SEP 8 WU N0001-76-C-0176 POLY-MRI-1410-80 NL ... · DvdM/Wu '52 11. PROaA LMENT. PROJECT. TASK...

IONOSPHERIC SCINTILLATION(U)SEP 8 0 M WU N0001-76-C-0176

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MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANDOARDS-1963-A

PolyechnicInt'tute

MRI LULA

POLY -MRI -1410-80

September 1980

QIONOSPHERIC SCINTILLATION

by

David M. Wu DTICSELEC T EM

DEC 0 4 1980J

DISTPIUTIOSTA AApproved for public releo o;

Distribution Unlimited

3 Prepared for

I~L. OFFICE OF NAVAL RESEARCH

Contract No. N00014-76-C-0176

8011 24 052

UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE Men. Date Emte.,r_________________

REPORT DOCUMENTATION PAGE ___________________

JI. REPORT MUMV19R Ia. 337 AccessioN NO: 3. aceCPIENTS CATALOG MNGUER

4. TITLE food5cJM S. TYPE OF REPORT A P90NOO COVERED

A/ IONOSPHERIC SCINTILLATION I 1 - Z

DvdM/Wu

'52 11. PROaA LMENT. PROJECT. TASKPolytechnic Institute of New YorkMicrowave Research InstituteRoute 110, Farmingdale, NY 11735

11. CONTROLLING OFFICE'NAMIE AND ADDRESS ---

Office of Naval Research800 N. Quincy StreetArlIitU_.y A_22217 0_____________

N. NI8 GY NAMIE G ADORIESSrII diflomelfrom Coaaoi'6111 Offes) is. SECUIaTy CLASS. @

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Approved for public release; distribution unlimited.

17. DISTRIIEUTION STATEMENT (of th*obetraee entered In~ Stock 3.It difloent tIma A0100)

IS. SUPPLEMENTARY NOTES

IS. ICEY WORDS (Conthnue on t*retg* codo It neesr and identify by block numer)

ScintillationsIonosphereTurbulence

20. _11 RACY (Contr nu rwrto aide It n.....i n d 1~11?mE~ by Neack tambio)A renorinalization technique, employed in the spirit of the formal theory

of scattering, is applied to the problem of ionospheric scintillation. Usingthe forward scattering approximation for wave propagation and a Markov ap-proximation for ionospheric fluctuations, one derives renormalized miomentequations descriptive of the wave statistics.-

The propagation of the two and four point wave sta tc through the ior.o-sphere are obtained using a slow quasiparticle distri atjon function S. For

DO ~ 14.3LASSIFIED,,3SECUT CLSIPICATI N OF THISOAGE(W- DOOM;

UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PA46(noam Ote Rafg.Mj

Z0. ABSTRACT - Continued

ransverse distance, one can obtain exact analytical solutions for the wavestatistics. For an arbitrary fluctuation spectrum, one can evaluate the Sfunction along a quasiparticle trajectory and thereby infer second momentsexactly and fourth moments approximately. Knowledge of the S function en-ables one to ascertain statistical properties, such as average intensity, twopoint intensity correlation and the scintillation index

The Scintillation index for the case of a plane wave propagating through theionosphere is studied for the case of an arbitrary power spectrum of iono-spheric fluctuations. Peaks superposed on a power law spectrum are found toincrease the value ofL S4A focusing effect on the %)vs. z variation is observ-ed in the strong scintillation limit.

An experimental power spectrum, corresponding to relative density fluctua-tions of about ZO% as obtained from in situ measurements by the AtmosphericExplorer-E satellite, is used to calculate the corresponding S(J for night timeequatorial scintillations. For an ionospheric slab model with a thickness of200 Km and an altitude ofiW Km numerical calculations yield, in approximateagreement with Dr. Basu, an almost saturated scintillation index of about1.1. ..

'i

UNCLASSIFIEDascUNITY CLASWPICA?IO OP ?W R0AObft Do Wm)

ACKO0WLEDGMENT

The research reported herein has been supported by the Officeof Naval Research under Contract No. N00014-76-C-0176.

This work is a portion of a Ph.D. thesis in Electrophysics atthe Polytechnic Institute of New York.

The author wishes to express his gratitude to Professor N.Marcuvitz, for his guidance and encouragement. He is also indebted4

to Dr. S. Barone and Dr. S. Basu for their helpful discussions and 'AccSsion For

DDc TABUnannotricedJustifi'(Xtiofl______

Ivn-t Codes

Avatl aid/orD1iSt special

774ik

TABLE OF CONTENTS

PageChapter I INTRODUCTION I

I. Motivation II. General Introduction and Review 2

Chapter 2 THEORETICAL ANALYSIS 7

I. Problem Description 7I. Theory of Multiple Scattering 11

M. Markov Approximation 21IV. The Equation for the Mean Field 23V. Parabolic Equation for the Mutual

Coherence Function 27VI. The Equation for Fourth Order Moment 30

Chapter 3 SOLUTIONS FOR THE MUTUAL COHERENCE FUNCTION 34

I. 3Slow" and "Fast" QusiparticleDistribution Functions 34

I. Exact Solution of Mutual CoherenceFunction for the Case H(J) a aj 2 40

III. Numerical Evaluation of Mutual CoherenceFunction for a Complicated Power Spectrum 55

IV. Conclusion and Discussion 93

Chapter 4 THE FOURTH MOMENT AND INTENSITY FUNCTION 101

I. Formulation of the Fourth Moment Equation 101II. The Exact Solution for the Case H(J) =aJ 2 103

Ill. Approximate Solution for the Fourth Moment 110IV. The Boundary Conditions and Steady State

Solution for Plane Wave Case 116V. Numerical Solution for M4 in Plane Wave Case 121

VI. Discussion and Conclusion 128

Chapter 5 THE INTENSITY FLUCTUATION AND SCINTILLATION

INDEX 145I. Definition of Intensity Correlation Function and

Scintillation Index 145

II. Numerical Solution of Scintillation Index forArbitrary Power Spectrum 148

III. Application and Discussion 158

Appendix 1 Renormalization Methods 161

Appendix 2 Convergent Series for VcG 171

Appendix 3 The Limit of Application of the Markov Approximation 181

Appendix 4 Convergence of the Green's Function 185

Appendix 5 Limits of Application of the Parabolic Equation 200

Appendix 6 Numerical Scheme for Solving M 4 208

Bibliography 210

i

TBEOF FIGURES

1Mutual Coherence Function at z u 0 52

Ia Amplitude of MCF at z a ll5 free space 53IC Aplitde f MC atz = .51 reespac. 5

Ib Amplitude of MCF at z a 21 free space. 53

2a. Amplitude of MCP at z 091.5 fre spce=5

2b. Amplitude of MCP at z a 0.; H( E =82 S

2c Amplitude of MCP at z a 1.5 H(t )-a 2 56

2d Amplitude of MCF at z 21.5 H( E )a t 2 57

3 Nrmai&zed MCF vsn. afor different values

of za collimated beamus in free space.* 58

4 Normalized MCF vs. B for different valuesof z; focused beans in free space.* 56

5 Normalized MCI Vs. 6 for different valuesof zo collimated beams H(E )a a t2* a-l. 59

6 Normalized MCP vs. e for different valuesof z; focused beams H( ~ ua ,aul. 59

7 Normalized MCP vas a for different valuesof a; H( £ .alas collimated beam. 60

8 Normalized MCI vs.,, for different valuesof a; H( ) aV s focused beam. 60

ga Normalized MCF vs. I for different valuesof beam width bi collimated beam* 61

9b Normalized MCF vs,, F for different valuesof beam width bi focused beam. 61

Wi

Fi Noinitial slow q.p. distribution functionfor collimated beam* 63

11 Collimated beam propagating in free space#x a 0. 75. 63

12 x a1.5 64

13 MC? Vs. & at different values of zj freespace; x a 0. 64

14 initial slow q.p. distribution functionfor focused beam * 65

15 Focused beam propagating in free space,z3a0.75. 65

.16 z Wl.5. 66

17 MC? Vs. at different values of z; 66

18 intensity function vs. x; collimatedbeam propagating in free space. 66b

19 Intensity function vs.* x; focused beampropagating in free space. 66

20a Power spectrum vs. k 70

2Gb Structure functions vs. *70

21a HQ~ vs. Z for spectrum (c). 71

21b H(~ vs. E for spectrum a~b~c. 71

22a S function for a collimated beam, z00 73

22b S function for a focused beam, z0O. 73

23a S function for a focused beamo 3.0.43753power law spectrum with p u4* 74

23b z a 0.875 75

23c z a 1.3125. 75

23d, z w 1.75. 75

ivi~'

-''~-W.

?i.No. Pg24a S function for a focused beams z -0.43751

power law -spectrum with peaks; 77

24b z a 0.875. 77

24c z = 1.3125. 77

24d az 1.75. 77

25a S function for a collimated beams z-0.4375;power law spectrum with peaks. XO=1. 79

25b z a 0.875. s0

25c x = 1.3125. so

25d a a 1.75. s0

26a S function for a collimated beam; z *0.4375;

power law spectrum with peaks, k'-3. al

26b z a0.875. 81

26c x a 1.3125. 82

264 z a 1.75. 82

27a z a 0.4375P kc a 4. 83

27b z aO.875, I 4, 83

27c za -1.3125, I a 4, 84

274 a .175, kQ a 4. 84

29a. (2 X)h 1 vs. z dotted line; 1(0.&) vs. xsolid lines collimated beam; Hua t21. 86

28b (2 YL )4vs. z; I(o,z) vs. x, focused beam.86

29% (2 C)19 vs. X1I(o X) vs. zj Gaussianspectrums col imat.d bean. 87j29b Focused beau case.* 67

V

ro.

(21E)h 1 vs. z; I(o,z) vs. zj collimatedbeam; power law spectrum, p a 4. 88

30b Focused beam case. 88

31a (2 -)Ji 1 vs. z;I(Oz) vs. zI Power lawspectrum with peaks, k' ul.collimatedbeam. 90

31b focused beam case. 90

32a (2 1 )4 1 vs. z; I(Oz) vs. zi collimatedbeam; power law spectrum with peaks. 91

32b focused beam case. k' 1. 91

33a Collimated beam case, k, u 3. 92

33b Focused beam case; k' a 3. 92

34 Quanlitative representation of S function;collimated beam case. 94

35 Focused beam case. 94

36 Quanlitative plot of structure functionin phase space. 97

37 S function of a collimated beam movingin a random medium characterized by H. 97


39a Quasiparticle density function, at x 0,in I space. collimated beam case. 98

39b Noramlized M(0, , z) vs. Z and z"collimated beam case.. 98

40a "Quasiparticle density function; focusedbeam case. 99

40b Normalized M(O, i , z) vs. and zsfocused beam case. 99

Vi

41 f(r 1 , r1 )vs. rL and r 2 pover spectrumvith p a 4. 126

42 M4 vs. r& and r, Splane wave case;pwr spectrum with p - 4, z a 0.1125. 126

43 z = 0.1725. 127

44 z a 0.24. 127

45 M4 vs. r& and r1 I z - 0.4275 from thebottom of the random slab, L u 0.24. 129

46 z - 0.6525. 129

47. z a 0.8775. 130

48 z " 1.1025. 130A AZ

49 (1) 1> vs. z; (2) I > VS. z; (3)S4 vs. z; collimated beam, H - a . 133

50 a • 5. 133

51 Focujed beam case, a - 1. 134

52 a a S. 134

53 A modulated collimated beam" propagatingin free spacel L a 0.25, a -,5; 135

54 A modulated focused beam case. 135

55 A modulated collimated beam casel a10. 137

56 A modulated focused beam case; a =10. 137

57 S4 vs. z for a - 2 to 16 with a step of 2;collimated beam in free space. 138

58 Focused beam case. 138A2

59 (1) 41>v. zo (2) 4.1> vs. zi (3)54 vs. zj a modulated collimated beam;L u 0.,, a a 5. 139


vii

Fig. No. Page61 A modulated collimated beam propaga-

tiung in free space, LwO.5, a-10. 140

62 A modulated focused beam case. 140

63 S4 vse z for a 2to 16, in step of 21collimated beam case, LuO.5. 141


65 S4 vs. z for L=0.125 to 1, in step of0.125; collimated beam; a =10. 143


51 4 vs. z; approximated solution. 144

68 54v.z; L 30.24. 144

69 54 vs. z; Gaussian spectrum with *r=101 150

70 54 vs. z; Gaussian spectrum with L0.06.15

71 f (rj. , r.,) vs. rj. and r. ; power lawspectrum, p =4. 152.

72 f(r 1 , .r, ) vs. r, arnd r, ; Von Karmannspectrum with p = 11/3. 152

73 f(z1 j g r2 ) vs. ru and rj ;,'Von Karmarnspectrum with a Gaussian peak, k-2. - 153

74 f(r , #rz ) vs. rt and rs ; Von Karmanispectrum with a Gaussian peak, k=3. 153

75 S4 vs. z; power law spectrum and Gaussianspectrum.* 155

76 S4 vs. zj Von Karmannr spectrum. 155

77SA vo. zp Von Karann spectrum with aGIuinsian peak. 156

7S 34 vs- 33 L=012 'r 151 Von Karmanraupsctru and Von Karmanri spectrum with aGaussian peak. 156

FigIlion

79 S4 vs. z; Von Karmann spectrum with aGaussian peak; L a 0.12, T .5,15., 157

80 S4 vs. z for power law spectrum* 157

81 In situ power spectrum. 160b

82 54 vs. 2 for a power spectrum in Fig. 81.160b

A2.1 v~g vs. g24V72> expansion a,b6 177

A2 *2 V~g vs. g'(v>I expansion c. 177

A2. 3 V~g vs.* g',LV'; Gaussian random variable.

A2*4 Vcg vs. gx'kqz.; V"' is a truncated17Gaussian.* 179

A2.5 Same as A2.4; ratio a 3.623. 180

A2.6 Same as A2.41 radio - 3.05. 180

A4.1 g vs. B;i v is uniformly distributed. 188

A4.2 The exact solution of gkr; v = 0.333. 188

A4.3 gkr vs. B;'V* is uniformly distributedover (-a, +a); /,V"% > 2.083. 189

A4.4 Exact solution; 4v, a 2.083. 189

A4.5 gkr vs. BI v is uniformly distributedover (-a, +a); I('v 8.333. 190

A4.6 Exact solution; i 8.333. 190

A4.7 gkr vs. B; V is Gaussian; (v 1)O= 1. 194

A4.8 Vcg vs. Bi is Gaussian; (,v,~o 1. 194

M4.g 1st approxiamtion of Fig. £4.8. 195

£4.10 2nd approximation of Fig. £4.8. 195

* ix

4. , - x ,sJ.

Fia. No.Page

A4.1 V~gvs. ; ~is Gaussian; 4v) > 0. 25. 19

A4.112 V 1s aprxii o ofFg A. 196

A4.12 2nd approximation of Fig. A4.11. 197

A4.14' Vcg vs. Bi V" is uniformy distributed overa range (-0.5, +0.5); 40.>= 0.083. 198

A4.15 1st approximation of Fig. A4.14. 198

A4.16 2nd approximation of Fig. A4.14. 199

ka

Chapter 1 INTRODUCTION

I. Motivation

Wave propagation in random media has been studied over

the last two decades. Heightened interest in this problem has

been mainly due to the large number of problems that arise in

radio physics, acoustice, plasma studies and certain other

branches of physics. From the physical viewpoint random wave

propagation can be analyzed at two levels: macroscopic, con-cerned with propagation in continuous random media such as

turbulent fluids and microscopic, concerned with the scatter-

ing of waves by randomly distributed scatterors such as elec-

trons, molecules, rain, blood cells. Wave propagation in

continuous random media, applies to such problems as thescattering of sound and ultrasound waves in sea water, lightscattering in the atmosphere, radio waves scattering in the

ionosphere and the twinkling of stellar images. Wave propa-

gation in discrete random media is of considerable interestfor such problems as molecular scattering of light, wave

scattered by the rain, and bioengneers may use the fluctua-

tion and scattering characteristics of a sound wave as a

diagnoistic tool. The abundance and variety of such problemshas stimulated development and refinement of statistical

methods for calculating wave propagation in a random medium.

This paper is mainly concerned with the wave propagationthrough an ionospheric random layer.

When a random wave from an extraterrestrial sources,such as a ra4io star or an artificial satellite, passesthrough the ionosphere, its wavefront and amplitude will be

distorted by the density fluctuation in the ionosphere plas-

ma. On propagating to the ground, the resulting phase varia-

tions cause interference to occur and a diffraction patternis set up across the ground. The resulting random variations

.......... ,

-2

of the electromagnetic field are termed scintillations. In

consequence, satellite based couuunications exhibit serious

performance degradation, the degradation being most serious

for propagation paths which transit the auroral and equatorial

ionosphere. The importance of a through investigation of this

problem is obvious.

II. General Introduction and-Review

Propagation models are required to provide satellite

comunication system and radar system designers with a means

to translate the available ionospheric statistics into cons-

traints for the specific systems that are being designed.

Approximate models have been proposed that apply under res-

trictive conditions. Analyses of propagation effects general-ly start with the scalar wave equation. If the dominantscattering irregularities have dimensions much larger than a

wavelength, depolaization effect can be neglected (1, 2).

Let the wave field be represented by E = E exp (-iot), where

w = 2f, f is the carrier frequency, t is the time, E is the

electric field vector and u is a component of E. If the

dielectric properties of the medium change slowly in time in

comparison with 1/f and slowly in space in comparison with

the wavelength X then with a monochromatic source, the

propagation of the wave through a random irregularity region

is governed by the scalar wave equation

a A A

+ LA 0(1.1)

A Awhere k , is the dielectric constant and both (A and

A Aare random variables. enters as a coefficient of the

aunknown wave function Lt * This is the root of all the mathe-

.3_

matical difficulties of the theory, since we don't know how to

find an exact solution of such a wave equation. It is necessa-ry to apply certain approximations that make use of the small

parameters such as the fluctuation of I , i.e. the deviation

from the mean value. It can also be the smallness of the wave-

length in comparison with the dimension of the inhomogeneities.In practice, two types of problems arise: the direct problem,in which one has to find the statistics of waves propagating

in the medium from the known statistics of the medium, and the

inverse problem wherein one draws conclusions about the proper-

ties of random inhomogeneities from the measured moments of the

field. Actually, these two problems are equivalbnt, .i.e. oneneeds to find. the relation between the statistice of the mediumand the wave field.

We shall assume for simplicity that the medium is on the

average homogeneous and stationary. Removing these two 'res-trictions does not give rise to any difficulties in principle.4

Under conditions usually encountered in ionospheric propaga-

tion, the energy is scattered into the forward direction and

(1.1) can be replaced by the parabolic ejuations

A A

2 + AL

where ken k3A u a U exp(ikoz), z is the directionof propagation, 'and Vt ,+ ,the transverse

Laplacian. Both the wave equation (1.1) and the parabolicequation (1.2) are stochastic. The wave properties of A

interest are the average, variance, and higher moment of LiIf and V^ were uncorrelated, these equations could be

Ag N.

-4-

solved easily. However, and U are generally correlated,

the approximations used to provide solutions to (1.1) and (1.2)

are all directed towards modeling the statistical relationship

between 9 and U . If E is small enough, then we cannaturally resort to the method of perturbations, and expand

in a power series in E , or more exactly in (P) Y),

A pertubation series solution to (1.2) can be formed thatseparates S and ' where U describes the fluctuation in

U V = + U * The first term in the perturbation series

for U is the Born or single-scattering approximation, whichapplies only when both & and U are small. When U is not

small, a large number of terms in the perturbation series mustbe summed. The nth term of the series describes the n-fold

scattering and contains the n-fold product F (r1 )... K(r,).A

Thus, in calculating the average , we liave to know the

moments < E (rs)... (r ))> of E of all orders. The reno-

malization schemes of multiple-scattering theory attempt to

solve this problem by a selective summation technique that

leads to readily evaluated expressions for the moment of U

of interest.

The Born approximation is shown to be valid when k • 1 issmall, where 1 is the characteristic scale of the turbulentmedium. As k. 1 increases, we must either take into accountthe higher order terms in the perturbation series or go overto other approximate methods which deal with multiple scatter-ing to some extent. Rytov proposed that an equation for "

1- V be used in place of (1.2). The equation for M isthen given by

-2 VP12 + i

K -jjj''' j -

-- 5 -. ----

the equation has seperated a and E so that the correlationbetween * and is not explicit. A perturbation seriesexpansion for W , representing the fluctuation in j canbe formed to solve the nonlinear equation ( in Vr) Thezeroth order equation represents free space propagation. Thelst order equation for I can be solved yielding the Rytovapproximation, which is also called the method of smooth per-turbations. The Rytov approximation is a weak scintillationapproximation that hold for small <9L>, where X. is the flu-ctuation of the logarithm of the amplitude, it has a widerrange of validity than the Born approximation.

