PROUD ELASTIC TARGET DISCRIMINATION USING LOW-FREQUENCY
SONAR SIGNATURES
by
Brenton Mallen
A Thesis Submitted to the Faculty of
The College of Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degree of
Master of Science
Florida Atlantic University
Boca Raton, Florida
May 2012
Copyright by Brenton Mallen 2012
ii
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ACKNOWLEDGEMENTS
I would first like to acknowledge my thesis advisor Dr. Pierre-Philippe Beaujean.
His constant direction and patience were instrumental in my journey throughout this
project. His guidance and advice allowed me to become a more focused and dedicated
engineer. I am very grateful for all your hard work and I thank you from the bottom of
my heart.
I would like to offer my gratitude to my committee members, Dr. Stewart Glegg
and Dr. Issac E. Elishakoff, for their assistance and support. I would also like to offer my
gratitude to Julia Gazagnaire for her guidance and advice.
I would like to thank my family for their support throughout this endeavor. To
my parents, grandparents and brothers, thank you for all your endless love and
encouragement. I could not have gotten this far without all of you. I would like to thank
my other brothers, Dan and Johnny for all the laughs and the sanity check that is game
night. I would also like to thank my officemates; Nicholas Waters, Mustapha Mjit and,
of course, my dear friend Gaultier for café sur le quai and the treasured French lessons.
Last, but not least, I would like to thank my beloved Jolene, whose unyielding and
relentless love has been my constant companion. Words cannot express my appreciation
and adoration. I could not imagine any adventure worth having without you by my side
iv
ABSTRACT
Author:
Brenton Mallen
Title:
Proud Elastic Target Discrimination Using Low-Frequency Sonar
Signatures
Institution:
Florida Atlantic University
Thesis Advisor:
Dr. Pierre-Philippe Beaujean
Degree:
Master of Science
Year:
2012
This thesis presents a comparative analysis of various low-frequency sonar
signature representations and their ability to discriminate between proud targets of
varying physical parameters.
The signature representations used include: synthetic
aperture sonar (SAS) beamformed images, acoustic color plot images, and bispectral
images. A relative Mean-Square Error (rMSE) performance metric and an effective
Signal-to-Noise Ratio (SNReff) performance metric have been developed and
implemented to quantify the target differentiation. The analysis is performed on a subset
of the synthetic sonar stave data provided by the Naval Surface Warfare Center – Panama
City Division (NSWC-PCD). The subset is limited to aluminum and stainless steel, thin-
v
shell, spherical targets in contact with the seafloor (proud). It is determined that the SAS
signature representation provides the best, least ambiguous, target differentiation with a
minimum mismatch difference of 14.5802 dB. The acoustic color plot and bispectrum
representations resulted in a minimum difference of 9.1139 dB and 1.8829 dB,
respectively
vi
DEDICATION
To my grandparents, Celeste Aurora and Luis Carlos Garcia.
A mis abuelos. Te amo con todo mi corazón.
PROUD ELASTIC TARGET DISCRIMINATION USING LOW-FREQUENCY
SONAR SIGNATURES
LIST OF FIGURES .......................................................................................................... xii
LIST OF TABLES ........................................................................................................... xiii
1.
INTRODUCTION ....................................................................................................... 1
2.
OBJECTIVE ................................................................................................................ 3
3.
SCIENTIFIC CONTRIBUTION ................................................................................. 4
4.
LITERATURE REVIEW ............................................................................................ 5
4.1. Target Classification................................................................................................. 5
4.1.1.
Machine Learning ......................................................................................... 5
4.1.2.
Traditional Methods ...................................................................................... 6
4.1.3.
Low-Frequency Target Response ................................................................. 8
4.2. Sonar Signature Representations ............................................................................ 10
5.
4.2.1.
Synthetic Aperture Sonar ............................................................................ 10
4.2.2.
Acoustic Color ............................................................................................ 12
4.2.3.
Higher-Order Spectral Analysis .................................................................. 12
SCIENTIFIC BACKGROUND ................................................................................ 14
5.1. Pressure Field for a Proud Spherical Target ........................................................... 17
viii
5.2. High-Frequency Sonar Systems ............................................................................. 18
5.2.1.
Rigid Spherical Scatterer ............................................................................ 18
5.3. Low-Frequency Sonar Systems .............................................................................. 19
5.3.1.
Elastic Spherical Scatterer (Using Membrane Approximation) ................. 20
5.3.2.
Elastic Response Using Thin-Shell Theory ................................................ 20
5.3.3.
Noise from Bottom Scattering .................................................................... 22
5.4. Signatures ............................................................................................................... 23
6.
5.4.1.
Synthetic Aperture Sonar ............................................................................ 23
5.4.2.
Acoustic Color ............................................................................................ 26
5.4.3.
Higher-Order Spectra .................................................................................. 26
METHODOLOGY .................................................................................................... 29
6.1. Simulated Dataset ................................................................................................... 30
6.2. Signature Image Generation ................................................................................... 32
6.2.1.
SAS Beamformed Images ........................................................................... 34
6.2.2.
Acoustic Color Images ................................................................................ 34
6.2.3.
Bispectrum Images...................................................................................... 36
6.3. Performance Evaluation ......................................................................................... 37
6.3.1.
Look-up Table ............................................................................................. 38
6.3.2.
Relative Mean-Square Error ....................................................................... 40
6.3.3.
Effective Signal-to-Noise Ratio .................................................................. 41
ix
