PERFORMANCE ANALYSIS FOR, AND INTERPRETATION...

PERFORMANCE ANALYSIS FOR, AND

INTERPRETATION OF, DATA FROM MASB SONAR

IN THE APPLICATION OF SWATH BATHYMETRY

by

Geoff Mullins

B.A.Sc, Queen’s University, 2001

M.A.Sc, University of British Columbia, 2003

a Thesis submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

in the School

of

Engineering Science

c© Geoff Mullins 2010

SIMON FRASER UNIVERSITY

Summer 2010

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Last revision: Spring 09

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Abstract

Multi-Angle Swath Bathymetry (MASB) sonar produces high-quality, fully-registered bathymetry

and imagery. This thesis presents a theoretical analysis of MASB sonar performance and example

survey data, thereby providing a foundation for quantitative results, and demonstrating its potential

as an effective survey tool.

The quality of bathymetric surveys depends on the ability of a sonar to estimate the location of

the bottom, therefore it is necessary to identify mechanisms that limit bottom estimation accuracy.

Signal cross-correlations between elements on a linear array are determined for several transmitted

pulse shapes for a general survey geometry. Decorrelation caused by geometrical mechanisms of

footprint shift and baseline decorrelation are identified in the correlation functions. Utilizing signal

correlations, the square root of the Cramer-Rao Lower bound (CRLB) on angle of arrival variance

is identified as a better performance measure than the standard deviation. It is shown that for short

pulse lengths, the dominant factors influencing performance are the footprint shift effect, the signal

to noise ratio, and the presence of multiple signals on the array. A new measure of performance,

error arc length (EAL), is defined using the CRLB, and EALs are plotted for a range of survey

conditions including various frequencies, array tilt angles and number of array elements for both salt

and fresh water surveys.

To demonstrate the capabilities of MASB sonar, a prototype system was designed, constructed,

and deployed. Pavilion Lake was chosen as one site for experimentation, as it is the focus of an

international astrobiological effort to examine microbialites. Using MASB surveys, deep water mi-

crobialites were discovered possessing a morphology unlike others examined in prior research at the

lake. Due to difficulties measuring ground truth information and operating on a moving survey

platform, bottom estimation accuracy could not be directly compared with theoretical EALs for

measurements at Pavilion Lake. Measurements taken at Sasamat Lake are presented which demon-

strate that, in principle, the EAL can be used to predict sonar bottom estimation performance. In

conclusion, the theoretical foundation presented, and the survey imagery and co-located bathymetry

produced with the prototype sonar, demonstrate the effectiveness of MASB sonar for shallow water

survey applications.

iii

Acknowledgments

I would like to first thank my senior supervisor, Dr. John Bird, for the insight and encouragement

that he has provided me over the course of the Ph.D. program. His advice and continuous support

were instrumental in the success of this research. I have learned from him that it is necessary to criti-

cally examine theory, simulation and experimental data in order to build a cohesive understanding of

a research problem. In addition I would also like to thank my committee, in particular Dr. Bernard

Laval, who helped me to understand the framework into which the sonar technology developed for

this research could be applied.

Friendships and collaborations cultivated in the Underwater Research Laboratory at Simon Fraser

University over the course of this Ph.D. have been instrumental in conducting this research. As such

I extend my gratitude to Pavel Haintz, Ying Wang, Jinyun Ren, Sabir Asadov, Steve Pearce and

Julian Mosely for their support during my graduate studies. Conducting research at Pavilion Lake

was made possible through the assistance of Harry Bohm, Mickey Macri, Linda Macri, Darlene Lim

and Alex Forest. Funded for this research was provided by the Natural Sciences and Engineering

Research Council of Canada, and the Canadian Space Agency.

Finally, I would like to thank my family for giving me the love and support to finish this thesis. In

particular I would like to thank my dad for instilling in me a curiosity about the natural world, and

having many late night talks about science. I would like to thank my mom for showing me that all

life’s troubles can be conquered using pure determination and will power. My sister Annie, though

far away, has always supported me during my studies and I would like to thank her. My son Hayden

has taught me how to learn again, and I have been inspired through watching him grow and learn.

Most of all I would like to thank my wife Melinda for her love and understanding over the past few

years, it was our shared love of the ocean that prompted me to begin this research. She has shown

me the meaning of patience, the importance of finishing what you start, and her encouragement was

what made this dissertation possible.

iv

Contents

Approval ii

Abstract iii

Acknowledgments iv

Contents v

List of Tables viii

List of Figures ix

Glossary xix

Dedication xxiii

Quotation xxiv

1 Introduction and Background 1

1.1 The Big Picture - Purpose and Objectives . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Goal 1 - Theoretical Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Purpose of Research and Summary of Accomplishments . . . . . . . . . . . . 3

1.2.2 Relevance of Prior Work and Accomplishments Explained . . . . . . . . . . 5

1.2.3 Direction of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7


1.3 Goal 2 - Demonstration of Prototype System . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Purpose of Research and Summary of Accomplishments . . . . . . . . . . . . 11


2 The Signal 16

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

v

2.2 Physical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Geometry of Simple Bathymetric Measurement . . . . . . . . . . . . . . . . . 20

2.3 Correlation and the Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Development of the Signal Correlation Integral

for a General Pulse Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.1 Pulse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.2 Measures of Comparison Between Pulse Functions . . . . . . . . . . . . . . . 38

2.5 Correlation - Specific Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.1 The Footprint Shift Effect for a Square Pulse . . . . . . . . . . . . . . . . . . 40

2.5.2 Development of the Full Signal Correlation for a Square Pulse . . . . . . . . . 43

2.5.3 Development of the Signal Correlation for a Matched Filtered Square Pulse . 51

2.5.4 Development of the Signal Correlation for a Finite Q Pulse . . . . . . . . . . 55

2.5.5 Development of the Signal Correlation for a Matched Filtered Finite Q Pulse 57

2.5.6 Development of the Signal Correlation for a Gaussian Pulse With Pulse Com-

pression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5.7 Uncorrelated Gaussian Noise Contribution . . . . . . . . . . . . . . . . . . . . 60

2.6 Summary of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3 Performance - Theory and Simulation 64

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Complex Multivariate Gaussian Signal . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3 The Fisher Information Matrix and The Cramer-Rao Lower Bound . . . . . . . . . . 71

3.4 The Use of CRLB Over Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . 78

3.4.1 Two Element Estimator PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4.2 Two Element AOA Estimator Variance . . . . . . . . . . . . . . . . . . . . . 81

3.4.3 Estimator Variance Compared to CRLB . . . . . . . . . . . . . . . . . . . . . 82

3.4.4 Difficulties with Estimating Standard Deviation . . . . . . . . . . . . . . . . . 85

3.5 Pre-estimation vs. Post-estimation- 2 Element Array Case Study . . . . . . . . . . . 88

3.5.1 Pre-Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.5.2 Post-Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.5.3 Comparison of Pre-Estimation and Post-Estimation . . . . . . . . . . . . . . 91

3.6 AOA Performance: Dependence on Waveform and Geometric Effects . . . . . . . . . 94

3.7 Angle of Arrival Estimation For Two Signals . . . . . . . . . . . . . . . . . . . . . . 107

3.7.1 Linear Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.7.2 Minimum Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.7.3 Minimum Variance Distortionless Response Beamforming . . . . . . . . . . . 124

3.8 Bottom Estimation Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

vi

3.8.1 Error Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.8.2 The Full Sonar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.8.3 Survey Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.9 Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4 Demonstration of a MASB Apparatus 144

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.2 System Design and Survey Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.3 Noise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.4 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.5 Survey of Pavilion Lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.6 Comparison of Performance with EAL . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.7 Summary of Chapter 4: An Alternative to Other Sonar Systems . . . . . . . . . . . 165

5 Summary of Conclusions, and Future Research 167

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.2 Future Directions for MASB Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Bibliography 172

vii

List of Tables

2.1 A summary of the centered pulse functions. The length of the SQ pulse is given by rsq,

which is the number of cycles times the wavelength of a single cycle. In the case of FQ

pulses, rsq is again used to represent the number of cycles that would be encountered

if q → 0, while c is the speed of sound in water. The MFSQ and MFFQ pulses are

matched filtered versions of the SQ and FQ pulses. Finally, the CG pulse, along with

both the variables a and rgc are defined in the accompanying text. A plot of these

pulse functions is given in Fig. 2.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 A summary of normalizations and relevant mean squared lengths. . . . . . . . . . . . 39

viii

List of Figures

1.1 3D sonar image showing orientation of sonar. . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Geometry of survey measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 The progression of pings used as slices to build a 3D map. Note that in this survey,

only 1 out of every 25 pings is displayed, so as to make the construction of a map

easier to interpret (ordinarily pings are packed much closer, so as to cover the full

bottom). The track of the sonar appears in magenta, and one image of the sonar is

displayed, so as to register the orientation. . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Geometry for sonar profiling measurement demonstrating what is considered to be

the region near nadir, and that which is considered to be non-nadir. . . . . . . . . . 23

2.4 Geometry for footprint. In this scenario, a pulse is transmitted from element 1, travels

outward, scatters off of the bottom and some of this scattered signal is backscattered

toward each of the array elements (in this case element i). Due solely to time of flight,

the signal received on different elements at the same time corresponds to slightly

different locations on the bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 The left figure demonstrates the geometry that leads to footprint shift between ele-

ments i and k. The right figure demonstrates the geometry where two elements see

the same part of the bottom at different time delays. Note, the footprint for each

element is not under consideration here, only the geometric conditions for the delta

function in Eq. 2.28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 An illustration of correlated and uncorrelated sections of the footprint as seen on two

separated elements, namely the footprint shift effect. . . . . . . . . . . . . . . . . . 26

2.7 Geometry demonstrating regions where 1 or 2 signals must be considered. . . . . . 28

2.8 Geometry for sloped bottom correction. . . . . . . . . . . . . . . . . . . . . . . . . . 29

ix

2.9 Waveforms considered in the course of this research. Shown are a SQ pulse of 20

cycles at 300 kHz (dashed black line), the match MFSQ with same parameters (black

line), a FQ pulse (red dashed line) with q of 10, and transmit length of 20 cycles

(time between start of transmission and beginning of exponential decrease), The cor-

responding MFFQ (red line), and finally a compressed gaussian pulse in green. A

sound speed of 1450 m/s was also assumed for all waveforms. . . . . . . . . . . . . . 39

2.10 The integration domain is the overlap of two footprints as seen by different array

elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.11 Autocorrelation (left) and cross-correlation d = λ2 (right) for a 300 kHz square pulse

of 20 cycles, tilt angle of 45◦ and altitude of 40m, as measured from 100 pings. In

left plot, the blue curve is for power received from primary signal alone, whereas the

black curve represents the contributions of signals from both in front of (primary), and

behind (secondary) the sonar. The dashed black line represents the simulation, and

shows good agreement with theory. The right plot shows primary contributions, blue

and green curves, to the real and and imaginary components of the cross correlation

respectively. The black and red curves are the real and imaginary components of the

combined primary and secondary signal correlations, and well represent the black and

red dashed lines which are the corresponding simulated data. . . . . . . . . . . . . . 50

2.12 The integration domain is again the overlap of two footprints as seen by different

array elements. However, the waveform must now be considered as piecewise over

three separate integration domains as illustrated here. . . . . . . . . . . . . . . . . . 51

2.13 Autocorrelation and Crosscorrelation for match filtered square pulse. The same pa-

rameters were employed as for Fig. 2.11. In left plot, the blue curve is for power

received from primary signal alone, whereas the black curve represents the contribu-

tions of signals from both in front of (primary), and behind (secondary) the sonar. The

dashed black line represents the simulation, and shows good agreement with theory.

The right plot shows primary contributions, blue and green curves, to the real and

and imaginary components of the cross correlation respectively. The black and red

curves are the real and imaginary components of the combined primary and secondary

signal correlations, and well represent the black and red dashed lines which are the

corresponding simulated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

x

2.14 Autocorrelation and Crosscorrelation for a FQ pulse. The same parameters were em-

ployed as for Fig. 2.11. In left plot, the blue curve is for power received from primary

signal alone, whereas the black curve represents the contributions of signals from both

in front of (primary), and behind (secondary) the sonar. The dashed black line rep-

resents the simulation, and shows good agreement with theory. The right plot shows

primary contributions, blue and green curves, to the real and and imaginary compo-

nents of the cross correlation respectively. The black and red curves are the real and

imaginary components of the combined primary and secondary signal correlations, and

well represent the black and red dashed lines which are the corresponding simulated

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.15 Autocorrelation and Crosscorrelation for a MFFQ pulse. The same parameters were

employed as for Fig. 2.11. In left plot, the blue curve is for power received from

primary signal alone, whereas the black curve represents the contributions of signals

from both in front of (primary), and behind (secondary) the sonar. The dashed black

line represents the simulation, and shows good agreement with theory. The right plot

shows primary contributions, blue and green curves, to the real and and imaginary

components of the cross correlation respectively. The black and red curves are the

real and imaginary components of the combined primary and secondary signal corre-

lations, and well represent the black and red dashed lines which are the corresponding

simulated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.16 Autocorrelation and Crosscorrelation for a CG pulse. The same parameters were em-

ployed as for Fig. 2.11. In the left plot, the blue curve is for power received from

primary signal alone, whereas the black curve represents the contributions of signals

from both in front of (primary), and behind (secondary) the sonar. The dashed black

line represents the simulation, and shows good agreement with theory. The right plot

shows primary contributions, blue and green curves, to the real and and imaginary

components of the cross correlation respectively. The black and red curves are the

real and imaginary components of the combined primary and secondary signal corre-

lations, and well represent the black and red dashed lines which are the corresponding

simulated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1 The return from uncorrelated footprints may be considered as separate snapshots of

the same ensemble if the geometry of measurements does not vary appreciably over

the region of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

xi

3.2 For a five element array, (all signal-to-noise ratios set at 10dB) the respective√

CRLBs

for the single signal estimation (red curve), two signals with unknown AOAs and

magnitudes (black curve), the same two signals with known signal to noise ratio (blue

curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3 Following Fig. 3.2, the√

CRLB (all signal-to-noise ratios set at 10dB) for all three

scenarios: 1 signal, 2 signals, 2 signals with known magnitude. From bottom to top

groupings are representative of 6,5,4 and 3 element arrays. . . . . . . . . . . . . . . 77

3.4 The probability that an estimate will be less than β multiples of the standard deviation

away from the true value is demonstrated to be dependent on snr. Thick solid curves

are for the angle probability density and signal-to-noise ratios 60, 50, 40, 30, and 20

dB, top to bottom. For reference, the overlaying dashed lines show what a similar

calculation for a Gaussian probability density with the same snr, and therefore same

standard deviation, would yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5 The probability that an estimate will be less than β multiples of the√

CRLB away

from the true value is demonstrated to be independent of snr. Thick solid curves

are for the angle probability density for the signal-to-noise ratios 60, 50, 40, 30, and

20 dB. (They lie on top of one another.) For reference the dashed lines are for a

Gaussian probability density with the same signal-to-noise ratios (hence same standard

deviation), top to bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.6 Convergence to the theoretical standard deviation for increasing numbers of simulated

snapshots, Ns. The ratio of the standard deviation of the electrical angle to√

CRLBα

is given as a function of the signal-to-noise ratio (thick solid black line). Dashed red

line is the ratio of the sample standard deviation to√

CRLBα for 1000 trials, dotted

cyan line, 10 000 trials, and dashed dotted blue line 100 000 trials, and solid magenta

line, 1 000 000 trials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.7 Geometry for pre-estimation technique, showing probability distribution of rs as cir-

cularly symmetric gaussian around rs. Note σ2z is used to illustrate projected variance

of a variable tangent to an arc centered at radius rs. . . . . . . . . . . . . . . . . . 89

3.8 Pre and post estimation results for both theory and simulation as a function of the

number of snapshots for SNR = 30, 20, 15 dB from top to bottom. The blue lines

represent pre-estimation and converge to one for an increasing number of snapshots,

such that the variance and CRLB become equal. The red lines are for pre-estimation

techniques and demonstrate that the ratio of variance to CRLB stays at the single

snapshot level even for increasing numbers of snapshots. The green and red asterisks

represent simulated results for post and pre-estimation respectively. . . . . . . . . . 93

xii

3.9 The SNR shown for both primary (x > 0) and secondary (x < 0) incoming signals,

in this case the level is set to 40dB at the maximum range for the x > 0 signal (in the

case of the square pulse, a 300 kHz pulse of 20 cycles was chosen). . . . . . . . . . . 95

3.10 Examining performance of AOA estimation for a SQ pulse. The SNR has been set to

40dB at the maximum positive range, and performance is given for 5 snapshots of a

20 cycles pulse recorded on a 3 element array. Green and red dashed lines are bounds

as determined by snr of the x < 0 an x > 0 signals alone. Black and and cyan dotted

lines are the bounds as determined by snr and footprint shift of the x < 0 an x > 0

taking into account the full 2 signal model, whereas the magenta dotted line is for the

bound as determined by snr and footprint shift of the x > 0 only. Green and blue

solid lines are for the full bound for x < 0 an x > 0 signals (again 2 signal model), and

the red solid line is for only the x > 0 signal. Note the dotted lines partially obscure

the solid lines, leading to the conclusion that baseline decorrelation plays little role in

the performance of AOA estimation for this survey scenario. . . . . . . . . . . . . . 96

3.11√

CRLBα for same resolution in range Fig. 3.10, only now it is a single snaphot of

a 100 cycle SQ pulse. As in Fig. 3.10, green and red dashed lines are bounds as

determined by snr of the x < 0 an x > 0 signals alone. Black and and cyan dotted

lines are the bounds as determined by snr and footprint shift of the x < 0 an x > 0

taking into account the full 2 signal model, whereas the magenta dotted line is for the

bound as determined by snr and footprint shift of the x > 0 only. Green and blue

solid lines are for the full bound for x < 0 an x > 0 signals (again 2 signal model), and

the red solid line is for only the x > 0 signal. Note the dotted lines are now visibly

separated from the solid lines, leading to the conclusion that baseline decorrelation is

present in the performance of AOA estimation for this survey scenario. . . . . . . . 97

3.12√

CRLBα as calculated for the MFSQ pulse, with far range SNR = 40dB, 3 elements,

and 5 snapshots. The solid green and blue curves are the full bound as calculated

including the effects of both signals, and the red curve is the bound using only the

x > 0 signal. The black asterisks are the approximation from Eq. 3.52. Note that the

black asterisks are obscuring the red curve almost completely for ranges greater than

broadside (i.e. x > 40m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.13 The√

CRLBα as plotted for the same scenario as in Fig. 3.12, however the num-

ber of array elements has been increased to six. Note a moderate improvement in

performance against the effects of noise, however no improvement against the effects

of footprint shift in the x > 0 signal. As in Fig. 3.12, the black asterisks again

are obscuring the red curve almost completely for ranges greater than broadside (i.e.

x > 40m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

xiii

3.14 The√

CRLBα as plotted for the FQ pulse, with the same survey geometry as used

in Fig. 3.12. The dotted green and red lines are the bounds calculated using only the

snr of the x < 0 and x > 0 signals alone. The solid green and blue curves are the

full bound as calculated including the effects of both signals, and the red curve is the

bound using only the x > 0 signal. Note a slight decrease in performance as compared

to the similar results calculated for the MFSQ pulse. . . . . . . . . . . . . . . . . . 102

3.15 The√

CRLBα as plotted for the MFFQ pulse, with the same survey geometry as used

in Fig. 3.12. The dotted green and red lines are the bounds calculated using only the

snr of the x < 0 and x > 0 signals alone. The solid green and blue curves are the

full bound as calculated including the effects of both signals, and the red curve is the

bound using only the x > 0 signal. Note a slight destabilization of calculation at far

range for x > 0, this is an artifact of performing numerical derivation on the long form

of the correlation function for the MFFQ pulse. . . . . . . . . . . . . . . . . . . . . 103

3.16√

CRLBα as calculated for the CG pulse, with far range SNR = 40dB, 3 elements,

and 5 snapshots. The solid green and blue curves are the full bound as calculated

including the effects of both signals, and the red curve is the bound using only the

x > 0 signal. The black asterisks are the approximation from Eq. 3.57. . . . . . . . 105

3.17 Two of the various angle of arrival estimation schemes. Though both schemes rely

on a calculation of weight vector ~w = [w1, w2, . . . , wn]T to minimize the squared er-

ror Jerror = e2 The linear prediction scheme operates on the sum of N − 1 elements

to effectively cancel out the remaining element. Alternatively, the minimum eigen-

value technique minimizes the total noise by looking for the lowest eigenvalue of the

covariance matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.18 A comparison of snr for both the theoretical value (in blue) and simulated signal (in

red) for the survey geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.19 The electrical phase of the simulated signal with noise (black dots), estimated from

one degree of freedom. The solid red curve is the expected electrical AOA. . . . . . 114

3.20 The electrical phase of the simulated signal with noise (black dots), estimated from

two degrees of freedom. The solid red curve is the expected electrical AOA. . . . . . 115

3.21 The error in electrical phase linear prediction estimates (black dots) for the SQ pulse (5

snapshots from multiple pings) as determined by taking the difference of the theoretical

AOA, and the estimated AOA from the simulated data (using one or two degrees of

freedom where appropriate). Also shown is 2√

CRLBα in green and −2√

CRLBα in

red for each of the rance cells in the simulation. . . . . . . . . . . . . . . . . . . . . 116

xiv

3.22 The error in electrical phase linear prediction estimates (black dots) for the MFSQ

pulse (5 snapshots from multiple pings) as determined by taking the difference of

the theoretical AOA, and the estimated AOA from the simulated data (using one or

two degrees of freedom where appropriate). Also shown is 2√

CRLBα in green and

−2√

CRLBα in red for each of the range cells in the simulation. . . . . . . . . . . . 117

3.23 The error in electrical phase linear prediction estimates (black dots) for the SQ pulse as

determined by taking the difference of the theoretical AOA, and the estimated AOA

from the simulated data (using one or two degrees of freedom where appropriate).

The 5 snapshots used for each estimate are taken from within a single ping, therefore

fewer estimates are shown than in Fig. 3.21. Also shown is 2√

CRLBα in green and

−2√


3.24 The electrical phase of the simulated signal with noise (black dots), estimated using

the minimum eigenvalue estimator with two degrees of freedom. The solid red curve

is the expected electrical AOA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.25 The error in electrical phase minimum eigenvalue estimates (black dots) for the SQ

pulse (5 snapshots from multiple pings) as determined by taking the difference of

the theoretical AOA, and the estimated AOA from the simulated data (using one or


CRLBα in green and

−2√


3.26 The error in electrical phase minimum eigenvalue estimates (black dots) for the MFSQ

pulse (5 snapshots from multiple pings) as determined by taking the difference of the

theoretical AOA, and the estimated AOA from the simulated data (using one or


CRLBα in green and

−2√


3.27 The electrical phase of the simulated signal with noise (black dots), estimated using

the MVDR technique. The solid red curve is the expected electrical AOA. . . . . . 126

3.28 The error in electrical phase MVDR estimates (black dots) for the SQ pulse (5 snap-

shots from multiple pings) as determined by taking the difference of the theoretical

AOA, and the estimated AOA from the simulated data. Also shown is 2√

CRLBα in

green and −2√

CRLBα in red for each of the range cells in the simulation. . . . . . 127

3.29 The error in electrical phase MVDR estimates (black dots) for the MFSQ pulse (5

snapshots from multiple pings) as determined by taking the difference of the theoretical

AOA, and the estimated AOA from the simulated data. Also shown is 2√

CRLBα in

green and −2√

CRLBα in red for each of the range cells in the simulation. . . . . . 128

3.30 The snr for a survey geometry using the sonar equation from Eq. 3.84. In this case a

300kHz MFSQ pulse MASB sonar with a 3 element array is tilted at 45◦ in salt water.

The red curve is for the x < 0 signal, and the blue curve is for the x > 0 signal. . . 134

xv

3.31 The black curve is the EAL corresponding to the same survey geometry in Fig. 3.30

(MFSQ pulse). The red curves display the EAL for the match filtered gaussian pulse,

and the green curves are for the compressed Gaussian pulse. In all curves the solid

lines are for the double angle region, the dashed are for the x > 0 signal only, and the

asterisks are for the approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.32 The EALs corresponding to a 6 element array, under the same survey conditions as

used in Fig.3.31. The black curves display the MFSQ pulse, the red curves display the

EAL for the match filtered gaussian pulse, and the green curves are for the compressed

Gaussian pulse. In all curves the solid lines are for the double angle region, the dashed

are for the x > 0 signal only, and the asterisks are for the approximations. . . . . . 136

3.33 The EALs under the same survey conditions as used in Fig.3.32, however the tilt

angle has been changed to 20◦. The black curves display the MFSQ pulse, the red

curves display the EAL for the match filtered gaussian pulse, and the green curves

are for the compressed Gaussian pulse. In all curves the solid lines are for the double

angle region, the dashed are for the x > 0 signal only, and the asterisks are for the

approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.34 The EALs under the same survey conditions as used in Fig.3.32, however the tilt

angle has been changed to 0◦. The black curves display the MFSQ pulse, the red

curves display the EAL for the match filtered gaussian pulse, and the green curves

are for the compressed Gaussian pulse. In all curves the solid lines are for the double

angle region, the dashed are for the x > 0 signal only, and the asterisks are for the

approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.35 The EAL for the 20 cycle MFSQ pulse for 100kHz, 200kHz and 300kHz in salt water

under the same survey conditions as used in Fig.3.33 (20◦ tilt angle). The black curves

display the EAL for 300kHz pulse, the red curves display the EAL for the 200kHz

pulse, and the blue curves are for the 100kHz pulse. In all curves the solid lines are

for the double angle region, the dashed are for the x > 0 signal only, and the asterisks

are for the approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.36 The EAL for a 300kHz MFSQ pulse MASB sonar with a 6 element array, tilted at

20◦ in salt water (black curves), and fresh water (red curves). The solid lines are for

the double angle region, the dashed are for the x > 0 signal only, and the asterisks

are for the approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.1 6 channel MASB system components. . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.2 Beampatterns for a six element receive transducer taken at 300kHz as measured in

the URL (units of dB are scaled such that the max of the beampattern is at 0dB).

The red points display the predicted cos2(θ) beampattern. . . . . . . . . . . . . . . 147

xvi

4.3 The filtered gps and digital compass data from one survey run on Pavilion Lake (shown

in left plot on scale of lake, and right plot in a smaller scale), red represents the edge of

the lake, blue arrows represent the heading given by compass data (here only one out

of every twenty points measured is displayed), which mostly obscure the green points

that are the track of the boat. Note that the easting and northing scales displayed

here are simply offset from the position of the DGPS base-station. . . . . . . . . . . 149

4.4 Refraction of various rays (A: apparent position, T: true position) for the stratified

sound speed profile encountered during surveying. The mixed layer above the ther-

mocline allows for rays to travel a sufficient distance before turning downward. A

constant sound speed was inferred for depths below the maximum measured value. 150

4.5 The analogous circuit representation for the transducer and setup for the amplifiers. 151

4.6 The total noise spectrum (in A/D units, offset 90dB above actual noise power) is

given in solid red, all other curves represent independent noise sources and have been

added in quadrature. For the noise behavior of the actual system (shown in cyan) it

should be noted that the values of the circuit components are not fully optimized for

the 205kHz transducer as the amplifiers were designed to be broad-band for use up to

400 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.7 Bathymetry (left) and imagery (right) are both useful in recognizing bottom features

such as these deep water microbialites. . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.8 Top Left: A new morphology the deepest water microbialite specimens, photographed

with an ROV. Top Right: Microbialites characteristic of finger-like features on side of

lake photographed by the author while scuba diving. Bottom Left: survey rigged pon-

toon boat, blue DGPS antenna located mid-boat on port side. Bottom Right: LBV150

ROV used to take top left photograph, shown with net for microbialite retrieval. . . 155

4.9 Maps constructed from two data sets taken of the same physical location are shown

in the upper left and right figures, although the surveys were taken from different

vantages, the purple track shows the path of the sonar (as indicated by having a

transducer and beampattern on the track). A full spectrum colormap is used to

demonstrate the subtle difference in target strength as the grazing angle, θG, vantage

is changed, with warmer colors represent higher return strengths. . . . . . . . . . . 157

4.10 Stitched map of imagery, outline of lake is given in blue. . . . . . . . . . . . . . . . 158

4.11 Averaged map of depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.12 Profile including a small deep microbialite mound in the foreground and a larger one

in the background. Bathymetry resolution is set to 1 m, while the imagery is resoled

at 8 cm. The scattering strength of the red regions is approximately 30dB greater

than dark blue regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

xvii

4.13 Depth measurements in the September 2005 300kHz survey, resolved at a 1m grid in

both easting and northing. Wall is in South-West region of the lake. . . . . . . . . . 160

4.14 Backscatter imagery measurements co-located with depth information from the Septem-

ber 2005 300kHz survey, resolved at a 1m grid in both easting and northing. Same

region as shown in Fig. 4.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.15 Profile taken at Sassamat Lake, with corresponding EAL as determined from the

CRLB for the MFSQ pulse using only noise and footprint shift for a single signal on

a six element array (ie. this ignores completely the surface multipath signals). . . . 163

4.16 The top image is a sector sweep of the Sasamat lake basin (note that backscatter

imagery measurements show sidelobe leakage due to conventional beamforming). The

bottom image is the corresponding EAS for the basin at Sasamat Lake. . . . . . . . 164

xviii

Glossary

Abbreviations

Abbreviation Full Meaning

MASB Multi-Angle Swath Bathymetry

CRLB Cramer-Rao Lower Bound

AOA Angle of Arrival

EAL Error Arc Length

URL Underwater Research Laboratory

SONAR SOund NAvigation and Ranging

DOF Degrees of Freedom

NOAA National Oceanic and Atmospheric Administration

CTD Conductivity Temperature Depth (probe)

SNR Signal-to-Noise Ratio

IHO International Hydrographic Organization

MUSIC Multiple Signal Classification

ESPRIT Estimation of Signal Parameters via Rotational Invariance Technique

MODE Method of Direct Estimation

BBS Bottom Backscatter Strength

AUV Autonomous Underwater Vehicle

ROV Remotely Operated Vehicle

GPS Global Positioning System

DGPS Differential Global Positioning System

PLRP Pavilion Lake Research Project

CSA Canadian Space Agency

CARN Canadian Analogue Research Network

MVDR Minimum Variance Distortionless Response

LP Linear Prediction

ME Minimum Eigenvalue

xix

Abbreviation Full Meaning

SQ Square (pulse)

MFSQ Match Filtered Square (pulse)

FQ Finite Q (pulse)

MFFQ Match Filtered Finite Q (pulse)

CG Compressed Gaussian (pulse)

FS Footprint Shift

BD Baseline Decorrelation

pdf Probability Density Function

tr Matrix Trace

TS Target Strength

FPGA Field Programmable Gate Array

TVG Time Variable Gain

SS Special Survey (defined by IHO)

FO First Order Survey (defined by IHO)

SO Second Order Survey (defined by IHO)

MBES Multi-Beam Echo Sounder

EAS Error Arc Surface

xx

Variables

Variable Full Meaning

χi Signal on element i

~χ Signal vector

N Number of array elements

si Narrowband complex gaussian signal

ni Narrowband complex gaussian noise

E{(·)} Expectation of (·)c Speed of sound in water [m/s]

r Range [m]

∆ri Range delay across array [m]

δri Range delay across between adjacent elements [m]

t Time delay [s]

θ Physical angle of arrival [rad]

γ Tilt angle of array [rad]

h Height of sonar above bottom [m]

d Array spacing [m]

x Horizontal range from sonar [m]

φ Angle at sonar constructed between nadir point and footprint location [rad]

R Covariance matrix

σ2 Variance

ρ Correlation coefficient

κ Correlation matrix

α Phase angle of arrival [rad]

λ Wavelength [m]

f Frequency [Hz]

snr, (SNR) The signal-to-noise ratio, (expressed in dB)

Et Energy in pulse

g(r) Pulse shape function

B(x) Bottom scattering function

|B|2 Bottom scattering strength

rsq Length of square pulse [m]

rgs Range spread of gaussian pulse [m]

rgc Range spread of gaussian pulse after compression [m]

q Quality factor

a Parameter for FQ and MFFQ pulse functions

b Parameter for chirp normalized gaussian pulse with complex envelope

cτ Compression ratio

xxi

Variable Full Meaning

r2 Mean squared pulse length [m]

Nsq Number of cycles in square pulse

u Equation simplification term for correlation functions

Be(θ) Beampattern

Z Complex impedance [ohm]

<{(·)} Real component of (·)={(·)} Imaginary component of (·)

∆f Bandwidth [Hz]

T Temperature [K]

kB Boltzmann constant [1.38× 10−23 J/K]

rfarfield Range to the far field [m]

CRLBα Cramer-Rao Lower Bound on variance for phase angle of arrival

K Number of plane waves impinging on array

f·(·) Probability density function for (·)F·(·) Cumulative density function for (·)

J Fisher information matrix

M Number of snapshots

J Jacobean

S Pre-estimation averaging function

Ψ Characteristic function

w Filter weight

~w Filter weight vector

~p Cross correlation vector

Pave Average output power of filter

ndof Number of degrees of freedom

z Complex filter root

~s(θ) Beam steering vector

λLagrange Lagrange multiplier

η Transducer efficiency

G Transducer gain

ts Target strength

ss Bottom backscatter strength [m−2]

P Transmit power [W]

α0 Range attenuation [dB/m]

θci Alongtrack beamwidth [rad]

θli Acrosstrack beamwidth [rad]

EAL Error Arc Length [m]

xxii

To Mom, Dad, Hayden and Melinda.

xxiii

There are still no [manned] submersibles that can go anywhere near the depth of the Mariana

Trench, and only five, including Alvin, that can reach the depths of the ”abyssal plain”-the deep

ocean floor- that covers more than half the planet’s surface. A typical submersible costs about

$25,000 a day to operate, so they are hardly dropped into the water on a whim, still less put to sea

in the hope that they will stumble on something of interest. It’s rather as if our firsthand

experience of the surface world were based on the work of five guys exploring on garden tractors

after dark. According to Robert Kunzig, humans may have scrutinized ”perhaps a millionth or a

billionth of the sea’s darkness. Maybe less. Maybe much less.”

— A Short History of Nearly Everything, Bill Bryson, 2003, [3] (pg. 279)

xxiv

Chapter 1

Introduction and Background

1

CHAPTER 1. INTRODUCTION AND BACKGROUND 2

1.1 The Big Picture - Purpose and Objectives

The Oceans of the world have been studied for as long as history records. Ancient mariners,

including the author’s own Polynesian ancestors were adept at reading the stars and weather for

navigation, and could predict large scale ocean currents long before the first conventional western

scientists began to understand the principles of ocean dynamics. Moving forward in time, one arrives

at such notable mariners and map makers as James Cook born 251 years and 3 days before the author,

whose bathymetric maps made using lead lines were more than adequate for the avoidance of hull

piercing reefs, and navigation of the globe. However, deployment of a plummet is time consuming

and difficult, and so more effective ways of mapping bathymetry remotely from a vessel were sought.

The quote by Bill Bryson that precedes this chapter illustrates the need for technologies to map the

seafloor in a manner that is high enough in resolution so as to understand better the underwater

surface of over half our planet. In addition, autonomous platforms may need to be employed. Manned

submersibles, as Bryson puts it, are no more effective at achieving this end then sending ”five guys

exploring on garden tractors after dark” to explore the land surface of the earth. One effective

method to achieve this underwater mapping goal is through the use of the topic most pertinent to

this thesis, namely the study of sonar, which in itself is long past its infancy (history suggests that

scholars as far back as Leonardo Da Vinci had a primitive understanding of the principles of source

direction estimation of vessels using listening devices underwater). So one begs the question, what

is the relevance of the current research? In what context do we seek to further improve upon the

cumulative work that has come previously?

To begin, it can be stated that the subject of this thesis is a form of sonar known as Multi-Angle

Swath Bathymetry (MASB). The word sonar, which originates as an acronym for SOund NAvigation

and Ranging, has come to describe almost any system that employs acoustics to probe the underwater

environment. MASB is a form of monostatic active sonar that uses a short pulse to obtain both high

resolution backscatter imagery of the bottom, and co-registered bathymetry. Range measurements

are derived from the time of flight of the transmitted pulse and the principle of interferometry

is utilized to estimate the angle-of-arrival (AOA) of incoming plane waves relative to an array of

acoustic receivers. It is the analysis and demonstration of such a system upon which the following

research is based. As quantitative performance analysis is at the core of optimization in engineering,

the purpose of this research is to refine the study of performance of AOA estimation for MASB

systems. To achieve this end requires a departure from the traditional paradigm which assumes

that confidence intervals can be based on the standard deviation of measurements. A new model

is presented that utilizes the Cramer-Rao Lower Bound (CRLB) on variance to better characterize

the performance of estimators of AOA. The CRLB is preferable over the standard deviation due to

the condition that the probability distributions of the AOA estimators do not always follow gaussian

statistics, instead there is more probability that the estimate will be in the tail region, away from


the mean value. In some cases, typically when the number of snapshots employed for an AOA

estimate is increased, the statistics tend to gaussian behavior as would be dictated by the central

limit theorem, and in these instances the CRLB will limit out at the estimator variance and the

original interpretation of confidence is restored.

However, to eventually examine the CRLB, first the model geometry had to be defined, and

signal statistics determined. The effects of different transmit pulses were accounted for, and the

correlation matrix for the receive array was constructed. Once this was accomplished, it was possible

to examine the performance of various estimators applied to the AOA problem. In addition, an MASB

system was designed and constructed in the Underwater Research Laboratory (URL) at Simon Fraser

University and field data was collected to examine the capabilities of this system.

As such the accomplishments of this thesis are twofold:

1. Constructed a theoretical framework for determining the accuracy of MASB.

2. Demonstrate the capabilities of a prototype MASB system.

Before continuing it should be noted that a significant portion of the research in this thesis was

presented in two journal papers([1],[2]) and two conference presentations (proceedings are found in

[28] and abstract given in [27]). However, advances made subsequent to those two papers are also

included in the following thesis. It should be mentioned as well that the reviewers comments for the

second paper [2] were useful in amending the research to consider performance analysis of additional

waveforms that are used in practical MASB deployment.

1.2 Goal 1 - Theoretical Performance Analysis

1.2.1 Purpose of Research and Summary of Accomplishments

All measurements of physical observable systems can be modeled as samples of one or more

stochastic processes. At the very least, all measurements fall prey to the presence of noise, be

it electrical or simply a limit on the precision of a measuring tool, this having the net effect of

making a measurement only reliable to a certain degree of accuracy. Each data point in turn reflects

an incidence of a random variable. Often what is desired is an estimate of a parameter from an

underlying model. For the purpose of the following research, the underlying model parameter of

interest is the AOA of a plane wave signal (a plane wave here includes any wave emanating from

the far field), which is determined from measurements of a MASB receiver array. In this case, the

AOA is not itself a random variable. However, because there are not enough degrees of freedom

(DOF) to determine all of the random variables (target signal strengths and random thermal noise),

the estimate of the AOA becomes a random variable. In conjunction, estimators of AOA are often


constructed from one or more data points and each estimator has an unique statistical distribution,

for which closed form expressions are often precluded due to complexity of the utilized estimator.

It should be stated that prior to this thesis there was no comprehensive description and analysis of

even a simple two element interferometric sonar performance. Though several of the geometric effects

that govern performance were described in some regard before this research there was no analysis

that encompasses all effects and compared their relative importance in conjunction with the effect

of having multiple signals present on an MASB array simultaneously. To this end the correlations

between signals on linear arrays are derived for five pulse shapes, including an expansive examination

of the assumptions required to make such calculations. Measures of comparison in length and energy

normalization for each of the pulse shapes are also calculated, so that the relative benefits of utilizing

these different pulses can be compared in subsequent analysis.

In addition there existed in the literature a misconception that the standard deviation defines a

confidence interval in which estimates of AOA will be located. It was determined in the course of

the research for this thesis that the Cramer Rao Lower Bound (CRLB) on variance is more relevant

in determining suitable confidence intervals than the variance itself. Preference of the CRLB to

the variance is also given for other reasons, which will be explained in the next section. This

thesis also extends the performance analysis to multiple element arrays and multiple signals, which

in itself demonstrates the benefits of using a MASB sonar over a simple sidescan sonar (which is

incapable of estimating AOA), or simple swath bathymetry system (which uses only two elements and

therefore cannot account for multiple signals). By also extending the analysis to multiple snapshots

for five different waveforms, it was possible to recognize that the performance (through calculation

of pertinent CRLBs) of the compressed Gaussian pulse is a good representative of the performance

for all the match filtered pulses considered (again, this has not been reported prior to this research)

and can be used to adequately define the best performance that a system is capable of achieving.

Using the CRLB and the sonar equation (to which much literature has already been devoted [42]

[4] [19], and as such is simply employed in this research), the concept of Error Arc Length, EAL, is

defined and demonstrated to be an intuitive measure of bathymetric survey uncertainty, and as it

does not rely on a variance based approach, and is more suitable as a performance indicator than the

depth uncertainty that has been utilized in previous work. Adherence of three specific estimators

of AOA to the EAL is tested through simulation (in this case a survey geometry is employed), and

found to be in agreement with the theoretical underpinning. In essence the first goal of this thesis,

is the construction of a general framework for predicting the performance that can be obtained by

an estimator for a given model geometry.


