, ‘‘Two Statistical Models for Radar Terrain Return,’’ IRETrans. Ant. Propag., Vol. AP-8, No. 5, 1960, pp. 242–246.

[46] Skolnik, M. I., (ed.), Radar Handbook, New York: McGraw-Hill, 1970.[47] Fuks, I. M., ‘‘To Theory of Radiowave Scattering at Rough Sea Surface,’’ Izv. VUZ’ov.

Radiophysics, Vol. 9, No. 5, 1966, pp. 876–885 (in Russian).[48] Kalmukov, A. I., et al., ‘‘Sea Surface Structure Influence for Spatial Characteristics of

Scattered Radiation,’’ Izv. VUZ’ov. Radiophysics, Vol. 8, No. 6, 1965, pp. 1117–1127(in Russian).

[49] Krulov, Y. M., (ed.), Wind Waves, Moscow, Russia: Foreign Literature Press, 1962 (inRussian).

[50] Kalmukov, A. I., et al., ‘‘Some Features of Microwave Sea Backscattering for Small GrazingAngles,’’ Preprint IRE NAS of Ukraine, No. 40, Kharkov, Ukraine, 1974, p. 38 (inRussian).

[51] Kwoh, D. S., and B. M. Lake, ‘‘The Nature of Microwave Backscattering from WaterWaves,’’ in The Ocean Surface, Wave Breaking, Turbulent Mixing and Radio Probing,Y. Toba and H. Mitsuyasu (eds.), Boston: Reidel, 1985, pp. 249–256.

[52] Kwoh, D. S., and B. M. Lake, ‘‘Identification of the Contribution of Bragg Scattering andSpecular Reflection to X-Band Microwave Backscattering in the Ocean Experiment,’’Proc. JGARSS’86 Symp., Zurich, 1986, pp. 319–325.

[53] Stepanenko, V. D., Radar Methods in Meteorology, Leningrad: Gidrometeoizdat, 1968(in Russian).

[54] Zuikov, V. A., G. P. Kulemin, and V. I. Lutsenko, ‘‘Special Features of MicrowaveScattering by the Sea at Small Grazing Angles,’’ Izv. VVZ - Radiofizika, Vol. 24, No. 7,1981, pp. 831–839 (in Russian).

[55] Dyer, F. B., and N. C. Currie, ‘‘Some Comments on the Characterization of Radar SeaClutter,’’ Int. IEEE/AP-S Symp., New York, 1974, pp. 323–326.

[56] Horst, M. M., F. B. Dyer, and M. T. Tuley, ‘‘Radar Sea Clutter Model,’’ Int. Conf. Ant.Propag., Part 2, London, 1978, pp. 6–10.

[57] Sittrop, H., ‘‘X- and Ku-Band Radar Backscatter Characteristics of Sea Clutter,’’ Proc.URSI, Berne, 1974, pp. 25–37.

[58] Michel, S., ‘‘Reflection of Radar Echo-Signals from the Sea Surface (Models and Experi-mental Results),’’ Zarubezhnaya radioelektronika, No. 7, 1972, pp. 13–26.

[59] Schooley, A., ‘‘Upwind-Downwind Ratio of Radar Return Calculated from Facet Statisticsof Wind Disturbed Water Surfaces,’’ Proc. IRE, Vol. 50, No. 4, 1962, pp. 456–451.

[60] Long, M., ‘‘Polarization and Sea State,’’ Electron. Letters, No. 5, 1960, pp. 483–485.[61] Balan, M. G., et al., ‘‘Polarization Characteristics of Microwave Scattering by Transient

Sea Waves,’’ Application of Millimeter and Submillimeter Radio Waves, Institute of RadioPhysics and Electronics, Acad. Sci. Ukraine, Kharkov, 1992, pp. 5–24 (in Russian).

[62] Kulemin, G. P., ‘‘Polarization and Spectral Features of Spiky Sea Backscattering,’’ SPIEInt. Symp. Radar Sensor Technology III, Orlando, FL, April 1998, Vol. 3, No. 395,pp. 112–122.

[63] Long, M. W., Radar Reflectivity of Land and Sea, 3rd ed., Norwood, MA: Artech House,2001.

References 225

[64] Ufimtsev, P. Y., Edge Wave Technique in Physical Theory of Diffraction, Moscow, Russia:Soviet Radio, 1962.

[65] Cherny, I. V., and F. A. Sharkov, ‘‘Remote Radiometry of the Sea Wave Breaking Cycle,’’Earth Research from Space, No. 2, 1988, pp. 17–28 (in Russian).

[66] Kulemin, G. P., ‘‘Sea Backscattering Model for Millimeter Band of Radiowaves,’’ Proc.8th URSI com. F Triennial Open Symp., Session 3, Aveiro, Portugal, September 1998,pp. 128–131.

[67] Croney, J., ‘‘Clutter and Its Reduction in Shipborne Radars,’’ Proc. Int. Conf. Ant. andPropag., London, England, Vol. 105, No. 2, 1973, pp. 213–219.

[68] Trunk, G. V., ‘‘Radar Properties of Non-Rayleigh Sea Clutter,’’ IEEE Trans. on Aerosp.Electr. Syst., Vol. AES-8, No. 2, March 1972, pp. 196–204.

[69] Kulemin, G. P., and V. I. Lutsenko, ‘‘On the Distribution Laws of Millimeter Wave SignalsScattered by the Sea Surface at Small Grazing Angles,’’ II Soviet Symp. on Millimeter andSubmillimeter Waves: Conf. Digest., Gorky, 1980, Vol. 1, pp. 293–294 (in Russian).

[70] Dyer, F. R., N. C. Currie, and M. S. Applegate, ‘‘Radar Backscatter from Land, Sea, Rainand Snow at Millimeter Wavelengths,’’ Adv. Radar Techn., London, England, 1985,pp. 250–253.

[71] Kulemin, G. P., ‘‘Spike Characteristics of Radar Sea Clutter for Extremely Small GrazingAngles,’’ SPIE Int. Symp. Radar Sensor Technology V, Vol. 4033, Orlando, FL,April 2000, pp. 129–138.

[72] Kulemin, G. P., ‘‘Spike Characteristics of Radar Sea Clutter for Extremely Small GrazingAngles (Part 2),’’ SPIE Int. Symp. Radar Sensor Technology VI, Vol. 4374, Orlando, FL,April 2001.

[73] Kulemin, G. P., ‘‘Microwave Sea Backscattering Features for Extremely Small GrazingAngles,’’ Modern Radioelectronics Progress, No. 12, 1998, pp. 17–47 (in Russian).

[74] Kulemin, G. P., and V. I. Lutsenko, ‘‘Special Features of Centimeter and Millimeter RadioWave Backscattering by the Sea Surface at Small Grazing Angles,’’ Preprint No. 237,Institute of Radio Physics and Electronics, Academy of Sciences of the Ukrainian SSR,Kharkov, 1984 (in Russian).

[75] Balan, M. G., et al., ‘‘Statistics of Envelopes of Microwave and Millimeter Wave SignalsScattered by Nonstationary Sea Waves,’’ Conf. Digest Representations and Processing ofRandom Signals and Films: II Soviet Conference, Kharkov, 1991, p. 199 (in Russian).

[76] Balan, M. G., et al., ‘‘Nonstationary Radar Reflections from the Sea in the MillimeterBand,’’ Conf. Digest Representations and Processing of Random Signals and Films: IISoviet Conference, Kharkov, 1991, pp. 82–83 (in Russian).

C H A P T E R 5

Microwave and MMW Backscattering byPrecipitation and OtherMeteorological Formations

5.1 Structure of Meteorological Formations

Millimeter-band radar systems offer several advantages over electrooptical systemsfor operation in the battlefield and for weapon guidance. Among these are all-weather operation, operation in smoke, and operation in dusted atmosphere.

To determine the radar characteristics of meteorological formations, we mustclassify them, in order to present the complex meteorological conditions of thepropagation path in terms of simpler phenomena.

A useful classification is based on grouping meteorological formations by sizeand the physical properties of their constituent particles, including the particlesphericity. Such a classification corresponds to definitions issued by the Interna-tional Meteorological Organization in 1956. According to it, all precipitations aredivided into liquid and solid ones, and in turn all liquid precipitations are dividedinto rain and drizzle.

Rain is water precipitation formed by drops with radius greater than 0.25 mm.Observations show that drops with radius greater than 3.5 mm become flat andbreak into smaller drops. The raindrop terminal velocity reaches 8–10 m/s.

Drizzle is rather homogeneous precipitation consisting mostly of drops withradii less than 0.25 mm. Drizzle intensity is not greater than I = 0.25 mm per hour,and terminal velocity through fixed air is less than 0.3 m/s. Consequently, allspherical water drops with diameters of 0.5–5.5 mm can be considered rain.

The geometrical characteristics of rain zones depend on the rain intensity andclimatical conditions in the local area, connected with geographical coordinates ofthis area. The horizontal and vertical extents of rain zones with different intensitiesare shown in Table 5.1 [1].

Rains in their precipitation zones are distributed nonuniformly, especially forrain intensities greater than 40 mm/hr.

227

228 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Table 5.1 Rain Geometrical Characteristics

Rain Intensity (mm/hr) Diameter (km) Height (m)2 300 4,0004 45 4,0008 35 8,000

16 20 8,00032 8 8,00064 1 8,000

Source: [1].

Showers and heavy rains are distinguished. Showers fall, as a rule, from nimbo-stratus clouds and are characterized by small to moderate intensities (less than 20mm/hr), insignificant temporal changes, and small drop sizes.

Heavy rains fall from cumulonimbus clouds. They are characterized by largeintensities (I ≥ 40 mm/hr), temporally changeable intensity, and comparativelylarge drop sizes.

The determination of rain intensity corresponding to a certain percentage ofobservation time is an important problem. Different precipitation rates are foundin different climatic regions. The probability (or percentage of time during whichrain of given intensity or more falls) is approximately described by the gammadistribution of form [2]

p =bn ? n

G(n + 1)? I n −1e −bI (5.1)

where G(n + 1) is the gamma function, and b and g are parameters taking intoaccount the climatic features of the region.

It is possible to use global rain models [3] for Earth’s regions for which thestatistical distributions of rain intensity are unknown. For the most simple Rice-Holmberg model [3], the percentage of mean yearly time p (I ), percentage duringwhich the rain intensity at a given station exceeds I mm/hr, is determined by

p (I ) =M087.6

[0.03be −0.03l + 0.2(1 − b ) (e −0.258l + 1.86e −1.63l )] (5.2)

where b = MG /M0 ; M0 and MG are the mean annual total rainfall and the meanrainfall per storm, respectively, in millimeters.

Heavy rains usually are 20% and moderate rains are 40%–60% of all raintypes. Taking into consideration that the mean rain duration usually is not lessthan 2 hours, the rain mean intensity over the ocean will be equal to 0.5–5.0 mm/hras a function of region.

The probability of rain at 100 mm/hr for seaside regions does not exceed5 ? 10−4 and the probability of rain at 10 mm/hr is 10−2; therefore, rain of moderate

5.1 Structure of Meteorological Formations 229

intensity (5 mm/hr and less) makes the main contribution to total annual precipita-tion.

One can find the rain distributions of certain intensity for different regions inmany papers [2, 4, 5]. The mean number of days with precipitations equivalentto rain intensities more than 0.1 mm/hr and the total annual precipitation foroceanic regions are shown in Table 5.2.

Note that even for a large number of rainy days, the mean rain probabilityusually does not exceed 10% because of the small rain duration. Considering this,the requirement on maritime radar to detect the objects in rain with intensity of1–5 mm/hr is justified. As a practical matter, maritime radar must be designed forrather extreme conditions, as one collision per 10 years is still too many, and mostships must operate in heavy snowfall as well as in rain.

Hail is formed, mainly in summer, in powerful convective clouds. The largesizes of cloud drops and high liquid water content in clouds assist in formation ofsolid ice layers on the ice particles falling through a supercooling part of the cloud.

Usually, the hail particles have the form of ice balls, but hemispherical, cone,and lentil-like shapes are also found. The particle sizes do not exceed some millime-ters but cases of hail with diameter up to 10–12 cm have been observed. The haildensity varies from 0.5 g/cm3 to 0.9 g/cm3, and density of small hail and ice grainsis approximately 0.3 g/cm3.

The particle size and precipitation intensity for hail formation and hail fallingchange as determined by rising airflow features. For continuous rising flow witha velocity of about 5 m/s, the hail diameter is constant some time after hail beginsto fall, and then a quick decrease is observed. If rising airflow is absent, the haildiameter quickly decreases in the first minutes of falling.

The hail surface is heated and melted for hail falling into air with positivetemperature. Formed water spreads around the hail surface, either as a thin waterfilm or as a water and ice mixture. We must recall that hail can form for internalcloud temperatures less than −(10 to 12)°C, and hail surface temperature for melting

Table 5.2 Precipitation Distributions for World Ocean Regions

Region Number of Days with Precipitation Total Annual PrecipitationNorway Sea 150 75Barents Sea 150 100Newfoundland 150 75North Sea 200 —Equatorial part of Atlantic Ocean 100–200 —Indian Ocean 100–200 —Equatorial part of Pacific Ocean 100–150 100Philippines 200 430Arabian Sea 50 —Antarctic region 250–300 —Source: [2].

230 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

is 0°C. The water film thickness on most hail surfaces at landing is fractions of amillimeter.

Usually, hail falls in narrow zones with width from fractions of a kilometerto 10–15 km and length from units to tens of kilometers. The period of hail fallingis small (typically 5–10 minutes). The annual number of days with hail is 2–3 daysin plain regions.

Cumulonimbus clouds with thunderstorm activity are very significant radarclutter sources. The thunderstorm centers depend on physical and geographicalconditions, weather, and season. Their maximal number is observed in June–July.The mean number of days with thunderstorm for some towns in the former USSRis shown in Table 5.3 [6].

Radar pulses reflect from the lightning channel boundaries in thunderstormsbecause of the large air refractivity factor gradients arising through intensive airheating in the channel and the high concentration of gas ions and free electronsin the discharge channel.

A cloud is the visual accumulation of condensation or sublimation productsof evaporated water at some height in the free atmosphere. In meteorology, sublima-tion is the transference process of evaporated water steam to its solid phase, skippingthe liquid phase.

Clouds are characterized by great variety of form and physical structure. Thebases of their classification are the forming conditions and morphological sign (i.e.,the outer shape of the cloud). For forming conditions, all clouds are divided intothree classes:

• Cumuliform clouds are strongly developed in vertical planes and compara-tively small horizontally sized ones. They are formed as a result of air-intensive rising (convectional movement).

• Undulating clouds are the great, horizontal layers that have the shape offleecy clouds, rollers, or banks.

• Stratus clouds are the layers in the shape of compact cover; their horizontalsizes exceed the vertical ones by some hundred times. They form as a resultof air that slows smooth rising movement.

For the cloud height, the clouds are divided into four classes: upper, middle,and lower tiers, and the cloud family of vertical development. The clouds of the

Table 5.3 Mean Number of Thunderstorm Days in June (Numerator) and July (Denominator)

Town Median Value Maximal ValueMoscow 6/7 13/16Kiev 7/6 14/16Odessa 7/5 15/14Simpheropol 6/6 14/16Source: [6].

5.1 Structure of Meteorological Formations 231

upper tier are disposed at heights more than 6,000m; the clouds of the middle tierare at heights 2,000–6,000m, and the clouds of the lower tier are at heights lessthan 2,000m. The vertical-development cloud foundations are placed at heights oflower tier clouds, and the tops are at the height of middle- or high-tier clouds.

For the external shape, all clouds are divided into 10 forms (or families) havingthe following names and abbreviations: cirrus (Ci), cirrocumulus (Cc), cirrostratus(Cs), altocumulus (Ac), altostratus (As), nimbostratus (Ns), stratocumulus (Sc),stratus (St), cumulus (Cu), and cumulonimbus (Cb).

For content, the clouds are divided into three groups: water (liquid-drop),consisting of water drops and supercooled drops at negative temperatures; freezing(crystal), consisting of ice crystals; and mixed, consisting of a mixture of supercooledwater drops and ice crystal.

The primary particles for cloud formation are, as a rule, liquid drops. Icecrystals form in a cloud if the cloud’s high part has rather lower temperature.Usually the cloud crystallization begins near the isotherm of −10°C and then canpropagate to the entire supercooled part of cloud.

Water clouds are found most often in summer, and ice clouds are found mostoften in winter. The clouds of mixed structure are not clearly seasonal.

The cloud vertical extent, generally, can reach 10,000m, and the horizontalextent can be up to 1,000 km for stratus clouds and only 10 km for cumulusclouds.

The water drop size distribution depends on cloud height; the drop mean sizegrows with altitude. The drop size distribution can be represented as

f (a ) =1

G(m + 1)mm −1 ?

am

rm +1 e −m (a /r ) (5.3)

where m is the half-width parameter and a is the drop radius.The microphysical cloud characteristics and their typical thickness are shown

in Table 5.4. The mean value of maximal water content depends on cloud typesand their power, and it varies from 0.1 g/m3 to 0.6 g/m3. The water content ofpowerful cumulus clouds can reach considerably greater values: in the Europeanpart of the former USSR, it can be 1.4–1.55 g/m3; for temperate and tropicallatitudes, it can be up to 4 g/m3 or more; and in the United States, the cumulonimbuscloud water content has been observed up to 20 g/m3 at heights of 5,500–7,500m.

Fog is the aggregate of water drops or ice crystals balanced in air, whichdecreases the visibility to 1 km or less. As a function of visibility range, one candescribe heavy fog corresponding to visibility less than 50m, moderate fogwith visibility from 50m to 500m, and light fog with visibility from 500m to1,000m. The fog water content changes within wide limits from thousandths to1.5–2.0 g/m3. With cooling, the fog water content increases, while the water content

232 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Table 5.4 Average Physical and Microphysical Characteristics of Clouds

Gamma-Distribution

Height of Lower Water Content ParametersType of Cloud Thickness (km) Boundary (km) (g/m3) m rCirrus 3 8.5 0.005 5 2Cirrocumulus 0.3 7 0.005 5 2Cirrostratus 1 7 0.005 5 2Altocumulus 0.5 4 0.1 5 2Altostratus 1 4 0.2 5 2Stratocumulus 0.5 1 0.1 5 2Stratus 0.5 0.4 0.1 5 2Nimbostratus 1 0.5 0.2 5 2Cumulus 2.5 1.1 0.2 6 3Cumulus congestus 3 1 1.2 6 3Cumulonimbus 7 0.7 1.0 6 1Source: [5].

of fog formed by evaporation from a water steam decreases with increasing tempera-ture. Experimental data show that the fog water content and visibility changesignificantly with height only at the upper and lower boundaries. Fogs are otherwiserather uniform in the vertical plane. The horizontal extents of fog can reach somehundred kilometers, with vertical extents up to 1,000m. The average verticalextent in Arctic Regions is 250m. The horizontal sizes of fog areas extend to130–180 km, and 20% of cases cover an entire sea (e.g., the Black Sea).

It is established that the number of drops per 1 cm3 changes from 0.5 to 93in advective fogs, from 50 to 860 for radiative fogs, and from 70 to 500 forevaporation fogs of medium intensity. The drop sizes change from fractions of amicron to some tens of microns, and average radius ranges from 2 microns to 18microns. The average number of days with fogs in world oceans is 50 days peryear. Fogs are often observed along coasts of the North Sea, Baltic Sea, Sea ofOchotsk, and in regions of Florida and California. In the region of Newfoundland,Canada, fogs take place up to 250 days per year, and they are also typical for theentire North Sea. They appear 30%–70% of the year in the Kara Sea, and 40%in Laptevs Sea. The average duration of fog is 6–8 hours, and this extends up totwo days for 1%–3% of all cases.

A special type of sea fog is sea vapor accompanied by strong wind, the durationof which can reach 80 hours. Such fog puffs occupy the vertical layer about 10m,impeding the detection of marine objects.

Sandstorm and dust-storm formations are distinguished by spatial extents of10–500 km2 and time durations of 3 hours to a few days, and by their chemicalcontent, geometrical parameters, and water content in aerosol particles. The useof the theoretical relations for radiowave attenuation in sand and dust cloudsdemands knowledge of particle shapes and sizes and their dielectric constant value

5.2 Atmospheric Attenuation 233

and size distributions. For practically used bands, including Q- and K-bands, onecan use the Rayleigh approximation, in which the particle shape does not affectthe wave attenuation. The estimates show that the Rayleigh approximation can beapplied in frequency bands up to 100 GHz.

Dust storms are characterized by the following particle sizes: a large fractionconsists of particles with diameters of about 0.01 mm, while a smaller fractionhas diameters of 0.001 mm or less [7, 8]. This aerosol density can be very large.There is about 10 mg/m3 of dust in the clear atmosphere, about 120 mg/m3 formoderately dusty atmosphere, and more than 200 g/m3 for a dust storm (i.e.,dustiness is greater than for clear air by as much as 107).

5.2 Atmospheric Attenuation

As it is well known, the received power Pr of a signal scattered by a target dependson both radar system parameters and terrain and environmental parameters. Radarsystem parameters include the transmitted power Pt , the frequency f (or wavelengthl ), and the antenna gain G. The other parameters are the two-way propagationfactor V 4, determined by terrain type, roughness, reflectivity and location withrespect to the radar, and atmospheric attenuation g (decibels per kilometer, oneway). Thus,

Pr =Pt G2l2s t

(4p )3r4 ? V 4 ? exp (−0.46gr ) (5.4)

where r is the range to target and s t is the target RCS.Radar signal attenuation in the atmosphere is determined by ionosphere absorp-

tion as well as molecular absorption in the tropospheric gases (i.e., water vaporand oxygen) and by attenuation in meteorological formations (i.e., rain, fog, clouds,and smog). The absorption factor in the ionosphere decreases quickly with increas-ing radar frequency, and absorption is significant at L-band and longer wavebands. For radar of X-band and shorter wave bands, the ionospheric attenuationis negligible and only tropospheric effects need to be analyzed.