6.3.4.
Statistical Averaging ................................................................................... 42
6.3.5.
Target Discrimination ................................................................................. 44
6.4. Performance Optimization ..................................................................................... 46
6.4.1.
7.
Minimum Threshold ................................................................................... 46
RESULTS .................................................................................................................. 50
7.1. Effective SNR......................................................................................................... 51
7.1.1.
Synthetic Aperture Sonar ............................................................................ 51
7.1.2.
Acoustic Color plot ..................................................................................... 52
7.1.3.
Bispectrum .................................................................................................. 54
7.2. Thresheld Effective SNR........................................................................................ 55
7.2.1.
Minimum Threshold ................................................................................... 55
7.2.2.
Synthetic Aperture Sonar ............................................................................ 57
7.2.3.
Acoustic Color plot ..................................................................................... 59
7.2.4.
Bispectrum .................................................................................................. 60
7.3. Overall Performance............................................................................................... 62
8.
CONCLUSION ......................................................................................................... 66
9.
APPENDIX ............................................................................................................... 69
A. Nomenclature ......................................................................................................... 69
B. SAS Results ............................................................................................................ 75
C. Acoustic Color plot Results .................................................................................... 81
x
D. Bispectrum Results ................................................................................................. 87
E. SAS Thresheld Results ........................................................................................... 93
F. Acoustic Color plot Thresheld Results ................................................................... 99
G. Bispectrum Thresheld Results .............................................................................. 105
H. Rigid Spherical Scatterer ...................................................................................... 111
I.
Elastic Spherical Scatterer (Using Membrane Approximation) ........................... 115
J.
Higher-Order Spectral Analysis ........................................................................... 118
10.
REFERENCES .................................................................................................... 121
xi
LIST OF FIGURES
Figure 1: Layout and dimensions of the target area with respect to the sonar.................... 7
Figure 2: Shadowgraph of ultra-short sound pulses propagating around ......................... 10
Figure 3: Considered Backscatter Ray Diagram. .............................................................. 14
Figure 4: Sonar Diagram (Top View). .............................................................................. 15
Figure 5: Back Propagation Block Diagram. .................................................................... 16
Figure 6: Synthetic Aperture Sonar Geometry ................................................................. 24
Figure 7: Return time-series signal from an aluminum spherical shell of radius 0.5m. ... 28
Figure 8: Complete SAS beamformed image of an aluminum spherical shell ................. 33
Figure 9: Cropped SAS beamformed image of an aluminum spherical shell................... 34
Figure 10: Sonar time-series echo of an aluminum spherical shell of radius ................... 35
Figure 11: Acoustic Color plot of an aluminum spherical shell with a radius of ............. 36
Figure 12: Bispectrum of the time-series response at zero degrees aspect of .................. 37
Figure 13: Examples of the correlated k-distributed background realizations. ................ 44
Figure 14: Acoustic Color SNReff as a function of measured and referenced .................. 48
Figure 15: Acoustic Color plot referenced image remaining signal energy as ................. 56
Figure 16: Acoustic Color plot measured image remaining signal energy as .................. 57
Figure 17: Rigid spherical shell diagram. ....................................................................... 111
xii
LIST OF TABLES
Table 1: Spherical Data. .................................................................................................... 31
Table 2: Possible feature index values. ............................................................................. 39
Table 3: SAS SNReff(2) standard deviation range results. ................................................. 52
Table 4: SAS SNReff(2) minimum and maximum mismatch difference. ........................... 52
Table 5: Acoustic color plot SNReff(2) standard deviation range results. .......................... 53
Table 6: Acoustic color plot SNReff(2) minimum and maximum mismatch difference..... 53
Table 7: Bispectrum SNReff(2) standard deviation range results. ...................................... 54
Table 8: Bispectrum SNReff(2) minimum and maximum mismatch difference. ................ 54
Table 9: SAS SNReff(2) thresheld standard deviation range results. .................................. 58
Table 10: Thresheld SAS SNReff(2) minimum and maximum mismatch difference. ........ 58
Table 11: Acoustic color plot SNReff(2) thresheld standard deviation range results. ........ 59
Table 12: Thresheld Acoustic color plot SNReff(2) minimum and maximum ................... 60
Table 13: Bispectrum SNReff(2) thresheld standard deviation range results...................... 60
Table 14: Thresheld bispectrum SNReff(2) minimum and maximum ................................ 61
Table 15: Target parameter combinations tested .............................................................. 63
Table 16: Overall target discrimination performances and ranges. .................................. 64
Table 17: SAS mean SNReff(1) results. .............................................................................. 75
Table 18: Complete SAS SNReff(1) standard deviation range results. ............................... 76
Table 19: SAS rMSE(1) results. ..........................................................................................77
xiii
Table 20: SAS mean SNReff(2) results. ...............................................................................78
Table 21: Complete SAS SNReff(2) standard deviation range results. ................................79
Table 22: SAS rMSE(2) results. ..........................................................................................80