The generally accepted model for ionospheric propagationthru a turbulent ionosphere has been the thin phase diffrac-tion screen model (3). The ionospheric irregularities perturbthe phase of the field at the layer and their effect upon thewave beyond the layer can be computed using diffraction theory.Mercier (4) and Briggs and Parkin (5) introduced a Gaussiancorrelation function to describe the phase fluctuation at thescreen and related all higher moments to the second moment.Spectral analyses of observed phase and amplitude fluctuationshave shown that the electron-density irregularities have apower spectrum that may be characterized by a power law shape.Rufenach (6) extended tha phase-screen theory using a powerlaw spectral shape for the irregularities.

For the weak scintillation case, the Born, Rytov and thethin phase screen approximations are all applicable. For

strong scintillation and a thick layer, the effect of multiplescattering on transionospheric signals has to be taken intoaccount.

Several tecbniques have been proposed to deal with the

-... I~10

-6-

strong fluctuation problem. They are the diagram method

(DeWolf 1968, 1977; Tatarski 1971; Frisch 1968) (I, 7, 8) the

integral equation method (Brown 1971, 1972), (9#10), including

the Dyson and the Bethe-Salpeter equations, the extended

Huygens-Fresnel principle (Kon 1970; Clifford 1974) (11, 12),

and the parabolic equation method (Tatarski 1971; Furustsu

1972) (1, 13). In this paper, we shall apply the parabolic

equation method to the case of radio wave on propagation

through an ionospheric slab. First the statistical moments of

waves are obtained inside the random slab. Then the free space

*propagation of resulting randomly modulated waves is analyzed

from the bottom of the slab to a ground receiver. The geometry

of this problem is shown in Fig. 1.

I.'U.,

. i.

-7-

Chapter 2 THEORETICAL ANALYSIS

I. Problem Description

Ionospheric, scintillation has been observed at frequen-- ) cies ranging from 10 lz to 6 GNz. The irregularities in the

ionosphere which produce the fluctuations are believed to bemainly in the F-region, the layer of strong irregularities is

often 100-500 ko thick ranging from 200-700 Xm in altitude.The electron density fluctuations are often of the order of a

fey tens of per cent and can be as high as 70 per cent. TheF-region irregularities are usually regarded as a stochastic

process and characterized by three dimensional random func-tions which are assumed spatially and temporally stationaryover the increments of interest, One defines the spatialcorrelation function BN(r") of the density function by

4 (N~ N()B < Na'py (2.1)

where r * (x,y,z) is the position vector, £" r s - Z is thecorrelation lag vector, 4 %;Ois the rum density fluctuation,and 4 > denotes the ensemble average. The relationship

* betwen the three dimensional power spectrum of the density

fluctuation fN(c) and the density correlation function isgiven by the Fourier transform pair:

Xt

ZJ > (2.2)( 2 n ) 3 ., $ -

t. -8-

BN (~ = ~ )e(2.3)

the power spectrum is often characterized by a wide range of

wave numbers, and hence designated as a wide-band process,whereas a narrow range of wave numbers associated with a Gau-

ssian spectrum is designated narrow-band process. The wide-band process is important since it is inferred from the in-situ results * In the F-region, the wide range of waveniuabers

corresponds to dimensions ranging from a few meters to hun-dreds of kilometers. If a Gaussian spectral shape is assumed,it is characterized by one dominant scale approximately equal.

to the Iresuiel wave number. (about 2k) - 1 ) (Brigg and Parkin1963).

Refering to Fig. 1, let us consider a time harmonic

radio wave incident on a region of ionospheric irregularitiesat z z o . L is the thickness of the slab. Inside theirregular region, the relative dielectric permittivity is

given by

(2.4)

where one models a collisionless ionospheric plasma byi a

Wp. (3)Eo J) : - oA (2.5)

2 ' (2.6) I- -Wp~.a r N ox)1(6W1 N0

W.( 5) is the plasma frequency for the background electrondensity profile, "(r)/N o is the percentage fluctuations ofthe electron density. In the following, we shall assume thatN(r) is a homogeneous, isotropic random field while N. is

taken as constant.

The wave first travels through the ionospheric slab oflength L, at z = zI the modulated wave enters the free region,and it is detected at z = zp . Therefore, there are tworegions: in the first region, the incident wave at z,,is deter-ministic and the outgoing wave at z1 becomes stochastic. Inthe second region, the entering wave at zj has a stochasticmodulation but the medium is deterministic,

Writing the scalar wave field as

E L e (2.7)

where the exp(-iwt) dependence is omitted, k 1Ee-4. K* , and(r) is the complex amplitude of the wave. Substituting (2.7)

into the wave equation (1.1), we obtain an equation for u (r)

-10-

L #0 0o (2.8)

2 2Awhere 1 _ - . If the complex amplitude u varies

markedly over distances of the order of the inhomogeneity scaleIAA

1, the second derivative a UA/ is of the order of 12

On the other hand, the term 2ik1,3/33Z in (2.8) is of theA Z

order of u/A1. Therefore, for X < 1, the term a t/W3 issmall compared to the first term in (2.8). Thus, one can re-

place VZA in (2.8) by the transverse Laplacian V U and

obtain the parabolic equation

2+ LA+ E (A o (2.9)

This is the starting point of our analysis, in the later

sections, we shall examine the statistical properties of u Athey are specified by the infinite set of correlation functions

A AN

ii~~~~~ (2. 10) **IAC~y

- 11 -

The problem is to manipulate equation (2.9) to obtain equations

for r mn in terms of the statistical properties of (r).

II. Theory of Multiple Scatterina

The general theory of multiple scattering has been deve-

loped by many authors. (Bourret 1961, Apresyan 1973, Tatarski

1971, Marcuvitz 1974, DeWolf 1979). The Greenis function me-

thod, which had previously been investigated in quantum field

theory, was applied to the problem of wave propagation in a

random medium. People have used the Dyson equation for the

average field, and the Bethe-Salpeter equation for the cova-

riance B =<A(r 1t) u(ra)>. However, one cannot derive such

closed equations by averaging the original differential equa-

tions for the field because moments of different orders are

coupled together. Bourret (14) was the first person to apply

the graph technique to scattering of waves in a continuous

fluctuating medium. He assumed that the parameters of the

medium fluctuate according to a normal law and are statistical-

ly independent of the sought field. This led to an approximate

expression for the effective average dielectric constant pro-

portional to the correlation function < g(r 1 )E (r a)> of the

medium. A more detailed derivation of the equations for the

average and 2 point correia-tion- assuming -a normal law for a

fluctuating medium is due to Tatarski, and to Frisch who

admitted deviations from the normal law.

The utility of series representations is dependent on

the rapidity of convergence. Operator techniques employed in

the spirit of the formal theory of scattering was shown by

IMarcuvitz (1974), to provide alternative and formally exact

representations of turbulent field quantities and their n-

point ensemble averages. In appropriate parametric ranges

these formally exact expressions are expandable into rapidly

I'//p

-12

convergent series. In this section, we shall apply this

technique to the average equation and investigate its conver-

gence.

In abstract the wave equation (2.9) can be written as

A

(L v =.V(2.11)

where, in a , z space the unperturbed operator Le . and

the perturbation V , whose average .V )= 0 , are represented

by

equation (7) is a generic operator equation for the field, with

the initial condition acting as a impulsive source applied at

z - 0, It is convenient to define a stochastic Green's func-.

tion such that

(L. V)G 1. (2.13)

where in 7 , z space

- J1* -- .. p W V

............................

-13-

and W x) is the Dirac delta function. For uniqueness, G will

be distinguished by the requirement

(2.1S)

If we introduce G oas unperturbed operator inverse to the

operator L* , and defined by

Lo oT - (2.16)

where the domain of La is such that

(2.17)

Aas is the case for G.

A

The ensemble average G of G is taken to satisfy

(L V )G 1 (2.18)

and depends on the "smoothed" scattering operator V. definedbelow. Eq. (2.18) provides a nonlinear defining equation for

G# the nonlinearity arising from the dependence of Vc on G. 4

'+- '

-14-

A stochastic operator Tc , with zero ensemble average,

is defined by

T c~+ GTr- (2-19)

which represents the multiple scattering in the smoothed back-

ground G. One can derive the expression for Tc in terms of

G and a power series in V to 4th order in V : (Appendix 2)

:r +V~ -<VTV> t CT C-4V- <¢e>-< r

- CTVCTV> <VV rVGt~ C-r > V >-C< C >C@ 2.0

and correspondingly the operator Vc = ( T ) is represented

as

t <V'-CT< V"'q V> T- > <V'WT , >CTV C V(2.20)

r 15

the expansions are in terms of the average operator G, which

in contrast to Go is nonsingular. They are more rapidly con-

vergent than the perturbative expansions in terms of G0.

As a simple examples used to study wave propagation in

a one dimensional random medium, let us consider a differen-

tial equation of the following type:

A

Cd (2.22)

If V(z) - Vl." (I+W(z)), where V is a real mean frequencyand (z) is a centered stationary and Gaussian random func-

tion of z, then it represents a randomly modulated oscillator

whose correlation function is

13 c 3-1) < ' 3€) 3 1)', >: . (2.23)

The model will be used as a check for the approximate expan-

sion in equation (2.21).

The Generic stochastic Green's function for (2.22) is

defined by

A

(2.*24)

.. .. ' ..c ... , , ...' -._

-16-

To find the average Green's function, we must calculate

where U(u) is the step function to insure the uniqueness of G.

Here SSV(Z)d- , being a linear functional of the centeredGaussian random function V(z), is a centered random variable

yhence

< > e 7-

(2.26)

Since

S) B~ -Z')(2.27)

We obtain the final result

CTU (-31) (2.28)

L-5rX4 CI

- 17-

If ve use the correlation function

(2.29)

the integration in Eq. (2,28) can be performed exactly to give

" e (2.30)

If we approximate VcG by taking the 1st term in eq. (2.21),:Li.e.

(2.31)

or

- ) U (3- ) J3 g(2.32)

Substitute eq. (2.30) into (2.32), one obtains

V - r P J(3 L)U(3-eVC2 (2.33)

since

G ( 1, 1) = - A. (2.34)

therefore

VCC - A Cq (3,3') U (I-S')(2.35)

the solution of eq. (2.24) can be easily obtained to give

2

CT,9=- e U( ~~ (2.36)

which is identical with the exact solution (2.30). It is an

interesting result and leads to the conclusion that i If Vis a centered stationary and Gaussian random function, with

a correlation function defined by (2.29), the first term ineq. (2.21) gives the exact expression for VcG o i.e. thehigher moments terms do not contribute to the expansion.

Furthermore, since

(2.37)

+ ...............

S19

it can be shown that in this particular case

(VG'> > (2.38)

Let us consider a more complicate correlation functions

e( - 1)- (2.39)

the integration in eq. (2.28) can be evaluated exactly to give

T, e T

for short-range correlation,

:e e- j, , '(-' i -1T 2.1)1

- T (33-') (.

1-;Tt- e

61T

-20-

Therefore

2T

-T -(2.42)

One can examine the 1st approximation in eq. (2.31) by substi-tuting (2.42) into the expression for VcGt

VC1 --C T , (2o43)

- Te -J (243)

vhile the exact solution for G gives rise to the expression

2 - .T (3 -3') (2.42)-~ eT

Compare (2.43) to (2.44), one observes that VcG <V"(GV">is agood approximation when the correlation range is small.

Thus, one concludes that for a centered Gaussian random

variable V with a delta correlated 2-point correlation func-

tion, VcG can be exactly expressed by the 1st term of theseries expansion (2.21), i.e. VcG <G7>G *CVG G. For

- 21 -

mall range correlation function, VcC; v. -<G > G is a goodapproximation. The representation (2.31) can also be applied

to the case of large scale correlation, but then the fluctua-tion % must be small. Otherwise# the higher moment terms

i have to be summed to get a good expansion.

II!.Markov ApDroximation

In order to proceed further, additional approximations

for the random medium statistics must be considered. A very

useful approximation is the Markov approximation which gives

correct results in cases of interest even in the limit ofstrong scintillations. The mathematical representation and

physical interpretation of the Markov approximation is basedon the following observation:the correlation of the dielectric

constant in the transverse direction has a direct bearing

on the transverse correlation of the field but the correlation

of the dielectric constant in the direction of the wave pro-

pagation has little effect on the fluctuation characteristic

of the wave. Hence, one can assume that (r) is delta corre-

lated in the direction of propagation,

dw (2.45)

i.e. one can treat wave propagation in a random medium as aMarkov random process. It is equivalent to assume that theturbulent eddies are like flat disks oriented normally to the

propagation path. Tatarski (1971) gives a derivation of thisapproximation which provides the limits of applicability of

the Markov approximation. For typical ionospheric parameters,

*i' a :%

22 -

f

these conditions are generally satisfied for an incident wave

with frequency about twice the maximum plasma frequency or

higher.

The relationship between the function A('t - E') and the

power spectrum W () is given by the followings

A IV)2JJ ( ) dw (2.46)

If the turbulence is isotropic, i.e. (k) * (k), then we

get

A (F- ) T2.ZS Y (f (F'-F')) ~R 4o, (2.*47)

A(C) is also related to the correlation function B( z,P )through

A .- '): i B M[,F-F') ;a* (2.48)

The limit of application of the Markov approximation was inves-

tigated by Tatarskli (1971) via variational techniques. We

derived identical results using the operator method as present

in Appendix 3.

-23-

IV. Equation for the Mean Field

In sumary, our analysis of strong scintillations will

based on the following assumptionst(1) The parabolic approximation. (limits of applicability of

the parabolic approximations will be derive in Appendix 4).

(2) (r) is delta correlated in the direction of propagation.

Let us take the ensemble average of parabolic equation

(2.9)

r (2..49)

The last term in (2.49) will be expressed as follows:

< E(r) q) Q51)> = ()<q >

which is valid if the incident field at z - 0 is deterministic.One notes that (r) - 29(r) is the fluctuation of the refrac-

AP A

tive index. To get a differential equation for ( (r)>, under

the assumptions (1) - (2), one obtains the following expre-

smion for Vca

v= < VTV> - < V °(.50)

- 24 -

If the fluctuation is a centered Gaussian random field

and characterized completely by the correlation function

E (r) U(r,)> , then (2.50) is exact.

To ascertain Vc, the unperturbed operator Gomust be found.

Therefore, we have to find the Green's function for the homo-

geneous equation

A

a i a = 0

* 2 2 (2.51)

Setting

~~0 J 4 ( ,1 (2.52)

One can write equation (2.51) as

A 2 AC) t_ 4_1 * APa ,2 Jo (2.53)

From (2,51), we define Go by

. +-__-r o _ (2.54)2 -l

- 25 -

Setting

=(2) 2 (2.55)

where

o<(2.56)

From (2.55), ie obtain

- -- (2.57)

- a (2.58)

For z~zl, ve note that

(2.59)

Using (2.57) and (2.58), one sees that

• - A(2.60) 1i- -- .... ../ I

-26-

One then observes from (2.59) and (2.60) that A = -i and

z(2.61)

From (2.54), one sees that for z-. z'

(2.62)

the above is also a good approximation provided k2 lz-z 1/2k,<<1.Therefore, by using (2.50), we obtain

2

,- oJ "(2.63)

where = /2. Setting z - ze we obtain

vccc0o (2.64)

Applying the Markov approximation, one obtains

-27-

VC A. 2' A (t) ~(2.65)

Therefore, the average equation (2.49) becomes

1 2 02 A(,o)<q->= 02 - k 2 (2.66)

V. Parabolic Equation for the Mutual Coherence Function

The parabolic equation (2.9) and its conjugate can bewritten ass

(2.67)

2A

U ~(3, f)O (2.68)

Multiplying (2.67) byp '(z,T') and (2.68) by qi^(z, ) and sub-tracting the second equation from the first, we obtaint

Y 0,

- 28 -

The mutual coherence function, < 'q) (z,? %) ()( f> is -

defined as the cross-correlation function of the fields 441andp i a direction transverse to the direction of propagation.

Using the operator notation described in section II, we

A I

revrite M in the form:

(L0 'Q M (2.70)

where

Lo A. +

a 2To

AA 8(2.71)

An unperturbed Green's function may be defined as

~2vL2- v

(2.72)

S -ere

- ..>!. .

-29-

Applying the samte analysis as described in section rV, one

finds approximately that

dw (2.73)

The smoothed scattering operator for the ensemble averaged

Green's function, defined as in (2.38) can be expressed by

Vc=K~rV> <Cv~ 0V>(2.74)

and wri14.ten ass'

(20.75)

Setting z - = ,one obtains

vr_ ) - 4,(20.76)

MI

-30-

Applying the Markov approximation, one obtains

(2.77)

Equation (2.70) then becomes

_j. 21A .~ [A ,)-A(5F-'] M

where

M2.3 (2.79)

VI. The Equation for Fourth Order MomentA

On considering the parabolic equation for f (z, P) ,-

(Z. ) (Jj1(z (p4*)zA , f4 ), respectively, we can obtain anddm

equation for

AA A

(2.80)

- 31 -

as in the form

v-I.--. = 0(2.81)

where

L. F2.) -n U,-7 - -

&(.

as before one defines an unperturbed Green's function defired

by

- 1-V

1W OW d ' (2.83)

Applying the same analysis as described in section IV, onesees that G. can be approximated ass

C% (

~ 'AI

-32-

Therefore,, the operator Vc for this case can be approximated

as

OwT~ &3 T3qc?

+ (2<.o5)

Upon application of the Markov approximation, one obtains

-A [2 Mo-~)- A ( 2 J,4)t-A~?-)(.6

t- A '~~-~By (2.28),:one then obtains the equation for the fourth moment

24 2 as:o

(2.87)

-33- 1

vhere

2A (,o)- Ac P- A- A(,

-tA (3 , f4-' 43

(2.89)

- -.' -.,

- 34 -

Chapter 3 SOLUTIONS FOR THE MUTUAL COHERENCE

FUNCTION

I. "Slow" and "Fast" Quasiparticle Distribution Functions

Equation (2.78) is a single mode type of wave problem

dependent on the statistics of the random medium. In order to

obtain the solutions for the mutual coherence function, we

shall define the following Fourier transform with respect to

the "fast" variable I = x, - x s , and the "slow" variable

x = X1 + x2/2. They are so called "quasiparticle distribution

function" or "Wigner distribution function" described independ-

ently by Marcuvitz (7$) and Bremmer (34). One introduces the

double Fourier representation.

AA A A ~(ALX±. IXI)

A AA

where = * (k 1 , z) and = (k& , z) are, respective-ly, the Fourier amplitudes of '.= Ji (x. , z) and %=P(x z , z).

Defining the "slow" ( x or k ) and "fast" ( f or k ) spatial

and wave number variables via

xl~AI z- AL2 P

and (3.2)

= , fitx +

- 35-

whence

and (3.3)

One rewrites equation (3.1) as

LPIS (3.4)

where

)27L (3.5)

is called the "fast" quasiparticle distribution function.

Alternatively, one writes equation (3.1) as

<-> (3.6)

where

• 4t (3.7)

-36-

and termed the "slow*' quasiparticle distribution functionalso called the ambiguity function.

The reality property F(k,x,z) = F * O,x~z) of the fast

quasiparticle d~f. follows readily from eq.(3.5); however,

positive definiteness is not in general assured. S(KE3)

is only real when q)( (X) is an even function which can be

proved using eq. (3.6). S(K E a 3), just as F~kpxpz), can be

negative for some value of k and .However, it should be

noticed that intergrals of .SF(&x,1)i~i 1is always positive. 2

It is interesting to observe the following properties of

F(ksx,z) and S(H( I~3(a) one notes that from eq.(3.4) and (3.6)

2 YE (3.8)

which suggests that S ( K,E, can be treated as a charac-

teristic function of F~kpxsz)t in Jk,x space, and vice

versa. The use of S( K pt , 3) or F~k,x~z) mainly dependent

on the nature of the problem to be solved.,

(b) for *0 and X .0 equations (3.4) and (3.5) yield

2 7E (3.9a)

-ell,

-37-

(3.* 9b)

whence on infers that

F eqcjx (3.10)

which suggests the identification of F~k,xgz) as a number

density of "fast" quasiparticles in lc,x phase space.

(c) Similary, for xc 0 and kc 0 and tp(x) is even, equations

(3.6) and ( .7), give

AJI14) 4 (3.11a)

2 12>(3.11b)

Which also suggests the identification of S( K I, as a

number density of "slow" quasiparticle in K, space.

also,

- 38 -

jSY 'E (3.12)

(d) One notes that

2 I "(3.13a)

35S (o,o, 3N, (.1b

where Nf(z) and Ns(z) denote the total number of "fast" and

"slow" quasiparticles respectively. Both Nf(z) and Ns(z)can be obtained from information about F or S.

(e) A coarser description of the dynamics of a quasiparticle

system views the overall system as a single Macro-particle.

It is of interest to find the average position, momentum,

sizep etc., of the overall system. The average coordinat-

es of quasiparticles are defined, in terms of the distribu-

tion functions F in kDx space, by

2

- 39 -

'( = N-(-D " 2 -.

(3.14b)

and in terms of S function in K , space

- N5C ) 2 (3.15a)

mI S

l<(3 N 5 CS SC' (3.15b)

Similarly, the higher moment averages

N2) (3.16a)

Ns 2. (3.16b)

- 40 -

It can be easily proved that

?n* YL

X(- n F (z3. - 17b)

Therefore, a macro-description of the overall system can

be found if either F or S is known.