Atmosphere gas attenuation is most significant at the resonance absorptionlines: 22 GHz, 182 GHz, and 340 GHz for water vapor and 60 GHz and120 GHz for oxygen. The most intense absorption takes place at frequencies above60 GHz, where for standard conditions at sea level the attenuation coefficientreaches 10–20 dB/km. Accordingly, radar frequencies for obtaining long rangesare chosen in transmission windows between the resonance absorption lines. Inthe millimeter band, these windows are near 35 GHz and 95 GHz. The averageddata for atmospheric absorption by gases in the frequency bands from 10–95 GHzare shown in Table 5.5 [9].

234 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Table 5.5 Atmosphere Gas Specific Absorption

Total Absorption forWater Vapor with Density (g/m3) Water Vapor Density

Frequency (GHz) 7.5 20.0 Oxygen 7.5 20.010.0 5.3 ? 10−3 7 ? 10−3 8 ? 10−3 0.013 0.01535.0 0.15 0.13 0.05 0.1 0.1895.0 0.2 0.52 0.045 0.25 0.57

Source: [9].

A water vapor density of 7.5 g/m3 is accepted as the standard for continentalareas with moderate climate, and a density of 20.0 g/m3 is normal for lower layersof the troposphere above oceans and seas of equatorial, tropical, and subtropicalzones and for coastal regions in these areas.

The absorption in atmospheric gases can be predicted, and for some cases onecan find seasonal maximum humidity values for total attenuation derivation withadequate accuracy.

Microwave attenuation in precipitation can be predicted only in a probabilisticsense because the duration and intensity of precipitation are the random processes.Estimation techniques have been developed for paths of different ranges, takinginto consideration the spatial and temporal inhomogeneity of precipitation.

Precipitation is the limiting factor for millimeter-band radar systems at rangesof more than several kilometers. The most serious source of attenuation is rain.The attenuation coefficient in decibels per kilometer for homogeneous rain can bedetermined as [10]

g = k ? I n (5.5)

where k , n are parameters depending on wavelength and temperature

k (l ) = a0 + a1l + a2l2; (5.6a)

n (l ) = b0 + b1l + b2l2; (5.6b)

a0 = 2.026, a1 = −3.759 ? 10−1, a2 = 1.949 ? 10−2,

b0 = 4.721 ? 10−1, b1 = 8.084 ? 10−2, b2 = 3.761 ? 10−3.

Based on the Marshall-Palmer raindrop size distribution model, Olsen et al.[11] tabulated a rain attenuation coefficient formula for the frequency band from1 GHz to 1,000 GHz in simpler forms that for temperature of 20°C are given by

5.2 Atmospheric Attenuation 235

g (dB/km) = 50.256I 0.9 for f = 30 GHz

0.412I 0.841 for f = 40 GHz

0.572I 0.792 for f = 50 GHz

(5.7)

In order to estimate the rain attenuation, a moderate rainfall rate of 4 mm/hrwas chosen as a reference because, as seen from [12], the rainfall duration for thisor more intense precipitation in Europe is less than 0.5% of total radar operationtime. In some cases, the radar efficiency estimate has been obtained for rainfallrate of 1 mm/hr, for which the rainfall time was not greater than 1.5% of totaloperation time in Europe (for areas with mean precipitation from 300 mm to 500mm per year).

Attenuation data in decibels per kilometer for MMWs in rains of these intensi-ties are shown in Table 5.6. Important information for rain attenuation influenceon radar operation is data on the degree of rain homogeneity within the radarrange because this determines the required value of radar power. The cell sizes fordifferent rain intensities are shown in Table 5.1. These data permit us to makean assumption about rain homogeneity at all radar ranges for low-RCS targetdetection.

The attenuation in fog depends on the liquid water quantity per unit volume(water content of this formation). The attenuation coefficient is

g f = kf ? Mf (5.8)

where kf is the specific factor of attenuation in dB(m3/gkm) and Mf is the watercontent, g/m3.

The values of kf factor for a temperature of 18°C are shown in Table 5.7.The fog water content can be approximately determined using the data in

Table 5.8 [13].Attenuation by dust storms, as with rain attenuation, is determined by the dust

particle dimension distributions and the dielectric constant of particles. Theoreticaland experimental investigations of dust parameters showed that:

Table 5.6 Attenuation Coefficient (dB/km) in Rain

Frequency (GHz) Rainfall Rate of 1 mm/hr Rainfall Rate of 4 mm/hr10.0 0.02 0.08637.5 0.25 1.095.0 0.6 3.0

140.0 0.7 3.2Source: [12].

236 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Table 5.7 Values of kf in Fog for Temperature of 18°C

Frequency (GHz) kf dB(m3/gkm)10.0 0.0515.0 0.11240.0 0.876

140.0 7.14Source: [4].

Table 5.8 Dependence of Optical Visibility on Fog Water Content

Water Content (g/m3) Optical Visibility (m)2 301 500.5 800.2 2000.1 300

Source: [13].

• The real part of the dielectric constant of dust aerosols does not practicallydepend on the water content of aerosols from 0% to 20–30% in the frequencyband of 3–37 GHz for all soil types.

• The imaginary part of the dielectric constant increases rapidly with increasingwater content up to 0.4 for water content of 4.3%.

• The approximate functions for dimension distributions of dust aerosolsare very different for different examples and can be presented as power,exponential, Gaussian, and lognormal functions. It is difficult to determinethe exact form of this distribution.

Data from experimental investigations of the microwave attenuation in duststorms are extremely limited. One can expect that for regions of Africa, Arabia,and Sudan, there are dry dust storms and the microwave attenuation in them israther small. Greatest attenuation is observed for rains, and this should be takeninto consideration.

5.3 Backscattering Theory

The theory of microwave scattering by precipitation and clouds is developed usingthe assumption on noncoherent volumetric scattering. The mean power of thescattered signal for this case is proportional to the effective backscattering area ofa unit of volume (or normalized volumetric RCS h ):

h = E∞

0

s (l , D ) ? N (D ) dD (5.9)

5.3 Backscattering Theory 237

where s (l , D ) is the RCS of a drop with diameter D, N (D ) is the number ofdrops with diameter from D to D + dD in the unit volume, and l is the wavelength.

The RCS of a single particle of spherical shape is derived from Mie’s formula[14]

s

pD2/4=

1

r2 | ∑∞

n =1(−1)n (2n + 1)(a 2

n − b 2n ) |2

(5.10)

where an and bn are coefficients determined by complex spherical Bessel and Hankelfunctions, r = 2pD /l .

Taking into consideration that the drop diameter is not greater than 0.01 cmfor clouds and fogs without precipitation and changes within limits from 0.01 cmto 0.6 cm with most probable value of 0.1 cm [15] for rain, the expression forRCS of a single drop in the millimeter band can be considerably simplified. Forr = 2pD /l << 1, it is easy to obtain

s (l , D ) ≈p5D6

l4 | m2 − 1

m2 + 1 | 2 (5.11)

where m =e − 1e + 2

; e is the water dielectric constant.

Taking into account the data on drop sizes, we reach the conclusion that (5.6)is applicable to drop RCS of fog, clouds, and light precipitations up to frequenciesof 300 GHz. For heavy rain, when the drop sizes are large, the use of (5.11) atfrequencies above 30 GHz leads to considerable errors, and it is necessary to usethe general formula (5.10).

The second factor necessary to estimate normalized volumetric RCS is the dropsize distribution. Theoretical investigations of precipitation do not permit us topropose a universal analytical formula for drop size distribution, and therefore weuse empirical distributions. The Marshall-Palmer distribution is most applicable

N (D ) = N0 ? exp (−LD ) (5.12)

where N0 = 0.08 cm−4; L = 41.1I −0,21 cm−1; and I is the precipitation intensity inmm/hour.

As the derivation shows, this distribution gives satisfactory results at frequenciesless than 100 GHz, while for higher frequency bands it is necessary to take intoaccount the contribution of small drops.

Often the Laws-Parsons distribution is applied

238 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

N (D ) =6Cn

p10n −9a−nI r −nbDn −4 exp (−a−n10nI −nbDn ) (5.13)

C = 72; a = 1.25; b = 0.199; n = 2.29; r = 0.867.It is not possible to take into account the drop size distribution for cloud

scattering because of the absence of experimental data on change in the distributionwith cloud height.

In radio meteorology, the integral characteristic of clouds and precipitationsis called the reflectivity

Z = ∑i

N (Di )D 6i DDi [mm6/m3] (5.14)

There are numerous data about its value for clouds and precipitation (e.g., in[4]). The normalized RCS is connected with reflectivity by

h =p5

l4 | m2 − 1

m2 + 2 | ? Z (5.15)

and can be used for estimations of different cloud types.Snow microwave backscattering is a more complex process. It is possible to

use for normalized RCS derivation of small snow crystals the relation from [14]

h = 0.11Wm ′kr2l4 Sm2 − 1

m2 + 2D2

(5.16)

where W is the quantity of ice in grams per m3, m ′ is the crystal mass in grams,r is the density, and k is the shape coefficient depending on the crystal shape; itsvalue is near unity. The crystals making the largest contribution require accountingof scattering particle shape.

Multiplier | m2 − 1

m2 + 2 | is about 0.9 for water and 0.2 for ice for microwaves and

the longwave part of the millimeter band, changing slightly with air temperaturechange. In the shortwave part of the millimeter band ( f > 40–60 GHz), it is necessaryto take into consideration the quick decrease of the real part of the water dielectricconstant with increasing frequency that takes place up to frequencies of about100 GHz. This leads to considerable differences in the frequency dependence ofthe normalized RCS from the law h ∼ l−4.

5.4 Experimental Results Review 239

5.4 Experimental Results Review

5.4.1 Precipitation Backscattering

Let us analyze the experimental data for microwave radar backscattering from theprecipitation and clouds [4, 14–17].

For the centimeter band, it is necessary first of all to note the satisfactorycoincidence of experimental data with the theoretical derivations using typical sizedistributions. The rain normalized volumetric RCS dependence on the precipitationintensity at wavelengths of 3.0 cm and 10.0 cm are shown in Figure 5.1. It is seenthat for increasing rain intensity from weak (I = 2.5 mm/hr) to heavy (I = 20mm/hr), the normalized RCS grows by about 20 dB. In [2], it is shown that thenormalized RCS can change by about 7–8 dB as a function of changeability of thesize distribution law N (D ) in rains of three types for I = const and l = const. Thetemperature change from 0° to 40°C does not lead to significant change of normal-ized RCS, and the value varied by less than 20%–25% for I = const.

The normalized RCS dependences on the radar wavelength for different precipi-tation intensity are shown in Figure 5.2. It is seen that rain of 10 mm/hr at 3 cmfor resolution cell volume of about 107 m3 produces clutter of about 1 m2, andthis value is comparable with fighter RCS.

Dry falling snow and rain of equal rate have practically equal backscatteringintensity. So, for snow crystals with weight of 1–2 mg and for rate of I = 10 mm/hr,

Figure 5.1 Volumetric normalized RCS dependence on precipitation intensity. (From: [12].)

240 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Figure 5.2 Volumetric normalized RCS of different precipitation intensity dependence on thewavelength.

the backscattering is comparable with backscattering from the rain of the sameintensity (i.e., at X-band, the normalized RCS of snow and rain do not differ).This is illustrated by Figure 5.1. In fact, the clutter from falling snow is considerablyless than that of rain. This is because the rainfall rate of 1 mm/hr corresponds toshower, and the same rate of snowfall correspondents to heavy snowfall, whichhappens considerably less infrequently. The snow reflectivity values increase whensnow was mixed with rain.

For the normalized RCS of rain in microwave band (1 cm ≤ l ≤ 10 cm), theempirical expression is [1]

h = 7 ? 10−12 f 40 I 1.6 [m−1] (5.17)

where f0 is the frequency in gigahertz.The theoretical and experimental results in millimeter bands show worse

agreement with theory. The greater dependence of normalized RCS on the dropsize distribution than in microwave bands is the reason for this, and this dependencecan greatly change within short temporal intervals.

The dependences of normalized RCS on rain intensity at wavelengths of8.0 mm (crosses) and 4.0 mm (points) and derivative dependences are shown inFigure 5.3(a) [16]. The experimental results are rather exactly described by theexpression

5.4 Experimental Results Review 241

Figure 5.3 Specific RCS of rain via (a) rain intensity and (b) frequency dependences of A and bcoefficients. (After: [12, 16].)

h = A ? I b (5.18)

where A = 0.18 ? 10−4, b = 1.2 at wavelength of 8.0 mm and A = 0.53 ? 10−4,b = 1.06 at wavelength of 4.0 mm.

Results of rain backscattering investigations presented in [17] are less than ourdata and data of Russian authors [2, 4, 16] by 5–10 dB. The possible reason ofthese differences is the different techniques of experimental investigations anddifferent rain distributions in different regions of Earth.

The analysis of experimental data from [2, 4, 16, 17] showed that for the rain-normalized RCS at the millimeter band, one can use (5.18); A and b coefficientsdependences on the frequency are shown in Figure 5.3(b).

For precipitation clutter within a radar resolution cell, one can use the relation

s ≈ hct02

u0w0 r2 (5.19)

where t0 is the radiation pulse duration; r is the range; and u0 , w0 are the antennapattern half-power widths in azimuthal and elevation planes. The results of rainclutter RCS at wavelengths of 2 mm, 4 mm, 8 mm, and 32 mm for typical radar

242 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

parameters of t0 = 1 ms, u0 = w0 = 0.01 rad are shown in Figures 5.4 and 5.5.The estimation of rain clutter RCS for other radar parameters is found by additionto the RCS obtained from these figures of the factor

K (dB) = 10Slgt

t0+ lg

uu0

+ lgw

w0D (5.20)

The results of total attenuation derivations are shown in these figures by thedotted lines; the scale for attenuation estimation is on the right coordinate axis.

It is known that the shape of large raindrops differs from spherical. As a result,the RCS of heavy rains is greater for horizontal polarization than for vertical bya factor 1.4.

The use of circular polarization leads to reduced scattering. At X-band, thisdecrease is approximately 18 dB for heavy rains (I = 15 mm/hr), 20–23 dB formoderate rains, and 30–35 dB for light rains (I = 3 mm/hr).

5.4.2 Cloud Backscattering

The microwave normalized RCS for clouds is lower by about 40 dB than forrain. According to [4], for the European part of Russia, the normalized RCS forcumulonimbus clouds with precipitation can be about 10−5 m−1 at the X-band; thedata is shown in Table 5.9.

It is seen from Table 5.9 that the meteorological formations that give the mostpowerful scattered signal are the clouds of the last three types. As a rule, theaveraged duration of backscattering from them is 1–2.5 hours, and it reaches 3–6hours for stratus. The scattering characteristics from hail and rain clouds differvisibly. Typically, the temperature at the upper boundary of hail clouds is, as arule, less than −30°C, and the normalized RCS is h = 5 ? 10−7 cm−1. The typicalprofile of hail clouds h (h ) differs from the analogous profile for heavy rain (seeFigure 5.6). The maximal values of h for hail zones are observed near the isotherm0°C, where the hailstones reach the maximal sizes and supply with water. In thesupercooled part of hail clouds up to heights of some kilometers above the zeroisotherm, h (h ) is about constant, and it then very quickly decreases with height.The value of h near the Earth’s surface is less than at heights near the zero isothermby about two orders on account of hailstone melting.

5.5 The Statistical Characteristics of Scattered Signals

The backscattered signals from meteorological formations are normally distributedbecause they are the superposition of large numbers of independent (or weakly

5.5 The Statistical Characteristics of Scattered Signals 243

Figure 5.4 Rain clutter RCS and attenuation at wavelengths of 2 mm and 4 mm. (From: [12].)

244 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Figure 5.5 Rain clutter RCS and attenuation at wavelengths of 8 mm and 32 mm. (From: [12].)

5.5 The Statistical Characteristics of Scattered Signals 245

Table 5.9 The Normalized RCS of Clouds

h (m−1)Cloud Type 10.0 cm 3.0 cmCirrus 2.5 ? 10−12 3 ? 10−10

Altocumulus 3.7 ? 10−12 4.5 ? 10−10

Altostratus 2 ? 10−12 3 ? 10−10

Stratocumulus 5 ? 10−12 6 ? 10−9

Stratus 2 ? 10−12 3 ? 10−10

Cumulus congestus 1.5 ? 10−10 2 ? 10−8

Stratonimbus 10−9 10−7

Cumulonimbus 7 ? 10−9 8.5 ? 10−7

Cumulonimbus with thunderstorm 5.5 ? 10−8 7 ? 10−6

Source: [4].

Figure 5.6 The normalized RCS of rain and hail on the height. (After: [4].)

correlated) components. This is why the amplitude fluctuation distributions areRayleigh and the RCS distributions are exponential

p (h ) =1

2s2h

? expS−h2

2s2hD (5.21)

where s2h is the normalized RCS fluctuation variance.

The generation of experimental results for signal amplitude fluctuation distribu-tions applicable to backscattering from meteorological formations [4, 14], wascarried out using dipole scatterer ‘‘clouds’’ having identical shapes and sizes and,consequently, equal radar reflectivity. For dipole clouds, the experimental distribu-tions matched the theoretical model satisfactorily. It is shown in [4] that analogous

246 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

results are obtained in precipitation. It was shown in [17] that the amplitudefluctuation distributions in the millimeter band conformed to the lognormal law,the rms does not depend on the rain intensity, and it grows with decreasing resolu-tion cell sizes, especially for high-resolution radar.

Power spectral analysis of the scattered signals was carried out in some papers[4, 14, 18, 19]. Most authors, particularly [18, 19], take the position that thebackscattering spectrum shape is described by a Gaussian shape to the level of−(30–40) dB. The precipitation power spectrum width at −3 dB reaches 140–150Hz at a 20-cm wavelength for wind velocity of 36 km/hr and increases in inverseproportion to the radar wavelength. It is shown in [18] that the spectrum shapeof backscattering intensity from dipole scatterer clouds is approximated rather wellby the Gaussian law and does not practically depend on the wavelength; thespectrum width increases in inverse proportion to the wavelength. That is,

G (F ) = G0 expS−aF 2

f 20D (5.22)

where G0 is the maximum spectral density, f0 is the radar frequency, F is theDoppler frequency, and a is the nondimensional coefficient characterizing thespectrum width (≈ 3 ? 1015 for rain clouds). As an illustration, the power spectraat the S- and X-bands are shown in Figure 5.7 [14].

In [19], it is confirmed that for small elevation angles (less than 6° ) at theX-band, the precipitation backscattering spectrum is mainly determined by the

Figure 5.7 Rain backscattering power spectra. (From: [14]. 1951 McGraw-Hill, Inc.)

5.5 The Statistical Characteristics of Scattered Signals 247

turbulence and the radial components of wind velocity having a Gaussian powerspectrum. Therefore it has, as a rule, a Gaussian shape in its intensive part, downto −(15 dB or 20 dB), and then falls down according to G (F ) ∼ F −6. For rain ofmoderate intensity, the spectral width in this band is 20–35 Hz at −3 dB. Thespectral shape does not change for wet snow; permitting us to repudiate the conceptof drop vibration for explanation of long tails in the spectra (in rains of moderateintensity, the frequencies are 80–100 Hz, and they are less by about one order forsnow). Authors explored the significant variations of the spectrum shape fromGaussian at levels below −40 dB, explaining this by drop fall-velocity variations;the variations at higher levels were explained by drop motion in the turbulent air.

The measurements of [19, 20] at centimeter and decimeter bands (wavelengthsof 4 cm and 35 cm) showed that the intensity (incoherent) power spectra of scatteredsignals from most meteorological formations are also rather well described by theGaussian curve (5.16). The analogous conclusion about the precipitation intensitypower spectral shape is made in [21] based on measurements carried out in themillimeter band of radiowaves. In some cases, bimodal spectra are observed, usuallyexplained by particles with different laws of fall velocities (e.g., drop-snowflakeor drop-hailstone).

The general data about the spectrum width of signal intensity fluctuations fordifferent meteorological objects obtained in [18] are shown in Table 5.10. Theminimal, mean, and maximal values of spectral width DF are given for objectsobserved at an elevation angle of 30°.

The spectral width is inversely proportional to wavelength l ; therefore, theexperimental data are presented in scale of (l /2) ? DF. This permits us to determinethe spectral width at any wavelength. The spectral width distribution for differentmeteorological objects is shown in Figure 5.8 [18].

These data show that the intensity fluctuations spectral width for thunderstormscan exceed the data for mean spectra obtained for the same days and heightsfor dipole scatterer clouds. There is no monotonous dependence between the spec-tral width and precipitation intensity. So, the spectra in rains with intensity of10–15 mm/hr are narrower than the spectra in rains with intensity of 2–4 mm/hr.

Table 5.10 Intensity Spectral Width for Some Meteorological Formations

Spectrum Width (l /2) ? DF (cm-Hz)Type of Formation Minimal Mean MaximalRain 30 45–60 110Zero isotherm and wet snow 25 40 50Clouds 20 35 65Clouds 30 45 75Cumulus clouds with positive temperature 35 75 125Thunderstorms and heavy rains 60 125 400Source: [18].

248 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Figure 5.8 Spectrum width histograms. (After: [18].)

In powerful cumulus clouds, the spectrum of fluctuations is considerably widerthan in clouds of other types.

As was shown in [19, 20], the intensive part of backscattering intensity spectrafrom clouds and precipitation in coherent radar can be approximated by a Gaussianfunction

5.5 The Statistical Characteristics of Scattered Signals 249

G (F ) = G0 expF−a(F − F0)2

f 20

G (5.23)

where F0 is the Doppler frequency shift that determined the wind drift; the rest ofthe parameters coincide with those in (5.16). For small elevation angles, the valueof F0 is determined as

F0 =2UH

lcos u

where UH is the horizontal component of wind velocity, and u is the angle betweenthe wind and the radar antenna pattern axis. The Doppler shift varies over widelimits.

In Table 5.11 some characteristics of clouds with precipitation are shown,including the Doppler frequency shift F0 and the parameter a.

The empirical formula from [22] for rain clouds determines the spectral widthat −6 dB as

DF =435 − 940

l, Hz (5.24)

This gives results coinciding with experimental data of other authors.The results of investigations carried out by the Institute for Radiophysics and

Electronics of NAS of the Ukraine [12] show that the backscattering spectral shapefrom cumulonimbus clouds and from rain changes considerably with elevationangle and time. There are both unimodal and bimodal spectra. The variations ofspectral density permit us to approximate the spectral shape as a Gaussian curve(5.22) modified by a power function, for which G (F ) ∼ F −6. The maximal spectralwidth is observed at the lower cloud edge, where the turbulent diffusion is maximal;the width is decreased higher in the cloud.