Table 23: Acoustic color plot mean SNReff(1) results. ........................................................81
Table 24: Complete Acoustic color plot SNReff(1) standard deviation range results. ........82
Table 25: Acoustic color plot rMSE(1) results....................................................................83
Table 26: Acoustic color plot mean SNReff(2) results. ........................................................84
Table 27: Complete Acoustic color plot SNReff(2) standard deviation range results. ........85
Table 28: Acoustic color plot rMSE(2) results....................................................................86
Table 29: Bispectrum mean SNReff(1) results. ....................................................................87
Table 30: Complete Bispectrum SNReff(1) standard deviation range results. .....................88
Table 31: Bispectrum rMSE(1) results. ...............................................................................89
Table 32: Bispectrum mean SNReff(2) results. ....................................................................90
Table 33: Complete Bispectrum SNReff(2) standard deviation range results. .....................91
Table 34: Bispectrum rMSE(2) results. ...............................................................................92
Table 35: SAS mean SNReff(1) thresheld results. ...............................................................93
Table 36: Complete SAS SNReff(1) standard deviation range thresheld results. ................94
Table 37: SAS rMSE(1) thresheld results. ..........................................................................95
Table 38: SAS mean SNReff(2) thresheld results. ...............................................................96
Table 39: Complete SAS SNReff(1) standard deviation range ............................................97
Table 40: SAS rMSE(2) thresheld results. ..........................................................................98
Table 41: Acoustic color plot mean SNReff(1) thresheld results. ........................................99
Table 42: Complete Acoustic color plot SNReff(1) standard deviation range ...................100
xiv
Table 43: Acoustic color plot rMSE(1) thresheld results. .................................................101
Table 44: Acoustic color plot mean SNReff(2) thresheld results. ......................................102
Table 45: Complete Acoustic color plot SNReff(2) standard deviation range ...................103
Table 46: Acoustic color plot rMSE(2) thresheld results. .................................................104
Table 47: Bispectrum mean SNReff(1) thresheld results. ..................................................105
Table 48: Complete Bispectrum SNReff(1) standard deviation range thresheld results. ...106
Table 49: Bispectrum rMSE(1) thresheld results. .............................................................107
Table 50: Bispectrum mean SNReff(2) thresheld results. ..................................................108
Table 51: Complete Bispectrum SNReff(2) standard deviation range thresheld ...............109
Table 52: Bispectrum rMSE(2) thresheld results. .............................................................110
xv
1. INTRODUCTION
The United States Navy has invested a great deal of resources over the decades to
the field of mine-countermeasures. This is due to the inherent danger in the presence of
underwater mines or unexploded ordnances (UXO) in the waterways. In fact, the amount
of mines and UXO present in the waters today poses a great threat in both the military
aspect and the humanitarian aspect. In order to locate and identify these hazardous
objects, sonar technology is used as sound can travel at much greater distances than light
underwater. Traditionally, high-frequency sonar systems have been used to generate high
resolution reflectivity images of the seabed.
This is possible because, at higher
frequencies, specular reflection is dominant.
From these images, size and shape
information can be extracted about an object in question (target). However, there is still a
great deal of ambiguity in the classification of the target.
In order to reduce this ambiguity, low-frequency sonar systems are considered.
There has been an increasing interest in the application of low-frequency sonar systems
in the field of mine-countermeasures (MCM) due to their ability to induce a resonant
response from the target.
This low-frequency target response is a combination of
specular reflection and a harmonic or resonant portion of the signal, where vital
information about the targets physical parameters lie. This additional information is
expected to improve the probability of successful target classification.
1
In this thesis, a method to use various representations of a target’s low-frequency
acoustic response, or signature, to discriminate targets is proposed and implemented. The
signature representations considered here are: a synthetic aperture sonar (SAS) intensity
image, an acoustic color plot, and a higher-order spectral image. It should be stressed
that the focus of this thesis is not the modeling of the target response, but rather the
analysis of the ability of the previously mentioned signature representations to
differentiate between targets of varying physical parameters.
The modeled target
response has been graciously provided by the Naval Surface Warfare Center Panama City
Division (NSWC-PCD).
2
2. OBJECTIVE
The information about a proud target, delivered by its elastic acoustic response, is
a function of the physical parameters of the target (i.e. size, shape, material type and
internal structure), the target’s location and orientation relative to the sonar, as well as the
intrinsic parameters of the sonar (in particular the frequency band of operation and
aperture) [1,4-16]. Therefore, the objective of this thesis is two-fold; to observe the
changes in the elastic acoustic response in different acoustic signatures representations
and to quantify the discriminatory ability of these acoustic signature representations to
distinguish targets of varying physical parameters. This has been accomplished through
the development and application of an effective signal-to-noise ratio (SNReff)
performance metric for each acoustic signature representation and the results compiled
into a look-up table. The look-up table may be applied to a classifier in order to increase
the probability of successful target classification, which is the ultimate goal of the minecountermeasures field.
3
3. SCIENTIFIC CONTRIBUTION
In the realm of mine-countermeasures, high-frequency sonar systems have been
the norm to extract information, or features, about a submerged object in question.
However, the high-frequency acoustic target response is limiting in its amount of
extractable features. The solution to the lack of target features is a shift to the lowfrequency domain; here, more information about the target can be extracted from the
target’s resonant, or elastic, acoustic response [1]. This is the area in which the work
presented in this thesis will contribute.
This thesis delivers an initial step into the
application of low-frequency sonar signature analysis in the field of minecountermeasures (MCM) to determine which of a series of sonar signature
representations allows for the best, least ambiguous, target discrimination. This is done