II. Exact Solution of Mutual Coherence Function for the Case

H( a f ( Cf. Furutsu, 13)The defining equation for the Mutual coherence function

has been derived in section (2.V). Introducing the "fast" and"slow" coordinates

= __ __= -,

2.

one obtains

x - -. TT.77 777;- 4 IV M2 , o

o 13Zi8

41

AAwhere M,(5 X ' 

We now consider the Fourier transform with respect to X., which

is exactly the "slow" quasiparticle d.f. defined in section I.The resulting differential equation for S ~ ,)becomes

- - + Hl J(3.19)

where

4- (3.20)

Equation (3.19) has the form of a collisional "kinetic"equation indicative of a distribution of "slow quasiparticle"in K , I space at a plane z. The "Characteristic" trajectoryalong which the "particles" move is defined by

Aqo cL, (3.21)

On a quasiparticle trajectory, one observes that

-H( ) - (3.22)

-42-

or

(3.23)

This implies that the number density of such "slow particles"

per unit volume of the phase space K , E decays along the

trajectory defined by equation (3.23).

For waves propagation in free space, the defining equa-

tion for S becomes

S <(0~), o), a)St S(3.24)

which provides the constancy of S on a quasiparticle trajectory.

To illustrate the applicability of the slow q,p view we

shall consider wave propagation through a random medium charac-

terized by H( E ) = a S2. As an initial condition, we choose

for analytical purposes a planar beam of the simple form

q(x, o)= e (3.25)

The slow quasiparticle distribution function at z = 0 follows

from the transform relation (3.6) as

.......

-43-

2bS~~~ ~ x ,o)= e

2 2 (3.26)

-x- 6e 4Ve 4

At a subsequent distance z the S function follows from (3.21)

and (3.23) as c

s= S ('. .(o), o) e

-O.J - S'j3z~cK,~) e o

.-H ) . • (3.27)

-* Ke e *j-= fK -0.(4b.4 3 E,

3~3

e

+L 2

',,.,_ ,--

-44-

For the beam wave case, the wave at the aperture z-O can

be assumed to have a Gaussian amplitude distribution with the

beam size WO and radius of curvature RO for the wave front.

Thus, at z=O the wave is given by

IIx (3.28)

where

r (3.29)

Introducing the normalized coordinates

-- = 2 - --

one rewrites (3.28) as

(.xa) 2 e (3.30)

Substituting into (3.6), one obtains (for simplicity, we will

drop the bars)

S .'

.:

- 45 -

e- ,.Kx: ci iti .f

z (3.31)

2 C t______T e 4at Yl

Using equation (3.23), one finds

(3.32)

-. (E2S - +) 4 *" K2 )

note that (3.32) can be reduced to (3.27) by setting r - 1

and o 0.

The Mutual coherence function at z can be calculated via

(3.6)

M2' 21It

=JeA~ 8 ~ C7 Rj (3.33)'.2 I

- 46 -

where

4 4 o(r'

2 2 4 r

whence on integration of (3.33), one obtains

_ S (.iX+ I C)?

' 2 (3.35)

where A,B and C are defined by (3.34).

Equation (3.35) is the exact solution for the Mutual

coherence function at a distance z. It is of interest to con-sider as a function of Z the beam size ratio, axial intensityratio, and the normalized MCF at x = 0 for the following cases:

(case a) For collimated beams (. 3i 0) propagating in free

space, one obtains from equation (3.34)

r2 (3.36)

4g&2

- 47 -

t

Then the beam size ratio (I.=O) is defined by

W2'C3) =l, -=O 2 4 - + ,"W 2(o) /.. (3.37)

increases with z.The axial intensity ratio, defined as the intensityof the wave at x - = 0, becomes

and decreases with z.The normalized Mutual Coherence function (NMCF) is animportant function for the measuiement of coherence.The integral of NMCF, cJ , gives ther'..,,)value of correlation scale. At x = 0, it is give by:

MC1 ,)= e C' "

M (o,o,3) (3.39)

(case b) For a focused beam (4>0) and a divergent beam(4£ < 0 ) propagating in free space, one has

t

-48 -

(3.40)

The beam size ratio is then

Wei(o) ( 3.41)

Whence for the focused beam, a minimum beam sizeI occurs at a distance z = z' such that

C = 0(3.42)

Applying equation (3.41), one obtains

-" t0 OCA (3.43)

and thus

i --

- 49 -

[1 f)WO J FL ,, ,.- Z ) )3.44)

Using equation (3.33) and (3.38), we have for the

axial intensity ratio

I(0) - drA 0~3.( I 231) (3. 45)

Irand the normalized MCF (x = 0) is then

M2(o,,3 = 4

(3.46)

We note that the correlation width, 2 rT (I-

is minimum when z - z 4 f , which implies

that the correlation length of a focussed beam reducesto a minimum at the focal point and retains its co-

herence beyond the focal distance.

(case c) In the presence of turbulent medium, a> 0, the beam

size ratio becomes

V.ae_

-50-

W. (0)

(3,47)

where "a' measures the strength of turbulence. For

the collimated beam, a(; = 0, the beam size ratio

reduces to t 1 ' C- *± -- which im-

plies a broadening of beam size with increasingdistance. If.(;>O, indicative of focused beams, the

minimum values of (3.47) will occur at

4 r (3.48)

If ot;<O, the beam focuses at negative values of z,

i.e. it diverges with increasing z. The axial inten-

sity ratio is

IWs

~ 2

For a divergent beam, p(<0 , the axial intensity ratio

decays with increasing turbulent strength or increasing

z. For a focused beam, .(, 0 , I(z)/I(O) will reach a

minimum value at z z' given by equation (3.48).

- 51 -

The normalized MCF given by at x = 0 is equal to

_e 771 r e

We observe that for collimated beams propagating in a random

medium, wherein .( = 0 and a 70, the wave will loss its cohe-

rence (the correlation length reduces) with increasing z.

For z>0, the focused beam will reduce its correlation length

before the focal point. After passing the focal distance, it

broadens until turbulent effects dominate at which point the

beam again loses its coherence. For the divergent beam case,

the wave will increase its correlation length and then lose

its coherence when the turbulent effect dominates.

To illustrate the evolution of Mutual Coherence Function,

plots of M(X , I ,J ) in the X , E space, for the case4= o(- 1,a a 0 representation of a focused beam propagating in free

space, are shown in Fig. 3.la-3.1b at distance z a 0.5, 1, 1.5,

2. For a initial MCF depicted in Fig. 1, the focusing effect

at z a 0.5 is apparent.

Figs. 2a-2b display the Mutual Coherence Function for a

focused beam in a random medium with a = kg = 1, = - 1, thebeam assumes a minimum width in x at z' according to equation

(3.48) and then spreads in the x direction. Because of the

turbulent effect, the MCF decorrelates as the wave is propaga-

ted and the width in space is reduced. The amplitude of

- 52 -

ii

- ..3~-

4.) *I.LzII n Oc~~4* ~-f ~-S

*

N~ 0I~.4 II0

3 ooU('I

II 4A1

b~

I:Ii f- f=4

* .4~1 -4, 1

C- U * fad

t

* I

-- S -

I I

EJ.t- S. 000000-01 g- 0. 0 Kgo S. I004

7.004.00

3.00

fifC. £a

-4 .7.7

1.00

-7.00 -3-60 0.0 3.50 7.00

Fig. la.Amplitude ofxMCF at z = 0.5; a focusedbeam propagating in free space;oiv = = 1.

diat. 1.00000 ke. 0.0 rea 5.000000.41k. .06000 kb- 1.000

Plutuat IC"i. Cuntion

7.004.00

3.00

3.-4.

1.00-.00. .S .

Fig. lb. Amplitude of MCF at z z 1; a focusedbeam propagating in free space; uL 3.

5. Lm -..

-54-

diets 1.50066 ke .0 .6.008s0hee 1.66666 kb- 1.00 K. .666.4

Muttust coh. function -

4.0

3.06

M. So

-7.66 -3.96 0.0 3.96 .6

Fig. 1c. Amplitude of MCF at z = 1.5; a focusedbeam propagating in free space; *Ir=oLA.= 1.

diets 2.66666 kgo .0 KOM wg.0666g.-el)cs 1.66666 kb- I.666e"

I'htut a"co. function

.. 0

4.00

3.00

1.0670

-7.00 -3.56 0.0 3.36 .0

Fig. id. Amplitude of ?4CF at z w 21 a focusedbeam propagating in free space; .uat1

- 55-

the MCF decays with increasing z. Normalized MCFs calculated

by use of equations (3.38) (3.45)-(3.50) are depicted in Figs.3-9. For the case of a collimated beam in free space, the

normalized MCF is broadened with increasing z (Fig.3). For

the focused beam, the correlation length reduces to a minimum

value at z = 0.5. Beyond the focal point, the beam starts tospread and increases its correlation length (Fig.4). Fig. 5

displays the normalized MCF for a collimated beam in a medium

with the turbulent strength a = 1; the correlation length

decreases as z increases. The same case for a focused beamis depicted in Fig. 6. Due to focusing, the width of norma-

lized MCF becomes narrower for small z, it then broadens for

some distance because of the diffraction effect, finally,turbulent effects reduces its coherence length. Fig. 7 and 8

shows the MCF different values of "a" for a collimated beam and

a focused beam respectively. In both cases, the MCF become

more decorrelated as the turbulence becomes stronger. The

effects of turbulence for different initial beam sizes are

shown in Fig. 9a-9b for collimated beam and focused beam. The

correlation length is larger for a wider beam size , as one

expects.

III. Numerical Evaluation of MCF for a Complicated Power

Spectrum

To obtain MCF for an arbitrary power spectrum, one can

apply the same procedure as described in section II. However,

because of the complicate integration involved in evaluating

the function H(E), it is not realistic to solve the problemanalytically. A numerical procedure via the quasiparticle

distribution function approach will be used to find the disireinformation.

It is necessary to develope a numerical scheme which cal-

culates the slow quasiparticle distribution function S(K ,1 , )

- . - .. w . - ~ -.-

-56-

dist. S. e8.- kg. ..00 Kg.- S.000.-01

Miutual coW. function~

7...

% 10

3.00

0.S-7."0 -3.50 0. 3.10 7.00

Fig. 2a. Amplitude of MCF at z =0.5; a focusedbeam propagating in a random medium with H(W=a~ 2 a = ,o o. 1dist- 1.e0e0 kg- 1.009~ Kgf- 6.00000.-41km- . 000e.. kb- 1.000

Mutu~al coh. function

7.00

MCx. C

-7.00 -3-90 0.0 3.90 7.

Fig. 2b. Amplitude of MCF at z =1; a focusedbeam propagating in a random medium with H(I)=all; a a= v oo 4;,..=1

-57-

d~atu 1506kgCu 1.0060O Kom S.0000,-O1he. 100606 kb- 1. see"Iutua coh. functiont

7.064.00

3.00

IICz. e4a.00

0.0-7.06 -3.50 0.0 3.50 7.06

xFig. 2c. z =1.5S.

dist* 8.000" kg- 1.00 Kg- 5.000ft-01k-1.0600s kb- 1.06096

Fhutual coh. (unction

4.00

- * 3.06

-7.00 -3.50 0.0 3. SO 7.06

Fig. 2d. z =2.

-58-

kg. 0.0 kt.- B.5*OGO Kt, .8.5OG-61k 0 0.0 kb- 1.00OO

Mu~tual coh. functiAon

0.0 1.50 3 . 4.50 6.00

Fig. 3. Normalized MCF vs. I for different valuesof z; a collimated beam propagating~ in free space.

ko.0. kt.- 2.50000 Kt.- 2.Se.GG.-41

Afuasl coh. function

1.00

0.00.0 .Se .00 .SO6.00

Fig. 4. Normalized MCF vs. 1 for different valuesof z; a focused beam propagating in free space.

- 59 -

g1.0"- ,4. O000 W t. 5.0006-el4e .0 kbe 1. 000-ut:at Cob. function

1.00

'?, S8s.41

° $. 00-el

61-0

0150 3.00 4.50 6.00

Fig. 5. Normalized MCF vs. t for different valuesof z; a collimated beam propagating in a randommedium with H( E ) = a 1, a = 1.

kg. 1,00000 kt- 4.00506 Ki.- 5.00006.-41ke- 1.00000 kb- 1. 60e0e

Iktual coh. function

1.00

7. SOW01

S.06 91-O

AS3o,,I

0..4" ..0 1 5 0 3.00 4.1;4 .00

Fig. 6. Normalized MCF vs. P for different valuesof z; a focused beam propagating in a random mediumwith H( ) a', a .

-60-

dia.. .6660 kg. 3 006 Kg- S50061Okes k. be 1.00666

Mlutual~ CoJ. tfunctiont

a. 5"6OlS

3.503.0.-50el0

0.6

Fig. 7. Normalized MCF vs.P, for different valuesof al a collimated beam propagating in a random mediumwith H( t ) = aI"

it-1.6606 ic,- 300660 Kv- S-000601e-1.66666 kb- 1. 00000

mutual coh. functxon for

1.00

7.50.-el

MC 8 ).-Me

0.0 1.50 3.0 4.30 6.00

Fig. 3. Normalized MCP vs.t for different valuesof a; a focused beam propagating in a random mediumwith H( ).a .'

- -A

-61 -

%ltuqal cok. funiction

h..0K-1.0003

7 0.-41

0.0 )IIS.0 .S 60

Fig 9a omlie.-ev.Ifr ifrntvle

of bemwdt ;a olmte-em rpgaigi

a random meiuwihHE)=aP4a

Fig Oat NomlzdM0 s o ifeetvle

of beam width bj a fcllmed beam propagatging iaarandom medium with H( )= a ', a'-- 1.

-62-

for a given S ( 1 4 , C ). The corresponding results for inten-sity, beam size ratio and normalized MCF can then be obtainedvia equations (3.6)'and (3.11). We shall first examine thecase of a collimated beam propagating in free space to checkthe accuracy of the algorithm being used. The z = 0 quasipar-ticle distribution function which follows from equation (3.31),i.e.

S .1,o) -4 e e _

(3.51)

is depicted in Fig. 10 for the case *v= 1, A,=O. The quasi-particles are evenly distributed in the intervals -7- 1 (+7,-7< K< +7. At a subsequent distance z the distribution func-

tion follows from (3.24) and (3.21) as

5(K,,') = 5 (K, -K', o)(3.52)

i.e. the quasiparticle density is redistributed when evolutesin 3' , as the quasiparticle spread in 8 the probability dis-tribution function (d.f) with coordinates S( K, -K&, 0), atz .0 "moves" to the new location S(K , , ") at z = zI. Theredistribution phenomena enable us to interpolate every pointin K , space from the initial d.f. to the final d.f. at z=z'numerically. Figs11-12 display the S function at z = 0.75,1.5 respectively. The quasiparticles with higher K "move"faster than those with smaller K . The resulting d.f. thusspreads along e directions. The MCF at x m 0 as a function of

Sfor distances z corresponding to the associated S(s,E, )

-63-

DIST= 6.6 KE= 2.90090 KBm 1.66666E-61 OUIOTHe 1 66988

0~n .6 1 OF INTER& .40.8 OMC 6c .8 4T* 9.37568E-03

4.6

K3.66

S( KA8,Z)

-7.6 -3.50 -4.7?E-6? 3.56 7.66

Fig. 10. Initial slow quasiparticle distribationfunction for collimated beam with at. = 1; z z 0.

DIST. 7 50080E-S1 KEx 2.0000 KB. 1.90000E-01 IITHe 1.0000

KE* 0.0 0 OFP 1hTifi Aa.0000 Us 8.0 KT* ).373OOE-03

4.00

3.06

S(K, E )

2.60

1.66

-7.68 -3.50 -4. 7 E-97 3.56 7669E

Fig. 11. S function in K ,tspace; z - 0.75.a collimated-beam propagating in free space.

-64-

DIST= 1.560860 KEw 2.00000 NoB. I OOOOE-0l *WIOTHs 1.00000KEZ 0.0 4 OF INTERz 40.0000 KCz 8.0 i(Tx 9.37508E-03

K

3.00

S(K,tE.Z)

2.00

0.0-7.66 -3.50 -4.77E-07 3.50 7.0

Fig. 12. S function in K space; z =1.5.a collimated beam propagating in free space.

DIST- 1.50009 KErn 2.00000 K8= I.0000E-01 #WIDTH= 1.00000KU. 0.0* OF INTERx 40.0000 KCx 0.0 KTz 9.37569E-03

8.00

6.0

4.00

2.66

-7.06 -3.50 -. 7E07 33 7.s0

Fig. 13. Y4CF vs. I at x n 01 a collimated beampropagating in free space.

-65-

DIST* 6G K Us 2.00600 KS' i.600E-Ol SI1T~14 8 *0W0

KErn 1.90080 # OF WHERA 40.0000 KCz 0.0 K73 1.37380E-03

7.04.00

3.66

S(K01E.Z)

-7.66 -3.50 -4.7?E-S? 3.56 7.66

Fig. 14. Initial slow quasiparticle distributionfunction for focused beam with etv. - aL= 13 z= 0.

DIST= 7,5000E-01 KErn 2.08080 K82 Z.OOOOCE-01 *&IIOTN- 1 0080

KEv 1.0006 # OF INTER* 40.004 KCS 8.0 KTa 9. r7569E-03

4.0

3.86

2.60

1.66

oi.6 -3.50 -4. 77E-9? 3.50 7.0

Fig. 15. S function in I( , space; z =0.75,a focus"d beam propagating in free apace.

-66-

DISTm 1.5060 KErn 2.000 VS*n 1.00000E-01 #WIDOTH* 1 6000SKErn 1.00060 * OF INTERrn 40.000a KCx 0.08 KTm 3.375OOE-63

4.00

K3.66

S( K t.Z)

-4.77E-0?

2.06

-7.0 -3.50 -4.77E-97 3.50 7.80

Fig. 16. S function Iin J( ~ space; z =15a focused beam propagating in free space.

DISTm 1.50006 KErn 2.0006 KR= 1.00000E-01 #W[DTH*n 1.esoe

KUs 1.0009 * OF INTERz 40.000 KCC. 0.0 Ki. 9.37588E-03

4.6

2.0

06-66 -3.56 -4.77E-0? 3.50 7.0

Fig. 17. MCF vs. I at x 01o a focused beam pro-pagating in free space.

=7_7~Qi%

-66-

01ST: 1.60808 KE: 2.0088 KB: 1.0000CE-01 #WID0TH= 1.00008

Kim 0.0 # OF INTER= 20.00 Ku 1.000O0E-01 KT- 1.25008E-02

4.66

3.66

6.80

-16.1 -5.83 6.0 5.63 i6.

Fig." 18. intensity function I(X,z) vs. x;a collimated beam propagating in free space.

01ST. i.86866 KE2 2.88880 KB. 2.00M8E-01 #IDTH= 1.080C0

Us. 1.80008 # OF INTER= 20.8000 KGz 1.OSBOGE-01 KTz 1.250OOE-02

4.86

3.08

A^2

2.00

9.6-1A -5.83 6.6 5.03 10.1

Fig. 19. intensity function I(x~z) vs. x;a focused beam propagating in free space.

M L.',,

- 67 -

are displayed in Fig. 13. The spreading effect increases the

coherence length as z increases. Fig. 14 depicts the initial

condition described by equation (3.51) vitholr= oi;= 1, a = 0i.e. focused beam in free space at z <0.5, the field reduceits coherence length because of the focusing effect, it then

spreads as z = 0.5, as described in section II. The S func-tion at z = 0.75, 1.5 are shown in Fig. 15, 16. Fig. 17 dis-plays the MCF as a function of I and z. The above exact re-sults have been used to check the accuracy of the algorithm

and found to be very good.

The Fourier transform of S(k , I, ) gives the MCFlM(x, , , L Setting =0, one obtains the intensity,

I(x , )=M(X, 0, ) as a function of x and z. Figs. 18and 19 depict the intensity function for collimated and focusedbeams propagating in free space. In the first case the inten-

sities spreads in the X direction as z evolves. The focusing

effect is observed in the second case and is shown in Fig.19.

For a wave propagating thru a random medium, characteriz-

ed by a power spectrum (A), the evolution of the S functioncan be obtained from the relations

S : ,. o) e 0

(3.53)

where, for the one dimensional case, H(.E ) can be representedas

-COs at) J(A) A(3.*54)

- 68 -

Along the quasiparticle trajectory, one can rewrite (3.53) as

•- j(33]

(3.55)

From knowledge of the power spectrum and the initial

distribution function, one can interpolate the S function

along the trajectory in accordance with (3.55). That is, the

S function spreads in E space and decays along the trajec-

tory with a structure function H( E ) dependence.

One finds equation (3.55) is different from eq. (3.52)

by a collisional (decaying) term exp -J}H[E-K(J-J')J '

It is thus important to observe the behavior of the H func-

tion for an arbitrary power spectrum. We shall choose three

different kinds of power spectrum and study the resulting

behavior of the H function:

(a) Gaussian spectrum

A, (3.56)

where -& is the center of the Gaussian spectrum and b is

the width T;

(b) Power law spectrum with spectral index P

+= A( .