Table 5.11 Spectral Parameters of Cloud Backscattering

Spectral Width (Hz) at LevelType of Spectrum l (cm) 10 dB 20 dB 30 dB F0 (Hz) aUnimodal 35 75 130 150 45 4.6 ? 1014

″ 35 78 100 140 40 1.9 ? 1014

″ 35 50 100 130 100 2.5 ? 1014

″ 4 55 100 250 240 1.8 ? 1014

″ 4 400 500 — 100 5.6 ? 1014

Bimodal 4 150 400 500 260 —″ 4 400 600 800 50 —Source: [12].

250 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

The spectral shape for nonrain clouds and fogs of small intensity is determinedby atmospheric turbulence and is very close to G (F ) ∼ F −5/3.

Thus, the conclusions can be drawn that the most applicable model for adescription of backscattering signal fluctuations is the Gaussian distribution; theamplitude fluctuations are described rather well by a Rayleigh distribution; andfor estimation of levels in the tails level, the lognormal distribution can be used.

The power spectral shape of backscattering signals is described by the Gaussiancurve (5.22) and for spectrum description at levels less than −40 dB, it is possibleto use a power function.

5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo)

5.6.1 Point Reflections

The origin of radar point reflections in the atmosphere can be reflections from so-called dot angels (the term angel is used in Russian literature) (i.e., closed areaswith large refractivity index gradients and with higher humidity and temperatureat their boundaries) that preserve these characteristics for long periods. The stabilityof these formations is determined by annular vortexes that are the air circulationinside angel, with velocity up to 7 m/s. The angels exist as the bubbles and streamsforming in a layer with unstable stratification; they are formed near the sea orland surface and then detach from it and rise. Stratification is an air temperaturevertical distribution determining the equilibrium conditions in the atmosphere thatreflects or prevents vertical movement of the air. For unstable stratification, thetemperature decreases with height and prevents atmospheric convection.

The mixed model of angel forming has been recently developed [23], in whichan angel is a rising stream with a bubble cap at the top; it is this bubble, mainly,that generates the scattered signal. Inside the unstable atmospheric layer, the angelarises with a small rising velocity of about 0.8 m/s. The vertical velocity decreasesupon reaching a stable layer, and it fluctuates in height with an average period of10–20 minutes.

Temperature and humidity measurements show that their diameters are, mainly,30–80m and can reach 300–500m.

For estimation of the total number of angel-echoes, it was proposed in [24]for marine radar and for moderate latitudes to use

NS ≈ 5(C − 160)uH uV (5.25)

where uH , uV are the antenna pattern width in horizontal and vertical planes,respectively, in degrees;

C =Pt G2l2h

(4p )3Pminq2Lp(5.26)

5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo) 251

where Pt is the transmitter power, G is the antenna gain, h is the microwavetransmission line efficiency factor, Lp is the power loss factor for processing inradar microwave sets, Pmin is the receiver threshold power, and q2 is the signal-to-noise ratio for target detection with requisite detection and false alarm probabilities.

It is seen that the number of angel-echoes increases with increasing radar energywhere angels with smaller RCS will be observed. Their mean concentration overland is 750 km−3 in layers of 1–3 km. The concentration, as a rule, decreases withincreasing height.

Seasonal and diurnal variation is observed for the appearance of these reflectors.In the former USSR territory, angels are detected from May to September withmaximal concentration in June–July [4]. The probability of their appearance growswith increasing air temperature and humidity. The diurnal variation of such reflec-tions is characterized by maxima of radar backscattering concentration, intensity,and height at 13–14 hours. As an illustration, in Tables 5.12 and 5.13, the dataobtained from [24] on angel concentration in height are shown as a function oftemperature and wind velocity.

It is seen that the concentration increases and the lower boundary of angellayer decreases with increasing temperature; the concentration and lower boundaryincrease with overland wind also increasing. Over sea, the angel-echo number isgreater at night in comparison with day.

Two models of angel backscattering are known: volumetric scattering fromturbulence and specular reflection from angel surfaces. For the volumetric scatteringmodel from air turbulence, it is necessary to keep in mind that the maximalprobable normalized volumetric RCS of such turbulence is 10−10 cm−1 for clearsky. Consequently, for angel diameters of 30–80m, their RCS does not exceedto 3 ? 10−7–6 ? 10−6 m2, which is lower than experimentally observed values.

For the specular backscattering mechanism, the angel RCS—more exactly, thebright point at its surface—depends on the surface curvature and the reflectioncoefficient; an approximation using the geometrical optics model giving

Table 5.12 Angel Concentration and Lower Height Dependence on Air Temperature

Temperature °C 12–14 15–17 18–20 21–23 24–26Mean concentration in layer of 150m 2.6 6.2 12.4 14.1 10.3Mean lower boundary of angel layer (m) 325 230 200 180 150Observation number 33 155 201 115 20Source: [24].

Table 5.13 Angel Concentration and Lower Height Dependence on Wind Velocity

Wind velocity (m/s) 0–5.5 7.5–13 15–20 22–28 30–35Mean concentration in layer of 150m 24.7 36.3 51.9 43.7 45.5Mean lower boundary of angel layer (m) 265 283 318 417 375Observation number 109 96 68 39 5Source: [24].

252 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

s = pr2 |Rx Ry | (5.27)

where r2 is the reflection coefficient and Rx , Ry are the main radii of curvatureat the point of specular reflection.

The point reflections from angels are explained by the reflections from semi-spherical thin layer zones with downwards-directed concave surfaces. The refractiveindex drop amounts to a few N-units and occurs in a layer with depth of aboutl /4. The reflection coefficient depends on the refractive index variance gradient. Forintermittent variance of refractive index at the boundary, the reflection coefficient is

r2 = Sn1 − n0n1 + n0

D2 =DN 2 ? 10−12

4(5.28)

where n1 , n0 are the refractive indexes on both sides of the angel boundary; andDN = (n1 − n0) ? 106 is the refractive index gradient.

For angel diameters of 30–80m and DN = 10 (such value can take place in theatmosphere), the derivation for the specular reflection model gives an RCS of order2 ? 10−8–1.6 ?10−7 m2. Measurements over the sea at the X-band [25, 26] showthat the real values of angel RCS are greater by 5–6 orders than the derivationsfrom (5.27), indicating the necessity for model elaboration.

For moderate latitudes and at wavelength of 5 cm for horizontal polarization,the RCS of the angels lay within limits of 10−6–10−2 m2, as illustrated by Figure5.9.

Measurements show the angel RCS independence on wavelength at S- andC-bands. For the X-band and millimeter bands, one can expect the RCS to decrease.

Figure 5.9 Angel RCS cumulative distribution. (After: [25].)

5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo) 253

The observations of angels carried out by the author at the S-band permittedus to find angels with maximal RCS up to 1 m2. It was noted that visible varianceof the RCS does not take place during an observation time of about one second.The reflected signals slowly fluctuate, sometimes disappearing in noise for 10–30seconds and rising again. The signal from angel at the S-band (on an A-type display)is shown in Figure 5.10(a); in Figure 5.10(b), the amplitude and Doppler frequencyvariance for signals from angels are shown at the C-band for a wind velocity of2.9 m/s.

The RCS distributions of the angel were approximated by a lognormal lawwith mean value of −42.5 dB/m2 and rms value of 5 dB.

Angel backscattering is distinguished by its small amplitude variance and practi-cally fixed position of its signal spectral line—the radial velocity variance did notexceed 1.5 m/s for 1 minute (i.e., it is coherent). In Table 5.14, the spectral widthand the correlation intervals are shown, as obtained at the X-band for incoherentradar operation. In Figure 5.11, the power spectra at the S-band [Figure 5.11(a)]and C-band [Figure 5.11(b)] are shown [12].

It is seen from these data that angels have the narrowest amplitude and powerspectra of all the meteorological formations considered here. The dependence of

Figure 5.10 (a) Angel backscattering at the S band (A-type display) and (b) temporal dependencesof amplitude and Doppler frequency of scattered signal.

254 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Table 5.14 The Correlation Intervals and Spectrum Width for Angel at X-Band

Correlation Interval (ms) Spectrum Width (Hz)Level

Object 0.5 0.1 0.5 0.1Angel:Clear sky 20 45 6 22Cloudy sky 12 26 12 41Source: [26].

Figure 5.11 Power spectra of angel backscattering at (a) the S-band and (b) the C-band.

spectral width on radar wavelength has the form DF ∼ l−1 (i.e., spectral broadeningis inversely proportional to the wavelength).

5.6.2 Backscattering from the Turbulent Atmosphere

The majority of works from the theory of radar signal scattering in a turbulentatmosphere is based on the works of V. I. Tatarsky [27]. According to these, zonesof refractive index microscale turbulent pulsations can be the reason for radarreflections in a clear sky.

It is known that only spectral components of turbulence forming spatial gridswith a size of l = l /2 can take part in generating backscattering. The normalizedvolumetric RCS can be determined as

h =p2

k4 ? Fn (k ) (5.29)

where k = 4p /l is the wavenumber, l is the wavelength, and Fn (k ) is the three-dimensional spectrum of refractive index fluctuations.

5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo) 255

According to Kolmogorov-Obukhov theory, for the inertial interval of turbu-lence limited by the outer L0 and inside l0 scales, the spectrum of pulsations canbe presented in form

Fn (k ) = 0.033C 2n ? k−1/3 ; k0 ≤ k ≤ km (5.30)

where C 2n is the value of refractive index fluctuation intensities and k0 = 2p /L0 ;

km = 2p /l0 .The outer scale of turbulence inertial interval L0 is approximately 10m [28]

and the value of l0 in an overland atmosphere layer is some millimeters or unitsof centimeters at a height of 10,000m.

For C 2n , the usual expression is [27]

C 2n = a2L4/3

0 Sdndh D

2

(5.31)

where a2 is the nondimensional parameter and dn /dh is the vertical gradient ofrefractive index.

The outer scale of the inertial interval can be determined as [28]

L0 = S eb D

1/2

(5.32)

where b is the vertical gradient of the average wind; and e is the velocity of turbulentenergy dissipation.

Then the expression (5.29) for the normalized RCS can be presented as

h = 0.38C 2n l−1/3 = 0.38a2e2/3b−2 Sdn

dh D2

l−1/3 (5.33)

It is seen from this expression that the radar reflections from atmosphericturbulence weakly increase with decreased wavelength. The clear air normalizedRCS variances as a function of wavelength, as obtained by the other authors andassembled (the points in Figure 5.12) are shown in Figure 5.12(a).

The dependence h (l ), obtained in a very dense layer of insects with concentra-tion of about 5 ? 106 m−3, is shown by the solid line. The values of C 2

n are shownon the right ordinate axis and are 10−15–10−14 cm−2/3, but for some rare cases theycan achieve values up to 10−12–10−11 cm−2/3. These data are obtained at greatheights above 300–700m.

256 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Figure 5.12 (a) The normalized RCS of turbulent atmosphere and (b) histogram of spectral widthdistribution. (From: [12].)

At low altitudes over the surface, one can expect considerable increase of thenormalized RCS for turbulent formation in the atmosphere for two reasons.First, the turbulent energy dissipation velocity quickly grows with decreasingheight. As seen from [29], for height decreasing from 100m to 1m, the turbulentenergy dissipation rate increases by more than two orders (from 6–8 cm2s−3 to2 ? 103 cm2s−3).

Second, the vertical gradients of the refractive index for overland layers, as arule, exceed the gradients in the free atmosphere not less than one order [30].

The backscattering power spectra from atmospheric turbulence are rather wideand are represented by G (F ) ∼ F −5/3 as was verified experimentally, particularlyin [26]. A histogram for spectra of different width of scattered signals in theturbulent atmosphere is shown in Figure 5.12(b).

In conclusion, we would like to note that the radiowave backscattering in thefree atmosphere turbulent formations forms the minimal level of radar clutter forhigh-energy radar systems.

References

[1] Edgar, A. K., E. J. Dodsworth, and W. P. Warden, ‘‘The Design of a Modern SurveillanceRadar,’’ Int. Conf. Radar: Present and Future, London, 1973, pp. 8–13.

[2] Krasuk, N. P., V. L. Koblov, and V.N. Krasuk, Influence of Troposphere and Surface onRadar Operation, Moscow, Russia: Radio and Svyaz, 1988 (in Russian).

References 257

[3] Ippolitto, L. J., ‘‘Atmospheric Propagation Condition Influence at Space CommunicationSystems,’’ Proc. IEEE, Vol. 69, No. 6, 1981, pp. 29–58 (in Russian).

[4] Stepanenko, V. D., Radar in Meteorology, Leningrad: Gidrometeoizdat, 1983 (in Russian).[5] Sokolov, A.V., and E.V. Sukhonin, ‘‘Millimeter Wave Attenuation in Atmosphere,’’ in

Science and Technics Results: ser. Radiotechnics, Moscow, VINITI, Vol. 20, 1980,p. 107–202 (in Russian).

[6] Stepanenko, V. G., and S. M. Galperin, Radiotechnical Methods of Hail Investigations,Leningrad: Gidrometeoizdat, 1983 (in Russian).

[7] Semenov, A. A., and T. I. Arsenyan, Microwave Attenuation in Sandy-Dusted Atmosphere,Preprint No. 4 (505), Inst. of Radiotechnology and Electronics, Academy of Science USSR,Moscow, 1989, p. 33 (in Russian).

[8] Arsenyan, T. I., and A. A. Semenov, ‘‘Attenuation of Microwaves in Sandy-Dusted Aerosol,Zarubeznaya Radioelktronika, No. 1, 1995, pp. 16–26 (in Russian).

[9] Kulemin, G. P., ‘‘Influence of Propagation Effects on Millimeter Wave Radar Operation,’’Proc. SPIE Radar Sensor Technology IV, Vol. 3704, April 1999, pp. 170–178.

[10] Malinkin, V. G., ‘‘Engineering Formula for MMW Attenuation in Precipitations,’’ III All-Union Symp. On Physics and Techn. MMW and subMMW, Gorky, September 1980,Thesis Reports, Moscow, Nauka, 1980 (in Russian).

[11] Olsen, R. L., D. V. Rogers, and D. B. Hodge, ‘‘The aRb Relation in the Calculation ofRain Attenuation,’’ IEEE Trans. Ant. Propag., Vol. AP-26, 1978, pp. 318–329.

[12] Kulemin, G. P., Backscattering of Microwaves and Millimeter Waves by Precipitationsand Other Atmospheric Formations, Preprint No. 287, Inst. Radiophysics and Electr.,AS Ukr.SSR, Kharkov, 1985, p. 34 (in Russian).

[13] Kulemin, G. P., and V. B. Razskazovsky, Millimeter Wave Scattering by Earth’s Surfacefor Small Grazing Angles, Kiev: Naukova Dumka, 1987 (in Russian).

[14] Kerr, D. E., Propagation of Short Radio Waves, Massachusetts Institute of Technology,Radiation Laboratory Series, Vol. 13, New York: McGraw-Hill, 1951.

[15] Borovikov, A. M., ‘‘Some Totals of Radar Observations for Powerful CumulonimbusClouds,’’ Trans. Central Aerological Observatory, Vol. 7, 1964, pp. 68–73 (in Russian).

[16] Vakser, I. X., ‘‘Rain Radar Reflection Measurements at Wavelengths of 4.1 and 8.15mm,’’ X All-Union Conf. on Radiowave Propag., Irkutsk, June 1972, Report Thesis,Part IV, Moscow, Nauka, 1972, pp. 76–79 (in Russian).

[17] Currie, N. C., F. B. Dyer, and R. D. Hayes, ‘‘Some Properties of Radar Returns fromRain at 9.375, 35, 70 and 95 GHz,’’ Rec. IEEE Int. Radar Conf., Arlington, VA, 1975,pp. 215–220.

[18] Gorelik, A. G., Y. V. Melnichuk, and A. A. Chernikov, ‘‘Correlation of Radar SignalStatistical Characteristic with Dynamic Processes and Micro-Structure of Objects,’’ Trans.Central Aerological Observatory, Vol. 48, 1963, pp. 38–47 (in Russian).

[19] Kapitanov, V. A., Y. V. Melnichuk, and A. A. Chernikov, ‘‘About Spectrum Shape ofPrecipitation Radar Signals,’’ X All-Union Conf. Radio Wave Propag., Irkutsk, July 1972,Report Thesis, Moscow, Nauka, Part II, 1972, pp. 373–376 (in Russian).

[20] Kivva, F. V., et al., ‘‘Spectral Characteristics of Meteorological Formation Backscattering,’’XII All-Union Conf. Radio Wave Propag., Tomsk, July 1978, Report Thesis, Moscow,Nauka, Part II, 1978, pp. 225–227 (in Russian).

[21] Sharapov, L. I., ‘‘Precipitation Radar Scattering Statistical Characteristics at MillimeterBand of Radiowaves,’’ Trans. 4th All-Union Meeting for Radiometeorology, Moscow,1975, pp. 21–23 (in Russian).

258 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

[22] Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988.[23] Skorer, R., Aerodynamica of Environment, transl. from English, A. Y. Presman, (ed.),

Moscow, Russia: Mir, 1980 (in Russian).[24] Edinger, J. G., and G. C. Holworth, ‘‘Angel Observations with AN/TPQ-6 at Santa

Monica,’’ Proc. 8th Weather Radar Conf., San Francisco, CA, 1960, pp. 132–142.[25] Gatkin, N. G., et al., Clutter Rejection in Typical Set of Signal Detection, Kiev, Russia:

Technics, 1971 (in Russian).[26] Gorelik, A. G., and L. N. Uglova, ‘‘Radar Characteristics of Clear Air Backscattering,’’

Izv. AS USSR, Physics Atmosph. Ocean, Vol. 4, No. 12, 1968, pp. 132–136 (in Russian).[27] Tatarsky, V. I., Wave Propagation in Turbulent Atmosphere, New York: Dover Publica-

tions, 1961.[28] Atlas, D., et al., ‘‘Optimizing the Radar Detection of Clear Air Turbulence,’’ J. Appl.

Meteor., No. 5, 1966, pp. 450–461.[29] Lumley, J. L., and H. A. Panofsky, The Structure of Atmospheric Turbulence, New York:

John Wiley, 1964.[30] Dorfman, N. A., et al., ‘‘Statistical Characteristics of Refractive Index in Over-Sea Layer

of Atmosphere,’’ Izv. AS USSR, Physics Atmosph. Ocean, Vol. 14, No. 5, 1978,pp. 549–553 (in Russian).

C H A P T E R 6

Sea and Land Radar Clutter Modeling

6.1 Land Clutter Modeling

6.1.1 Initial Data

The purpose of this chapter is the development of the mathematical models (algo-rithms and software) of land and sea radar clutter over a wide band of frequenciesfor different types of surface and polarizations.

In the first stage, the main directions of modeling, as well as the model character-istics and principles of its development, are determined. The common principlesfor the formal model design are considered, taking into consideration the influenceof additive clutter on radar operation.

The most serious attention is paid to formal description of the modeling pro-cesses. There is a large volume of experimental data, and, in fact, the physicalmodel of microwave land and sea backscattering is absent due to complexity ofthe backscattering processes and the variety of surface types. This requires thedevelopment of empirical models for land and sea clutter on the basis of experimen-tal results obtained by the author and other authors, and the estimation of theiraccuracy.

There are some principles for the computer modeling of external factor influenceon radar operation. The first approach is the most simple and consists of generationof stochastic number group sequences that are the digital equivalent of clutter. Inthe simplest case, this digital sequence imitates the fluctuating amplitude or thequadrature components of the signal scattered from the land or sea surface. Itsrms value is functionally determined using the normalized RCS of clutter, thetransmitter power, the antenna gain, and the range to the target. The normalizeddistribution of fluctuations and their spectra (or autocorrelation functions) corre-spond to the chosen clutter model. The problem of modeling the external factorinfluence for such an approach reduces to an appropriate choice of the clutterstatistical parameters and the generation of digital sequence with required statisticalproperties. Such a model is the most simple and economical in exploitation. Itpermits modeling the radar operation in real time. It can be augmented by thetarget model.

259

260 Sea and Land Radar Clutter Modeling

The second approach is distinguished from the previous one by the modelingof the physical mechanism of radiowave backscattering from the Earth’s surface.

The first level of this approach is the input data including the data bank,describing the relief and the vegetation cover of the region and the meteorologicalconditions.

The next level forms the scattering surface model for surface facets usinggeometrical optics to form the backscattering signal with its corresponding normal-ized RCS. Here the relief of scattering facets can be represented by smooth approxi-mating functions.

The third level consists of models that consider the influence of radar clutteron radar system operation. Having this type of data, one can model the processesof detection, automatic tracking, and classification of targets. The realization ofsuch a model requires very large operative memory and high speed of computeroperation.

As a compromise, the model can be used in which the formal descriptionincludes propagation conditions and radar response to external factors. For exam-ple, radar clutter modeling can be carried out according to the following scheme:selection of the scattering surface facet, estimation of its RCS with the use ofmicrorelief and vegetation data, and estimation of spectral parameters with theuse of wind velocity data.

The radar detection range is mainly determined by the land, sea, and precipita-tion clutter. As a basis for clutter map creation, digital topographical maps can beapplied that permit, at the first stage, modeling the illuminated and shaded areasof the surface. One has to take into consideration the radar height, the curvatureof the surface, and the vegetation height. For small grazing angles, the incorrectestimation of the heights can lead to significant errors in detection range prediction.The shadowed facets do not practically contribute to the scattered signal, becausethe diffraction field is very weak. As result, one can obtain the illuminated zonesof land (or sea) by using digital topographical maps.

The basis of this solution is the development of digital map database that is apart of geoinformatic system of a country. The main principles of database develop-ment include:

• The hierarchy of the informational base;• The ability for its continuous development and updating;• Quick access to separate data and any data subset;• The ability to automatically reorganize data storage.

The data banks for clutter modeling have to contain:

• Surface relief data;• Climatical data;• Vegetation cover data.

6.1 Land Clutter Modeling 261

The surface topographical data are used jointly with meteorological and climat-ical data.

They are transformed to the radar clutter map at the next stage, using landand sea clutter models.