through the development and implementation of performance metrics to observe the
target differentiation ability of three sonar signature representations (synthetic aperture
sonar, acoustic color plot and bispectrum). The targets considered are proud elastic
spherical shells.
4
4. LITERATURE REVIEW
The research performed within this thesis stems from many different areas in the
field of acoustic target scattering and signal processing. However, the focus lies on sonar
signature representations and their application within the field of MCM to increase the
probability of successful target classification. The following literature review gives an
overview of the recent work that has been performed in using sonar signature
representations as a means for target feature extraction and target classification with the
intent to show from where the work presented in this thesis is built. The proceeding
sections start with an overview of the end-user application of the extracted sonar
signature information in the form of classification. Following this, a look into what
impacts the target’s acoustic response in the low-frequency, elastic domain. Lastly, a
look into which sonar signature representations have been chosen, and a look at how they
have been applied in the fields of feature extraction and target classification will be
discussed.
4.1. Target Classification
4.1.1. Machine Learning
Successful target classification is key to any mine hunting sonar system and is
broken down into two main components: feature extraction and classification. The focus
5
of this thesis deals with the feature extraction component. Here, features are considered
to be elements within the acoustic target response that differentiate the target from all
others. Ideally, one would want the features extracted from the image to be completely
distinguishable from all others in order to make the job of the classier trivial. However,
this is not the case and this is where machine learning comes into play [2].
The authors explain, in [2], that the main goal of machine learning is to use the
features, extracted from the data, to minimize the occurrences of false negatives and false
positives, while increasing the probability of successful target classification.
False
negatives are instances when a target has been labeled as “not a mine” when in fact it is a
mine. False positives are the opposite and are inherently less consequential. The labels:
“not a mine” and “mine” are called classes and, in the problem at hand, these are the only
two possibilities available. Therefore, it is essential that a number of differentiating
features be used to help improve the probability of successful target classification.
4.1.2. Traditional Methods
High-frequency sonar systems have been used throughout the past to produce
high resolution, photo-like images of the seafloor. These images are used to locate and
identify objects in question laying on the seafloor. As was mentioned before, it is the
extractable features that are of greatest importance when trying to classify an object in
question (target).
Reference [3] gives a description of classical image processing
techniques to extract target feature information from these high resolution sonar images.
The method is called segmentation and involves the isolation of specific regions of
interest, which can be seen in Figure 1.
6
Figure 1: Layout and dimensions of the target area with respect to the sonar
location. [3]
The segmentation method is used to determine features such as; height of the
target, width of the target, length of the shadow and width of the shadow. The separation
between the shadow regions and the highlighted regions is determined by edge detection.
The location of the edge of the target and/or shadow (both along-track and cross track
directions) is denoted by a sharp change in the intensity from one pixel to the next and
can be defined as
Gr  n   I p  n p  1  I p  n p  ,
(4.1)
where Gr is the gradient field of the acoustic image, Ip is the intensity of the pixel, and np
is the pixel number which goes up to N being the total number of pixels in each slice of
the image.
7
The gradient equation was used to identify the location of the edges of the target
and the shadow zone. When these locations are known, as well as the height of the sonar,
one can calculate the height of the target and shadow as well as the width of the target
and shadow. Figure 1 shows the geometry and relative location of the target and shadow
zone with respect to sonar location. The height of the target, h can be calculated using
the equation;
h
H S
rS
(4.2)
Where H is the size of the no-echo zone which is directly below to the sonar
(nadir), S is the length of the shadow zone, and r is the distance from the sonar to the
target. [3]
4.1.3. Low-Frequency Target Response
It is helpful to gain a background in the acoustic response of elastic targets to
understand the contributing vibration and waves to the resonant response observed in the
sonar signatures. Much work has been performed in the area that is known as Resonant
Scatter Theory (RST) on the influence of physical target parameters on the target’s
acoustic response under ideal conditions. When considering a thin shell target, as is done
in [1], the main physical target parameters of influence are the target’s material density,
its radius (a sphere or cylinder is considered here) and its wall thickness.
These
parameters have a direct effect on the target’s resonant frequencies, both in their location
and their width (duration).
8
Another main contributor to the structure of the elastic response is a phenomenon
known as mid-frequency enhancement.
This occurrence has been observed by the
authors of [4] and [5]. It is noted that this phenomenon occurs due to a phase matching
of the circumnavigating elastic waves. In other words, as the elastic waves travel around
the target, they match phase at the near and far ends of the target’s perimeter. This
causes a buildup of wave energy and, in turn, causes an increase, or enhancement, in
certain regions of the frequency response. This can be seen in the acoustic color plot
signature representation where the middle frequency region is composed of relatively
large, in magnitude, peaks. Figure 2 gives a visual of these circumnavigating waves as
the envelope a spherical target. The waves denoted in Figure 2 are composed of Franz
waves, anti-symmetric (A0) and symmetric (S0) lamb waves. The waves denoted as I and
SR are the incident and specular reflection waves, respectively.
9
Figure 2: Shadowgraph of ultra-short sound pulses propagating around
water-immersed steel shell [6].
4.2. Sonar Signature Representations
The following sections discuss the sonar signature representations used within this
thesis and their implementation in recent work in the field of target recognition.
4.2.1. Synthetic Aperture Sonar
The first sonar signature representation considered is the synthetic aperture sonar
(SAS) representation. SAS is the first choice of signature representation because of its
strong presence in the field of target classification and because the data used in this thesis
is modeled as if it was collected from a SAS system [7,8,9,10].
10
The authors in [7] begin with a discussion of the low-frequency scattering of a
solid aluminum cylindrical target. It is mentioned that a main contributor to the elastic