- - - . . .-,'-- ....... . . . . . . . . .

- 69 -

where &L= i/L is the wavenumber associate with the

outer scale Lo and F is the spectral index.

(c) Compound spectrum

~ C[ 1 ... )P x2 (3.58)

is the superposition of spectrum (a) and (b).

The power spectrum and associated H function are shown

in Figs. 20a and 20b. Curve 1 in Figs. 20a, b represents (f)and H( E ) for spectrum type (b). The Gaussian bump power spec-

trum and corresponding H function are depicted in curves 2,3,4

with *' equal to 1,2,3 respectively, and with a = 0.3, b = 0.2.

One observes that the additive Gaussian bump contributes an

oscilltory structure to the power spectrum. For instance, .' =3imposes an oscilltory structure with three peaks located at

= /3, X , 5 1/3.

The oscillatory behavior implies a higher randomness in

the medium. As a result, the wave is more decorrelated by the

random field and a stronger scattering effect is encountered.

Fig. 21a displays the Amplitude dependence of the Gaussian

bump for a = 0.3, 0.5 and 0.7 from lower curve to higher curve.The superposed behavior of spectrum (a) and (b) is shown in Fig.

21b for the case *1=1, a = 0.3, b = 0.2.

It is of interest to evaluate numerically the S function

at a distance ; using equation (3.55). Once the S function

is obtained, one can easily find the intensity function I(xz)

and correlation function M(O, , 1 ) at x 0 from

- 70 -

OIST" 2.13281 WIDTH= 1.0ee Ala 1.00809 GAMIMA- 1.6069.

GCTR, 3.00098 GWID, 2.000B9E-01 GAMP, 3.0800E-01 INOEX- 2.00000

POWER SPECTRUM WITH A GAUSSIAN BUMP

I .e8

?.56E-Ot

5.91E-01

2

2.51E-01 -1

1. 48E-03 I

8.8 1.25 2.59 3.75 5.88

Fig. 20a. (k) vs. k; curve 1. stands for spectrum (b)

curves 2,3,4ffor spectrum (c) with k' = 1,2,3 respect-ively; p = 4, a = 0.3, b = 0.2.

DISTx 2.13281 WIDTHO 1.89909 AIm 1.8888 GAMMA* 1.6080

GCTR= 3.80e09 G1ID= 2.0080E-91 GAMP- 3.0068E-01 INOEX- 2.0909

POWER SPECTRUM WITH A GAUSSIAN SUMP

3.9803.

2.25 2

•.. .. .

1.50

7. 56E-B1-'. o

-7.66 -3.58 -4.??E-07 3.58 7.80

Fig. 20b. H( ) vs. L ; Curve 1,2,3,4 correspond toJ(k) of Fig. 20a.

- 71 -

01STa 2.13261 MIDTHa 1.8666 Ala i.0e6ee GAMMA. 1.996

GCTRz 2.989 e GID 2.e00eOE-0l GAMP- 7.600eeE-81 INDEX- 2.66008

POWlER SPECTRUM N A GAUSSIAN BUMP

3.66

tI

2.25 2

H

i 1.50 .

7. SE-el

8.66.6 1.25 2.50 3.75 5.86

Fig. 21a. H(E ) vs. I ; spectrum (c) with a = 0.3(curve 1), a = 0.5 (curve 2 ), a = 0.7 (curve 3).

OIST" 2.13281 IlDTH= 1.0880 Al. 1.066 GAMMA- 1.09600

GCTRa 1.0000 GID 2.0898E-e1 GAMP- 3.e00eeE-61 INDEX= 2.06686POWER SPECTRUM WITH A GAUSSIAN BUMP

3.66

3

2.25

H

1.56

7. SeE-el

-14.0 -7.68 -9.54E-07 7.0 14.0

Fig. 21b. H( 1.) vs. I I curve 1 for spectrum (a),curve 2 for spectrum (b), curve 3 for spectrum (c).

. . , . . , . ,. .. : % >. ,,,: ...'.'

- 72 -

S2 7E (3.59)

J 2 7C (3.60)

The correlation scale of the field is defined by

S(0,0,J3) (3.61)

and hence the knowledge of M (0, , z) permits one to eva-luate the correlation scale 1. We shall first present somenumerical evaluations of the S function by means of eq. (3.55).The significance of intensity and correlation length for bothfocused and collimated beams will be discussed later.

In the following numerical evaluations, we have used thenormalized quantities: z -0 z/kLe , x -4 x/L. ,' r= k4 Lo' A(0).

where ke is the wavenumber of the incident field and Le isthe outer scale of the random field. Figs. 22a and 22b depictthe S function at the boundary z = 0 for the collimated beamand focused beam, respectively. Figs. 23a-d depict the S func-

-73-

:T's O-e WIDTH% 1.00090 Als 8.0 oZ

~CTRx 0.13 GWION 1.00000 CAMP. 140096 11dDEXZ 200.1

;AUSSiIA SPECTRUM

4.00

3.80 F KA-S

-?.ea -3.50 -4.7?E-07 3.50 ..00

Fig. 22a. S function for a collimated beam with1L.l z=O0.

DIST= 0.0 WIDTH= 1.00080 AIX 1 0888 GAMHA. 10.0661CTRz 3.00080 GWID= 2.0800OE-01 GAMPs 3.088E-01 INDEXa 2.0066

POWER SPECTRUM WITH A GAUSSIAN eUMP

4.8

SM-

2 .00

-7.66 -3.50 -4.77E-07 3.50 7.60

Fig. 22b. S f unction for a focused beam withS1; z =0.

-74-

OtjT* 4.3750@E-01 IHTHe 1.0060 Ala 1.0060 GAMMA. 18.8999

ICTRa 3.00990 GIED. 2.00066E-01 CAMPz 3.00OE-81 INDEX= 2.00000

POWIER LAW SPECTRUM

4.09

KA3.66

S

2.66.

2 .00

-7.0 -3.50 -4.77E-07 3.50 7.0a

Fig. 23a. S function for a focused beam propa-gates thru a random medium with a power law spectrum,p =4, 'I = 1 0 , cv. = *(A:= 1; z = 0.4375.

01STa S.75G00E-01 WIDTH 1.000 Ala 1.66660 GAMMA- 10.008GCTRa 3.0600 GWNzD 2.0900@E-01 GAMPu 3.08000E-01 INDEX= 2.09660

POWER LAW SPECTRUM

7.6004.68

KA32 MA 3.99

-7.00 -3.50 -4.77E-07 3.50 7.99

Fig. 23b. z* 0.875.

-75-

DIST= 1.7125e W1O1N= 1.00006 Ate 1.00060 GAfMAm 1006f£CTRu 3.88690 GWIDm 2.0089E-01 GAMPs 3.0800CE-01 INDE'As .0000oPOWlER LAW SPECTRUM

4.88

KA3.0

2.06

-7.06 -3.50 -4.77E-07 3.50 7.0

Fig. 23c. z=1.3125.

DISTs 1.75000 WIDTH. 1.8009 Ala 1.0000 GAMMA. 10.006

ICTRz 3.0000 G1410a 2.000OOE-01 GAMlP* 3.00000E-01 INDEX= 2.00000

POWER LAW SPECTRUM'

4.08__________7.08_

KA

3.00

2.0

1.60

-.9 -3.50 -4 77E-07 3.50 7,00

Fig. 23d. Z 1.75.ii

-76-

tion at z = 0.4375, 0.875, 1.3125, 1.75 for focused beam pro-

pagating thru a random medium with power law spectrum definedby the spectral index p = 4, or = 10, tr-- 1. One observes that

the quasiparticle d.f. spreads in the E direction and decaysaccording to the H function as it redistributes. If a Gaussian

bump is imposed with its center located at k' = 3(GCTR), width

b = 0.2(GWID), amplitude a = 0.3 (GAMP), the S function is dis-torted and sharpened compared to the case of the simple power

law spectrum. These effects are displayed in Figs. 24a-24d.As discussed earlier, the location of the Gaussian bump deter-

mines the number of peaks in the H function. In order to illu-strate the effect of these peaks on the S function, we shall

examine the case of a collimated beam propagating thru a randommedium with the composite spectrum (c). The center of thebump is chosen to be located at k' = 1, 3,4, respectively. For

I" = 10, GWID = 0.2, GAMP = 0.3, p = 4, one displays the nume-

rical solutions in Figs. 25a-25d, figs. 26a-26d and figs. 27a-

27d. At z = 0.4375, the S function is less distorted for k'-lthan it is for k; = 3 and 4. At z = 1.75, the S function for

k' = 3 and 4 clearly shows the effects of the H function peaks.One observes that the presence of these peaks in the H function

greatly reduces the correlation length in the direction.

It is easier and clearer to discuss the independent

effects of different spectrum structure by means of the inten-sity function and correlation length associated with the propa-gating waves. Substituting the numerically evaluated S func-

tion into equations (3.59)-(3.60), one obtains the desired

information for I(x,z) and . . Fig. 28a depicts collimated

beams propagating thru a random medium with H( L )m T1.

Curve 1 represents the axial intensity and correlation lengthfor the case of wave propagation in free space. Turbulent

effects for ar 0.5 and 1 are depicted in curves 2 and 3. (In

Figs. 28-31 we use the same curve index for different values of

-77-

OKST= 4.3750GE-91 WIDTH* 1.900 Alz 1.00000 GAMMA. 10.60666CTRZ 3.60066 GWIO. 2.0086E-81 GAMP. 3.00666E-01 INDEXn 2.0066POWER SPECTRUM WITH A GAUSSIAN BUMP

4.06

KA

S 2 0

1.60

6.6-?.Go -3.59 -4.?7E-67 3.56 7.68

Fig. 24a. S function for a focused beam propagatesthru a random medium characterized by spectrum (c);kl=.3, b = 0.2, a = 0.3,.=el = 1; z = 0.4375.

DZ3Tu 8.750E-0l WIDTI~u 1.00886 Ala 1.00660 GAMMAN 16.60604CTR= 3.009 GWI~a 2.600E-01 CAMP% 3.08900E-01 INDEX= 2.06866POWER SPECTRUM WITH A GAUSSIAN SUMP

7.66

4..,.

3.99

-7.86 35 -4?7E-6? 3.50 9

Fig. 24b, z -0.875.

-78-

DISTs 1.31250 WIDTH= 1.00880 Alm 1.0000 GAMMAs 10 0800£CTRu 3.0890 GWIOx 2.080G0E-01 GAMPs 3.08000E-01 INDEXx 2 0800POWER SPECTRUM WITH A GAUSSIAN BUMP

4.00

KA3.8

S2.8

8.8

-7.88 -3.50 -4.77E-87 3.50 7.80

Fig. 24c. z =1.3125.

DIST* 1.75000 WIDTHm 1.0008 AIX 1.00000 GAMMAa 10.see.

tCTRa 3.08850 GWIDx 2.000@E-81 GAMPz 3 OOSE-01 INDEX2 2.00000


7.004.08

KA3.99

1 .00

-7.89 -3.50 -4.7,E-07 3.50 7,80

Fig. 24d. z 1.74.

-79-

43750GE-01 WIDTHe 1.80000 A~IR 0.0 AfMi1

~ 0800 GWlDx 2-DOOOOE-01 GA4MPc 3 OOOOOE-01 INDEXT~ 2,V

P3WER SPECTRUM1 WITH A GRaUSAN SUMP

4.00

S

2.88

-7.90 -3.50 -. E-07 3.50 6 0

Fig. 25a.S function for a collimated beam propagates thrua random medium characterized by spectrum (c);k' 1,a = 0.3, b =0.2, p = 4, Or = 10; z = 0.4375.

2;,Tn 8.739GOE-el WIDTH= 1.00080 AIv 8.0 NMuI.e

.2-'.T F- 0030 GRID2 2.080E-0I GAMP= 3.0080E-0.1 H'DE-* IOGWEP SPECTRUjM WITH A GA.USSIN CIJNP

4.0 Pie_____

S

2.00

aee

Fig. 25b. z 0.875.

-80-

4'X 1T IO8 GJJID= 2.000OOE-01 CAMfP= .3.00060E-fj I 11 E'.:POWER~ :iFECTRU IJ TH Aq GA.USS!AM BUMHP

4 80 ___________

2. o.

-? ~ -3.50 -4.?EJ 3.50) 70

Fig. 25c. z =1.3125.

.1 .=1.7oo IDTH= 1.9eoe A4[= a36 GAMMASi~ :e.eloa

ji:;Fz -0130A cuWtD 2.000OE-01 GA9IPu 0000*E-0: &POiE*-*= 2 Zae

?-CNER SPECTRUM W4ITH A~ CG.J)SIeI BUMHP

4..7E

-700 -304.?TE-07 7 SO5.0t

Fig. 25d. z 1.75.

-81-

01STx 4.3750E-01 MIOTHw 1.9000 Ala *.6 GAMMAS 19.6696

$CTRS 3.6000 GWID*~ 2.00099E-91 GAMPa 3.00060E-01 [HOEXn 2.060M

POllER SPECTRUM 111TH A GAUSSIAN BUMIP

4.80

3.9

S

1.00

0.0-7.60 -3.56 -4.77E-07 3.36 7.0

Fig.26a.S function for a collimated beam propagates thrua random medium characterized by spectrum (c); k' = 3,a = 0.3, b = 0.2, p = 4, Or = 10, z = 0.4375.

01ST-s.50El WIDTH= 1.000 Al2 0.0 GAMMAS 10.008£CTRZ 3.0060 GWID= 2.0OGOE-01 GAMP* 3.00000E-01 INDEX- 2.00000POWlER SPECTRUM WITH A GAUSSIAN BUMNP

4.00_________

KA

S

-4.77

-7.00 -3.50 -4.77E-97 3.50 7.00

Fig.26b. z 0.875.

-82-

DIST- 1.31250 IDITH= 1.00000 Alm 0., GAMMA= 10.0800

6CTR- 3.00000 GWID= 2.0000E-01 CAMP- 3.0000@E-01 INDEX= 2.00000


4.00 __________

3.80

2.00

1 .00 _____________________

0.8--7.06 -3.50 -4.77E-87 3.50 7.00

Fig. 26c. z =1.3125.

DIST= 1.75000 WIDTH= 1.00080 Ala 0.8 GAMMA= 10.0000

9CTRz 3 00000 GWIDm 2.00MOE-01 GAMPm 3.000GOE-01 INDEXz 2.00000


4.00

KA3.00

S-4.7?7E-0

2 00

1.0 4a0.0 __________________

-7.00 -3.50 -4.7?E-07 3.50 7.00

Fig. 26d. z =1.75.

AD-AQ92 720 POLYTECHNIC INST OF NEW YORK FARMINGDALE MICROWAVE R--ETC F/6 4/1IONOSPHERIC SCINTILLATION(U)SEP 80 0 M WU N0001-76-C-0176

UNCLASSIF IED POLY-MR-1410-80NLm hhhhIIhEIIIIEm/l/////////lEllllllEElllEI-I/I///E////EImlluuuuuulluu-IIIIIIIIIIIu

2.2

lI-E--

1111.25 f1.4 311.6lf-III lIE Hhll

MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANDARDS-1963-A

-83-

DIST2 4.37300E-01 WdIDTH= 1.00080 At= 0.8 GAMMAw 10.0000

6CTR* 4.00908 GWIC,= 2.60OSGE-01 GAMP- 3.00000E-01 INDEX& 2.00000


4.0

-7.88 -3.50 -4.77E-07 3.50 7.e@

Fig. 27a. S function for a collimated beam propagatesthru a random medium characterized bt spectrum (c);k'= 4, a = 0.3, b = 0.2, p = 4, vY= 10, z = 0.4375.

DIS'Ta 3.7500eE-01 W10T14 1.00000 At= 0.0 GAMNAx 10.@90

£CTRu 4.000 GwiDs 2.00080E-81 GAMPa 3.0080OE-01 INDEX* 2.080


4.8

S

2.80

-700 -350 -4.77E-07 3.58 7.00

Fig. 27b. z =0.875.

rpm-,-_<

-84-

DIST* 1.31250 UIOTHe 1.08080 Ala 0.0 GAMMA= 18.8980

ICTRm 4 00008 GU40= 2.80800E-al GAMPs 30090E-01 INDE~m 2,00900


4.00

-4.77?E-2.08

;: 00 -35 -4.7E-0 3.:8 .80 ee:BI

POW1ER SETUWIHAGAUSSIAN SUMP

4.0

KA

3 go

S-4."47E-8

2.00 _________

1.00

0.0-

Fig. 2?d. z 1..

- 85 -

). One observes that the axial intensity I(O,z) decays as

the wave propagates while the correlation length decreases due

to turbulent effects until about z = 0.3; diffraction effects

then become important and coherence is partially restored.

After further propagating turbulent effects again dominate and

the wave is decorrelated. For a focused beam propagating in

free space with ar = 1, 44=1, the intensity at z = 0.5 willreach a maximum and its correlation length a minimum as shown

in the curve 1 of Fig. 28b. If a turbulent medium is present,

the focal distance is reduced due to the scattering effect im-

posed by the turbulent medium. Curves 2,3 of Fig. 28b display

the decorrelation process contributed by both focusing and

turbulent phenomena from z = 0 to z = 0.4. At z T= 1.2, the

turbulent effect dominates and reduces the correlation lengthas z increases. Figs. 29a,b display the same effect for theGaussian spectrum. Comparing these curves with the square law

spectrum, one observes that the Gaussian spectrum has a small-

er effect on the wave statistics than the square law spectrum.

It is of interest to consider a power law spectrum for the

ionospheric random medium. We shall choose the spectrum type

(b) with p = 4 and make use of numerical calculations of the

S function to evaluate the axial intensity function I(o,z) and

correlation length 1. The results are depicted in Figs. 30ab.

Fig. 30a displays the case of collimated beam for 'r= 0.5 and 1.The turbulent effect slightly reduces the intensities (Note that

this is a weak turbulence case), and the correlation length de-

creases compared to the case of free space (curve 1). For the

case of focused beams, depicted in Fig. 30b, one finds that

scattering effects decrease the focal length; the wave is also

diffracted and decorrelated as in the previous case. The spec-

trum of a real random medium is not a smooth function in general.

Experimental results show that the power spectrum could posses

a number of "bumps" in various frequency ranges. We shall illus-

trate the effect of such bumps by simply adding a typical

- 86 -

OISTs 2 1323! WIDTH- 1.00000 AIa O.A GAMMAS 8.o

SCTR= 0 6 GUID% 1.000 GAt1P, 1.6006 IHOEX- 2.00099

2.00o ..'

e*1

.... ... ... ... ... ... ... ... ...

1.. ..... 3' o . . .. .. . . . .....oo ., o-, - °o .°- o . o o... . ... . . .

5.eE-01

6.8 I I II

0.0 5.33E-01 1.07 1.68 2.13j7

Fig. 28a. 4J 1 vs. z dotted line; I(O,z) vs. zsolid line; curve is free space; curve 23'r= 0.5;curve 3: O" = I; collimated beam with ei-= 1.H(A) = "E~ "

DI-Ta 2 13281 WIDTHS 1.00000 AI- 1.002O GAMMAS 5.00990E-01GCTR: 0.0 GIDw 1.08000 GAMP= 1.:86090 INOEX- 2.90086

I , il . ". ...........1 .50

S.99E-01

6.6 5.33E-01 1.e7 1.60 2.13Fig. 28b. focused -beam case; o o ,(

-87-

DISTu 2.13201 MIOTH- 1.09660 Ain 0A GAMMAa 8.0

GCTRw 6.0 GMIO. 1.9606 GAMP-U 1.66666 INDEXa 2.6606

GAUSSIAN SPECTRUM

1.56 ................

....... .. .

1.66

6.0 5.33E-01 1.6? 1.66 2.13z

Fig. 29a. 12-4 1 vs. z dotted line; I(O,z) vs. zsolid line; (1.) free space; (2) -r= 0. 5; (3)u 1;Gaussian spectrum, k' 0, A = 1, b =1.

DISTz 2.13281 MIWTH t.66660 Ala 1.6U6, GAMMAB 1.66666

GCTRw 0.6 61110. 1.00660 GAMPR V900906 INDEXz 2.9909

GAUSSIAN SPECTRUM

..... ................1.56

3.NE-6 I

9.S 5. 33E-91 1.67 1.66 2.13

Fig. 29b. focused beam case; 04r 01;~ 1

-88-

GISTS 2.13281 WIDTHS 1.60066 At. 04GAMMAS 1.09069

GCTRw 6.6 GNIDS 1.S0666 GANIP- 1.09,68 INOEXw 2.6696

POWlER LAW SPECTRUM

5.09E-01

0.2.

6.6 5.33E-81 1.67 1.66 2.13z

Fig. 30a. WE 1 vs. z dotted line: I(0,z) vs. zsolid line; (1 ) free space; (2) P = 0. 5; (3) ' 1;Pavwer law spectrum with-p = 4.