The following problems are solved in this chapter:

• The development of models for different types of the land and the sea clutterfor different wind velocities (sea wave heights);

• The elaboration of the principles of topographical and vegetation map con-struction for estimation of radar clutter intensity.

The input data for simulation are set at the stage of clutter signal forming.They characterize the basic factors determining the parameters of clutter fromdifferent types of terrain and sea surface. Because clutter properties are determinedby a large number of factors having different physical sense and value, it is expedientto divide the data for simulation into functional groups that integrate the parametersaccording to their semantic contents.

While setting the input data, it is also necessary to define the ranges of theiracceptable values. The value control of the entered parameters enables the softwareproduct reliability to increase, drawing attention to the inapplicability of mathemat-ical models of radar clutter for some frequency bands, observation angles, andother parameters, as well as avoiding mistakes caused by incorrectness of enteredvalues.

Input data for radar clutter simulation according to their functional groups,including acceptable values of assigned parameters, are given next. Radar parame-ters comprise the first group, the initial parameters describing the radar positionare in the second group, and data describing the land and sea surface parametersand weather conditions are in the third group.

Mathematical models of clutter are developed for sea surface and basic typesof land surface. The surface types for which the simulation is possible are givenin Figure 6.1 with their classification.

While solving the task of computer simulation of clutter signal and designingthe appropriate methods and algorithms of simulation, it is important to satisfythe following requirements:

1. To apply simulation algorithms and methods that ensure a high degree ofcorrespondence to clutter statistical characteristics in accordance with themodel data;

2. To provide relative simplicity of simulation algorithm implementation;3. To provide acceptable computation costs in computer simulations of

clutters;4. To minimize the degree of user intrusiveness in the process of simulation;5. To provide the ability to visually and numerically control the parameters

of the generated sequences (samples) of clutter signal.

262 Sea and Land Radar Clutter Modeling

Figure 6.1 Classification of land surface types.

Apart from these requirements, the selection of simulation methods and algo-rithms is essentially determined by the properties of the clutter signals and by thepeculiarities of simulation tasks.

The material presented in this chapter is the result of model development atthe Institute for Radiophysics and Electronics of the National Academy of Scienceof Ukraine. Part of the results was published in [1–4].

6.1.2 Peculiarities of Land Clutter Simulation

Let us consider the basic peculiarities and properties of clutter signals backscatteredfrom land that follow from a mathematical model of backscattering given in Chapter2 and determine the selection of simulation methods and algorithms. For suchpeculiarities, it is first necessary to consider the possibility of representing the clutteras the sum of fluctuating and stable components.

The stable component for each realization in azimuth can be represented as avector with random phase and amplitude dependent on the RCS section of reflectorssteady in time

6.1 Land Clutter Modeling 263

x st (i, Rm , k ) = Ast (Rm ) ? e iwst (Rm ,k ) (6.1)

where Ast (Rm ) and w st (Rm , k ) denote the amplitude and phase of stable compo-nent, respectively. As follows from (6.1), Ast depends only on range to the surfaceelement. The values w st for different ranges Rm and the k th clutter realization arethe independent random values characterized by uniform pdf within the interval[0, 2p ]. This assumption is stipulated by the fact that the stable component ofclutter is formed as the result of coherent superposition of some number of reflectorsstable in time and belonging to the same radar resolution cell. As a result, the vectorcorresponding to the signal x st has random directions with equal probabilities. Theclutter stable component xst does not change for the signal sample sequence inazimuth and, therefore, does not depend on the radiation pulse index i.

The fluctuating component of clutter signal backscattered from land can berepresented as a complex stochastic process having the property of stationarity [5].Let’s consider the basic statistical and power characteristics of the fluctuatingcomponent, describing it as a stochastic process. The real and imaginary parts ofcomplex signal (x fl

Re (i, Rm , k ) and x flIm (i, Rm , k )) are statistically independent

and are characterized by the normal distribution with zero mean and identicalvariances s

2fl proportional to the RCS of reflectors unstable in time. Within the

framework of the mathematical model of land clutter, it is also possible to considerthe sample sequences of clutter corresponding to different ranges Rm statisticallyindependent, as are also the samples of fluctuating components obtained for differ-ent k th realizations.

The correlation function r (t ) describes important power characteristics ofclutter signal and describes the relationship between random samples of fluctuatingcomponents of clutter for fixed resolution element

r (t ) = r (i − j ) = m {x fl (i, Rm , k ) ? x fl ( j , Rm , k )} (6.2)

where m{?} denotes the average for an ensemble of realizations. Another powercharacteristic, strictly connected with correlation function according to Wiener-Khinchin theorem [5], is the power spectrum of clutter signal. Within the frameworkof the mathematical model of reflections from land, the power spectrum of thefluctuating component is characterized by the parameters from (2.64), such as themaximal value of spectral density, the spectrum width, and the power exponent.

Formation of the sample of the clutter stochastic process from a given resolutioncell at range Rm consists of the following stages:

1. Initialization of the pseudorandom number generator by the value corre-sponding to the kth realization the modeled stochastic process;

2. Formation of fluctuating components of clutter signal for the range Rm andthe k th realization:

264 Sea and Land Radar Clutter Modeling

x flRe (i, Rm , k ), x fl

Im (i, Rm , k ) i = 1 . . Naz

3. Formation of the stable component of the clutter signal according to (6.1);4. Obtaining total clutter at the output of phase detector as a result of adding

the fluctuating and stable components.

Simulation of the fluctuating component of clutter consists of the formationof two realizations of the stochastic process with given statistical and powerfulproperties (see Chapter 2) that correspond to signal real and imaginary components.For this purpose, we use the standard method—at the first step, independentpseudorandom values with uniform distribution are generated. Then they are sub-jected to linear and nonlinear transformations with the purpose of obtaining therequired statistical properties [6]. The generation of samples of stochastic processis realized in several stages (see Figure 6.2):

1. The determination of the length of pseudorandom value sequence Nprs (thenumber of stochastic process samples) used for formation of clutter signalwith the given power spectrum;

2. The generation of two sequences of uncorrelated samples x udRe , x ud

Im (whitenoise) with uniform pdf in the interval [0. . .1];

3. The obtaining of uncorrelated samples of stochastic process x ndRe , x nd

Im withnormal (Gaussian) pdf, zero mean, and variance equal to unity s2 = 1 as

a result of the nonlinear transformation of the samples x ndRe , x nd

Im of theinitial process with uniform pdf;

4. The formation of stochastic process samples x knRe , x kn

Im with the given powerspectrum and normal distribution;

5. The truncation of the realization length of clutter signal to Naz samplescorresponding to the required sample size in azimuth);

6. The formation of clutter x flRe , x fl

Im , with given intensity as a result of the

transformation of samples x cnRe , x cn

Im .

For generation of uncorrelated samples with uniform pdf as well as for simula-tion of the clutter stable component, a pseudorandom number generator is used.

The forming of sequence of uncorrelated samples X ndRe , X nd

Im with normal distribu-tion law can be carried out in two ways.

The first simulation method is based on use of the central limit theorem,according to which the sum of a large number of independent random variables hasapproximately Gaussian distribution [5, 6]. Most standard mathematical programscontain built-in generators for random numbers with normal distribution. Anadvantage of this method is its comparative simplicity.

6.1 Land Clutter Modeling 265

Figure 6.2 Algorithms for simulation of the fluctuating component of land clutter.

The drawback is the need to use a certain number (as a rule ≥ 12) of pseudo-random values (usually with uniform distribution) for generation of one valuewith normal distribution. Besides, the maximum/minimum value of pseudorandomvalues obtained by this method is limited to the sum of maximum/minimum valuesof initial pseudorandom values.

For generation of the correlated samples of clutter with a given power spectrum,zero mean and variance s

2nd = 1, we use a linear transformation of the initial

(uncorrelated) sequence of samples X ndRe , X nd

Im (see Figure 6.1, block 4). In thiscase, according to the central limit theorem, the distribution law of samples remains

266 Sea and Land Radar Clutter Modeling

normal [5]. For solving the tasks of practical simulation, two ways are widelyapplied: one based on application of methods of linear filtering and a method ofcanonical decompositions [7].

The method of a linear filtering is based on expression determining the powerspectrum of clutter at the filter output where this filter has the transfer functionK ( jv )

Fz (v ) = |K ( jv ) |2 ? Fj (v ) (6.3)

where |K ( jv ) | is the frequency characteristic of the filter, and Fj (v ) and Fz (v )denote the power spectrums of stochastic process at the input and output of linearfilter, respectively. In cases where the stationary stochastic process with Fj (v ) =const (white noise) and normal distribution enters the filter input, the powerspectrum at its output according to (6.3) has the power spectrum described by thesquare of its frequency characteristic |K ( jv ) |2.

While using this method of linear filtering for generation of discrete samplesequence, the simulation task consists of the creation of a digital filter with thegiven frequency response. To obtain the sequences of correlated samples, digitalfilters with finite impulse response and filters with infinite impulse response canbe used. For their design, the standard methods of digital filter design with givenfrequency response [8] can be used.

One serious drawback of the linear filtering method is the impossibility in somecases to ensure precise conformity of filter frequency response square to the givenpower spectrum; therefore, the given method can be considered an approximation.For example, the power spectrum (2.64) can be formed precisely for values ofspectrum parameter n = 2, 4, 8, . . . using the Batterworth filter of the orderp = 1, 2, 3, . . . , respectively [8]. For other values of spectrum parameter n , theprecise conformity of power spectrum to the desired spectrum cannot be provided.

Taking this drawback into account, for simulation of stochastic process withpower spectrum (2.64), we have selected the method of canonical decompositions.The essence of this method consists of representation of the simulated stochasticprocess j (t ) by canonical decomposition [7]. One drawback of the Karhunen-Loeve transform is the considerable difficulty of solving the equation for stochasticprocesses with a power spectrum that is not rational (and the power spectra ofclutter from land and sea surfaces are in this class). Because of this we used asimpler approach for realization of the canonical decomposition method based ondecomposition of stochastic process into Fourier series [7, 9]. The advantage ofthe method of simulation that uses Fourier-series expansion is the benefit in compu-tation efficiency in contrast to the linear filtering method. A considerable reductionof computational load in execution of the transformation can be obtained usingFFT algorithms [8, 9]. An additional benefit of this method is the simultaneous

6.2 Sea Clutter Modeling 267

formation of two mutually uncorrelated sequences of pseudorandom values

x knRe (n ) and x kn

Im (n ) and the economical use of initial sequences of uncorrelated

samples x ndRe (n ), x nd

Im (n ).This method of clutter correlated sample formation is realized in block 4 of

Figure 6.1. Along with the initial sequence of uncorrelated samples for realizationof this method, it is also necessary to set the size of the sample array Nprs and the

factors h2k that determine the power spectrum form.

The algorithm for calculation of power spectrum factors h2k is realized in blocks

7 and 8, and it consists of two stages. The first stage presumes the obtaining ofpower spectrum samples for a clutter signal sampled in time. Thus, the spectrumof the continuous signal is set by (2.64). The spectrum parameter Go determinesthe spectral density value for zero frequency, and it can be set arbitrarily. Condition-ally, let us consider Go = 1. As follows from the Shannon sampling theorem, ifthere is discretization of the continuous signal, spectrum aliasing takes place for aperiod determined by the sampling rate. As the result, the spectrum of the sampledsignal is distorted by multiple superposition of the continuous signal spectrumdisplaced in the frequency domain. As a rule, these distortions show themselvesby increasing the level of high-frequency spectrum components. In [10], it is shownthat in case of stochastic process discretization, the power spectrum behaves in asimilar manner.

The second stage of power spectrum factor calculation (Figure 6.2, block 8)consists of normalizing the values hk in such a manner that for the correlatedstochastic process (obtained by the decomposition method for Fourier series), one

has to ensure the variance value s2kn = 1.

The final simulation stage is the formation of the clutter signal x flRe (n ),

x flIm (n ) with the required intensity.

6.2 Sea Clutter Modeling

6.2.1 Peculiarities of Sea Clutter Simulation

In contrast to clutter backscattered from land surfaces, sea clutter is characterizedby a number of peculiarities and properties that lead to the necessity of introducingadditional stages and complicating the clutter simulation algorithm. For instance,the backscattering from crests of breaking sea waves and spray results in theappearance of spikes in the scattered signal. Because of this, the distribution lawof clutter differs from Gaussian. As shown in Chapter 4, the most satisfactoryapproximation of clutter distributions at the output of a quadrature detector isthe compound normal distribution

268 Sea and Land Radar Clutter Modeling

wcn (x ) = (1 − g )1

√2ps2cn

expS−x2

2s2cnD + g

1

√2pk2s2cn

expS−x2

2k2s2cnD(6.4)

where g and k2 are the parameters describing the properties of spikes (see Chapter

4), and s2cn characterizes the variance of fluctuations. To obtain samples of a clutter

signal that has a compound normal distribution, the simulation algorithm can usethe method of nonlinear transformation of stochastic processes with Gaussiandistribution [7].

Another distinctive peculiarity of sea clutter is the absence of reflectors thatare steady in time and, consequently, the absence of stable clutter component. Thecalculation of RCS values for the sea is also executed in an essentially differentmanner.

The power spectrum of real and imaginary components of clutter signal fromsea differs from the power spectrum of land clutter by the presence of a meanDoppler frequency caused by the motion of particular surface scatterers. This valuedepends on wind speed, wave direction, and radar operation frequency, and it ischaracterized by parameter F0 . The spectrum shift results in the appearance of asignal fluctuation correlation function for azimuthal samples with Doppler fre-quency of F0 . In this case, the condition of stochastic process stationarity for thesequence of azimuthal sample is not valid as well; this obstacle requires additionallya correction to the parameters of the compound normal distribution (6.4), whilesimulating the clutter.

6.2.2 Algorithm of Sea Clutter Simulation

As in the case of land clutter, the simulation of real and imaginary components ofsea clutter is executed in several stages. Let us consider the algorithm for generatingthe kth realization of clutter signal samples for sea clutter case in the range Rm .As noted earlier, the real and imaginary parts of the clutter signal backscatteredfrom the sea do not contain stable components. This enables us to eliminate fromthe algorithm of simulation the stage of stable component formation, as well asthe stage of summing the stable and fluctuation components. Thus, the algorithmof clutter signal simulation is simplified, and it can be represented by the generalizeddiagram presented in Figure 6.3.

The simulation process includes the following stages:

1. Initialization of the pseudorandom number generator by the value corre-sponding to the kth realization of the modeled stochastic process;

2. Formation of the clutter signal for the k th realization range Rm : xRe (i,Rm , k ), xIm (i, Rm , k ), i = 1 . . . Naz .

6.2 Sea Clutter Modeling 269

Figure 6.3 Algorithm of the simulation of sea clutter.

270 Sea and Land Radar Clutter Modeling

As follows from a comparison of Figures 6.2 and 6.3, the differences betweenthis algorithm and that for the fluctuation component of land clutter consist inthe presence of additional stages of simulation represented in Figure 6.3 by blocks9–12. Common blocks for both diagrams of simulation are the blocks 1–8. There-fore, only the blocks 9–12 will be considered in detail.

The algorithm of sea clutter simulation according to the scheme in Figure 6.3consists of the following stages:

1. Defining the length of the pseudorandom sequence of samples Nprs (thenumber of stochastic process samples) used for formation of the cluttersignal with the power spectrum (4.79);

2. Finding the function y = f (x ) of nonlinear transformation of the stochasticprocess with Gaussian distribution law for given parameters g and k ofcompound normal distribution (6.4);

3. Calculating the factors h2k of the power spectrum (4.79) at F0 = 0 and their

normalization (realized in units 7 and 8);

4. Correcting the factors h2k of the given power spectrum of the stochastic

process according to the nonlinear transformation y = f (x ) and obtaining

power spectrum factors hn2k for initial normal stochastic process;

5. Generating two sequences of samples of the stochastic process x knRe , x kn

Im

with normal pdf and the power spectrum determined by the factors hn2k

(realized in blocks 2–4);

6. Truncating the realization length of the clutter signal x knRe , x kn

Im up to Nazsamples (i.e., to the number of samples corresponding to clutter signal inazimuth);

7. Forming the samples x ksRe , x ks

Im of the stochastic process with the powerspectrum having the central frequency F0 = 0 and compounding the normaldistribution as the result of nonlinear noninertial transformation y = f (x )

of the samples x knRe , x kn

Im ;

8. Transforming the power spectrum for the stochastic process x ksRe , x ks

Im by

Doppler frequency F0 and forming the samples x dsRe , x ds

Im ;9. Forming the clutter signal xRe , xIm with given intensity as the result of

transformation of the samples x dsRe , x ds

Im .

At the initial stage of simulation, the calculation of the number Nprs of samplesof the stochastic process generated at subsequent stages with given power spectrumand normal distribution law is executed in block 1. The value Nprs is determinedaccording to peculiarities of the simulation of correlated stochastic processes withnormal distribution using the Fourier series expansion method. In this case, the

6.2 Sea Clutter Modeling 271

number of samples Nprs is larger or equal to the number Naz of samples of themodeled signal in azimuth. The selection of the value Nprs is realized according tothe algorithm used for simulation of land clutter. The parameter F0 at this stageof simulation is supposed equal to zero.

The next simulation stage is the search for nonlinear transformation y = f (x ),permitting transformation of the Gaussian distribution of the initial stochasticprocess into the compound normal distribution. The modulating transformationresulting in the required change of the distribution law of the stochastic processcan be applied for formation of the stochastic process with power spectrum (4.79)with central frequency F0 ≠ 0 at one stage of simulation. To provide a high accuracyof conformity of the distribution law of the modeled stochastic process to thecompound normal distribution law, the preliminary correction of given parametersof g and k of the compound normal distribution is realized. As a result, theparameters gF , kF of the distribution of the samples of the stochastic process

x dsRe , x ds

Im (see Figure 6.2), ensuring minimum value of goal function, are assumed.As seen, the algorithms of clutter simulation from land cannot be directly used

for clutter formation for the sea clutter case. For example, the method of canonicaldecompositions [7, 9, 11] allows getting the stochastic process with a power spec-trum of practically any kind—in particular, the power spectrum defined by (4.79).At the same time, the method of canonical decompositions is based on lineartransformation of initial (uncorrelated) sample sequences, and by virtue of thecentral limit theorem it does not allow us to form the pseudorandom sample withcompound normal distribution (6.4). For the method of canonical decompositions,the sequence of samples with given correlation properties is formed as a result ofsummation of a large number of pseudorandom numbers [9], and, in the case ofa normal distribution law of the initial sequence samples, the obtained signalsamples also have Gaussian distribution. If the distribution law of the initialsequence differs from normal, by virtue of the central limit theorem the obtainedsamples will have a distribution slightly different from a Gaussian one [9].

To form the sea clutter signal, the methods of simulation of non-Gaussianstationary stochastic processes [7] can be used. In this case, the stochastic processcan be described by either a multidimensional distribution or structurally as atransformation from random variables and determined functions. In the first case,the problem of simulation can be solved as a problem of forming the realizationof a random vector with given multidimensional distribution. For this purpose, wecan apply either the multidimensional method of Neumann or the method basedon use of the conditional probability density. In the second case, the probabilisticprocess is set parametrically. Its simulation consists of forming the realizations ofrandom variables and their subsequent transformation.

The practical use of these two simulation methods is significantly limited byproblems arising in generation of stochastic process realizations with large lengths.

272 Sea and Land Radar Clutter Modeling

Besides, while carrying out the experimental observations, it is rather difficult(and in some cases impossible) to get multidimensional laws of random vectordistributions or to set the stochastic process parametrically.

In the case where the one-dimensional (marginal) probability law and powerspectrum are known, for non-Gaussian stochastic process generation, it is possibleto use a nonlinear transformation method. At the first stage of this method, theformation of stationary stochastic process jx (t ), which has some specific powerspectrum and the normal distribution, is made. Then, the stochastic process samplesare subjected to nonlinear transformation without inertia y = f (x ), which transformsthe initially normal pdf wx (x ) of the process jx (t ) into the given pdf wy (y ) of theobtained process jy (t ). As known, such a transformation always exists [5, 7].Beside pdf transformation of the initial stochastic process, the nonlinear transforma-tion also results in the power spectrum changing. Consequently, the correlationfunction corresponding to it also changes. Let us denote rx (t ) the correlationfunction of initial stochastic process jx (t ). Then, as the result of nonlinear transfor-mation, the stochastic process jy (t ) will have the correlation function ry (t ), dif-fering from rx (t ) and connected with it by

ry = w (rx ) (6.5)

The form of relationship w (rx ) is determined by the nonlinear transformationy = f (x ). The correlation functions of the initial process jx (t ) is selected so thatafter transformation (6.5), we get the stochastic process with the given correlationfunction (power spectrum). For finding rx (t ), it is necessary to execute the inversetransformation

rx = w−1(ry ) (6.6)

where w−1(ry ) is the function inverse to the function w (rx ).Thus, the simulation of the stochastic process using the method of nonlinear

transformations consists of the following stages:

1. Finding the nonlinear transformation y = f (x ) using the given pdf wy (y );2. Obtaining the dependence y = f (x ) for given function ry = w (rx );3. Finding the inverse function w−1(ry ) and correlation function rx (t ) of

initial process jx (t );4. Forming the normal stochastic process jx (t ) with correlation function rx (t );5. Obtaining the stochastic process with required characteristics as the result

of nonlinear transformation of the initial stochastic process jy = f (jx ).

Next, we suppose that the nonlinear transformation y = f (x ) is a monotonicallyincreasing function. Obviously, this requirement is satisfied for the case of transfor-

6.2 Sea Clutter Modeling 273

mation of the normal distribution law into the compound normal law (6.4). Alsoconsider the modeled and initial stochastic processes as having zero mean andvariance s2 = 1. The assumption that the clutter signal mean equals zero followsfrom the mathematical model of sea backscattering. The condition s2 = 1 is acceptedas a matter of convenience for simulations. The required variance of the cluttersignal is set in the final simulation stage as a result of multiplying the sample valuesof the modeled sequence by a derived constant.