response of the target is a surface elastic wave referred to as a leaky Rayleigh wave. This
is a wave that is present in the target-fluid half space and it is guided around the target,
circumnavigating it. The wave is called leaky because energy “leaks,” or propagates
outward, into the fluid medium as the wave’s phase velocity exceeds the fluid’s sound
velocity. The authors continue their study in a small scale experiment to observe the
response of the target as a SAS image. The result is a target response image that is noted
to be composed of three distinct sections: a specular reflection section (occurring first), a
section that is composed of the elastic response and a section that is composed of the
multipath response. It is noted that, along with the leaky Rayleigh waves, the elastic
portion of the signal is also composed of Franz waves and whispering gallery waves, both
of which are also circumnavigating waves.
The authors also observed the data collected during a pond experiment
(PONDEX10) that was performed at the Naval Surface Warfare Center – Panama City
Division (NSWC-PCD) in 2010. The target responses are composed of three sections
(called a triplet structure by the author in [8]), which include the specular response
followed by the overlapping of the elastic and multipath responses. This structure was
also observed by the authors of [9] and [10]; however, it is their observations of the
spectral response as a function of target aspect angle that is of interest.
11
4.2.2. Acoustic Color
The second sonar signature utilized in this thesis is the acoustic color plot. This
representation is a frequency spectrum as a function of target aspect angle and has been
implemented in [9,10,11]. Both [9] and [10] observe the acoustic color plot on the data
collected at the PONDEX10 experiment. The acoustic color plot representations of
several targets are utilized to make a superficial comparison between targets. It is also
noted that frequency enhancement can clearly be seen in this signature representation. It
appears useful for separating frequency contributor, such as; in this case, helical and
meridional waves since the targets are cylindrical.
An experiment similar to PONDEX10 was performed in 1998 and it is described
in [11]. The authors discuss the use of frequency versus aspect angle processing of a
cylindrical shell target as a method of target identification. It is shown that the acoustic
color plot is composed of quasi-hyperbolic and quasi-parabolic curves centered at either
broadside or endfire, which are a result of diffraction and elastic features. It should be
noted that these features would not be present in the case where the target is spherical.
The authors state that these features produce a patterned texture to the image and that this
pattern is not evident in the response of a natural object. This allows for the distinction
between natural and man-made objects; the first step to target classification, target
identification.
4.2.3. Higher-Order Spectral Analysis
The final sonar signature representation considered in this thesis is the higherorder statistic, the bispectrum. The bispectrum has been implemented in the field of
12
target classification in [12,13]. In both instances, the bispectrum is used as a means to
extract additional features from the target response that is missed when using lower-order
statistical analysis. The appeal behind using higher-order statistics is that they produce a
processed signal that is invariant to scaling, rotation and translation. This is very helpful
in mine hunting as the relative position between the sonar and the target may change
from one measurement to the next.
References [12,13] both contain a large list of statistical features that have been
extracted from the target’s bispectrum and trispectrum. It is also noted in [13] that the
higher-order spectral (HOS) analysis allows for phase information to be retained that
would otherwise be lost in traditional spectral analysis; furthering the list of unique,
extractable features. A list of 64 bispectral and 64 trispectral features has been compiled
and passed through a classifier [12] and the performance was a 90 percent mine detection
rate with a 10 percent false alarm rate. In other words, 10 percent of the mine-free
images were classified as containing a mine.
13
5. SCIENTIFIC BACKGROUND
The work presented in this thesis is based upon a SAS model whose resulting
signature data is provided by NSWC-PCD. The synthetic sonar stave data are generated
using a physics-based sonar model.
The considered sonar signature representations
(SAS, acoustic color plot and bispectrum) are produced from this data. The following
sections discuss the physics of the wave-target interaction considered in the model, as
well as, the formulation of the different sonar signature representations.
Figure 3: Considered Backscatter Ray Diagram.
14
Figure 4: Sonar Diagram (Top View).
The diagrams shown in Figure 3 and Figure 4 aid in the description of the
formulation of the sonar system considered. A source moves in the y-direction shown
and insonifies a perfectly smooth, thin-shell, spherical target in contact with the seafloor.
Three frames of reference are used to establish the governing equations: a fixed frame at
the origin O  x, y, z  , a fixed frame at the center of the target T  x, y, z and a moving
frame at the center of the source S  x, y, z  . In addition to the insonification of the
target, the source signal also insonifies an area of the bottom, denoted as dA . Certain
simplifying assumptions have been made. These assumptions include: (a) the target is
perfectly smooth and isotropic (i.e. azimuthal invariance), (b) the source has a directional
factor Hs(θ,φ), where θ and φ are the polar and azimuthal angles from the source center,
15
and (c) the only contribution from the bottom is backscattering (i.e. the boundary
conditions along the edge of the target are the same everywhere). It should be noted that,
here, the direct target path, the direct ground path and the target-bottom bounce path (the
bottom-target bounce arrives at the same time) are the only contributors to the multipath.
Source Signal, s(t)
Propagation:
1. Cylindrical spreading for SAS and Side-Scan
2. Spherical spreading for color plots and mid to lowfrequency rigid and elastic response
Rigid Back
Propagation:
Equation (5.7)
Elastic Back
Propagation:
Equations (5.9), (5.12)
Bottom Back
Propagation:
Equation (5.16)
Pse  t , R,  
Ps  t , R, 
Psc  r ,  
Pb  t ,  ,  
+
Received Signal, P(t)
Figure 5: Back Propagation Block Diagram.
As shown in Figure 3, the incident signal takes various paths. In fact, more paths are
taken than shown in the figure; however, these are the paths considered here. Each path
has a modulating effect on the incident signal and these modulations are broken down in
Figure 5 and described in further detail in the following sections.
16
5.1. Pressure Field for a Proud Spherical Target
This section provides a derivation of the scattered pressure field from the target
shown in Figure 3. It begins with a description of the rigid response, followed by the
elastic response (using both membrane theory and thin-shell theory) and the contribution
of the bottom as an acoustic backscatterer. The pressure wave, incident to the target is
P0  t , ,   
 t
P0 it kr r 
e
H s  ,   rect 
T
r
 p