GIST= 2.13261 WIDTH- 1.86066 Ala 1.66968 GAMMAS. 6.6GCTRu 6.6 GhIlDS 1.66868 CAMP- 1.66,88 INOEXa 2.66666

POWlER LAW SPECTRUM

2.66

........

1 .56

5.66GE-01

2

6 .6 3.33E-61 1.67 1.66 2.137

Fig. 30b. focused beam case; ckv- -1.

- 89 -

Gaussian bump on the power law spectrum, i.e. a compound spec-

trum as described in eq. (3.58). Fig. 31a, 31b depict the

effect of a Gaussian bump on the axial intensity function and

the correlation length for collimated and focused beams res-

pectively. The parameters are chosen as : Or= 0.5 and 1.

GCTR = 1, GWID = 0.2, GAMP = 0.3 and p = 4. The results show

that the addition of this Gaussian bump causes strong scatter-

ing phenomena. We have presented, for the weak fluctuation

cases Y = 0.5, 1, the effects of turbulence in the medium

As a comparison, we shall now display in Figs. 32-33 some re-

sults for the strong turbulence case, 'Y= 5 and 10. A colli-

mated beam propagates thru a random medium with a composite

spectrum formed byaLGaussian bump located at GCTR = 1, GWID =

0.2 GAMP = 0.3, for the cases 'r= 5 and 10. Curves 2,3 in Fig.

32a represent the intensity and correlation length for = 5

and 10; they shows that for strong turbulence case the corre-

lation length decreases rapidly and a long tail appears at

large Z. The axial intensity drop appreciably and approachsa small constant value. For the focused beam case, the focal

length is greatly reduced for large values of Wr . We shall

expect, as the turbulent strength is very strong, that the

focusing effect will not be present and the wave will decay

as z increases. These results are depicted in Fig. 32b. If

a Gaussian bump is located at k' = 3, the scattering effect is

stronger than for the previous case. Figs. 33a and 33b dis-

play data for a collimated beam and focused beam propagating

thru a random medium described by a composite spectrum with

k' = 3. One observes that decay is much greater for the in-

tensity and that decorrelation phenomena are more evident

than for the case k'=l

-90-

01 STs 2.13281 MIOTHs 1.60660 Ala OA.4 GAMlMA= 0.0GCTRu 1.9000 GMIDa 2.00066E-01 GAl1F";3.06668E-0i INDEX- 2.6000POWlER SPECTRUM WITH4 A GAUSSIAN4 BUMP

2.90

................................................. .........

1.56 .

Fig. 31a. rijl1 vs. z dotted line; I (0,z ) vs

DIST* 2.13281 WIDTH* 1.66660 Alm 1.9900 GAMMA= 6.0

6CR .00 GW5 .000E-0 1.67 -1.60 O-9 INE.12300POWE SPCTRU WIH AGAUSIANBU3

31b. f cused b.am ..s.... ..........

-91

DIST. 2.13281 WIOTHa 1.89090 Ala 0,4 GAMMA- 5."00O

CCTRa 1.66606 GI~ 2.8SeeeE-61 GANPIb 3.SSOOOE-01 INDEX. 2.066


2.04

1.59

5.66E-...........

6.8 3.33E-01 1.97 1.66 2.13zFig. 32a. fl- 1 vs. z dotted line; I(0,z) vs. zsolid line; (1) free space; (2) rY' 5; (3) -Y- = 10;Power spectrum with a Gaussian peak, k' = 1, a = 0.3,--b = 0.2, p = 4, collimated beam case.

DISTa 2.13291 WIDTH= 1.09990 Ala 1.98.090 GAMMA= 5.06989

GCTRx 1.eoeee GWW-a 2.009E-01 GAflPz 3.'8009E-01 INDEX* 2.00009

POWER SPECTRUMI WITH A GAUSSIAN BUMP

2.9Ol

.....................

Fig. 32b. focused beam case; dy. 1.

-92-

DISTu 2.13281 WIDTHe 1.6089 Ala BA"* GMNAu 5.00GCTRn 3.9800 GWIb. 2.06008E-01 CAhF' 3.000OOE-01 INDEX. 2.66600POWER SPECTRUM WITH A GAUSSIAN BUMP H

2.09 1..'

1.56

5~~.. ..99E01.............

............. ... . .. ................ .

6.9 z.3-1 18 1.60 2.13

Fig. 33a. r= 1 vs. z dotted line; I(0,z) vs. zsolid line; (1) free space; (2) -r 5; (3) -Y' 10:,Power spectrum with a Gaussian peak, k' =3, a =0.3,-b = 0.2, p = 4, collimated beam case.

DIST= 2.13281 WIDTH.x 1.00000 At- 1.00J000 GAMMA= 0. 0GCTR= 3.0080 GUID= 2.90000E-01 GAMPz 3.:800900E-01 1IJOEX= 2.00808POWER SPECTRUM WITH A GAUSSIAN BUMP

2.08

.............. ........... .............. ............

6.6 3.33E-01 1.0? 1.60 2.13

Fig. 33b. focused beam case; eor = 1.

-93 -

IV. Conclusion and Discussion

A wavepacket or beam can be viewed as a finitely extend-

ed moving and deforming system of "Quasiparticles". The Four-

ier transform of the Mutual Coherence Function w.r.t. either

the "fast" or "slow" coordinate is defined respectively as the

"fast" or "slow" quasiparticle distribution function F or S.

For wave propagation in a random medium, it is usually more

convenient to utilize the "slow quasiparticle" distribution

function S. The knowledge of the S function provides the de-

sired information for two point statistics of a stochastic

wave. For example, the correlation length lat any value z,

defined in equation (3.61), can be interpreted as the total

*numer of q.p. Ns(z) divided by the q.p. density at = 0.

Also, the axial intensity I(0,z) may be represented by the

slow q.p. density at 0 = . To obtain the 2 point statisti-

cs at any distance z, one has only to observe the q.p. flow

in K ,E space as z increases. For the case of a collimated

beam propagating in free space, one considers, at z = 0, a

system of quasiparticles evenly distributed between -7 < ,

1< +7. Qualitatively, we shall use the "area" of a circular

"point" to represent the number density of slow q.p.; thus at

z = 0 the q.p. are distributed in K , space as shown in

Fig. 34. As z increases, those q.p. with higher X "move"

faster than those with smaller X . As a result, the q.p.

density at E = 0 decreases as z increases and hence the axial

intensity I(0,z) decreases. Since the total number of q.p is

conserved ih free space propagation , i.e. Ns(z) is constant,

the correlation length increases as q.p. move along the trajec-

tory.

For a focused beam propagating in free space, the q.p.

at z = 0 are distributed in , space with a maximum along

an axis deviating from the = 0 axis as shown in Fig. 35.

94 -

0* - o Fig. 34.

4-0 4- ) 4-0

00--

inr. Fig. 35.

0'0-

Fig. 34. Quanlitative representation of slow q.p.distribution function in K , Z~ space; collimated beant-in free space. Fig. 35. focused beam in free space.

- 95 -

One observes that the q.p. density reaches a maximum at aplane z = zf, when those q.p. with larger density reach the

center line E = 0. i.e. the axial intensity is maximum at

the focal distance zf. The conservation of the total num-

ber of q.p. in free space then implies a reduction of the

correlation length from z = 0 to z = zf. As the q.p. flow

moves beyond the focal point, those q.p. with lower density

move to the center and the axial intensity decreases as z

increases; Correspondingly, this implies an increase in the

correlation length.

For a wave propagating in a random medium characteri-

zed by the structure function H( t), one just simply leads

to a decay of the wave field in the K , space as depicted

in Fig. 36. The structure function acts as an "absorber"

which absorbs part of the q.p. according to its strength.

For z >0, Fig. 37 depicts for a collimated beam at z = 0,

how the q.p. move in a random medium characterized by H(E ).As z increases those q.p. with smaller density move toward

the center while some of them are absorbed by the structure

function. The resluting q.p. density at E= 0 is decreased

and is smaller than in the case of free space. One notes

that the total number of q.p. is not conserved in this case.

The ratio of Ns(z) and I(O,z), which determines the correla-

tion length 1, will decrease as z increases for the strong

turbulent case; however, in the weak turbulence case, it will

first decrease and then increase as z increases at least for

a short distance (Note that the q.p. at K = 0 will never

move). For the case of a focused beam propagating in a ran-

dom medium, one depicts the q.p. flow thru a random medium

characterized by'H( E ) as in Fig. 38.

Because of turbulence in the medium, the q.p. withlarger density are absorbed before reaching the center E = 0.

':71

- 96 -

As a result, the q.p. density at 0 creates a maximum as a

result of q.p. which initially have less density. Thus, the

focal length (zf) is smaller compared to the free space case.

The correlation length in this case is reduced from z 0 to

z = zf. Beyond the focal point, the q.p. density, at = 0,

is reduced and the ratio of Ns(z) and I(0,z), which yields 1,

will again decrease for the strong turbulent case or decrease

and then increase, as z increases at least for a short dis-

tance, for the weak turbulent case.

The qualitative discussion presented above is suitable

for an arbitrary random field. The form of the structure

function H( E.) will determine the q.p. density profile. To

illustrate this effect, we will present some numerical re-

sults for a wave propagating in a random medium characterized

by the spectrum (C). Fig. 39a depicts the q.p. density func-

tion at x = 0 in E -space, i.e. M(O, gE , z), for the case of

collimated beams propagating in a turbulent medium with spec-

trum (C). We choose the parameters GCTR = 3, GWID = 0.2,GAMP = 0.3 and p = 4. One notes that the axial intensity

decreases as z increases and the deformation of M(O, E , z)

reflects the behavior of the decay field. The normalized

correlation function, WPE is shown as a function of z

in Fig. 39b. The area under each curve represents the corres-

ponding correlation length 1. For I'= 10, which represents

a strong turbulent case, one sees that the correlation width

decreases as z increases.

The corresponding phenomena for the focused beam case

is displayed in Fig. 40a and 40b. One notes the focusing and

defocusing effects in Fig. 40a and the associated curves for

the normalized MCF in Fig. 40b.

The choice of an symmetric function for the input field

JL

- 97 -

Fig. 36. Quanlitative plot of structure functionH(E) in space.

o-**

I'I

04 0'.C- ( 0-0 --

4

4Q ) #-O 0A -

Fig. 37. S function of a Fig. 38. S function of acollimated beam moves in a focused beam moves in a randomrandom medium, medium.

-98-

OISTx 1.75080 WIDTH= 1.69088 Ala 0.8 GAMMAa 10-0000 *

GCTR= 3.00080 GWIO= 2.00000E-01 GAMP= 3.00000E-01 INDEX- 2.0009


4.00

2.00

0.08I

-7.00 -3.50- -4.77E-07 3.50- 7.00

Fig. 39a. q.p. density function, at x = 0 ,in

space; plotted as a function of E and z; collimatedbeam propagating thru ar random medium with spectrum (c)ik'= 3, a z: 0.3, b = 0.2, p = 4, -r 10.

GIST= 1.75000 WIDTHx 1.00080 Ala 0.8 GAMMA= 10.0000

9CTRz 3.0000 G&IIDz 2.00000E-01 CAMPz 3.00000E-01 INDEXz 2.00000


4.00

M (0, ) 8.75E-O2.00

S. 00

9.S 0-.e -3.50 -4.77E-07 3.50 7.00

Fig. 39b. Normalizel q.p. density function, MCOL241(0,.)vs. t and zj the area under each curve representsthe corresponding correlation length 1.

-99-

DISTs 1.75000 WIDTH= 1.90080 Alm 1.00880 GAMMA= 10.6666

GCTR- 3.0860 GWIOm 2.0000GE-01 GAMP2 3.0@080E-01 INDEX- 2.00006


4.88

3.6

2.68

1.8

9.6-7.06 -3.58 -4.77E-07 3.58 7.00

Fig. 40a. q.p. density function, at x = 0, in space;focused beam propagating thru a random medium withspectrum (c), kc' = 3, a = 0.3, b = 0.2, p = 4, -y' 10.

DIST- 1.M580 WIDTH= 1.80088 AIX 1.00800 GAMMA= 18.0888

£CTR= 3.08688 GW~ 2.8000DE-01 CAMP. 3.00000E-01 INDEX= 2.00880


M(O,M (0,) 8.75E-02.0

-7.66 -3.58 -4.77E-07 3.56 7.08

Fig. 40b. Normalized q .p. density function, MOL/MG,9)vs. z; the area under each curve gives 1.

-100-

leads to a real S function. It provides a simpler approach to

obtaining 2 point statistics especially for numerical evalua-

tions. The exact solution of the Mutual Coherence Function

for the case H( E ) = a t2 is solved and presented in section

II. For an arbitrary spectrum, a numerical scheme has been

used to evaluate the S function as shown previously in section

III.

For simplicity the above analysis has been limited to

the two (zx) dimensional case. The extension to three dimen-

sions is straightforward but requires more computation. For

a random medium which is inhomogeneous in z, one simply adds

a z dependence to the H function in equation (3.55) and all

of the calculational procedures will then be eunchanged.

- 101 -

Chapter 4 THE FOURTH ORDER MOMENT AND INTENSITY

FUNCT ION

I. Formulation of the Fourth Moment Equationt

The parabolic equation for the 4th moment

(4.1)

may be derived for the case of statistically homogeneous fluc-

tuation as follows: (cf. Tatarskii, 1; Ishimaru, 25)

- - (r2 ... - ,)+& ia M =?zd 6R (4.2)

where

" 3H HH (3,

and

I,|H .,

..... ....... , ! r - - - - . . ,Cod (4,.4)::=,..i. .:-, - ..... _ ~

- 102 -

Introducing the new variabless

+ 2 &

X 2 *

.l : ,-Y,' , X2 (4.5)

2 2

then V, + , _ -V : 2('7tV?+Vr 1,) and equation (4.2)

takes the form

(4.6)

where the function f, expressed in terms of the new variables,

is independent of the center coordinate R and

H +

~~+ H

- i-i c3, ig )-H (3, i-tz.) (4.7)H +.

-103-

Equation (4.6) and (4.7) together with the boundary conditions

on M4 at z = 0 determine the corresponding fourth order statis-

tics of the field in the random medium. To date, no analytical

solutions have been derived for equation (4.6), except for the

special case H( E ) = a 1 2, which has been solved by Furutsu(13)

using functional techniques.

In section 2, we shall extend the "quasiparticle distri-

bution function" concepts to a 4-point correlation spectrum,

i.e. the Fourier transform of 4-point correlation function, to

obtain an analytical result for the case H( ) = a E 2. An

approximate solution for arbitrary power spectrum using 4-point.

Slow spectrum will be derived in section 3. In the last section

of this chapter, we will present a numerical scheme to solve the

fourth moment equation (4.6) for an arbitrary power spectrum.

II. The Exact Solution for the Case H( ) = a E2

We shall define the F and S functions corresponding to the

fourth moment M4, viz:

F M 4 , lftZ.V

(4.8a)

(4.8b)

-104-

where R and rl are denoted as the "slow variables", and 9andr3j as "fast variables". One observes that F and S form a 4-

fold Fourier transform pair with respect to R, r, and k, , r

(4.9)

Furthermore, the "macroparticle coordinates" can be obtained

by the following relation:

+ 3 4n3 +l ~

2.(A: (4.11)

- (~r )2 t 5 r ) 2

r,+r) r - r.

-105-

Substituting (4.11) into (4.6) and using (4.8b), we .can derive

the defining equation for S as,

(4.12)

The characteristic trajectory equations corresponding to equa-

tion (4.12) are

*0 0

(4.13)

Equation (4.12) can be described as the slow quasiparticle

distribution function in a two-dimensional phase space spanned

by koR , krz, , . long the trajectories, the S function

obeys

S ,,[ ..,n )= 5(. sr, ,,, v o), a) ,, L

e (4.14)

which is essentially the same expression as described in the

2-point correlation case except that 6-fold spatial directions

are involved. For simplicity, we shall consider only the two

dimensional case which can be extended to the 6-dimensional

-106-I.'

case without great difficulty.

For incident wave at z = 0 of the form

9) (x, o)= e 2~~~

the four point correlation function can be calculated as

M4 ( X,,X',XZ, o) < 

_ 4 z x, (, 2+ I. X, 4 ,+ x (-) -L id; (i2x2.z)

2 2

e

Z Z 4f4) 1~ ~ (4.16)e 2or2~

where kM and kr-, are normalized by k 0 . Substituting (4.16)

into (4.14) and appling the trajectory equations defined by

(4.13), we get

if -o107-

The fourth moment defined by (4 5) can be derived by substitu-ting (4.17) into (4.8b). For example, let us consider aspecial case when r1 = P = 0z = , then

4 r) e(4.18)

where

2o~r,,23)

1 3

The- fourth moment defined b,,y +15)__ ca-n, b der ve by-su-bst

t i n g 4.. 17) i n t o ( 4-. . . 8 b)-. . . .. ., u s c s i d r a

specal cse wen r r.,= 0,the

-108-

and ,S2 '

r~l (' ,oo, , ) 9 e ,___,,_,__

_ o r , e 4 (4.19) i

For the ionospheric propagation case described by Fig.1.1,

the S function obtained in equation (4.17) will be used to ob-

tain the distribution function at the bottom of the random slab

at z = z,. The quasiparticles will redistribute while propaga-

ting in free space and obey the following equation:

(4.20)

The trajectory equations becomes

~~)= r~L~)t ~(4.21)

where z. is the location of the receiver on the ground. One

notes that

A U!

- 109 -

(4.22)

using (4.17), (4.22) and (4.21), one obtains

r, j r2 ( 3 2) 3L-2)

_' - , ( - ,3) ] z

1-t --A- 32)]j

[ I "" ( r_ n2)]1 (4.23)

where A z z2 - z, is the distance from the bottom of the slab

to the ground. For the special case P = r, = r. = 0 the S func-

tion can be calculated from (4.23) as:

S (fee,, 0, ,. o ,

__ 1 (4.24)

aLr

a 2. *

'I.L. - ..

-110-

and j i!

JM* 0z__ 0 0SoA

S- 1 -14 13 (4.25) K

where

A~ t~

= ...L t . + L2 (4.26)

Oi rz ± -

313.

One thus concludes that if the 4-point statistics of the wave

field is given at z = 0, the corresponding 4th order statisticsat the ground can be obtained from the z evolution of the qua-siparticle distribution function from equation (4.14) and (4.22).

III. Approximate Solution for 4th Moment

The defining equation for the 4th moment can be rewritten

as _

I. .-

H r H(eI (4.27)

M4L

~- 111 -

Inserting (4.8b) in (4.27), one obtains

A k+r

2 -~fR+ Ar r)-- 4,e --

(4.28)

C1 dr,

where

Onr..,) H (r +t 4 + H (ry±? + ~>- 2.

2 .

since

(4.29)

one derives from (4.28) after some manipulation

r r' ) Jn C 3 (4.30)

e -and

-+M

. r. ., ) . &.

L . i - . ........ .... .... I ' I

I." I %]r li 'l#

- 112 -

From (4.28) and (4.31) we readily derive the integralequation for S ass

S - (4.32)

Equation (4.32) contains an extinction term H(r1 + )+H(r,- )which will reduce the quasiparticle distribution function whileredistributing it in kR , ? ,k, , r. space. The "scattering

term" on the right hand side of the equation describes the con-tribution to the total S function at the "momentum coordinates"

ki , r, from a "momentum coordinates" k% , k' - ,r The

complicated form of equation (4.32) makes it impossible to ob-

tain a solution in analytical form. We shall consider firstthe simplest case of a plane wave, for which obviously 7,M4 =0.

V7 M4 can also eliminated from the equation, ( ? is only aparameter) by setting it to zero because in this case,? = 0,

the points ?, , ,' , ,, fa' lie in the plane z at the vertices

of a parallelgram centered at the point R and with sides, -17' =

5' - *= r2 9,-',' ?.' - ,.= Therefore equation (4.27)takes the form

3 +3)(4.33)

• ' .J .:'':....-.......

- 113 -

where

( , -,- , ,} ) =2 H ,,) 2 H (rO) H (r +Y',9) -tI -r.) . .

(4.34)

one notes from equation (4.5) that

- ( -,(4.35)r2 T-[ c-

which suggests that r, and ra can be treated as "slow" and

"fast" variables respectively. In this case, one encounters

the same degree of difficulty using either the "slow spectrum

S" or the "fast spectrum F" to solve equation (4.33). In

order to maintain the consistency, we shall adopt the "slow

spectrum procedure" as described previously.

Following the same steps as in (4.27)-(4.32), one obtains

the defining equation for S function for the plane wave case as,

(4.36)= r

-114-

On the trajectory defined by

= -s- =Q

(4.37)

equation (4.36) becomes

2 (4.38)4

where

(4.39)

The solution of (4.38) has the form

:, (1%Ar , ) '), ) r2 (4.40)

• ___________..,-. .. >.