For finding the transformation y = f (x ), let us use the condition of equalityof cumulative density functions of random samples of the initial and modeledprocesses,

Wy (y0) = Wx (x0) (6.7)

where y0 = f (x0), Wy (?), and Wx (?) are the cumulative distribution functions ofstochastic processes jx (t ) and jy (t ). As the stochastic process jx (t ) is Gaussian,Wx (?) is determined from the expression

Wx (x ) = 0.5 ? FFXx /√2 C + 1G (6.8)

where F(?) is the error function

F(x ) =2

√p Ex

0

e −z2dz (6.9)

If the stochastic process is preset by pdf wy (y ), (6.7) can be reduced to [5, 7]

wy [ f (x )] ?df (x )

dx= wx (x ) (6.10)

where

wx (x ) =1

√2pe −x2 /2 (6.11)

The dependence y = f (x ) is found by solving (6.7) or (6.10). If it is not possibleto find the function y = f (x ) analytically, the solution can be obtained by computernumerical methods as a table of values of y = f (x ). In this case, an interval ofpossible values of argument x is restricted by the limits for which the probabilityof x exceeding them is negligible.

When the transformation y = f (x ) is found, the relation (6.5) between correla-tion functions of initial jx (t ) and transformed jy (t ) processes can be determined.

274 Sea and Land Radar Clutter Modeling

According to its definition, the correlation function can be derived as an expectationof product jy (t ) ? jy (t + t ) = f [jx (t )] ? f [jx (t + t )]. Then the correlation functionry (t ) of the transformed process is determined by

ry = w (rx ) = E∞

−∞

E∞

−∞

f (x1) ? f (x2) ? wx (x1 , x2 , rx ) dx1 dx2 (6.12)

=1

2p√1 − r2xE∞

−∞

E∞

−∞

f (x1) ? f (x2) ? e

x21 − 2rx x1x2 + x

22

2(1 − r2v ) dx1 dx2

where wx (x1 , x2 , rx ) is the two-dimensional pdf corresponding to the Gaussiandistribution, and x1 and x2 are the values of the initial stochastic process at timeinstants displaced from each other by the value t . Px is the coefficient of correlationbetween x1 and x2 .

Direct use of (6.12) for obtaining the dependence ry = w (rx ) is, as a rule,problematic because the integral cannot always be calculated in a closed form.Besides, if using the computer to solve (6.7) and (6.10), the function y = f (x ) isset as a table; this does not allow us to get the correlation function ry (t ) analyticallyon the basis of (6.12). As shown in [5], the solution of (6.12) can be simply obtainedif one represents the function w (rx ) as power series. To find the coefficients ofthe power series, it is proposed to use the decomposition of the two-dimensionalpdf wx (x1 , x2 , rx ) into series using orthogonal Hermittian polynomials [12].Then, the required dependence can be obtained as

ry = ∑∞

m =0C 2

mrm

xm !

(6.13)

The coefficients Cm can be found as [5]

Cm =1

√2p E∞

−∞

f (x ) ? Hm (x ) ? e −x2 /2 dx (6.14)

where Hm (?) are Hermittian polynomials.Note that because initial and transformed stochastic processes have variance

s2 = rx (0) = ry (0) = 1 and zero mean, the coefficient C0 = 0. Consequently, thefollowing equality is valid

∑∞

m =0

C 2m

m != 1 (6.15)

6.2 Sea Clutter Modeling 275

After getting the power series coefficients Cm , for finding the correlation func-tion rx of the initial stochastic process, it is necessary to solve (6.13) with respectto rx . The inverse function rx = w−1(ry ), as a rule, cannot be found analytically,and it is expedient to get the solution of (6.13) by numerical computer methods[7]. In this case, because we get a numerical solution to this equation, the functionw−1(ry ) is set by a table.

We would like to note the peculiarities of practical use of the nonlinear transfor-mation method for simulation of sea clutter. As it is mentioned in [7], this methodcannot be used for generation of stochastic processes with the correlation functionof definite type because the solution of (6.13) with respect to rx (t ) does not alwaysexist. However, when the correlation function is nonnegative, the solution of thisequation always exists [7]. Let us consider the power spectrum of sea surface clutterdefined by (4.79). Its difference from the spectrum of land clutter (2.64) consistsin the presence of F0 , the center frequency. In this case, the value of power spectraldensity is maximal at center frequency F0 . The correlation function correspondingto the power spectrum (4.79) can be represented as

ry (t ) = ry (t )* ? cos (2p ? F0 ? t ) (6.16)

where ry (t )* is the correlation function corresponding to spectrum (2.64). From(6.16), it follows that the function ry (t ) has a fluctuating character with frequencyF0 , equal to the center frequency of the spectrum. Obviously, no stochastic processwith spectrum (4.79) can be obtained as a result of initial process nonlinear transfor-mation. As shown in [5], the nonlinear transformation results in spectrum wideningand the appearance of the local extreme in frequency being a multiple of the centerfrequency F0 . At the same time, the method of nonlinear transformation can beused for the formation of a stochastic process with power spectrum (2.64), whichcorresponds to spectrum (4.79) with the center frequency F0 = 0. In this connection,the formation of a sea clutter signal with spectrum (4.79) can be done in twostages:

1. Generation by the nonlinear transformation method of a sequence of sto-chastic process samples with a compound normal distribution and powerspectrum (2.64);

2. Transformation of power spectrum (2.64) of stochastic processes by themodulation method [5] to the form (4.79).

If approximating the experimental distributions of clutter signal by the com-pound normal distribution (6.4), the weighting coefficient g does not exceed thevalue 0.1, and the ratio of variances of distribution component k2 is within theinterval 10–20. Obviously, for typical values of parameters g and k obtained forexperimental data, the distribution does not practically depend on time and differs

276 Sea and Land Radar Clutter Modeling

only slightly from the compound normal distribution (6.4). The degree of confor-mity of the sample distribution law of the modulated stochastic process jF (t ) tothe compound normal distribution can be additionally increased as the result of aprecorrection of the pdf parameters of initial stochastic process j0(t ).

The values of parameters g F and kF ensuring minimum difference of the distri-bution law of process jF (t ) from the compound normal law with parameters gand k are determined by criterion

Ew = E∞

−∞

[wcn (x , g , k , s2cn ) − wF (x , g F , kF , s2

cn ,F )]2 dx (6.17)

As unknown values, the parameters g F and kF , Ew are accepted to providethe minimum of goal function (g F , kF ). In Table 6.1, for different g and k of thecompound normal distribution (6.4), the estimated values of total square error Ew(6.17) are presented.

The corrected values g F and kF ensuring goal function minimum are also given.From the presented data, it follows that the deviation of the distribution law ofsamples of the formed stochastic process from a compound normal distribution isincreased with increase of values g and k .

At the same time, for maximum values g = 0.1 and k2 = 20, the generalquadratic errors are rather small (of order 10−6) and the distribution law deviationfrom the given one can be neglected. It is also necessary to mention that thedifference between the corrected distribution parameters g F and kF and the givenones is not great.

6.3 Clutter Map Development

6.3.1 Initial Data for Modeling

As shown in Chapter 2, the land clutter intensity characterized by its normalizedRCS depends on a number of factors. The surface relief and vegetation type exertthe primary influence. The alternation of the different vegetation types causes a

Table 6.1 Corrected Parameters of Compound Normal Distribution and the Values of GoalFunction Corresponding to Them

Typical Parameters of pdf Corrected Parameters of pdf Total Square Errorg k2 gF k2

F Ew0.01 10 0.00705 14.01 4.49 ? 10−9

0.01 20 0.00660 30.06 1.34 ? 10−8

0.1 10 0.07220 13.91 5.62 ? 10−7

0.1 20 0.06768 29.82 1.99 ? 10−6

6.3 Clutter Map Development 277

mixed character of land areas with clutter. As result, the conditions for targetdetection are changed.

Analysis shows that, for most tasks, it is necessary to model surface areas withdimensions not less than 10 km. As a rule, a plain surface of such dimensions hasa height variation greater than 100m and is described by contour intervals of10–20m. These contours can have gaps or be absent. As long as the surface areahas not less than 5–10 contours, it is possible to apply the simple and quick techniqueof mean heights for restoration, characterized by the simplicity of realization andsmall restoration time.

The minimal initial data for modeling are:

• The set of surface contours;• Radar characteristics (e.g., operation frequency, antenna height, and coordi-

nates);• The map of the land surface (e.g., grass, forest, and concrete);• Wind velocity and direction.

As additional data, a map of land surface heights, a map of atmosphericprecipitations, or a soil map can be used.

6.3.2 Software Input and Processing Components

The necessity of a digital relief model for compiling masking maps, in its turn,conditions the need for input, processing, and (perhaps) storing data on heightsand vegetation. The software in question provides for inputting and processingsuch information from both ready electronic maps (the data export function fromthe exchange format of the MapInfo package) and printed maps.

In order to prepare a digital relief model (raster image of a given locality ona given scale, with the brightness of every pixel designating the height of the reliefin it), the information found on topographic maps about the lines of equal heights(isolines) is used. These data are exported into the internal data format from readyelectronic maps or recognized from a scanned image of a topographic map. In thelatter case, the information on equal height lines is highlighted in the image throughstipulated color, brightness, and area; vectored; saved in the internal format; fin-ished manually (removing, for example, breaks and wrongly recognized lines); anddigitized. Data exported from ready electronic maps may also be edited manually(correction of mistakes).

While rendering a digital relief model, three alternative algorithms can be used—the iterative algorithm based on discrete cosine conversion [13], the algorithm basedon the Delone triangulation [14], and the one based on smoothing filters. The firstalgorithm ensures a more exact relief rendering with a more natural view, butit’s unstable towards error-containing incoming data (line breaks and incorrect

278 Sea and Land Radar Clutter Modeling

digitizing). The second algorithm shapes a less exact relief rendition; nevertheless,it is highly stable for erroneous data (able to function correctly even with fragmentsand dotty data). The third algorithm outruns the other two in terms of speed andcan be used for quick approximate digital relief renditions and, as with the secondalgorithm, when processing erroneous data.

For creating maps of vegetation, the information found on topographic mapsabout the area outlines and their types is used. Data can be exported from readyelectronic maps (the exchange format of the MapInfo package) and subsequentlyedited manually (correction of downloaded mistakes).

This software possesses mechanisms for storing data in separate sheets oftopographic maps (equal height lines, vegetation areas, raster image of the sheet)and synthesizing data for a given locality on their basis (lacing and highlightingalgorithms).

Figure 6.4 presents an enlarged structural scheme of the software being dis-cussed depicting in greater detail the modules responsible for input and processingdata on heights and coating maps.

The interface is arranged in a way that allows us at any time to gain access toany of the shown modules so the user does not have to stick to a rigid sequenceof actions when inputting and processing data. This enables us, when setting atask, to easily distribute bits of work between various users specializing, for instance,only in scanning and interlacing topographic map sheets or only in correcting dataon contour lines, and to carry out processing of data on an incomplete package(e.g., only scanning of topographic map fragments and their storage for futureprocessing).

In the following sections, we will get down to a more detailed description ofthe modules requiring attention.

6.3.3 Raster Image Processing Module

In the process of scanning topographic map fragments (as a rule, a scanner will notentirely accommodate map sheets), pieces of images are obtained having nonlineardistortions conditioned by paper folds and paper unevenly resting on the scannerworking surface and distortions of deviation of the vertical axis of the map sheetfrom the vertical axis of the scanned image due to the user’s inaccuracy in placingthe map sheet onto the scanner. These distortions must be removed prior to sheetinterlacing and starting to highlight information on contour lines. If cheaper scannermodels are used, an image brightness correction might also be needed to enhanceits subjective visual quality.

The procedures of interlacing images, highlighting an image fragment into aseparate file, and rotating present no difficulty from the algorithmic point of view;still, the large size of images in processing—dozens and hundreds of megabytes—creates certain technological impediments. As a rule, the image size is far bigger

6.3 Clutter Map Development 279

Figure 6.4 Structural scheme of inputting and processing data on heights and vegetation mapsas well as topographic map raster images.

280 Sea and Land Radar Clutter Modeling

than the size of the random access memory (RAM) available, and downloadingthe whole image into the RAM leads to the operational system creating a virtualmemory on the hard drive, which increases the processing period by hundreds andthousands of times. In order to solve that problem, this software buffers the imageto be processed, so at each moment of time only part of it is found in the RAM.

In most cases, engaging this approach enables us to minimize spending non-scheduled time, though it complicates the application scripts to some extent andinflates the size of the execute files.

The geometrical image correction procedure is nonlinear conversion on imagepixel coordinates aiming at eliminating nonlinear distortions of scanning and distor-tions of the rotation angle. In this procedure, initial and resulting coordinates offour user-designated reference points of the image are employed.

6.3.4 Automatic Highlighting of Contours on the Raster

The procedure of automatic highlighting of contours on the raster consists of arigid sequence of steps, though selection of parameters is needed at every step and,consequently, this step will be repeated several times, which leads to the moduleof automatic highlighting of contours on the raster.

The initial data for the module of automatic highlighting of contours on theraster will be raster full-color (24-bit true color) BMP-format images. As resultingdata, files of passports (this file contains detailed data contents, its sequence, andits size in bytes for each line), and vectors of contour lines are compiled. Aftersubmodule initiation, which is part of the module of automatic highlighting ofcontours on rasters, intermediate data may be both raster images saved in BMPfiles (modules from the linear filter to the removal of large objects with line color)and vector images saved in text files (modules from the vectoring line raster imageto the offshoots removal). The key module here is the one of vectoring line rasterimage, which ensures the transition from a raster image to vector and, thus, unlikemany other modules, cannot be deleted in the process of line highlighting. Anotherkey module is the one of line narrowing, the algorithm of which will be examinedlater. The linear filter and the vector border underline modules, as a rule, arenecessary when working with most scanners and printed topographic maps,although there are cases when they will not be needed (the image quality will meetthe recognition requirements without resorting to them), so they might be dropped.The line fragments, cycles, and offshoots removal modules actually serve the pur-pose of saving subsequent manual work of correcting automatic recognition mis-takes and might also be dropped if those mistakes are few in number.

Now let us move to a more detailed description of submodules of the moduleof automatic highlighting of contours on the raster.

6.3 Clutter Map Development 281

The linear filter module, together with the module of vector underlining ofborders, are designed to eliminate the effect of black line lamination into severallines of various colors, due to which part of them might erroneously be colorclassified as contour lines. The functions of the vector filter are described in [15].It is seen that the line lamination into several lines of various colors has beenneutralized, and now it can easily and fully be color classified.

In the submodule of contour color and brightness highlighting, image pointsnot belonging to contours are sifted off. We can use all sorts of point color interrela-tions R , B, and G as the operator-designated sifting conditions (e.g., B > 0.65R )as well as the summary brightness value of a point R + G + B (e.g., R + G + B <10).

The median filters submodule sifts off separate small-sized and shapeless con-gestions of points mistakenly color selected and left there. Upon the operator’schoice, up to three median filters can be employed simultaneously, with the windowsize adjustable and the sequence statistics selectable. This will filter the image thenumber of times defined one by one. Activating the second and third median filtersis not obligatory, and it is defined by the operator. Practically, good results areobtained, for example, by engaging in series two median filters with the windowsizes 5 and 3, sequence statistics 10 and 4, and filtering repetition in the course offour iterations.

As an alternative to recognizing lines on scanned topographic maps and theirsubsequent manual editing, this software enables data export on contour lines fromready electronic maps. The exchange format of the well-known MapInfo packagehas been chosen as a data format to be exported from. When data is exportedfrom the exchange format of the MapInfo package, it is automatically bound togeocoordinates. When highlighting contour lines on raster and editing them manu-ally, geocoordinates binding must be done manually through a special module ofthis software. The geocoordinates binding file is a text file where all the data isplaced in one line and parted by spaces. The data contains the coordinates indegrees of North latitude and East longitude of the map’s top left-hand corner (thecoordinates will be real numbers) and the number of pixels in one latitude degreeand one longitude degree. The number of pixels per one degree of latitude orlongitude depends on the chosen scanner resolution and on further scale changesof the scanned raster image.

This system of binding data to geocoordinates possesses two essential advan-tages. First, the point coordinates are not directly bound to geocoordinates, sowhen the data about the geographic coordinates is unavailable, no fictitious geo-coordinates for points need to be registered. Second, adding a point to or removingone from the isolines data file will not affect the file of geocoordinates binding,which could not be avoided if, for instance, the coordinates for two corners of thesheet (the top left-hand and the bottom right-hand) were recorded there.

282 Sea and Land Radar Clutter Modeling

Editing data on coating maps is fully identical to editing data on contour linesand is carried out by the same module. The only difference is that instead of theheight value for a contour line, the coating-type value is recorded in the case of acoating map.

6.3.5 Steady Algorithm of Surface Recovery from Contours

Initial data for steady algorithms of surface recovery is the list of contours in avector form. The coordinates of segment tops making contours are set as geographiccoordinates of latitude and longitude. The contours can have gaps, passing orpartially to miss. A scale of repaired surface is arbitrary one. The recovery hasthree levels:

• Recovery of normals (perpendiculars) to contours;• Filling of interspace by mean altitudes;• Smoothing of irregularities and discontinuities.

Recovery of normals to the contour segments is necessary for decreasing anerror of surface recovery. Recovery of altitudes is made as follows:

1. Coordinates of a normal to a section are evaluated.2. Three interceptions of normal with other contours (their altitude) are

searched.3. If not less than three altitudes have different values, surface altitudes between

contours restore them by the two-dimensional spline for four points. Theintervals between contours are filled by reference points, which are takeninto account at the following stages of the surface recovery. The altituderecovery by this method has a great error rate but allows us to restore thesurface with initial data of different quality, in view of neighboring contours.

At the filling, the mean altitude takes into account both the altitude of contoursand the altitude of points obtained earlier.

The filling is made until three points with a different altitude will cover. Afilling depth is the mean value between the first and second covered points. Afterfilling, the surface looks like the domains of filling have a staircase, and the altitudeof the steps is proportional to the quality of the initial contours.

The stepwise surface is completely unacceptable for the calculation of shadedzones and simulation; therefore, it is necessary to receive a regular surface withthe help of filtering. The analysis has shown that the best results for the givenmethod of surface recovery reach the Gaussian pyramid-shaped filter. It is necessaryto set the depth of smoothing manually, depending on the features of particularrelief.

6.3 Clutter Map Development 283

6.3.6 Simulation of the Absolute Reflectivity

For calculation of the absolute value of normalized RCS, three parameters areneeded: the grazing angle, the radar frequency, and the surface type. The generalsimulation algorithm permits us to evaluate the values of the absolute reflectivityfor all facets of the modeled area. Having the values of the absolute reflectivity atany point of the land surface, radar parameters, and weather conditions makesit possible to calculate the fluctuating and steady components of the absolutereflectivity.

To decrease the effect of the recovery relief errors, the grazing angle is calculatedfor a surface segment whose size is M × N pixels, and the radar altitude is set asabsolute altitude at sea level. The size M and N are set depending on the qualityof relief recovery, the model for minimum segment 7 × 7 facets is presented inFigure 6.5. In this figure, the brightness determines the normalized RCS for differentareas of this fragment. The dark areas in the figure correspond to shaded terrains.The increasing of radar height leads, as a rule, to decreasing of shaded areas.

As result of the sea and land clutter mathematical model development, thefollowing results are obtained:

Figure 6.5 Reflectivity for radar height 20m, M = N = 2.

284 Sea and Land Radar Clutter Modeling

• The algorithms are developed, and the land clutter mathematical model iscarried out for a wide variety of surface types with and without the vegetationand for the different wind velocities on the basis of experimental investiga-tions of the clutter statistical characteristics.

• The database has been developed for land clutter signals for real cluttersignals obtained at on-land radars.

• Algorithms have been developed and the sea clutter mathematical modelhas been carried out for different wind velocities and sea states and formotionless radar.

References

[1] Kulemin, G. P., A. A. Kurekin, and E. A. Goroshko, Radar Clutter Modeling, CollectedArticles, Kharkov Military University, Kharkov, Ukraine, Vol. 2, No. 28, 2000,pp. 59–65 (in Russian).

[2] Kulemin, G. P., A. A. Kurekin, and E. A. Goroshko, ‘‘Radar Clutter with Non-GaussianDistribution Modeling,’’ Radiophysics and Electronics, Vol. 7, No. 1, 2002, pp. 56–67(in Russian).

[3] Kulemin, G. P., and E. A. Goroshko, ‘‘Land Clutter Estimation in Airplane Pulsed DopplerRadar,’’ 2nd Int. Conf., CD Trans., Kiev, Ukraine, National Aerospace Academy,October 2000 (in Russian).

[4] Kulemin, G. P., and E. V. Tarnavsky, ‘‘Modeling of Radar Land Clutter Map for SmallGrazing Angles,’’ URSI General Assembly, Amsterdam, August 2002, to be published.

[5] Levin, B. R., Theoretical Basics of Statistical Radio Engineering, Moscow, Russia: SovietRadio, 1969 (in Russian)

[6] Knut, D., Art of Programming for Computers, Moscow, Russia: Mir, 1977 (in Russian).

[7] Bikov, V. V., Digital Modeling in Statistical Radio Engineering, Moscow, Russia: SovietRadio, 1971 (in Russian).

[8] Rabiner, L., and B. Gold, Digital Signal Processing Theory and Applications, Moscow,Russia: Mir, 1978 (in Russian).

[9] Yaroslavsky, L. P., Digital Signal Processing in Optics and Holography: Introduction toDigital Optics, Moscow, Russia: Radio and Communications, 1987 (in Russian).

[10] Gribanov, Y. I., and V. L. Malkov, Spectral Analysis of Stochastic Processes, Moscow,Russia: Energia, 1974 (in Russian).

[11] Ermakov, S. M., and G. A. Mihailov, Statistical Modelling, Moscow, Russia: Science,1982 (in Russian)

[12] Ango, A., Mathematics for Electrical and Radio Engineers, Edition of K. S. Shifrin,Moscow, Russia: Science, 1965 (in Russian).

[13] Ponomarenko, N. N., V. V. Lukin, and A. A. Zelensky, ‘‘The Iterative Procedure ofRendering Digital Relief Model on Isogram Map Using Discrete Cosine Conversion andHistogram Filtering,’’ Aviation and Space Techniques and Technologies, Kharkov, Russia:Kharkov Aviation Institute, 2000 (in Russian).

