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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
All rights reserved. However, in accordance with the Copyright Act of Canada, this work
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limited reproduction of this work for the purposes of private study, research, criticism, review
and news reporting is likely to be in accordance with the law, particularly if cited appropriately.
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
1.2.4 Direction of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Goal 2 - Demonstration of Prototype System . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Purpose of Research and Summary of Accomplishments . . . . . . . . . . . . 11
1.3.2 Direction of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
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√
CRLBα in red for each of the range cells in the simulation. . . . . . . . . . . . 118
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
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. . . . . . . . . . . . 122
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
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. . . . . . . . . . . . 123
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
CHAPTER 1. INTRODUCTION AND BACKGROUND 3
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
CHAPTER 1. INTRODUCTION AND BACKGROUND 4
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.
CHAPTER 1. INTRODUCTION AND BACKGROUND 5
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
CHAPTER 1. INTRODUCTION AND BACKGROUND 6
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.
CHAPTER 1. INTRODUCTION AND BACKGROUND 7
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
CHAPTER 1. INTRODUCTION AND BACKGROUND 8
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).
1.2.4 Direction of Chapter 3
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.
CHAPTER 1. INTRODUCTION AND BACKGROUND 9
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.
CHAPTER 1. INTRODUCTION AND BACKGROUND 10
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
CHAPTER 1. INTRODUCTION AND BACKGROUND 11
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.
CHAPTER 1. INTRODUCTION AND BACKGROUND 12
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
CHAPTER 1. INTRODUCTION AND BACKGROUND 13
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.
1.3.2 Direction of Chapter 4
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
CHAPTER 1. INTRODUCTION AND BACKGROUND 14
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
CHAPTER 1. INTRODUCTION AND BACKGROUND 15
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
CHAPTER 2. THE SIGNAL 18
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.
CHAPTER 2. THE SIGNAL 19
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.
CHAPTER 2. THE SIGNAL 20
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
CHAPTER 2. THE SIGNAL 21
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
CHAPTER 2. THE SIGNAL 22
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
CHAPTER 2. THE SIGNAL 23
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.
CHAPTER 2. THE SIGNAL 24
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)
CHAPTER 2. THE SIGNAL 25
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.
CHAPTER 2. THE SIGNAL 26
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)
CHAPTER 2. THE SIGNAL 27
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
CHAPTER 2. THE SIGNAL 28
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.
CHAPTER 2. THE SIGNAL 29
Figure 2.8: Geometry for sloped bottom correction.
CHAPTER 2. THE SIGNAL 30
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.
CHAPTER 2. THE SIGNAL 31
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)
CHAPTER 2. THE SIGNAL 32
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)
CHAPTER 2. THE SIGNAL 33
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.
CHAPTER 2. THE SIGNAL 34
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
CHAPTER 2. THE SIGNAL 35
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.
E{sks∗i }'EtE{|B|2}∫ ∞
−∞g(rc − r1i − kδri)g∗(rc − r1i − iδri)e
4πjλ (k−i)δridxi (2.32)
CHAPTER 2. THE SIGNAL 36
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.
E{sks∗i }'EtE{|B|2}∫ ∞
−∞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]).
CHAPTER 2. THE SIGNAL 37
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
4arsq−2c+2ce−2arsq
c
−rsq ≤ r ≤ 0
4rsqa−2ce2ar
c −4ra+ce2a(−rsq+r)
c +ce−2a(r+rsq)
c
4arsq−2c+2ce−2arsq
c
0 ≤ r ≤ rsq
c(−2e−2ar
c +e−2a(r+rsq)
c +e−2a(−rsq+r)
c )
4arsq−2c+2ce−2arsq
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.
CHAPTER 2. THE SIGNAL 38
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.
CHAPTER 2. THE SIGNAL 39
−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.
CHAPTER 2. THE SIGNAL 40
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)
CHAPTER 2. THE SIGNAL 41
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)
CHAPTER 2. THE SIGNAL 42
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)
CHAPTER 2. THE SIGNAL 43
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)
CHAPTER 2. THE SIGNAL 44
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:
CHAPTER 2. THE SIGNAL 45
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
CHAPTER 2. THE SIGNAL 46
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.
CHAPTER 2. THE SIGNAL 47
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.
CHAPTER 2. THE SIGNAL 48
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.
CHAPTER 2. THE SIGNAL 49
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).
CHAPTER 2. THE SIGNAL 50
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
horizontal range [m]
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.
CHAPTER 2. THE SIGNAL 51
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.
CHAPTER 2. THE SIGNAL 52
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:
CHAPTER 2. THE SIGNAL 53
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.
CHAPTER 2. THE SIGNAL 54
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
horizontal range [m]
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
1MFSQ Pulse
horizontal range [m]
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.
CHAPTER 2. THE SIGNAL 55
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.