115

Substituting (4.37) into (4.40), one obtains r

fit, r

S J(4.41)

H cr 3j J- kn3)J

Equation (4.41) together with the z = 0 boundary condition,

S(o, ra , kr, ) = (kr,), gives the solution for the S func-

tion at distance z. The first approximate solution can be

obtained by substituting the z = 0 value of the S function

into equation (4.39), whence one obtains

fl-cs A~~3 ~ 442)

and from equation (4.41)

S (A-.,r-. 3) - ( .) - H ,, +

41,17S3A

@ COS AIr (r 2. - :k(3')J (4.43)

4-s

I . *4

- 116-

One notes that the Fourier transform of equation (4.43), which

gives the approximate solution of M4, is not a symmetric func-

tion of r, , r& . This violates the symmetry property of M4 .

Therefore, it is logical to consider the Fourier transforma-

tion with respect to r, , i.e. via the fast spectrum procedure.

Let us define

" "(4.44)

I:

One obtains a solution for F similar to that described in (4.43),

except for the interchange r, -* r, and kr+*kr " The approxi-

mate solution of M4 can then-be written as

" r.'5 (4.45)

F Or, -

which maintains the symmetry character of M4 with respect to

r rz .(cf. Tatarskii, 1)

IV. The Boundary Conditions and Steady State Solution for Plane

Wave Case

We note that when the pairs of point (?i, ?,j) and

move an infinite distance apart. It is obvious that the field's

corresponding to these pairs of points become statistically in-

dependent, thus one has

- 117 -

at 1r1 1'.

2.

similary, we have

at Ir, -

N4 (r, n,3) = I M 1~ ($,r) 2~ " (4.47)

we now consider the properties of the function f(rI ,ra , z) in

equation (4.34). One observes that f(ri ,r, z) is positive

everywhere except at r, = 0 or ra = 0, f(r, , r.) = 0. We

therefore consider the term kez/4 f(rL ,ra ,z) as a decaying

term which tends to make the function M4 decay, while the 2nd

term in equation (4.33) is a diffraction term which will diff-

use the value of M4 among the transverse coordinates. With

the diffraction term only, equation (4.33) can be written as

M4 - A. M4= (4.48)

W, ra

-118-- I18

the solution of (4.48) can be written as

iJ M (rr'i,M, 2'r" ) ~ (2 )' * ("9" ) e(4.49)

Thus at z>O the value of M4 will be redistributed as descri-

ed by the above equation.

One also observes that due to the decay term, M4 will

decay to zero when z >> 1 for all rl and r2 except near

r, = 0, r, = 0. However, because of the diffraction term the

value of M4 near r, = 0 or r,= 0 will be redistributed to all

other points (r, , r. ) where f(r, , r, ) = 0. Therefore one

observes that M4 will decay to zero for large distance z unless

M4 is of the form (cf. Lee and Jokipii, 68,69.)

M* t 3) ll,(A1 2 ( ) t ± 3 ((4.50)

For this range the diffraction term is zero and M4 will reach a

asymptotic solution. In order to satisfy the boundary condi-

tions in equation (4.47), we must have

at I =

(4.51)

i:iI

1191

~~- 119 - I

therefore

2

M, - l(,) ± - 11(3M 2

(4.52)

M2.

at ~I~~o

27n

T 'l 3 (3) (4.53)

therefore

in1 ~ IM2

I n 2 ( , v 1 O (4 54 )

Substituting (4.54) into (4.52), one obtains

I M2 t 2 (4.55)

-120-

For the plane wave case, we obtain the following expression I

+,-e (4.56)

After the wave leaves the random slab with thickness z, and

propagates in the free space region, we have the differential

equation:

*~ (4.57)

the boundary conditions for M4 are given by equations (4.46).

However, M2 appearing in equation (4.46) now takes the form

M= (,r)= M2(,,r)M(4.58)

since in free space M2 (z,r) is unchanged.

The solution of (4.57) can be written immediately in

terms of the value of M4 (z, r. , r 2 )g

; ~~~~~ ~~~~~ ..' ., ..; ..;: T ... .! .., '•

I.

M4 (r, 2 3)=A- - r

for z >0, the value of M4 is redistributed among the trans-

verse coordinates until it reaches a steady state, if such

exists. One finds that at steady state, M4 obeys

-IM 2 ,, 2 '°(4.60)

V. Numerical Solution for MA in Plane Wave Case

The general solution of M4 is too complicated for compu-

tation. In order to simplify the problem and still keep the

amin features of the solution, we shall take the four points

?, 2 , ', a' on the z = constant plane along a straight line

such that r, and r2 in equation (4.33) become scalar. Intro-

ducing the dimensionless variables

__ - r'- .Q(4.61)

.41

where . is a parameter related to the scale size of the irre-

gularities. When written in terms of these variables, eq. (4.33)be come s

rl r2(4.62)

where

(F, AF.) 6-A(1.

A (4.63)

one observes that Y' = k;2 A(O) is a dimensionless quantity.

For convenience, we shall drop all the "bars" and rewrite (4.62)

as

(4.64)

Since M4 is a complex field, it is convenient to divide it into

real and imaginary components by writing M4 = MR "f-M.. One

obtains two coupled partially differential equations

-123-

M [tM; MR

~ MR-rYM~(4.65)

To obtain a unique solution of eq. (4.65), one must specify the

boundary condition at z = 0 and the boundary conditions given

by equation (4.46), (4.47). In practice, we cannot apply the

boundary conditions at r1 , r3 -- @0 because this would re-

quire an infinite number of mesh points. One can simply trun-

cate at appropriately large values of r, , r. . The numerical

scheme used will be presented in Appendix (V.).

The solution of eq. (4.65) is dependent on the choice of

power spectrum f (k). A simple and very commonly used form is

the Gaussian

TE (4.66)

then 2

(4.67)

• - -- -.. . . . . .-.. ....... . . .. , .

- 124-

and

(4.68)

however, in situ measurements indicate a more realistic form

is given by the modified power law spectrum defined through

U2L~) e e'~/..E.) (1 + 2(4.69)

with kL,<( km. This spectrum is flat for k < kL6 , is a

power law with index -p for kL,<k < km , and is exponenti-

cally decay for k >km. LO = 1/kL, is the outer scale and 10=

1/km is the inner scale of the irregularities. In the iono-

sphere case, usually 2< p< 4 and p = 11/3 corresponds to the

modified Kolmogorov spectrum (Von-Karmann Spectrum).

For the Kolmogorov spectrum, the power spectrum can be

written as

-2 Y60-33 Lo (+ fqL.)- e(4.70)

with

...

- o ~ . ' -

- 125 -

if L).>lo one obtains

H () A (o)-A C')

(4.72)

where KY denotes a modified Bessel function of the second

kind.

If we choose,Le = 1 = 300m, and C4

the value of P L4 F5 ranges from about 8.5 for f/fp = 100to sbout 86.5 for f/fp = 10. The value of in eq. (4.65) is

equal to 0.0987 P for the Von Karmann spectrum, 0.16 P for thepower law spectrum with p = 4, and 0.2821 P for the Gaussianspectrum.

We shall adopt the power spectrum given by equation (4.69)

with p = 4 and present a numerical solution for the fourth mo-3 1ment. The value of Or= 4K.LA(0) has been chosen to be 3.5.

Fig.41 depicts the normalized function f(r, , r a ), defined by

equation (4.63), as a function of r, , r2 . One recalls that

f(r ,rz ) represents a decaying factor which is positive every-

where and equal to zero when r1 and/or r& = 0. The initial

field M4(r, , r1 ,0)=l is not shown in the figures. The evolu-

tion of the 4th moment at z = 0.1125, 0.1725 and 0.24 is shown

'4

* . . %. .,.$ *

-126-

VON KARMANN SPECTRUM F(R1,R2)

36.80

22.5

F( RI,R2)

7.56

8.8-4.68 -2.69 6.6 2.06 4.86

RI

Fig. 41. f (r1 , r. vs. r, and r2 ;Power spectrum withp = 4 .

Zz 1.1250SE-e1 GAMMAO 3.50008 2

4.00

2.56

1.25

6.25E-01

6.6-4.00 -2.086 R 6.6 2.00 4.88

Fig. 42. M4 (r, , , z) vs. r, and rx at z =0.1125;a plane wave propagating in a random medium with apower law spectrum, p =4,.df 3.5

-127-

Zu 1.?23S8E-01 GAMIMA= 3.50000

2. A58jjjjj

6.25E-81

9.8-4.00 -2.96 6.8 2.00 4.80

RIFig. 43. M4vs. r. and r2 at z =0.1725.

Z'a 2.40COE-01 GAMM~A= 3.56000

4.00

2.56

1.8

1.25

-4.00 -2.00 0.6 2.90 4.00RI

Fig. 44. M4 vs. r, and r2 at z =0.24.

- 128 -

7

128.in Fig.42 - Fig.44. One observes that the field decays every-

where except near the neighborhood of r, and/or r= 0. The

redistribution phenomena, caused by the 2nd term (diffraction

term) in equation (4.64), forces the points near r, and/or

r z = 0 "move" to the decaying region. Therefore, if the evo-

lution continuous to a large distance, the final field will

tend to reach a steady state as shown by equation (4.55).

However, we are not interested very large distance propaga-

tion in this paper. The extend of the normalized in random

slab will be choosen such that L = 0.24 and take the solution

will be evoluated at z = z', whence the wave will continue to

propagate from the bottom of the slab to the ground. Fig.4548 depict the free space propagation at z = 0.4275, 0.6525,

0.8775 and 1.1025. The diffraction effect, resulting from

the interference of the distorted wave front, tends to focus

and defocus the field. This focusing and defocusing phenome-

na will only occur for strong turbulence case, which will be

shown clearer in next chapter.

VI. Discussion and Conclusion

The 4 point slow quasiparticle distribution function,

defined as the 6 dimensional Fourier transform of the 4 point

correlation function w.r.t. the slow coordinates R and r., is

used to obtain the 4 point wave statistice. Pictures of the

slow q.p. flow are essentially the same as those in the 2point correlation case except we are now dealing with a four

dimensional phase space kg , R, kv , r, . An individual slow

q.p. moves along the trajectory defined by eq. (4.13) with"momentum" kR in the F direction and kr in the r1 direc-tion. The structure function is, in general, dependent not

noly on the "coordinates" and r2 but also dependent on the

-129-

Zu 4.2?59E-01 GANIIAu *.0

2.58 9

1.25

6.25E-91

-4.08 -2.66 6.8 2.88 4.00Ri

Fig. 45. M4 ( r,, r1 ,z) vs. r, and rz at a distancez = 0.4275 from the bottoxr 1 of thle slab; L =0.24.

Zz 6.5250E-01 GANI A= 8.80

2.56

1.88

9.

* 1.25

6.25E-01

0.0-4.00 -2.66 6.0 2.06 4.00

RI

*Fig. 46. M4 vs. rL and r. at a distance z =0.6525from the bottom of the slab.

-130-

Zu 0.?7759E-01 GAMrnAu 08.

2.56

1.88

6.25E-61

-. .4v,, -2.88 0t .8 2.8 40

Fic. 47. M4 vs. r4 and r. at z =0.8775 from tlebottom of the slab.

ZZ 1.10259 GAMMA=u 8.8 S

2.56

1.88

6.61.25

6.25E-01

-4.06 -2.08 9.0 2.80 4.00RI

Fig. 48. M4 vs. r, and r2 at a distance z =1.10215

from the bottom of the slab.

-131-

tH

"coordinate" r, associate with the "momentum" kr. For the

special case, H( ) = a 2, one finds that the structure func-

tion is dependent only on , and of the form f(r, , r. , 7 ) =

aj7

Knowledge of the S function will provide all of the 4

point field statistics. For example, the second moment of the

field intensity can be obtained as follows:

). (2)T (4.73)

Therefore, the second moment of the intensity can be viewed as

the total number of slow q.p.'s located at R = ri = r a = = 0.

The 2 point correlation function of the intensity at R

0 can be obtained as follows:

eM 0 , , 4 .

(4.74)

One notes that when = 2 = R 0 0, then rj= x1- x2= x1- X2

xl' - xZ.

The Scintillation Index S4, defined as the normalized

LL) W .-

-132- i

I.

variance of intensity, i.e. S4 =41 -<1U/, can then be

obtained from the S function using eqs. (4.73) and (4.74). As

an illustration, we shall calculate the 1st and 2nd moments of

intensity and the associated S4 for the case H( ) = a E, using

the S function. Fig. 49 and 50 depict<I),41I2and the corres-

ponding S4 for a collimated beam propagating in a random medium

characterized by H( t ) = a twith a = 1 and a = 5 respectively.

(In the following figures, curve 1 stands for 41>, curve 2 for

<J2> and curve 3 for S4 .) One observes that both and (Z1>

decrease as z increases, while the scintillation index increa-

ses monotonically with z. The case of a focused beam propagat-

ing in a random medium is represented in Figs. 51, 52 wither=

o= 1, a = 1 and 5 respectively. Focusing and defocusing' <^2

effects in (Z)and <Iare clearly revealed in these figures.The scintillation index is greater in the case of collimated

beams in the range z < 2, approximately; for z > 2, focused beams

display a smaller scintillation index than the collimated beams.

One thus concludes that for short distance propagation, a colli-

mated beam is to be perferred while a focused beam is perfer-

able for long distance wave propagation in a random medium.

In ionospheric propagation, we are interested in finding

the received signal intensity and its associated S4 at ground

level. A deterministic wave signal is randomly modified by theionosphere and when the resulting modulated wave front enters

the free space, between a random ionospheric slab and the gr-

ound, diffraction modifies the wave statistics. In order to

show this effect we present results for 4 I.> , < and S4 .

Figs. 53 and 54 depict the case ar = 1, a =5 with the thick-

ness of the random slab being L = 0.25 for collimated beams

and focused beams respectively. One observes that the focused

beam displays saturation at z 2 0.75, where z measures the dis-

, * . @ t

' . , . , , . • . " - " -+

' .+ . . . . , " % -

- 133 -

IS? AND 1'140 MOMENT OF I FcI~H X (r )2- If,: A000=~ 0.0 GAMMA=t 1.00900

2.0A8

i g. 1 S49.()<J s ;()4 v .z 3 4v -Z

II

2..0020030040

0.0 1.000' 2.01 T. CIO1A 4. 00l0

2.00

Fi.5.1aeasFg54.eceta 5

1.0 . r -

.Fi. . sa as Fig 49 exep a . 5..

1l34-

)S" Alin 2110 1IONENT OF I FOP F:I)=.xl2AR; I.00000 A:I= 1.0O00 GAMMA=i 1.00000

2.0

1 .50

1I-2,S4

1.80

F .0 .6.8 .00 tO.4.0

Fi.51. C1) <^>vs. z; (2) 41 > vs.(3) S4 vs. z; a focused beam propagati.ng in a

I ST AND 2140 MOMENT OF I FCP 14( *:I )zr 1: ) 21 .00000 Al= 1.00000 GAMMA~= 5.06000

.. . .. ............. .......... .........

DO 2..00 4.00Fig. 52. same as Fig. 51. except a 5.

-135-

1 2.04

...... .....

0.8 1.88 2 .00I 3.00 4 .100-A>vs Z; A

Fig. 53. (1) Iv. (2) 41I>vs. z ; (3) S4vs. z; a modulated collimated beam propagatingin free space; L = 0.25, a = 5, c,-= 1 .

!3?' AND 2MND MOMIENT OF I FOaR Hkc:'l~~~i)AR= 1.00000 AI 1.0009Q GRIMf1= 5.00000 SLA~B= 2.5000'-A

2. E8

J11-2,S4

1.0

.... ..................................................0..............

6 0 1.00 2. ('0 3.4.0z

Fig. 54. a focused beam propagating in free space;0.=20.25, a =5, al; ~= I.

-136-

tance from the bottom of the random slab to the receiver. For

z smaller than approximately 1.8, the collimated beam yields a

greater S4 than the focused beam. However, as z 1.75 the fo-

cused beam displays a smaller S4 . Figs. 55 and 56 for a = 10

display the same effects. One obtains a higher scintillation

index and observes a saturation at z t 0.75 for the focused

beam case. It is of interest to plot S4 vs. z for different

turbulent levels in the ionospheric slab. Figs. 57 and 58 de-pict S4 vs z for a = 2 to 16, in steps of 2, for the collimated

beam and the focused beam, respectively. One observes that,

for the focused beam, S4 always saturates at z = 0.75. At z v1.75 the focused beam start to display a higher scintillation

index than the collimated beam.

When the slab thickness increases to 0.5, the random wave

fields emerging from the bottom of the ionosphere are increas-

ingly more turbulent. The corresponding S4 in this case is high-A A2

er than for the case L = 0.25. Figs. 59-62 depict .4I> 4iz >and S4 vs. z for a = 5 and 10 respectively ; one observes that

the focused beam saturates at z L 0.6 approximately. The focu-

sed beam has a smaller S4 than the collimated bean in the rangez greater than 1.5. The influence of the turbulent strength a

on the scintillation index is shown in Fig. 63 and Fig. 64 for

a collimated beam and a focused beam , respectively. The valueof a ranges from 2 to 16 in steps of 21 one notes that at z 2

0.6 the scintillation index of a focused beam saturates for all

values of a. At a distance z T 1.5 the focused beam starts

to have a smaller S4 for all values of a.

The wave front of the wave emerging from the bottom ofthe random slab is greatly dependent on slab thickness. As a

comparison, we present calculations of S4 vs. z for different

values of thickness ranging from L = 0.125 to 1 with increments

of 0.125 as depicted in Fig. 65 and Fig. 66. One observes that

for the focused beam case the saturation distance decreases as

-137-

I'ST AN'14 110 MOMf~ENT OF I FOR H( :-* "-=A* % ) .*Z.

AR. 1 QOOLIO Ala 0.0 rAR1Ii= 10.0000 SLAB= 2.O60E-:J

11-2,S4

0.0 1.80 2.00 3.00 4.00-Fig. 55. ( A 2>s A%

()4)s -7(2) (I)vs. z (3) S4 vs-z; Fmodulated collimated beam propagating in freespace, L =0.25, a z1O,olr= 2,

S7 AND 2ND0 NMnEtT OF I FOR 4i(:W =~:,-2'R= 1.00000 Al= 1.00000 Gildifl4= 10.0003 SLAC= 2.500CE-;'i

5-GOE-01

..........0.9 .......... .0.8 1.60 2.00 3.00 4.00

Fig. 56. a modulated focused beam propagating'in free space, L =0.25, a =10, =af*.= 1.

.............

-138-

ARC 1.0'000 Al- 0'.0 SsB25~~E

2.08

1.58 14

S4 1210

8

2

Fig. 57. s4 z for a 2 to 16, in step of 2;a modulated collimated beam pronagating; in frcaspace, L =0. 25, O1~r 1.-

; 1.IL'C~0 AI .Q~D'30 SLAB= 2.50000'E-01

2.00

14S14S4 12

10LOO 8

6

------ ------ 4

* .. 80 .0E0 o 1 so 2 OCI

Fig. 58. S4 vs. z for a 2 to 16, in step of 2;a modulated focused beam case.

......... . . ...-.

-139-

1 0411D 2 1D rIC1i1ENT OF I FOP H x z )=if.'*: x i4'a 1.oooo i= 13.0GH1IMW 5.100000 SLAB= 5e0'E~

2.00

1.50

1.1I-2,S4

, 33

Fig.~~~~~~... 59j1lI)s.z 2 .I~s 3

0.0 1.00 .0 3.-00 4.100

Fig 60. saeazg 9; a modulated focusmaed beamprmacgase. i respcL=05,a=5 v

'ZTAN f.mEITOF FR (XI;%A- M'1--.-- - 2

AI= .0000 CUIN 5.0000 SLA 5 1? ' 100

- - 140-

!ST Ak' 210 MlOMENT CIF I FOR~ H(-1'-.l 1 "I).iz j .00000 Alz 0.03 GetMfl,= 10.0000 SL'E- 5.0000E-Cl

.00

1. oe iIt

802...................... ....... ...........

Fig. 61. (1) vs2 >; vs(3) S4 vs. z; a modulated collimated beam tr-ot0n.-gating in free space, L =0.5,. a 10,,4r= 1

ST ANJD 'NO MOMENT OF I FOP H(XI"=j i~iR- 00O0= ~1=.006300 S~~~i:1.01 LAB= ~G1~E1~

2.88

1.503

* (A122.S4

1.00

5BOE-91

.........................................0.8

8.0 zFig. 62.'same as Fia. 61.; a modulated focusedbeam case.

-141-

;Rx~') 41i= 0.0 SLAB= 5.0006CE-01

a 16

2.800

1.50

S4

1 .00

.00a

Fig. 63. S4 vs- z for a 2 to 16, in step of 2;a modulated collimated beam propagating in free-space. L = 0. 5, dU, .

Vs FOR HN =N '=1.010 A!= 1 .00000 S r .'C0E 1

a 16

34

2.00

a 2

0.0 5 0s3E-01 1.00 1.503 (1zFig. 64. S4 vs- z for a =2 to 16, in step of 2;a modulated focused beam propagating in free space,L 0 .5 ,ar= 01; 1.

-142-

L increases. The distance at which the focused beam displays

a greater S4 than the collimated beam decreases as L increases.