 ,

(5.1)
where P0 is the pressure amplitude at 1 meter from the source, r is the distance from the
source (here, spherical spreading is assumed),  is the angular frequency, kr is the
 t
radiated wave number, H s is the directional factor of the source transducer and rect 
T
 p



is the transmitted pulse envelope of duration Tp .
In order to bring the response of the target and bottom into the same coordinate
system as the source, a change in frame of reference must be made. This is done by
setting the following:
FO :  O, x, y, z 
(5.2)
FS :  S , x, y, z  ;
y   y  yS
(5.3)
where, OS  yS
FT : T , x, y, z   ;
 x, y, z   x, y, z   OT   z, y, z   TO,
17
(5.4)
where FO , FS , and FT are the coordinate frames for the fixed origin, the moving sonar,
and the fixed target, respectively. The subscripts, T and S denote locations relative to the
source and target, respectively. The following sections discuss the response of the target,
to the incident wave (5.1), in the FT frame and can be translated into the FS frame using
(5.2-5.4).
5.2. High-Frequency Sonar Systems
5.2.1. Rigid Spherical Scatterer
A complete theoretical discussion regarding the rigid scattering from a spherical
target can be found in Appendix H. Under assumption (a), the azimuthal dependence
may be neglected in the direction of wave arrival. From here, it is shown, in the general
case that the rigid surface acceleration distribution becomes [14]:
s ( ) 
kPi

 (2n  1)i
n 0
n
Pn (cos  ) jn (kRT ),
(5.5)
and rigid scattered pressure becomes
ps ( RT , )   P0  (2n  1)i n Pn (cos  )
n 0
jn (kR)
hn (kRT ).
hn (ka)
(5.6)
Here, k is the wave number, RT is the distance from the target to an arbitrary point outside
the sphere, P0 is the pressure amplitude of the incident wave,  is the density of the fluid,
Pn denotes the m=0 Legendre polynomial, jn denotes the Bessel function of the first kind,
hn denotes the spherical Hankel function of the first kind, and the primes denote the
respective derivatives.
18
To adapt the general solution in (5.6) to the problem at hand, the incident wave in
(5.1) must be incorporated into the scattered pressure. This leads to the rigid scattered
response to become [14]
ps (t , RT , ) 

 t  (5.7)
P0 it kr r 
j  (kR)
e
(2n  1)i n Pn (cos  ) n
hn (kRT ) H s  ,   H T  ,   rect  

T 
r
hn (ka)
n 0
 p
where H T denotes the directivity of the scatter from the target. It is assumed that the
source and receiver are one in the same and therefore have the same directivity.
5.3. Low-Frequency Sonar Systems
In the case of low frequency sonar systems, an entirely different echo is received.
This echo not only contains the specular reflection response from the target (as in the
high frequency sonar system) but it also contains the elastic, harmonic response of the
target. In high-frequency systems, the transmitted signal reflects off of the target while in
low-frequency systems, the signal penetrates the surface of the target. This is why
specular reflection is the dominant portion of the high-frequency response and a high
reflectivity is observed. As the wave passes into the structure of the target, it causes the
target to vibrate. This vibration is made up of a combination of various types of surface
and body waves. The surface waves consist of: Rayleigh waves, Franz waves and
Stoneley waves. The body waves consist of Lamb waves. The Rayleigh wave group is
associated with the lowest-order Love wave group and is a shell-borne surface wave and
is denoted by S 0 . The Franz wave is also known as the creeping wave and is a fluidborne wave that circumnavigates the target. The Stoneley wave is a fluid-borne surface
19
wave that circumnavigates in the fluid medium on the surface of the shell. These waves
can be seen in Figure 2. [1]
5.3.1. Elastic Spherical Scatterer (Using Membrane Approximation)
As was mentioned in the previous section, a more complete discussion of the
elastic scattering of a spherical target can be found in Appendix 0. Here, the azimuthal
dependence maybe neglected as well.
It is shown that the scattered pressure is a
superposition of the rigid scatter response with the elastic scatter response [14]:
pse ( RT , ) 
ikr R
i  cPe
i
kRT

(2n  1)i n Pn (cos  ) 
c
jn (ka) 