1.2.2 Relevance of Prior Work and Accomplishments Explained

In examining the problem of angle estimation performance, one must not only consider various es-

timation procedures, but also the physical geometric factors that limit the intrinsic performance such

as footprint shift and baseline decorrelation. Accuracy of interferometric sonar has been examined

in several key articles, [8] [9] [18] and most recently [23] prior to the research in this thesis. A report

produced by NOAA, [14], also compared a commercial interferometric system, with a multibeam

system on the basis of experimental output, and not underlying theory. It is the goal of this thesis

to address the theory that underlies the performance of MASB systems.

It was demonstrated in [18] and recognized in [23] that baseline decorrelation does not significantly

degrade AOA estimation for short pulse lengths, such as those used in the surveys performed over

this research (a pulse of 20 cycles at 300kHz is considered short). In the following research, the

effect of baseline decorrelation will be accounted for in the calculation of signal correlations for

various waveforms, and an example will be given to illustrate the conditions under which baseline

decorrelation will play a role in performance of AOA estimation. In [23] the effects of footprint shift

were shown to be a limiting factor in angle estimation. In this thesis it is demonstrated that the

effect of footprint shift is dependent on the pulse shape that is transmitted and the spacing of array

elements, but not the number of array elements that are used. Additionally, the analysis presented in

[23] does not present a derivation of the relevant shift length, and the value used in that prior work is

in error and different from the derivation given in this thesis (in this regard the derivation of such a

length may be considered new research). Aside from footprint shift, uncorrelated thermal noise plays

a role in limiting MASB performance, especially at further ranges where path loss from spherical

spreading has diminished the signal strength. Finally, in the case of interferometric sonar where

multiple angles of arrival are estimated, the number of angles can adversely influence the estimation

accuracy. This last area is one focus of this research, and although past literature has emphasized

the unique methods by which multiple angles may be estimated ([20], [21]), the performance of AOA

estimation under the scenario of multiple signals has not been addressed in previous literature on

MASB sonar.

Although it is the research in this thesis that identifies the CRLB as useful in the context of con-

fidence intervals for AOA estimation and the subsequent definition and interpretation of the EAL,

much prior work has been done regarding methods for calculating and approximating the CRLB for

the direction finding problem (notably [40] and associated references, also including but not limited

to [36] for multiple signal characterization (MUSIC) and maximum likelihood estimators, [34] for

unknown correlation between multiple signals, [35] for singular information matrices and [32] for

asymptotically large arrays). In [37], an approximation is given for the CRLB which includes two

closely spaced incoming plane waves, taking into account not only the angle and magnitude variables


of incoming signals, but also the correlation of the two signals. In fact, many of the techniques used

in these works emphasize approximation of the CRLB for large arrays, whereas for the research

presented in this thesis array sizes remain manageable (typically 6 elements or less for practical

systems). Since the signal correlations presented in this research represent new achievements, the

resulting covariance matrices, Fisher Information Matrices and CRLBs also represent new results.

In addition, the CRLB is used to estimate the performance of source location estimators in cir-

cumstances where signal propagation is that from one-way bi-static sonar, and much research has

been published on this subject for multiple sources in complex multi-path environments (one such

example is [31]). However much of the aforementioned research is for much larger arrays than are

used in MASB, and tend to be more relevant to lower frequency implementations with emphasis on

a more broad-band approach to the AOA problem. It should be stated again that in the context

of narrow-band high resolution mono-static sonar such as MASB systems, there is no pre-existing

literature that recognizes the relationship of CRLB to confidence interval, which is an accomplish-

ment of the research presented in this thesis. In addition this research accounts for the effects of

transmitted pulse-shape, tilt angle and multiple incoming signals. Pulse shapes included in this

research include not only a square pulse, which was one of the pulses considered previously in the

context of interferometric sonar in [23] (however a general signal correlation was not presented in

prior research), but extend the square pulse to include the effects of a matched filtering. In addition,

a pulse that exponentially rises and falls to steady state is analyzed along with its match filtered

counterpart, and a compressed gaussian pulse is considered. Finally, it should be re-iterated that the

performance measure, Error Arc Length, is unique and is both defined and adopted in this research

as the appropriate performance measure for MASB systems.

Here it is useful to make a distinction between MASB and other interferometric sonar. Relative

phase bathymetric sidescans (commonly called interferometric sonar) generally only estimate one

AOA using either two elements or averaged combinations of estimates from several two element

sets. In contrast, MASB devices use the properties of a phased array to separate multiple incoming

signals simultaneously, and therefore require a minimum of three elements (for two signals). In

general MASB utilizes signal processing techniques that require a minimum number of elements that

is greater than the number of signals. Only linear arrays of uniform spacing will be considered for

the present analysis, however in principle the methods presented here can be applied to sparse, or

even non-uniform arrays of arbitrary orientation. It should also be noted that although the analysis

in this thesis has been applied to MASB sonar, it is as valid in any application where antennas are

used in an interferometric system, where in particular pulses with short range extent after processing

are transmitted at a known carrier frequency and received on an array.


The work included in chapters 2 and 3 addresses a pervasive problem in industry with regards

to general overstatement of the accuracy of both interferometric based sonar and multi-beam tech-

nologies. Because there is not yet an accepted model of bathymetric performance for MASB-type

systems, companies often report values for the accuracies of their products that exceed the limita-

tions imposed by the geometry, physics and signal processing presented in this research. As there

are several commercial MASB systems that have become available to consumers during the course

of this work (such as the Benthos C3D sonar), it is expected that future research on performance of

these systems will need to adopt a more robust model such as the error arc length standard. Further-

more, as the technical specifications regarding accuracy of bathymetric measurements required for

surveying standards are set by the International Hydrographic Organization (IHO), other accuracies

related to surveying such as transducer motion (mainly roll, pitch, tilt) must also be accounted for in

order for a system to be approved for deployment in hydrography. However these factors are beyond

the scope of the present research.

1.2.3 Direction of Chapter 2

Chapters 2 and 3 are presented to achieve the first goal stated in the previous section, namely to

create a theoretical framework by which the performance of MASB systems can be predicted and

used to optimize survey geometries and parameter choices. In chapter 2 the objective is to derive

the cross-correlation function between signals on various elements of a linear array for five distinct

transmit waveforms. First a model survey geometry is defined. This simple geometry consists of a flat

bottom, and a MASB array set at a known tilt angle. Anisotropic environmental considerations, such

as vertical sound speed stratification, are neglected in this simple model, as it will be demonstrated

in chapter four that these refractive effects can be compensated for through the use of instruments

such as conductivity-temperature-depth (CTD) profiler. The process of constructing a bathymetric

map is described, and various regions of the seafloor are shown. These include bottom areas wherein

two bottom return signals need be considered, as well as a description of common sonar terms such

as nadir, endfire and broadside. Signal formalism, leading to the construction of a correlation matrix

for multiple signals and uncorrelated gaussian noise is then presented. The condition of having

zero correlation between angle-separated signals is also assumed, and the geometry required for this

condition is described. The signal-to-noise (snr) ratio is also defined in this chapter, and for any two

elements the utilization of effective snr is proposed to compare the relative importance of competing

geometric effects and noise, with respect to performance.

Under the condition of uncorrelated gaussian bottom scatter (this holds for a correlation length

less than the pulse length) an expression for the cross correlation of a general waveform is developed.

This development using gaussian bottom scatter is also chosen as it allows for comparison with other

work in the literature, such as [23], and has been shown in [18] [41], [11] to model bottom conditions


adequately in many cases. Following this condition five waveforms are defined, these being the square

pulse, the match filtered square pulse, a pulse portraying exponential rise and fall to steady state (this

pulse closely approximates real sonar pulses in water having been modified by a transmit transducer),

the match filtered version of this pulse, and a compressed gaussian pulse. The compressed gaussian

pulse is derived from a chirp normalized gaussian pulse. In addition measures of comparison between

pulses, namely the mean squared pulse length and the pulse energy normalization are calculated and

tabulated with the normalization condition for each of the pulses considered.

Beginning with the square pulse, various approximations are employed to determine the decorre-

lation between elements due to footprint shift. Following this, the general closed form correlation

between receive elements is calculated for the square pulse, again under certain approximations. This

general form encompasses not only footprint shift, but also baseline decorrelation and the effect of

considering two elements not located at the phase center of the array. Interpretation of these results

is also presented. A simulation is employed to verify the closed form expression for the general cor-

relation, including the effect of having bottom returns from two distinct signal regions. In a similar

manner the same expressions are calculated in turn for each of the pulses previously defined. A final

comment is then given on the addition of uncorrelated gaussian noise to the signal model. It can

be noted at this time that the general results for the practical sonar pulse and its match filtered

counterpart have not as yet appeared anywhere in the literature (not even in [1] or [2], which give

account of most of the research included in this thesis).


Chapter 3 utilizes the results of the signal correlations for various waveforms presented in chapter

2 in order to examine the theoretical performance of an MASB system through the use of the CRLB.

First, the concept of the signal as being complex multivariate gaussian is introduced and justified.

The Fisher Information Matrix and resulting CRLB are then defined for both the one and two signal

scenarios. The form of the Fisher Information Matrix is simplified by the condition that the signal

vector is zero mean and multivariate gaussian in nature. Examples are provided to demonstrate that

the single signal scenario requires only consideration of AOA as an estimation parameter (in the

Fisher Information Matrix ), and that for two (or more) signals both the AOA and signal strength

for each signal must be considered. In addition it is shown that the effect of having a second signal

on an array influences how well the first signal can be estimated. Arguments are also presented in

the early sections of Chapter 3 to clarify that for the survey geometries considered in this thesis

correlations between incoming plane waves can be neglected, instead of being included as nuisance

parameters in the calculation of the Fisher Information Matrix.


To justify use of the CRLB over the standard deviation (which has been used in the past to

describe performance of AOA estimation [23]), the case of a single plane wave impinging on a simple

two element array is examined. The probability distribution for a single AOA phase estimator

between two elements is first determined, and subsequently the standard deviation is computed. By

plotting the probability of a phase estimate falling within a number of standard deviations from the

mean it becomes apparent that the traditional confidence interval of two standard deviations does

not always account for 95% of the estimates. Furthermore it is shown that the probability of falling

within a set number of standard deviations is highly dependent on the snr of the signal considered. In

contrast the probability of an estimate falling within a number of√

CRLB from the mean is robustly

independent of snr, meaning that it is in fact the appropriate parameter for defining a confidence

interval. This is an important result, as industry standards (set by the IHO) are currently based

on a confidence interval of two standard deviations, a standard that not only depends on the value

of the snr, but is difficult to estimate using data, whereas the CRLB is calculated directly from

the underlying signal model. It is also demonstrated that the standard deviation is dominated by

the presence of long tails in the probability distribution and therefore a few outlier points dominate

estimation of the standard deviation using either simulations and measurement. The√

CRLB has an

added bonus, in that it converges to the standard deviation in the circumstances where the standard

deviation is relevant, namely when multiple snapshots are employed.

One further justification for use of the CRLB is that it is an estimator-independent measure. Cal-

culation of the probability density function of an estimator is often prohibited due to the complex

nature of most estimation routines. For arrays larger than two elements, and for multiple snapshots

many estimation routines contain transformations such as matrix inversion or search algorithms,

making direct calculation of a closed form probability distribution difficult, if not impossible. Dif-

ferent estimators can often be utilized to measure the same physical quantity and in general the

performance of each estimator will vary, yet all can be compared to the CRLB. However, a caveat

must be stated that the variance of an estimator is not guaranteed to converge to the CRLB, and

the CRLB does not often provide insight as to how to form an efficient estimator.

Next, the AOA CRLB for various survey geometries is calculated and plotted for each of the

waveforms considered in the second chapter. These scenarios include, but are not limited to, ex-

amining the effect of increasing pulse length (demonstrating when baseline decorrelation begins to

play a role in AOA estimation performance), and the effect of increasing array size (demonstrates

that the uncorrelated noise is mitigated through the use of more elements, but footprint shift is

only mitigated to an extent corresponding to the widening separation of the two outermost array

elements). An approximation technique for the CRLB is also presented using the effective snr, and

demonstrated to effectively emulate the full CRLB in scenarios where only one signal is present on

the array. The effect of different signal-to-noise ratios is also investigated within this framework.


One significant result is the observation that the performance of several of the waveforms is nearly

identical. Specifically, the match filtered square pulse and match filtered exponential rise / fall to

steady state pulse behave in a way that is extremely close to the compressed gaussian when relative

pulse normalization and mean squared length are taken into account. This is an important result

because it allows for the compressed gaussian to be analyzed in place of the other pulses for sub-

sequent analysis. In general the square pulse is shown to perform the worst of the pulse functions

considered with respect to footprint shift, whereas the match filtered pulses and compressed gaussian

demonstrate the best performance (the exponential rise / fall to steady state pulse performs only

slightly worse than the best pulses).

To compare the CRLB to various estimators, it was necessary to choose a set of estimation algo-

rithms. One requirement of any viable estimation routine is that it must be able to employ multiple

snapshots (a snapshot is a single measurement, which in principle belongs to a larger ensemble

of possible measurements). The first consideration was whether to choose those that employed

pre-estimation averaging which uses multiple snapshots in a single estimator, or post-estimation

averaging which simply averages the single snapshot estimator. To make this determination, the

simple two element array was re-visited. For the simple scenario with a high signal to noise ratio,

the variance of the pre-estimation AOA estimator is shown both in theory and simulation to converge

to the CRLB as the number of snapshots is increased, which is an new result not found in the prior

literature. However, for post-estimation averaging the ratio of the variance to the CRLB remained

constant for increasing numbers of snapshots, which is not a new result, as the variance and CRLB

are both proportional to the inverse of the number of snapshots. In light of these last two results,

an emphasis was placed on examining estimators that use pre-estimation.

Though there are a large number of estimators that have been used historically for angle estimation

(references for methods such as ESPRIT, MUSIC, MODE, as well as the chosen algorithms can all

be found in [40], [36]), the three chosen for comparison in this research are minimum variance

distortionless response beamforming (MVDR), linear prediction and minimum eigenvalue analysis.

Reasons for choosing these three algorithms include performance (ascertained from use in practical

survey systems), speed of computation, lack of bias and ease of implementation. Computational

efficiency is of practical importance in any MASB system, as real time display of data is beneficial

in survey scenarios. A useful estimator should also be designed to be unbiased over a useful range

of snr. Though the CRLB does not aid in the creation of these estimators, and does not indicate

inherent limitations due to bias, it allows a benchmark by which all may be assessed.

To compare the performance of different estimators, simulations are employed. For many of these

estimators, performance in a multiple signal environment depends on the relative values of the signal

strength and phases of the incoming signals. Unfortunately, bias can be introduced under certain


circumstances such as low snr in the cases of linear prediction and MVDR (if two or more AOA are

present), or in the case of minimum eigenvalue analysis if the modeled covariance matrix is slightly

different from the true covariance matrix. Bias can also be incurred if two or more signals approach

each other in the angular domain. In general it is safe to conclude that signals separated in angle

by at least one Rayleigh beamwidth [40] do not tend to influence each other detrimentally. One

final effect that can negatively impact AOA estimation is the condition of having too many DOF

in the estimator. This can result in mathematical degeneracy in the AOA estimates for multiple

signals. In such circumstances as described in the previous few lines the simulation and analysis of

all possible combinations of incoming signal phases can be time-consuming. It will be endeavored in

this research to at least cover scenarios that are common to the defined survey geometry. Examining

a simulated flat bottom (equivalently any moderately smooth bottom can be used), the percentage of

AOA estimates to within the confidence interval defined by the error arc length (around the known

bottom position) will be tabulated as a check on estimator performance for the various estimators

using simulated MASB data for all waveforms.

Following the comparisons of the CRLB on AOA for various pulse shapes, the error arc length,

EAL is introduced as the correct bathymetric performance measure for MASB (it is simply the arc

swept out by ±2√

CRLB in the physical angle domain, at the range corresponding to the location

of the bottom) with the intuitive advantage that it has the same length scale and units as depth

estimation uncertainty. After implementing the full sonar equation, the EAL is plotted for several

survey scenarios including the effects of attenuation in both fresh and salt water, tilt angle of the

array, and frequency.

1.3 Goal 2 - Demonstration of Prototype System

1.3.1 Purpose of Research and Summary of Accomplishments

Chapter 4 demonstrates the capabilities of a small six element MASB system, deployed in fresh-

water for the purpose of mapping microbialite structures. In doing so, this chapter proves the utility

of an MASB system in a survey environment in which either conventional sidescan or two element

interferometry would fail to provide both high resolution backscatter imagery, and a bathymetric

survey. A 3D sidescan sonar, such as the MASB sonar used in this research, differs from conventional

sidescan in that a multiple element receive array is utilized to estimate arrival angles of multiple in-

coming plane waves. Bathymetry information is calculated using the range of a target and angle of

arrival (AOA) as previously described. The desired bottom signal can be separated from multipath

signals which would otherwise corrupt imagery of the bottom. Additionally, the narrow along track

beamwidth and short transmitted pulse length allow for a high resolution image of the lakebed.


To accurately measure high frequency acoustic bottom backscatter strength from the seafloor or

lakebed many physical effects of the measurement system and physical environment must be taken

into account. The two most common commercial systems employed for this type of measurement are

multibeam and sidescan systems. Beam modulation can limit the performance of multibeam systems

and such effects are characterized in [16] where the authors demonstrate that it is difficult to achieve

an absolute sensitivity accuracy greater than 3 dB in most cases (this in principle eliminates any

possibility of applying bottom characterization techniques [5]). In comparison sidescan measurements

of BBS are not affected by beam modulation, but cannot correct for multiple incoming plane waves

(including multipath signals such as surface reflections). Additionally, as angle of arrival information

is not computed in sidescan systems, there is no capacity to compensate for either transmit or receive

beampatterns. It should also be noted that sidescan systems are completely unable to measure or

correct for the grazing angle dependence of backscatter.

One issue with using MASB for surveying is that there is typically a gap in the data at nadir (this

bottom location will be defined later in chapter 2). However, it should be noted that multibeam

systems can extract useful bathymetric information near nadir by employing narrow beamforming

techniques. However, multibeam systems often cannot fully account for the specular component

of the bottom reflection in compensation for grazing angle effects, and therefore any estimation of

bottom scattering strength is rendered ineffective, leading to range artifacts under the track of the

survey vessel. Also the density of swath coverage in a multibeam sonar can be lower than for MASB

systems, and at distant ranges, if the angles of adjacent beams are not adjusted properly, they

ensonify regions that can be spaced unevenly, resulting in additional complications for surveying.

As the 3D sidescan maintains a small aperture (only slightly larger than sidescan), and uses

less channels than multibeam systems (hence fewer electronic components), it is an ideal solution

for small platform applications (such as autonomous underwater vehicles AUV). The instrument

designed for this research uses one transducer for transmission of a narrow band pulse, and six

transducer elements for reception. Surveying can also be performed at a variety of depths including

near surface boat mounts, which typically vex sidescan and simple interferometric systems due to a

multipath environment.

Pavilion Lake, in British Columbia, was chosen as a survey testbed for two reasons. The primary

reason is its importance in analogue space research, having been identified by the Canadian Space

Agency as a part of the Canadian Analogue Research Network following the work of [12], which

originally reported unique microbialite structures in Pavilion Lake. Through this work the applica-

bility of sonar to problems as diverse as the search for life beyond the planet Earth is demonstrated.

It is demonstrated that this mapping technique can benefit not only research at Pavilion Lake, but

has the potential to improve research at a number of analogue research sites, both nationally and


internationally. The second reason is that Pavilion Lake is acoustically interesting. Both the micro-

bialite formations and the soft sediment regions provide for acoustically diverse lakebed with mean

backscattered signal levels that vary in excess of 30dB. In addition, the complexity of the many

mound structures in the lake, provide a survey scenario that is well suited to test the performance

of the MASB system under varied and complicated geometries.

Surveys included in this thesis were performed on several occasions between June 2005 and Septem-

ber 2006 using systems operating at 200 → 300 kHz (the latter of these surveys forms the basis for

the data in chapter 4, as the lake was less stratified). Comparing various visual ground truth results

(using techniques such as diver observations, remotely operated vehicles and GPS referenced drop

camera observations) with backscatter imagery, the distribution of microbialites, as recorded with

MASB sonar was examined. Microbialites are located not only around the lake perimeter walls, but

also on the slopes of the mound features within the lake as well as sparse isolated patches within

the basins. The sonar images of microbialites taken at Pavilion Lake facilitated the discovery of

structures with previously unknown morphologies at depths up to 55m, which are deeper than those

previously recorded in [12]. As these structures are sparsely distributed, and located deeper than is

possible with available scuba diving techniques, detection through methods other than sonar would

have proved difficult. Samples taken from these deep sites are morphologically different from sam-

ples collected at shallower depths, including the deepest samples obtained and analyzed in [12], and

so represent a unique discovery made using MASB. An example of co-located sonar imagery and

bathymetry is given in Fig. 1.1 showing large lobe-like microbialite patches along the walls of the

lake. Owing to the enormous size of the patches of microbialites, and the relatively short range vis-

ible through diver or underwater camera observations (excluding picture mosaics) the large patches

are ideally suited for MASB imaging to detect and display with large-scale distribution. MASB

measurements have also revealed acoustic clutter areas associated with rock slides, human instigated

wreckage, and other bottom features that are assisting in characterizing the lake. It should be noted

that although sidescan, multibeam and MASB sonar data has been collected for some limited regions

of the lake in previous years, no geographically referenced data was recorded prior to the research

contained in this thesis.


Chapter 4 begins by describing the the prototype MASB system designed at the URL. Included in

this section is the system schematic of the whole survey apparatus, including position and orienta-

tion sensors. The various component devices are listed, with applicable performance specifications.

Calibration data showing beampatterns of the receive transducers is displayed, and they can be ob-

served to be fairly smooth and agree well with a theoretical beampattern, making beam effects easy

to correct for in the recorded data. Though not shown, correction due to the transmit beampattern


Figure 1.1: 3D sonar image showing orientation of sonar.

is equally trivial (in practice it is often receive arrays that experience cross-talk effects producing a

rippling of the beampattern, however this effect has been mitigated through extensive experimen-

tation in transducer design). As bathymetric data must be corrected for refraction in the water

column, the CTD profile recorded during surveying is also presented, and the apparent and true

position of various rays are shown. Following this analysis system noise is modeled and accounted

for using a circuit representation of the receiver and relevant electronics. It is observed that the

noise present in the system during surveying is fully accounted for in the circuit model, with the

dominant noise source being due to the impedance of the circuit equivalent of the transducer ele-

ment. Arguments against the utilization of time variable gain are also made. Following the system

description, the data processing required for surveying is examined. In this respect the dominant

noise is irreducible. The only other noise source that could be lowered slightly (a couple dB) would

be to replace the broadband amplifiers circuits in the present system with narrowband amplifier

circuits that are matched to each specific transducer set. However, in practice this is avoided so

that the same amplifier electronics can be applied to several different transducer arrays, with only

minimal loss in snr being the trade-off.

A charting and navigation program that integrated GPS data with previous sonar navigation

measurements was also created, and demonstrated to run in real time with the use of only a laptop

and a conventional handheld GPS unit. This tool proved useful in assisting other researchers at

Pavilion lake, an example of which was performing AUV deployment to sites of interest underneath

the lake ice in winter. Through use of GPS at the lake, it has been determined that the accuracy to

which one of these units can be trusted (approximately 3-10m with wide area augmentation system

engaged, as on existing PLRP Garmin e-trex legend GPS units) is sufficient to locate targets on

the lake bed that are of comparable size to the GPS accuracy. Sonar survey data was also pivotal

in coordinating Deep-Worker missions held at the lake in subsequent years since the surveying was


completed.

Unfortunately, in order to realize the goal of direct comparison with the accuracy model presented

in chapters 2 and 3, more stable survey platform is required. The main conditions that prevented

a direct comparison of theory and experiment fall into two categories: sonar position/orientation

uncertainty, and ground-truthing. It was found that the orientation sensors available to the URL

were insufficient at capturing the sonar pitch and roll to a high enough degree as to make the actual

MASB AOA estimation the dominant mechanism in performance analysis. A pontoon boat mounted

setup, such as was employed in chapter 4 is subject to the effects of wave and wind action, making

small, but abrupt motions. Accounting for this motion using the present position and orientation

sensors proved impossible. A stationary mount is also not ideal to prove the results of chapter 3, as

the imaging is then limited to a single instance of scatterers, namely the same bottom is continuously

imaged. The desired data is that of a relatively featureless bottom, where adjacent pings can be used

to emulate multiple instances of the same bottom character. In the future it will be necessary to

use either a more accurate sensor, or deploy the MASB on a more stable platform, such as an AUV.

Aside from providing a smoother transition through the water, an AUV mount has the benefit over

a boat mount of being able to operate lower in the water column, away from both surface multipath

effects and below the sharp sound speed gradients which occur at thermoclines (or other sound speed

stratification). As to the problem of performing ground truthing, again it is preferential to have a

large surface of constant depth and composition. To ground truth some points in Pavilion Lake a

lead line method was attempted, in conjunction with Differential GPS. However during subsequent

attempts at sediment coring it was revealed that the section that had been identified as most ideal

for ground truthing and lead lined in Pavilion Lake due to its fairly constant composition was in fact

very unconsolidated sediment (so unconsolidated that every attempt to gravity core a sample failed),

and in all likelihood the lead line had been sinking to an indeterminate depth in the sediment.

However, the associated error arc length is shown for data collected within a single ping taken at

a different location, Sassamat Lake. Although many problems are acknowledged with the Sassamat

experiment, the EAL is at least qualitatively consistent with the spread observed in the AOA of

the bottom position over a large range. The concept of EAL is then expanded to encompass an

entire surface, and such a surface is produced for a sector scan of Sassamat Lake. The results of

chapter 4 indicate that MASB systems can provide an alternative to conventional sidescan in many

applications, supplying not only high density imagery, but also co-located bathymetry (with the

advantage that these two quantities can be resolved at different scales).

Chapter 2

The Signal

16

CHAPTER 2. THE SIGNAL 17

2.1 Introduction

At its core, the performance analysis of a sonar is derived from the understanding of the statistics

of the signals on the receive elements. To develop the signal model, the system must be broken

down into inputs (from both predictable and stochastic processes) and signal transformations. The

inputs for MASB are simply the backscattered signals from the underwater environment due to a

short transmitted pulse, and the Johnson thermal noise (a discussion of this is included in latter

sections of this thesis) present on each array element. The backscattered component of the signal

is comprised of the initial waveform, transformed by both by the physical environment (such as

spreading or refraction) and any signal processing that is necessary, for example matched filtering.

One important aspect of MASB applied to surveying, is that the signal present on the receiver at any

given time is comprised of a sum of contributions from an extended target (area of bottom usually)

as well as any other point or extended targets. The thermal noise in comparison is characterized

by the impedance and temperature of the piezoelectric receive element and pre-amplification circuit,

and is uncorrelated from element to element. The signals across an array can then be combined in a

signal vector, such as in Eq. 2.1, and as such will form the basis of the following analysis. It should

also be noted that physically the recorded signal χi, is the voltage induced on a piezoelectric element

i by the pressure on that element at a given time.

~χ(N×1) =

χ1

χ2

...

χN

=

s1 + n1

s2 + n2

...

sN + nN

(2.1)

In Eq. 2.1, an N element array is portrayed, with narrowband complex gaussian signal on element

i, si, and ni representing the uncorrelated complex gaussian noise recorded on element i. The

statistics of the individual signals si and noise ni will be discussed in the course of this chapter,

however it should be emphasized at this point that these representations are at complex baseband.

The analysis is mainly concerned with determining the approximations and conditions under which

the expectation E{sks∗i } can be computed, and how they relate to the transmitted pulse shape and

geometry of the ensonified environment.

In an ideal Multi-Angle Swath Bathymetric sonar, a pulse is transmitted from a transmit element,

spreads spherically as it travels away, scatters off a target (energy is spread in various directions),

and some portion of the original energy is received on at least one receive element. Though the

initial transmitted pulse length might not be short, the effective pulse length after filtering is applied

will be short in these systems, so as to maximize the resolution of the sonar imagery. This simplified

model, though illustrative, neglects many of the finer details of a sonar system. It is these details that


will be examined in this chapter. The analysis presented within this chapter addresses the effects

of several real world phenomena such as footprint shift, baseline decorrelation, the effects of having

signal contributions from two separated sections of the sea bottom (in principle this can be applied

to multiple signals from various directions), as well as the effects of gaussian noise, the transmit

and receive beampatterns and various contributions from the sonar equation. For this research

the MASB sonar transducers under examination form filled, linear receive arrays with constant

element-to-element separation (though the core concepts developed here can easily be applied to any

general phased array). The waveform transmitted is composed of an envelope function and a carrier

frequency. The carrier frequencies used in this research are between 100 → 300 kHz, however this

range of frequencies is merely a reflection of the practical frequencies used in surveying and can be

scaled appropriately for higher and lower frequency applications.

The first section defines the specific geometry for a sonar profiling measurement. This geometry

includes a single transmit element and allows for multiple sonar elements to be utilized for receive.

The physical model used in a MASB general survey scenario is described for the case of an array

tilted at a known angle. The ensonified area of the sea or lake floor, known as a footprint, for the

sonar measurement is also calculated for a given range cell.

In the second part of this chapter the signal formalism will be further developed and the covariance

matrix is described. The construction of a covariance matrix that includes not only the signal

correlations between multiple incoming plane waves, but also includes a gaussian noise contribution

is formally described, as it will be required in chapter 3.

Section three covers the correlation of signals corresponding to a general waveform. An uncorre-

lated bottom scattering function is assumed, and reasons for this to be the case are presented. The

definition of a normalized pulse function is given, and two normalized pulses are presented, namely

the square (SQ) pulse, and the finite Q (FQ) pulse. The matched filtered versions of both pulses

are also considered (MFSQ and MFFQ). In addition a compressed gaussian pulse is also described.

Furthermore the mean squared pulse length for each of the pulse functions is calculated, as this will

be required for comparison of the pulses in the analysis of chapter 3.

The fourth part of this chapter extends the results of the general correlation function to the 5 spe-

cific transmit waveforms defined in the third section. The first of these waveforms is the SQ pulse,

which is the simplest for which results can be derived analytically and interpreted. Additionally, the

SQ pulse is used to set the approximations that allow for an analytical closed-form solution to the

correlation. Simulations are then employed to confirm the proper behavior of the correlations, en-

suring that the approximations are valid under conditions similar to those utilized in surveying. The

correlation function of each signal is dependent on the shape of the originally broadcast waveform.


In considering all waveforms, general insights are gained as to the physical effects that effect decor-

relation across an array, such as footprint shift and baseline decorrelation, with the former playing

an important role in determining the performance of MASB (this will be demonstrated in Chapter

3). Also in this section, the effects of a secondary bottom return are demonstrated to significantly

effect the signal correlation through the use of both theory and simulation. Finally the contribution

of Gaussian noise, which is uncorrelated between different receive elements is discussed in the fifth

section.

The results of this chapter indicate that the signal correlations of a MASB system can be charac-

terized through the geometry of the environment, taking into account the specific waveform under

consideration. This chapter sets up the theoretical framework to compare the relative significance

concerning the effects of footprint shift, baseline decorrelation, uncorrelated noise on the elements,

and the incorporation of multiple signals in a single covariance matrix. In addition simulations are

used to confirm that the approximations presented are valid.


2.2 Physical Geometry

2.2.1 Geometry of Simple Bathymetric Measurement

Sonar Footprint - The Random Signal Component

In this section, the geometry of a bottom mapping system that utilizes MASB is described. Swath

bathymetry is a survey technique wherein a sonar moves along a track and images across the track

(i.e. perpendicular to the direction of travel). Each of these across-track slices is called a ping, and

by combining adjacent pings, one builds a 3D representation of the bottom geometry. To visualize

this scenario Fig. 2.1 demonstrates the relation of the profile to the sonar (In this case a 6 element

array is utilized). Following this, Fig. 2.2 demonstrates how multiple pings are employed to build up

the bottom image.

The systems under consideration fall under the category of monostatic sonar, which simply refers

to the condition that the transmitter and receivers operate from approximately the same location

(mounted on the same platform). In addition the model is prepared for a MASB system operating

in the far field, however relaxation of this condition will be discussed in the following section.

Fig. 2.3 illustrates that the pulse first propagates out, spreading spherically, and intersects the

bottom at nadir (see location 1), then travels along the bottom out into the non-nadir region (passing

through location 2, and then at a later time location 3 which although not shown will also consist

of a corresponding signal from the left side of the receive array). It can be observed that at first

the bottom return represents only one signal at the nadir point (again location 1), but the return

quickly resolves itself into two separate signals as it continues propagating (some time before it

reaches location 2). As previously mentioned, the short pulse length of the MASB system delimits

the footprint length, and this small angular extent facilitates resolution of both these signals.

To examine the geometry for mapping the bottom using a MASB system, Fig. 2.4 presents two

receive elements of an array located at 1 and i, where a finite length pulse is transmitted from

location 1 (here location 1 also serves as the transmit element). The pulse then travels outwards in

all directions, with power distribution given by the transmitter’s beampattern. At a given instant,

the pulse has traveled along the path shown as r1i to the bottom at xi, and ensonifies a portion of

the bottom known as the footprint area, and is scattered in all directions (with some directivity that

is not necessarily omni-directional). The portion of signal that is scattered back towards the sonar

along path r2i is appropriately called the backscatter, and is a function of the bottom backscatter

strength, the incident signal strength (at this part of the transmission, the signal can be considered

an acoustic wave) and the the directivity of the bottom at that location. It should be noted again

that bottom scatter is what makes si random (in Eq. 2.1). The sonar footprint refers to the ensonified

area on the sea bottom (or any extended surface) taken at a single instant, and is delimited both by


Footprint

Along-track Beamwidth

Single Element Beampattern

6 Element Array

Figure 2.1: Geometry of survey measurement.

the shape of the pulse in the acrosstrack dimension and by the narrow alongtrack beamwidth. Ideally

the strength of the signal from the bottom is the combination of the bottom scattering strength,

beam and pulse shape over the area of the bottom. However, as the alongtrack footprint dimension

does not change appreciably between the leading and trailing edges of the footprint in the acrosstrack

direction, the geometry can be reduced to a two dimensional model, in which contributions along

the alongtrack direction are accounted for over the length of the across track profile (In this research

this is a simple process in which the width of a footprint is given by the half maximum strength of

the narrow alongtrack beam as pictured in Fig. 2.1).

To simplify the model, a homogeneous medium of constant sound speed c is assumed for the

propagation of signals (the effects of a stratified sound speed gradient are ignored for the present


Figure 2.2: The progression of pings used as slices to build a 3D map. Note that in this survey,only 1 out of every 25 pings is displayed, so as to make the construction of a map easier to interpret(ordinarily pings are packed much closer, so as to cover the full bottom). The track of the sonarappears in magenta, and one image of the sonar is displayed, so as to register the orientation.

analysis, as they can be measured and accounted for through the use of a Conductivity Temperature

Depth, CTD, probe in actual experimental measurements, as demonstrated later using measured

data). For the purposes of analysis, all regions of the sonar profiling model geometry are considered

to be in the far field (this will be discussed later in this chapter).

The path length of the signal to the ith element is denoted by 2ri and is the sum of r1i and r2i

in Fig. 2.4. For the derivations provided in this work the range ri is considered to be associated

with half the delay t from transmission of the center of the pulse to the time of measurement,

namely ri = ct2 . However, as is explained later, use of the range variable is simply a mathematical

convenience, and not truly range in the proper sense. Another note regarding Fig. 2.4, the geometry

for only a specific instant of delay is shown, the footprint would consist of looking between the delays

associated with both the leading and the trailing edges of the pulse, as seen by each element. To

explore the consequence of having an extended target such as a footprint, Fig. 2.5 will now be used.

At one instance in time, the signal sampled on each element is for a slightly different bottom location

as indicated by the left side of Fig. 2.5, where receive elements i and k see different parts of the

bottom, note the transmitter is at location 1. This slight difference in footprint location is known


x →nadir region non−nadir region

h 1 1 2 2 3

Figure 2.3: Geometry for sonar profiling measurement demonstrating what is considered to be theregion near nadir, and that which is considered to be non-nadir.

as footprint shift, and is responsible for decorrelating the signals received on any two elements.

However, as the footprint is continuous over a small area, the right side of Fig. 2.5 shows that one

single point on the bottom will be seen on both receive elements as long as the path differences are

not longer than what is dictated by the pulse length. To further illustrate, if the backscatter on

element i from xi is weighted by a specific portion of the transmit pulse, the backscatter from xi on

element k will be weighted by a portion of the transmit pulse that is slightly shifted in time, ∆t.

The associated shift in time (for the two way path) between the two receivers can also be expressed

as an equivalent range delay across the array, ∆ri = c∆t2 . Fig. 2.6 also shows how the correlated and

uncorrelated sections of two footprints are seen on separated receive elements at a single moment.

In order for the signals to be correlated at all, the footprint and by association the pulse length must

be wide enough that it exceeds the effect of this path difference. Footprint shift will be considered

in the following sections for various waveforms.

For the development of the signal models to follow, it is necessary to determine ∆ri, as this relates

to the decorrelation between spaced array elements. From a geometric standpoint the range delay

across the array in Fig. 2.4 is defined by Eq. 2.2.


0 xi

h

γ

φi

di

i

1

ri

r1i

r2i

Figure 2.4: Geometry for footprint. In this scenario, a pulse is transmitted from element 1, travelsoutward, scatters off of the bottom and some of this scattered signal is backscattered toward eachof the array elements (in this case element i). Due solely to time of flight, the signal received ondifferent elements at the same time corresponds to slightly different locations on the bottom.

∆ri = ri − r1i(2.2)

Examining Fig. 2.4 also allows for derivation of a constraint equation between r1i, r2i, and ri,

namely Eq. 2.3.

r1i + r2i = 2ri (2.3)

Using Pythagoras’s equation, it is simple to find the lengths r1i and r2i.

r21i = x2

i + h2 (2.4)

r22i = (h + di cos(γ))2 + (xi − di sin(γ))2

= h2 + 2dih cos(γ) + d2i cos2(γ) + x2

i − 2dixi sin(γ) + d2i sin2(γ)

= x2i + h2 + d2

i + 2dih cos(γ)− 2dixi sin(γ)

(2.5)


0 xi

xk

h

γ

φiφ

k

di

dk−d

i k

i

1

r1k

r1i

r2i

(A)

r1i

+ r2i

= r1k

+ r2k

r2k

0 xi=x

k

h

γ

φi=φ

k

di

dk−d

i

(B)

k

i

1

r1i

=r1k

r2i

r2k r

1i+ r

2i ≠ r

1k+ r

2k

Figure 2.5: The left figure demonstrates the geometry that leads to footprint shift between elementsi and k. The right figure demonstrates the geometry where two elements see the same part of thebottom at different time delays. Note, the footprint for each element is not under consideration here,only the geometric conditions for the delta function in Eq. 2.28.

By re-arranging Eq. 2.4, x2i =

(r21i − h2

)can be substituted into the right side of Eq. 2.5 and

similarly by re-arranging and squaring Eq. 2.3, r22i =

(4r2

i − 4rir1i + r21i

)can be substituted into the

left side of Eq. 2.5. The result of these substitutions is Eq. 2.6.

(4r2

i − 4rir1i + r21i

)=

(r21i − h2

)+ h2 + d2

i + 2dih cos(γ)− 2dixi sin(γ)

4r2i − 4rir1i =d2

i + 2dih cos(γ)− 2dixi sin(γ) (2.6)

Eq. 2.6 can be re-arranged so that r1i is isolated on the left side, as portrayed by Eq. 2.7. Note

that no approximations have been employed thus far.

r1i =ri +−d2

i − 2dih cos(γ) + 2dixi sin(γ)4ri (2.7)

Examining the form of Eq. 2.7, there are several key points to note. First, the magnitude of d2i in

the bracketed term will be substantially smaller than 2h cos(γ) for all values of gamma less than 90◦

(at which point the last term in the brackets may also be larger than d2i ). Secondly, in examination of

Fig. 2.4 several ratios can be equated to trigonometric terms, namely hri

= cos(φi) and xi

ri= sin(φi).

Utilizing the afore mentioned relations, Eq. 2.7 can be re-represented as Eq. 2.8.


Figure 2.6: An illustration of correlated and uncorrelated sections of the footprint as seen on twoseparated elements, namely the footprint shift effect.

r1i ≈ri +di

2(sin(φi) sin(γ)− cos(φi) cos(γ))

≈ri − di

2cos(φi + γ)

(2.8)

Examining the form of Eq. 2.2, Eq. 2.8 is used to provide a closed form for the variable ∆ri, which

is incidentally the same path difference that is expected in most interferometric measurements under

the far field assumption.

∆ri ≈di

2cos(φi + γ) (2.9)

Eq. 2.9 holds to excellent agreement for any signal where ri À d (which is almost always the case

in surveying). In addition it should be understood that φ is a function of x, and therefore of r. It

then becomes apparent that the range delay itself is a function of the range. In addition to Eq. 2.2,

two similar relations between r1i, r2i, ri and ∆ri can be determined from Eq. 2.3 using Eq. 2.9.