References 285

[14] Shikin E. V., A. V. Boreskov, and A. A. Zaytsev, The Basics of Computer Graphics,Moscow, Russia: Dialog-MIFI, 1993 (in Russian).

[15] Kurekin, A. A., et al., ‘‘Adaptive Nonlinear Vector Filtering of Multichannel RadarImages,’’ Proc. of SPIE Conference on Multispectral Imaging for Terrestrial ApplicationsII, Vol. 3119, San Diego, CA, July 1997, pp. 25–36.

C H A P T E R 7

Clutter Rejection in MMW Radar

7.1 Influence of Propagation Effects on MMW Radar Operation

7.1.1 Introduction

MMW land-based and maritime radar systems are applied widely for weapon andmissile control, and this chapter will be concerned with short-range, very-low-altitude radar applications such as battlefield radars. This can be explained by thefact that the range, angle, and velocity resolution of MMW radar systems is betterthan for analogous systems in the centimeter band and that the reserve and stabilityto radio countermeasures are higher. The success in solving low-altitude targetdetection and tracking problems is determined, mainly, by propagation effects.Among these are multipath propagation attenuation and attenuation due to precipi-tation (i.e., rain, fog, or snow) that limit the maximum detection range. Theproblems of attenuation and backscattering of MMW in precipitation are consid-ered in detail in Chapter 5. The precipitation influence on land-based radar opera-tion is less important in the microwave band, and it is necessary to take intoconsideration this limiting factor in the MMW band at ranges more than fewkilometers. The essential advantage of MMW-band radars is the small influenceof multipath attenuation in comparison with radars in the centimeter band.

Multipath propagation is the propagation of a wave from one point to anotherby more than one path. For radar, it usually consists of a direct path and one ormore indirect paths by reflection from the land or sea surface or from large manmadestructures. In this situation, there is simultaneous or near-simultaneous receptionof waves that have reached the receiving antenna by direct and reflected paths.Depending on the relative phases and amplitudes of the several simultaneouslyreceived components, the result is a composite electromagnetic field that can benear zero or as much as twice that received by the direct path only. Consequently,multipath propagation can lead to attenuation of the electromagnetic field in com-parison with free space propagation.

There is also a second problem limiting the application of MMW radar systems.This is the clutter from the land or sea surface and volume clutter from suchscatterers as precipitation, the latter increasing in the MMW band and limiting

287

288 Clutter Rejection in MMW Radar

the use of these radar frequencies. There are many papers and books [1–4] in whichthe influence of these propagation effects on radar operation is discussed separately,but the joint estimation of terrain and precipitation clutter and precipitation attenu-ation leads to more accurate determination of radar target detectability and avail-able detection range.

The joint influence of these effects on MMW radar operation, in particular onmaximal attainable detection range and target detectability, is considered in thischapter. The parameters of MMW radar are compared with parameters of analo-gous X-band radar, and the comparison is carried out for two situations: an antennaaperture that is constant with frequency change and an antenna gain that is constantwith frequency change (i.e., a change of antenna aperture area proportional to thesquare of wavelength takes place).

7.1.2 Multipath Attenuation

There are two reasons determining the total attenuation of signal in microwavesand millimeter bands. One of them is the multipath propagation over the Earth’ssurface.

For estimation of the multipath effect, the propagation factor V is used, as arule, as a function of the heights of radar hr and target ht and the rms roughnessheight sh [4]. For propagation over sea and land without vegetation for smallgrazing angles, the propagation factor can be presented in a form

|V | = √1 + r2 − 2r ? cos2pl

d ≈ √1 + r2s − 2r s cos

2pl

d ; (7.1)

sin c ≈ c =hr + ht

r; d ≈

2hr htr

Here r = r0 r s rv where r0 is the Fresnel reflection coefficient, r s is the specularscattering factor, and rv is a vegetation factor, depending on the presence ofvegetation on the land surface. The Fresnel reflection coefficient r0 does approach−1 at low grazing angles. The minus sign in (7.1) implies that the phase angle ofthe reflection coefficient is exactly p , which is true only for very low grazingangles. The vegetation greatly weakens the specular reflection even for microwavefrequencies. So, at the S-band, the vegetation factor for grass with height about10 cm and for grazing angle of 2° equals about 0.8, and for height of grass about50 cm, it did not depends on the grazing angle and equaled 0.1–0.4 [4]. At afrequency of 35 GHz for field with short grass, the vegetation factor rv was about0.5, and at frequencies of 98 GHz and 140 GHz, its value was 0.17–0.24 [4].

The specular scattering coefficient r s is

7.1 Influence of Propagation Effects on MMW Radar Operation 289

r2s = expF− S4psh sin c

l D2G (7.2)

The derivation of |V | as a function of range shows that a multipath structurein field strength as a function of height appears at relatively long ranges becauseof increase in r s resulting from the reduced grazing angle c . This structure isreduced for r s ≤ 0.3—that is, for ranges less than

rs ≅sh ? (hr + ht )

0.12l(7.3)

where rs values for radar and target heights of 6m and 2m and sh = 0.25m and0.1m, shown in Table 7.1.

It is seen that for minimal target heights, the multipath structure of the electro-magnetic field is practically destroyed in the shortwave part of the MMW bandfor ranges less than 1.5–3.0 km. This permits us to neglect the multipath losses atthese ranges, while at the X- and Ka-bands the multipath attenuation must be takeninto consideration.

In the dual-path propagation assumption, one can determine propagation factorvalues for available values of the specular scattering coefficient. The probabilitythat the propagation factor is less than some value V is determined as

T (V ) =1x

arccos1 + | r s |2 − V 2

2 | r s |(7.4)

For dual-path propagation, we obtain greater probabilities of deep multipathattenuations than for natural terrain. This is because for the natural terrain theamplitude pdf is closer to Rician, and, besides, it is necessary to take into consider-ation the scattered electromagnetic field attenuation by the antenna pattern.

The multipath attenuation for real terrain paths is different for smooth andbroken terrain. For broken terrain, the experimental data are the following [4]: atKa-band, the electromagnetic field in the interference minimum is more than 7 dBbelow that for free space, while at W-band the difference is less than 6 dB. These

Table 7.1 The Ranges in Kilometers Within Which Specular Reflection Is Destroyed

hr = ht = 6m hr = ht = 2mFrequency (GHz) sh = 0.1m sh = 0.25m sh = 0.1m sh = 0.25m

10.0 0.33 0.83 0.11 0.2737.5 1.24 3.12 0.41 1.0395.0 3.51 8.8 1.17 2.92

140.0 5.18 12.9 1.72 4.31

290 Clutter Rejection in MMW Radar

values are smaller than ones derived from (7.1). The smooth surface is rather closeto a plane over comparatively small areas (dimensions less than several hundredmeters). For larger areas, a very gently sloping roughness influence becomes notice-able. This leads to an increase in grazing angle in comparison with that derivedusing the assumption of a plane surface for all paths, resulting in decreased r s .The specular scattering coefficients and derived multipath attenuation factors thatare less than these values 90% of the time are presented in Table 7.2. It is necessaryto note that the data of V in Table 7.2 are, on average, less than those obtainedfrom (7.1). Therefore, the derivative data with use of (7.1) for radar detectionrange estimations leads to some increase of multipath attenuation.

7.2 Influence of Rain and Multipath Attenuation on Radar Range

Let us evaluate the influence of the effects discussed earlier on use of MMW radarfor low-altitude target detection by land-based systems with antennas a few metersabove the surface. It is worthwhile to compare radars in different bands for thesame two conditions used in Section 7.1: the antenna aperture (and antenna areaSA ) is constant with frequency change, and the antenna gain GA is constantwith frequency change (i.e., an inversely proportional change of antenna aperturedimensions takes place).

For SA = const we consider the dependence of the path loss coefficient A onfrequency [5]

A =Pr

Pt s t S 2A

=V 4

4p4 S cf D

2 ? 10−0.2g r (7.5)

where g is the attenuation factor in precipitation.Here Pr is the receiver power, Pt is the transmitter power, s t is the target RCS.

Practically, this coefficient determines the energy potential of the radar in conditionsof multipath and rain attenuation for constant SA . The dependence of A on fre-quency in rain and multipath conditions is presented in Figure 7.1. While for light

Table 7.2 The Scattering Coefficient and Multipath Attenuation V 20.9 for Different Paths and

Frequencies (hr = 4m, ht = 4–6m)

Paths with Vegetation Paths with Smooth SurfacePath Typeand Arable Land (Snow, Ice, and Sand)

Frequency (GHz) 35 95 140 35 95 140r s 0.6 0.2 0.1 0.8 0.6 0.4

V2

0.9 , dB −6.6 −1.8 −0.9 −9.3 −6.6 −4.0

7.2 Influence of Rain and Multipath Attenuation on Radar Range 291

Figure 7.1 Coefficient A versus frequency for hr = ht = 6m, and for rainfall rates of 1 mm/hr (solidlines) and 4 mm/hr (dashed lines); sh = 0.1m (curves 1, 2, 5, and 6), sh = 0.25m(curves 3, 4, 7, and 8).

rains (I = 1 mm/hr), the shortwave part of the MMW band is preferable, formoderate rain the use of frequencies above 50 GHz is not beneficial.

The performance of the MMW band compared to the X-band for constantantenna aperture can be done using the factor

Cs = S Vf

V10D4 ? S10

f D−2

? 10−0.2r (g f −g 10 ) (7.6)

here Vf , V10 are the multipath attenuation factors and g f , g10 are the attenuationcoefficients at frequencies f and 10 GHz. The examples of derived values of Cs asa function of range r are shown in Figure 7.2 (for sh = 0.1m and hr = ht = 6m).

It is seen that for ranges of 2–3 km, the MMW band has visible advantagesover the X-band, especially for smooth paths. For ranges of about 5 km, rainattenuation is the prevailing factor, which is why the advantage of the shortwavepart of the MMW band is seen only for light rain, while for moderate rain theKa-band has insignificant advantage with respect to the X-band and shortwavepart of MMW band.

For light rains, the MMW band is more effective than the X-band, and formoderate rains only (I ≥ 4 mm/hr) the advantages of MMW appear for ranges lessthan 3.0–3.5 km.

The estimation of MMW efficiency the for second case (when GA = const) canbe expressed as

292 Clutter Rejection in MMW Radar

Figure 7.2 Factor CS versus range at frequencies of 35 GHz (solid lines) and 95 GHz (dashed lines)for rainfall rates of 1 mm/hr (curves 1 and 3) and 4 mm/hr (curves 2 and 4).

B =Pr

Pt G 2A s t

=V 4S c

f D2

(4p )3r4 ? 10−0.2g r (7.7)

The derivation results for sh = 0.1m and hr = ht = 6m are shown in Figure7.3(a). The comparative estimation of MMW-band advantages with respect to theX-band for this case are given by the factor

CG = S Vf

V10D4S10

f D−2

? 10−0.2r (g f −g 10 ) (7.8)

The derived dependences CG = f (r ) for sh = 0.1m and hr = ht = 6m are shownin Figure 7.3(b). As seen from Figure 7.3, the longwave part of the MMW bandhas advantages in comparison to the X-band radar at ranges less than 1.5 km. Inall conditions, the MMW band is less effective than the X-band at ranges greaterthan 3–3.5 km, and at smaller ranges only the frequency band 20.0–50.0 GHz issomewhat more effective than the X-band.

7.3 Influence of Land and Rain Clutter on Radar Detection Range

The second basis for frequency choice in radars for low-altitude, land-based targetdetection is the land clutter. As is well known, the total land clutter RCS for pulsed

7.3 Influence of Land and Rain Clutter on Radar Detection Range 293

Figure 7.3 Factors (a) B versus frequency for rainfall rates 1 mm/hr (solid lines) and 4 mm/hr (dashed lines)and for sh = 0.1m (curves 1, 2, 5, and 6) and sh = 0.25m (curves 3, 4, 7, and 8), and (b) CGversus range at frequencies of 35 GHz (solid lines) and 95 GHz (dashed lines) for rainfall ratesof 1 mm/hr (curves 1 and 3) and 4 mm/hr (curves 2 and 4).

radar (the term pulsed radar includes systems using pulse compression) can bedetermined as

s cl ≅ct02

ru0s 0(c , f ) (7.9)

where t0 is the processed pulse duration, u0 is the azimuth beamwidth, and s 0 isthe normalized RCS of land.

The s 0 values can be determined, for example, from the model for differentterrain types presented in Chapter 2. According to that model, the normalized RCSis a function of grazing angle and radar frequency only. All various land territoriesare classified into eight general terrain types. The coefficient values for differentterrain types are shown in Table 2.13.

For two limiting cases discussed earlier, we obtain the total RCS of land clutteras

s cl ≅ 0.03A1 S9p D

A2

?ct02

?hA2

Lu? S f

10DA2 −1

? r1−A2 for SA = const (7.10)

s cl = A1 S9p D

A2

?ct02

? u0 ? hA2 ? S f10D

A3

? r1−A2 for GA = const

294 Clutter Rejection in MMW Radar

where Lu is the antenna aperture size in horizontal plane in meters, f is the fre-quency in gigahertz, u0 is the beamwidth in radians, and hr and r are the radarantenna height and range in meters, respectively.

For different terrain types we have different dependences of total RCS on range:

• For quasi-smooth surfaces (e.g., concrete or surfaces with snow), RCSdecreases rapidly with increasing range, so, for range change from 0.2 kmto 5.0 km, the clutter RCS decreases by about 30 dB for concrete and20–22 dB for surfaces with snow;

• For surfaces with vegetation (e.g., forest and grass), the clutter RCS doesnot practically depend on range (i.e., s ≈ const);

• For country and town areas, a small increase of clutter RCS takes place forincreasing range; its growth is about 7 dB for range increasing from 0.2 kmto 5.0 km.

The frequency dependences of land clutter total RCS are different for twolimiting cases of antenna size choice:

• For GA = const and at frequencies from 10.0 GHz to 100 GHz, the cluttertotal RCS increases with increasing frequency by up to 20.0 dB for quasi-smooth surfaces and 3.0 dB for urban terrain;

• For SA = const, the total RCS of quasi-smooth surfaces increases with increas-ing frequency by up to 10.0 dB for concrete; at the same time the RCSdecreases for rough surfaces. Its decrease is 7.0 dB for urban areas and4.0 dB for terrain with vegetation.

As an example, the RCS dependences on frequency are presented in Figure 7.4for GA = const (curves 1 and 2) and for SA = const (curves 3 and 4) and for tworadar heights (hr = 2.0m and 6.0m). At the right side of Figure 7.4, the minimaldetectable target RCSs are presented for conditions that detection takes the placefor single pulse with detection probability D = 0.9 and false alarm probabilityF = 10−3.

Besides land clutter, the volumetric precipitation clutter is important in theMMW band. It is characterized by a normalized volumetric RCS h that is a functionof radar operation frequency and rainfall rate I given in millimeters per hour. Asseen from Chapter 5, its value can be presented as

h = A ? I b (7.11)

The A and b dependences on frequency in the band 10.0–200.0 GHz are shown,for example, in Figure 5.3(b).

7.3 Influence of Land and Rain Clutter on Radar Detection Range 295

Figure 7.4 The clutter total RCS and minimal detectable target RCS (right axis) of land terrain withvegetation versus frequency for GA = const and SA = const and for hr = 2m and 6m.

The total RCS of rain clutter can be represented as

scr = 5ct02

r2 ? u0 ? f0 ? h ( f , I ) for GA = const

0.09ct02

?1

Lu Lfr2 ? f −2 ? h ( f , I ) for SA = const

(7.12)

Here w0 is the beamwidth in the vertical plane (in radians), Lw is the antennaaperture size in the vertical plane (in meters), while u0 and Lu are the correspondingvalues in the horizontal plane. As we can see from (7.12), the total RCS of rainclutter rises quickly with increasing range. The dependences s cr versus range areshown in Figure 7.5. They are derived for rainfall rates of 1.0 mm/hr, 4.0 mm/hr,and 10.0 mm/hr and for two frequencies—Figure 7.5(a) for GA = const and Figure7.5(b) for SA = const. At 40.0 GHz and a rainfall rate of 4 mm/hr, the total RCScan be 1 m2 for a range of 5.0 km, exceeding the rain RCS at 10.0 GHz by17–18 dB.

The frequency dependence of the rain RCS has different forms for the twolimiting cases. For GA = const, the RCS rises with increasing frequency, but thisgrowth is slowed at frequencies above 40.0–50.0 GHz, as is seen from Figure7.6(a). For SA = const, the total RCS is reduced at frequencies above 20.0–30.0GHz, and in the short part of the MMW band (above 100 GHz), its values are

296 Clutter Rejection in MMW Radar

Figure 7.5 The clutter total RCS and minimal detectable target RCS (right axis) versus range atfrequencies of 10.0 (solid lines) and 40.0 GHz (dashed lines) for rainfall rates of1 mm/hr, 4 mm/hr, and 10 mm/hr for (a) GA = const and (b) SA = const.

greater by 4–10 dB than at 10.0 GHz, as seen in Figure 7.6(b). The aggregate effectof land and rain clutter leads to increasing the total clutter RCS in comparisonwith that at the X-band at ranges beyond 1.0 km for moderate rain (with intensityof I ≤ 4 mm/hr), even for cases of constant antenna aperture. This makes worsethe clutter input power in mobile target indication (MTI) systems of MMW radars.Thus, for fine-weather conditions, the MMW radars with SA = const have smallerlevels of land clutter than X-band radars. However, rain clutter is dominant forlight rains (with intensity of about 1 mm/hr) at ranges beyond 1.0–2.0 km. Thismakes noticeably worse the clutter problem in MMW band radar systems incomparison with analogous systems at the X-band.

Taking into consideration the multipath and rain attenuation and the land andrain clutter, MMW radar is better when applied to land-based and low-altitude

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 297

Figure 7.6 The clutter total RCS and minimal detectable target RCS (right axis) versus frequency (a) atranges of 5.0 km (solid lines) and 2.0 km (dashed lines) for GA = const; (b) at range of 5.0 kmfor SA = const.

target detection at ranges less than 2.0–3.0 km. The application of microwaveradar is preferable for all-weather conditions at greater ranges.

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar

7.4.1 General Notes

One of the demands on radar for detection and tracking of low-altitude targets ishigh land clutter rejection. The main source of interference for the majority ofradars operating over the land or sea surface is the clutter caused by backscatteringfrom the surface, because the clutter power is significantly greater than the receivernoise power. The clutter echoes from a rough sea and from some types of terrainhave many characteristics like those of thermal noise; they are randomly fluctuatingin both amplitude and phase. But the spectrum is often much narrower than thatof white or quasi-white noise (i.e., the clutter can be correlated, either partially ornearly totally, for times of the order of the typical period of signal integration).

298 Clutter Rejection in MMW Radar

The use of the millimeter band for such radars leads to a need to obtain precipitationclutter rejection, too.

The use of stationary random processes with Gaussian pdf and the powerspectra of white noise type as clutter models is a significant limitation for mostpapers in which the statistical theory of target detection has been developed [6–8].This model is properly applied, as a rule, only for target detection in receiver noise,and the calculations of detection characteristics for this case are developed in detail[9].

As is well known [10], the qualitative indexes of optimal detection in theGaussian noise don’t depend on the signal waveform but only on its energy relativeto the spectral density of noise power. For extended correlated clutter, the signal-to-noise ratio is determined by the ambiguity function of the radiated signal. It ispossible to develop different techniques for modulation spectrum design; the signal-to-clutter ratios (SCRs) are different for different signals.

In this chapter, the land and rain clutter rejection is estimated for some typesof radiated signals that are widely used in radar systems in centimeter and millimeterbands [11–13]. Among the most often used signals, one can note the periodicuncoded pulse sequence, ensuring range resolution and velocity indication of mov-ing targets, pulsed-compression signals exemplified by pulsed signals with linearfrequency modulated or phase-coded pulsed sequences, unmodulated continuoussignals, and continuous signals with sinusoidal frequency modulation.

7.4.2 Land and Sea Clutter Rejection

Let us estimate the radar signal modulation required for high rejection of land andsea clutter. Clutter rejection is determined, first of all, by the characteristics of landand sea clutter. An empirical land model for normalized RCS is developed inChapter 2, obtained from the experimental investigations in bands from 3 GHzto 100 GHz for grazing angles less than 45°. The empirical model for sea clutteris developed in Chapter 4; this model takes into account the scattering from sprayformed by sea wave breaking and propagation in the boundary layer of the atmo-sphere with the enhanced refractivity over the sea. These models are used for clutterrejection estimation.

It was noted earlier that the SCR depends on the transmitted signal ambiguityfunction. For ambiguity functions produced by a short individual pulse, with widespectrum, the shift in Doppler frequency between target and clutter scattered signalis insignificant. Then the target velocities are usually ambiguous, and only theranges of interest are unambiguous. The SCR for pulsed signals of this type withoutintrapulse phase modulation is

q =s tsc

≈2s t

cts 0(r )ru0(7.13)

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 299

where s t is the target RCS, sc is the clutter RCS, s 0(r ) is the normalized clutterRCS as function of range r, t is the pulse duration, and u0 is the azimuthal antennapattern width at the level of −3 dB. For typical radar parameters of t = 0.05–0.2ms, u0 = 1°, target RCS s t = 1 m2, s 0 = −30 dB, and r = 5 km, the SCR determinedfrom (7.1) is equal to −(8 to 14) dB. High target detection probability in theseconditions cannot be ensured.

Velocity and high range resolution can be obtained using transmitted signalswith line spectra. These periodic signals with arbitrary interperiod modulation haverange resolution, and in the most of cases it is impossible to obtain unambiguousvelocity and range indication simultaneously. One can form a signal with wideseparation between spectral lines to provide unambiguous range determination[10]. Such signals, formed by n continuous sinusoidal oscillations, suffer fromclutter accumulation from all of detection ranges. For pulsed sequences, the esti-mates of land and sea clutter rejection were done in [11, 12]. The main conclusionsof this analysis are the following. For uncoded periodic pulsed signals, the SCR isdetermined as

q =s t

ct2

u0r ∑∞

l =0

s0(r )[1 − (l0 − l )cTr /2]

(7.14)

where Tr is the pulse repetition period and l0 = [2r /cTr ] is the integer part of therange to unambiguous interval ratio, l = 0, 1, 2, . . .