The approximate solution for the 4th moment M4 is pre-

sented in Section III. In (4.43), we have replaced the true

slow q.p. distribution function S( z, r1 , kr- k') on the

right in (4.41) by the initial unattenuated q.p. distribution

function S( 0, r2 , k - k'). It is expected that the solution

so obtained will give an exaggerated result for the intensity

fluctuation. To illustrate this fact, we shall calculate thescintillation index for the plane wave case and compare it

with the exact solution obtained numerically. The random slab

will be characterized by a power law spectrum with p = 4. Fig.

67 depicts the scintillation index calculated from the approxi-

mate solution (4.43) for the cases L = 0.06, 0.12, 0.24. The

general behavior of scintillation index agrees with its well

known feature. Fig. 68 shows the comparison of the approxima-te solution (solid line) with the exact solution (dash line).

As expected, one observes an overestimated intensity fluctua-

tion in the approximate solution.

Boundary conditions and the steady state solution arediscussed in Section 4. We have applied these boundary con-

ditions in the numerical evaluation of the 4th moment as pre-

sented in Section 5. The general solution of S4 is too compli-

cated for computation. In section 5, we computed the solutionof M4 for a special case wherein the 4 points x1 , x1 ', x2 ,

lie along a straight line on the z = constant plane. The nu-

merical evaluation of M4, for a plane wave propagating thru

a random slab with L = 0.24 and characterized by a power lawspectrum with p = 4, is presented in Section 5 for z = 0.4275,

0.6525, 0.8775 and 1.1025(Fig. 45-48). One observes that as

the field evolution is continuous by extrapolated to a largedistance, the final field will tend to reach a steady state as

shown by eq. (4.55).

-143-

kf- GAMMA&t~ 10 0080L

34

1.0')

L =0. 125

8. 588-0 0 1.50 .00

Fig. 65 .S 4 vs. z for L 0.125 to 1, in step of0.125; a modulated collimated beam propagating infree space, a =10, o'y= 1.'

4 U0S. ' FOR H(XP=A--1) .2.~= .O0'00 '1.00000 GAMMA~= 10.0000

L- 1

%4

L =0.125

0.0 .OOEO1 .03 ~0 2.00

Fig. 66. S4 vs E or L =0.125 to 1, in stop ofo.125; a modulated focused beam case, a =10.

.....................-.-.. .. .. .

,. 7 77 .C

-144-

1.20

9.88E-81

G'OeE-91 .

* 0.8 3.GGE-01 6.80E-01 9.OGE-81 1.28

Fig. 67. S4 vs. z calculated from the approx.solution (4.43); (1) L =0.06; (2) L 0.12;(3) L = 0.24.

S .eeE-01

3-OSE-St

US.

Fig 68. S4 vs. z, L =0.24; solid lines approx.solution; dotted lines exact solution.

low_

-145-

Chapter 5 THE INTENSITY FLUCTUATION AND SCINTILLATION

INDEX

I. Definition of Intensity Correlation Function and Scinti-

llation Index

If the new variables (4-5) are introduces, the function

M4(r, r, z) is expressed in terms of the wave fields by

the formula (for ? = 0 )

A r

M. 1P R (+ y'j q(~;2, )y(R+ r, )q,(~L2 )

(5.1)

where in this case, a= F• -?1 2A-

The 2nd moment intensity function is defined by

- - (5.2)

which is identical to the fourth moment M4 when ?1 = , = 'or rY = 0.

One also recalls that

- 146-

A A

M~ (3X, < LP <'4 , xt fo)' -4)

(5.3)

A

M2, , < ~,)>

(5.4)

where 2 One defines an intensitycorrelation function as

A A A

Br 3 ) = (3I 30,1 (332)>-(5.5)

I

and one observes from equations (4.5) that rz--, R! -t .z-,

a R- -L , and x = R. Therefore

r, A

13 Mt -) (3, R-= + )I (,- '

A A7

- -. R

-147-

For the plane wave case, one takes into account that M2 isindependent of the slow coordinate and write

Thus, M4(z, r*, 0) enables one to determine the correlationfunction BI(Z• ,,,?) = B(z, r, ).

Al -

The scintillation index S4 is defined as

A_< A >Z

54 A >2 >Z(5.8)

which plays an important role in the scintillation problemand is used as a measure of fluctuation level. For the case

H(s a 12 , one obtains from (5.8) and (4.25)

M4(',0 0 , 3)= -f- 6 e 4

M2 (-- -,-3.-- - -- e2) (5.9)

2i

-148-

where

-- -

C4' t 4Z

(5.10)

z 2.

the scintillation index can be calculate from (5.8) as

- <12 -4I>4

- M (o,oo,o, - (ov,) i(5.11)

M2z o 0,

II. Numerical Solution of Scintillation Index for Arbitrary

Power Spectrum

In our computation, we shall first choose a Gaussian

power spectrum. The corresponding correlation function is

also Gaussian. The main reason for choosing such a spectrum

is to check with the results obtained from the thin phase

screen approximation (4). For the Gaussian spectrum, the

7(r1 ,r,) in equation (4.63) is given by

- 149 -

2I2I

y r2r) Z2t +~ e 2=rz- ) -_____

2. (5.12)

Using (5.12) in the numerical calculation to obtain the scin-tillation index, we shall present some computational results.Fig.69 depicts the behavior of the scintillation index as a

function of z (from the bottom of slab to the receiver) for

the case"(= 10. Curves 1,2,3 correspond to slab thickness

L = 0.06V 0.12, 0.24 respectively. For L = 0.06, corresponds

to a thin phase screen, we observe that S4 increases monoto-

nically with z, and finally reaches a constant value. The

results agree very well with the thin phase screen theory.

As L increases, corresponding to a thick random slab, the

scintillation index grows. One notes that a focusing pheno-

menon occurs when L = 0.24, the scintillation index reaches

a maximum before it settles to some saturation level.

It has been indicated that, in the thin phase screen th-

eory, focusing occurs only in the strong turbulence case(rms

phase fluctuation =M4L in our case). Our results shows the

same behavior and should be better, since we take into ac-

count the thickness of the slab.

Scintillation index curves as a function of z for the

case L = 0.06 with different values of or , equal to 5, 10,

30 are shown in Fig.70. The curve on the top indicates that

a focusing behavior appears for strong turbulence case even

though the random slab is very thin.

,..'i

-150-

G.AUSSIiAN SPECTRUIICA,*IMA= 10

1.29)

6.OeE-eI

8. .6 3.0@E-01 6.OOE-0i1 9.e0E-01 1.20z

Fig. 69. S4 vs. z; Gaussian spectrum withdrY 10;

(1) L = 0.06; (2) L =0.12; (3) L =0.24.

~LnSPe.:rumLc..00(Y

3.000-Ol

* 0 3. o0.-ol a. e0.-Ol 9.006-0l 1 all

Fig. 70. S4 vs. z; Gaussian spectrum, L =0.06;

(1 V 5; (2) OY= 10; (3) or= 30.

151 -

The following types of power spectrum will also be con-

s idered

(a) Power law spectrum with power index p 4

rTE 2) ( kZ*L z)2 (5.13)

(b) Von-Karmann spectrum p 11/3

_____ >_ L63

7E , 7" (-L) (' 1. +! L<, ) ' (5.14)

(c) Von-Karmann spectrum with a Gaussian bump

F ) g>L*

the calculated function f(r, . rz) for cases (a)and b)are shown

in Figs.71 and 72. Figs.73, 74 depict the function f(r1 ,r2 )

of case (c) for k = 2, )<I= 0.5, a = 0.5 and 1Z = 3, )<2= 0.5,

a = 0.5 respectively. The scintillation index vs. z for case

(a) is displayed by the dashed curve in Fig. 75 forcr= i0/7 ;

the higher curve corresponds to L = 0.24 and the lower curve

to L = 0.06. As a comparison, we also show the corresponding

values for the Gaussian power spectrum which are displayed by

solid curve in Fig.75. One notes that the scintillation index

associated with the power law spectrum is larger immediately

- N. .. .

-152-

VON KARHANHH SPECTRUM F(R11 R2)

38.80

22.5

F( R I R 2)..............

7.584

-4.08 -2.88 RI8 2.08 4.88

Fig. 71. ff(rj , r,,~ vs. r arnd r. Power lawspectrum with p = 4.

VON KARIIAtI4 SPECTRUM F(R1PR2)

38.80

22.5

F( RI, R2

15.8

7.580

-4.8 -.88 8.82.89 4.00- RI

Fig# 72. f(rS , r1 ) vs. r and r. Von Karmarnspectrum with p =11/3.

-153-

V'wn Karmnn~ Spectrum F~rl.rB)'Ant Karmanns Spectrum with G Us.- petak

20 .00000 wid.- S.00 4ps. COO06e-01

F~rI.r2)

-4.00 -2. 00 0.0 2.00 4. e-

ti hFi~g. 73. F(r± r~z ) vs. rand rz Von Karmann

spectrum with a Gaussian bump, k=2, K L 0.5, a =0.5.

'JiOn IKarmazun Spectrum with G * ealxO* 3.00000 wid..

30.0

F(rl,rB)

75so

0.0-4.90 -2. 00 0.0 2.00 4.00

ti

Fig.74. f(r,. , r~ ) vs. r ±and r. Von Karmann

spectrum with a Gaussian bump, k=3, )(Z= 0.5, a =0.5.

- 154 -

below the slabe and become smaller at large distance. Fig. 76

shows the scintillation index vs. for case (b) with L = 0.12

and = 5 (lower curve), r = 15 (higher curve). The distinc-

tion between strong and weak turbulence is obviously related

by the focusing phenomenon. The Von-Karmann spectrum with an

addition Gaussian bump greatly changes the behavior of scin-

tillation index and is shown in Fig.77. Curve 1 depicts the

case k = 3,'r= 5, L = 0.12 and curve 2 displays the case k = 2,= 5, L = 0.12. One concludes that the location of the Gau-

ssian bump has a significant effect on the scintillation index.

The focusing behavior occurs more than once in both curves,

which indicates the additive bump increases the distortion of

the wavefront emerging from the bottom of the slab. Fig. 78

ccj pares the different behavior of the scintillation index

for the case (b) and case (c) and for the parameters r = 15,

k = 2, L = 0.12. Focusing occurs in both curves. At J- ,

spectrum (c) has a larger S4 than spectrum (b). Fig. 79 de-

picts the behavior of S4 for case (c) with k 2 and 0= 5, 15

respectively. One notes that even for O= 5, the scintilla-

tion index vs. z shows a focusing phenomenon which is not

observed in case (b) with 7= 5. This apparently indicates

that the additive Gaussian bump causes a strong scatteringeffect. In order to show the effect of the power index P, we

observe, for the case L 0.24, ke Lg<E>= 35.44, the scintilla-

tion index vs. z for p = 4 and p = 11/3 as displayed in Fig. 80,

the solid line being for p = 4 and the dash line for p = 11/3.

One finds that the scintillation index vs. z has the same cha-

racter in both cases but increases with the larger power index.

I

1 '

-155-

,NFISPLA54'\G

t1Y*1/2S1.2IX GFOP

1.20)

S4

6. OSE-81

0.0 3. CCE-8 I 6. OeE-e I 9.OOE-81 1.28

Fig. 75. S4 vs. z; dotted linest (1) Power law

spectrum, p = 4, L =0.06; (2) Power law spectrum,

L = 0.24; solid lines: (1) Gaussian spectrum, L=U.,n Krmann Spctrumr 0.06, (2) L 0.24.-

.4

1.00el4.20*'41 7. 800'41 1.08 13

Fig. 76. S4 vs. z for Von Karmarin spectrum;(1.) L - 0. 12, r 51 (2) L =0. 12, Or= 15S.

t. - ., - ,. -.

5QjnLtation aIndex ye. Z (Von Karmairn Spactrum)&A-in ICrmann Spectrum wlJ~ C&USOL&A Puak.j;ima. 5 00000

*V.1V811.Rm1x gfk-&1v

1.20

'4 2

Fig. 77. S4 vs. z for Von Karmann spectrum with

a Gaussian bumnp; (1) = 3, Y- = , L = 0. 12;

(2) k= 2, r=f 5, L =0. 12.

IS-000 5.0000"--03. kom a. eeeeo .0"00

1.80

9."o4

.

Fig. 78. S4 vs. z ;()Von Karmanl spectrum,

or= 15, L = 0.12, p a 11/3; (2) Von Karmafln spectrum

with a Gaussian bump, =2, Y = 15, L 0.12, p =11/3.

-157-

k-.- 3. 00000 wlda 6. 000ft.-6l amp- 6.8008ea-61r~zpaLZ'G

'vfk-lv

l(2)

S.00-0l1

2.0ee-61

Fig. 79. S4 s for Von Karmarin spectrum witha Gaussian bump, I~=2,-r=~ 5, L=0.12 ,(curve 1);I= 2,-r 15, L 0.12, (curve 2).

'\FISPLA57'\G

'&Y+1/2*1.2MXGFK-1U

1.28

9.SNE-O1J.()S4.. .

6.SOE-6112)

3. 66E-6 1

66 3.OBE-01 6.80E-01 9.8SE-6I 1.26z

Fig. 80. S4 vs.- z for power law spectrum;(1) L = 0.24, 'V' = 8.86, p 4; (2) L =0.24,

=8.86, p =11/3.

-158-

III. Application and Discussions

A power spectrum, obtained from the Atmospheric Explorer

- E (AE-E) satellite in situ measurements (70), is used to

calculate the corresponding S4 for the night time equatorial

scintillation. The variance of N/19 is estimated to be 0.207.

We have normalized the power spectrum such that the integral

of the fluctuation powerS (k) dk over the observed frequency

range is equal to Z(N/N)2 >. For computational purposes, we

assume a flat spectrum for irregularity wavelength greater

than 5 km and truncate the high frequency tail for irregularity

scales smaller than 0.4 km. The resulting power spectrum is

depicted in Fig. 81. In order to compare the theoretical model

with the experimental results, we have chosen an incident wave

frequency 137 MHz, f-o = .9 MHz and 1 = 600 m such that the

parameter Or= 1/4 ko 12A(0) in equation (4.65) is equal to

52.73. The propagation distance z is normalized to -103 km ;i.e. measures distance in 103 km.

Radar observations suggest that the ionosphere can be

replaced by a random slab with an effective thickness -200 km.

The distance from the bottom of the random slab to the ground

is taken to be about 300 km. Applying these parameters to the

defining equation for the fourth moment and using the numerical

scheme described in Appendix 6, one obtains a plot of the scin-

tillation index vs. z as shown in Fig. 82. A scintillation

index as high as 0.35 is predicted at the bottom of the random

slab. As the wave leaves the random slab and propagates in

free space, diffraction phenomena develop. The scintillation

index increases, passing through a maximum value at a distance

-250 km from the bottom of the slab, and then saturates. At

ground ( z = 300 km from the bottom of the slab), one obtains

a scintillation index of 1.14, which corresponds to a peak to

-159-

[peak fluctuations of approximately 32 dB.

Experimental results obtained by S. Basu et al. indicate

a scintillation index of 0.6 at 137 MHz, which is lower than

our analytical results. For a density fluctuation of 20% in

the equatorial region, the experimental value, S4 = 0.6, seems

a little lower than expected. In private communication from

Dr. Basu, he has informed us that his experimental result may

be too low because of instrument saturation.

Possible explanations of this discrepancy might bet

(1) The dynamic range of VHF receivers are limited to about

16 dB - 25 dB. A scintillation index as high as 1.14,

corresponding to peak to peak fluctuations of 32 dB, is

beyond the capacity of such receivers.

(2) The in situ power spectrum was obtained at an altitude of

250 km. By extrapolation to magnetic field conditions at

the scintillation measurement site, Basu et al. suggest

an ionospheric structure -200 km in extent about 300 km

above the earth. The accuracy of this model assumption

is not evident.

(3) In our theoretical work, we have assumed a flat spectrum

for X >-5 km and have truncated the tail of the power

spectrum for A14 0.4 km. This assumed power spectrum is

also a possible source for The above discrepancy.

(4) From the investigation in section II, one concludes that

S4 increases as the thickness of the random slab increases.

In the numerical evaluation, we have assumed the thickness

of the random slab to be -200 km which may be larqer than

- 160 -

the real effective thickness of the ionospheric slab.

In spite of the above discrepancy, one notes that the

numerical results for the case of a Gaussian power spectrum

are in good agreement with the results obtained by Mercier

(4). For the case that L = 0.06 which corresponds to the

thin phase screen case, one obtains a saturated S4 which

agrees with the value, S4 = 1 - exp(-2 .L), obtained by Mer-

cier (4). Our numerical scheme provides better agreementwith thin phase screen theory than the scheme proposed by

Liu et al. (60). Therefore, the proposed propagation model I tand numerical scheme should also provide satisfactory results

for the case of an arbitrary power spectrum. The only real

difficulties are proper location of the random slab and es-

timation of its effective thickness.

For further study, we shall apply the theory and numeri-

cal scheme presented herein to the multiple frequency, multi-

dimensional, inhomogeneous, etc. cases. Consideration of an

anisotropic power spectrum and of regime represent nonlinear

effects in the strong scattering areas for future investiga-

tion and may provide further insights for the theory of iono-

spheric scintillations.

I . -. . .. I

-160b-

1.48 .

1.12.

S.07"0.0 6.87.-Sl 1.25 1,88 2.51

Fig. 81. in situ power spectrum vs. k;

S4

S 546--ol

8.00.-01 3 00.-Ol 4.00a-Ol S. 006-e 1 00ey-al

Fig. 82. S4 vs. z for a power spectrum depictedin Fig. 81.

- 161 -

APPENDIX 1

Solution of ( Le- V ) f = 0 where LO is the an unperturbed

operator and V is a random perturbation operator with 4 V > =0.

To solve this equation, it is convenient to define a stochastic

Green's functions

~(A1 .1)

For example, in r, t space

A AA A

S,) f ( r' V) d r'dk'

A

For uniqueness, the domain of G will be defined by the require-

ments

It is convenient to introduce the coherent unperturbed operator,

Go defined by

L. c, -- . (A1.2)

LJ 4

-162-

The perturbation operator in equation (A1.1) can then be treat-A

ed as the scattering part of the propagation.

At By equation (A1.1), we have adjoint operators, L. VGt which are defined by

= c~(0 -V)(A1.3)

Therefore,

(AI..4)

Similarly, we have

Lo = C~ O o.(A1.5)

Multiplying (A1.4) by G and using (Al.5), we obtain

IV A

since G* L0 = 1, we have

dim k

-163-

A= A

where we have defined (16

A A

v CT T G- (Al.6a)

The operator T represents multiple scattering by the stochas-

tic perturbation V in the background G0 . By using (Al.6),

we obtain:

(Al.7)

Substituting (A1.7), into (Al.6a), we obtain

A A -

T v CrTo v I CT v)(Al.8)

or

A A

(A1.9)

- 164 -

Note that the inverse operator can be expanded in a Neumann

series:

=" 6 + oV + Ci. V ....(AI.10)

but it displays secular divergence that limits its range of

applicability.

By using (A1.6), (AI.8) and (AI.9), we obtain ensemble

average relations as follows:

CT 4- o + Cro V CT * To CoCTo (A1.11)

where

>

A

T < T > (A1.12)

vr - T ro

where Vc is a smoothed scattering operator in a background G.