∆ri = r2i − ri (2.10)

2∆ri = r2i − r1i (2.11)


Though the derivation is not shown, for completeness it is also useful to provide a value for the

location of the footprint on the bottom, xi, given a height h and tilt angle γ and element spacing

di. The resulting relationship is given in Eq. 2.12 (note, this expression can be found using Eqs. 2.3,

2.4 and 2.5).

xi =di sin (γ)

(4 ri

2 − di2 − 2 hdi cos (γ)

)± 2 ri

√(4 ri

2 − di2) (

4 ri2 − 4 hdi cos (γ)− 4 h2 − di

2)

8 ri2 − 2 di

2 (sin (γ))2

(2.12)

As demonstrated by the geometry shown in Fig. 2.7, Eq. 2.12 also indicates that there are two

solutions for horizontal range. The solution with the positive square root is the to the right of the

transducer in Fig. 2.7, whereas the other solution (negative root, located left of the transducer in

Fig. 2.7) will be encountered later and considered as a separate signal. Eventually as the pulse

travels out beyond the point at endfire to the array, here represented by −xlim in Fig. 2.7, the

returning signal beyond this range is due to just the one positive root signal. In the following work

it is demonstrated that the effect of having a second signal is detrimental to the estimation of the

primary signal, and so it would seem logical to set the tilt angle to zero. However, this would result

in the estimation of signals near nadir at endfire to the array, where little energy is transmitted

and sensitivity of the receive elements is low. In addition, near enfire the beampattern is rapidly

changing, which will aversely effect estimation. Therefore a tilt angle must be chosen to mitigate

the performance loss consequent of having two-signals, with the performance loss of estimation near

endfire.

To further clarify that this analysis has been defined in a manner suitable for a multiple element

array, it can be noted that for the ith element, the spacing is simply di = id, as is the case of most

interferometric systems. For the following research a value for the interarray spacing d is chosen to

be d = λ2 , however it should be noted that this is not a limitation on the analysis, and other array

spacings can be employed. Since di = id, it is also convenient for the following analysis to define an

adjacent element range delay, δri, using Eq. 2.13 such that ∆ri = iδri.

δri ≈d

2cos(φi + γ) (2.13)

Though a flat bottom is assumed, Fig. 2.8 illustrates that for a non-flat bottom, a local flatness

can be applied, and projected, such that a new value for h is computed to be h′. Similarly the tilt

angle must be also be adjusted. This process requires the estimation of a local grazing angle from

AOA estimates around the range cell in question, and is somewhat recursive (using AOA estimates

to find the performance of the estimate on AOA), however in principle this method could be iterated


x →xlim

−xlim

h

γ

2 signal region 1 signal region

Figure 2.7: Geometry demonstrating regions where 1 or 2 signals must be considered.

for robustness. In practice it is often found that the estimates of AOA are sufficient to compute the

local grazing angle, and as such these adjustments to h and the tilt angle are possible. It should

also be noted that as the curvature of the bottom is increased the local grazing angle becomes more

suspect, and eventually undulations of length on the order of the footprint will be difficult to handle.


Figure 2.8: Geometry for sloped bottom correction.


2.3 Correlation and the Covariance Matrix

The covariance matrix R(N×N) for the signal vector given in Eq. 2.1 is square and of size N ×N ,

and is defined by Eq. 2.14 (note that all signals and noise are uncorrelated, E{skn∗i } = 0 for all i, k).

R(N×N) = E{~χ(N×1)~χH(N×1)

}

=

E{s1s∗1}+ E{n1n

∗1} E{s1s

∗2}+ E{n1n

∗2} · · · E{s1s

∗N}+ E{n1n

∗N}

E{s2s∗1}+ E{n2n

∗1} E{s2s

∗2}+ E{n2n

∗2} · · · E{s2s

∗N}+ E{n1n

∗N}

......

. . ....

E{sNs∗1}+ E{nNn∗1} E{sNs∗2}+ E{nNn∗2} · · · E{sNs∗N}+ E{nNn∗N}

(2.14)

For signal ι, the corresponding contribution Rι(N×N) to the full covariance matrix is given by

Eq. 2.15.

Rι(N×N) =

E{sι1s∗ι1} E{sι1s

∗ι2} · · · E{sι1s

∗ιN}

E{sι2s∗ι1} E{sι2s

∗ι2} · · · E{sι2s

∗ιN}

......

. . ....

E{sιNs∗ι1} E{sιNs∗ι2} · · · E{sιNs∗ιN}

=

2σ2ιs 2σ2

ιsρι12e−jαι · · · 2σ2

ιsρι1Ne−j(N−1)αι

2σ2ιsρι21e

jαι 2σ2ιs · · · 2σ2

ιsρι2Ne−jαι

......

. . ....

2σ2ιsριN1e

j(N−1)αι 2σ2ιsριN2e

j(N−2)αι · · · 2σ2ιs

= 2σ2ιs

1 ρι12e−jαι · · · ρι1Ne−j(N−1)αι

ρι21ejαι 1 · · · ρι2Ne−jαι

......

. . ....

ριN1ej(N−1)αι ριN2e

j(N−2)αι · · · 1

= 2σ2ιsκι

(2.15)

Here αι is the electrical phase angle associated with an incoming plane wave signal ι, (i.e. the

phase difference between adjacent array elements for signal ι) and ριik is the correlation between

elements i and k for signal ι. The variance of the complex gaussian signal contains σ2ιs from both

real and imaginary components, hence the variance is given by 2σ2ιs. In Eq. 2.15 κι is known as

the correlation matrix for the signal ι alone and is useful in subsequent calculations that require the

derivative of the covariance matrix with respect to signal power. Since the angle of arrival of the

signal is θι, then for interarray spacing d and wavelength λ, αι is given by Eq. 2.16.


αι =2πd

λsin(θι) (2.16)

Assuming uncorrelated zero mean gaussian noise across the array results in expectations for the

noise of E{nin∗k} = δik2σ2

n (here again both the real and imaginary components of ni contribute σ2n,

and δik is the Kronecker delta function equal to 1 when indexes match and zero otherwise), and so

the contribution to the covariance matrix associated with noise alone is given by Rn in Eq. 2.17.

Rn(N×N) =

2σ2n 0 · · · 0

0 2σ2n · · · 0

......

. . ....

0 0 · · · 2σ2n

(2.17)

For K uncorrelated signals on the array, the full covariance matrix is the sum of all signals

contributions and noise, hence Eq. 2.18.

R(N×N) = Rn(N×N) + ΣKι=1Rι(N×N) (2.18)

Examining the case of having a single signal on the array will allow for further development, and

so the index ι can be dropped for the time being.

R(N×N) =

2σ2s + 2σ2

n 2σ2sρ12e

−jα · · · 2σ2sρ1Ne−j(N−1)α

2σ2sρ21e

jα 2σ2s + 2σ2

n · · · 2σ2sρ2Ne−jα

......

. . ....

2σ2sρN1e

j(N−1)α 2σ2sρN2e

j(N−2)α · · · 2σ2s + 2σ2

n

= 2σ2s

(1 +

1snr

)

1 ρnρ12e−jα · · · ρnρ1Ne−j(N−1)α

ρnρ21ejα 1 · · · ρnρ2Ne−jα

......

. . ....

ρnρN1ej(N−1)α ρnρN2e

j(N−2)α · · · 1

= 2σ2s

(1 +

1snr

)κn

(2.19)

The correlation matrix κn includes both signal and noise, and so the components of this matrix

will be used in the following analysis to illustrate the contributions to decorrelation between array

elements under various influences. The signal-to-noise ratio snr is defined by snr = 2σ2s

2σ2n

where ρn is

the correlation coefficient associated with noise alone.

ρn =snr

1 + snr(2.20)


The use of capital letters, i.e. the form SNR, will be used in the following analysis to denote

the decibel form of the signal-to-noise ratio, SNR = 10 log10(snr). It will be useful in further

developments to also acknowledge that Eq. 2.20 can be equivalently stated as Eq. 2.21.

snr =ρn

1− ρn(2.21)

The two relations presented in Eqs. 2.20 and 2.21 between the correlation and snr will be useful

in later analysis. For instance, following Eq. 2.21, the correlation brought about by geometric effects

for any two elements (i.e. by forming a two element array using arbitrary elements i and k) can

also be expressed as a signal-to-noise ratio, snrgeo (subscript geo is short for geometric), where ρn

is replaced by ρik. An equivalent signal-to-noise ratio snre can be then be defined by the product

ρe = ρnρik. Expanding snre leads to Eq. 2.22.

snre =ρe

1− ρe

=ρnρik

1− ρnρik

(2.22)

By inserting Eq. 2.20 in Eq. 2.22, , and by acknowledging that the correlation ρik is dependent on

geometric conditions (therefore ρik = snrgeo

1+snrgeo), a more illustrative expression can be obtained for

the equivalent signal-to-noise ratio.

snre =snr

1+snrsnrgeo

1+snrgeo

1− snr1+snr

snrgeo

1+snrgeo

=snrgeosnr

(1 + snr)(1 + snrgeo)− snrgeosnr

=1

1snrgeosnr + 1

snr + 1snrgeo

(2.23)

For high snr and snrgeo Eq. 2.23 can be well approximated to Eq. 2.24.

snre ≈ 11

snr + 1snrgeo

(2.24)

It will later be demonstrated that the dominant geometric effect is footprint shift, with the corre-

sponding signal-to-noise ratio, snrfs, and so for any two elements Eq. 2.24 becomes Eq. 2.25 (this

high snr form is also used in [23]).

snre ≈ 11

snr + 1snrfs

(2.25)


From the form of Eq. 2.19 it is evident that the effect of uncorrelated noise from element to

element has a different effect on performance than ρik. Specifically, ρn is the same for all pairs

of elements across the array, whereas ρik depends on the element spacing. It will be subsequently

demonstrated that ρik represents the decorrelation from geometric considerations such as footprint

shift and baseline decorrelation, with the former being the dominant effect for short pulse lengths in

the geometry considered [18] [1]. In addition, it will be shown that the larger the spacing between

elements (i.e. as |k − i| increases), the lower the correlation. Therefore, as the number of elements

on the array increases, the net performance increase against footprint shift becomes only slightly

incremental, with the performance increase of each element being lower than that of the effect of

adding the element before it. It will also be shown that the geometry of the mapping scenario

decorrelates the signals in such a way as to make the correlation matrix slightly non-Hermitian.

Though the magnitudes of the off-diagonal terms are symmetric, |ρik| = |ρki|, the phases will be

slightly rotated in a non-Hermitian manner.

The assumption is also made that all incoming signals are uncorrelated, which is true in practical

survey applications. Often secondary signals represent the effects of multi-path reflections from the

sea-surface or bottom, yet if the depth of the sonar is greater than the pulse length multiplied by the

number of snapshots employed, then the difference in total path lengths of the original signal and

reflected signal will be great enough to have the two signals uncorrelated. For short pulse lengths this

condition is assured. In addition the process of reflection serves to further decorrelate the signals.

As the performance of a sonar against noise depends on the snr, it should also be noted that

the performance is thus dependent not on the absolute signal level or absolute noise level, but the

relative behavior of these quantities.

Now that the form of the covariance matrix has been defined, it is necessary to examine the specific

correlation functions for various transmitted waveforms. Though the correlations will be different

for each waveform, the same general method can be applied to arrive at the signal correlations. The

specific details of these calculations will be covered in the following sections.


2.4 Development of the Signal Correlation Integral

for a General Pulse Function

A derivation of the signal correlation integral is made in this section for a general pulse function.

The general pulse consists of two parts, a scalar representing the energy in the pulse,√

Et, and a

pulse shape function, g(r) (where r is defined as ct2 , for some time t) and is centered at r = 0. Under

this definition, r is mathematical convention and not truly range since this definition implies that

half of the pulse has already been transmitted at zero delay.

The signal on element i can be found through inspection of Fig. 2.4. Using the convention of

integration over the domain of the bottom (ie. in x) one can show that the signal at element i is

given by:

si =∫ ∞

−∞

√Etg(rc − r1i + r2i

2)B(xi)e

2πjλ (r1i+r2i)dxi (2.26)

In Eq. 2.26, B(xi) is the bottom scattering function and rc is the range from the center of the

array to the center of the pulse on the seafloor. Note that xi is defined by the total range r1i + r2i,

and therefore the integration domain is defined by an expanding ellipse with the transmitter and

the receive elements at the foci. Even with separate elements being located along the same line,

individual ellipse shapes will be slightly different. This subtlety of the geometry is accounted for in

the following mathematical derivations (next section), and are demonstrated to fall within certain

approximations that allow for the ellipses in question to be represented as expanding circles. It is

also useful to observe that the different time delays for xi and xk will have an effect of decorrelating

the two signals through what is known as the footprint shift effect ([23] and [1]).

Utilizing Eq. 2.26, the signal correlation between elements i and k is given by:

E{sks∗i }=Et

∫ ∞

−∞

∫ ∞

−∞g(rc − r1k + r2k

2)g∗(rc − r1i + r2i

2)E{B(xi)B∗(xk)}e 2πj

λ (r1k+r2k−r1i−r2i)dxkdxi

(2.27)

Here the ranges of xi and xk include both positive and negative values. Therefore, the above

convention will include contributions from the bottom on both sides of nadir for non-zero tilt angles.

The maximum extent to which the signal from the negative x side will need to be considered is

located on the bottom at endfire to the array.

To eliminate one of the integrals in Eq. 2.27, one must first define the expectation of the bottom

scattering function. Following [38], one can assume a scattering function for the bottom that is

uncorrelated over its flat surface, with x1 and x2 being horizontal ranges. The random component of


this signal is solely dependent on the bottom backscattering function. This is equivalent to viewing

the bottom as composed of a uniform collection of point-like scatterers, each with its own spatially

independent complex gaussian amplitude.

E{B(xk)B∗(xi)} = E{|B|2}δ(xi − xk) (2.28)

This assumption of Eq. 2.28 should hold for most conditions, provided that the footprint of the

waveform on the bottom is sufficiently larger than the correlation length associated with any bottom

features.

Since ∆ri = iδri (the equivalent is true for index k), it is possible to insert Eqs. 2.11, 2.13 and

2.28 into Eq. 2.27 yielding Eq. 2.29.

E{sks∗i }'EtE{|B|2}∫ ∞

−∞

∫ ∞

−∞g(rc−r1k−kδrk)g∗(rc−r1i− iδri)δ(xi−xk)e

4πjλ (r1k+kδrk−r1i−iδri)dxkdxi

(2.29)

To evaluate one of the integrals in Eq. 2.29 through use of the delta function, it is useful to express

r1i, r1k, δri and δrk as functions of xi and xk (for instance φi = arctan(

xi

h

)).

E{sks∗i } 'EtE{|B|2}∫ ∞

−∞

∫ ∞

−∞g

(rc −

√h2 + x2

k − kd

2cos(γ + arctan(

xk

h))

)

× g∗(

rc −√

h2 + x2i − i

d

2cos(γ + arctan(

xi

h))

)δ(xi − xk)

× e4πj

λ

(√h2+x2

k+k d2 cos(γ+arctan(

xkh ))−

√h2+x2

i−i d2 cos(γ+arctan(

xih ))

)dxkdxi

(2.30)

Using the properties of the delta function to eliminate one of the integrals yields:

E{sks∗i } 'EtE{|B|2}∫ ∞

−∞g

(rc −

√h2 + x2

i − kd

2cos(γ + arctan(

xi

h))

)

× g∗(

rc −√

h2 + x2i − i

d

2cos(γ + arctan(

xi

h))

)

× e4πj

λ

(√h2+x2

i +k d2 cos(γ+arctan(

xih ))−

√h2+x2

i−i d2 cos(γ+arctan(

xih ))

)dxi

(2.31)

Re-expressing the integrand in terms of variables r1i and δri, Eq. 2.31 can be simplified to Eq. 2.32.


−∞g(rc − r1i − kδri)g∗(rc − r1i − iδri)e

4πjλ (k−i)δridxi (2.32)


For simplicity, the index i can be dropped from Eq. 2.32, and so r1i → r, δri → δr. By making the

substitution r =√

x2i + h2, one arrives at the form of the correlation that will be used to evaluate

all of the specific pulse waveforms, namely Eq. 2.33.


−∞g(rc − r − kδr)g∗(rc − r − iδr)e

4πjλ (k−i)δr r√

r2 − h2dr (2.33)

To integrate Eq. 2.33, one needs to consider that δr and the pulse function itself are both functions

of r. Therefore, to further the analysis, it is necessary to consider the various waveforms.

At this stage in the analysis it is important to note several points. Only two approximations

have been used in this analysis, namely the form of δr developed in Eq. 2.13, which is accurate for

all values of r in the far field, and the second approximation is the form of the bottom correlation

in Eq. 2.28. In addition it is useful to note that the order of i and k should be consistent in any

subsequent calculations. It will thus be necessary to consider all possible cases of i and k, namely

i = k, i < k and i > k.

2.4.1 Pulse functions

In the following development several waveforms are considered, specifically the square (SQ) pulse,

the matched filtered square (MFSQ) pulse, a pulse that rises and falls exponentially (this will be

known as a finite q pulse or FQ pulse and is a good representation of a practical sonar pulse as it

accounts for the transformative properties of sonar transducers), a match filtered FQ pulse (MFFQ)

and finally the compressed gaussian (CG) pulse. In the case of the FQ and MFFQ pulses Q refers

to the quality factor q which can be defined as the number of cycles of the carrier frequency that it

takes to achieve 95.7% of the maximum pulse value. This behavior is equivalent to the exponential

envelope of an underdamped second-order circuit operating at resonant frequency f , and requires

the parameter a = 2πf2q . The limiting form of the FQ pulse is found in the SQ pulse (limit of q → 0).

The SQ and FQ pulse functions are defined in such a way that the shape function is normalized in

in the square integrable sense, i.e.∫∞−∞ |g(r)|2dr = 1. All the pulse functions are defined such that

they are centered about zero, meaning that∫∞−∞ r|g(r)|dr = 0. Even though the FQ, MFFQ and the

Gaussian pulses are essentially long in extent, it is evident that away from the central pulse, noise

will eventually exceed the transmitted pulse contributions and it will be shown subsequently that all

the pulses can be characterized by a mean squared length.

The first 4 waveforms are self evident by their descriptions, and are formulated in piecewise manner

in Table 2.1. However to realize the CG pulse, it is necessary to begin with a chirp normalized

Gaussian pulse with a complex envelope, as given in Eq. 2.34 (note this is the same form used in

[38], pg. 290, and in [2]).


Name Pulse Shape

SQ gsq(r) =

0 r ≤ − rsq

21√rsq

− rsq

2 ≤ r ≤ rsq

2

0 rsq

2 ≤ r

MFSQ gmfsq(r) =

0 r ≤ −rsqr+rsq

rsq−rsq ≤ r ≤ 0

−r+rsq

rsq0 ≤ r ≤ rsq

0 rsq ≤ r

FQ gfq(r) =

0 r ≤ − rsq

2 − c2a√

2a

(1−e−

2arc− arsq

c−1

)

√2rsqa−c+ce

−2rsqac

− rsq

2 − c2a ≤ r ≤ rsq

2 − c2a

√2a

(e−

2arc

+arsq

c−1−e−

2arc− arsq

c−1

)

√2rsqa−c+ce

−2rsqac

rsq

2 − c2a ≤ r

MFFQ gmffq(r) =

c(−2e2ar

c +e2a(r+rsq)

c +e2a(−rsq+r)

c )

4arsq−2c+2ce−2arsq

c

r ≤ −rsq

4rsqa−2ce2ar

c +4ra+ce−2a(r+rsq)

c +ce2a(−rsq+r)

c


c

−rsq ≤ r ≤ 0

4rsqa−2ce2ar

c −4ra+ce2a(−rsq+r)

c +ce−2a(r+rsq)

c


c

0 ≤ r ≤ rsq

c(−2e−2ar

c +e−2a(r+rsq)

c +e−2a(−rsq+r)

c )


c

rsq ≤ r

CG gcg(r) = e

(−r2

4r2gc

)

Table 2.1: A summary of the centered pulse functions. The length of the SQ pulse is given by rsq,which is the number of cycles times the wavelength of a single cycle. In the case of FQ pulses, rsq

is again used to represent the number of cycles that would be encountered if q → 0, while c is thespeed of sound in water. The MFSQ and MFFQ pulses are matched filtered versions of the SQand FQ pulses. Finally, the CG pulse, along with both the variables a and rgc are defined in theaccompanying text. A plot of these pulse functions is given in Fig. 2.9.


gg(r) =(

1πr2

gs

)1/4

e−

(1

2r2gs−jb

)r2

(2.34)

In Eq. 2.34 rgs determines the range extent of the pulse, b is a variable that represents the

magnitude of the frequency sweep over the pulse. It is easily confirmed that∫

g∗g(r)gg(r)dr = 1 (for

rgs > 0, which is always the case). There are several features that make gaussian pulses interesting.

First it can be observed that the gaussian pulse represents the limiting form of matched filtering,

as a gaussian pulse convolved with itself results in another gaussian pulse. In addition analysis

of gaussian forms often allows for closed form solutions to be realized. Matched filtering is then

employed to realize the form of the CG pulse in Table 2.1, as demonstrated in Eq. 2.35.

gcg(r) = e−

(1+64b2

r4gs

c4

)r2

4r2gs = e

−r2

4r2gc (2.35)

Here, the ratio of the ranges of the pulse before and after compression is related by the compression

ratio cr = rgs

rgc=

√1 + 64b2 r4

gs

c4 .

Having now ascertained the forms of the five waveforms that will be considered in this research,

Fig. 2.9 displays the various shapes for values of the pulse parameters relevant for MASB surveying

(for instance the carrier frequencies typically used in this research are between 100 → 300 kHz, and

the speed of sound in water is assumed to be approximately 1450 → 1500 m/s).

2.4.2 Measures of Comparison Between Pulse Functions

For means of comparison, two characteristic measures must be computed for each waveform, these

are the mean squared range which defines the pulse extent in the water (the resolution), and the

relative energy between pulses. To compute the mean squared pulse length r2, a new scaling for

each pulse must be employed, where normalization consists of setting the area under each modified

pulse to one, g(r) is defined by the relation∫∞−∞ g(r)dr = 1 (note that this definition is for real,

non-negative pulses). The mean squared length is then given by Eq. 2.36.

r2 =∫ ∞

−∞r2g(r)dr (2.36)

In addition, as the normalizations of the various pulse functions differ (only the non-filtered wave-

forms are normalized to one), it is also important to examine the term∫∞−∞ |g(r)|2dr. This nor-

malization scales the energy in the transmitted signal, and so for meaningful comparisons of the

performance of different waveforms, both the energy and resolution must be considered. The effect

of matched filtering on noise will also be considered later. A summary of the various mean squared

pulse lengths and pulse normalizations is presented in Table. 2.2.


−0.1 −0.05 0 0.05 0.1 0.150

0.5

1

1.5

2

2.5

3

3.5

4Normalized Pulse Waveforms and Matched Filtered Waveforms

Range [m]

Figure 2.9: Waveforms considered in the course of this research. Shown are a SQ pulse of 20 cyclesat 300 kHz (dashed black line), the match MFSQ with same parameters (black line), a FQ pulse (reddashed line) with q of 10, and transmit length of 20 cycles (time between start of transmission andbeginning of exponential decrease), The corresponding MFFQ (red line), and finally a compressedgaussian pulse in green. A sound speed of 1450 m/s was also assumed for all waveforms.

Name∫∞−∞ |g(r)|2dr r2

SQ 1 r2sq

12

MFSQ 2rsq

3

r2sq

6

FQ 1 r2sq

12 + c2

4a2

MFFQ 15c3−60c3e2arsq

c +45c3e4arsq

c +12c2rsqa+32a3r3sqe

4arsqc −24c2rsqae

2arsqc −48c2rsqae

4arsqc

12a(4r2sqa2e

4arsqc −4crsqae

4arsqc +4crsqae

2arsqc +c2e

4arsqc −2c2e

2arsqc +c2)

r2sq

6 + c2

2a2

CG rgs

√2π 2r2

gc = 2r2gs

c2r

Table 2.2: A summary of normalizations and relevant mean squared lengths.


2.5 Correlation - Specific Waveforms

2.5.1 The Footprint Shift Effect for a Square Pulse

The footprint shift effect was first proposed to contribute to the decorrelation of signals in MASB

systems in [23]. However, results in the following research differ slightly from the previously reported

results, as mentioned in the previous chapter, because the value of the relevant shift length along

the bottom used in [23] is in error. It will be shown in this section that the form of the decorrelation

for footprint shift is simply the overlap of the waveform integrated with a shifted version of the

waveform. It should also be noted that the correlation calculation performed in this section for

footprint shift alone will be required in the following section to derive a more general correlation

function for the square pulse (by extension the results will also be applied to all subsequent pulses).

To begin, the expression for δri from Eq. 2.13 must be substituted into the exponential term of

Eq. 2.32.

E{sks∗i } ' EtE{|B|2}∫ ∞

−∞g(rc − r1i − kδri)g∗(rc − r1i − iδri)

× e4πj

λ ((k−i) d2 cos(γ+φ(xi)))dxi

(2.37)

A very simple approximation may then be made to solve for the correlation relating to footprint

shift, E{sks∗i }f , namely that cos(γ + φ(xi)) can be expanded around xi = xc which corresponds

to the center of the pulse on the bottom. Keeping only the first two terms of the expansion yields

Eq. 2.38.

cos(γ + φ(xi)) ≈ cos(γ + φ(xc))− 1h

sin(γ + φ(xc)) cos2(φ(xc))(x− xc) (2.38)

Inserting Eq. 2.38 into Eq. 2.37 results in Eq. 2.39.

E{sks∗i }f ' EtE{|B|2}e2πj(k−i)d

λ cos(γ+φ(xc))

∫ ∞

−∞g(rc − r1i − kδri)g∗(rc − r1i − iδri)

× e−2πdj

λh ((k−i) sin(γ+φ(xc)) cos2(φ(xc))(x−xc))dxi

(2.39)

From [18], it is seen that the exponential term in the integral leads to baseline decorrelation,

and the corresponding contribution to decorrelation is small for short pulses, over the range of x

considered in the survey model. Hence, the baseline decorrelation term can be ignored for the time

being in Eq. 2.39 (it will be derived later in the analysis for the full signal correlation function of the

SQ pulse) and the correlation simplifies to Eq. 2.40.

E{sks∗i }f ' EtE{|B|2}e2πj(k−i)d

λ cos(γ+φ(xc))

∫ ∞

−∞g(rc − r1i − kδri)g∗(rc − r1i − iδri)dxi (2.40)


To further the calculation of the signal correlation for the footprint shift effect it is possible to

use the relationship x2i = r2

1i − h2. This leads to the differential condition dxi = r1i√r21i−h2

dr1i. By

dropping the index i and redefining r1i → r it is now necessary to examine the Taylor expansion of

β(r) = r√r2−h2 about the point r = rc.

β(r) ≈ β(rc) + β′(rc) (r − rc) + O((r − rc)

2)

≈ rc√r2c − h2

− h2

(r2c − h2)

32

(r − rc) + O((r − rc)

2) (2.41)

In Eq. 2.41, O((·)η) refers to all terms of order η in (·). To determine the region of xc for which

the zeroth order term well approximates the function, the magnitude of the ratio of the first order

term (here order refers to power of (r − rc)) to the zeroth order term should be small.

∣∣∣∣∣∣O

((r − rc)

1)

O((r − rc)

0)

∣∣∣∣∣∣=

∣∣∣∣∣h2

(r2c − h2)

32

(r − rc)

√r2c − h2

rc

∣∣∣∣∣

=h2 |r − rc|

rc (r2c − h2)

(2.42)

By definition,√

r2c − h2 = xc. In addition, for the case of the square pulse of length rsq centered

about rc, it is observed that |r − rc| ≤ rsq

2 . For the first order term to be considered negligible it is

desired that∣∣∣∣O((r−rc)

1)O((r−rc)

0)

∣∣∣∣ < 0.1, and therefore limits are imposed on the range of xc in order for this

approximation to be considered valid. The relation rc ≥ h holds for any signal to be present (notice

that for minimum xc near nadir rc ≈ h), therefore the following useful inequality for xc is produced.

hrsq

2x2c

≤0.1√

5hNsqλ

2≤|xc|

(2.43)

In Eq. 2.43, rsq = Nsqλ2 . It can be similarly shown that for the matched filtered pulse |r − rc| ≤

rsq, which results in√

5hNsqλ ≤ xc. Though these are loose bounds on xc, they are useful in

demonstrating that the approximation holds for values of xc quite near to nadir. The zeroth order

approximation is inserted back into Eq. 2.40 through the relationship dxi ≈ rc√r2

c−h2dr = dr

sin(φc).

Dropping index i and again r1i → r, yields Eq. 2.44.

E{sks∗i }f ' EtE{|B|2}sin(φc)

e2πj(k−i)d

λ cos(γ+φ(xc))

∫ ∞

−∞g(rc − r − kδr)g∗(rc − r − iδr)dr (2.44)


It should be noted that the form of Eq. 2.44 will hold in principle for any of the waveforms

considered in this research, and is a general correlation function for computing the influence of

footprint shift. By examining Fig. 2.10, the proper integration limits can be discerned (for the case

of k > i) and as the SQ pulse function has simply a constant value, 1√rsq

, the integration is easily

performed in a few steps. In order to see the overlap, the pulse functions relevant on each element

have been plotted as functions of the relevant range, both shifted and mirrored in the range variable.

Figure 2.10: The integration domain is the overlap of two footprints as seen by different arrayelements.

The resulting SQ pulse signal correlation for the effect of footprint shift alone (including both

cases of i < k and i > k) is given in Eq. 2.45.

E{sks∗i}sqf=EtE{|B|2}

sin(φc)ej(k−i)α

(1−|k−i| |δrc|

rsq

)(2.45)

Following Eq. 2.19, Eq. 2.45 leads to a form of the decorrelation for footprint shift, Eq. 2.46, which

will be utilized in the following section to calculate a closed form solution for the general correlation

function of an SQ pulse.

ρsqf = 1−|k−i| |δrc|rsq

(2.46)

In Eq. 2.46 it should also be noted that the correlation is zero if |k−i||δrc| > rsq. In accordance

with Eq. 2.21, an equivalent signal-to-noise ratio can be defined for the footprint shift effect, and is

given by Eq. 2.47.

snrsqf =rsq

|(k − i)δrc| − 1 ≈ rsq

|(k − i)δrc| (2.47)


2.5.2 Development of the Full Signal Correlation for a Square Pulse

To evaluate Eq. 2.33, which is the more general case for signal correlation than just considering

the effect of footprint shift, for the case of a square pulse (and by extension to the other waveforms

under consideration) two approximations must be made. The first approximation was considered in

the previous section and resulted in the conditions under which r√r2−h2 ≈ rc√

r2c−h2

= 1sin(φc)

, which

will also be valid in Eq. 2.33. The second approximation requires the expansion of δr(r)) around

the same point, r = rc. As this term appears both in the exponent of the integrand (specifically

j 4πλ (k−i)δr(r)) of Eq. 2.33, and in the pulse function g(·), it would seem that a single approximation

should be applied to all instances of δr(r). However, this is unnecessary due to differences in scales of

the exponent and g(·). The phase term changes with δr(r) on the scale of only a single wavelength,

and so both zeroth and first order terms in (r − rc) are required for an adequate approximation.

In comparison, the pulse function is a low pass function at baseband in the frequency domain on

the scale of rsq, and as such, it will be demonstrated that contribution of the first order term is

negligible, and only a zeroth order term need be considered.

To begin this approximation, one takes a Taylor expansion of the term δr(r) = d2 cos(φ(r) + γ)

about r = rc. In a similar manner to the previous approximation, only the zeroth, δr(rc), and first

order terms in (r − rc) (specifically δr′(rc) (r − rc)) are retained.

δr(r) ≈ δr(rc) + δr′(rc) (r − rc) + O((r − rc)

2)

≈ kd

2cos(φc + γ)− u(r − rc) + O

((r − rc)

2) (2.48)

In Eq. 2.48, k is the number of interarray spacings separating the two receive elements, and u is

defined as:

u =d

2sin(φc + γ) cos(φc)√

r2c − h2

(2.49)

To examine the necessary conditions such that only the zeroth order term can be used to approxi-

mate δrc, by considering a two element array formed by elements i and k, the contribution to snrsqf

for each shift term must be calculated using Eq. 2.47 and Eq.2.25. From Eq. 2.47, the signal-to-noise

ratio for footprint shift is given by:

snrsqf ' rsq

|mδr(r)|' rsq∣∣md

2 cos(φc + γ)−mu(r − rc)∣∣

(2.50)


Here m = k − i, namely the number of interarray spacings d separating the two receive elements.

Examining Eq.2.25, which is used for high signal-to-noise ratios, as is required to see any effect from

the first order term, snre is given by:

snre =1

1snrn

+ 1snrsqf

=1

1snrn

+ |mδr(r)|rsq

=1

1snrn

+∣∣∣md cos(φc+γ)

2rsq− mu(r−rc)

rsq

∣∣∣,

(2.51)

where md cos(φc+γ)2rsq

and mu(r−rc)rsq

are the contributions from the zeroth and first order terms re-

spectively. In order to neglect the first order term, it must be shown to be less influential that

the contributions of either the zeroth order term (keeping in mind that the absolute value brackets,

which shall be dealt with accordingly) or thermal noise, to snre. For the sum or difference of the

zeroth and first order terms, one can write∣∣∣md cos(φc+γ)

2rsq− mu(r−rc)

rsq

∣∣∣ ≤∣∣∣md cos(φc+γ)

2rsq

∣∣∣ +∣∣∣mu(r−rc)

rsq

∣∣∣.For the purpose of considering the worst case scenario with regards to the first order contribution,

it is useful to note that snre is bounded by:

snre ≥ 11

snrn+

∣∣∣md cos(φc+γ)2rsq

∣∣∣ +∣∣∣mu(r−rc)

rsq

∣∣∣

≥ 11

snrn+ 1

snrsqf0+ 1

snrsqf1

,

(2.52)

where snrsqf0 =∣∣∣ 2rsq

md cos(φc+γ)

∣∣∣ and snrsqf1 =∣∣∣ rsq

mu(r−rc)

∣∣∣ are signal to noise ratios representing

the contributions of the zeroth and first order terms respectively. As the lowest of snrn, snrsqf0

and snrsqf1 will dominate snre, it is necessary to determine under what circumstances snrsqf1 can

be neglected. Using |r − rc| ≤ rsq

2 once again, it can be shown that snrsqf1 has a minimum value

snrsqf1 min, and it is given by Eq. 2.53.

snrsqf1 ≥∣∣∣∣

4xc

md sin(φc + γ) cos(φc)

∣∣∣∣ ≥∣∣∣∣4xc

md

∣∣∣∣ = snrsqf1 min (2.53)

Comparing thermal noise to snrsqf1 min, a value of xc can be determined, beyond which the first

order term is guaranteed to be insignificant to snre, due to the effects of thermal noise alone. For

positive xc:


snrsqf1 >snrn

4xc

md>snrn

xc >mdsnrn

4

(2.54)

The objective now shifts to consider for which constraints on xc is snrsqf0 < snrsqf1. Taking the

ratio of the two signal-to-noise terms for positive xc, with rsq = Nsqλ2 yeilds:

snrsqf1

snrsqf0>

snrsqf1 min

snrsqf0

>4xc

Nsqλ|cos(φc + γ)|

(2.55)

In order for snrsqf1 to be neglected, (2.55) should be greater than some constant C. Expanding

the cos(φc +γ) term in (2.55), and using the identities cos(φc) = hrc

and sin(φc) = xc

rc, then to neglect

snrsqf1:

xc

(h

rccos(γ)− xc

rcsin(γ)

)>

λNsqC

4(2.56)

In (2.56) on the left hand side, the term within the brackets is zero at broadside, and xc is zero

at nadir. Between these two limits exists a region for which the inequality holds, and snrsqf1 can

be neglected (it should also be noted here that a similar region exists beyond broadside, however

in that region it has already been demonstrated that at the very least snrsqf1 > snrn). One now

considers the effect of various tilt angles. For γ = 0, (2.56) becomes the following.

hxc

rc>

λNsqC

4(2.57)

Noticing that xc

rc= sin(φc), (2.57) becomes sin(φc) >

λNsqC4h . In this case, φc is very small, and so

sin(φc) ≈ tan(φc) = xc

h . This sets the minimum xc for the influence of snrsqf1 to be neglected for

γ = 0 to be:

xc >λNsqC

4(2.58)

Now for γ > 0 a range of xc must be determined over which snrsqf0 > snrsqf1, since cos(φc + γ)

in (2.55) can equal zero for a finite value of xc (unlike the case for γ = 0 where xc is infinite at

broadside and the region of interest is therefore only bounded on one side). As tilt angles larger than

45◦ are unlikely to be employed in swath mapping scenarios, the bounds for the region of interest


are determined for this angle. If γ = 45◦ then cos(γ) = sin(γ) = 1√2. Substituting these values into

(2.56) yields the following condition.

xc√2rc

(h− xc) >λNsqC

4(2.59)

Since only values of xc between 0 and broadside are being considered for the region of interest,

h < rc <√

2h and using the upper bound to obtain the tighter inequality, (2.56) becomes

xc (h− xc) >λNsqCh

2. (2.60)

It is then straight forward to solve the quadratic equation defined by (2.60) and therefore determine

the limits, xc min 2 and xc max 2, of the region for which the zeroth order term dominates over the first

order term.

xc min 2 =h

2− h

2

√1− 2λNsqC

h' λNsqC

2

xc max 2 =h

2+

h

2

√1− 2λNsqC

h' h− λNsqC

2

(2.61)

The approximations in Eq. 2.61 are valid in general as 2λNsqCh ¿ 1. Therefore, the first order

term is insignificant compared to the zeroth order term if xc min 2 < xc < xc max 2. On the other

side of broadside, the zeroth order term will eventually dominate again over the zeroth order term

for values of xc > h + λNsqC2 . So due to the zeroth order term alone, the first order term may

be ignored for most values of xc excluding a small region near nadir, and another near broadside.

However, it has already been demonstrated that thermal noise may be larger than the first order

term in the both region near broadside and beyond, and so the first order term may be neglected in

such circumstances.

Consequently, under the above set of conditions the approximations presented are valid. Denoting

δr(rc) as δrc and with the aid of Eqs. 2.48 and 2.49, Eq. 2.33 becomes:

E{sks∗i}=EtE{|B|2}

sin(φc)ej(k−i)α

∫ ∞

−∞g∗(rc−r−iδrc)g(rc−r−kδrc)e−j

4π(k−i)uλ (r−rc)dr. (2.62)

By examining Fig. 2.10, the proper integration limits can be discerned (for the case of k > i) and

as the pulse function is simply a constant, 1√rsq

, the integration is easily performed in a few steps.


E{sks∗i}sq =EtE{|B|2}rsq sin(φc)

ej(k−i)αrc

∫ rc−kδrc+rsq2

rc−iδrc− rsq2

e−j4π(k−i)u

λ (r−rc)dr

=EtE{|B|2}rsq sin(φc)

ej(k−i)α −λ

j4π(k − i)u

×(ej

4π(k−i)uλ kδrc− j4π(k−i)ursq

λ −ej4π(k−i)u

λ iδrc+j4π(k−i)ursq

λ

)

=EtE{|B|2}rsq sin(φc)

ej(k−i)α(1+(k+i)u

2 ) λ

2π(k − i)u

× sin(

2π

λ(k − i)u(rsq − (k − i)δrc)

)

(2.63)

Multiplying the numerator and denominator of (2.63) by (rp−(k−i)δrc), and using the interchange

of k and i to arrive at the solution for i > k, the above equation can be rearranged to determine the

general form of the correlation function for a square pulse.

E{sks∗i}sq =EtE{|B|2}

sin(φc)ej(k−i)α(1+

(k+i)u2 )

(1−|k−i| |δrc|

rsq

)sinc

(2λ

(k−i)u(rsq−|(k − i)δrc|))

(2.64)

The form of Eq. 2.64 leads to several insights as to the physical phenomena that contribute to

the total decorrelation of signals across an array for the SQ pulse. First, the complex exponent in

the exponential term is the electrical AOA for a single array spacing d, multiplied by the number of

interarray spacings k− i. However there is also an additional term, (k+i)u2 , because the phase center

of the two elements i and k may not be the center of the array. For small arrays (such as those

usually employed in MASB that contain 6 elements or less) away from nadir, u is small and this

factor plays little role in the decorrelation. It should also be noted that the presence of this term is

the only effect that makes the correlation matrix R non-Hermitian.

The sinc(·) in Eq. 2.64 is what was described in [18] and previously dismissed in the analysis of

the previous section as baseline decorrelation. It is a result of the change in phase shift over the

extent of the correlated part of the bottom. This term is exactly the same as that determined by

[18], with the exception of the |(k − i)δrc| term subtracted from the pulse length. This is due to

the consideration that the integration is only over the correlated portion of the footprint, which

was shown through the effects of footprint shift to get smaller as the array elements are separated.

Baseline decorrelation begins to effect signal correlations as the pulse length is increased, however it

is not of concern for short-pulse, high resolution sonars.