It is enough to limit oneself by value of lmax = [2rmax /cT ] for summing in thedenominator of (7.14) because the clutter power contribution from the surfacecells at ranges exceeding the maximal range of the target rmax is significantly lessthan the backscattering from the surface cell under the target and from cells atshorter ranges. For unambiguous target range determination (rmax ≤ cTr /2 andl0 = 0), the clutter power is a sum of backscattering from the surface cell underthe target and from the cells situated at ranges greater than the radar maximalrange. Taking into account that the clutter power decreases with the range increaseproportionally to r−3 for land and to r−7 for the sea, the calculation of clutter fromranges of r > rmax is usually not necessary.

Range ambiguities appear for high pulse repetition frequencies (PRFs) that areoften necessary for effective clutter rejection by MTI systems. In this case, theclutter power is a sum of backscattering from the radar cell containing the targetand from the closer cells. As a result, the SCR degrades and for ranges close tor = lcTr /2 the target observation is impossible due to transmitter leakage andreceiver saturation by transmitted pulses.

Upon first consideration, it may appear that shortening the pulse durationwould increase SCR; however, it is necessary to consider spatial clutter spikes for

300 Clutter Rejection in MMW Radar

which the RCS can reach large amplitudes. For this spatial-temporal structure, theSCR is better than predicted by (7.14) at some ranges, while for other ranges thisratio is worse.

The use of wideband signals permits an increase in clutter rejection. In particu-lar, for pulses with linear frequency modulated or phase-coded signals, the gain inSCR compared with an uncoded pulsed sequence using the same transmitted pulsewidth is K = qw /q , where K = tDf is the compression ratio, Df is the spectrumbandwidth, and qw is the SCR for wideband signals.

If the medium contains a large number of scatterers, the resulting signal hasrandom noiselike characteristics. In addition, if the range to target is large, it ispossible to ignore the dependence of clutter level on range when calculating theperformance of the pulse compression waveform. Then the resulting signal can bepresented as a stationary and Gaussian random process for which the results ofworks [8–11] can be applied. In this case, the resulting clutter power density perresolution cell is decreased when the range resolution of radar increases. Theadvantage of range resolution is limited by the target dimensions as well as thechange of the clutter statistical characteristics that leads to an increasing ofthe false alarm level.

For signals with equal range resolution, the opposite situation is observed. Thenormalized RCS of clutter is practically constant in the radar resolution cell atlong ranges for uncoded pulsed signals of short duration. For a wideband signalwith equal range resolution Dr0 but longer transmitted pulse width t the cell sizescorresponding to the transmitted and processed pulse widths are determined by

Dr =Kct0

2= KDr0 (7.15)

where Dr is the cell size corresponding to the transmitted pulse width t and Dr0is the cell size of the processed pulse width t0 . Because K = tDf = t /t0 , it is clearthat t0 and Dr0 must refer to the resolution of the processed output, whilet = Kt0 and Dr = KDr must refer to the longer transmitted pulse.

When the radar energy potential is increased by increasing the transmittedpulse width without decreasing the range resolution (i.e., when the widebandtransmitted pulse width is equal to t = Kt0 , where t0 is the processed pulse width),and when tc /2 becomes a significant fraction of the target range r, the clutternormalized RCS will not be constant in the bounds of the transmitted pulse. Inthis case the use of wideband, long-pulse signals leads to decreased SCR. For twodependencies of the normalized RCS on range, typical for grazing angles [s 0 ∼ r−4

for sea because for sea clutter the propagation factor starts to vary directly withgrazing angle if the grazing angle is below the critical one, and s 0 ∼ r−1 for landaccording to (7.10)], the clutter power is [12]

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 301

Pc = 5163

AK12 + b2

(4 − b2)3 for land

256AKS4

3+ b2D (12 + b2)

(4 + b2)6 for sea

(7.16)

Here

A = PtG 2l2u0s 0(r )Dr0

(4pr )3 (7.17)

is the clutter power in a radar resolution cell at range of r for an uncoded pulsedsignal with duration of t0 , b = Dr /r = KDr0 /r is the ratio of transmitted pulsewidth to range delay.

For the thumbtack ambiguity function with residue level d , the clutter is distrib-uted uniformly on the ‘‘range-velocity’’ plane. This reduces the signal-clutter ratioin comparison to that for an uncoded pulsed sequence. The losses for this case are

L =qpc

q= 5

163

AK12 + b2

(4 − b2)3 for land

256AKS4

3+ b2D (12 + b2)

(4 + b2)6 for sea

(7.18)

The dependence of SCR losses for pulse-compression signals as a function ofb, the relative transmitted pulse width, is shown in Figure 7.7. It is apparent fromthese results that the transmitted pulse width should be restricted to a small fractionof the target time delay, b << 1.

For residue level determined as d = K−1 (this is typical for noise and noise-likesignals), the losses are significant when the radar cell dimensions are comparedwith the range to target r0 . If the residue level of is d = K−1/2, then the SCR isconsiderably worse than that for uncoded pulsed sequences at all values of b.

Thus, the use of extended transmitter pulses with pulse compression in radaroperating over land and sea surfaces does not improve the clutter rejection incomparison with an uncoded pulsed sequence of equal range resolution, and itmay significantly increase the input clutter, especially for short-range radar. Thisobstacle limits the use of long pulses with pulse compression for increasing of thetarget detection range in clutter when the transmitted power is limited.

302 Clutter Rejection in MMW Radar

Figure 7.7 SCR losses as function of relative pulse width.

Let us discuss the clutter rejection for a CW signal without modulation. Theclutter power at the receiver input is determined for CW radar by integration alongthe entire surface within the beam. As is shown in [14], for conditions of thenormalized RCS constancy (s 0 = const) and the radar antenna pattern approxi-mated by

G (w , u ) = G0Ssin pu /u0pu /u0

D2Ssin pw /w0pw /w0

D2 (7.19)

where w0 , u0 are the −3-dB beamwidths in the elevation and azimuth. The SCRat the output of a Doppler bandpass filter with rectangular frequency response,lower cutoff frequency F1 , and upper cutoff frequency F2 , and for horizontalelevation beam axis is

qcon =s t h 2

r

s 0r4 ?2p2 ln 2

u0w20

?F1F2

DF (F2 − F1)(7.20)

where hr is the radar antenna height and DF is the clutter spectrum width.As an illustration, the dependence of qcon on the range r is shown in Figure

7.8 for hr = 3m, s t = 0.02 m2, s 0 = −25 dB, and w0 = u0 = 1°. The lower cutoff

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 303

Figure 7.8 The dependence of SCR on range for CW, frequency-modulated CW, and pulsed signals.

frequency of the filter corresponds to the target velocity of 150 m/s and the filterbandpass width is equal to F2 − F1 = 1 kHz. For comparison, the dependence ofq (r ) is presented in the same figure (curve 3) for coherent pulsed radar with pulsewidth of t0 = 0.2 ms and an MTI filter. The target and clutter parameters are thesame as for the CW radar. It is seen that the SCR exceeds unity only at ranges lessthan several hundred meters for the CW signal, and the reliable detection of targetswith small RCS can be ensured only at ranges less than 500m. This ratio equalsto −(40–60) dB at ranges of 3–4 km, which determines the requirements on landand sea clutter rejection in MTI systems.

When we use CW radar, the surface areas near the radar create the maincontribution to clutter. A considerable decrease of this contribution is provided byapplication of a sinusoidal frequency modulated continuous signal. This resultsfrom the appearance of a number of harmonics at the radar input because ofsinusoidal modulation. Each of them is modulated by a signal of Doppler frequencyproportional to [15]

In (r ) = InSDfFm

sin2prlm

D (7.21)

where

Df is the deviation of the modulation frequency.

Fm is the modulation frequency.

lm = c /Fm is the modulation wavelength.

In is a Bessel function of order n.

304 Clutter Rejection in MMW Radar

For use of third harmonics of the modulation frequency and for the antennapattern of (7.19), it is easy to obtain the clutter power at the receiver input

dPrc =Pt G 2l2s 0u0

(4p )3r4 ? I 23 SDf

Fmsin

2prlm

D ? r dr du (7.22)

One can determine the full power of clutter using the condition for small

grazing angles w ≈hrr

Prc =Pt G 2l2s 0u0

2(4p )3 ? #∞

hr

sin4 phrw0r

Sphrw0r D

4 ?

I 23 SDf

Fmsin

2prlm

Dr3 dr (7.23)

Let us divide the integrand in (7.23) into two parts, choosing some range R0as a boundary so that the maximal power of clutter from the ranges r < R0 , whereR0 = hr /w0 , is

max Prc =Pt G 2l2s 0u0

(4p )3 ?p6

12? SDf

FmD6S w0

phrD5 R8

0

l6m

if r < R0 (7.24)

and

Prc ≅Pt G 2l2s 0u0

(4p )3 ?0.32

lm R0if r > R0 (7.25)

The ratio of the clutter power from the near and far zones is determined from(7.24) and (7.25) as

PrcnearPrcfar

≅p6

3.84? SDf

FmD4S w0

phrD4 R9

0

l5m

(7.26)

The estimations show that for antennas with narrow elevation beamwidths,this ratio is considerably less than one. For lm = 6 ? 104 m, Df /Fm = 4, w0 = 10−2,and hr = 10m, it is approximately 3 ? 10−4. Therefore, the clutter from the nearzone is considerably attenuated.

For this case the SCR is

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 305

qFM =3.12s t lm hr

s 0r4u0w0? I 2

3 SDfFm

sin2prlm

D (7.27)

The dependence of qFM on the range for the same radar parameters as for thecontinuous signal and for lm = 20 km and Df /Fm = 4 is shown in Figure 7.8 (curve2). The analysis of these results shows that, first of all, frequency modulated CWradar obtains higher SCRs in comparison with unmodulated CW at ranges greaterthan 1.5–3.0 km. The gain reaches 15–17 dB and is greater for broader beamwidthsand for lower radar heights. Really,

PrcFMPrc

≅5hr

lm w0(7.28)

The use of frequency modulated CW reduces the dynamic range of targetsignals at the expense of their attenuation by the modulating function when thetarget range is decreased.

7.4.3 Rain Clutter Rejection

Let us estimate the precipitation clutter rejection for transmitted signals of thetypes considered earlier. The SCR for an uncoded pulsed sequence and forct /2 << r can be represented, as for land and sea clutter (7.14), by

q =s t exp (−2gr )

ct2

hu0w0r2 ∑∞

l =0

exp (−2gr )[1 − (l0 − l )cTr /2r ]

(7.29)

where g is the precipitation attenuation coefficient and h is the precipitationnormalized volumetric RCS.

It is necessary to have in mind from the summation in the denominator of(7.29) that SCR decreases quickly as target range increases. For the determinationof unambiguous target range (rmax ≤ cT /2, l0 = 0) the contribution of the reflectingareas at ranges r > rmax is commensurable with the clutter power from the pulsedvolume containing the target.

With range ambiguity, the backscattering from the areas closer to the targetand the backscattering from the volume containing the target add. This results ina significant reduction in the SCR. The quantitative presentation of the dependenceq on range for rain intensities of 1 mm/hr and 4 mm/hr is given in Figure 7.9.Curves 1 and 2 correspond to the regime of unambiguous target range determinationfor rmax = 3 km, curve 3 corresponds to the regime of ambiguous target rangedetermination with additional scattered power from the volume closer to the target

306 Clutter Rejection in MMW Radar

Figure 7.9 The SCR dependence on range for pulsed signal and for (1) I = 1 mm/hr and(2–4) I = 4 mm/hr (s t = 1 m2, t0 = 0.1 ms, w0 = u0 = 10 mrad, V = 1).

(l = 1), and curve 4 takes into account the backscattering from volumes closer thanthe target as well as from volumes beyond the target.

The dependences of q on the radar frequency for pulsed signals, shown inFigure 7.10, permit us to reach some conclusions. First, transition to the millimeterband is accompanied by considerable decrease of the SCR in comparison withvalues obtained in the X-band, amounting to 20 dB at 35 GHz. This decrease ismost visible in rains with intensity less than 4 mm/hr. The value of q decreasesinsignificantly in the shortwave part of the millimeter band ( f = 95 GHz). For rainintensities less than 4 mm/hr, these SCR decreases are less than 6 dB in comparisonwith values at 35 GHz. For rain intensities more than 16 mm/hr, the SCR doesnot change over all MMW bands.

For the unmodulated CW, the main contribution to the clutter power at thereceiver input is from areas near the radar. Let us discuss this problem in moredetail, taking for the boundary between the near and far zones the range

r =ka2

2(7.30)

where k = 2p /l is the wave number and a is the radius of the antenna aperture.

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 307

Figure 7.10 The SCR dependence on radar frequency for pulsed signal (s t = 1 m2, t0 = 0.1 ms,w0 = u0 = 10 mrad, r = 5 km, V = 1): (1) I = 1 mm/hr, (2) I = 4 mm/hr,(3) I = 16 mm/hr.

For reflector antennas in the near zone, defined by b =ka2

2r≥ 1, the antenna

gain G depends on the coordinates of the observation point and, in to a firstapproximation, is determined by

G ≈2(ka )2

1 + b F1 −(ka )2 sin2 u

4 G (7.31)

where u is the angle between the antenna electrical axis and the direction to theobservation point.

The expression (7.31) is valid only near the antenna electrical axis where thecondition

Ska sin u2 D2 << 1

is fulfilled, for an exponential illumination of the aperture, and for ka >> 1. Thenthe rain clutter power from the near zone with spatial homogeneity is

Pcnear =Pt l2h

(4p )3 EV

G

r4 dV =Pt l2h

(4p )3 EV

2(ka )2

1 + b2 F1 − Ska sin u2 D2G dV

r4 (7.32)

where the volume element dV in the spherical system of coordinates is

308 Clutter Rejection in MMW Radar

dV = r2 sin u ? dr ? du ? dw

Let us divide the space on two areas of integration (see Figure 7.11). The clutterpower from area 1 is

Pcnear1 ≈564

?Pt l2hk (ka )2

(4p )2 (7.33)

and from area 2 is

Pcnear2 ≈p10

?Pt l2hk2a

(4p )2 (7.34)

The clutter power from the far zone for identical antenna patterns in bothplanes is

Pcfar =Pt l2G2

0h

(4p )3u0

ka2 , u0 = w0 (7.35)

and the ratio of clutter from the near zone to the total power is

a =PcnearPcS

=1

1 + 0.48 Sla D

2 =1

1 + 0.48u20

(7.36)

(i.e., it is inversely proportional to the square of beamwidth). Then for narrowdirectional antennas (a ≈ 1), the total scattered power is practically determined bythe near zone of the antenna (for u0 = 30 mrad a = 99.9%).

Figure 7.11 The areas of integration.

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 309

The ratio of signal to rain clutter is

q cont =PsPc

≅8G0ls t

5p2hr4 =25.6a2s t

plhr4 (7.37)

It is seen from analysis of (7.37) that in the millimeter band, values of q contdo not practically change with wavelength shortening for the condition a = const(constant aperture area) because the wavelength decrease is balanced by increasein the normalized volumetric RCS.

For proportional decrease of the antenna aperture with wavelength shorteningand for the condition of Go = const, the SCR decreases in conformity with thecharacter of h (l ) dependence.

The dependences of q on range for different rain intensities and for the conditionDA = 2a = 0.6m are shown in Figure 7.12. It is seen that the SCR differs by lessthan 3–5 dB between 35 (curve 2) and 75 GHz (curve 3).

One method of rejecting rain clutter from the near zone is the use of frequencymodulated CW. When using the first harmonic of the modulating frequency wefind

PnearFM ≈Pt l2h

(4p )2 ?(ka )4a2x2

512(7.38)

where x = mvm /c, vm = 2pFm , m is the modulation index.

Figure 7.12 The continuous signal SCR dependence on range for rain intensity of 4 mm/hr atwavelengths of (1) 3 cm, (2) 8 mm, (3) 4 mm, and of 1 mm/hr at wavelengths of(4) 3 cm and (5) 8 mm.

310 Clutter Rejection in MMW Radar

It is seen from comparing (7.38) with (7.33) that the use of frequency modula-tion permits us to considerably attenuate the scattering level from the near zoneeven for application of the first harmonics in radar operation. The increasing ofharmonic number leads to further attenuation of the scattering level.

The clutter power from the far zone is

PfarFM ≈ gPt l2G 2

0 u0

(4p )3 ?pc

4vmx2 (7.39)

The signal to clutter ratio is

qFM ≈8g

?s t lm

(2p )2m2hr4u20

? I 21 Sm sin

vm rc D (7.40)

Here lm is the wavelength of modulation.Supposing that the target is at the range corresponding the maximum of the

Bessel function, one can simplify (7.40) to

qFM ≈2.8g

?s t lm

(2p )2m2hr4u20

=2.8g

?s t

pm2hr3u20

(7.41)

If the antenna aperture is constant with change in wavelength (7.41) can betransformed to the form

qFM ≈11.2

(2p )2g?

a2s t lm

m2l2hr4 (7.42)

It is seen from (7.42) that for a constant aperture, the SCR increases with thedecreased wavelength. As an illustration, the dependences q (r ) are presented inFigure 7.13 for the following radar parameters: lm = 3 ? 103 m, s t = 1 m2, u0 =30 mrad. The values of h have been used for rain of the same intensity as in Figure7.12 for the continuous signal. The result of comparing Figure 7.12 with Figure7.13 shows that sinusoidal frequency modulation permits us to improve the SCRby 40–50 dB in comparison with the continuous signal. This results from the rainclutter rejection in the near zone.

Thus, the analysis of land (sea) and rain clutter rejection in the MMW bandfor several signal types permits us to make the following conclusions.

The use of periodic uncoded pulsed sequences with unambiguous target rangeis preferable. Otherwise, a considerable decrease of SCR takes place due to back-scattering contributions from the ambiguities closer than the target.

References 311

Figure 7.13 The continuous signal with frequency modulation SCR dependence on range (fors t = 1 m2, lm = 3 ? 103 m, m = 1, w0 = u0 = 30 mrad) for rain intensities of 4 mm/hrat wavelengths of (1) 3 cm, (2) 8 mm, and (3) 4 mm, and of 1 mm/hr at wavelengthsof (4) 3 cm and (5) 8 mm.

The use of complex signals with pulse modulation permits us to increase theclutter rejection if the processed pulse width is less than the uncoded pulse width.Otherwise, the clutter accumulation from the large area or volume illuminated bythe transmitted pulse leads to decrease of the SCR and increases the clutter powerfrom the side lobes of the ambiguity function.

For CW signals, the clutter power from the near zone increases and frequencymodulated CW signals do not provide clutter rejection to levels obtained for thepulsed sequence.

References

[1] Skolnik, M., Radar Handbook, New York: McGraw-Hill, 1990.[2] Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988.[3] Nathanson, F. E., J. P. Reilly, and M. N. Cohen, Radar Design Principles, 2nd ed., New

York: McGraw-Hill, 1991.[4] Kulemin, G. P., and V. B. Razskazovsky, Scattering of Millimeter Waves by Earth’s Surface

for Small Grazing Angles, Kiev: Haukova Dumka, 1987 (in Russian).[5] Kulemin, G. P., ‘‘Influence of the Propagation Effects on Millimeter Wave Radar Opera-

tion,’’ SPIE Conf. Radar Sensor Technology IV, Vol. 3704, Orlando, FL, April 1999,pp. 170–178.

312 Clutter Rejection in MMW Radar

[6] Woodward, P. M., Probability and Information Theory with Applications to Radar,Oxford: Pergamon Press, 1953; Dedham, MA: Artech House, 1980.

[7] Cook, C. E., and M. Bernfeld, Radar Signals: An Introduction to Theory and Application,New York: Academic Press, 1967; Norwood, MA: Artech House, 1993.

[8] Berkowitz, R. S., (ed.), Modern Radar: Analysis, Evaluation, and System Design, NewYork: John Wiley, 1965.

[9] Marcum, J. I., ‘‘A Statistical Theory of Target Detection by Pulsed Radar,’’ IRE Trans.,Vol. IT-6, No. 2, 1960.

[10] Bakut, P. A., et al., Problems of Radar Statistical Theory, Moscow, Russia: Soviet Radio,1963 (in Russian).

[11] Kulemin, G. P., ‘‘The Clutter Rejection in Short-Range Radar with Wideband PulsedSignals,’’ PIERS Workshop on Advances in Radar Methods, Baveno, Italy, July 20–22,1998, p. 73.

[12] Kulemin, G. P., ‘‘Clutter Rejection in Short-Range Radar with Uncoded and WidebandSignals,’’ J. Electromag. Waves and Applications, Vol. 14, No. 2, 2000, pp. 245–260.

[13] Kulemin, G. P., ‘‘Land and Rain Clutter Rejection in Millimeter Band Radar with Continu-ous and Pulsed Signals,’’ SPIE Conf. Radar Sensor Technology VII, Orlando, FL, April2002, Vol. 4744, 2002 (to be published).

[14] Kulemin, G. P., and V. I. Lutsenko, Detection and MTI Features by Near-Range Radarwith Use of Some Signal Types, Preprint No. 136, Inst. Radiophys. Electr., Kharkov,Ukraine, 1979, p. 28 (in Russian).

[15] Skolnik, M., Introduction to Radar Systems, New York: McGraw-Hill, 1962, p. 100.

About the Author

Gennadiy P. Kulemin received a diploma degree in radiotechnic engineering fromKharkov Polytechnic Institute in 1960, a Ph.D. in 1970, and a doctor of sciencedegree in electronic systems in 1987.

Dr. Kulemin has been an assistant professor and senior tutor on the ElectronicSystems Faculty of the Kharkov Aviation Institute. Since 1966, he has worked atthe Institute of Radiophysics and Electronics of the Ukrainian National Academyof Science. Presently, he is a principal scientific researcher within the MillimeterRadar Department and a professor at Kharkov Military University. His main areasof interest are backscattering from targets, land, and sea; radar remote sensing ofthe Earth in microwave bands; and millimeter band radar systems research. Hehas been investigating the experimental and theoretical aspects of these problems.

Dr. Kulemin is the author of Scattering of Millimeter Radiowaves by the Earth’sSurface at Small Grazing Angles, edited in Russian, and has written more than200 publications on scattering problems in microwaves and radar efficiency. Heis a member of the Academy of Science of Applied Radioelectronics, a member ofCommission F of URSI and the Ukrainian National Committee of URSI, and amember of the IEEE.