CHAPTER 2. THE SIGNAL 56
0 20 40 60 80 100 1200
1
2
3
4
5
6
7
8
9
10FQ 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
10FQ pulse
horizontal range [m]
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.
CHAPTER 2. THE SIGNAL 57
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
horizontal range [m]
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
horizontal range [m]
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.
CHAPTER 2. THE SIGNAL 58
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.
CHAPTER 2. THE SIGNAL 59
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
horizontal range [m]
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
1CG Pulse
horizontal range [m]
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.
CHAPTER 2. THE SIGNAL 60
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
CHAPTER 2. THE SIGNAL 61
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.
CHAPTER 2. THE SIGNAL 62
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
CHAPTER 2. THE SIGNAL 63
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 66
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 67
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 68
angle at a given range, in order to compute the uncertainty in depth the EAL must be projected in
the vertical axis.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 69
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 70
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 71
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 72
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 73
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 74
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 75
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,
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 76
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 77
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 78
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).
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 79
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 80
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 81
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 82
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 83
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 84
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 85
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 86
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 87
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 88
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 89
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 90
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 91
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 92
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 93
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 94
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 95
−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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 96
−40 −20 0 20 40 60 80 100 1200
1
2
3
4
5
6
7
8
9
10Square Pulse
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 97
−40 −20 0 20 40 60 80 100 1200
1
2
3
4
5
6
7
8
9
10Square Pulse
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 98
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 99
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 100
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
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 101
−40 −20 0 20 40 60 80 100 1200
0.5
1
1.5Matched Filtered Square Pulse
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 102
without any matched filtering.
−40 −20 0 20 40 60 80 100 1200
0.5
1
1.5Finite Q Pulse
Distance Along Bottom (m)
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,
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 103
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
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 104
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 105
−40 −20 0 20 40 60 80 100 1200
0.5
1
1.5Compressed Gaussian
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 106
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 107
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 108
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 109
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 110
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 111
~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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 112
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 113
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
Distance Along Bottom (m)
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 114
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 115
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
Range from Source(m)
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 116
−40 −20 0 20 40 60 80 100 120−60
−40
−20
0
20
40
60
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 117
−40 −20 0 20 40 60 80 100 120−20
−15
−10
−5
0
5
10
15
20
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 118
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
Distance Along Bottom (m)
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 green and −2
√CRLBα in red for each of the range cells in
the simulation.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 119
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 120
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 121
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 3.24: 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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 122
−40 −20 0 20 40 60 80 100 120−60
−40
−20
0
20
40
60
Distance Along Bottom (m)
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
√CRLBα in green and −2
√CRLBα in red for each of the range cells
in the simulation.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 123
−40 −20 0 20 40 60 80 100 120−20
−15
−10
−5
0
5
10
15
20
Distance Along Bottom (m)
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
√CRLBα in green and −2
√CRLBα in red for each of the range cells
in the simulation.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 124
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)
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 125
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 126
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
Range from Source(m)
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 127
−40 −20 0 20 40 60 80 100 120−60
−40
−20
0
20
40
60
Distance Along Bottom (m)
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 green and −2
√CRLBα in red for each
of the range cells in the simulation.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 128
−40 −20 0 20 40 60 80 100 120−20
−15
−10
−5
0
5
10
15
20
Distance Along Bottom (m)
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 green and −2
√CRLBα
in red for each of the range cells in the simulation.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 129
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 130
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 131
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 132
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 133
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 134
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
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 135
−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
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 136
−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
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 137
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
Distance Along Bottom (m)
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 138
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
Distance Along Bottom (m)
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 139
0 50 100 150 200 2500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Distance Along Bottom (m)
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 140
0 50 100 150 2000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Distance Along Bottom (m)
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.
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 141
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 142
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
CHAPTER 3. PERFORMANCE - THEORY AND SIMULATION 143
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 146
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 147
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 148
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)
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 149
−2000 −1000 0 1000 2000 3000−3000
−2500
−2000
−1500
−1000
−500
0
500
1000
1500
Easting [m]
Nor
thin
g [m
]
−200 0 200 400 600 800
−1300
−1200
−1100
−1000
−900
−800
−700
−600
−500
Easting [m]
Nor
thin
g [m
]
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
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 150
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)
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 151
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
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 152
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
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 153
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
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 154
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 155
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).
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 156
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 157
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 158
Figure 4.10: Stitched map of imagery, outline of lake is given in blue.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 159
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 160
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 161
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
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 162
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).
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 163
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).
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 164
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 165
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.
CHAPTER 4. DEMONSTRATION OF A MASB APPARATUS 166
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
CHAPTER 5. SUMMARY OF CONCLUSIONS, AND FUTURE RESEARCH 169
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.
CHAPTER 5. SUMMARY OF CONCLUSIONS, AND FUTURE RESEARCH 170
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
CHAPTER 5. SUMMARY OF CONCLUSIONS, AND FUTURE RESEARCH 171
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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