Finally, the remaining term, located before the sinc(·) in Eq. 2.64 is easily recognizable as arising

from footprint shift, as was demonstrated in Eq. 2.45, having been previously described in [23]

(although the calculation of the effect is different here, as the former used an incorrect calculation

for δx, here rectified in the δrc term). It is the result of different elements seeing slightly different

footprints at a single instant, as pictured in the left side of Fig. 2.5. The decorrelation due to this

term increases toward endfire as δrc grows, and is minimal at broadside to the array. For a SQ pulse

this effect can play a significant role, however it will be demonstrated in the following sections that

for other pulse shapes the footprint shift effect can be greatly mitigated. In essence the footprint

shift effect is the worst for a SQ pulse.

As the physical effects can be separated for the SQ pulse, as well as for other pulses, it is possible to

examine the contribution of each phenomena to performance loss. This examination is demonstrated

in the next chapter. In addition, it is necessary to consider the effect of having a second signal on

the array as demonstrated in Fig. 2.7, for horizontal ranges of |x| < xlim. The degree of correlation

across the array is negatively influenced by the addition of multiple signals, and the number of

degrees of freedom of an angle estimation procedure must be increased (the maximum degrees of

freedom of an array is one less than the number of elements, and represents the number of signals

that can be distinguished). The topic of degrees of freedom is covered in greater detail in Chapter

3, where methods of performance estimation are defined.

Finally it should also be noted that the effects of transmit and receive beampatterns must also

be included in any meaningful analysis of signal correlation. For the present development a root

cosine pattern is employed to model the beampatterns of real systems. The beamwidth of such a

system is approximately 120◦, and is at the upper end of realizable patterns (most used in practical

systems fall between 60◦ and 120◦, for example data collected from actual transducers is given later

in chapter 4 and the beamwidth is shown to be approximately 100◦). Therefore for a physical AOA

of θ, the directivity Be(θ) used in the theoretical analysis is scaled to maximum of one and defined

by Eq. 2.65.

Be(θ) =

{ √cos(θ) |θ| ≤ π

2

0 otherwise

}(2.65)

To account for the beampattern in subsequent analysis, the expression EtE{|B|2} in Eq. 2.64

must be multiplied by (Be(θ))4 to account for both the two way path for each signal (i.e. both

transmitter beampattern and receiver beampattern), and the fact that two elements are considered

in the correlation.

To demonstrate that Eq. 2.64 does in fact correctly model the correlation of signals across an array,

a simulation was created following the simple geometry modeled in the first section of this chapter.


For this and subsequent figures, the signal was normalized so that a pulse length on the bottom

yielded a power return of one, EtE{|B|2} = 1, and other effects such as terms of the sonar equation

will be examined in the next chapter. The bottom was modeled as a line of randomly distributed

scatterers (uniform probability density function) with an density of 300 scatterers per meter. This

value was chosen to be greater than the number used in [23], so as to avoid the scenario of using

too few scatterers to adequately model the bottom. Each scatterer was given a complex gaussian

amplitude, and these amplitudes were uncorrelated, independent from scatterer to scatterer along

the bottom. The effect of each scatterer was added to a data set at every range that it influences,

building a full ping. For each ping a completely new set of scatterers was generated, and 100 pings

were simulated. A result of this simulation for a tilt angle of 45◦ is shown in Fig. 2.11, where good

agreement is seen between the simulated autocorrelation and cross-correlation (separation d = λ2 ).

The real and imaginary components of the total signal are given in red and black respectively. As

expected, the received power is highest near nadir and decreases to a value near one. Blue and green

curves represent the real and imaginary component of the primary signal, which account for most

of the total signal. It can also be observed that the because the tilt angle is 45◦, and the altitude of

the sonar is 40m, the secondary signal contributes only out to a range of 40m because of the beam

pattern of the individual array elements. Finally, no noise is added to either the autocorrelation or

the cross correlation (even if noise were added, it is uncorrelated between elements, and therefore is

not expected to influence the cross correlation).


0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9

10SQ Pulse

horizontal range [m]

Cro

ss C

orre

latio

n (d

=0)

0 20 40 60 80 100 120−10

−8

−6

−4

−2

0

2

4

6

8

10SQ Pulse


Cro

ss C

orre

latio

n (d

= λ

/ 2)

Figure 2.11: Autocorrelation (left) and cross-correlation d = λ2 (right) for a 300 kHz square pulse

of 20 cycles, tilt angle of 45◦ and altitude of 40m, as measured from 100 pings. In left plot, theblue curve is for power received from primary signal alone, whereas the black curve represents thecontributions of signals from both in front of (primary), and behind (secondary) the sonar. Thedashed black line represents the simulation, and shows good agreement with theory. The right plotshows primary contributions, blue and green curves, to the real and and imaginary components ofthe cross correlation respectively. The black and red curves are the real and imaginary componentsof the combined primary and secondary signal correlations, and well represent the black and reddashed lines which are the corresponding simulated data.


2.5.3 Development of the Signal Correlation for a Matched Filtered Square

Pulse

As was demonstrated in the previous section, footprint shift plays large role in the decorrelation of

signals across an array for a SQ pulse, however it will be shown in this section, and in later sections

that this effect is greatly reduced for various other waveforms. For example, if a SQ pulse is match

filtered, then the resulting shape of the waveform is a MFSQ pulse which is triangular in shape,

as shown in Fig. 2.9. Therefore, the effect of footprint shift on such a system will be that of the

decorrelation of a triangular waveform. This is now investigated.

The two approximations made in the previous section, leading to both Eq. 2.44, and Eq. 2.62, are

valid for the case of a matched filtered square pulse (with only slight broadening to the unapplicable

domain of xc near nadir, as mentioned earlier). In this regard, the signal correlation integration is

performed in a similar fashion. However there are two regimes under which the integration must be

considered. These can be recognized most easily by examining the overlap of the pulse functions in

the integrand, as in Fig. 2.12. In most swath bathymetry systems |(k − i)δrc| < rsq. This can most

effectively demonstrated by considering that the pulse length is usually longer than the array width.

If instead |(k− i)δrc| > rsq, then only the center overlap region in Fig. 2.12 would remain, and the

limits would need to be altered accordingly.

Figure 2.12: The integration domain is again the overlap of two footprints as seen by differentarray elements. However, the waveform must now be considered as piecewise over three separateintegration domains as illustrated here.

Consider the first case |(k− i)δrc| < rsq, as pictured in Fig. 2.12. The signal correlation calculated

for a matched filtered square pulse using Eq. 2.62 is given below.


E{sks∗i}mfsq=EtE{|B|2}r2sq sin(φc)

λ3

43π3|k − i|3u3ej(k−i)α(1+

(k+i)u2 )[4 sin

(2π|k−i|u(2rsq−|k−i||δrc|)

λ

)

+8πu|δrc|(k−i)2

λcos

(2π(k−i)u(2rsq−|k−i||δrc|)

λ

)−8 sin

(2π|δrc|u(k−i)2

λ

)

− 16π

λ|k−i|u(rsq−|k−i||δrc|)cos

(2π|δrc|u(k−i)2

λ

)]

(2.66)

In the case of the autocorrelation k = i, and E{sis∗i}mfsq can be calculated by either applying

L’Hopital’s rule to (2.66) and calculating the third derivatives of the numerator and denominator,

or alternatively one can just perform the original integration knowing the weight given by pulse

normalization. Both methods yield:

E{sis∗i}mfsq=

EtE{|B|2}sin(φc)

2rsq

3(2.67)

Unfortunately, the effects of footprint shift and baseline decorrelation do not separate in Eq. 2.66

as they did in the case of a SQ pulse, and so to examine the scenario where u → 0 (i.e. neglecting

the effect of baseline decorrelation) either L’Hopital’s rule must be applied to Eq. 2.66, and the third

derivatives of the numerator and denominator are required, or the original integration is performed

without any exponential term in the integrand by way of Eq. 2.44. In either calculation the signal

correlation becomes:

E{sks∗i }mfsqf =EtE{|B|2}

sin(φc)ej(k−i)α 2rsq

3[1− 3

2(k − i)2

δr2c

r2sq

+34|k − i|3 |δrc|3

r3sq

]. (2.68)

It is interesting to note that unlike the case of the footprint shift in the SQ pulse, the correlation

in Eq. 2.68 does not drop to zero over the range of |(k− i)δr| < rsq. This is because the MFSQ pulse

is non-zero over twice the length of rsq.

For completeness, a development is given for the scenario of when either a large array is employed,

or an extremely short pulse. Here the relevant condition is that |(k − i)δr| > rsq. In Fig. 2.12,

the pulses separate such that only the center region will contribute to the final signal correlation.

The development is the same as that above, and so the resulting correlation under these conditions,

E{sks∗i }mfsq, is given by:


E{sks∗i }mfsq=

EtE{|B|2}r2sq sin(φc)

(λ3

43π3|k − i|3u3

)ej(k−i)α(1+

(k+i)u2 )[4 sin

(2π

λ|k − i|u(2rsq − |(k − i)δrc|)

)

− 8πu|k − i|λ

(2rsq − |(k − i)δrc|) cos(

2π

λ|k − i|u(2rsq − |(k − i)δrc|)

)]

(2.69)

Finally, if there is no contribution from baseline decorrelation, then the signal correlation is given

by:

E{sks∗i }mfsqf=

EtE{|B|2}sin(φc)

ej(k−i)α 2rsq

3

[2

(1− |k − i|

2|δrc|rsq

)3]

(2.70)

To compare the decorrelation due to footprint shift for the MFSQ pulse with those obtained earlier

for the SQ pulse, it is again useful to examine the equivalent snr due to footprint shift. Following

the methods leading to Eq. 2.46, the decorrelation for the MFSQ pulse is given by Eq. 2.72 (note

the relative effects of normalization have been scaled appropriately from Table. 2.2).

ρmfsqf = 1− 32(k − i)2

δr2c

r2sq

+34|k − i|3 |δrc|3

r3sq

(2.71)

By assuming that |(k − i)δr| ¿ rsq, as in the case of most survey geometries, the cubic term in

Eq. 2.72 can be ignored, and using Eq. 2.21, the equivalent snr for the footprint shift is found to be:

snrmfsqf =23

r2sq

(k − i)2δr2c

− 1 ≈ 23

r2sq

(k − i)2δr2c

(2.72)

Comparing Eq. 2.72 to Eq. 2.47 results in Eq. 2.73.

snrmfsqf ≈ 32snr2

sqf (2.73)

In decibel form Eq. 2.73 is given by SNRmfsqf ≈ 2SNRsqf − 1.76. To examine this in practical

terms, for a 20 cycle pulse rsq = 20λ2 , the maximum value of δrc with an array spacing of d = λ

2 is

δrc = λ4 . The corresponding values of equivalent signal-to-noise due to footprint shift are SNRsqf =

16 dB and SNRmfsqf = 30.3 dB, thus the detrimental effect of footprint shift is greatly reduced by

matched filtering.

To confirm the results given in Eqs. 2.68, 2.66, a simulation was again performed. The same

number of scatterers and pings were generated as in Fig. 2.11. The results are shown in Fig. 2.13,

and demonstrate good agreement between the theoretical and simulated autocorrelation and cross-

correlation.


0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1MFSQ Pulse


Cro

ss C

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latio

n (d

=0)

0 20 40 60 80 100 120−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1MFSQ Pulse


Cro

ss C

orre

latio

n (d

= λ

/ 2)

Figure 2.13: Autocorrelation and Crosscorrelation for match filtered square pulse. The same param-eters were employed as for Fig. 2.11. In left plot, the blue curve is for power received from primarysignal alone, whereas the black curve represents the contributions of signals from both in front of(primary), and behind (secondary) the sonar. The dashed black line represents the simulation, andshows good agreement with theory. The right plot shows primary contributions, blue and greencurves, to the real and and imaginary components of the cross correlation respectively. The blackand red curves are the real and imaginary components of the combined primary and secondary signalcorrelations, and well represent the black and red dashed lines which are the corresponding simulateddata.


2.5.4 Development of the Signal Correlation for a Finite Q Pulse

As was previously mentioned, the FQ pulse represents a closer approximation to the performance of

actual sonar transmit transducers than a SQ pulse. This is due to the use of a quality factor which

models the damped driven resonator behavior of the acoustic piezoelectric transmitter in MASB

sonar. In a similar manner to the SQ and MFSQ pulses, the signal correlation for the FQ pulse can

be calculated from Eq. 2.62, by using the form of the FQ pulse shape in Table. 2.1. Again a piecewise

approach must be used to carry out the calculation, which can be most adequately performed using

a symbolic math program to help track and simplify terms. The result of this calculation is given in

Eq. 2.74 for the case of k 6= i, and |(k − i)δrc| < rsq.

E{sks∗i }fq =− 4 EtE{|B|2} ej(k−i)α (1+1/2 (k+i)u)ejnrca2[2 sin (1/2 n (rsq − kdrc + iδrc)) n2c2

+ 16 j sin (1/2 n (rsq − kδrc + iδrc)) nca− 32 sin (1/2 n (rsq − kdrc + idrc)) a2

+(n2c2j − 4 nca

)(−e1/2

nkdrc cj−jncidrc−jnrsq c−4 akdrc+4 aidrc

c + e−1/2(rsq−kdrc+idrc)(4 a−jnc)

c

− e1/2nkdrc cj−jncidrc+4 aidrc+jnrsq c−4 akdrc

c + e−1/2(rsq+kdrc−idrc)(4 a−jnc)

c )]

× e1/2 jndrc (k+i)e1/2−2 jncrc a−jnc2+4 rsq a2

ca (sin (φc))−1

n−1[−12 jnca2rsq e2rsq a

c

+ 6 jnc2ae2rsq a

c − 6 jnc2a− 2 n2c2rsq ae2rsq a

c + n2c3e2rsq a

c − n2c3 + 16 a3rsq e2rsq a

c

− 8 a2ce2rsq a

c + 8 a2c]−1 (4 a− jnc)−1

(2.74)

In Eq. 2.74 it is convenient to use the recurring term n = 4π(k−i)uλ to shorten the form of the

equation. For the case of k = i, one can again examine Eq. 2.62, with the simplification that the

exponential in the integrand is zero. Furthermore, as the pulse is normalized, the integral is simply

one, and the form of the autocorrelation is given by Eq. 2.75.

E{sis∗i}mfsq=

EtE{|B|2}sin(φc)

(2.75)

As was found for the case of the MFSQ pulse, the effects of FS and BD cannot be extracted

separately upon inspection from the general correlation. Calculation of the correlation corresponding

to FS only can in principle be performed through Eq. 2.44, however for the current analysis it is

not necessary (the performance of the FQ pulse will be shown to be similar to the MFSQ and CG

pulses). As in the case of the previous two pulses, a simulation was created using the same number

of scatterers and pings as were generated as in Fig. 2.11 and Fig. 2.13. The results are shown in

Fig. 2.14, and demonstrate good agreement between the theoretical and simulated autocorrelation

and crosscorrelation.


0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9

10FQ pulse


Cro

ss C

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latio

n (d

=0)

0 20 40 60 80 100 120−10

−8

−6

−4

−2

0

2

4

6

8

10FQ pulse


Cro

ss C

orre

latio

n (d

= λ

/ 2)

Figure 2.14: Autocorrelation and Crosscorrelation for a FQ pulse. The same parameters wereemployed as for Fig. 2.11. In left plot, the blue curve is for power received from primary signal alone,whereas the black curve represents the contributions of signals from both in front of (primary), andbehind (secondary) the sonar. The dashed black line represents the simulation, and shows goodagreement with theory. The right plot shows primary contributions, blue and green curves, to thereal and and imaginary components of the cross correlation respectively. The black and red curvesare the real and imaginary components of the combined primary and secondary signal correlations,and well represent the black and red dashed lines which are the corresponding simulated data.


2.5.5 Development of the Signal Correlation for a Matched Filtered Finite

Q Pulse

The correlation for MFFQ pulse was calculated using symbolic mathematics software, namely

Maple 11. The correlation needed to be constructed in a piecewise manner, similar to the previous

piecewise waveforms. Unfortunately the closed form of the signal correlation is far too long to be

included in this text, and as such does little to provide any insight as to the contributions of various

geometric effects that contribute to decorrelation. However, the calculated signal correlation can be

plotted and compared with a simulation as with the previous waveforms. The simulation was created

using the same number of scatterers and pings as were generated as in Fig. 2.11 and Fig. 2.13. The

results are shown in Fig. 2.15, and again demonstrate good agreement between the theoretical and

simulated autocorrelation and crosscorrelation.

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1MFFQ pulse


Cro

ss C

orre

latio

n (d

=0)

0 20 40 60 80 100 120−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1MFFQ pulse


Cro

ss C

orre

latio

n (d

= λ

/ 2)

Figure 2.15: Autocorrelation and Crosscorrelation for a MFFQ pulse. The same parameters wereemployed as for Fig. 2.11. In left plot, the blue curve is for power received from primary signal alone,whereas the black curve represents the contributions of signals from both in front of (primary), andbehind (secondary) the sonar. The dashed black line represents the simulation, and shows goodagreement with theory. The right plot shows primary contributions, blue and green curves, to thereal and and imaginary components of the cross correlation respectively. The black and red curvesare the real and imaginary components of the combined primary and secondary signal correlations,and well represent the black and red dashed lines which are the corresponding simulated data.


2.5.6 Development of the Signal Correlation for a Gaussian Pulse With

Pulse Compression

While a true Gaussian pulse is not realizable because of its infinite extent in time, it is useful

to consider the Gaussian pulse in the context of swath bathymetry. As was mentioned earlier, the

gaussian pulse is not only the limiting form in terms of match filtering, but it is also amenable to

closed form analysis, and therefore is useful in gaining insight into relationships between the factors

that influence performance.

In the same manner as for the previous pulse functions, Eq. 2.62 is used to derive the signal

correlation for the compressed gaussian pulse. The resulting equation is given in Eq. 2.76.

E{sks∗i }cg =EtgE{|B|2}

sin(φc)rgs

√2π

cτej(k−i)α(1+

(k+i)u2 )e

− (k−i)2δr2c2τ8r2

gs e− r2

gsu2

2c2τ (2.76)

Much like the SQ pulse, the effects of footprint shift and baseline decorrelation can be separated

for the case of a CG pulse. Using Eq. 2.40, the correlation function for the CG pulse corresponding

to only footprint shift, E{sks∗i }cgf , is calculated to be given by Eq. 2.77.

E{sks∗i }cgf =EtgE{|B|2}

sin(φc)rgs

√2π

cτej(k−i)αe

− (k−i)2δr2c2τ8r2

gs (2.77)

The two separated real exponential terms in the signal correlation Eq. 2.76 can thus be identified

as the contributions from FS and BD. This will be of use in the next chapter, where the relative im-

portance of FS and BD will be compared in the context of angle estimation performance. Comparing

the expression for E{sks∗i }cg with E{sks∗i }sq from Eq. 2.64, one notices that the complex exponential

term is the same, however the remaining real terms for footprint shift and baseline decorrelation are

different.

To validate Eq. 2.76, as in the case of previous waveforms, a simulator was constructed with the

same parameters as used in Fig. 2.11, to confirm the closed form of the autocorrelation and cross

correlation functions for the CG pulse (here cτ was set to one, and as such no pulse compression was

utilized). The results of this simulation are given in Fig. 2.16, and show excellent agreement with

the theoretical signal correlation.


0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CG Pulse


Cro

ss C

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n (d

=0)

0 20 40 60 80 100 120−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1CG Pulse


Cro

ss C

orre

latio

n (d

= λ

/ 2)

Figure 2.16: Autocorrelation and Crosscorrelation for a CG pulse. The same parameters wereemployed as for Fig. 2.11. In the left plot, the blue curve is for power received from primary signalalone, whereas the black curve represents the contributions of signals from both in front of (primary),and behind (secondary) the sonar. The dashed black line represents the simulation, and shows goodagreement with theory. The right plot shows primary contributions, blue and green curves, to thereal and and imaginary components of the cross correlation respectively. The black and red curvesare the real and imaginary components of the combined primary and secondary signal correlations,and well represent the black and red dashed lines which are the corresponding simulated data.


2.5.7 Uncorrelated Gaussian Noise Contribution

In addition to the signals received on the elements, there is also noise that is present on each

of the receivers. Noise can be from both acoustic and electrical sources. However, for the high

frequency sonar systems considered in this research, noise will be considered as arising only from

thermal noise on the receiver elements. Thermal noise [33] appears on all elements of the array. This

noise is considered to be uncorrelated between elements of the array, and hence will only appear on

the autocorrelation of each element which is along the diagonal of the covariance matrix (ie. only

contributes to E{χiχ∗i }). The characteristics of thermal noise are assumed to be white and complex

gaussian, with the same value of variance 2σ2n on all elements of the array (again the variance has

contribution from both the real and imaginary components of the noise). It will be assumed that for

the instrumentation used in this research, all array elements have the same noise contribution (this is

verifiable for practical survey systems). In practice, several other mechanisms can contribute to the

interference on a sonar array, including cross-talk between elements and unaccounted extra signals

on the array (for instance multipath returns that include several reflection or scattering events, and

as such are too week to be detected as an impinging signal on the array). These additional effects are

only mentioned here for transparency, however they will not be considered in the present analysis.

The absolute value of the noise variance will be discussed again later in chapter 4, in the context

of the experimental system used in this research. For the present it is enough to maintain that the

noise is uncorrelated between the elements and can at least be approximated as having equal variance

on all array elements for well made systems. Thermal noise arising from an impedance Z, which

represents the equivalent impedance of the entire receive circuit (not just the transducer element),

is given by Eq. 2.78

E{nin∗i } = 2σ2

n = 2(4kBT<{Z}∆f) (2.78)

where kB is Boltzman’s constant and T is simply the temperature of the receiver measured in

Kelvin and <{Z} represents the real component of the complex impedance Z. In Eq. 2.78 case ∆f

can be considered to be the bandwidth of the filter that is applied to the final data set, as the noise

level is found to be fairly constant over narrow bandwidths.

Finally it should be noted that the exact cause and value of noise in this research is not necessarily

required, as it can often be measured. In addition, exact calibration of sonar transducers in the

context of absolute target strength is rarely done in practice for MASB systems. Calibrated targets

are not often available, and the exact method of calibration must be performed in the far field

which is a difficult task for highly directional transducers [13]. MASB systems in particular use

long thin transducers which consequently have extremely narrow along-track beamwidths, and thus

measurements must be performed at far ranges. If D is the maximum length of the transducer in


question, the far field is given in [43] as rfarf ield > 2D2

λ (it is also recognized that other definitions

of the far field exist, for example [44] gives the far field as rfarf ield > πD2

λ ). Thus for a 0.46m

piezoelectric element resonant at 205kHz the far field range is rfarf ield ≥ 57.8 m. In comparison, a

0.30m 300kHz transducer needs only 36m to be in the far field. However to find an acoustic test bed

of these distances is quite daunting, and the process of aligning fixed mounts to do the calibration

properly can be vexing. Thus one is often more concerned with the ratio of signal to noise than the

level of either the signal or noise.


2.6 Summary of Chapter 2

The goal of this chapter was to define the signals encountered in MASB sonar surveying corre-

sponding to several waveforms. In so doing there have been several accomplishments that are unique

to the research in this thesis. In order to achieve the aforementioned goal, the physical geometry of

a sonar measurement was first defined. In particular, the details of a mechanism known as footprint

shift were explained. Although this effect was not discovered during the course of this research, prior

work in [23] was predicated upon a shift in the footprint that does not concur with these results.

Therefore the recognition and proper calculation of the shift in footprint can be considered original

in the context of this research. Although the value δri is equivalent to what is often encountered in

other interferometric measurements, its formulation here is unique.

Following the physical geometry, the signal formalism for MASB is presented, including the defi-

nition of the relevant covariance matrix. The matrix algebra in itself is not unique to this research,

and the use of an effective snr to model the combined decorrelation due to the combination of noise

and geometric effects was used in [23] in the context of simple interferometric sonar. However, the

recognition of a high snr approximation to combine geometric decorrelation (for instance footprint

shift) with gaussian noise for larger arrays (by consideration of individual element pairing) is unique

to this research, and its subsequent application to the problem of performance estimation for angle

of arrival in the following chapter will emphasis the true importance of this method.

Five waveforms were chosen for analysis in this thesis, two unfiltered pulses (SQ and FQ wave-

forms), and three filtered pulses (MMSQ, MFFQ and CG waveforms). In particular the SQ pulse

was chosen because of its simplicity, and its susceptibility to the footprint shift effect. The second

pulse considered was a natural extension of the SQ pulse is its match-filtered counterpart, MFSQ

pulse. The FQ and MFFQ pulses were defined and analyzed to address waveforms that are repre-

sentative of what current MASB systems employ in practical survey applications. Finally, the CG

pulse was addressed because firstly it was determined to represent the limiting form of any filtered

pulse (being that a Gaussian filtered Gaussian pulse remains Gaussian) and secondly it lends itself

readily to closed form analysis. To examine these different waveforms, two measures of comparison

were defined in this research, the relative pulse normalization and the mean squared pulse length.

The recognition of these measures is unique to this thesis as is the subsequent tabulation for the

waveforms that are considered.

Several authors have previously attempted to address the various effects that contribute to the

performance of angle estimation in MASB systems (notably [8] [9], [18] and [23] ), but none have

defined correctly the correlation function between the signals on separated elements, including both

baseline decorrelation and footprint shift for any of the waveforms in this thesis (only prior work was

for the SQ pulse in [23], however those results are in error due to the use of an incorrect footprint


shift separation). This represents a significant improvement in the analysis of swath bathymetric

systems, as the signal correlations are the basis for performance analysis. In addition, also presented

in this chapter are comprehensive explanations of the approximations used to calculate the signal

correlations. Of distinction in this chapter is the recognition that for the SQ and CG pulses, the

effects of baseline decorrelation and footprint shift can be separated in the closed form of the signal

correlation. This will have consequences in the next chapter, when the contributions of each geometric

effect is evaluated in the context of angle estimation performance.

Chapter 3

Performance - Theory and

Simulation

64

CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 65

3.1 Introduction

Since all estimators are functions performed on data, they are at the very least subject to ran-

domness in the measurement variables. As such, the estimates become random variables themselves,

and are therefore each described by a probability density function (pdf). An important point to

make when considering estimator behavior is that the parameter being estimated does not need to

itself be a random variable. For instance, in the AOA problem only the bottom scattering strength

(E{|B2|}) and the thermal noise are random, the actual AOA, α, is a non-random variable. However

the estimate of AOA is itself a random parameter, α.

In examining the probability that an estimate will be within a given interval, the analytical

confidence can be computed only if the pdf of the estimator is known. This confidence interval

is generally taken about some mean value. It is most beneficial if this mean value represents an

unbiased estimate of the desired variable, however, in the case of some estimators under non-ideal

conditions such as low snr or unfavorable geometry, an estimator for the AOA can become biased.

The second moment of the probability distribution about the mean is the variance (i.e. second central

moment), which fully captures the performance of an estimator for a gaussian random variable. This

is useful, because the central limit theorem ensures that the pdf of the estimate of the sum (or

average) of many random variables tends to the gaussian distribution. However, in the case of a low

number of snapshots (instances of a measurement) that can be combined coherently in a way other

than straight averaging, for the AOA estimation process, the central limit theorem is not met. In

addition, as can be seen in [25], some of the distributions of estimators for AOA can have significant

weight in the tail regions. It will be demonstrated in this chapter that the standard deviation of an

estimator does not define a useful confidence interval, as the probability that an estimate falls within

a number of standard deviations from the mean becomes highly dependent on the snr. Finally, for

the performance analysis of most useful estimators in the AOA finding problem, it is generally found

that the calculation of the probability distribution is prohibitively difficult for arrays larger than two

elements and for more than one snapshot.

Aside from measuring performance using the analytically derived pdf, which is different for each

estimator, there also exists estimator independent measures of performance. One of these is the

Cramer-Rao lower bound (CRLB), which sets a lower limit on the variance of an estimator. In this

research, the variable of interest is the AOA which has an associated variance σ2α, and the lower

bound on the variance of the AOA is CRLBα. To obtain the largest bound, which is the most

useful, one must take into account all unknown parameters ( see [39]). In the case of K plane waves

(where K is greater than one) impinging on a N element array (here K ≤ N − 1), one must include

at least two parameters for each plane wave, the AOA and the signal-to-noise ratio. In addition

[37] and associated references also include the cross-correlations between the incoming signals as


additional parameters. Inclusion of more parameters generally will result in a higher CRLB, which

is potentially tighter to the actual variance. However for this research there are only up to two plane

waves considered, and for reasons discussed both earlier and in the following work the signals can be

considered in large part uncorrelated. It should also be emphasized that in the following research the

CRLBα is itself the quantity of interest, and will play an important role in determining confidence

intervals for estimator performance, regardless of whether or not the CRLBα is much lower than the

variance. The applicability of this use of the CRLBα will be discussed and expanded upon later in

this chapter.

For the MASB configuration considered in this thesis, the correlation between two signals can

be generally disregarded for the following reasons. For the downlooking configuration, two signals

come from separate locations on the bottom and so the condition in Eq. 2.28 ensures uncorrelated

signals. In the sidelooking configuration, for a single snapshot with a short pulse length (in this

case the pulse length must only be less than half of the transducer depth), the footprint is typically

much shorter than footprint shift between the surface image receiver, and true receiver, unless the

bottom is right at surface, in which case signals are generally indistinguishable (only one angle is

estimated). In addition, the physical process of surface reflection also serves to decorrelate the phase

of the reflected signal from the incident signal under most practical real-experimental conditions.

In order to increase the performance of an estimator, multiple snapshots can be employed (a

snapshot is a sample taken at a single range cell). However, unlike simulations where multiple pings

can be generated and thus multiple snapshots taken from the same range cell on different pings, in

experimental measurements a single range cell sampled across adjacent along-track pings may not

yield suitable multiple snapshots. This is because for a given range cell the location on the bottom

might not possess similar bottom characteristics such as composition, and slope (changes in roll,

pitch, and heading of the transducer induced by survey platform motion are also responsible for

the differences in adjacent pings) and hence cannot be considered to belong to the same statistical

ensemble. The distance between the same range cell in adjacent pings is not guaranteed to be

the same from ping to ping, as angular changes in heading will swing ping direction accordingly.

Alternatively, across track distance between range cells is always the constant, and can be far less

than the inter-ping distance (this is limited in principle by the two-way return time of a transmitted

pulse at the maximum observable range). There is also greater phase coherence between range

cells in a single ping, than between the same range cell in adjacent pings. Therefore, in order to

utilize multiple snapshots coherently in an estimator, multiple snapshots can be taken as consecutive

uncorrelated footprints as shown in Fig. 3.1. In which case there exists a slight possibility of having a

surface multipath being partially correlated with the bottom return, however, in the field, the effect

of water surface roughness serves to further decorrelate the signal. In principle it is not inconceivable

to have partially correlated surface multipath and bottom returns, but for the scenarios considered


in this research any correlations between multiple signals are considered negligible.

Figure 3.1: The return from uncorrelated footprints may be considered as separate snapshots of thesame ensemble if the geometry of measurements does not vary appreciably over the region of interest.

The results of this chapter indicate that the performance of a MASB system can be characterized

through knowledge of the measurement geometry, through the use of the CRLBα. It will be demon-

strated that the CRLBα is calculable without requiring multiple measurements, and that this is a

preferable performance indicator to the standard deviation, which is often not calculable explicitly

(ie. the pdf is incalculable), cannot be estimated reliably from a small number of measurements,

and ultimately is too dependent on snr to be of benefit in many circumstances. By computing the

CRLBα for the variance of an AOA estimator, including the incorporation of multiple snapshots,

the consequences of the various geometric effects (namely footprint shift and baseline decorrelation)

presented in the previous chapter will be examined. In addition, the effect of having multiple in-

coming plane waves are also examined. These effects will be examined for each of the waveforms

presented in chapter 2.

Following the theory based analysis in the first half of this chapter, simulations are employed

to model the various survey geometries, and several estimators are applied to the simulated data

sets. The merits of estimation techniques that employ pre-estimation for multiple snapshots will be

compared with those that use post-estimation, with the former demonstrated to be preferential in

most practical scenarios. Finally the confidence limits imposed by the CRLBα will be tested by

using three different pre-estimation averaging techniques on simulated data.

Finally, it will be demonstrated that the error arc length, EAL, which is based on the CRLBα,

is a useful and intuitive measure of performance for swath bathometry techniques. As the EAL

represents the confidence than a bottom measurement will fall within the swing of a specific arc


angle at a given range, in order to compute the uncertainty in depth the EAL must be projected in

the vertical axis.


3.2 Complex Multivariate Gaussian Signal

In order to progress on the course of performance analysis, one must re-examine the signal from

the perspective of a probability based approach. Narrowband bottom backscatter from a sea or lake-

bed may be considered the coherent sum (real and imaginary components) of contributions from a

number of discrete complex scatterers. This assumption was made for the signal model for three

reasons. First, there is evidence that some backscatter is of this type in [18]. Second, other authors

[23] have used this model, and therefore, results can be compared. Third, the analysis is possible

with this model, and it is not as yet with other models (though trends determined with this scatterer

model may hold in other models). If enough scatterers are ensonified in a single footprint, then by

the central limit theorem the measured signal on any element may be though of as distributed as a

complex gaussian signal (si in Eqs. 2.1 and 3.1). In addition, complex gaussian noise (ni in Eqs. 2.1

and 3.1) is added to the signal, which further asserts the assumption of a measured complex gaussian

signal (in this case χi in Eq.2.1). Other models for backscatter do exist, and it should be noted that

there is some evidence, [17], to suggest that in certain physical environments, where the number of

discrete scatterers ensonified is itself a random process (in this case a Poisson process), the resultant

random variable has a K-distribution in magnitude and a uniform distribution in phase. However the

simpler scenario of a complex gaussian distribution has been demonstrated to correspond to many

environments, such as a silt bottom.

Assuming the central limit theorem may thus be considered valid, and each element experiences a

complex gaussian signal (complex gaussian refers to gaussian in-phase and quadrature components

with complex covariance R), Eq.2.1 can be equivalently stated using variables χci and χsi which

denote the cosine (in-phase) and sine (quadrature) components of the narrowband signal on receiver

i. As previously stated, this signal also incorporates the effects of uncorrelated gaussian noise. The

signal on a N element array as defined in Eq. 2.1 must be re-expressed as that of a complex gaussian

signal vector, with the form given in Eq.3.1.

~χ(N×1) =

χ1

χ2

...

χN

=

s1 + n1

s2 + n2

...

sN + nN

=

χ1c + jχ1s

χ2c + jχ2s

...

χNc + jχNs

(3.1)

As the signals si on each of the elements are uniformly distributed in phase (note this is for absolute

phase, relative phase is of course the basis for interferometry in these systems), the expectation of

their mean is E{si} = 0. Similarly, the mean of the complex gaussian noise is zero for both real and

imaginary components. Therefore, the mean µi of each element is given by Eq. 3.2.


µi = E{χi} = E{χic + jχis} = E{si}+ E{ni} = 0 (3.2)

The probability density function of ~χ(N×1) is by definition the zero mean complex multivariate

gaussian probability density function for an N element array, and is given in [25] as Eq. 3.3.

fχc1,χs1,χc2,χs2,...,χcN ,χsN(χc1, χs1, χc2, χs2, . . . , χcN , χsN ) =

e−~χH

(N×1)R−1

(N×N)~χ(N×1)

πN det∣∣R(N×N)

∣∣ (3.3)

In Eq. 3.3, R(N×N) is taken from Eq. 2.14. The form of Eq. 3.3 will be examined in detail later for

the case of estimation of AOA of a plane wave on a two element array. It can be stated that the general

pdf fχc1,χs1,χc2,χs2,...,χcN ,χsN(χc1, χs1, χc2, χs2, . . . , χcN , χsN ) is the basis for computing the pdf of an

AOA estimate, and therefore describing the confidence interval for which an AOA estimate is most

likely to be found. However, in the case of most estimators for AOA the estimation procedure leads

to intractable transformations of the pdf, either via the jacobean or through integrations that are not

solvable by even the leading symbolic math programs. The scenarios for which analytical performance

(via confidence limits) of AOA estimators are not tenable is the topic which the remainder of this

chapter is devoted. The alternative to estimator dependent performance is to choose an estimator

independent measure such as the CRLBα, which is developed in the following section.


3.3 The Fisher Information Matrix and The Cramer-Rao

Lower Bound

Though a thorough development of the CRLB is given in [40] (also associated references), a brief

outline of the CRLB is given to establish the conventions employed in this research. The CRLB

requires Fisher’s information matrix. For the sake of convention, the Fisher Information Matrix will

be represented by the variable J . For a complex gaussian process, such as outlined in the previous

section, the (i, j)th component of J is given by Eq. 3.4.

Jij = Mtr

[(R−1

(N×N)

∂R(N×N)

∂νiR−1

(N×N)

∂R(N×N)

∂νj

)]+ 2M<

{∂µ(N×1)

∂νi

H

R−1(N×N)

∂µ(N×1)

∂νj

}(3.4)

In Eq. 3.4, νi represents the parameters to be estimated or nuisance parameters, M is the number

of snapshots, <{·} represents the real component of the term in parenthesis, tr is the matrix trace

and µ(N×1) is the mean of the vector χ(N×1) for a N element array. For a complex gaussian zero

mean signal the second term is eliminated, and Eq. 3.4 is simplified to Eq. 3.5.

Jij = Mtr

[R−1

(N×N)

∂R(N×N)

∂νiR−1

(N×N)

∂R(N×N)

∂νj

](3.5)

The CRLB is found by taking the inverse of J . In the case of multiple parameter estimation,

the individual bound on variance for parameter νi fall on the diagonal of the CRLB, which is

mathematically represented by Eq. 3.6.

CRLBii = (J−1)ii (3.6)

For the problem at hand, two electrical angles are estimated if |x| < xlim and only one needs to be

considered if |x| > xlim. These AOA estimates represent the parameters of interest. In addition, the

power in the respective signals must be considered as nuisance parameters because they are unknown

in an a-priori manner, and must be simultaneously estimated along with the AOAs. In the case of

only one signal, not knowing the signal power does not matter because the cross terms in the Fisher

information matrix are small (except very close to nadir where the power is changing very quickly).

However, for two signals, the backscatter energies must be included. For reasons listed in the first

section in this chapter, any cross-correlation between the two signals is neglected (for both angle and

signal strength). In addition the noise level is assumed known, though even if it were not, one would

only need to consider a bound on the relevant snr for each signal instead of the signal strengths.

Therefore the following parameter vectors are employed.


νi =

{[2σ2

s1, 2σ2s2, α1, α2] |x| < xlim

α1 |x| > xlim

}(3.7)

Again, as in previous sections, α1 is the electrical angle for backscatter from positive x, α2 is the

electrical angle for backscatter from negative x, and 2σ2s1 and 2σ2

s2 are the respective signal strengths

at the receivers. This analysis can also be extended to the case of more signals on an N element

array (where the number of signals must be less than N − 1) because the principle of superposition

allows for the covariance matrices of uncorrelated signals to simply be added into a single covariance

matrix.

One Signal

To provide a simple example, in which the cross terms in the Fisher information matrix are

negligible for the single signal scenario, and to demonstrate a calculation of the full CRLB, the

case of a two element array with one plane wave signal is examined. It is useful to present a simple

calculation of the CRLBα, specifically since for most of the waveforms and array sizes considered

later in this research a closed form CRLBα is not presented due to extremely long formulas or the

necessity of performing the derivative with respect to the variable α numerically. Note that the plane

wave in this case is not constrained by the condition that it is due to a pulse or the physical model

constructed in chapter 2. Instead, this plane wave can be simply thought of as a complex gaussian

signal of strength 2σ2s , emanating from some location in the far field and impinging on a two element

array at AOA α. Complex gaussian noise of strength 2σ2n is added as well. This model will be useful

in later sections of this chapter as well, and so its consideration is insightful in more than just the

context of this present calculation. The signal model is constructed following Eq. 2.1.

[χ1

χ2

]=

[sejϑ + n1

sejϑ+jα + n2

](3.8)

where s is a Rayleigh distributed amplitude of a signal representing a plane wave impinging on the

array, ϑ is the uniformly distributed random electrical phase variable over (−π, π] associated with the

incoming wave, α is the electrical phase difference corresponding to the AOA of the incoming wave.

Both n1 and n2 are random uncorrelated complex gaussian noise variables with real and imaginary

variances of σ2n, which gives the aforementioned full variance of 2σ2

n. The covariance matrix for this

simple scenario, R(2×2) , is given by Eq. 2.19 with N = 2, and correlation coefficients set to one due

to the simple plane wave nature of the model which is independent of a specific survey geometry.