313

Index

A factor, dependence, 37factor, experimental, 40Absolute reflectivity, 283–84fog, 235Acoustic waves, 52frequencies, 41Aircraftgas, 233–34freedom degree number for, 60maximal, 39RCS, structure contributions, 12maximal total, 40remote sensing, 157–59microwave, 234Ambiguity functionmicrowave total, 37SCRs and, 298multipath, 288–90thumbtack, 301rain, 235Angel backscattering, 251temporal dependence, 41amplitude variance, 253

Autocorrelation function, 102, 105power spectra, 254

envelope, 105–6at S-band, 253

exponential, 144, 145, 148Angel-echoes, 250, 251

Fourier transform of, 102, 107Angel RCS cumulative distribution, 252

Gaussian surface, 144, 145, 148Antenna aperture, 310

of plowed field, 153Asphalt

of roughness, 153dielectric constant, 89, 146, 147

of scatterer velocities, 106spectral density, 147 sea roughness, 181surface roughness estimates, 146 spatial surface, 186surface statistical characteristics, 147 Azimuthal dependence, 218

Atmosphere, turbulent, 254–56Attenuation, 233–36

Bangular dependencies, 40atmospheric, 233–36 Backscatteringcoefficient, 234 angel, 251, 253, 254data, 235 of birds/insects, 13determination, 233 cloud, 242dust storms, 235–36 cloud, spectral parameters, 249

from forest, 130, 132factor, 36

315

316 Index

Backscattering (continued) microphysical characteristics, 231, 232nonrain, spectral shape, 250from grass, 129, 130, 132

of human body, 12 normalized RCS of, 245physical characteristics, 232land, 89–132

microwave, electromagnetic field, 54 stratus, 230undulating, 230from natural turbulence, 54

precipitation, 239–42 water drop size distribution, 231See also Meteorological formationsrain, 239, 241

sea models, 189–93 Clutter mapscontour highlighting, 280–82snow, 114–18, 239

soil, modeling, 138–45 development, 276–84image interlacing, 278from sonic perturbations, 41–55

spectra, 213–21, 251 initial data, 276–77inputting/processing data on, 279from swamp, 131

from SWF, 54–55 raster image processing module,278–80theory, 236–38

from turbulent atmosphere, 254–56 software input/processing components,277–78from vegetation, 118–20

Beaufort scale, 172 for vegetation, 278Clutter modeling, 259–84Biological objects, power spectra, 70

Birds land, 259–67map development, 276–84density in flocking places, 17

distribution by altitude, 17 sea, 267–76Clutter power, 304, 308mass, RCS dependence on, 13, 15

power spectra of, 73 for CW signals, 311from far zone, 310RCS of, 16

scattering pattern, 14 ratio, 304Clutter rejection, 287–311velocity of, 71

Bistatic RCS, 80 for CW signal, 302general notes, 297–98Boiling surf formation, 177

Bubble bursting, 177 land, 298–305rain, 305–11

C sea, 298–305ConcreteCanonical decompositions, 271

Chi-square distribution, 57, 63 dielectric constants of, 89normalized RCS for, 110Clouds, 230–32

backscattering, 242 Cone cylinder targetsRCS cumulative functions, 77classes, 230

cumulonimbus, 230 RCS distributions, 64scattering pattern, 8defined, 230

formation, 231 Continuous-wave (CW) radars, 28clutter contributions, 303height, 230–31

Index 317

clutter power, 311 Diffuse scattering coefficient, 77–78Dirac function, 75, 129clutter rejection for, 302

parameters, 28 Doppler frequencies, 221shift, 249SCR on range dependence for, 303

unmodulated, 306 spectrum width determination, 71Drizzle, 227Contours

automatic highlighting of, 280–82 Dual-channel polarization, 157Dust storms, 232–33color and brightness highlighting, 281

surface recovery from, 282 attenuation, 235–36formation, 232See also Clutter maps

Copolarizations, 201ECorrelation factors, 197

Correlation function, 263, 274 Echo(s)angel, 250, 251Correlation radius estimate, 145

Cross-correlation function, 196 angular dependencies, 31duration as function of filter low-bandCross-polarizations

components, 221 frequency, 32instantaneous, 33power spectra, 220

total scattered signal, 201 power spectra, 62–72power spectrum analysis, 31

D power spectrum at different momentsof time, 33Depolarization

coefficients, 197–98 power spectrum shape, 32sea, fluctuations, 83coefficients, mean, 198

degree, 197 tails, 32Effective front width, 46for horizontal polarization, 200

of scattered signals, 123–26, 197–202 Effective radar cross section, 46Electromagnetic field reflectionin spikes, 212

weak, 198 from sound wave package, 49from SWF, 43Dielectric constants

of asphalt, 146, 147 Electromagnetic field scattering, 49Erosion stateof concrete/asphalt, 89, 146

of corn leaves, 96 classification, 167situ measurements of, 168dependence on volumetric moisture,

90 Euler’s constant, 7Explosion(s)dry snow, 92, 93

for grain, 97 bands, RCS of, 30experimental attenuation factors, 40normalized RCS and, 111

soil, 91 fluctuation intensity reduction, 26–27gas-like products refractivity, 21in sound wave field, 47

water, 237 products, 22radar reflections from, 28–34wet snow, 92

318 Index

Explosion(s) (continued) duration distributions, 213mean duration of, 212RCS estimation, 23

refractive index fluctuations, 26 probabilities, 211spectral width, 220spatial-temporal characteristics, 23–28

turbulence local characteristics, 24 Gas attenuation, 233–34Gas-like products, 35volume, 34

volume, dimension measurements, 30 Gas wakefluctuations spectrum, 27

F radar observation results, 32Facet model, 190–91 radar reflections from, 28–34

small grazing angles and, 190–91 Gaussianuses, 190 curve, 249

Fast Fourier transform (FFT), 153, 154 distribution, 106, 264Foam law, 246

bubble structure, 177 random process, 210dependences, 176 surface, autocorrelation function, 144,distribution parameters, 176 145formations, 176 Green’s theorem, 99See also Sea Ground control points (GCPs), 162

Fog, 232attenuation, 235 Hsea, 232

Hail, 229–30Forests

fall zones, 230backscattering, 130, 132

formation, 229experimental values of model

normalized RCS of, 245parameters for, 98

surface, 229as homogeneous scatterers, 94

See also Meteorological formationsnormalized RCS, 119, 122

Helicoptersnormalized RCS vs. frequency, 123

freedom degree number for, 60RCS, 119

power spectra of, 66See also Vegetation

Hermittian polynomials, 274Fourier-series expansion, 266

HumansFourier transforms

moving, power spectra, 71autocorrelation function, 102, 107

power spectra, 70fast (FFT), 153, 154

swimming, power spectra, 72Fresnel coefficients, 100Friction velocity, 188

IG Image superimposing, 159–63

Inflatable boats, amplitude distributions,Gapscentral frequency, 220 62

Index 319

J input data, 261peculiarities of, 262–67Jet propulsion enginesrequirements, 261as intensive sound field, 53simulation algorithms, 265sound radiation spectrum, 50surface model, 260JONSWAP spectral model, 181See also Clutter modeling

K Laplace function, 83Laws-Parsons distribution, 237–38Karhunen-Loeve transform, 266Linear filtering, 266Karman’s constant, 172Logarithmic Gaussian law, 82Kirchoff’s method, 4Lognormal law, 246gently sloping surfaces and, 99Long-life reflections, 29normalized RCS estimations with, 100Lorentz-Lorentz relationship, 47Kirchoff’s model, 139Low-altitude targets, 3Kolmogorov-Obukhov theory, 255Low-RCS airborne vehicle detection, 2

LMLand

classification of, 89 Magnetron oscillators, 39Marine targetsclutter rejection, 298–305

forest, 94, 98 oscillations, 69power index, 70grassy terrain, 96

object power spectra, 66–67 power spectra, 68RCS distributions, 60objects, RCS of, 9

radar range influence, 292–97 spectrum width, 70Marshall-Palmer formula, 193roughness parameters, 90, 92

snow, 92, 93, 94 Marshall-Palmer raindrop sizedistribution, 234, 237soil, dielectric constants, 91

surface types, 262 Meteorological formations, 227–56clouds, 230–32vegetation, 93

Land backscattering, 89–132 drizzle, 227dust storms, 232–33power spectra, 128

power spectrum model, 101–8 fog, 232grouping, by size, 227RCS models, 95–101

scattering surface, 103 hail, 229–30rain, 227–29, 241–44simplified models of, 98

Land clutter modeling, 259–67 sandstorm, 232structure of, 227–33clutter stochastic process, 263

correlation function, 263 thunderstorms, 230Microwave attenuation, 234database development, 260

fluctuating component, 263 causes, 35total, 37initial data, 259–62

320 Index

Microwave scattering, sound cover coefficient and, 120perturbations from, 47–51 dependence, 108–9

Millimeter wave (MMW) radar, ix dependence on incidence angle, 79advantages, 287 dependence on radar wavelength, 111bands, 9 dependence on relative moisture, 112clutter rejection in, 287–311 dependence on wavelength, 100efficiency estimation, 291 dielectric constant and, 111limitations, 287–88 dual-polarization ratio of, 138measurements, 207 estimation of, 100MTI systems of, 296 estimation with Kirchoff’s method,performance, 291 100propagation effects on, 287–90 forests, 119, 122

Mobile target indication (MTI), 296 frequency dependence, 111, 121Multichannel image processing, 159–66 frequency dependence for dry/wet

filtering methods, 163–66 snow, 118image superimposing, 159–63 Gaussian model vs., 128

Multichannel method of hail, 245application of, 150 models, 120–23capabilities analysis, 140 for nadir radiation, 109, 112efficiency, 145–50 pdf, 128, 206measurements, 141 of quasi-smooth surface, 108–9

Multichannel radar images of rain, 245after adaptive nonlinear vector on range, 300

filtering, 166 ratios for cross-polarized reception,filtering methods, 163–66 157

Multipath attenuation, 288–90 reflectivity and, 238estimation, 288 of rough surface, 99radar range influence, 290–92 for rough surfaces without vegetation,for real terrain paths, 289 109–14

Multiple surface reflection, 78–84 sea, 193–97Multiplicative noise, 158 of snow, 114

surface scattering, 116N

of turbulent atmosphere, 256Nadir radiation volumetric scattering, 116

RCS for, 109, 112 wind-dependent saturation, 203rms roughness height, 109 See also RCS

Nonlinear transformation, 272Normalized RCS, 108–23

Oangular dependences, 112, 113, 114,Orbital speed, 215203Orbital velocity, 221of clouds, 245

for concrete, 110 Oxygen absorption factor, 35

Index 321

P echo, 62–72forest/grass backscattering, 130Peaks, 184–89Fourier-transform of autocorrelationmean duration, 188

function, 103mean number of, 188for GAZ-63 truck, 68See also Seaof helicopter, 66Phillip’s generalized constant, 183in high-frequency region, 69

Phillips-Miles model, 178intensity, 219

Phillip’s spectrum, 181, 182for L-200 airplane, 65

Pierson-Moscovitch spectrum, 179,of land backscattering, 128

187–88land clutter, 129

Plasma parameters, 36 of land objects, 66–67Plunging, 174 of marine targets, 68

asymmetry coefficients of, 175 model, 101–8defined, 174 of moving humans, 71See also Sea waves rain backscattering, 246

Point reflections, 250–54 of scattered signals, 128–32angel-echoes, 250, 251 of seagull, 73from angels, 252 sea surface returns, 214origin, 250 shape of backscattering signals, 250

Poisson’s law, 187 swamp backscattering, 131Polarization ratios, 142 of swimming humans, 72

analysis, 142 for tank, 67sea ice, 124 width, 67, 69, 70for vegetation, 126 at X-band, 64–65

Polarization(s) Precipitationdual-channel, 157 backscattering, 239–42HH, 149, 161, 162, 196, 207 clutter, 241horizontal, 199, 200 distributions, 229of scattered signals, 123–26 intensity, 239vertical, 198, 220 microwave scattering by, 236VV, 149, 161, 163, 196, 207 volumetric normalized RCS of, 240

Power density, 78 See also Drizzle; Rain; SnowPower series coefficients, 275 Pressure jumpPower spectra illustrated, 43

of angel backscattering, 254 power spectra of, 44from atmospheric turbulence, 256 at SWF, 41of backscattered signals, 218 Probability density function (pdf), 1backscattering, of land surfaces, amplitude, 56

101 chi-square, 57of biological objects, 70 experimental, 57

of normalized RCS, 128, 206cross-polarizations, 220

322 Index

Probability functions Rangeland influence on, 292–97amplitude, 207

of signal instantaneous values, 208 multipath attenuation influence on,290–92Propagation effects, 287–90

Pseudorandom number generator, 263 normalized RCS on, 300rain clutter influence on, 292–97Pulsed radar, 28, 293

Pulse repetition frequencies (PRFs), 299 rain influence on, 290–92SCR dependence on, 303

Q total RCS on, 294Quasi-smooth surfaces Rayleigh distribution, 250

dependence on radar wavelength, 111 Rayleigh modelnormalized RCS of, 108–9 defined, 56

experimental results and, 59R Rayleigh targets, 56

RCSRadar cross section. See RCSRadar reflection aircraft, 12

of air targets, 10from explosion and gas wake, 28–34long-life, 29 bird mass dependence, 13, 15

of birds, 16mechanisms, 18–23from shock wave front, 41–47 bistatic, 80

bounds, 16Radar tail, 32Rain, 227–28 clutter total, 295, 296, 297

crosswind, 29attenuation, 235backscattering, 239, 241 decrease methods, 13

derivation, 4backscattering power spectra, 246clutter, 241, 242 electromagnetic wave polarization

dependence, 6clutter RCS, 243–44clutter rejection, 305–11 estimation model, 14

evaluation, 5distributions, 229geometrical characteristics, 228 experimental, 7

explosion, 20global models, 228heavy, 228 explosion bands, 30

fluctuations, 1, 77intensity, 229mean probability, 229 of insects, 16

of land objects, 9moderate, 228normalized RCS of, 245 of man, 13

of marine vessels, 7, 8in precipitation zones, 227probability, 228–29 mean, of land targets, 11

mean values, 30radar range influence, 290–91RCS, frequency dependence, 295 median value, 8

models, 3–7See also Meteorological formationsRandom access memory (RAM), 280 in point of reception, 72–73

Index 323

probability functions, 73, 74–75 intensity, 27spatial-temporal spectra shape, 28rain, 295

range dependence, 9, 10 temporal, 24, 26Refractivityof real targets, 7–17

relationships, 4 coefficient, 44–45of combustion products, 22rough estimation of, 4

of small marine targets, 9, 11 Regression analysis, 219Rice’s formula, 187surface scattering, 116

of SWF front, 46, 47 Rough surfacesasphalt, 146target, 3–17

total, 78 characteristics determination, 155field values, 153volumetric scattering, 116

wavelength dependence, 7 growth of, 110measurement, 152See also Normalized RCS

RCS distributions, 75 normalized RCS for, 109–14scattering models, 139anchored sphere, 63

cone cylinder targets, 64 spatial spectrum, 144jet airplanes, 59

Spiston engine, 59quantiles, 77 Sandstorms, 232

Scattered signalsSwerling models vs., 84Reflected signal ratio, 50 depolarization of, 123–26, 197–202

fluctuations, 126Reflection coefficientas function of distance to pressure intensity, 137

mean power of, 236peak, 46numerical derivation of, 50 power spectra analysis of, 246

power spectra of, 128–32for plane surface, 99Reflection(s) by spray, 200

stable component, 127mean RCS for, 80point, 250–54 statistical characteristics of, 126–28,

242–50radar, 250–56from semi-spherical thin layer zones, temporal characteristics, 127

total power spectra in, 212252specular, 251 Scatterer velocities, 106

Scattering patternsurface, multiple, 78–84Reflectivity of bird species, 14

calculation, 4absolute, 283–84normalized RCS and, 238 of cone-cylinder bodies, 8

of Convair-900 aircraft, 5Refractive focusing, 38Refractive index, 256 of man, 14

SeaRefractive index fluctuationsfor explosion products area, 27–28 clutter rejection, 298–305

324 Index

Sea (continued) shadowing of, 185slope variance, 186echo fluctuations, 83

foam, 176, 177 statistical description, 181Sea wavesfog, 232

heavy, shadowing/peaks in, 184–89 asymmetry of, 174breaking process, 175ice, 124

normalized RCS, 193–97 height and slope angle, 173plunging, 174, 175spray, 177, 193

state, 220 secondary, 182slope variance, 182Sea backscattering

models, 189–93 spilling, 174Shadowing, 184–89radar spike characteristics, 209–13

spectra, 213–21 dependence of, 186grazing angle vs., 187Sea clutter

absence of reflectors, 268 mean function, 185for small grazing angles, 191characterization, 267

RCS model, 202–6 zone, 185, 186Shock wave front (SWF)signal formation, 271

statistics, 206–9 backscattering from, 54–55electromagnetic field reflection from,total RCS, 184

Sea clutter modeling, 267–76 43expansion, 44algorithm illustration, 269

algorithms, 268–76 front, RCS of, 46, 47parameter values, 43algorithm stages, 270

initial stage, 270 pressure difference, 42pressure jump at, 41peculiarities, 267–68

search for nonlinear transformation, radar observation of, 43reflections, 41–47271

simulation process, 268 reflections, detection of, 51turbulent atmosphere intersection,of stochastic process, 272

See also Clutter modeling 43width, 45Sea roughness

autocorrelation function, 181 Shock wave ionized front (SWIF), 2expansion, 20cause of, 179

characteristics, 171–84 high temperature, 19propagation law, 18determination, 171

features for small grazing angles, radius, 18spherical, 19171–89

Gaussian statistics, 182 temporal dependence, 19Shock wave propagation, 47, 51power spectrum, 179

Sea surface Side-looking radar (SLAR), 137, 161Signal amplitude, in point of reception,backscattering from, 184

complexity, 171, 184 72–73

Index 325

Signal-to-clutter ratios (SCRs), 298, 310 radar measurement results, 155–57RCS vs. incidence angle, 156dependence on range, 303

at Doppler bandpass filter output, 302 set and technique of measurement,150–51losses, as function of relative pulse

width, 302 state, situ measurements of, 168state classification, 167losses, dependence, 301

pulse duration and, 299 statistical/agrophysical characteristics,151–55signal ambiguity function and, 298

for uncoded pulsed sequence, 305 Soil moisture, 113content determination, 140for wideband signals, 300

Small grazing angles content sensitivity, 141measurements at reference points, 151facet model and, 190–91

scattering elements near wave crests, weighted, 151Sonic perturbations, radar backscattering216

shadowing for, 191 of, 41–55Sound absorption coefficient, 51two-scale model for, 192

Small perturbation model, 139 Sound oscillations, 52Sound perturbationsSnow

air-snow boundary, 116, 117 atmospheric pressure variation during,48backscattering, 114–18, 239

cover, 117 electromagnetic field reflection, 49first phenomenon, 49–50dry, 92, 93, 94, 117

normalized RCS of, 114, 117 microwave scattering from, 47–51Sound wavewet, 92, 117

See also Precipitation energy loss, 51intensity, 51Soil

backscattering modeling, 138–45 propagation, 47, 51Spatial grids, 254clay content, 152

cultivation methods, 152 Spectral widthhistograms, 248dielectric constants, 91

drying, 151–52 intensity, 247proportional to wavelength, 247normalized RCS dependencies, 115

parameters, estimation of, 137–50 Specular reflection, 251Specular scattering coefficient, 288–89RCS variations due to moisture, 138

remote moisture determination, 142 Spheroid, geometry, 201Spikesspatial correlation radius, 155

Soil erosion central frequency, 220characteristics, 209–13aircraft remote sensing, 157–59

determination from ratio images, depolarization in, 212duration distributions, 212, 213166–68

experimental determination, 150–59 maximal, 211mean number of, 210from multichannel remote sensing

data, 157 probability, 210–11

326 Index

Spikes (continued) Targetsacceleration, 63probable duration, 211

spectral width, 220 cone cylinder, 8, 64nonfluctuating, 81statistics, 210

See also Sea backscattering Rayleigh, 56statistical characteristics of, 55–72Spilling, 174

Spray, 177 velocity, 63Target statistical models, 55–58contribution, 204

effect, 193 analysis, 55experimental pdf and, 57signal scattering by, 200

volume normalized RCS of, 193 Swerling, 57–58Target to surface, 84See also Sea

Square-law detection, 131 Thumbtack ambiguity function, 301Trucks, power spectra, 68Statistical characteristics, 55–72

diffuse scattering surface influence on, Turbulence, 52Turbulent atmosphere, 254–5672–78

echo power spectra, 62–72 backscattering from, 254–56normalized RCS of, 256estimation of, 142

models, 55–58 power spectra, 256Two-scale model, 191–92real, 58–62

of scattered signals, 126–28, 242–50 defined, 191for small grazing angles, 192surface influence on, 72–84

Stratification, 250Superimposed images, 159–63 U

interpolation of, 161Upwind-to-crosswind ratio, 196

multiplicative noise, 161Super-refraction, 203

VSurface recovery, 282Swerling models, 57–58, 73 Vector filtering methods, 165

Vector sigma filters, 165–66illustrated, 58models 1 and 2, 74, 75 advantage, 166

noise-suppressing efficiency, 166models 3 and 4, 74RCS distributions vs., 84 Vegetation, 93

backscattering from, 118–20use of, 57Synthetic aperture radar (SAR), 137 clutter maps for, 278

normalized RCS seasonal dependence,T 119

penetration depth for, 115Tanks, power spectra, 67Target RCS, 3–17 polarization ratios, 126

RCS angular/frequency dependencesmodels, 3–7real, 7–17 and, 118

rough surfaces without, 109–14See also RCS

Index 327

Volumetric water content effective incidence angle and, 80of corn leaves, 96 lower height dependence on, 251for grain, 97 mean, 172

in sea roughness, 171Wsea wave height dependence on, 173

Wedge, geometry, 199 spike mean number dependence onWeibull distribution, 128 range for, 189Wiener-Khinchin theorem, 263Wind velocity, 69

Zcritical, 173Zacharov-Philonenko-Toba spectrum,dependence of single foam formation,

176 183