On using (A1.11), (AI.12), one obtains:

! l

1 )65-

T Vc Cr CT* ( I- r. V)'C, 0 :T

v~(:~c~i0Vc)(Al. 13)

or

V (I +CqVV (Al. 14)

VC +VC C, VC

Multiply (A1.11) by LO and make use of (Al.5), we obtain

L oCTVc.CT ( Lo - Vr) C

= c~C~oVc)(Al. 15)

Equation (A1.15) provides a nonlinear defining equation for G,

the nonlinearity arising from the nonlinear dependence of Vc

on G. Using (A1.6) and (A1.11), we obtain

A

G~ T CQ a 4TT (A1.16)

whereA

i~TT

Wiwi

-166-

From (A1.9), we obtain

(A1.17)

Using (Al.11) and (A1.12), we have

CT a C I + eVC)- CT+± CT VC) G (Al.18)

Therefore, we can express (A1.16) as follows

S(I .I9 -

(A1.19)

A

(Al. 20)

where Tc, the multiple scattering in the smooth background G,

is defined by

rc== (.+Vc&C) T t -C Vr1

(AI.21)

-167-

On averaging (A1.4), we obtain

<(Lo-V) CT >-= A (Al.22)

Comparing with (A1.15), one infers

=A

(Al. 23)

* From (A1.20), we have

=A<c V CT t> V T-eC >

(A1.24)

We would like to observe the physical meaning o2 (A1.20) as

follows from (A1.4), we can write G in the renormalization

from as

A

(L-V -C) A =(A1.25)

Therefore, the scattering is now caused by a stochastic per-A

turbation V, = V - Vc of the coherent renormalized back-

-168-

ground operator (Lo Vc ).Multiplying (A1.25) by G, one

obtains

A A A

(A1.26) I

where we have defined

A AVCT~ L (A1.27)

On using (Al.26), one obtains

A ^ - I

TC V C (

- , Tt( ,C (A1.28)

or

Tc V(ICT V ,

From (A1.24), one finds that

A

Vc 'V, GT C > (A1.29)

By using (A1.27), one obtains

-169-

-A A A A

TC V, +V, aV,

A A A

V, + V, V, +~ (Al. 30)

4 (Al.31)

S inceA

V, =VVC

Therefore

v Cr V CV> +(Al.32)

+ (Al.33)

on expanding (Al.21) and making use of (Al.20) in (A1.17), we

obtain

~~~~ ~ C~~~Vc (Al.34)

Elm,.

-170-

- < Tr ,v>(A.35)V

Substituting (A1.24) in (A1.34) and making use of (A1.35), we

obtain

(Al.36)

vc V CT V -< T V

(A1.37)

- 171 -

APPENDIX 2

In order to find a rapidly convergent series for VC G,

one considers the following possible expansions;

(1) Expansion as

S < > +..........

as described in Appendix 1.

(2) Expansion b,

since

VC < ci e. >(A2.2)

and

T,= + Vc C)- T C1 + CT cv)-(A2.3)

one obtains

Vc -Ci v )-' > C T vC) -

(A2.4)

where

.. . .

-172-

T + Vt~V<V V>+v Tc. V

< SV ~CT CTV. > (A2. 5) I.

One multiplies eq. (A2.5) by VG and takes the ensemble average

of this equation. Using equation (A2.3), one then obtains4

(assume >~1 . >= 0)

(1+- GrVc)'l

From equations (A2.5) and (A2.3), one observes that

Therefore, one finally obtains the expansion

V-(1I V rV TV& > (2 + CTVc)

(3) Expansion C

From equation (A2.3) and (A2.5), one obtains

-1 c

-~ i.V~cC ~(Vi-V>) +-<VC)> (A2.9)

ifi .:&A4- .LA

-173-

Therefore, from equation (A2.2) and (A2.9), one hast

One rewrites equation (A2.10) as

Vc ~vGr Tec> C I+ Vc C) 170v^-,t < 1701& 7>}

wher -r V> (A2 11)J

where 0

Th trm< VV'',V- > q.V~ .2(a ewitn a 2.1

<~ ~ ~ ~~. c...-' > I t

-174--

In order to reduce the complexity, we shall examine

the convergence of VCG for the equation of the form

AA

A. (A2.15)

One also assumes that is a pure stochastic variable indep.

of r and z. The defining equation for the average Green's

function is

A. > (A2.16a)

or

A (A2.16b)

Taking the Fourier transform of (A2.16b), one obtains the

algebraic equation,

-= I (A2.17)

or

(A2.18)

+ .4

- -... J

- 175 -

In order to evaluate g, one requires information about VC .

The most appropriate expansion for VC g will be determined via

numerical investigation.

(1) For expansion a, we define the following approximations:

1st approx.t V = < t >

2nd approx.t = .± < o2

3rd approx.: is 2n-1 apwprox . + O-

4th approx. is pridw es O

For the case that ng---Vis uniformly distributed between -a

and a, one finds that V. g tends to converge as shown by the

solid line of Fig. 1. Note that -

1st approximation is valid when 4 - o .2

2nd approximation is valid when <V5 > 0.

3rd approximation is valid when V2 s> O .8

(2) We define the following approximations for expansion b:

1st approx, V4Vr

2nd approx. Vc. VIs) afrw j ?

3rd approx, Vc.'VK- + +adap'i~f Ivr~~~'

4th approx. Va 3dl APPr63e.J + <

- 176 -

as shown by the dotted curves of Fig. 1, one observes that

for the case of a uniform distribution, expansion b tends to

converge; however, not as fast as expansion a converges.

(3) Approximations for expansion C are,

1st approx. Vog = (i-t Vc ) 'O<T >

2nd approx. V' C --

3rd approx. % I>

4th approx.

For the case of a uniform distribution, this expansion shows

fast convergence. The 3rd approximation falls right between

the 3rd and 4th approx. of expansion a, wherein is locatedthe exact solution (Fig.2).

-z

(+Vv ) >

<IV (> <_V1.

a-2

1 (14)~) ± +- 4)

1.177

F 4 U%

1.00

0.0 S.000-01 1.00 1.Se 2.00

Fig. A2.l1. V .cgv 2V> ;-.Solid line, expansion a;dotted line: expansion b.

I.S4 (4)

(1)

.0 S. "-01 1.0 1.50 2. Oe

g2,c.Q2

Fig. A2.2. V~g vs. 924,7,>; expansion cs(1) 1st approx.; (2) 2nd approx.; (3) 3rd approx.;(4) 4th approx.;

AD-AU92 720 POLYTECHNIC INST OF NEW YORK FARMINGDALE MICROWAVE R--ETC F/S 4/1IONOSPHERIC SCINTILLATION' (U)SEP 80 D M WU N0001B-76-C-0176

UNCLASSIFIED POLY-MRI-141080 O

I ENDfllfflllff"Dff 181

i1111 .0 ,~2.0

11111-25 4 1111.

M ICROCOPY R LSOLU II(N S CHARI

NAIINAL IINI AU OIANDAHR, l b A

-178-

If n obeys a Gaussian statistics with a distribution func-

tion of the form: -cm) Ae

one finds that all expansions described above lead to poor

convergence. As an example, we calculate the 1st, 2nd and

3rd approximations for expansion c and depict it in (Fig.3).

Let us observe the exact solution of G for equation (A2.15),

when n is Gaussian, we have

2.

As a power series expansion, one writes

If the series is truncated after a finite number of terms,

we have

The "secular" behavior of G in this case explains somewhat

why "momen' expansions for Gaussian statistics are not

suitable.

Consider a Gaussion statistics truncated at tial namely:

(n) A e- ( 0') 7The results of numerical calculations of Vc are shown in

Fig. 4 - Fig. 6. One observes that the convergence of the

series expansion is dependent on the ratio C/<ii>

where <7t > denotes the variance of '. For Fig. 4 to Fig. 6,

we use a1%;2' =4.59796, 3.62336, 3.05054 respectively, and

find that a smaller ratio gives a better convergence.

_ _ _ _ _ _ _ I- ~- .*

-179-

3.50

(3)

a. S

VC9

* (2)

R. 83w-07*0.0 7.SO-01 l.Se 2.85s 3."*

Fig. A2.3. Vcg *vs. g3~ n is Gaussian.Expansion ct (1) 1st approx.;(2) 2nd approx.;/ (3) 3rd approx.;

2.0 800 30 4.00

g 1Jv3Fig. A2.4 Vcg vs. gz-L%>; n-obeys a Gaussian statisticstruncated at ±jai all /.4 I > 4.598. (expans~on c)

-180-

*(3)

VCg (2)

1.3

10, l8Cga 00C 3.3 4.0

Fig. A2.5. same as Fig. A2.4 except a2'/ 4 ;>, 3.623.

a. 46

1.35 (3)

V~g (2)

1.3

-310-CS0.0 1.", a . 3 3.0.

g 4v.)Fig. A2.6. same as Fig. A2.4. except a-a n 3.05.

- 181 -

APPENDIX

The limit of application of the Markov approximation

Let us consider corrections to the Markov approximation

when a finite radius of correlation in the propagating direct-

tion is taken into account. Let 9(3,?) be a sharp peaked

function of the argument 3-j' and < (5)i("''> =V.

The parabolic equation for the random field LA.(L,) is

written as:

LA (Sf 2 A%

e " - (A3.1)

Taking the Fourier transform of eq. (A3.1) with respect to

and averaging, one obtains

0%0- + < A.(3ff> -. (A3.2)

Applying the renormalized series expansion described in

Appendix 1, we have

If the correlation length in the z direction is very small

such that the 1st term in the series expansion of eq. (A2.1)

dominates, we can approximate (A3.3) as followings

-182-

V. 4, Js B-.,o2' Z€,.) c ,,"="- 4.-,, -.' 4--,.

One rewrites (A3.2) as

2& ~ A A0

(3,?) L~(,A ) e q? ? 0J'd5~ (A3.5)CT , <- 1)>

assuming the field .(5,? is statistically homogeneous,

so that its two dimensional spectral density satisfies therelations

Applying (A3.6) one rewrites (A3.5) after some manipulations,

as followss

(A3.7)

Since the function F(z-z ),kt, is a peaked function of theargument z-z', one infers that the main contribution to the

integral over z' lies in the neighborhood of z z'. We

shall expand the functions G and <U in powers of z-z', viz:

AI

- 183 -

(3-3,)2

<OC~~',~~)>9 <4(, ,3 ) - 3,__a < UA

+ 2 'i+--..

Substituting (A3.8) into (A3.7), one obtains after some te-

dious calculations

In the coordinate representation, this equation takes the

form

AL-() <~,1 A'-. 1 ' .>

flo z A M A2. (A3. 10)

wee..... < ,!)>

where

..............................,--~ -. .. -

- 184 -

do (A3. 11)

if~ Ae f.(31P)I 1 Np44t

(A3.12)

To use the Markov approximation, the terms on the right side

in eq. (A3.10) have to be small compared to the corresponding

terms on the left in this equation. Thus, one can write

thes conditions as follows:

(1) 4 A2, 0>

(2) A.2.2 o < < I(A3.13a)

A similar analysis can be performed for the higher moment

case.

lll7

i - 185 -

APPENDIX 4

In Appendix 2, we examined several expansions for VcG

and used numerical methods to test their convergence. We

assumed that the imaginary part of G was very small compared

to its real part, i.e. G is approximately real. This is not

generally true. Hence all of the results in Appendix 2 only

apply to ranges wherein G is real.

Behavior of G w.r.t. k3(=B+ie )t

We are dealing with the simple case defined by the equa-

tion:

A=

(A4.1)

where V is a random number. Taking the Fourier transform of

(A4.1), one obtains the algebraic equations

A

(A4.2)

Df" v (A4.3)

ThereforeA A

Real part of ( - i A4.4a)

-~ 4

-186-

1mg. part of 94 (A4.4b)

If the probability distribution function F(v) of V is given,

ve can evaluate the average Green's function exactly:

and (A4.5)

A 0.0 A

<~~~ ~~ (-v> 9 cA(lr Fsr c

on the other hand, we can expand Vcj*in the following way:

+i 6~.> 74V 9-> ,7 (A4.6)

The first approximation for 41k>is defined by the follow-

ing equation:

(& < *~ &> & (A4.7)

nIi

- 187-

The 2nd approximation is defined by:

(A4.8)

etc.

We shall examine the approximate solutions of (A4.7) and

(A4.8) and compare these solutions with the exact solution

(A4.5) by numerical methods.

(a) For the case that V is uniformly distributed over a rangefrom -a to +a, one obtains for the 2nd moment of V

3and for the 4th moment of V a

V >vSubstituting these moments into eq. (A4.7) and (A4.8) and

using contour integral techniques, we evaluate I k as a

function of B and depicted in Fig.(A4.1) to Fig.(A4.6).

Explanations of Figures:

Fig A4.1:

variance = 0.333

dotted line = 1st approx. of I kr

solid line 2nd approx. of 9 kr

Fic. A4.2:

The exact solution of 9 kr for V)= 0.333

Fig. A4.3 i

variance = 2.083dotted line = 1st approx of I kr

solid line = 2nd approxof 0 kr

Fig. A4.4s

The exact solution of icr for 4V> = 2.083

- 188-

over a range (-za, +a ); i =~ 0.333; 1st approx.dotted line; 2nd appr~qx. solid line.

gkr

a _S 0 B 00 S 0 10

Fi.A..Teeatslto-fgc v>=033

-189-

gkr 034

R Q .-' -5 O4. e

4.Q.-1B

-e.0 4.4 TheO exc1ouio0f4rfr V> 203

* B

- 190 -

. , ' . . .. ..

v'r- 8.33312

4 0e0-01 _

- . S 0B

Fig. A4.5. g vs, B; v is uniformly distributedover a ranget-a,'+a); 4v"> = 8.333; 1st approx.dotted line; 2nd approx. solid line.-8 3MI2

-10r 33331

S~ .)aO

-4. ' -'1 -

-4 ee- . .... . .. 1

B I

Fig. A4.6. The exact solution of gKr for <v = 8.333.

-191-

Fig. A4.51

variance = 8.333

dotted line = 1st approx.of kr

solid line - 2nd approx. of kr

Fig. A4.6s

The exact solution of I kr for LV>= 8.333

Conclusions

For a r.v.v with a uniform distribution the approxima-

tion procedure displays quite good convergence.

(b) For the case that 7 obeys a Gaussian statistics, one ob-

tains for the even moments of V:

ZV )= 1 3 5 ... (2n-1) Zv 2>

Substituting these moments into (A4.7), (A4.8) and apply-

ing the same technique as in (a), we can evaluate k as

a function of B. The results are shown in Fig. (A4.7).

Explanation of Fig. (A4.7)s

variance = 1

dotted line = 1st approx of 9 kr

crossed line (x) = 2nd approx of I kr

solid line = 3rd approx of I kr

thin line = exact solution of kr

Conclusions

We observe that these approx. procedures do not show

convergence. However, we shall consider some other obser-

vation before making any definite conclusions.

Since Vc can be evaluated exactly from the following inte-

grals

(A4.9)

*AT." .. .A . ....

192

We shall compare the real and img. parts of VcG for the 1st,

2nd and 3rd approximations, respectively, with the exact re-

sult.

For the case that V obeys a Gaussian statistics, we display

a few plots of these results: ( write gk as G

Fic.A4.8s (exact result)

variance = 1

curve(l) = real part of VcG

curve(2) = img part of VcG

curve(3) = absolute value of VcG

Fig. A4.9s

variance = 1

curve (1) = real part of VcG

curve (2) = img part of VcG

curve (3) absolute value of VcG

Fir. A4.10variance 1


curve(2) = img part of VcG

curve (3) = absolute value of VcG

Fig. A4.11,

For variance = 0.25



curve(3) - absolute value of VcG

Fig. A4.12:

For47. & 0.25

curve ( 1) = real part of VcG

curve (2) = img. part of VcG


Fig. A4.13t

for -V> = 0.25

curve (1) 2 real part of VcG

curve (2) = img. part of VcG

J

- 193 -


The above observations show that one can not get a good

approx. for VcG by taking a finite number of terms in eq.

(A4.6) for a Gaussian statistics. We conclude that the

"moment" expansion for VcG is not suitable for a Gaussian

statistics.

For the case that V is uniformly distributed over the range

from -0.5 to +0.5, we plot VcG both exactly and for several

approximations, viz,

Fig. A4.14: (exact result)

variance = 8.333 x 10-2

curve (1) real part of VcG

curve (2) img part of VcG


Fic.A4.15i (1st approx.)

variance = 8.333 x 10- 2



curve (3) absolute value of VcG

Fig. A4.16:, (2nd approx)

variance 8.333 x 10 -2

curve (1) - real part of VcG

curve (2) - img part of VcG

curve (3) f absolute value of VcG

We observe that eq. (A4.6) provides a rapidly convergent

series expansion for VcG when V obeys a uniform statistics.

-194-

gir 3.0 2

Fig. A4.7. gc vs. B? v is Gaussian; 4V) 1.(1) ... 1st approx . (2) xooc 2nd approx.;(3) _ 3rd approx.; (4) thin lines exact solution.

Vcg

* -3.c3 -1.50 3 . ie. . .00B

Fig. A4.a. v is Gaussian; -(v> 11 (1) real partof Vcgj (2) Img. part of Vcg: (3) absolute value

of V4 g.

-195-

B.&.o 3e

Fig. A4.9. 1st approx. of"Fig. A4.8-.

0.0

Fig. A4. 10. 2nd' approx. of Fig. A4.8.j

-196-

Vcg

BFig. A4.11. v is Gaussian; ;(1) real partof Vcg; (2) 1mg. part of Vcg; (3) absolute valueof Vcg. <V"',> 0.25.

war- 8.500-1

V'cg

B 1..0 3.

Fig. A4.12. 1st approx. of Fig. A4.11.

- 197 -

I

I I * -3 I* 'a -I *1

P12 *.~ S

I-'.... q ULQ

* * I* lb

LAJ

r'JI

'A

/9

0

I-.

I-'

I

-198-

a~e .3333.-qa

1.63

-6606dr46 Ol" )0

Fig. A4.14. V g vs. S; vis uniformly distributedovera rage -0.5, +0.5); (1) real part of Vcg;

(2) 1mg. part of Vcg; (3) absolute value of Vcg.

S 3.333w-e-

1.68

Vcg

Fig. A4.15. 1st apprgx. of Pig. A4.14.

4 0l

0 0 A

-200-

APPENDIX 5

Limits of Application of The Parabolic Equation

The parabolic equation differs from the complete scalar wave

equation by a term a u/z . If one treats e u/ as a

small perturbation term and writes

AA

rft a4 3 - (A5.1)

and supposes that u2 is of the same order of smallness as

Substituting (A5.2) into (A5.1) and equating groups of

terms of the same order, one obtains

A A

2 t U2CA]i,.,o-.' .Pv.,,+ + " - (A5.3) ' i

't7

-201

with the boundary conditions

AA

LAI, ( 0, = t OL tA 0

one finds on taking the ensemble average of (A5.3)

t Y < LA> t fe, 4 EL =

(A5,4a)

2 Au, > (A5.4b)

Using the renormalization procedure described in Appendix 1

one writes equation (A5.4a) as

on using of the approximation

V~U (A5. 5)

LA (3s?') ~ 'lyand the assumption that

(A5.6)

dv-

-202-

One obtains, after some manipulation,

A (o)- .- -(-3,9)

(A5.7)

8

Therefore, we can rewrite eq. (A5.4a) as follows:

z fe0 z + LU, + Ao) (A5.8)

Similarly, eq. (A5.4b) can be rewritten as

21IA2 IV 2. IA2+A()A

+C) - - (A5.9)

Let us consider the case of a plane wave incident upon a ran-

dom medium such that

L40 (?) Crvs tont U t

One obtains, from equation (A5.4a), that

A

U, c ) =9e (AS.10)

=LLoe -0L

where A (o)

4

-203-

equation (A5.4b) then takes form

2 z 23 3

The solution of eq. (A5.11) corresponding to the boundary con-dition u,(0) = 0 can be calculated as follows:

set

z=e (A5.12)

One obtains from eq. (A5.5)

L2 L2'(0 + I2Az (A5.13)

Therefore

S~)2?Aoe-

= .Z*.Az ) ( A5.1.5)

In this case

A ()= LL, L)+ LA2 ) +

-01 . J (A5.16)

I!/sV

-204-

The field u,should be considered only in the region

where it is marking different from zero, i.e. for OCJS 1, thus,

we obtain the condition for a small correction term,2

i.e.

0 V c4 « 1(A5.17)

If condition (A5.17) is valid then the correction term in eq.

(A5.16) will be small compared to its first term. For the

higher moment case, the calculation is quite complicate. For

example, in the 2nd moment case, one considers the corrections

to M2 ( z, Z, * ) = ( z, ) *( z, u If u = u + U.+

one has

< , ( k) (. (3, f.L)> +AA

M ') t1 (3j• ().>s(8) (2) *(A5.18)

-M2 + rV " + " M 12) +...N2 + 2 2

Applying the Markov approx. and some very tedious calculations,

one can obtain equation for M 2 , M2 , .. as follows:

22 =,0

" + cA ( I'';M t -

-

- 205-

2 rA (o) -A

~i- --~v4 - ~A.~o) ~ .&~ 2(ID)j

(AS. 20)

4I 4f 1 (A5. 21)

If the parabolic approximation is used, the correction terms

M 2) and ML) * have to be small compared to the Ist term Ma.2 (a) (2)4

One obtains from eq. (A5.20) and (A5.21), let + M 2 )wen

the following equation for M2 I

2. 14 (?+2

- - 72 7f-1 v-2.

Let us now consider the case of a plane incident wave, one

obtains

J'-' .. . '. .[ . .. , - ."

- 206 -

(A5. 23)

In this case

M ( , J(A5.24)

Moreover

2!

-0

2. (A5.25)

VZ a,. ~) ~ = 2 l2~rMl

Equation (A5.22) thus takes the form

+ A* H 1 (2)(A5.26)

f (o

from (A5.23), one finds

I-i

- 207-

4?

2 23

2 (A5.27)

Therefore

I M ( f)= '-I e ' 49- - -

_- 2321r Y (A!5. 28)

this leads to the conditions

then

22

- 208 -

Appendix 6

The equations to be solved are:

M_. _ 3-Mz - or it(A6.1)

One writes

(A6.3)

and

a r2()j I) a (A6.4)

0' *i

-209-

The difference equations, to the 2nd order accurracy in z,area

Ma (31'A3) M. r 1 r + M,

4,AY A Y:~

and

tA12 3) + rvl Ci r(l~r~IzM (rf 4) N

A,(A.6

erf A3 )M"A1

wheredr.1 forward difference operator w.r.t. r.

d =backward difference operator wortt. rer;£dr.= center difference operator wor~t. r;,

The boundary conditions for M4 are described in equations(4.46) and (4.47).

lll~jjl iliiiiiiiiliiiiil~l~il~illllililI IIIIIIIIIIII IIIII~ i,

- 210-

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SEP 8 WU N0001-76-C-0176 POLY-MRI-1410-80 NL ... · DvdM/Wu '52 11. PROaA LMENT. PROJECT. TASK...

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