R(2×2) = E{

~X(2×1)~XH

(2×1)

}=

[2σ2

s + 2σ2n 2σ2

se−jα

2σ2sejα 2σ2

s + 2σ2n

](3.9)


For this simple case, where the mean is zero, the Fisher’s information matrix is then calculated

from Eq. 3.5 analytically to be:

J =

tr

[R−1

(2×2)

∂R(2×2)

∂2σ2s

R−1(2×2)

∂R(2×2)

∂2σ2s

]tr

[R−1

(2×2)

∂R(2×2)

∂2σ2s

R−1(2×2)

∂R(2×2)

∂α

]

tr[R−1

(2×2)

∂R(2×2)

∂α R−1(2×2)

∂R(2×2)

∂2σ2s

]tr

[R−1

(2×2)

∂R(2×2)

∂α R−1(2×2)

∂R(2×2)

∂α

]

=

[4

(2σ2s+σ2

n)2 0

0 2(σ2s)2

σ2n(σ2

n+2σ2s)

] (3.10)

Taking the inverse of J results in a CRLB of:

CRLB =

(2σ2s+σ2

n)2

4 0

0 σ2n

σ2s

+ 12

(σ2

n

σ2s

)2

=

[(2σ2

s+σ2n)2

4 0

0 1snr + 1

2

(1

snr

)2

] (3.11)

The derivatives used in Eq. 3.10 reveal the placement of the relevant bounds in Eq. 3.11. Con-

sequently, for the two element array with one signal and a single snapshot, the bound on variance

for the signal strength is CRLBσ2s

= CRLB(1, 1) (with known noise level this bound is easily trans-

formed to a CRLB on the signal-to-noise ratio), and the bound on the variance of the AOA as

CRLBα = CRLB(2, 2). Unsurprisingly, each of the bounds carries with it the appropriate units

of the variance on which it specifies a lower bound. In addition, the two bounds are independent

(the off-diagonal terms are zero), so in retrospect the calculation of either bound could have been

performed independent of the other. For M independent snapshots, the bounds are divided by M ,

as this is the case for the corresponding variance of the average of this number of snapshots.

CRLBα,M =1M

[1

snr+

12

(1

snr

)2]

(3.12)

In general, the CRLBα on AOA for M snapshots and an N element array, using the N × N

extension of the simple one signal covariance matrix presented in Eq. 3.11 is shown in [40] (pg. 946)

and is given by Eq. 3.13.

CRLBα,M,N =1M

(6

N(N2 − 1)snrn+

6N2(N2 − 1)snr2

n

)(3.13)

For Eq. 3.13, it should be noted that the signal-to-noise ratio snrn in this simple model is only

for noise, and neglects the geometric decorrelation effects discussed in chapter two. In addition, it

can be observed that the bound presented in Eq. 3.13 is independent of α. Also, the second term


in Eq. 3.13 dominates at low snrn and the first term dominates at high snrn. Since the discussion

in this thesis is primarily concerned with high snrn situations, the CRLB for this scenario is well

approximated by the first term, and can be expressed by CRLBα,M,N .

CRLBα,M,N ≈ 6MN(N2 − 1)snrn

(3.14)

Two Signals

For two signals, the number of variables to be estimated increases to 4, as given by the first line of

Eq. 3.7. Unlike the single signal case shown in Eq. 3.10 where the variables are independent, for two

signals, the entire Fisher information matrix (shown in Eq. 3.15) must be calculated and inverted to

get the full CRLB, and consequently the desired quantities CRLBα1 and CRLBα2 . For the sake of

brevity, the correlation matrix will be represented by R instead of R(N×N) .

J =

tr[R−1 ∂R

∂2σ2s1

R−1 ∂R∂2σ2

s1

]tr

[R−1 ∂R

∂2σ2s1

R−1 ∂R∂2σ2

s2

]tr

[R−1 ∂R

∂2σ2s1

R−1 ∂R∂α1

]tr

[R−1 ∂R

∂2σ2s1

R−1 ∂R∂α2

]

tr[R−1 ∂R

∂2σ2s2

R−1 ∂R∂2σ2

s1

]tr

[R−1 ∂R

∂2σ2s2

R−1 ∂R∂2σ2

s2

]tr

[R−1 ∂R

∂2σ2s2

R−1 ∂R∂α1

]tr

[R−1 ∂R

∂2σ2s2

R−1 ∂R∂α2

]

tr[R−1 ∂R

∂α1R−1 ∂R

∂2σ2s1

]tr

[R−1 ∂R

∂α1R−1 ∂R

∂2σ2s2

]tr

[R−1 ∂R

∂α1R−1 ∂R

∂α1

]tr

[R−1 ∂R

∂α1R−1 ∂R

∂α2

]

tr[R−1 ∂R

∂α2R−1 ∂R

∂2σ2s1

]tr

[R−1 ∂R

∂α2R−1 ∂R

∂2σ2s2

]tr

[R−1 ∂R

∂α2R−1 ∂R

∂α1

]tr

[R−1 ∂R

∂α2R−1 ∂R

∂α2

]

(3.15)

Determining the derivatives of R with respect to the power parameters is achieved by noticing

that the form given in Eq. 2.19 allows for Rι to be expressed as simply the signal strength of signal

ι multiplied by the corresponding correlation matrix.

Rι = 2σ2sικι (3.16)

For the full R, the principle of superposition allows for various covariance matrices each from

independent underlying processes to be added. Therefore using Eq. 3.16 and ∂Rn

∂2σ2sι

= 0, the derivative

of the summed covariance with respect to each of the component signal strengths is simply given by

Eq. 3.17.

∂R

∂2σ2sι

=∂Rι

∂2σ2sι

= κι (3.17)

The derivative with respect to the electrical AOA is most easily determined numerically. The

change in Ri is calculated for a small change in electrical angle αi at each point along the bottom,

and the derivative is found from Eq. 3.18.

∂R

∂αi=

∆Ri

∆αi(3.18)


In Eq. 3.18, ∆Ri represents the associated element-wise change in Ri for the given change ∆αi.

The numerically determined derivatives are found to be robust with respect to the value of ∆αi

used (except in the case of the MFFQ pulse where exceptionally long computation resulted in some

calculation instability for far values of range), with a suitable value of ∆αi = 0.0003π being chosen

for the given research.

With the definitions for the correlation matrix and derivatives, J is determined through Eqs. 3.5,

3.7, as in Eq. 3.15 for the case of |x| < xmin, and Eq. 3.10 for |x| > xmin. As the CRLB is the inverse

of the Fisher Information matrix, the elements corresponding to the various variables from Eq. 3.7

will be accordingly laid out. As such, for |x| < xmin the (3, 3) and (4, 4) entries in the 4× 4 CRLB

matrix correspond to CRLBα1 (return from x > 0) and CRLBα2 (return from x < 0) respectively.

In the case of |x| > xmin it was demonstrated above that the form of J is just a 1× 1 matrix and so

CRLBα1 is just 1J .

As opposed to the approximation methods used in [37], current computational methods allow for

full CRLB calculations, and approximate bounds are no longer necessary. The full calculation of

CRLBα is now a simple matter with any mathematics software such as Matlab. However it should

be noted again that one simplification is made for the current development in that there is assumed

to be no correlation between the two incoming signals. If measurements are taken in a multi-path

environment, such that a secondary signal is a multi-path return of the first signal (perhaps a surface

reflection of the returning signal), then for longer transmitted pulses there will be correlated signals.

It is also the case that when multiple uncorrelated range cells are employed as multiple snapshots,

the signal from one range cell can be correlated with the multi-path from one of the other range

cells. However, as alluded to earlier, in the experimental situation encountered here, the transmitter

is not only placed at a depth far greater than the number of snapshots multiplied by the pulse length

(creating a longer path for the second signal), but the reflection surface is often rough enough to

de-cohere the phases of the reflected signal. This effectively decorrelates the signals, allowing for the

analysis to be representative of real experimental conditions.

Although the number of terms to include in the full version of the CRLBα for multiple signals is

unfortunately too high to write out in a closed form in this thesis, interpretation of results is directly

possible from a few simple plots. To illustrate the most significant consequence of having two signals

on an array, one needs only to examine Figs. 3.2 and 3.3. In all figures, all signals have been given

a signal-to-noise ratio of 10dB. In Fig. 3.2, three different bounds are given for a five element array.

The red curve is the bound that would be encountered in estimating the AOA for one signal on a

five element array, and being the smallest bound is also the least useful. The largest bound is the

black curve, which is the full√

CRLBα1 on the standard deviation of the AOA for incorporating

the effects of two signals, separated by the angle apparent on the α-axis. This bound is the largest,


and one can see that only when the two AOAs are >∼ 1 [rad] does the bound even approach the 1

signal limit. Finally, the blue curve represents the bound if both signal strengths are known a priori,

only the AOAs of the two incoming signals are unknown. This bound is not as useful as the center

bound for single snapshot measurements, however, if the bottom type and slope are known then this

model may be more applicable (indicating that one might do better at AOA estimation if bottom

is known). In Fig. 3.3, the same process used in Fig. 3.2 is repeated for arrays ranging from 3 to 6

elements, and it is apparent that the same observations hold true for all of the arrays. In estimating

more than just the AOAs of the two plane waves impinging on the array, the effects of estimating the

magnitudes can be seen most clearly in the comparison of the black and blue plots in Fig. 3.2. The

reason why the bound diverges when α1 → α2 in the black plot of Fig. 3.2 is that the the estimation

of the signal strengths mathematically degenerate when the angles are close together.

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

α1−α

2 [rad]

CR

LBα 1

1/2

Figure 3.2: For a five element array, (all signal-to-noise ratios set at 10dB) the respective√

CRLBsfor the single signal estimation (red curve), two signals with unknown AOAs and magnitudes (blackcurve), the same two signals with known signal to noise ratio (blue curve).

It is pertinent at this stage to outline the direction for the next few sections, in order to reach

the desired goal of determining a method of performance analysis for AOA estimation. Now that

the relevant variables required for the Fisher Information matrix have been defined, and their effects

on the CRLBα demonstrated in part, the next objective is to establish that there is a link between

the confidence interval for the AOA variable’s measurement, and the CRLBα. To accomplish this,

a simple estimator will be examined for the two element array, with one signal. The probability

distribution for the AOA estimator will be determined for this model in a unique way. Next, a


0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

α1−α

2 [rad]

CR

LBα 1

1/2

Figure 3.3: Following Fig. 3.2, the√

CRLB (all signal-to-noise ratios set at 10dB) for all threescenarios: 1 signal, 2 signals, 2 signals with known magnitude. From bottom to top groupings arerepresentative of 6,5,4 and 3 element arrays.

calculation is made of the probability that an estimate of AOA is less than γ√

CRLBαs away from

the unbiased mean value, in effect forming a confidence interval. This interval will be compared

to a comparable interval being formed by using the standard deviation. By demonstrating that√CRLBα is in fact the more relevant variable to set confidence intervals, the performance of various

waveforms will then be examined, including an investigation of the effects that limit AOA estimation

performance. Using simulation, three AOA estimators will then be compared to the√

CRLBα. To

establish bottom estimation performance, a relation will be given from√

CRLBα to the error arc

length (EAL). Several examples of the EAL for various survey geometries (tilt angles, frequency,

ocean vs fresh water attenuation, etc.), and waveforms will then be plotted to demonstrate both

geometric effects (such as footprint shift and multiple signals) and the relative performance of various

waveforms.


3.4 The Use of CRLB Over Standard Deviation

3.4.1 Two Element Estimator PDF

In the simplest of interferometric measurements, a two element array is sampled, with a single

complex Gaussian narrowband random plane wave incident. This is the same model considered in

the previous section for a simple calculation of the CRLB. Part of the reason for investigating

the single snapshot in detail is that a probability density for the angle estimator is calculable and

therefore the true variance can be determined. Also, the problems associated with a single snapshot

shed light on problems likely to be encountered with multiple snapshots and multiple array elements.

In this section, the two element scenario will be used to set up distinctions between models and

estimators. A probability density function for a simple AOA phase difference estimator will be

derived, and the variance of this estimator will also computed. To begin analysis, the narrowband

signals (both in-phase and quadrature components include the effects of uncorrelated gaussian noise)

on the two receivers are taken from Eq. 3.1, with N = 2.

~χ(2×1) =

[χ1

χ2

]=

[χc1 + iχs1

χc2 + iχs2

](3.19)

Since variables χc1, χs1, χc2 and χs2 in Eq. 3.19 represent gaussian signals, the probability dis-

tribution of these variables is given by using N = 2 in Eq. 3.3. This is also known as the complex

bivariate normal probability distribution and is given by Eq. 3.20.

fχc1,χs1,χc2,χs2 (χc1, χs1, χc2, χs2) =e−~χH

(2×1)R−1

(2×2)~χ(2×1)

π2 det∣∣R(2×2)

∣∣ (3.20)

To continue the analysis it is useful to express the covariance matrix in Eq. 3.20 as a function of

the snr as shown in Eq. 3.21.

R(2×2) = E{~χ2~χH

2

}= 2σ2

n

[1 + snr snre−iα1

snreiα1 1 + snr

](3.21)

For the two element array and all other measurement scenarios it is important to again make the

distinction between an model parameter, measurement variable and estimators for model parameters.

The model parameter has a value that may or may not be random. However, as mentioned earlier,

estimates are formed from measurements, which are always random, because they contain noise and

possibly other random model parameters. One instance of a measurement is called a snapshot, and

estimators can be formed from one or more snapshots. Estimators may, or may not actually represent

model parameters, or may just represent some combination of model parameters that is required to

fill out the full probability domain space (this will be demonstrated in the following example).


In this case, one estimator choice for the electrical AOA is given in Eq. 3.22, with α = ∠(χ1χ∗2),

where the domain α ∈ (−π, π] is chosen so as to avoid ambiguity. In addition the three variables

ϑ ∈ (−π, π], s ∈ [0,∞) and r ∈ [−s,∞) are also estimated. The variables s and r are chosen such

that they are unit-less, and when squared have the same relevant scaling as snr. In this case r in

particular represents the choice of an estimator that has no obvious meaningful relationship to a

model parameter, and is in essence a representation of noise inherent in the model. In this research

the convention of using a (·) will be used to denote the variable as an estimator. The various

relationships between all the measurement parameters (χc1, χs1, χc2, χs2) and estimator variables

are expressed in Eq. 3.22.

~X(2×1)

(s, ϑ, φ, r

)=

χc1

(s, ϑ, α, r

)+ iχs1

(s, ϑ, α, r

)

χc2

(s, ϑ, α, r

)+ iχs2

(s, ϑ, α, r

) =

√2σ2

n

[sei(ϑ+α)

sei(ϑ) + rei(ϑ)

](3.22)

The estimation of α presented in Eq. 3.22 is not unique and other variations on the r term can

be considered to give the correct answer, however this particular parametrization yields a readily

solvable form, and Jacobean that is free of any dependance on ϑ, an arbitrary angle. It also reflects

the structure of a single plane wave of impinging on the array (with the r term here representing

the influence of noise). Finally, it should be noted that any two points on the complex plane can be

represented through this estimator.

By separating the real and imaginary components of Eq. 3.22, a Jacobean, J(2×2) , is found for this

transformation of variables in Eq. 3.23.

|J(2×2) | =∣∣∣∣∂χc1, χs1, χc2, χs2

∂s, ϑ, α, r

∣∣∣∣

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

det

√2σ2

n

cos(ϑ− α

)−s sin

(ϑ− α

)−s sin

(ϑ− α

)0

sin(ϑ− α

)s cos

(ϑ− α

)s cos

(ϑ− α

)0

cos(ϑ)

− (s + r) sin(ϑ)

0 cos(ϑ)

sin(ϑ)

(s + r) cos(ϑ)

0 sin(ϑ)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= (2σ2

n)2s (s + r)

(3.23)

Using the above result (which it can be noted is always positive), and the covariance matrix from

Eq. 3.21, a restatement of the probability distribution in Eq. 3.20 is given in Eq. 3.24.


fs,ϑ,α,r

(s, ϑ, α, r

)=

e− ~XH

(2×1)R−1

(2×2)~X(2×1)

π2 det∣∣R(2×2)

∣∣ |J(2×2) |

=s (s + r) e

−(2+2snr−2snr cos(α−α))(s2+sr)−r2(1+snr)1+2s1

π2(1 + 2snr)

(3.24)

There are a few observations that are immediately evident from the form of the probability density

function in Eq.3.24. First, there is no dependence on σ2n, which simplifies the analysis. Second, the

distribution of α is symmetrically distributed about α which indicates that there is no bias inherent

in the angular estimator. Finally, one notices that there is no dependence on ϑ, which means that

to integrate out the arbitrary angular variable one needs only to multiply through by the range 2π.

fs,α,r (s, α, r) =e− ~XH

(2×1)R−1

(2×2)~X(2×1)

π2 det∣∣R(2×2)

∣∣ |J(2×2) |

=2s (s + r) e

−(2+2snr−2snr cos(α−α))(s2+sr)−r2(1+snr)1+2s1

π(1 + 2snr)

(3.25)

In a similar way, one can integrate out the magnitude variables r and s1.

fα (α) =∫ ∞

0

∫ ∞

−s

fs,α,r (s, α, r) drds (3.26)

By integrating out the length variables s and r, using a symbolic math program such as Maple,

one arrives at a form of the pdf for α.

fα (α) =1 + 2snr

2π (1 + 2snr + snr2 − snr2 cos2(α− α))

+snr cos(α− α) (1 + 2snr)

(π + 2 arcsin

(snr cos(α−α)

1+snr

))

4π (1 + 2snr + snr2 − snr2 cos2(α− α))32

(3.27)

Though in a different form, Eq. 3.27 is the same result as Eq. 3.28 which is found in [25] (pg. 404,

with snr = k01−k0

, therefore k0 = ρn = snr1+snr , as in Eq. 2.20), by a slightly different transformation

of variables.

fα (α) =1− ρ2

n

2π (1− ρ2n cos2(α− α))

32

[ρn cos(α− α)(arcsin(ρn cos(α− α)) +

π

2) +

√1− ρ2

n cos2(α− α)]

(3.28)


The probability distribution fα (α) describes the probability of the phase difference between two

elements. To further the analysis in the next section, where the variance of the AOA is calculated, it

is useful to modify the phase estimate so that it is centered around the true phase, namely α0 = α−α.

The resulting pdf is given by Eq. 3.29.

fα0 (α0) =1− ρ2

n

2π (1− ρ2n cos2(α0))

32

[ρn cos(α0)(arcsin(ρn cos(α0)) +

π

2) +

√1− ρ2

n cos2(α0)]

(3.29)

This probability density differs greatly from the normal (gaussian) density although it does have

a characteristic bell shape. The first difference to be noted in passing is that the angle probability

density must be understood as being modulo 2π, so for large variances comparing with the normal

density is meaningless. For small variances the angle probability density has much larger tails than

the normal density and the remaining probability is more tightly grouped around the mean. These

two observations have consequences in the following sections, when the standard deviation is shown

to ill-represent the performance of angle estimates.

3.4.2 Two Element AOA Estimator Variance

The analytical variance of the AOA estimator α can be calculated for Eq. 3.29, by using Eq. 3.30.

σ2α = σ2

α0=

∫ π

−π

(α0)2fα0(α0)dα0 (3.30)

Since the pdf is an even function about α0 = 0, this integral will be twice the value taken over the

range (0, π). Using integration by parts, this equation can further be reduced to Eq. 3.31.

σ2α =

[2α2

0Fα0(α0)]π

0− 4

∫ π

0

α0Fα0(α)0dα0 (3.31)

where the form of the cumulative density function, Fα0(α0), is from [25] and is given by Eq. 3.32

(it was already stated that the pdf in Eq. 3.27 is equivalent to the development in [25]).

Fα0(α0) =12

+α0

2π+

ρn sin(α0) arccos(−ρn cos(α0))2π

√1− ρ2

n cos2(α0)(3.32)

An intermediate step in the integration yields Eq. 3.33.

σ2α =

π2

3+ (arccos(ρn))2 − 1

π

∫ π

0

(arccos(−ρn ∗ cos(α0)))2dα0 (3.33)


After some manipulation one arrives at the desired result for the phase variance, which can also

be derived from the expression for variance of phase difference given in [26] (page 411).

σ2α =

π2

3− π arcsin(ρn) + arcsin2(ρn)− 1

2

∞∑n=1

ρ2nn

n2(3.34)

Finally, it should be noted that a closed form for the series in Eq. 3.34 could not be determined,

and convergence of the numerical computation of this series is slow for high snr (eg. SNR > 30dB).

3.4.3 Estimator Variance Compared to CRLB

The next step in examining the performance of angle estimation is to determine if the CRLBα is

actually a good representation of how well the angle of arrival can be determined. The CRLBα is a

bound on the variance of an unbiased AOA estimator but there is no guarantee that the bound can

actually be attained. Angle estimation is intrinsically non-linear and therefore establishing closeness

to the CRLBα is difficult especially for low sample support. In addition, since bottom estimation

must be accomplished with only a few samples to maintain high resolution, asymptotic closeness

of the standard deviation to the CRLBα is of little use even if it could be proven analytically. So

how then is performance to be characterized? Instead of using the CRLBα as an estimate of the

standard deviation, the CRLBα itself will be proposed, and demonstrated to perform effectively, as a

measure of probability suitable for defining confidence intervals. Usually in the frequentist tradition

of statistics, one is accustomed to dealing in large part with gaussian (also called normal) variables

because the central limit theorem ensures that the sum (or average for many samples) of any random

variables tends toward a gaussian variable. As such, confidence intervals for the estimates of variables

are fully characterized by the standard deviation, meaning that the probability of an estimate lying

a specific distance from the mean will be only dependent on the standard deviation. Alternatively,

in this section it will be demonstrated that the standard deviation of a phase measurement is not the

best way to interpret the accuracy of a phase measurement for scenarios where only a few snapshots

are usually available, and in its stead the CRLBα can be utilized effectively. To establish the

preference of the CRLBα over standard deviation for determining performance of angle estimation,

the limiting case of the single snapshot estimator will be analyzed.

In this approach to the problem, it is extremely fortunate that the probability density function

for the simple estimation scenario exists in an easy to manipulate closed form, which can be used

to compare performance by probability bounds with that of the estimator variance. One of the

properties of the angle pdf that makes this approach attractive is that much of the probability is

close to the mean implying that for a single sample there is a high probability that the angle will be

close to the mean. So for a given snr, a probability that a sample will be within so many degrees of

the mean can be determined.


Specifically, referring to Fig. 3.4 where the probability of the angle estimate being less than so

many standard deviations from the mean is shown, it is evident that the initial probabilities are

higher than those for the normal probability density function and vary with signal-to-noise ratio.

In estimating the electrical angle it can be said that for reasonably high signal-to-noise ratio, the

estimate will be within approximately one standard deviation 90% of the time, while for the normal

density it is only 68% of the time. Note as well the strong dependence of the confidence value on snr

for the true pdf, whereas for a gaussian pdf the probability is independent of the snr (all gaussian

dashed curves in plot lie on top of each other), and therefore the traditional frequentist statistical

interpretation of confidence intervals using multiples of the standard deviation applies (ie. the 95%

confidence interval is ±2σα for a gaussian pdf).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β (multiples of σα)

P(

|α| <

β σ

α)

SNR Dependence

True pdf SNR = 20dB Gaussian pdf SNR = 20dB True pdf SNR = 30dB Gaussian pdf SNR = 30dB True pdf SNR = 40dB Gaussian pdf SNR = 40dB True pdf SNR = 50dB Gaussian pdf SNR = 50dB True pdf SNR = 60dB Gaussian pdf SNR = 60dB

Figure 3.4: The probability that an estimate will be less than β multiples of the standard deviationaway from the true value is demonstrated to be dependent on snr. Thick solid curves are for theangle probability density and signal-to-noise ratios 60, 50, 40, 30, and 20 dB, top to bottom. Forreference, the overlaying dashed lines show what a similar calculation for a Gaussian probabilitydensity with the same snr, and therefore same standard deviation, would yield.


Practically, the curves in Fig. 3.4 show angle estimates will generally be more tightly grouped

around the means than if they were distributed normally with the same variance, but there will

be outliers that are required to make the variances equal. So although the true variance might be

significantly larger than the CRLBα, a large percentage of the estimates may well be within the

standard deviation described by the bound.

The density describing the angle estimates for one snap-shot is particularly interesting in this

regard. In Fig. 3.4 the probability is plotted as a function of multiples of the standard deviation. If

probability is plotted as a function of multiples of√

CRLBα the result shown in Fig. 3.5 is obtained.

In other words, the probability of being within a multiple of the√

CRLBα is independent of the

signal-to-noise ratio, if the signal-to-noise ratio is reasonably high, say above 20 dB. Note that the

curves for the normal density are now not independent of snr.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β (multiples of CRLBα1/2)

P(

|α| <

β C

RLB

α1/2 )

SNR Independence

True pdf SNR = 20dB Gaussian pdf SNR = 20dB True pdf SNR = 30dB Gaussian pdf SNR = 30dB True pdf SNR = 40dB Gaussian pdf SNR = 40dB True pdf SNR = 50dB Gaussian pdf SNR = 50dB True pdf SNR = 60dB Gaussian pdf SNR = 60dB

Figure 3.5: The probability that an estimate will be less than β multiples of the√

CRLB awayfrom the true value is demonstrated to be independent of snr. Thick solid curves are for the angleprobability density for the signal-to-noise ratios 60, 50, 40, 30, and 20 dB. (They lie on top ofone another.) For reference the dashed lines are for a Gaussian probability density with the samesignal-to-noise ratios (hence same standard deviation), top to bottom.


The reason that the probability of being within a multiple of the√

CRLBα is independent of snr

follows from the function describing the probability density function for high signal-to-noise ratio.

The probability density function for high signal-to-noise ratio near the mean can be approximated

by fα0 (α0) in Eq. 3.35.

fα0 (α0) ≈√

snr

(2 + snrα20)

(3.35)

For high snr, the CRLBα is given by Eq. 3.14 where N = 2 and M = 1 for this simple scenario,

which results in CRLBα ≈ 1snrn

. Therefore, scaling the the phase AOA by√

CRLBα through the

equation α0 = γ√

CRLBα ≈ γ√snr

, and making the appropriate change of variables for Eq. 3.35

(including the relevant Jacobian) yields a pdf in the form of Eq. 3.36.

fγ (γ) ≈ 1(2 + γ2)3/2

(3.36)

In Eq. 3.36, fγ (γ) is independent of signal-to-noise ratio, which makes it an ideal scale factor to

determine confidence intervals. In addition the pdf resembles the Students t-distribution with two

degrees of freedom. However, it is obviously not the Students t-distribution because the interval for

that density is the entire real line whereas the interval here is ±π at most. Strictly speaking the

density resembles that of a Students t-distribution near the mean only. For high signal-to-noise ratio

this interval (near the mean) contains most of the probability. Moreover, the variance for a Students

t-distribution with two-degrees of freedom is infinite which is certainly not the case here because the

variance is bounded by the integration limit of ±π. Therefore, by employing the CRLBα for high

signal-to-noise ratio, probability bounds for the error are obtained from Fig. 3.5. Specifically, errors

in the electrical angle estimates for a single snaphsot phase estimator will be less than 2√

CRLBα

for 81% of the time. The case of multiple snapshots will be examined later, as the choice of pre-

estimation or post-estimation plays a role in the confidence percentages, and can result in tendency

toward more gaussian statistics, with ±2√

CRLBα tending towards the more familiar 95% confidence

interval.

3.4.4 Difficulties with Estimating Standard Deviation

In addition to snr independence, another factor to consider in contrasting the performance of

AOA estimation using either the standard deviation or the√

CRLBα is the number of independent

snapshots that are required to form a good estimate of the standard deviation. Ultimately this will

provide a second indication that√

CRLBα is a more reasonable scale factor on which to base per-

formance than the standard deviation, which is typically applied in conventional statistical practice.

Obviously the preferable form of the standard deviation in any scenario is that of a closed form solu-

tion to an analytical calculation from the analytically derived probability density function. However


the closed form standard deviation for most estimators is rarely available (unlike the CRLBα which

it should be emphasized again is estimator independent), and consequently multiple snapshots are

required to form an estimate of the standard deviation. It will be demonstrated here that the very

process of forming a robust estimate of the standard deviation (again for the simple two element

array) can be misleading if too few snapshots are employed, especially at high snr which is often

the case in practical applications. Even under ideal circumstances, convergence to a single standard

deviation is slow and is dependent on the value of the snr, and it is unlikely that a sufficient number

of snapshots will be available to form such an estimate under any practical survey scenario.

The difficulty of estimating the standard deviation of the AOA can be most easily illustrated by

using a simulation of the two element array in comparison with the predicted theoretical value (in

this case scaled by√

CRLBα). This procedure involves programming a simulator to produce the

same signal statistics as the model suggests and determining the sample variance of the estimator.

Usually a certain large number of trials (for example 100 as employed in [23]) are performed in hopes

that the variance will be well estimated. However, the goodness of the variance estimator depends

to a large degree on the underlying probability density of the angle estimations. In what follows it

is shown that for a single snapshot of a two-element array the sample variance is typically a poor

representation of the real variance.

As a result of the nature of the angle probability density in Eq. 3.29, it can be expected that the

variance is difficult to estimate accurately even with a large number of trials. The larger tails of the

density results in low probability events contributing significantly to the variance, and therefore the

number of trials has to be large enough to capture the behavior of these events. Fig. 3.6 illustrates this

effect by plotting the ratio between the true standard deviation and the square root of the CRLBα as

a function of signal-to-noise ratio. Also plotted is the ratio between the estimated standard deviation

and the square root of the CRLBα for 1000, 10 000, 100 000 and 1 000 000 trials.

The general character of the estimated ratio is that it follows the true value to a certain signal-to-

noise ratio and then breaks off at that level and fluctuates rather wildly after that. The explanation

is that as the signal-to-noise ratio increases, the angle probability density function gets narrower

but the tails are still significant. Therefore, as the signal-to-noise ratio increases the large variations

become less probable but they still contribute significantly to the variance. Hence a larger number

of trials is required to capture the true variance as the signal-to-noise ratio increases.


0 10 20 30 40 50 600

1

2

3

4

5

6

SNR [dB]

σ α CR

LBα−

1/2

Comparison of Theory to Simulation

Theory N

s = 1000

Ns = 10000

Ns = 100000

Ns = 1000000

Figure 3.6: Convergence to the theoretical standard deviation for increasing numbers of simulatedsnapshots, Ns. The ratio of the standard deviation of the electrical angle to

√CRLBα is given as a

function of the signal-to-noise ratio (thick solid black line). Dashed red line is the ratio of the samplestandard deviation to

√CRLBα for 1000 trials, dotted cyan line, 10 000 trials, and dashed dotted

blue line 100 000 trials, and solid magenta line, 1 000 000 trials.


3.5 Pre-estimation vs. Post-estimation- 2 Element Array

Case Study

Continuing on with the simple scenario of estimation of an AOA of a single plane wave on a

two element array, it is now necessary to examine the case of utilizing multiple snapshots. To this

end it is informative to examine the case of two basic angle estimators. These estimators fall into

two separate categories: pre-estimation, where multiple snapshots are combined before the AOA

is estimated; and post-estimation, where AOAs are determined for individual snapshots, and then

averaged. The signal model is constructed for a given snapshot k by extending Eq. 3.8 beyond the

single snapshot model.

[χ1k

χ2k

]=

[skejϑk + n1k

skejϑk+jα + n2k

](3.37)

where sk is a Rayleigh distributed amplitude for the kth snapshot of a signal (representing a plane

wave impinging on the array), ϑk is the uniformly distributed random electrical phase variable over

(−π, π] for snapshot k, and α is the electrical phase difference corresponding to the AOA of the

incoming wave. Both n1k and n2k are random uncorrelated complex gaussian noise variables with

variances 2σ2n (again this represents contributions from both real and imaginary components).

For M snapshots, the first estimator combines the signals and sums over the snapshots before the

angle is computed. This can be represented as αpre = ∠(S), where S =∑M

k=1 χ∗1kχ2k. The second

estimator averages the individual phase estimates of the snapshots, αpost = 1M

∑Mk=1 ∠(χ∗1kχ2k).

For the purpose of performance analysis it is thus important to examine the predicted behavior of

variances the of the two different estimators.

3.5.1 Pre-Estimation

To arrive at the variance for the the pre-estimation technique, one must first expand the S term.

S = ejαM∑

k=1

s2k +

M∑

k=1

ske−jϑkn2k +M∑

k=1

skejϑk+jαn∗1k +M∑

k=1

n∗1kn2k (3.38)

Recognizing that the final sum term in Eq. 3.38 is far smaller than the other terms for reasonably

high SNR it can be ignored. Now a tactic must be employed whereupon one must first solve for the

variance of the phase estimate under the condition that the coefficients can be considered constant

and known, represented here by sk. Therefore the second and third terms in Eq. 3.38 are now sums

of the products of assumed known sk and unknown independent complex gaussian terms. Given that

the phase shifts e−jϑk and ejϑk+jα both represent the effect of the uniformly distributed random


phase variables, ϑk, on circularly symmetric complex noise terms (which means that that the angular

phase component of each noise term is itself a uniform random phase variable) it is simply convenient

to redefine the noise. Note that in the previous statement, α is neglected, as the effect of a uniform

distribution in phase, with any subsequent shift in phase is simply another uniform distribution in

phase. Therefore e−jϑkn2k → n2k and ejϑk+jαn1k → n1k, where n1k and n2k are both circularly

symmetric complex gaussian variables with the same variance as the originals, namely 2σ2n. Defining

rs =∑M

k=1 s2k, and re-assessing S now gives Eq. 3.39.

S ≈ ejαrs +M∑

k=1

skn2k +M∑

k=1

skn∗1k (3.39)

Figure 3.7: Geometry for pre-estimation technique, showing probability distribution of rs as circu-larly symmetric gaussian around rs. Note σ2

z is used to illustrate projected variance of a variabletangent to an arc centered at radius rs.

It is then necessary to recognize that the sums of gaussian variables in Eq. 3.39 are themselves

gaussian variables, with additive variance σ2n

∑Mk=1 s2

k, and so the total variance σ2S for both the real

and imaginary parts of S must account for the two sum terms in Eq. 3.39, and is given by:

σ2S = 2σ2

n

M∑

k=1

a2k = 2σ2

nrs (3.40)

Examining Fig. 3.7, it is useful to look at the variance in the direction given by the projection

arrow at rs. As this direction represents simply a rotation of the real and imaginary axes, and since

the random contribution to S is circularly symmetric around the position represented by the end


of rs in Fig. 3.7, the component variance in the direction indicated by the arrow at this position,

σ2z , will simply be the same as the variance in either the real or imaginary axes, namely σ2

z = σ2S .

Now what is desired from this analysis is the variance of the angle estimator, σ2αpre

. Assuming high

snr gives σz < rs, such that the variance of the estimator under the assumption of known sk, here

denoted by σ2αpre|rs

, can be thought of as the ratio of the projected variance σ2z and rs.

σ2αpre|rs

≈ σ2z

r2s

=2σ2

nrs

r2s

=2σ2

n

rs(3.41)

Knowing the conditional variance σ2αpre|rs

, one arrives at the unconditional variance σ2αpre

through

the relation given by Eq. 3.42.

σ2αpre

=∫ ∞

0

σ2αpre|rs

frs(rs)drs (3.42)

Where frs(rs) is the pdf of rs. Previously, rs was defined as the sum of the square of Rayleigh

variables, sk, so it can equivalently be stated that rs =∑M

k=1 bk where bk = s2k. The pdfs of the

Rayleigh variables are defined as fsk(sk) = sk

σ2se−s2k2σ2

s , and knowing that the square of a Rayleigh

variable is an exponential random variable, the pdf of bk is defined as fbk(bk) = 1

2σ2se−bk2σ2

s .

One method to arrive at the desired pdf of frs(rs) is to first calculate the characteristic function

Ψrs(ω). For M snapshots, the characteristic function of bk, Ψbk(ω), must be raised to the power M .

In Eq. 3.43 the characteristic function of bk is defined, and Eq. 3.43 follows for Ψrs(ω).

Ψbk(ω) =

∫ ∞

0

12σ2

s

ebk(iω− 1

2σ2s)=

11− i2ωσ2

s

(3.43)

Ψrs(ω) = [Ψbk(ω)]M =

[1

1− i2ωσ2s

]M

(3.44)

One can then recognize from [30] that Eq. 3.44 is the form of the characteristic function for a

chi-square random variable, and has the following pdf:

frs(rs) =rM−1s

(2σ2s)MΓ(M)

e−rs2σ2

s (3.45)

where Γ(·) is the Gamma function. Solving Eq. 3.42 using Eq. 3.41 and Eq. 3.45, one can determine

σ2αpre

(again with only the assumption of high SNR).

σ2αpre

≈∫ ∞

0

2σ2n

rs

rM−1s

(2σ2s)MΓ(M)

e−rs2σ2

s drs =1

(M − 1)2σ2

n

2σ2s

=1

(M − 1)snr(3.46)


It can also be noted that the snr represented in Eq. 3.46 can just as meaningfully be snre which

was developed to describe the relationship between any two elements in Eq. 2.24. It is now relevant

to compare σ2αpre

with the high snr form of the bound given in Eq. 3.14 (N = 2 for this simple

scenario), namely CRLBα,M ≈ 1Msnr .

σ2αpre

CRLBα,M

≈ M

M − 1(3.47)

In Eq. 3.47, the condition M > 1 is implied. The ratio in Eq. 3.47 yields the interesting result that

for pre-estimation averaging, the variance of the estimator approaches CRLBα,M as M increases.

This will be discussed in more detail in the next section.

3.5.2 Post-Estimation

For post-estimation averaging, the variance of the estimator is simply the sum of the variances

of the single snapshot estimations. Fortunately, the variance of a single phase estimate (i.e. M =

1) is known from Eq. 3.34. The variance is the same for each snapshot, and averaging results in a

reduction of the estimator variance for one snapshot, by a factor of M , the number of snapshots.

σ2αpost

=1M

[π2

3− π arcsin(ρ) + (arcsin(ρ))2 − 1

2

∞∑n=1

ρ2n

n2

](3.48)

Similarly, the CRLBα,M is reduced by a factor of M because the Fisher information is multiplied

by M . Therefore, the ratio of the estimators variance to the CRLBα,M remains constant at the

value for a single snapshot although the estimators variance decreases by a factor of M .

3.5.3 Comparison of Pre-Estimation and Post-Estimation

The two multi-snapshot estimation techniques are compared in Fig. 3.8, where the ratio of the

standard deviation to√

CRLBα,M is plotted as a function of M for three levels of signal-to-noise

ratio given by 15, 20 and 30dB (from bottom to top respectively). The red and blue lines are the

predicted ratios for post and pre-estimation averaging respectively. The green and red asterisks are

simulated results for post and pre-estimation averaging respectively. It can be seen that the ratios for

both cases follow the predicted curves with the ratio for pre-estimation averaging approaching one

for increasing M . From Fig. 3.8 and Eq. 3.47 it is concluded that the number of snapshots processed

does not have to be very large before the CRLB is achieved for practical purposes using the pre-

estimation technique. Most of the gain for all three of the SNRs considered is obtained in the first

five snapshots for which the ratio for the variances is 1.25 and the ratio for the standard deviations


is√

1.25 = 1.118. Specifically, the test statistic becomes essentially Gaussian for increasing numbers

of snapshots and therefore the√

CRLBα,M is a reliable estimate of the variance.

It should be re-iterated at this point in the analysis that for most estimators (especially multiple

snapshot estimators for which the complexity of calculation most often increases with increasing

M), closed form expressions for the variance, such as those given by Eq. 3.46 and Eq. 3.48 are not

often possible to derive in an analytical equation. If, as in the case of the pre-estimation technique

under the assumption of high SNR, the test statistic does tend to Gaussian behavior, then not

only does the CRLBα provide insight as an estimate of variance, but the traditional interpretation

of confidence intervals is restored, with ±2√

CRLBα representing the %95 confidence interval one

would expect for Gaussian statistics.

Although pre-estimation averaging seems to have a decided advantage over post-estimation averag-

ing, post-estimation average should not be too quickly dismissed. The performance of pre-estimation

averaging is greatly affected by the validity of assumptions made concerning the underlying random

variables. The gain achieved in Fig. 3.8 assumes that the snapshots are drawn from identical distri-

butions. If however one of the snapshots is substantially greater than the others, it will dominate the

pre-estimation average and the resulting variance will be close to that for a single snapshot. Post-

estimation averaging does not suffer from this effect because one angle will not dominate another in

the same way. Therefore, post-estimation averaging is more robust in unequal snapshot situations.


0 5 10 15 201

1.5

2

2.5

3

3.5

Number of Snapshots

σ α CR

LBα−

1/2

Comparison of Theory to Simulation

Figure 3.8: Pre and post estimation results for both theory and simulation as a function of thenumber of snapshots for SNR = 30, 20, 15 dB from top to bottom. The blue lines represent pre-estimation and converge to one for an increasing number of snapshots, such that the variance andCRLB become equal. The red lines are for pre-estimation techniques and demonstrate that the ratioof variance to CRLB stays at the single snapshot level even for increasing numbers of snapshots.The green and red asterisks represent simulated results for post and pre-estimation respectively.


3.6 AOA Performance: Dependence on Waveform and Geo-

metric Effects

It is at this point in the research that all the foundations have been laid to demonstrate how the

decorrelation associated with geometric effects and the presence of multiple incoming signals effect

the performance of AOA estimation for various waveforms using√

CRLBα. In previous sections

2√

CRLBα was shown to provide confidence limits for the case of a two element array. Here a

similar case will be presented for arrays larger than two elements, using up to two signals. In this

section√

CRLBα will be plotted to demonstrate the relative influence of footprint shift, baseline

decorrelation, noise and the presence of multiple signals, and later in this chapter various estimators

for larger arrays, that also employ multiple snapshots, will be compared with the confidence interval

set by 2√

CRLBα. It should be prefaced at this point in the analysis that there are far too many

combinations and permutations of waveforms (including different pulse lengths), signal to noise

ratios, geometries and array sizes (not to mention many other variables) to be individually presented

in this work. Therefore, it is the goal of this research to present the framework under which specific

systems and survey geometries may be analyzed, and present several examples that illustrate some

important considerations for choosing survey parameters.