,
2
hn (ka)
(ka) hn (ka)( Z n  zn )  (5.8)
n 0


for kRT  n 2  1
where c is the sound speed of the fluid, Zn is the shell impedance and zn is the radiation
impedance. As was done for (5.7), the general result in (5.8) must be adapted for the
incident wave given in (5.1). This results in a scattered pressure field in the form [14]:
pse (t , RT , ) 
 t 

i  cP0eikR eit kr   (2n  1)i n Pn (cos  ) 
c

j
(
ka
)

H

,

H

,

rect




  . (5.9)

 n
 s
T
kRT r
hn (ka)
(ka) 2 hn (ka)( Z n  zn ) 
n 0

 Tp 
for kRT  n2  1
5.3.2. Elastic Response Using Thin-Shell Theory
A thin shell is defined as a body that is bounded by two curved surfaces whose
distance from each other is very small compared to all other dimensions. The middle
surface is defined as the surface which lies in the middle between both bounding
surfaces. The distance between the two bounding surfaces is known as the thickness, h.
20
Shells may be thought of as a generalization for plate theory since plates are a special
case of shells in which the radius of curvatures is infinite. In this case, linear elasticity is
assumed to hold, which constitutes the assumptions that the material is isotropic,
homogeneous, and displacements are small. Also, it is assumed that shear deformation
and rotary inertial effects are neglected, and the thickness is held constant.
Sabine [15] demonstrated a derivation of the equations of motion for the vibration
theory of a thin spherical shell. What is important for this thesis is the description of the
normal mode solutions. It is shown, for the transient incident case that the backscatter
pressure is described by
2r 1
Psc (r , , ) 
a 2

 G ( x) f

( , x)e
ix (  ar )
dx.

(5.10)
.
ct

This expression is normalized in that x  ka    a and   where c is the sound
a
c
speed of the surrounding fluid,  is the angular frequency, a is the radius of the sphere,
and t is time. Also, G(x) is the spectrum of the incident pulse, f ( , x) is the normalized
form function of the steady state response of an elastic, air-filled, spherical shell and is
given as
2 
 Pn (cos  )(2n  1)Tn ( x)
ix n 0
j  ( x)
c
Tn ( x)  n
 2
hn ( x) x hn ( x) 2 ( Z n  zn )
f  ( , x) 
(5.11)
Taking (5.10) and introducing the incident wave, (5.1) the scatter response becomes:
21
2r 2
Psc (r , , )

a

 t 

P0 eit kr r  
ix (  ar )
dx  H s  ,   H T  ,   rect   .
  G ( x) f  ( , x)e
T 
2
 

 p
(5.12)
5.3.3. Noise from Bottom Scattering
As shown in Figure 3, an area of the surface becomes insonified by the source and
in turn emits backscatter which is interpreted as noise from the seafloor. The bottom
scatter can be expressed in terms of the incident wave with a time delay, t B and bottom
directivity H B
Pb  t , ,   
 t
P0 i t tb kr r 
e
H

,

rect



s
r2
 Tp

 H B  ,  ,  ,  B , cB ,  , c, dA  .

(5.13)
The directivity of the bottom is shown to be a function of not only the direction of the
incoming wave but also a function of the waves angular frequency  , the bottom’s
density  B and sound speed cB , the fluid density  , the fluid sound speed c, and the
insonified area dA .
Assuming that the incident angle to the bottom is the same as the scatter angle,
Lambert’s Law [14], defined as
I s   Ii sin 2   dA,
(5.14)
where I s is the scatter intensity, I i is the incident intensity, and  is a proportionality
constant. Plugging in the relationship between intensity and pressure,
22
P2
I
,
c
(5.15)
the scattered pressure field from the bottom can be expressed as
1

 Pb  2  2
PB (t , ,  )    B cB   sin   .
 c 


(5.16)
5.4. Signatures
As discussed in Section 4.2, the sonar signature representations chosen are the
synthetic aperture sonar, the acoustic color plot and the bispectrum. The following
sections discuss these signatures and how they are developed for their implementation
within this thesis.
5.4.1. Synthetic Aperture Sonar
The following development of the synthetic aperture system model has been
adapted from [16]. The complete derivation is beyond the scope of this paper, so that
only the main points are provided. Several assumptions have been made. The model is
developed in two-dimensions. It is assumed that the platform is moving in the same
plane as the target (Figure 3 and Figure 4) so that the range to the target is slant range.
The conversion to ground range is straight forward and a function of the grazing angle.
The system being modeled is a broadside strip-map, multiple receiver, synthetic array
system and assumes that the start-stop approximation is valid: the platform transmits and
receives each pulse at the same location before instantaneously moving forward to the
next pulse location [16]. The mathematical model developed here ignores the effects of
medium turbulence, refraction and multipath. However, it is important to note that the
23
simulated data generated for this analysis includes multipath from the seafloor, as
depicted in Figure 3. It is assumed that the target reflectivity is constant over all viewing
angles.
The array is a single element transmitter collocated with a six element receiver.
As seen in Figure 6, transmitter and receiver array coordinates (u,t) coincide with the
image coordinates (x, y). Here we assume that u = 0 at the center of the scene. The
platform travels at a velocity of vp and repeatedly transmits a linear frequency modulated
(LFM) pulse at uniformly spaced intervals determined by the total number of elements in
the array and the element size. The complex LFM pulse is given by
 t  j   t  K t 2 
p m  t   rect   e 0 c ,
c 
where Kc is the LFM chirp rate and
is the center frequency.
Figure 6: Synthetic Aperture Sonar Geometry
24
(5.17)
Given the above assumptions, the received echoes as a function of time t and
cross-range u, eem(t,u) take the form [16].
eem  t , u  
 ff  x, y   A  t , x, y  u 
xy
t
2 
 2 2
pm  t 
x   y  u   dxdy
 c