As in the case of chapter 2, a survey geometry was chosen with an altitude of 40m and tilt angle of

45◦, such that there are two signals present on the array up to 40m of horizontal range, and one signal

present for horizontal ranges beyond this point. This geometry has also been chosen to represent

a survey scenario where the effects of having a second signal are clearly evident in plots of the full

horizontal range (as opposed to smaller tilt angles where it becomes difficult to plot the results in

an illustrative and meaningful manner). The estimation of more than one signal necessitates the use

of arrays with greater than two elements (the maximum number of signals, or degrees of freedom,

is one less that the number of array elements required, therefore 3 elements or more are required to

estimate two simultaneous signals). The effect of using fewer or more degrees of freedom than signals

present will be examined in a later section using specific estimators. It should also be noted that as

in the case for the figures comparing the theoretical and simulated pulse correlations in chapter 2

the energy of the pulse on the bottom was normalized such that EtE{|B2|} = 1 (implementation of

the full sonar equation will be shown in the next section). However, noise is added to the signal for

the figures presented in this section, and the snr was set such that the level is 40dB at the maximum

range in the direction of the primary signal as demonstrated in Fig. 3.9.

To begin,√

CRLBα for the square pulse will be examined. In Fig. 3.10 a 20 cycles square pulse

was utilized on a 3 element array with 5 snapshots. For x < 0, the full√

CRLBα is represented by

a solid green curve, and is partially obscured by the black dotted line, which is the bound as if it

were determined by noise and footprint shift (including the influence of the x > 0 signal). Note that


−40 −20 0 20 40 60 80 100 120−10

0

10

20

30

40

50

60

Distance Along Bottom (m)

SN

R [d

B]

Figure 3.9: The SNR shown for both primary (x > 0) and secondary (x < 0) incoming signals, inthis case the level is set to 40dB at the maximum range for the x > 0 signal (in the case of the squarepulse, a 300 kHz pulse of 20 cycles was chosen).

since the full and approximate bounds are overlapping, it can be stated that the effects of baseline

decorrelation (which are included in the full bound) are negligible. Also on the x < 0 side of the

plot is the bound for the x < 0 signal alone determined solely by the snr for the x < 0 signal (in

the absence of footprint shift or secondary signal effects) using Eq. 3.14, which is represented by the

dashed green line. On the positive x side is the full bound as calculated including the influence of

the second signal (represented by a blue solid curve), and is partially obscured by the approximate

bound (cyan dotted curve) which includes only the effects of footprint shift (for both signals) and

noise. The full bound for the x > 0 signal only (neglects second signal) is the red solid curve,

which is overlapped by an approximate bound for the same signal that only includes the effects of

footprint shift and noise, the a magenta dotted curve. The bound calculated using only the snr of

the x > 0 signal is given by a dashed red line. In short all of the√

CRLBα curves agree well with

the counterpart plots that neglects the significance of baseline decorrelation, and only at broadside

(x = 40m), where the effects of footprint shift are negligible, that the bound using only snr becomes

the dominant mechanism.


−40 −20 0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9

10Square Pulse


CR

LBα1/

2 (D

egre

es)

Figure 3.10: Examining performance of AOA estimation for a SQ pulse. The SNR has been set to40dB at the maximum positive range, and performance is given for 5 snapshots of a 20 cycles pulserecorded on a 3 element array. Green and red dashed lines are bounds as determined by snr of thex < 0 an x > 0 signals alone. Black and and cyan dotted lines are the bounds as determined bysnr and footprint shift of the x < 0 an x > 0 taking into account the full 2 signal model, whereasthe magenta dotted line is for the bound as determined by snr and footprint shift of the x > 0 only.Green and blue solid lines are for the full bound for x < 0 an x > 0 signals (again 2 signal model),and the red solid line is for only the x > 0 signal. Note the dotted lines partially obscure the solidlines, leading to the conclusion that baseline decorrelation plays little role in the performance ofAOA estimation for this survey scenario.

Very near nadir and at nadir in Fig. 3.10 and subsequent figures, the primary and secondary signals

are nearly equal and produce a broad angular return, making it impossible to estimate either; hence,

the√

CRLBα increases dramatically. As the limit in x is reached in the negative direction, the

secondary signal fades because of the beam pattern, and the√

CRLBα increases again dramatically

because the snr for that signal goes to zero.

In contrast, Fig. 3.11 shows a scenario with the same across track resolution as the scenario in

Fig. 3.10 , however a longer pulse of 100 cycles is used, with only a single snapshot. All curves

are shown to represent the same scenarios as given in Fig. 3.10, however now the effects of baseline

decorrelation become evident in the mismatches between the various full bounds, and those that only

use noise and footprint shift (for each of the various corresponding curves). The only region that


−40 −20 0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9

10Square Pulse


CR

LBα1/

2 (D

egre

es)

Figure 3.11:√

CRLBα for same resolution in range Fig. 3.10, only now it is a single snaphot of a100 cycle SQ pulse. As in Fig. 3.10, green and red dashed lines are bounds as determined by snr ofthe x < 0 an x > 0 signals alone. Black and and cyan dotted lines are the bounds as determined bysnr and footprint shift of the x < 0 an x > 0 taking into account the full 2 signal model, whereasthe magenta dotted line is for the bound as determined by snr and footprint shift of the x > 0only. Green and blue solid lines are for the full bound for x < 0 an x > 0 signals (again 2 signalmodel), and the red solid line is for only the x > 0 signal. Note the dotted lines are now visiblyseparated from the solid lines, leading to the conclusion that baseline decorrelation is present in theperformance of AOA estimation for this survey scenario.

seems to remain unaffected by the change between the two scenarios is the single signal√

CRLBα

in the positive x direction out beyond broadside. Generally, it can be found for any high resolution

bathymetry systems that employ short pulses or short effective pulses after compression, the effects

of baseline decorrelation are minimal, as demonstrated here for the SQ pulse. This effect can also

be verified by calculating the equivalent snr associated with baseline decorrelation using Eq. 2.22

and the correlation expression for baseline decorrelation alone and by comparing it with those for

footprint shift alone (determined similarly) and thermal noise. For short pulses, the equivalent snr

for baseline decorrelation is quite high and, therefore, does not contribute significantly to the angle

estimation error. It should however be mentioned that the current analysis is done for a nominally

flat bottom. In the case of highly sloped geometries, where as the sonar footprint widens (such as

near nadir points) and the vantage of the sonar subsequently becomes wider in aspect angle, the


baseline decorrelation may influence the√

CRLBα. Having demonstrated that the effect of baseline

decorrelation is much smaller than either noise, footprint shift, or the effect of multiple signals for

the short pulse lengths considered in the survey geometries used in MASB, it will now be neglected

for the extent of the remaining research, although the full bounds for√

CRLBα will be used unless

otherwise stated.

Unlike baseline decorrelation, it was seen in Fig. 3.10 that footprint shift does contribute signif-

icantly to angle estimation error, especially when the signal arrival angle is not near broadside for

short pulses. If there is just one signal on the array, it may be possible to employ the use of time

delays to compensate for footprint shift, however if a second signal is present on the array, then

any time delay corrections applied to the first signal will alter the second signal accordingly. In this

thesis, time delay compensation will not be considered due to the added complication of secondary

signal effects, although it may prove a productive avenue for future exploration.

Although the full contribution from footprint shift to√

CRLBα must be taken into account (i.e.

the full covariance matrix must be computed) to properly examine effects on performance of AOA

estimation, it is possible to develop a reasonable approximation that is suitable for many practical

situations. Specifically, as the thermal snr increases, the Fisher information reaches a limit because

of the decorrelation caused by footprint shift, which increases with element spacing across the array.

This limit can have several plateaus that depend on a number of factors including the mathematical

form of the the decorrelation (i.e., related to pulse shape), the number of elements in the array, and

the thermal snr. However, the first plateau is consistently the Fisher information associated with

footprint shift for a two element array with the inter-element spacing of the array, regardless of the

array size. Though the full closed form of the CRLBα can be calculated in the figures up until this

point, it can be stated that the closed form expressions for such bounds are long and their as such

become unmanageable for simple interpretation. An approximation on the other hand can be useful

for quick reference, and knowing that the Fisher information is sensitive to the footprint shift in

the manner stated above, there is an opportunity to approximate the effects of thermal noise and

footprint shift for an arbitrary array with one signal present.

The essence of the approximation at hand is to consider that for a linear filled array, the effective

snr against footprint shift corresponds to a simple two element array, while the effect of noise can be

mitigated by considering the filled array. To investigate this principle, the MFSQ pulse will first be

examined. For high SNR with the MFSQ pulse, the snr against footprint shift for adjacent elements

is given by Eq. 2.72 with (k − i) = 1, as demonstrated in Eq. 3.49.

snrmfsqf ≈ 23

r2sq

δr2c

(3.49)


For the contribution of thermal noise that is uncorrelated between array elements, the snrmfsqn is

the ratio of the autocorrelation of the MFSQ pulse to the noise level, and requires the use of Eq. 2.67.

snrmfsqn =EtE{|B2|}2σ2

n sin(φc)2rsq

3(3.50)

Now it is necessary to illustrate a feature of the CRLBα for a two element array as given by

Eq. 3.12 under the condition of high SNR. Since CRLBα is simply the inverse of snre (in this case

snre can be substituted for snr, because there is only one off-diagonal element in the covariance

matrix), then by utilizing Eq. 2.25 one sees that the CRLBα is the sum of the bounds for thermal

noise, ˜CRLBα,n, and footprint shift, ˜CRLBα,f .

˜CRLBα ≈ 1Msnre

≈ 1Msnr

+1

Msnrfs

≈ ˜CRLBα,n + ˜CRLBα,f

(3.51)

Since it was stated earlier that the effect of footprint shift across an array depends on the inter-

array footprint shift for a two element array, and as Eq. 3.14 was already demonstrated to properly

represent the effect of noise in Figs. 3.11, 3.10 in the case of the SQ pulse (and similar plots can

be produced to demonstrate the same result for the MFSQ pulse) it stands to reason that a good

approximation for the N element array would be to add the CRLBα as determined by noise alone

for an N element array, to the bound for footprint shift (for the two element array) in a manner

similar to Eq. 3.51. Therefore by extension, Eq. 3.51 becomes Eq. 3.52 for the N element array.

CRLBα,M,mfsq ≈ 6MN(N2 − 1)snrmfsqn

+1

Msnrmfsqf(3.52)

To test the approximate bound,√

CRLBα was calculated for the same survey scenario previously

used for Fig. 3.10. The 20 cycle pulse length from the SQ pulse bound in Fig. 3.10 was multiplied

by 1√2

and the pulse energy was multiplied by 2rsq

3 as is required from ratios of the normalized pulse

shape and mean squared length for the MFSQ pulse with that of the SQ pulse, as listed in Table 2.2

(note the correction for pulse normalization results again in the value of SNR = 40dB at the far

range). The resulting√

CRLBα is plotted in Fig. 3.12, where the dotted green and red lines are the

bounds calculated using only the snr of the x < 0 and x > 0 signals alone. The solid green and

blue curves are the full bound as calculated including the effects of both signals, and the red curve

is the bound using only the x > 0 signal. The black asterisks are the approximation from Eq. 3.52

using the footprint shift from a two element array and the noise performance from the three element

array. It should be noted that the approximation tracks the single signal bound quite well down to


ranges much less than the bottom at broadside to the array, so if there were only one signal this

approximate bound would work well.

−40 −20 0 20 40 60 80 100 1200

0.5

1

1.5Matched Filtered Square Pulse


CR

LBα1/

2 (D

egre

es)

Figure 3.12:√

CRLBα as calculated for the MFSQ pulse, with far range SNR = 40dB, 3 elements,and 5 snapshots. The solid green and blue curves are the full bound as calculated including theeffects of both signals, and the red curve is the bound using only the x > 0 signal. The blackasterisks are the approximation from Eq. 3.52. Note that the black asterisks are obscuring the redcurve almost completely for ranges greater than broadside (i.e. x > 40m).

It is now useful to illustrate the effect of moving to a larger array, on√

CRLBα, which can be best

interpreted by considering the approximation in Eq. 3.52. In Fig. 3.13 the same survey geometry

is used as in Fig. 3.12, however the size of the array has been increased to 6 elements. There is an

improvement in performance against noise alone (i.e. the dashed line is suppressed to a degree of1

N(N2−1) as evident in the second term in Eq. 3.52), however it is clear that there is little improvement

against footprint shift as expected from the first term in Eq. 3.52 which is independent of the array

size. Again it should be noted that the scenario here is for high SNR, so as to make the expressions

easier to interpret, however similar expressions can be constructed for more moderate SNR if the

relevant scenario arises.

It is now important to examine the same survey scenario for each of the remaining pulse shapes.

It should be noted that for several of the pulses, the signal correlation functions are calculated for a


−40 −20 0 20 40 60 80 100 1200

0.5

1

1.5Matched Filtered Square Pulse


CR

LBα1/

2 (D

egre

es)

Figure 3.13: The√

CRLBα as plotted for the same scenario as in Fig. 3.12, however the number ofarray elements has been increased to six. Note a moderate improvement in performance against theeffects of noise, however no improvement against the effects of footprint shift in the x > 0 signal.As in Fig. 3.12, the black asterisks again are obscuring the red curve almost completely for rangesgreater than broadside (i.e. x > 40m).

post-filtering or post-compression pulse shape, which demonstrates that footprint shift is related to

the post-filtering or post-compression pulse shape, and is independent of the pre-compression pulse

shape, except insofar as that affects post-compression. For the FQ pulse, the 20 cycle pulse length

from the SQ pulse bound in Fig. 3.10 was multiplied by√

1− 3c2

(a2r2sq) as is required for the ratio of

the mean squared length for the FQ pulse with that of the SQ pulse, as listed in Table 2.2. However

as both pulses are normalized appropriately, no change in normalization was required for the FQ

pulse relative to the SQ pulse (again this leads to the value of SNR = 40dB at the far range). A q

of 20 was also chosen for this pulse. The results of this calculation are presented in Fig. 3.14 where

the dotted green and red lines are the bounds calculated using only the snr of the x < 0 and x > 0

signals alone. The solid green and blue curves are the full bound as calculated including the effects

of both signals, and the red curve is the bound using only the x > 0 signal. No approximation is

employed for this figure and it is apparent that the performance of the AOA estimation using the√CRLBα for the FQ is only slightly worse that that associated with the MFSQ pulse in Fig. 3.12

which is interesting because the FQ pulse represents a sonar pulse that is transmitted and received


without any matched filtering.

−40 −20 0 20 40 60 80 100 1200

0.5

1

1.5Finite Q Pulse


CR

LBα1/

2 (D

egre

es)

Figure 3.14: The√

CRLBα as plotted for the FQ pulse, with the same survey geometry as used inFig. 3.12. The dotted green and red lines are the bounds calculated using only the snr of the x < 0and x > 0 signals alone. The solid green and blue curves are the full bound as calculated includingthe effects of both signals, and the red curve is the bound using only the x > 0 signal. Note a slightdecrease in performance as compared to the similar results calculated for the MFSQ pulse.

To investigate MFFQ pulse, the 20 cycle pulse length from the SQ pulse bound in Fig. 3.10 was

multiplied by√

12 − 3c2

(a2r2sq) as is required for the ratio of the mean squared length for the FQ pulse

with that of the SQ pulse, as listed in Table 2.2. The pulse normalization was adjusted as well using

the long expression for the MFFQ pulse also listed in Table 2.2. As in Fig. 3.14 a q of 20 was used

for the MFFQ analysis. The resulting√

CRLBα calculation for the MFFQ pulse are presented in

Fig. 3.15 where the dotted green and red lines are the bounds calculated using only the snr of the

x < 0 and x > 0 signals alone. As in previous figures, the solid green and blue curves are the full

bound as calculated including the effects of both signals, and the red curve is the bound using only

the x > 0 signal. As in the case of the FQ pulse no approximation is given. In showing the plot

for the MFFQ pulse, it is necessary to address the apparent noise in the calculation at far range,

which is an artifact of the numerical calculation of the derivative of the correlation functions for

the MFFQ pulse with respect to the phase AOA. As mentioned earlier, the exact expression for the

correlations and corresponding covariance matrix are long and relatively uninformative equations,


however the calculation of the final bound itself illustrates that the the performance is quite similar

to that achieved for the MFSQ pulse.

−40 −20 0 20 40 60 80 100 1200

0.5

1

1.5Matched Filtered Finite Q Pulse


CR

LBα1/

2 (D

egre

es)

Figure 3.15: The√

CRLBα as plotted for the MFFQ pulse, with the same survey geometry as usedin Fig. 3.12. The dotted green and red lines are the bounds calculated using only the snr of thex < 0 and x > 0 signals alone. The solid green and blue curves are the full bound as calculatedincluding the effects of both signals, and the red curve is the bound using only the x > 0 signal.Note a slight destabilization of calculation at far range for x > 0, this is an artifact of performingnumerical derivation on the long form of the correlation function for the MFFQ pulse.

Finally, the CG pulse can be examined. It will be shown that it is useful in considering the CG

pulse to again develop an approximation for the performance due to footprint shift, and the same

arguments can be used that resulted in Eq. 3.52. Therefore what remains to be determined for the

CG pulse is the effective snrcgf due to footprint shift, and snrcgn which is the result of noise. First

it can be stated for the CG pulse, that the correlation due to footprint shift alone, ρcgf , is taken

from Eq. 2.77 and is given by Eq. 3.53.

ρcgf = e− (k−i)2δr2c2τ

8r2gs (3.53)

Furthermore, for high SNR (above 10dB), no pulse compression (cτ = 1) and adjacent elements

(i.e. (k − i) = 1), ρcgf is well approximated by the expansion of the exponential in Eq. 3.53, up to


and including the first order term, as given in Eq. 3.54.

ρcgf ≈ 1− δr2

8r2gs

(3.54)

Following the method used for the MFSQ pulse in Eq. 2.72, yields snrcgf as given by Eq. 3.55.

snrcgf ≈ 8r2gc

δr2c

=23

r2sq

δr2c

= snrmfsqf (3.55)

Therefore the effect of footprint shift on either the MFSQ pulse or the CG pulse is approximately

the same for high SNR. As in the case of the MFSQ pulse, to examine the effects of thermal noise

on CRLBα for the CG pulse, snrcgn must be calculated by taking the ratio of the autocorrelation

in Eq. 2.76 to the noise level (to adjust for no pulse compression, cτ can be set to one).

snrcgn =EtE{|B2|}2σ2

n sin(φc)rgs

√2π

cτ(3.56)

Therefore, under the assumption that the effects of footprint shift are again dominated by the

interarray spacing, and the thermal noise contribution to the bound is given again by Eq. 3.14, the

approximation to CRLBα for the one signal scenario on a multiple element array is analogous to

Eq. 3.52, and yields:

CRLBα,M,cg ≈ 1M ∗ snrcgf

+6

MN(N2 − 1)snrcgn(3.57)

To demonstrate that the approximation in Eq. 3.57 is valid, and to investigate the performance of

the CG pulse, the same survey scenario used earlier in Fig. 3.10 is modeled for the CG pulse. The 20

cycle pulse length from the SQ pulse bound in Fig. 3.10 was multiplied by√

12 as is required for the

ratio of the mean squared length for the CG pulse with that of the SQ pulse, as listed in Table 2.2.

The pulse normalization was adjusted as well using rgs

√2π as listed in Table 2.2. The resulting√

CRLBα calculation for the CG pulse are presented in Fig. 3.16 where the dotted green and red

lines are the bounds calculated using only the snr of the x < 0 and x > 0 signals alone. As in

previous figures, the solid green and blue curves are the full bound as calculated including the effects

of both signals, and the red curve is the bound using only the x > 0 signal. In a similar manner to

Fig. 3.12, the black asterisks are the approximation from Eq. 3.57, using the footprint shift from a

two element array and the noise performance from the three element array, and are shown to be in

good agreement to the single signal bound.

The performance of the CG pulse demonstrated in Fig. 3.16 is very similar to what was previously

observed for the MFSQ and MFFQ pulses. In addition, performance of the CG is only slightly better

than the performance demonstrated for the FQ pulse. In light of the similarities of performance, the


−40 −20 0 20 40 60 80 100 1200

0.5

1

1.5Compressed Gaussian


CR

LBα1/

2 (D

egre

es)

Figure 3.16:√

CRLBα as calculated for the CG pulse, with far range SNR = 40dB, 3 elements, and5 snapshots. The solid green and blue curves are the full bound as calculated including the effectsof both signals, and the red curve is the bound using only the x > 0 signal. The black asterisks arethe approximation from Eq. 3.57.

results of the CG pulse will be chosen to represent the presumed behavior of AOA performance for

the various pulse functions in subsequent sections. The choice is made to use the CG pulse, and not

one of the others for two reasons. First, the form of the correlation for the CG pulse is relatively

more manageable than forms developed for the other pulse functions. Second, the cg pulse has the

bonus of being able to employ pulse compression.

Several points should also be re-iterated regarding the use of the approximations in Figs. 3.12,

3.16. The approximations for both MFSQ and CG pulses hold for x > 40 m but departs from the true

bound for x < 40 m. For this scenario, the tilt angle is 45◦, so the double-signal region extends along

x to the altitude of the sonar which is 40 m. For x < 40 m, where two backscatter signals are being

received, the approximation to the bound is not expected to hold because it was developed under

the assumption of there being only one signal. The approximation, however, faithfully represents

the bound that would be obtained if the second signal were not present even for x < 40 m (red

solid curve in Figs. 3.12, 3.16). The fact that the approximation follows the solid line to the

dashed line at x = 40 m indicates that the approximation accurately represents the error due to


thermal noise because at this distance along the bottom, footprint shift is zero as the array is at

broadside. Therefore, it is concluded that the simple bounds represented by the approximations in

Eqs. 3.52, 3.57 are valid whenever there is only one backscatter patch. If a second backscatter signal

is present, the actual bounds can be expected to be higher than the approximations. Therefore, the

approximations may still be considered as lower bounding the actual performance, although they

may not be used to establish confidence limits in the same way that the actual bounds can.

In conclusion, for small arrays, short effective pulses, and high SNR, the major contributor to

electrical angle estimation error is footprint shift. As SNR decreases, footprint shift becomes less

of an issue, and the error caused by thermal noise dominates. The effect of footprint shift is not

mitigated for the levels of SNR considered in this section by increasing the number of elements in

the array, however, the effect of thermal noise on angle estimation accuracy is always reduced by

adding array elements.


3.7 Angle of Arrival Estimation For Two Signals

Having now described CRLBα, it is necessary to re-visit the proportion of AOA estimators falling

within ±2√

CRLBα of the expected electrical angle for 2 AOA and arrays with greater than 2

elements. As was discussed earlier, physical systems are described by models and the process of

taking data from the system and using it to extract a measurement of a model parameter (in

this case α, the phase AOA) is known as estimation. The estimate of the model parameter does

not necessarily represent an unbiased or precise measurement. Several different estimators can

be employed to estimate the same model parameter. In addition, multiple snapshots can also be

incorporated into an estimator to improve the performance for a single estimate, at the cost of

lowering the number of uncorrelated estimates that can be generated from a particular data set.

One can choose an optimum estimator for AOA estimation based on many factors including bias

and speed of computation however they must all be examined for performance. Estimators have their

own distributions, which are dependent on the distribution of the data, and the transformations of

variables used in the estimation procedure.

As MASB sonar consists of a filled array of transducers, and plane waves impinging on the array

give rise to various electrical phase differences on the array elements, a signal processing technique

must be employed to extract the AOA of the incoming plane waves. There are various techniques that

can be used, but the performance of each must be measured against an independent measure, which

for this research is the Cramer-Rao lower bound (CRLB). In this section there are three estimators

that are employed in the estimation of angle of arrival (the model parameter) from a linear array of

equally spaced elements. These are Linear Prediction (LP), Minimum Eigenvector (ME) Analysis

and Minimum Variance Distortionless Response (MVDR) Beamforming, all of which are examples of

post processing estimators that are implemented as filters. Furthermore, the simultaneous estimation

of multiple signals is determined by the number of degrees of freedom in the filter (i.e. the maximum

number of angles that the system is able to distinguish, or number of null angles the filter can steer).

Sometimes, if low numbers of signals are present, degrees of freedom can be sacrificed to increase

averaging and a more precise estimate of the AOA can be made. Usually this is accomplished by

examining sub-arrays, with each producing a pseudo-snapshot of the impinging signals (pseudo-

snapshots are created by sub sampling the array into smaller sub-arrays). Each pseudo-snapshot is

at a slightly shifted position, and contains some different elements which are subject to physically

different surroundings (e.g. physical coupling with each other and receiver housing), consequently

the use of sub-arrays averages across the array and alleviates the need for precise calibration. For

bottoms with slowly changing slope, one can also average over consecutive uncorrelated snapshots (i.e.

non-overlapping footprints). The simulated signals that were introduced for various waveforms in

chapter 2 will be used in this section as a specific survey example so as to examine the aforementioned

estimators. In actuality, the combinations and permutations of array sizes, degrees of freedom, and


survey geometries are too many to display here, and therefore an example will be given to illustrate

how to benchmark several estimators against the CRLB. This set of estimators is just a subset of

the much larger group of AOA estimation techniques that are available, however, they will suffice

to demonstrate the utility of the CRLB as a performance indicator. It should be emphasized at

this point that the focus of the research in this thesis is the performance bound, not the actual

estimators. The topics of too many, or too few degrees of freedom in an estimator, as well as

subjects such as estimator bias are only touched upon in this section, as the various combinations of

such contributions to estimator performance are too many to be covered in the current research. The

aforementioned issues could well be the starting points for further investigation in future research.

It should be noted that the estimation routines implemented in this section (LP, ME, MVDR)

are established techniques in signal processing, and have all been used in prior work and applied to

the problem of AOA estimation. However, since the particular calculations of the CRLB presented

in this thesis represent new achievement in this field (including the correlation functions for each

waveform, and combination of physical effects that have been considered in this survey geometry), the

performance of each of these estimators in comparison to the bound are new results. It should also be

premised that many variations exist on each of these estimation routines (such as the implementation

of thresholds for acceptable root placement in the LP and ME techniques). Due to the existence

of several variations on these estimators, a brief derivation and description of each techniques are

given in the following sections, so that the exact implementation of each estimator is clear. Some

variations on theses estimators may improve performance, however each of these variations warrants

its own investigation, and may be the subject of further work. It should however be recognized that

if an estimator achieves performance that is close to the CRLB, there is no requirement to improve

upon that estimation routine (at least with regards to AOA performance, other considerations such

as speed of implementation might be improved upon).

3.7.1 Linear Prediction

One feature of the linear prediction estimation algorithm is that AOAs can be determined explicitly

from a signal, which allows for speedy computation. This is a driving principle behind both the

techniques of linear prediction, and minimum eigenvalue analysis. In addition, for each of these two

techniques the power coming out of the estimation filter is minimized subject to some constraint.

The linear prediction filter works on the principle of coherent prediction of a single element signal,

by using a weighted combination of signals on other array elements (here weight values are given

by wi). As a rule, the number of weighted array elements required in this process is equal to the

number of plane wave signals on the array (e.g. in the left filter in Fig. 3.17 three elements are used

in the weighted part of the filter to cancel out three signals). By using the same number of weighted


elements as there are signals one may find that extra elements on an array may again be used as

extra pseudo-snapshots for the estimation of the covariance matrix. Although linear prediction has

been used previously as an AOA estimator (see [40] and associated references), a brief derivation

is presented here so that the exact implementation is specified (several variations on this method

exist).

Figure 3.17: Two of the various angle of arrival estimation schemes. Though both schemes relyon a calculation of weight vector ~w = [w1, w2, . . . , wn]T to minimize the squared error Jerror = e2

The linear prediction scheme operates on the sum of N − 1 elements to effectively cancel out theremaining element. Alternatively, the minimum eigenvalue technique minimizes the total noise bylooking for the lowest eigenvalue of the covariance matrix.

Allowing for pseudo-snapshots to be taken across the array (in the following set of equations three

pseudo-snapshots are taken from a six element array for every snapshot), and for M snapshots, one

can construct Eq. 3.58.

χ∗2 [1] χ∗3 [1] χ∗4 [1]

χ∗3 [1] χ∗4 [1] χ∗5 [1]

χ∗4 [1] χ∗5 [1] χ∗6 [1]

χ∗2 [2] χ∗3 [2] χ∗4 [2]

χ∗3 [2] χ∗4 [2] χ∗5 [2]

χ∗4 [2] χ∗5 [2] χ∗6 [2]

...

χ∗2 [M ] χ∗3 [M ] χ∗4 [M ]

χ∗3 [M ] χ∗4 [M ] χ∗5 [M ]

χ∗4 [M ] χ∗5 [M ] χ∗6 [M ]

w1

w2

w3

=

χ∗1 [1]

χ∗2 [1]

χ∗3 [1]

χ∗1 [2]

χ∗2 [2]

χ∗3 [2]

...

χ∗1 [M ]

χ∗2 [M ]

χ∗3 [M ]

(3.58)


To gain perspective on the expected behavior of this filter one can examine the case of having only

a three element array (e.g. χ1 [n] , χ2 [n] , χ3 [n] in the left side of Fig. 3.17). For this scenario one

expects two degrees of freedom in the array, and so may include two separate zero mean complex

signals ( A1 and A2) in the model, with variances given by 2σ21 and 2σ2

2 respectably.

~χ = A1

1

eiα1

ei2α1

+ A2

1

eiα2

ei2α2

+

n1

n2

n3

=

dcross

χ2

χ3

=

[dcross

~χsub

](3.59)

In Eq. 3.59, ~χsub is a subset of the full signal vector. It is a relatively straightforward procedure

to then calculate the expected value of the cross-correlation vector ~p = E { ~χsubd∗cross}, as defined in

Eq. 3.60.

~p =

[E {χ2d

∗cross}

E {χ3d∗cross}

]=

[E

{(A1e

iα1 + A2eiα2 + n2

)(A∗1 + A∗2 + n1)

}

E{(

A1ei2α1 + A2e

i2α2 + n3

)(A∗1 + A∗2 + n1)

}]

=

[2σ2

1eiα1 + 2σ22eiα2

2σ21ei2α1 + 2σ2

2ei2α2

]

(3.60)

A sub-set of the correlation matrix, Rsub is also required to calculate the filter weights.

Rsub = E{~χsub~χ

Hsub

}

=

[E {χ2χ

∗2} E {χ2χ

∗3}

E {χ3χ∗2} E {χ3χ

∗3}

]

=

[2σ2

1 + 2σ22 + 2σ2

n 2σ21e−iα1 + 2σ2

2e−iα2

2σ21eiα1 + 2σ2

2eiα2 2σ21 + 2σ2

2 + 2σ2n

]

= 2σ2n

[ρ1 + ρ2 + 1 ρ1e

−iα1 + ρ2e−iα2

ρ1eiα1 + ρ2e

iα2 ρ1 + ρ2 + 1

]

(3.61)

In Eq. 3.61 the signal to noise ratios are ρ2 = 2σ21

2σ2n

and ρ2 = 2σ22

2σ2n. To then obtain the optimum

weight vector, ~w0 , for the linear prediction estimator, Eq. 3.62 is employed.


~w0 = R−1sub~p

=4σ2

n

det (Rsub)

[ρ1 + ρ2 + 1 −ρ1e

−iα1 − ρ2e−iα2

−ρ1eiα1 − ρ2e

iα2 ρ1 + ρ2 + 1

][ρ1e

iα1 + ρ2eiα2

ρ1ei2α1 + ρ2e

i2α2

]

=

(ρ1+ρ2+1)(ρ1eiα1+ρ2eiα2)−(ρ1e−iα1+ρ2e−iα2)(ρ1ei2α1+ρ2ei2α2)(ρ1+ρ2+1)2−(ρ1e−iα1+ρ2e−iα2 )(ρ1eiα1+ρ2eiα2 )

(ρ1+ρ2+1)(ρ1ei2α1+ρ2ei2α2)−(ρ1eiα1+ρ2eiα2)(ρ1eiα1+ρ2eiα2)(ρ1+ρ2+1)2−(ρ1e−iα1+ρ2e−iα2 )(ρ1eiα1+ρ2eiα2 )

=

[ρ1ρ2eiα2+ρ1ρ2eiα1+ρ1eiα1+ρ2eiα2−ρ1ρ2e−iα1+i2α2−ρ1ρ2ei2α1−iθ2

1+2ρ1ρ2+2ρ1+2ρ2−ρ1ρ2e−iα1+iα2−ρ1ρ2eiα1−iα2

ρ1ρ2ei2α2+ρ1ρ2ei2α1+ρ1ei2θ1+ρ2ei2α2−2ρ1ρ2eiα1+iθ2

1+2ρ1ρ2+2ρ1+2ρ2−ρ1ρ2e−iα1+iα2−ρ1ρ2eiα1−iα2

]

(3.62)

Taking the limiting case of high signal to noise ratios ρ1 À 1 and ρ2 À 1 , the optimum weight

vector simplifies to Eq. 3.63.

~w0 =

[eiα2+eiα1−e−iθ1+i2α2−ei2α1−iα2

2−e−iα1+iα2−eiα1−iα2

ei2α2+ei2α1−2eiα1+iα2

2−e−iα1+iα2−eiα1−iα2

]

=

(2eiα1−eiα2−ei2α1−iθ2)+(2eiα2−eiα1−e−iα1+i2α2)2−e−iθ1+iα2−eiθ1−iα2

eiα1+iα2(e−iθ1+iα2+eiα1−iα2−2)2−e−iα1+iα2−eiα1−iα2

=

[eiα1 + eiα2

−eiα1+iα2

](3.63)

By forming a polynomial using Eq. 3.64, it is seen that the polynomial can be factorized to give

roots that are on the unit circle, and in the directions of the signals.

(z2 −z −1

)

1

w01

w02

= z2 − w01z − w02 =

(z − eiα1

) (z − eiα2

)= 0 (3.64)

The motivation for using this filter can now be interpreted to steer a null in the direction of

the incoming plane wave signals, so as to best reduce the mean squared power (this is only an

applicable interpretation in high signal-to-noise limit as established earlier). It should be noted that

at low signal-to-noise ratio, the filter can drive the weight magnitudes to zero, so the direction of

incoming signals becomes irrelevant, rendering the angle estimation procedure useless. Since the

sub-set of the correlation matrix is estimated by Rsub (which can be estimated by averaging over

multiple snapshots), and the cross-correlation vector by ~p (again this can be averaged) one obtains

a simplified method of determining the optimum weight vector, as in Eq 3.65.


Rsub~w0 − ~p = 0

~χsub~χHsub

~w0 − ~χsubd∗cross = 0

~χsub

(~χH

sub~w0 − d∗cross

)= 0

~χHsub

~w0 = d∗cross

(3.65)

If multiple snapshots are to be used in conjunction with Eq. 3.65 using the psuedo-inverse, [10],

on a matrix with rows comprised of many measurements of χHsub will give the minimum-norm, least-

squares solution for the optimum weights. As in the previous case where two signal directions are

estimated, under the high signal-to-noise ratio approximation for an array with ndof degrees of

freedom one can generally factor the polynomial with weight coefficients to produce:

zndof − w01zndof−1 − . . .− w0(ndof−1)z − w0ndof

=ndof∏y=1

(z − eiαy

)= 0 (3.66)

Though the linear prediction filter is easy to implement it does become biased at low signal-to-noise

ratio as demonstrated above for the three element array. However the robustness of this filter will be

examined in the proposed research because it is not only used in the only existing commercial version

of a MASB sonar (Benthos Corporation’s C3D sonar) but has proven to be useful in practical survey

scenarios during the course of this research. In addition, the closeness of the roots on the right side

of Eq. 3.66 to the unit circle in complex space can be compared to a threshold so as to eliminate poor

estimates (or extraneous roots corresponding to extra degrees of freedom in the estimator), however

this variation is not included in the current implementation of LP. The placement of extraneous

roots in the LP routine was examined in [6] and [22], wherein it was demonstrated that extraneous

roots migrate away from existing signals, and to a mean value located within the unit circle. For

example, in the case of ndof = 2 with only one signal on the array, the root corresponding to the

extraneous degree of freedom is located at an angle opposite the root corresponding to the signal

(180 degree separation in phase) and at a mean value of 0.5, which is easily identified through the

implementation of a radial threshold to the unit circle (in this case threshold is a chosen value less

than 0.5). In general, the greater the number of extraneous roots, the closer the extraneous roots

will be to the unit circle, necessitating the need for smaller thresholds.

Linear Prediction Examined Through Simulation

To test the linear prediction estimator of electrical AOA against the CRLBα, it is first necessary to

determine the electrical angle for the simulated data given in Fig. 2.11 (chapter 2). The parameters

for the survey geometry are again set to be: tilt angle 45◦, frequency 300kHz, altitude 40m, and

20 cycle SQ pulse. In adding noise to the simulated data, the case of high signal to noise is again

examined, and SNR = 40dB at the maximum range (for x > 0 signal), as demonstrated in Fig. 3.18


with simulated data in red, and theoretical snr in blue (note that the theoretical data, is the same

level as in Fig. 3.9). In Fig. 3.19, a simulation corresponding to a SQ pulse waveform is utilized, and

estimates are obtained using five snapshots (taken for the same range cell from different simulated

pings) of a three element array with one degree of freedom in the linear prediction routine (i.e. two

subarrays of two elements spaced at d). The solid red line represents the actual angle of arrival that

corresponds to the bottom, and the simulated data is given in black. As can be seen from the solid

line, there are signals from two angles for ranges from 40 m to about 56 m, and beyond that only

one angle. The larger angles past 127◦ correspond to the secondary signal (x < 0 signal), and the

smaller angles to the primary signal (x > 0 signal). Because only one degree of freedom is being

used to estimate the electrical angle, the angles corresponding to the weaker secondary signal are

completely lost. Nevertheless, the presence of the secondary signal produces a noticeable bias on the

angle estimates for the primary signal in the region where the secondary signal exists. The uniform

random scatter of the angle estimates for ranges less than 40 m is due to the presence of simulated

thermal noise which has no preferred angle. For ranges greater than 40 m, the zero of the polynomial

for estimating the angle migrates to the arrival angle because of the presence of the signal, which is

much larger than the noise.

−40 −20 0 20 40 60 80 100 1200

10

20

30

40

50

60


SN

R [d

B]

Figure 3.18: A comparison of snr for both the theoretical value (in blue) and simulated signal (inred) for the survey geometry.


0 20 40 60 80 100 120−100

−50

0

50

100

150

200

Range from Source(m)

Ele

ctric

al P

hase

[deg

rees

]

Figure 3.19: The electrical phase of the simulated signal with noise (black dots), estimated from onedegree of freedom. The solid red curve is the expected electrical AOA.

Fig. 3.20 is for the same situation as Fig. 3.19, but two degrees of freedom are used to estimate the

angle instead of just one. Again, there is a uniform random scatter of angle estimates for ranges less

than 40 m because of noise. The density of these scatters is greater than in Fig. 3.20 because there

are twice as many estimates. For ranges between 40 and 56 m, both degrees of freedom migrate to

the signal angles; hence, there is no longer a uniform scatter of angle estimates. For ranges greater

than 56 m, the secondary signal disappears, and only one degree of freedom is needed; one of the

degrees of freedom migrates to the signal angle. In Fig. 3.20, the angles for the secondary signal are

estimated as well as those for the primary signal. The reason the spread is greater for the secondary

signal is that the snr is much lower than for the primary because of the cosine beam pattern and

the secondary signal being near endfire. More importantly, there is no longer a noticeable bias in

the angle estimates for the primary signal. Therefore, it is concluded that the angle estimation

procedure should have degrees of freedom consistent with the angle arrival structure. This is a

general rule of thumb that will be adopted in the estimators that follow in this section, as it was

mentioned earlier that the number of combinations of scenarios, including both too few and too many

degrees of freedom is vast and deserves its own exploration in the future. For swath bathymetry in

the absence of multi-path, a two signal scenario is generated whenever the array is tilted from the

horizontal. Hence, at least two degrees of freedom are required to accurately estimate angles in the


region |x| < xlim, where xlim was defined in Fig. 2.7. One further note regarding Fig. 3.20, it is seen

that in the case of using two degrees of freedom in the region beyond 56m where there is only one

signal present, the implementation of the Linear Prediction algorithm tends to place the estimate of

the second signal away from the estimate of the first signal. This is the behavior mentioned earlier

that is predicted for extraneous roots in [6] and [22].

0 20 40 60 80 100 120−100

−50

0

50

100

150

200


Ele

ctric

al P

hase

[deg

rees

]

Figure 3.20: The electrical phase of the simulated signal with noise (black dots), estimated from twodegrees of freedom. The solid red curve is the expected electrical AOA.

To illustrate the spread in electrical angle estimates with respect to the CRLBα, the known

absolute electrical angle was subtracted from the estimates to produce estimates centered around

zero. Fig. 3.21 shows these estimates with the ±2√

CRLBα (upper and lower bounds given in

green and red respectively). This result is for the same scenario as depicted by the ±2√

CRLBα in

Fig. 3.10. In processing the simulated signals the choice was made to only use the same number of

degrees of freedom as signals present on the array, therefore two degrees of freedom were used from

nadir out to a horizontal range of 40 m (xlim in this case) and then one degree of freedom was used

out to the maximum horizontal range. The percentage of points inside ±2√

CRLBα is about 85%,

indicating that the linear prediction method used, while not producing results equal to the CRLB

(i.e., 92% → 93% for five snapshots as indicated by Fig. 3.8), does produce results that are useful.


−40 −20 0 20 40 60 80 100 120−60

−40

−20

0

20

40

60


Ele

ctric

al A

ngle

Err

or [d

egre

es]

Figure 3.21: The error in electrical phase linear prediction estimates (black dots) for the SQ pulse (5snapshots from multiple pings) as determined by taking the difference of the theoretical AOA, andthe estimated AOA from the simulated data (using one or two degrees of freedom where appropriate).Also shown is 2

√CRLBα in green and−2

√CRLBα in red for each of the rance cells in the simulation.

Fig. 3.22 displays the angle estimation accuracy for a MFSQ pulse for the same scenario as for

Fig. 3.21. The bounds were calculated using a MFSQ pulse derived from the 20 cycle SQ pulse, and

the noise level was set accordingly after the matched filtering. Since signal-to-noise ratio against

thermal noise was 40 dB or higher, the dominant mechanism for angle estimation error was footprint

shift. After matched filtering, however, the effect of footprint shift is reduced, and the effect of

thermal noise comes into play. In terms of performance against the√

CRLBα, matched filtering

yields results consistent with those obtained previously for a square pulse, in that the angle estimates

are within ±2√

CRLBα about 85% of the time. Also, the angle estimates are significantly more

accurate because of the reduction of the footprint shift effect.

Finally, it should be noted that in practical survey geometries, to achieve multiple snapshots for an

estimator, it is not often the case that adjacent pings can be used (due in part to changing bottom

geometries and vessel orientations), therefore multiple range cells must be utilized in the fashion

displayed in Fig. 3.1. In the previous figures, where the five snapshots processed were taken from

five separate pings at the same range. This method of obtaining the five snapshots is useful for


−40 −20 0 20 40 60 80 100 120−20

−15

−10

−5

0

5

10

15

20


Ele

ctric

al A

ngle

Err

or [d

egre

es]

Figure 3.22: The error in electrical phase linear prediction estimates (black dots) for the MFSQpulse (5 snapshots from multiple pings) as determined by taking the difference of the theoreticalAOA, and the estimated AOA from the simulated data (using one or two degrees of freedom whereappropriate). Also shown is 2

√CRLBα in green and −2

√CRLBα in red for each of the range cells

in the simulation.

analysis because the model is exactly the one proposed by the equations. In practical applications,

this method of averaging snapshots is equivalent to along-track averaging where the along-track

distance between pings is large enough that the scatterers are not correlated from ping to ping.

While this form of averaging may be applicable in some applications, typically the separate array

snapshots must be obtained from the same ping. For analysis with a single ping, range samples

are chosen far enough apart that they are not correlated, and these samples are used as separate

snapshots. Fig. 3.23 shows the same situation as in Fig. 3.21, but the five snapshots are chosen

from a single ping. In the simulation samples were spaced 0.48 m apart and therefore are much

farther apart then the 0.1m pulse length (calculated for two-way path), and so adjacent samples are

uncorrelated (in real data the sample separation is much smaller, and so sufficient sample spacing

must be used, however uncorrelated samples can be taken from a much smaller separation than used

here in the simulation). Therefore, by obtaining the snapshots from the same ping, range resolution

is sacrificed resulting in fewer independent angle estimates. In Fig. 3.23, it is evident that there are

fewer angle estimates than in Fig. 3.21, but they are still mostly within ±2√

CRLBα, approximately


82% versus the 85% obtained with snapshots obtained from separate pings. Therefore, it is concluded

that obtaining the snapshots from the same ping reduces range resolution, but the angle estimates

are almost as accurate.

−40 −20 0 20 40 60 80 100 120−60

−40

−20

0

20

40

60


Ele

ctric

al A

ngle

Err

or [d

egre

es]

Figure 3.23: The error in electrical phase linear prediction estimates (black dots) for the SQ pulseas determined by taking the difference of the theoretical AOA, and the estimated AOA from thesimulated data (using one or two degrees of freedom where appropriate). The 5 snapshots usedfor each estimate are taken from within a single ping, therefore fewer estimates are shown than inFig. 3.21. Also shown is 2


√CRLBα in red for each of the range cells in

the simulation.


3.7.2 Minimum Eigenvalue Analysis

An alternative to the linear prediction filter design is one in which all inputs are weighted to

minimize the output power. As in the case of linear prediction, minimum eigenvalue analysis has

been used previously as an AOA estimator (see [40] and associated references), a brief derivation is

presented here so that the exact implementation is specified (again several variations on this method

exist). The form of this filter is that of the right side of Fig. 3.17. To begin, the output power from

the filter Pave is given by Eq. 3.68.

Pave = E{

~wH ~χ(~wH ~χ

)H}

= ~wHE{~χ~χH

}~w

= ~wHR(N×N) ~w

(3.67)

Using Eq. 2.18, and imposing the constraint ~wH ~w = 1 on Eq. 3.67 yields Eq. 3.68.

Pave = ~wH(ΣK

ι=1Rι(N×N)

)~w + 2σ2

n, (3.68)

If there are fewer signals than degrees of freedom available in the array, nulls in the weight vector

can be steered to eliminate all signals, which leaves a minimum power Pmin corresponding to the

noise alone.

Pmin = 2σ2n (3.69)

Equating Eq. 3.67 to Eq. 3.69 gives an expression for the minimum eigenvalue of R(N×N) , namely

Eq. 3.70.

~wHR(N×N) ~w = 2σ2n ~wH ~w

~wH(R(N×N) ~w − 2σ2

n ~w)

= 0

R(N×N) ~w − 2σ2n ~w = 0

(3.70)

Therefore, the minimum eigenvalue is just the noise power, 2σ2n, as stated previously. To determine

the corresponding filter weights, the normalized eigenvector must be calculated. For the case of two

signals, and a three element array, the signal was determined to be Eq. 3.59 and R(3×3) is thus given

by Eq. 3.71.

R(3×3) = E{~χ~χH

}

=

2σ21 + 2σ2

2 + 2σ2n 2σ2

1e−iα1 + 2σ22e−iα2 2σ2

1e−i2α1 + 2σ22e−i2α2

2σ21eiα1 + 2σ2

2eiα2 2σ21 + 2σ2

2 + 2σ2n 2σ2

1e−iα1 + 2σ22e−iα2

2σ21ei2α1 + 2σ2

2ei2α2 2σ21eiα1 + 2σ2

2eiα2 2σ21 + 2σ2

2 + 2σ2n

(3.71)


The eigenvector corresponding to the minimum eigenvalue of Eq. 3.71 can be computed, and is

given by Eq. 3.72.

~w =1√

4 + 2 cos (α1 − α2)

[1 −e−iα1 − e−iα2 e−iα1−iα2

]H

(3.72)

Eq. 3.72 is reminiscent of the weight vector found in the high signal to noise ratio limit for the

linear prediction estimator (multiplied by a normalization constant to ensure that the constraint

condition is satisfied). By re-scaling the weight vector so that the first component is 1, one finds

that the same polynomial expression given in the linear prediction approach. In a similar fashion

the above development can be extended to a larger array, with more signals.

Due to the eigenvalue / eigenvector calculations, the minimum eigenvalue AOA estimation tech-

nique demonstrated above is more complicated in practical implementation than the simple linear

prediction scheme. In addition, the calculation of smallest (non-zero) eigenvalue is computationally

expensive.

Minimum Eigenvalue Analysis Examined Through Simulation

The same simulation data that was presented earlier for linear prediction estimation was processed

using the minimum eigenvalue estimation technique. Having outlined much of the physical interpre-

tation for the previous estimator, the emphasis in this section will be on the challenges specific to

the minimum eigenvalue AOA technique. Fig. 3.24 shows the AOA estimation using the minimum

eigenvalue technique with two degrees of freedom for the same survey geometry as was demonstrated

in Fig. 3.20 (three element array). The first difference that can be noted is that the minimum eigen-

value technique does not appear to place the estimate of the second signal away from the estimate

of the first signal in the region beyond 56m, where there is only one signal on the array (there is one

extra degree of freedom).

The extra degree of freedom becomes problematic in comparing AOA estimates to the CRLBα for

both the SQ pulse and MFSQ pulse. In Fig. 3.25, the minimum eigenvalue method was implemented

similar to Fig. 3.21, however there is not an obvious choice for reducing the number of degrees of

freedom in the region beyond 40m of horizontal range. The technique that was chosen in this region

of Fig. 3.25 was to first compute the AOAs corresponding to the two degrees of freedom, and then

by beamforming in each of the estimated directions, and chose the AOA that corresponded to the

greater power as being the ”correct” AOA. For the SQ pulse, in the region |x| < 40m where two

signals are present, approximately 77% of the estimates fell between ±2√

CRLBα, whereas for the

full plot only 70% came between the bounds. In either case, performance was lower than in the

linear prediction estimates, however, it is particularly notable that the extra degree of freedom in

the |x| > 40m one signal region seems to degrade the estimator.


0 20 40 60 80 100 120−100

−50

0

50

100

150

200


Ele

ctric

al P

hase

[deg

rees

]

Figure 3.24: The electrical phase of the simulated signal with noise (black dots), estimated usingthe minimum eigenvalue estimator with two degrees of freedom. The solid red curve is the expectedelectrical AOA.

In Fig. 3.26 the MFSQ pulse was examined, and a similar result to Fig. 3.25 was observed in the

|x| > 40m one signal region. Fig. 3.26 utilizes the same parameters as Fig. 3.22, with the difference

being only the estimation technique. For the MFSQ pulse in the region |x| < 40m where two signals

are present, approximately 82% of the estimates fell between ±2√

CRLBα, whereas for the full plot

only 69% came between the bounds. Again, the performance of the minimum eigenvalue technique

was worse than the linear prediction, however in the region where the number of signals equaled the

degrees of freedom, the performance was only marginally worse than the linear prediction estimation.

AS the scope of this section is simply to demonstrate the utility of the√

CRLBα in benchmarking

estimator performance, modifications to this particular estimator to increase performance will not

be discussed. It should be noted that mismatch in number of signals and degrees of freedom is a

problem that effects many AOA estimation techniques, and will be investigated again in the next

estimation routine, Minimum Variance Distortionless Response Beamforming.


−40 −20 0 20 40 60 80 100 120−60

−40

−20

0

20

40

60


Ele

ctric

al A

ngle

Err

or [d

egre

es]

Figure 3.25: The error in electrical phase minimum eigenvalue estimates (black dots) for the SQpulse (5 snapshots from multiple pings) as determined by taking the difference of the theoreticalAOA, and the estimated AOA from the simulated data (using one or two degrees of freedom whereappropriate). Also shown is 2



in the simulation.


−40 −20 0 20 40 60 80 100 120−20

−15

−10

−5

0

5

10

15

20


Ele

ctric

al A

ngle

Err

or [d

egre

es]

Figure 3.26: The error in electrical phase minimum eigenvalue estimates (black dots) for the MFSQpulse (5 snapshots from multiple pings) as determined by taking the difference of the theoreticalAOA, and the estimated AOA from the simulated data (using one or two degrees of freedom whereappropriate). Also shown is 2



in the simulation.


3.7.3 Minimum Variance Distortionless Response Beamforming

When examining the power incident on an array, one can weight the input variables such that

there is a minimum variance on the output of the beamformer. Therefore, if a preferential direction

is chosen for a beamformer, the weight vector is chosen such that it steers nulls in the beam pattern

to minimize the effect of signals coming from other directions (as dictated by the correlation matrix),

while keeping the signal strength in the desired direction at unity (or some set gain). This technique

is known as Minimum Variance Distortionless Response beamforming (MVDR) and requires use of

a constraint equation Eq. 3.73 on the weights (which were defined previously in Eq. 3.67). The

derivation provided below is a standard technique following [15] and is presented for completeness,

so that the exact implementation of the MVDR estimator in this thesis can be distinguished from

any variants on the method.

~wH~s (α0)− g = 0 (3.73)

In Eq.3.73 the gain of the beam along the direction α0 (electrical angle) is fixed at g, and the

beam steering vector for an array of length N is given by Eq. 3.74.

~s (α0) =[

1 ejα0 ej2α0 ... ej(N−1)α0

]H

(3.74)

By using the method of Lagrange multipliers (here the multiplier is λLagrange), and the above

constraint condition, one defines the mean squared error, Jerror.

Jerror = ~wHR~w + λLagrange

(~wH~s (α0)− g

)(3.75)

Taking the gradient of the minimum mean squared error with respect to ~w and setting it to zero

(i.e. finding the minimum) yields Eq. 3.76.

∇~wJ = 2R~w0 + λLagrange~s (α0) = 0 (3.76)

Eq. 3.76 is rearranged to solve for the optimum weight vector w0.

~w0 = −12λLagrangeR

−1~s (α0) (3.77)

Substituting Eq. 3.77 into the constraint equation Eq. 3.73 provides a value for g.

g = ~wH0 ~s (α0) = −1

2λLagrange~s

H (α0)(R−1

)H~s (α0) (3.78)


From Eq.3.78 the Lagrange multiplier can be determined.

λLagrange = − 2g

~sH (α0)R−1~s (α0)(3.79)

Setting the value of the gain g = 1, the corresponding weight vector is then given by Eq. 3.80.

~w0 =R−1~s (α0)

~sH (α0) R−1~s (α0)(3.80)

Finally, the minimum mean squared error, Jerror min, can be determined using ~w0 and the covari-

ance matrix.

Jerror min = ~wH0 R~w0 =

~sH (α0)(R−1

)HRR−1~s (α0)

(~sH (α0) R−1~s (α0))2 =

~sH (α0)R−1~s (α0)

(~sH (α0) R−1~s (α0))2 =

1~sH (α0) R−1~s (α0)

(3.81)

Thus, by leaving the electrical angle α0 → α unfixed one can scan through Jerror min and find the

directions that display the maximum power (will appear as peaks in this function). MVDR gives

a far narrower power response in the angle domain than a conventional beamformer. However, one

limitation of the MVDR procedure is that the number of nulls that can be steered into the beam is

two less than the number of elements in the array ([15] page 21).

The procedure for processing the data requires that a sweep of the angle be made for each calculated

value of the correlation matrix. In each sweep the local maximums must be found in the spatial

power spectrum. Computationally this can be a tedious procedure. Angles that are close in phase

space can also influence each other, causing a bias in the final estimated angle(s).

One application where this has already been realized is in the implementation of a regulated

MVDR beamformer (see [15] pg 409), which has the advantage of slowing any fluctuations of the

spatial power spectrum.

MVDR Examined Through Simulation

The same simulation data that was presented earlier for linear prediction estimation and the

minimum eigenvalue estimation technique was processed using the MVDR estimation technique.

Having outlined much of the physical interpretation for the linear prediction estimator, the emphasis

in this section will be on the challenges specific to the MVDR AOA technique. Fig. 3.27 shows the

AOA estimation using the MVDR technique for the same survey geometry as was demonstrated

in Fig. 3.20 (three element array). In the same manner as was observed with the linear prediction

estimation, MVDR places the estimate of the second signal away from the estimate of the first signal


in the region beyond 56m, where there is only one signal on the array, unlike the minimum eigenvalue

technique, where estimates were distributed randomly in electrical phase.

0 20 40 60 80 100 120−100

−50

0

50

100

150

200


Ele

ctric

al P

hase

[deg

rees

]

Figure 3.27: The electrical phase of the simulated signal with noise (black dots), estimated using theMVDR technique. The solid red curve is the expected electrical AOA.

In Fig. 3.28 the error in electrical phase MVDR estimates are presented for the simulated SQ

pulse. Similar to the linear prediction estimator, 83% of the estimates fell within ±2√

CRLBα of the

theoretical electrical phase. Although two angles are estimated in the single signal region, |x| > 40m,

beamforming was used for each of the estimated directions and the AOA that corresponded to the

greater power was chosen as being the ”correct” AOA (this is similar to the minimum eigenvalue

estimation). Unlike the minimum eigenvalue technique, the number of estimates between the bounds

in the two signal region did not differ significantly from the one signal region.

Fig. 3.28 presents the error in electrical phase MVDR estimates for the simulated MFSQ pulse

utilizing the same parameters as Fig. 3.22. The number of estimates that fell within between

±2√

CRLBα was 82%. Similar to Fig. 3.28, and the linear prediction estimation technique there

was little difference between the one and two signal regions. It is again notable, that although the√CRLBα provides a benchmark against which estimators can be measured, it provides little insight

as to how estimators can be improved.


−40 −20 0 20 40 60 80 100 120−60

−40

−20

0

20

40

60


Ele

ctric

al A

ngle

Err

or [d

egre

es]

Figure 3.28: The error in electrical phase MVDR estimates (black dots) for the SQ pulse (5 snapshotsfrom multiple pings) as determined by taking the difference of the theoretical AOA, and the estimatedAOA from the simulated data. Also shown is 2


√CRLBα in red for each

of the range cells in the simulation.


−40 −20 0 20 40 60 80 100 120−20

−15

−10

−5

0

5

10

15

20


Ele

ctric

al A

ngle

Err

or [d

egre

es]

Figure 3.29: The error in electrical phase MVDR estimates (black dots) for the MFSQ pulse (5snapshots from multiple pings) as determined by taking the difference of the theoretical AOA, andthe estimated AOA from the simulated data. Also shown is 2


√CRLBα

in red for each of the range cells in the simulation.


3.8 Bottom Estimation Performance

3.8.1 Error Arc Length

It should be noted at this point in the analysis that the√

CRLBα on phase AOA is not itself

an intuitive measurement, considering that the objective of angle estimation in this research is to

determine the performance on bottom estimation in the context of the bathymetric surveys of lake

or sea-bed. To this point the accuracy of angle estimation for extended targets such as the bottom

is discussed from the point of view of determining the phase, or electrical AOA. A shift is now made

to the physical angle and ultimately bottom location accuracy. To arrive at an useful performance

measure it is first necessary to determine the corresponding bound√

CRLBθ on the estimate of

physical angle θ. To achieve this end, the relation between the phase and physical angles of arrival in

Eq. 2.16 must utilized in conjunction with the procedure of taking the CRLB under a transformation

of variables, such as given by [40] (pg. 929). The resulting bound on physical angle of arrival is given

by Eq. 3.82.

CRLBθ =(

∂α

∂θ

)−2

CRLBα

=(

λ

2πd cos(θ)

)2

CRLBα

(3.82)

Subsequently, a performance measure in the context of surveying is better understood if it is a

distance parameter. In this case the error in space can be considered to be the swing of arc that

corresponds to ±2√

CRLBθ on either side of the bottom location. The radius of the arc, rc, is given

by time of flight of a signal emanating from the position of the sonar. This quantity is referred to as

the error arc length (EAL). Strictly speaking the EAL will sweep out an arc that is not a vertical

range accuracy, as the pivot point is at the sonar, however, the component in the vertical direction

will always be smaller than the EAL. The EAL is represented mathematically as the product of

range and the aforementioned bound via Eq. 3.83.

EAL = 2rc

√CRLBθ (3.83)

Finally, it can be noted that sonar surveys standards are set by the International Hydrographic

Organization (IHO), which define that the quality of a bathymetric survey, see [14], is defined as a

specific uncertainly in depth, which is strictly vertical position of bottom. However, in utilizing the

EAL, the actual error in depth or distance along the bottom depends on the geometry. The EAL

is at right angles to the range vector at the point on the bottom. In other words, the estimated

bottom falls within ±EAL along an arc centered at the true location with the same probability that

the physical angle falls within ±2√

CRLBθ or that the electrical angle falls within ±2√

CRLBα.


Therefore, interpretation of the EAL must be made in light of the IHO standards, and understood

to trace out an arc, with only one axis of projection in the vertical plane. In which case it can be

noted that the depth uncertainty will always be less than the EAL, with the EAL providing more

insight as to how the estimation uncertainty is distributed, and should prove to be a useful asset in

assisting hydrographers in planning their surveys.


3.8.2 The Full Sonar Equation

Following previous results in this chapter, the three main contributions to the√

CRLBα, and

consequently to the EAL, are the effects of footprint shift, the effect of having multiple incoming

signals present on the array, and thermal noise. The effect of footprint shift is fully dependent on

the term δrc (given by Eq. 2.13), and is determined by geometry alone. The effect of having two

signals has to be dealt with on a case by case basis, with the knowledge that performance decreases

as angular spacing decreases between multiple signal AOAs. To examine the effect that thermal

noise will have on performance, the snr must be calculated. In order to determine the snr that

corresponds to a scenario that is typical of a swath bathymetry survey it is necessary to implement

the sonar equation (note, in practice snr can be measure reliably, and the sonar equation here is used

strictly to model and contrast performance of several different survey geometries and conditions).

For the purpose of the research presented here, a linear sonar equation was used, from which the

signal to noise ratio is given by Eq. 3.84, and all parameters will be briefly explained in the following

few paragraphs (detailed explanation of the contributions to the sonar equation is not the scope of

this research and have been previously developed in many texts on sonar and acoustics such as [4]).

snr =λ2η1η2G1G2ts(ss)

(4π)2r410

−2α0r10

P

kBT∆fB4(θ) (3.84)

In Eq. 3.84, B(θ) is the beampattern, which as mentioned earlier is given by B(θ) =√

cos(θ). The

transducer efficiencies of the transmitter and receiver are given by η1 and η2, which are calculated

in practice from the relative impedance of a transducer measured in water to the impedance of

the same transducer in air. A value for the transducer efficiencies was chosen to be 0.2, and is

based on measurements of physical transducers used in practice of swath bathymetry. Gains for the

transducers are given by G1 and G2, and are calculated using the alongtrack beamwidth, θli , and

the acrosstrack beamwidth θci via the formula Gi = 4πθliθci

(note that beamwidths are in radians).

For swath bathymetry applications, the along track beam width is typically small, and the across

track beam width is large (fan beam). For the results presented in this thesis, the along track

beamwidth expressed in degrees is 1◦, and the across track beam width is 120◦ which is consistent

with a cosine beam pattern in this plane. The resulting gain is 343.8 or 25.4 dB. The λ2 term is

also required in conjunction with the gain and efficiency of the receiver to fully model the effective

area of the receiver. To model the effects of spreading, two factors of 1(4π)r2 must be applied in the

sonar equation. This represents spherical spreading in both in the path from the transmitter to the

bottom, and the path back from the bottom to the receiver.

The target strength, ts(ss), is calculated by multiplying the footprint area by the bottom backscat-

ter strength, ss (a value of 10−3 per m2 is chosen for the modeling in this chapter, which empirical

measurements exist [42] showing this can be anywhere from a bottom of silt to sand, depending on


composition and grain size). Footprint area is calculated by approximating the footprint as a rectan-

gle of acrosstrack length given by√

r2pulse/ sin(φc) (where

√r2pulse is the mean squared pulse length)

and alongtrack length xcθli (where θli is given in radians). An allowance is also made in ts(ss) for

dependence of scattered power on grazing angle, which for the purposes of this research is given by a

simple approximation known as Lambert’s Law (empirical data supporting this law is found in [24]).

This law is stated in [42] via the definition: ”power is assumed to be scattered proportionately to

the sine of the angle of scattering” (grazing angle). Just for completeness, it should be noted that

many other alternative grazing angle dependance models exist, for instance some are presented in

[29].

The range attenuation of the signals in water is given by α0, and is expressed in dB per meter.

The value of the attenuation varies depending on whether the measurement is performed in fresh or

salt water (for a more detailed discussion of the effects that contribute to attenuation such as shear

and bulk viscosity, and the effects of ionic recombination of various substances in salt water, see

[19]). In this chapter, values of attenuation are again taken from [42], and are given for salt water

as: 300kHz α0 = 0.063 dB/m, 200kHz α0 = 0.047 dB/m, 100kHz α0 = 0.030 dB/m. In addition, for

fresh water at 300kHz, α0 = 0.013 dB/m.

The transmit peak power power in the sonar is P , and though not explicitly stated, is taken in

ratio with the noise power to determine the snr. In Eq. 2.78, the noise voltage value depends on the

temperature, real component of impedance and bandwidth. However, since the formulation of snr

here compares signal power to noise power it is useful to note that the noise power is independent

of impedance, Pnoise = σ2n

<{Z} = 4kBT∆f . In this model the value of ∆f = 15000Hz, which is the

frequency divided by the number of cycles of a square pulse, f/Nsq, for a 300kHz square pulse with

20 cycles. In Eq. 3.84 the factor of 4 noted in Eq. 2.78 is eliminated in the formulation of the sonar

equation by accounting for only the rms noise power delivered to a matched load as seen by the

receiver.


3.8.3 Survey Scenarios

The snr given in Eq. 3.84 will now be applied to the earlier calculation of CRLBα so as to generate

the EAL (from Eq. 3.83) for practical survey geometries, under modeled ocean and lake conditions.

As demonstrated previously, the performance in terms of the CRLBα for the MFSQ, FQ, MFFQ

and CG pulses is similar (for cτ = 1). In addition, these pulses all perform better than the SQ pulse.

Therefore, to examine the expected performance of MASB systems under practical survey scenarios,

the MFSQ, and CG pulses will be considered. In addition, the CG pulse will be considered under

both the cases of having no compression, and with a compression ratio of cτ = 10.

In Fig. 3.30, the snr for both x > 0 and x < 0 signals is displayed for a 3 element 300kHz

sonar, tilted at 45◦, and used in salt water. The pulse here is a MFSQ pulse of 20 cycles in

length. It is evident from the snr that the signal under consideration has high value of snr at close

ranges, dropping below 20dB only beyond 100m for the x > 0 signal. Using the snr in Fig. 3.30,

the corresponding EAL is calculated in Fig. 3.31 (for MFSQ and CG pulses with and without

compression). The black solid curve in Fig. 3.31 represent the value of the EAL corresponding to

the MFSQ pulse in Fig. 3.30, where both signals are taken into account. The dashed black line

represents the EAL for the MFSQ pulse when only the x > 0 signal is taken into account, and the

black asterisks are for the simple approximation given in Eq. 3.52 (which was stated previously to

represent the effects of thermal noise across the array and footprint shift between adjacent elements).

The curves reach a minimum at broadside to the array (x = 40m) as the contribution from footprint

shift is absent at this point on the bottom, and the effect of thermal noise is low because the range is

not yet too great and the snr is still high (also only one signal needs to be estimated). The shape of

the curves in the double angle region, |x| < 40m, is observed to be influenced by the effects of having

to estimate two signals, as the solid curve is noticeably higher than both the one signal EAL and

the approximation. The break and dramatic increase in the curves at nadir is due to the breakdown

of the CRLBα. At far range, signal levels eventually drop (mainly due to range spreading and range

attenuation terms in the sonar equation) such that eventually thermal noise dominates and the EAL

increases accordingly. Tracking the black curves in Fig. 3.31 very closely are the corresponding EALs

for the CG pulse in red, where there is no pulse compression and the root mean squared pulse length

is the same as the MFSQ pulse. For the purposes of the current section, this pulse will be referred

to as the match filtered gaussian pulse. The solid red curve represents the EAL for the case of two

signals, the dashed red curve is for only the x > 0 signal and the approximations from Eq. 3.57 are

given as red asterisks. The improvement in performance between the black and red curves (i.e. lower

EAL) represents a gain against thermal noise that occurs for the match filtered gaussian pulse. The

green solid curve is the EAL for the compressed gaussian with cτ = 10, considering the effects of both

x > 0 and x < 0 signals. It should also be noted that the compressed root mean squared length is set

to the corresponding length for the MFSQ pulse described above. The dashed green curve represents


the EAL for only the x > 0 signal and the asterisks are for the corresponding approximation again

given by Eq. 3.57 (although the gain is now cτsnr). The behaviors of the gaussian pulse both with

and without compression are similar to those described above for the MFSQ pulse, with adjustments

for the relative gain in snr as already described. Furthermore, the color and line style conventions

described above are also used to represent the various EAL curves in the following Figs. 3.32, 3.33,

3.34, however as will be described later, various parameters will be changed for each figure.

−40 −20 0 20 40 60 80 100 120−10

0

10

20

30

40

50

60

70


SN

R

Figure 3.30: The snr for a survey geometry using the sonar equation from Eq. 3.84. In this casea 300kHz MFSQ pulse MASB sonar with a 3 element array is tilted at 45◦ in salt water. The redcurve is for the x < 0 signal, and the blue curve is for the x > 0 signal.

The reference level of EAL performance in bottom estimation is set at 1m for this research, it can

be noted for a three element array that the MFSQ and uncompressed gaussian pulse achieves this for

−10 < x < 80m, except for about an 8m break around nadir. For the compressed the performance

range in the x > 0 direction increases to 105m.

Fig.3.32 shows the EAL for the same survey geometry as Fig.3.31, however the number of element

in the array has been increased from 3 to 6 (here color and linestyle are repeated from Fig.3.31,

corresponding to same pulses and approximations). It is seen that by increasing the number of

elements there is gain over thermal noise, so the EALs are closer to the limit imposed by footprint

shift. The region for which the EAL is less than 1m now extends from −18 < x < 101m for the


−40 −20 0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Err

or A

rc L

engt

h (m

)

Figure 3.31: The black curve is the EAL corresponding to the same survey geometry in Fig. 3.30(MFSQ pulse). The red curves display the EAL for the match filtered gaussian pulse, and the greencurves are for the compressed Gaussian pulse. In all curves the solid lines are for the double angleregion, the dashed are for the x > 0 signal only, and the asterisks are for the approximations.

MFSQ and cτ = 1 gaussian pulse (again with a gap around nadir, where estimation of two signals

breaks down). In the case of the compressed gaussian, the range extends even further to 128m (not

shown on plot). Therefore, increasing the number of elements significantly increases the effective

performance range by mitigating the effects of thermal noise.

Fig.3.33, the EALs are again presented for a six element array, however now the tilt angle has

been lowered from 45◦ to 20◦. The double angle region now only extends −14.6 < x < 14.6m. The

simple approximations now hold for the MFSQ pulse for x > 14.6m, however they break down and

do not agree well for the gaussian (for x < 20m) and compressed gaussian pulses (for x < 50m) due

to very high snr. This breakdown can be attributed to the multiple element array actually having

some gain against footprint shift in scenarios of high thermal snr, and therefore the approximation

will actually be higher than the actual value of the EAL as demonstrated in Fig.3.33. It should

be noted that for this tilt angle broadside to the array occurs at a horizontal range of 110m. At

broadside thermal noise has already become the dominant mechanism of decorrelation, and thus the

effects of footprint shift disappearing cannot be discerned. The range for an EAL < 1m now extends

from 4 < x < 110m for the MFSQ and uncompressed Gaussian pulses, whereas it has increased to


−40 −20 0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Err

or A

rc L

engt

h (m

)

Figure 3.32: The EALs corresponding to a 6 element array, under the same survey conditions asused in Fig.3.31. The black curves display the MFSQ pulse, the red curves display the EAL for thematch filtered gaussian pulse, and the green curves are for the compressed Gaussian pulse. In allcurves the solid lines are for the double angle region, the dashed are for the x > 0 signal only, andthe asterisks are for the approximations.

142m for the compressed gaussian pulse.

Fig.3.34 shows the EALs for a tilt angle of zero degrees, which has no double angle region. The

contributions to the performance are a combination of footprint shift and thermal snr, the latter

of which dominates at small x because the beampattern of the transmit and receive transducers is

a shape that is tapered to zero at endfire to the array (in this case pointing at the bottom). The

distance for which the EAL is now less than 1m extends from 15 < x < 105m for the MFSQ and

uncompressed gaussian pulses, and 9 < x < 138m for the compressed gaussian.

The previous set of figures showed the EAL for variation in tilt angle, with the performance at

far range being dominated by the thermal snr. For the simple propagation model employed in

Eq. 3.84, two range dependent mechanisms reduce the signal strength, namely spherical spreading

and attenuation. If the frequency is lowered, the attenuation is decreased therefore the performance

at greater range is improved. For an 6 element array tilted 20◦, Fig. 3.35 shows the EAL for a 20

cycle MFSQ pulse at three different frequencies, 300kHz (black), 200kHz (red) and 100kHz (blue).

Again the solid lines at x < 14.6m represent the bound for the two signal region, the dashed curves


0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Err

or A

rc L

engt

h (m

)

Figure 3.33: The EALs under the same survey conditions as used in Fig.3.32, however the tilt anglehas been changed to 20◦. The black curves display the MFSQ pulse, the red curves display the EALfor the match filtered gaussian pulse, and the green curves are for the compressed Gaussian pulse.In all curves the solid lines are for the double angle region, the dashed are for the x > 0 signal only,and the asterisks are for the approximations.

represent the EAL for the x > 0 signal only and the approximations are given by the asterisks for

each of the frequencies. In the near horizontal range, the EALs are all similar, meaning in this

case that the bounds are dominated by the effects of footprint shift, however as range is increased

eventually the thermal snr dominates the performance. As attenuation is higher at higher frequency,

the effects of thermal noise will become more evident at nearer range, and as frequency is lowered

the performance improves out to further ranges. In the case of Fig. 3.35, as in Fig. 3.33, broadside

is at 110m, which represents the point at which footprint shift is at a minimum. However both

the 300kHz and 200kHz signals become dominated by thermal snr before this point, with only the

100kHz signal displaying a dip in EAL at this horizontal range. The distance along the bottom for

which the EAL is less than 1m extends from x = 5m for all frequencies, out to 110m for 300kHz,

147m for 200kHz and 230m for 100kHz. It should also be noted that the resolution changes with

frequency as well (provided pulses are the same number of cycles as is the case here). For example,

the range resolution of the 100kHz sonar is 13 the 300kHz sonar, therefore to achieve improved EAL

performance, range resolution must be sacrificed. Also the larger array required for the 100kHz sonar

is 27 times heavier than that required for the 300kHz sonar, and requires an increase in the range


0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Err

or A

rc L

engt

h (m

)

Figure 3.34: The EALs under the same survey conditions as used in Fig.3.32, however the tilt anglehas been changed to 0◦. The black curves display the MFSQ pulse, the red curves display the EALfor the match filtered gaussian pulse, and the green curves are for the compressed Gaussian pulse.In all curves the solid lines are for the double angle region, the dashed are for the x > 0 signal only,and the asterisks are for the approximations.

that may be considered the far field.

Finally, Fig. 3.36 demonstrates the relative performance associated between surveying in salt to

surveying in fresh water for a MFSQ pulse of 20 cycles at 300kHz. The tilt angle is again set at 20◦.

The black set of curves is the for salt water, and is the same set of curves as the black curves in

both Fig. 3.35 and 3.33, namely the solid lines are for the double angle region, the dashed are for the

x > 0 signal only, and the asterisks are for the approximations. The red set of curves in Fig. 3.36

corresponds to the fresh water survey scenario, and the minimum distance for the EAL < 1m

is again 5m, however the acceptable range now extends to 165m (much better than the 109m in

salt). Therefore, if the application is fresh water, a higher resolution, smaller, lighter sonar may be

desirable (i.e. a higher frequency sonar in fresh water would give equivalent EAL performance to a

lower frequency unit in salt water), for it yields significant range.


0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Err

or A

rc L

engt

h (m

)

Figure 3.35: The EAL for the 20 cycle MFSQ pulse for 100kHz, 200kHz and 300kHz in salt waterunder the same survey conditions as used in Fig.3.33 (20◦ tilt angle). The black curves display theEAL for 300kHz pulse, the red curves display the EAL for the 200kHz pulse, and the blue curvesare for the 100kHz pulse. In all curves the solid lines are for the double angle region, the dashed arefor the x > 0 signal only, and the asterisks are for the approximations.


0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Err

or A

rc L

engt

h (m

)

Figure 3.36: The EAL for a 300kHz MFSQ pulse MASB sonar with a 6 element array, tilted at 20◦

in salt water (black curves), and fresh water (red curves). The solid lines are for the double angleregion, the dashed are for the x > 0 signal only, and the asterisks are for the approximations.


3.9 Summary of Chapter 3

The goal of this chapter was to derive a method for predicting the performance of MASB sonar for

the application of AOA estimation and consequently bottom location estimation. The performance

measure chosen was the CRLB, and investigation was made of the factors that influence AOA

performance, such as the shape of the transmitted waveform and decorrelation arising from physical

effects. One useful feature of this method was that it was independent of any estimator that might

be used to compute AOA for incoming signals on a MASB array, relying instead on the underlying

signal model. In order to demonstrate that the CRLB was a suitable performance measure, several

simple calculations needed to be performed.

First, the complex gaussian signal model is re-examined, and the pdf for the signal was defined.

Using the same signal model, the Fisher Information matrix and CRLB were calculated for the case

of either one or two simple plane waves impinging on a linear array. Through this simple calculation

it was demonstrated that in the case of a single signal, only the AOA of the signal needs to be

considered to determine performance, whereas in the case of two signals both signal strengths and

AOAs need to be considered. These two results were standard calculations and previously appeared

in the literature [40]. In order to demonstrate the influence of having a secondary signal on the

estimation of the AOA for a primary signal, the bound was plotted for three cases: no secondary

signal; a secondary signal with all signal strengths unknown; and a secondary signal where all signal

strengths are known. Included in this plot were the results for 3,4,5 and 6 element arrays. There

was a modest increase in performance for the case where all signal strengths are known, and it

was proposed that this may be an avenue for exploration in future research. In the case where

all signals are unknown, the bounds diverge as signals become closer, indicating that the AOAs

of the two signals become ambiguous, such that the estimation of either AOA becomes impossible

(although this does not rule out the possibility of reducing the model to instead perform a single

AOA estimation).

Following the calculation of the CRLB, it was necessary to provide a justification for use of the

CRLB as a performance measure. To accomplish this, the pdf of the phase AOA for a single

plane wave signal on a simple two element estimator was first calculated. Though the method of

arriving at the AOA pdf was new, the result was equivalent to [25] and associated references. In

addition, the pdf of the AOA estimator was used to calculate the variance of the AOA estimator

(also determined through different methods in [25]). After calculation of the variance was performed

most of the subsequent calculations and plots in this chapter are unique to the research in this thesis.

Next, the probability that an estimate of phase AOA was less than a number of multiples of the

standard deviation was plotted, demonstrating that the standard deviation does not provide a useful

confidence interval for the phase AOA (highly dependent on the snr). Conversely, the probability


that an estimate of phase AOA was less than a number of multiples of the√

CRLBα was plotted,

and it was demonstrated that√

CRLBα provides a useful confidence interval for the phase AOA as

all curves are independent of the snr. This is an important result, that is unique to the research

in this thesis, as it sets the framework for using the CRLB as a performance measure. Following

this result, it was also demonstrated that if the standard deviation of the AOA estimate was to be

estimated using simulated data (this was relevant because for most AOA estimators a closed form

for the variance is not easily calculable), for high snr the data converges to the actual value only

after an extremely large number of estimates are utilized. In practice, there would rarely be enough

estimates to have this value converge for even moderate to low snr. This provides another motivation

for further investigating the CRLB as the useful performance measure.

In order to compare phase AOA estimates that incorporate multiple snapshots, an investigation was

performed, again on the simple two element array, to determine whether pre-estimation averaging

or post-estimation averaging was preferential. In the case of pre-estimation averaging, a closed

form for the variance in the high snr limit was first calculated. This calculation is considered a

new result, not found in the prior literature. Alternatively the post-estimation variance on the

other hand was a simple calculation that was known prior to investigation. In order to determine

the preferred averaging technique, the two methods were plotted as a function of the number of

snapshots for several values of snr. The result of this plot was that the standard deviation of the

pre-estimation technique converges nearly to√

CRLBα after only 5 snapshots, whereas the post-

estimation technique remains at a set multiple of√

CRLBαs regardless of the number of snapshots

employed. This result, which was new to the literature, indicated that pre-estimation averaging

was preferable. In addition, this result suggests that for increasing snapshots, the variance of the

estimator converges to the bound, and therefore restores the traditional frequentist interpretation of

confidence intervals (yet another reason to utilize the CRLB as a performance measure).

In light of the determination that the CRLB was superior to the variance in describing AOA per-

formance, 2√

CRLBα was chosen as the proper performance measure (because possible convergence

to the standard deviation under multiple snapshot scenarios restores the traditional confidence inter-

val), and an investigation was performed to examine the performance of the five waveforms for which

correlation functions were developed in chapter 2, under various survey conditions. The first result of

this investigation was a demonstration that for the short pulses typically employed in MASB, base-

line decorrelation does not play a significant role in governing performance. This result was achieved

by showing that the the performance of an estimator using 5 snapshots for a 20 cycle SQ pulse was

not hindered by baseline decorrelation, whereas if a single 100 cycle SQ pulse were instead used

then the effects of baseline decorrelation begin to manifest. Given that footprint shift and thermal

noise are both demonstrated to be the dominant contributors to performance degradation, a simple

high snr approximation was developed to predict the performance that would be achieved for AOA


estimation if only one signal were present on the array. This approximation was used for the MFSQ

pulse and tracks the performance of the single x > 0 signal contribution to very near nadir. The

effect of having a second signal on the array was also shown to be detrimental to performance for all

waveforms. The SQ pulse was demonstrated to perform worse than all other waveforms, and it was

also recognized that the MFSQ, FQ, MFFQ, and CG pulse performance was very similar, leading to

the conclusion that only the MFSQ and CG pulses need be considered in subsequent calculations.

The MFSQ pulse was chosen, because the corresponding correlation function (especially under the

approximation of no baseline decorrelation) was easy to manipulate, and the CG pulse was chosen

because it allows investigation of pulse compression.

In order to compare the CRLB to various estimation routines, simulated data was produced. The

three estimators chosen to be benchmarked against 2√

CRLBα were linear prediction, minimum

eigenvalue analysis, and minimum variance distortionless response beamforming. Specific imple-

mentations were derived for each of the estimators. Of the various techniques, linear prediction

performed the best, with between 82% → 85% of the estimates falling between the performance

bounds depending on whether multiple snapshots were drawn from a single ping, or multiple pings.

The important result was that estimators can be benchmarked against the CRLB, and it was noted

that the expansive list of AOA estimators and various corresponding implementations was not the

focus of this research (just demonstrated that it was a useful tool).

Finally, in order to provide a more intuitive performance measure for bottom estimation, the Error

Arc length was defined. This measure gives the arc that corresponds to the radial vector from the

sonar to the bottom being swung through an angular range of ±2√

CRLBα. The significance of the

EAL was that it expresses the error as a distance measure, and it was expected that 80% to 85%

of the estimates will be at least this close to the true bottom location using the signal processing

methods outlined earlier. In order to achieve the most meaningful results, the full sonar equation

was utilized to determine the signal strength that would be seen on the sonar for several different

sets of survey parameters. The EAL corresponding to the MFSQ and CG pulses (with and without

compression) was calculated for a range of parameters including several frequencies, tilt angles, array

sizes and range attenuation that would be present in both salt and fresh water. Performance against

footprint shift and thermal noise were also discussed in the context of each set of survey parameters.

It was demonstrated that thermal noise becomes the dominant mechanism for performance loss as

the range was increased. The results of these investigations indicated that there was a trade-off

between resolution and maximum achievable range in the context of frequency, with lower frequency

sonar achieving greater range at the cost of lowered resolution. In general a larger array will perform

better against thermal noise, however its gain against footprint shift was only marginal as additional

elements are added. Finally, performance was seen to improve for operation in fresh water.

Chapter 4

Demonstration of a MASB

Apparatus

144

CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 145

4.1 Introduction

Sidescan sonar is used in many applications that do not require a measurement of bathymetry. The

image produced using sidescan represents the target strength (TS) captured in the acoustic beam at a

given time delay, which is used to calculate range. This energy is representative of both the TS of the

bottom, as well as surface reflections (in the shallow water environment), other multipath signals and

water column targets. In interpretation of sidescan imagery, often corruption from multiple signals

cannot be removed entirely. In addition, the location and orientation of the sidescan towfish relative

to the boat is uncertain, which will also add error to the estimated location of bottom features.

In examining bottom composition it may also be useful to have a record of bottom backscatter

strength (BBS), which sidescan alone is incapable of measuring. Only with bathymetric information,

and by eliminating multipath contributions, can one obtain an estimate of the BBS, which requires

both ensonified area and grazing angle of a pulse with the bottom.

An alternative to conventional sidescan is 3D sidescan sonar, such as the Multi-Angle Swath

Bathymetry (MASB) system used in this research (see [1] [2] [20] [21]). It differs from conventional

sidescan in that a multiple receive element array is utilized to both separate, and estimate arrival

angles of multiple incoming plane waves (see [1] and associated references). Bathymetry information

is calculated using the range of a target and angle of arrival (AOA) estimation. With co-located

bathymetry and backscatter data, a calculation of BBS can be performed, enhancing the image of

the bottom, and making image interpretation less subjective. The removal of multipath signals also

aids in the image correction.

As the MASB sonar only requires a small aperture (just slightly larger than sidescan), and uses

less channels than multibeam systems (hence fewer electronic components), it is an ideal solution for

small platform applications (such as autonomous underwater vehicles AUV). Surveying with MASB

sonar can also be performed at a number of depths (due to multipath elimination), making it possible

for a wider range of surveying alternatives such as surface boat mounts (covenient for shallow water),

and AUVs.

Pavilion Lake, located near the town of Cache Creek, British Columbia, Canada, has been used as

a survey testbed for the prototype sonar, mainly due its relevance as a site in the Canadian Analogue

Research Network (CARN). The microbialite formations within Pavilion Lake [12] are of significance

to analogue researchers due to their size, morphological variations, and ability to thrive in a cold

freshwater environment. In the research at Pavilion Lake, 3D sidescan surveying has resulted in high

resolution imagery of the backscatter and a useful bathymetric survey, using only orientation and

positioning data from inexpensive components and a fixed mount on a pontoon boat.


4.2 System Design and Survey Apparatus

The MASB system utilized for this research was designed and constructed in the Underwater

Research Laboratory (URL) at Simon Fraser University, and a schematic of the system is displayed

in Fig. 4.1. A field programmable gate array (FPGA) is used both to set the gain of the amplifiers,

and transmit two pulse control signals to the H-bridge circuit at the top of Fig. 4.1 (frequency and

number of pulse cycles are loaded into FPGA at start of every survey session). The H-bridge then

produces a high voltage pulse at the carrier frequency. The FPGA also provides the clock signal to

an 8 channel 16 bit A/D converter so that sampled data is always taken at the same delay intervals

following the ping transmission. The result of this feature is that the signal remains coherent over

multiple pings and the sonar is capable of measuring temporal coherence of the physical signal

environment.

The 16 bit A/D converter has ∼ 90dB of dynamic range, eliminating the need for time-variable

gain on the receive amplifiers, providing system noise does not fall to the level of bit noise in the

A/D converter. One benefit of this type of simplification is that the functional requirements of the

system are reduced. As the output signal-to-noise ratio will never be higher than the value at the

output of each receive transducer element, the amplifier and receiver setup was designed such that

the ratio of the signal strength to the total noise is optimized over the ranges of both frequency

( 100 → 400 kHz) and transducer impedances for which the sonar is designed to operate.

Figure 4.1: 6 channel MASB system components.


The URL constructed a beampattern measurement system for calibration of experimental trans-

ducers, with one transducer used as a fixed transmitter, and another transducer is rotated to measure

receive sensitivity. However, as the acoustic test-bed in the URL is less than 5m in length, far-field

measurements are only available for the across track beampattern (this corresponds to a pattern

around the axis aligned with longest dimension of element). Therefore, the total sensitivity of the

transducers has to be calibrated in another fashion and estimates of the target strength are scaled

arbitrarily. However, the relative backscatter strength between targets is correct. Both transmit

and receive transducers have been designed so that their beampatterns have as little distortion as

possible (beampatterns for a 6 element 300kHz receive transducer are given in Fig. 4.2), and the

receive transducers have also been designed to have as little crosstalk as possible.

Figure 4.2: Beampatterns for a six element receive transducer taken at 300kHz as measured in theURL (units of dB are scaled such that the max of the beampattern is at 0dB). The red points displaythe predicted cos2(θ) beampattern.

As shown in Fig. 4.1, the instrument uses one transducer for transmission of a narrow band pulse

train, with six transducer elements used for reception. The receive array is a filled linear array with

interarray spacing of λ2 or less, to avoid the complications of ambiguous signal arrival directions.

The survey geometry is shown in Fig. 2.1, where the ensonified area of the sonar is determined by

the alongtrack beamwidth, range, pulse length and grazing angle with the bottom. In using only 6

elements the total aperture width is guaranteed not to exceed 3λ, and the length of the aperture can

be decided for each application and is of similar length to sidescan systems. This is a much smaller

aperture width than multibeam echo sounders, and so this system can be used in the future for small

platform applications such as an Autonomous Underwater Vehicle (AUV) where space is limited.


For a complete surveying apparatus, it was required to integrate a compass and global positioning

systems with the MASB sonar to provide sonar position and orientation data, which is demonstrated

for a single survey run in Fig. 4.3. The digital compass module is a flux gate magnetic compass

(Honeywell HMR 3000), and was calibrated once installed on the pontoon boat. It was utilized for

heading information only, as pitch and roll data proved unnecessary with the stable platform of a

pontoon boat under the influence of only low wind and small waves. The two GPS units used in the

course of this research were the Garmin XL12 and a Trimble 5700/5800 differential GPS (DGPS)

system on loan from the Canadian Space Agency. Inaccuracies associated with the compass data

and conventional GPS can cause substantial increases in the uncertainty of the measured bottom

position, however the DGPS processing produces position accuracy usually less than about 0.05m.

For both the compass and conventional GPS modules, filtering routines were developed to smooth

data based on the physical limitations imposed by moving and turning the survey boat (for instance

the Garmin GPS data was filtered by using the UTM Easting and Northing as real and imaginary

components of a position vector that could then be low-pass filtered). Filtering improves both the

precision and the accuracy in the compass data, but the accuracy of the conventional GPS data

cannot be improved by averaging, as the mean value of the GPS is offset from the true position

due to atmospheric distortions. The compass mean accuracy can also be systematically off of the

true values of heading, pitch and roll if the unit is improperly calibrated (note that the calibration

routine must be performed, while unit is mounted to boat to avoid magnetic offsets intrinsic to the

actual survey vessel).

Finally, in order to correct survey measurements for refraction, the vertical sound speed profile

was estimated using a Seabird SBE19 conductivity-temperature-depth (CTD) profiler. Data was

collected using only the downcast, and a low-pass filter was used to smooth the data. CTD sensor

displacements were aligned before the data was processed to derive the sound speed as seen in

Fig. 4.4. The data shown are from September 22 2005, taken in the deepest basin of Pavilion Lake.

Multiple casts were performed at various points in the lake to ensure that corrections for refraction

are relatively close over the entire lake. As such the deepest profile is used, with a constant sound

speed assumed below the deepest measurement (measured profile does not change appreciably below

45m).

For the transmission of a plane wave traveling from one fluid, with sound speed c1 incident at

an angle of θi to the normal vector of the fluid interface, to a second fluid, with sound speed c2,

refraction changes the transmitted ray angle, θt , via Snell’s Law, Eq. 4.1.

sin(θi)c1

=sin(θt)

c2(4.1)


−2000 −1000 0 1000 2000 3000−3000

−2500

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0

500

1000

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Easting [m]

Nor

thin

g [m

]

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]

Figure 4.3: The filtered gps and digital compass data from one survey run on Pavilion Lake (shownin left plot on scale of lake, and right plot in a smaller scale), red represents the edge of the lake,blue arrows represent the heading given by compass data (here only one out of every twenty pointsmeasured is displayed), which mostly obscure the green points that are the track of the boat. Notethat the easting and northing scales displayed here are simply offset from the position of the DGPSbase-station.

To correct for refraction, the profile was divided into layers of constant sound speed. Eq. 4.1 was

implemented at the interface of all layers, and for the full range of angles incident to the array. Then

the layer thickness was reduced until the refracted correction converges to within a predetermined

acceptable error threshold.

4.3 Noise Analysis

In [33] one can see that the variance of the thermal noise voltage, σ2n, (also known as Johnson

noise) for an impedance Z can be found by considering the real component of the impedance, <{Z},the temperature of the impedance, T , and the frequency band over which noise is being observed,

∆f .

σ2n = 4kBT<{Z}∆f (4.2)

The amplifier and receiver setup, as shown in Fig. 4.5 was designed such that the ratio of the signal

strength (Vsignal in Fig. 4.5), to the combined noise sources (given as circles in the circuit diagram)

is maximized over the ranges of both frequency (≈ 100 → 400 kHz) and transducer impedances for

which the sonar is designed to operate. This actually limits the ability to optimize for one individual

transmit frequency, however the snr for the designed circuit is not more than a couple dB below the


Figure 4.4: Refraction of various rays (A: apparent position, T: true position) for the stratified soundspeed profile encountered during surveying. The mixed layer above the thermocline allows for raysto travel a sufficient distance before turning downward. A constant sound speed was inferred fordepths below the maximum measured value.

optimized value.

As the output signal-to-noise ratio will never be higher than the value at the output of the trans-

ducer element, when choosing the values of the circuit components one should ensure that the noise

contributions to the output are less than those intrinsic to the transducer circuit, so that the output

signal-to-noise ratio is optimized. For multiple amplifier circuits such as in Fig. 4.5 the lowest noise

amplifier should be placed first.

In performing noise analysis, the noise spectrum of a complex circuit (or as in this case a few sub

circuits such as the transducer and front end of the first amplifier stage) can be simplified by looking

at only the real component, <{·} , of the total combined circuit impedance Ztotal. This means that

some groups of circuit elements can be reduced and Eq. 4.2 becomes:

σ2n = 4kBT<{Ztotal}∆f (4.3)


Figure 4.5: The analogous circuit representation for the transducer and setup for the amplifiers.

The use of the total combined circuit impedance in Eq. 4.3 is made possible due to the feature

that the noise generated by one resistive circuit element is uncorrelated with the noise generated

by all the other elements, i.e for circuit elements i, j, with corresponding noise voltages Vni , Vnj the

expectation E{VniVnj} = σ2nj

δij where δij is the Kroneker delta function. The result of Eq. 4.3

is that the root mean square voltage contributions (at the output) from ι circuit elements can be

combined in quadrature, as shown in Eq. 4.4.

σ2n =

ι∑

j=1

σ2nj

(4.4)

An example of the combined (solid red curve) and contributing component noise spectrums for

a set of circuit parameters (corresponding to the circuit given in Fig. 4.5) are shown in Fig. 4.6.

The experimental noise spectrum is accounted for by the individual noise sources given in Fig. 4.6

for a range of frequencies encompassing those required for the experimental measurements (i.e.

≈ 100 → 400 kHz). The experimental noise data agrees well with the predicted values in Fig. 4.6.

One consequence of the design process was that it demonstrated that time-variable gain in the

lab MASB system is not only an unnecessary feature, but also diminishes performance. When a

system is designed such that the dominant noise source comes from the actual sensor impedance,

amplification is performed on both the signal and noise, so the ratio of the two remains constant

(therefore nullifying any perceived improvement). The only stipulation is that the thermal noise

cannot fall to levels close to what would otherwise be bit noise in the A/D converter. Consequently


0 1 2 3 4 5 6

x 105

−60

−50

−40

−30

−20

−10

0

Frequency Hz

Vrm

s

Example of noise measurement for 205kHz transducer June 01 Pavilion Lake

measuredV totalV amplifierV AV BV 1V 2V 3

Figure 4.6: The total noise spectrum (in A/D units, offset 90dB above actual noise power) is givenin solid red, all other curves represent independent noise sources and have been added in quadrature.For the noise behavior of the actual system (shown in cyan) it should be noted that the values of thecircuit components are not fully optimized for the 205kHz transducer as the amplifiers were designedto be broad-band for use up to 400 kHz.

it is often true with commercial systems that improperly corrected TVG can lead to range dependent

artifacts in processed data. One benefit of the simplification of removing TVG is that the functional

requirements of the system are reduced.

4.4 Data Processing

To ensure that the front end electronics, and the analogue to digital data acquisition system were

performing as designed, it was desired to keep all of the raw data, sampled at 8 times the carrier

frequency (far higher than the minimum Nyquist sampling requirements). Keeping raw data also

allowed for various data processing routines to be implemented and tested in post-processing. First

an approximate matched filter is applied to the raw data, taking into account the length of the


transmitted pulse. Next the in-phase and quadrature components (obtained from samples of the

band-passed data which is then decimated) were combined to create complex data.

An estimate of the angle of arrival (AOA) of the incoming signal could be obtained, either for each

range cell, or through the combination of adjacent range cells, as outlined in the previous chapter.

Though several estimation routines were implemented, including linear prediction angle estimation

and minimum variance distortionless response angle estimation, all seemed to perform similarly with

regards spread of AOA. However, the linear prediction routine could be implemented to process data

the fastest, and so it was selected. Though the maximum number of degrees of freedom for each

estimator for a single snapshot was one less than the number of elements in the array (for an array

of 6 there are only 5 degrees of freedom), the number of degrees of freedom in the angle estimation

could be lowered by using multiple sub arrays of the full array. The best results were obtained when

the number of degrees of freedom were limited to the number of signals impinging on the array.

Once the AOA is estimated, corrections for refraction, range attenuation in fresh water, and

beampattern were implemented. A simple bottom tracking algorithm was then used to eliminate

angle estimates from surface reflections, water column targets and other extraneous random points.

Multiple tracks are maintained when the bottom estimate is lost due to factors such as acoustic

shadow regions. Implementing a correction for grazing angle in this complex environment has not

yet been successful. The difficulties associated with implementing a grazing angle correction are

largely due to the geometry of the microbialite structures in Pavilion Lake, and will be discussed

further in the next section.

4.5 Survey of Pavilion Lake

Pavilion Lake, located near the village of Cache Creek in the interior of British Columbia, is a

part of the Canadian Analogue Research Network, a program run by the Canadian Space Agency.

The microbialites located within Pavilion Lake offer researchers an opportunity to study a modern

analogue to ancient dendritic reef structures [12]. Pavilion Lake is characterized as a dimictic lake

with annual ice cover, which provides the best survey opportunities in the fall, when the surface

layer is mixed to a depth of at least 10m, and refraction downward toward the bottom does not limit

the range of the sonar. The weather is also fairly calm in fall, and so surveying can be performed

from a pontoon boat (lower left picture in Fig. 4.8), without requiring pitch and roll corrections.

Pavilion Lake is a fiord lake, with 3 basins, the deepest basin has a depth of approximately 60m.

The unconsolidated sediment that covers large portions of the basins is fine (attempts to sample

with a gravity corer failed and became lost in the water column upon retrieval), and the microbialite

features that line the walls of the basins are rough, with a hardness that varies as a function of


depth. Rock slide areas are also present on the lake walls, as the lake resides in a canyon with steep

cliffs on either side.

In [12] various morphologies of microbialites were identified consisting of shallow (10 → 20m),

intermediate (20 → 30m), and deep water (> 30m) structures. Distinctions in friability, porosity,

and morphology were used to characterize and distinguish the various structures. During a survey

taken in June of 2005, several regions of high signal return were observed at depths exceeding 50m

in the central basin of the lake, which are shown here as the central mounds in Fig. 4.7. The signals

were of similar mean backscattered strength to the deep water microbialites located along walls of

the lake basins (the difference in signal return strength between the sediment and the microbialites

can be as great as ≈ 30dB, for the deeper water morphologies). Images also displayed a distinct

boundary separating sediment from high clutter, a feature similar to lobe-like regions identified as

microbialites along the lake walls (shown in Fig. 4.9) and distinct from the areas where slides had

occurred (which appeared to have less well defined boundaries, seen in in upper right and lower left

corners of Fig. 4.7).

Figure 4.7: Bathymetry (left) and imagery (right) are both useful in recognizing bottom featuressuch as these deep water microbialites.

Investigation with a DeepSea Power and Light drop camera identified these features as a form of

microbialite, distinct from the other three morphologies observed in [12] (one of which is shown in

the upper right picture in Fig. 4.8) and a Seabotix LBV150 remotely operated vehicle (ROV) was

used to image (the upper left picture in Fig. 4.8) and recover small samples (the lower right picture

in Fig. 4.8). Prior to the use of MASB sonar, these microbialites have not been identified in any of

the sonar surveys performed in the lake.


Figure 4.8: Top Left: A new morphology the deepest water microbialite specimens, photographedwith an ROV. Top Right: Microbialites characteristic of finger-like features on side of lake pho-tographed by the author while scuba diving. Bottom Left: survey rigged pontoon boat, blue DGPSantenna located mid-boat on port side. Bottom Right: LBV150 ROV used to take top left photo-graph, shown with net for microbialite retrieval.

It was also reported in [12] that the structures at intermediate depths vary in scale from decimeters

to meters in diameter, and in deep water they vary from centimeters to meters in diameter. With

these undulations of the bottom on length scales approximately equal to that of a single sonar

footprint, bathymetry estimates can become artificially smoothed. Within a single footprint a variety

of grazing angles may be represented. A consequence of this geometry is that correction for grazing

angle is not possible. In addition, each of the morphologies of microbialites displays specific physical

characteristics, (such as column features, and branching) that likely alter the dependence of the

backscatter on grazing angle. Eventually it may be possible to measure the dependence of backscatter

on grazing angle for large patches of microbialites. However, further analysis is required to determine

if this feature can be used to help map out the distribution of the different morphologies over the

area of the entire lake (in addition to other analogue research sites).


The scale of microbialite patches is displayed in Fig. 4.9 with coverage in many narrow lobe-like

zones extending down the walls of the lake and terminating in the deeper water. These microbialites

generally rise above the surrounding sediment by heights of up to a meter or more, hence the grazing

angle of many footprints is largely unknown, especially around the boundaries of these features.

When regions of the lake have been mapped from multiple viewpoints, such as the location shown

in Fig. 4.9, it becomes apparent that with no viable correction for grazing angle, the backscatter

imagery from overlapping survey runs cannot easily be combined. Therefore, to display large scale

features in the lake from multiple survey runs, imagery maps have been generated through stitching

data sets together, as displayed in Fig. 4.10 which shows a large portion of Pavilion Lake. In

contrast, estimates of the bathymetry can be averaged from multiple survey runs to improve the

overall accuracy of the final measurement as in Fig. 4.11. Bathymetry measurements and imagery

measurements can also be resolved at different scales, and an example of this is given in Fig. 4.12.

In practice, it is often useful to resolve the imagery on a much smaller scale than the bathymetry.

Examining a small region of the lake wall located in the South-West region of the lake, from a

plot of the depth only, as in Fig. 4.13, it is difficult to get a sense of either the undulation in depth

of the microbialite fingers (See Fig. 4.8 top right) which tend to grow slightly above the lake wall,

or any change in composition of the lakebed. False lighting can improve the interpretation of the

depth-only bathymetric map in Fig. 4.13, however, without foreknowledge of the contours of the

ensonified region, it is difficult to place the lighting in such a way as to best view features of the

survey (i.e. information is often overlooked). However both the undulation of the microbialites, and

the changes in composition are evident in Fig. 4.14 which is a plot of co-located bathymetry and

imagery.

Finally it should be noted that if the end-user application of the MASB survey is to obtain imagery

representing prominent features of the bottom, and bathymetric resolution is only required on a scale

comparable to the accuracy of a conventional GPS, then there is no necessity to use DGPS. The final

factors that affect resolution of survey maps are the sonar ping rate and survey vessel speed. With

the present capabilities of the prototype URL system, for a range of 120m and operating frequency

of 300kHz, approximately two pings per second can be recorded, and vessel speed is maintained

between 2 → 4 knots. These parameters indicate that one to two pings are taken every meter.


small θG large θG

small θG→ weaker return large θG→ stronger return

Figure 4.9: Maps constructed from two data sets taken of the same physical location are shown inthe upper left and right figures, although the surveys were taken from different vantages, the purpletrack shows the path of the sonar (as indicated by having a transducer and beampattern on thetrack). A full spectrum colormap is used to demonstrate the subtle difference in target strength asthe grazing angle, θG, vantage is changed, with warmer colors represent higher return strengths.


Figure 4.10: Stitched map of imagery, outline of lake is given in blue.


Figure 4.11: Averaged map of depth.

Figure 4.12: Profile including a small deep microbialite mound in the foreground and a larger onein the background. Bathymetry resolution is set to 1 m, while the imagery is resoled at 8 cm. Thescattering strength of the red regions is approximately 30dB greater than dark blue regions.


Figure 4.13: Depth measurements in the September 2005 300kHz survey, resolved at a 1m grid inboth easting and northing. Wall is in South-West region of the lake.

Figure 4.14: Backscatter imagery measurements co-located with depth information from the Septem-ber 2005 300kHz survey, resolved at a 1m grid in both easting and northing. Same region as shownin Fig. 4.13.


4.6 Comparison of Performance with EAL

Unfortunately, in order to realize the goal of a direct comparison between data and the accuracy

model presented in chapter 3, a more stable survey platform was required than was realizable at the

Pavilion Lake site. The main conditions that prevented a direct comparison of theory and experiment

fall into two categories: sonar position/orientation uncertainty, and ground-truthing. It was found

that the orientation sensors available to the URL were insufficient at capturing the sonar pitch and

roll to a high enough degree as to make the actual MASB AOA estimation the dominant mechanism

in performance analysis. A pontoon boat mounted setup, is subject to the effects of wave and wind

action, making small, but abrupt motions. Accounting for this motion using the present position

and orientation sensors proved impossible.

A stationary mount is also not ideal to prove the results of chapter 3, as the imaging is then

limited to a single instance of scatterers, namely the same bottom is continuously imaged. The

desired data is that of a relatively featureless bottom, where adjacent pings can be used to emulate

multiple instances of the same bottom character (perhaps a deep water measurement of a featureless

section of the continental shelf). In the future it will be necessary to use either a more accurate

sensor, or deploy the MASB on a more stable platform, such as an AUV. Aside from providing a

smoother transition through the water, an AUV mount has the benefit over a boat mount of being

able to operate lower in the water column, away from both surface multipath effects and the more

extreme sound speed gradients that are most often found near the surface of lakes in the climate of

British Columbia. As to the problem of performing ground truthing, again it is preferential to have

a large surface of constant depth and composition. To ground truth some points in Pavilion Lake, a

lead line method was attempted in conjunction with Differential GPS. However during subsequent

attempts at sediment coring it was revealed that the section that had been identified as most ideal

for ground truthing and lead lined in Pavilion Lake due to its fairly constant composition was in fact

very unconsolidated sediment (so unconsolidated that every attempt to gravity core a sample failed),

and in all likelyhood the lead line had been sinking to an indeterminate depth in the sediment.

Although not ideal, an experiment using a nearly stationary mount was attempted at Sasamat

Lake, located in Port Moody British Columbia. It is a man-made lake approximately 0.5 km in

diameter, with a bottom composition of mud, small gravel rubble and organic debris such as sunken

tree branches. Data collection began during the winter of 2004, when the prototype MASB system

shown in Fig. 4.1 (without the digital compass and GPS modules) was taken to Sasamat Lake and

mounted on a floating pier (with calm weather conditions, this mount was nearly stationary), from

which various measurements were undertaken by rotating a fixed mount sonar in a manner similar

to sector scan. On February 7, 21 and 25, experiments were performed to compare accuracy of AOA

estimation in comparison to the CRLB. At this point in the winter, temperatures were such that the


lake was assumed to be minimally stratified, thus no CTD was used. Also, as no ground truthing was

performed at this site the performance of the angle estimation routine can only be used to illustrate

qualitative compliance to the bound.

Fig. 4.15 shows a survey profile and the corresponding EAL performance of a linear prediction

estimator for one AOA on a six element array. The transmitted signal was a FQ pulse of 20 cycles

with q = 10, and an approximate matched filter was implemented in Matlab using a hamming filter

of appropriate width. The use a MFSQ pulse to model this scenario can be justified on the basis that

MFFQ pulse performance is close to MFSQ performance, and the calculation of FS contribution is

known for the MFSQ pulse, whereas it is not a simple matter for the MFFQ pulse. Five snapshots

(taken as adjacent uncorrelated snapshots in the single ping) were used in this analysis for both

imagery resolution and bathymetry resolution, as such the ping profile is not as nice as the one given

in Fig. 4.12. The EAL was calculated using the footprint shift corresponding to a MFSQ pulse,

and the signal to noise ratio as calculated from the measured noise level, and the signal strength

estimated using the output of a conventional beamformer steered to the AOA of the bottom return.

It should be mentioned that the choice of modeling only a single AOA ignores completely the surface

multipath AOA, which will surely impact the performance of such a system. This is particularly

the case at far ranges, where the surface multipath AOA and direct return AOA are approaching

each other in phase space, and when they come within a Rayleigh beamwidth in separation some

extra spreading of the data is noticed, as is evidenced beyond 100m. Without ground truthing it was

impossible to measure the true number of estimates that fell within the EAL confidence interval. This

is because the act of fitting a function (such as a polynomial) to the bottom shape ensures that at

the very least the mean estimates fall within the bound, and depending on the fitting methodology

(for example the degree of the polynomial used, or the local length scale that the fit is applied),

fewer or greater numbers of estimates can be included. Another objection that should be addressed

is that the geometry of the ping profile includes a significant region that falls nearer than would be

considered the far field for this apparatus. However, it can be noted that the qualitative agreement

of the scale of the EAL to the spread in data is encouraging.

Applying the same analysis to a full sector scan at Sasamat Lake yields Fig. 4.16. Instead of a

simple EAL, an entire Error Arc Surface, EAS, has been constructed (bottom of Fig. 4.16). This

could be used as a possible tool to assist in masking and thresholding techniques that are practiced

in another sonar discipline, bottom classification (for instance, data corresponding to an area where

the EAS is high can be eliminated from analysis). Here the estimate of error in depth is compared to

standard benchmarks for International Hydrographic Organization (IHO) special surveys (SS), first

order (FO) and second order (SO) surveys (requirements taken from [14] for 25m depth).


Figure 4.15: Profile taken at Sassamat Lake, with corresponding EAL as determined from the CRLBfor the MFSQ pulse using only noise and footprint shift for a single signal on a six element array (ie.this ignores completely the surface multipath signals).


Experimental data: sector sweep at Sasamat lake, B.C.Co-located Bathymetry and Sidescan Imagery

Sonar Location

Single Ping Profile

Error Arc Surface 5 Snapshots

Acoustic Shadow Region

Error Arc Length

Figure 4.16: The top image is a sector sweep of the Sasamat lake basin (note that backscatterimagery measurements show sidelobe leakage due to conventional beamforming). The bottom imageis the corresponding EAS for the basin at Sasamat Lake.


4.7 Summary of Chapter 4: An Alternative to Other Sonar

Systems

There are several advantages that MASB sonar has over conventional sidescan systems. Aside

from the benefits of being able to measure bathymetry, MASB systems reduce the contributions from

multipath and multiple signals. In addition, compensation for beampatterns of both the transmit

and receive transducers cannot be performed in sidescan systems, whereas MASB sonar computes

the angle of arrival of incoming signals and can correct for the sensitivity of the beampattern. With

bathymetry measurements, MASB is capable of computing the grazing angle in survey geometries

where the bottom does not vary on a scale comparable with a single footprint.

In many applications where sidescan is used, the addition of bathymetric information can lead

to discoveries that might otherwise have been overlooked, such as the new deep water microbialites

discussed earlier. Though sidescan imagery might show the shape of deep mound features, the

absence of bathymetry data hides their true significance. At best, if the deep water structures had

been located in the image of a conventional sidescan system, multiple passes of the structure would

have to be made to position the mounds directly beneath the survey vessel, and thereby estimate

their depth.

It is also advantageous to have a boat mounted sonar when surveying the the central region of

Pavilion Lake. It is demonstrated in Fig. 4.11 that the lake has a complex bathymetry, with many

mound features on both large and small scales. When sidescan surveys over the same region have

been performed, the towfish altitude must constantly be adjusted to avoid hitting the microbialites,

which must be preserved as a condition of doing research in the lake. To locate features captured

with sidescan data, the location and orientation of the towfish must also be known. Alternatively,

by using a boat mounted MASB system, the position and orientation of the sonar can be known to

a reasonable degree of accuracy, using at the very least a flux gate compass and conventional GPS.

Multibeam echo sounders (MBES) do have an advantage over MASB systems in estimating the

bottom location near nadir, as the footprint size for MASB sonar becomes larger than the beam

of a MBES system. However it was noted in [16] that the corrections required to extract useful

information on backscatter imagery from MBES data are not reliable in beams steered close to nadir.

Consequently, the bottom coverage of a MASB system should be comparable to MBES for imaging,

over the same region of swath away from nadir. Furthermore, the density of imagery measurements

is higher in MASB sonars than many MBES systems, which are limited by the maximum number of

beams that can be formed.


Due to minimal electronics components required, and small aperture size, future versions of MASB

sonar can be manufactured for marginally more than the cost of a sidescan, and the small physical

size of MASB sonar makes it an ideal candidate to deploy aboard small platforms, such as AUVs.

In conclusion, the use of 3D sidescan can be implemented with a smaller aperture than multibeam

systems, and as an alternative to conventional sidescan in many sidescan applications. The imagery

produced by 3D sidescan, when used in conjunction with co-located bathymetric data (not necessarily

taken at same resolution), gives good swath coverage in survey applications with shallow water and

complex geometries.

Chapter 5

Summary of Conclusions, and

Future Research

167

CHAPTER 5. SUMMARY OF CONCLUSIONS, AND FUTURE RESEARCH 168

5.1 Conclusions

In this thesis, a detailed analysis of the bottom estimation performance of MASB sonar was

presented. A simple survey geometry was defined and both general and closed-form expressions for

the cross-correlations of the received backscatter across a receive array were determined for several

waveforms. Closed-form expressions clearly show the contributions that the various error mechanisms

make to the correlation. From the correlations, the CRLB for estimating the electrical angle was

determined. It was shown through geometry, simulation, and the CRLB that when the array is

tilted there is a double angle region that requires two angles be estimated rather than just one if

the bottom location is to be correctly estimated in this region. Estimating two angles requires an

array with at least two degrees of freedom (three or more elements) as opposed to the one degree of

freedom available with a simple two element relative-phase array.

A determination was made that pre-estimation averaging for AOA estimation perform better than

post-estimation averaging for a simple model, and as such, pre-estimation routines were chosen to

investigate the more complex survey geometry. It was shown through simulation that angle estimates

made with a simple linear prediction algorithm fall within 2√

CRLBα approximately 82% → 85%

of the time, which is slightly lower than the 92% → 93% expected for five snapshots if the estima-

tion procedure attained the CRLB. Two other estimation techniques, namely minimum eigenvalue

analysis and minimum variance distortionless response beamforming were also employed for angle

estimation, with performance that was slightly lower than that achieved for linear prediction. Never-

theless, the results were robust and sufficiently accurate for practical applications. The percentages

for linear prediction were similar for both the square pulse and match filtered square pulse, whether

the five snapshots were taken from consecutive uncorrelated pings or adjacent uncorrelated range

bins in the same ping. The bottom estimation performance related to a match filtered Gaussian

pulse and Gaussian chirp pulse was compared with that obtained with a match filtered square pulse.

Little difference in performance was found, except when the angle estimation accuracy was thermal

noise limited and then the chirp pulse performed better because of the higher resultant snr.

A simple approximation procedure for predicting estimation performance was developed that in-

cludes the major sources of error for a high resolution MASB sonar, namely, footprint shift and

thermal noise. This simple procedure applies in the single-angle region, and comparisons with the

full calculations showed it to be quite accurate. For the double angle region, the form of the CRLB

that includes two signals and their respective energies must be used. Examples of bottom estimation

performance were given for different array sizes, tilt angles, frequencies, and water types. Perfor-

mance was presented in the form of the error arc- length (EAL), which is an arc perpendicular to

the range vector from the sonar to the point of interest on the bottom. The EAL is interpreted as

the arc length that 2√

CRLBα for the electrical angle represents at the range of interest. The actual


orientation of the arc with respect to the bottom depends on the range and angle. It was determined

that the influence of footprint shift was dominant at closer ranges because the thermal snr was high.

However, at further ranges the influence of thermal noise dominated the bottom location accuracy.

The influence of footprint shift was highest for zero tilt angle. Large tilt angles resulted in a double

angle region but reduced the effect of footprint shift. Nevertheless, the effect of footprint shift was

relatively minor when determining the practical range of operation since the main error source at the

further ranges was thermal noise. Moving to lower frequencies to reduce acoustic attenuation helped

mitigate the effects of thermal noise but reduced the resolution. Also, changing from saltwater to

freshwater, where the attenuation is less, improved performance at further ranges.

Following the development of MASB performance analysis, a description is given of a prototype

system developed in the Underwater Research Laboratory at Simon Fraser University. The design

and construction of this MASB system is described in detail, including schematics, beampattern

measurement, and setup of the full survey apparatus. The process of refraction correction using

measured CTD data is also shown. Measured noise in the MASB system is shown to be in good

agreement with the predicted value, as calculated from the circuit model. Surveys of Pavilion Lake

are presented, highlighting the high resolution imagery of the MASB system, along with co-located

bathymetry. A new discovery was made at the lake due to the imagery and co-located bathymetry

provided by the prototype MASB, namely a new deep water morphology of microbialite in Pavilion

Lake. Due to the excellent swath coverage of the prototype MASB sonar, these features were iden-

tified, despite being sparsely located on the lake bottom. The co-located bathymetry measurements

of the microbialites allowed for the recognition that they represented a form of microbialite that

existed deeper than any morphology previously identified.

In attempting to compare the performance of the prototype MASB system with the EAL devel-

oped in chapter 3, it was recognized that a moving platform contributes additional error to bottom

estimation. In the attempt to resolve this, a nearly stationary platform at Sasamat Lake is chosen

for further examination of the prototype system. Despite the lack of ground truthing data, and a

survey geometry that differs from the geometries examined in chapter 3, reasonable agreement with

the approximate EAL for a single signal geometry is achieved (performance approximation is based

only on the contributions of footprint shift and thermal noise). The concept of an Error Arc Surface

(EAS) was presented, and an example is given using data from a sector scan of Sasamat Lake.

In summary, the results of this thesis provide tools for evaluating the potential accuracy of present

day MASB sonar for bottom topographic surveys. In addition, the research presented in this thesis

demonstrated the capabilities of a prototype MASB system in the application of swath bathymetry.


5.2 Future Directions for MASB Analysis

There are many avenues for further development in the study of MASB performance that arise

from the results of this thesis. To begin, it can be recognized that only a select set of geometries

have been considered in the course of this research, and often real survey scenarios can be much

more complex. In addition to the signals arising from separated sonar footprints on the sea or lake

bottom, other signals will often impinge on a sonar array in practical survey scenarios. Both point

and extended water column targets are issues to consider when undertaking MASB surveying, and

their impact on bottom estimation warrants investigation at some later date. Similarly, for shallow

water mapping, multipath returns of sonar signals from the water surface and bottom should be

considered and added to the analysis of performance. These signals can be added to the framework

presented in this thesis, however it should be recognized that it is likely that there may not be

enough degrees of freedom in small arrays to fully separate all signals. In actuality it may even be

preferential to treat signals that are closely spaced as if they represented a single signal.

In order to obtain experimental data that can be directly compared with the theoretical framework

presented in this thesis, a deep water survey over a flat, uniform bottom might be considered. This

would require the application of MASB to a tow-fish, ROV or AUV survey platform. It is expected

that this experiment would yield data that should be directly comparable with the simple geometry

presented in chapters 2 and 3, as it would eliminate the surface multipath contribution to the data

that limited the performance confirmation in chapter 4. It should also be recognized that in order to

achieve the proposed correct data, extensive ground truthing (for at least depth information) would

also need to be performed, as well as the integration of highly accurate position and orientation

information sensors for the survey platform. A deep water experiment is also suggested here because

it would eliminate the high degree of ray refraction that is often experienced in near surface water

due primarily to temperature stratification.

The waveforms that were used in the theoretical analysis contained in this thesis were not all

implemented in the prototype system, and so an extension to the work in this thesis would be to

try various waveforms in real survey systems. It can also be recognized that the set of waveforms

used in this research does not encompass all possibilities, and other waveforms exist that might be

examined both theoretically, and in practice. Since the compressed gaussian pulse displayed the best

performance of all the waveforms examined in this thesis, there should be research into implementing

real pulses that utilize pulse compression in future MASB systems.

Estimator performance benchmarking using the CRLB or EAL can also be extended from the

current analysis. As mentioned earlier, many more estimators of AOA can be found in the literature

(such as those in [40]), and each can be examined using the same methodology presented in chapter

3. In particular, the issues of both too many or too few degrees of freedom present in an estimator


should be investigated (for even the estimators presented in this thesis). It would be useful to know

if there is a general method of reducing or increasing the number of degrees of freedom for a given

estimator, such that the bottom estimation performance is maximized (i.e. optimizing estimation

for the desired signal under the influence of interfering signals).

In addition to the afore mentioned examples, which are direct extensions to the current line of

research, there are many avenues of exploration for MASB that have yet to be realized. One such

example is the application of MASB to bottom classification or characterization. Both the high

density swath imagery recorded by MASB systems and the co-located bathymetry (including the use

of performance bounds) might be of great utility when examining the dependance of bottom type on

grazing angle of backscatter (to improve on measurements such as those found in [24] and subsequent

literature). In addition, many other classification routines can be implemented using MASB, such as

those employed in [7] where multibeam sonar was used to classify the bottom using a purely statistical

approach, and clustering of bottom patches was based on self-similarity. Even without implementing

such routines, it was possible in this research through simple visual inspection of the survey data to

identify areas of Pavilion Lake that represented both microbialite and non-microbialite bottom types.

In the future a more quantitative technique is desired, with remote sensing that could be automated

to distinguish different bottom types. Generally, remote characterization methods not only save

time and resources in comparison to the arduous ways in which divers and submersibles are required

for bottom sample extractions, but also do not damage any ecosystems in which measurements are

required.

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