(5.18)
where ff  x, y  is the complex reflectivity distribution of the insonified area, A  t , x, u 
is the spatial-temporal amplitude response of the combined transmitter and receiver
apertures, and
t
represents the convolution with respect to time. The wave number
algorithm is used to generate the synthetic aperture image. Key to this technique is the
derivation of the two-dimensional Fourier transform via the principle of stationary phase
[16]. If FFx (k x , k y ) represents
x  ff ( x, y) in the wave number domain, the synthetic
aperture image can be expressed in the angular frequency vs. cross-range wave number
  ku  domain as
EEm  , ku  

Pm   A  ku 
jk
where
FF (k , k )
x
x
(5.19)
y
 is the Stolt mapping operator. The synthetic aperture image is obtained using
an efficient implementation of the conversion from   ku to x  y , as explained in [16].
25
5.4.2. Acoustic Color
The elastic response of a submerged target can provide some unique target
information when represented in the frequency-aspect domain, which is called an
acoustic color plot [17,12,3]. Acoustic color is simply a target strength plot of frequency
versus aspect. In this thesis, it is assumed that both the sphere and the loading fluid are
homogeneous and isotropic.
The acoustic color of the target
is the distribution, scaled in dB re 1Pa,
of the returned acoustic pressure as a function of angular frequency and aspect angle.
Given a received signal eem  t , u  in Pa, using time windowing and expressed as a
function of ω and u,
EL  , u  
   2 2


20 log  Ft eem  t 
r0  u 2 , u  w  t   
   c

 

(5.20)
In this thesis, a Tukey window w(t) with steep roll-off is used. The acoustic color
EL  ,  is obtained using a change of variable,
u
.
 r0 
  tan 1 
5.4.3. Higher-Order Spectra
Higher-order statistics (HOS) show great promise in the field of minecountermeasures in that they uncover signature information that would otherwise be lost
using solely traditional statistics [18]. HOS allows for the retention of phase information,
26
the detection of non-linearities and deviations from Gaussianity within the signal [18].
However, the most important and useful feature of HOS, for target classifications
purposes, is its invariance to translation, rotation and scaling. Theoretically, this allows
for the same HOS image to be representative of the same target regardless of the target’s
location or orientation.
This could mean the difference between a successful
classification and a false negative when passed to a classifier. Here, we focus on the
well-documented bispectrum and for a brief derivation see Appendix J, as the
formulation of this higher-order statistic is beyond the scope of this thesis.
The bispectrum is calculated using the direct Fourier transform method [19]. The
direct method differs from the indirect method in that the latter implements the Fourier
transform post auto-correlation while the former takes the Fourier transform of the
observed data. The bispectrum is calculated by taking the received echo in the slant
range-aspect angle domain, eem(r,u) and setting u to zero degrees aspect. This zero
aspect angle corresponds to the center of the target (i.e. broadside). Figure 7 shows the
real part of the returned echo, eem(r,0), for an aluminum spherical shell of radius 0.5
meters and wall thickness that is 5% of the radius.
27
Figure 7: Return time-series signal from an aluminum spherical shell of radius 0.5m.
The slant range, r is converted to the one-way travel time, t (in seconds) by using
t
2r
,
c
(5.21)
where c is the sound speed of the fluid. Figure 7 shows the real part of the returned echo
at zero degrees aspect, eem(r,0) for an aluminum spherical shell of radius 0.5 meters and
wall thickness that is five percent of the radius. If we define
X m (i ), i  1, 2
(5.22)
as the Fourier transform of eem(r,0), its bispectrum is [12,13]
B(1 , 2 )  X m (1 ) X m (2 ) X m* (1  2 ).
28
(5.23)
6. METHODOLOGY
To gain a better understanding of the procedure of classifying a target, consider
the scenario where a sonar operator has acquired a low-frequency acoustic signature from
a suspicious object.
This measured target signature is contingent upon the target’s
physical parameters. In order to classify the target, the procedure proposed here is to take
the measured signature and compare it, on a case-by-case basis, to a collection of
signatures contained within a library of target signatures from various, known target
parameters. The signatures in the library are indexed by a series of variables (Section
6.3.1) to organize the data as well as to allow for a correspondence between the target
signature and the indexed physical parameters for target identification. When a close
match is found, the corresponding target parameter indices can be retained and used to
classify the measured, unknown target.
This is the job of a classifier.
Here, the
effectiveness of each target signature to produce discriminating signatures is analyzed in
order to determine their feasibility to be applied to the library comparison part of the
classification procedure (Section 6.3).
This procedure is further explained in the
following sections, beginning with an overview of the considered dataset, followed by the
generation of each target signature representation and then the performance evaluation
procedure that is used to quantify each signature’s ability to differentiate between targets.
29
6.1. Simulated Dataset
The datasets used in this thesis were generated and provided by the Naval Surface
Warfare Center, Panama City Division (NSWC-PCD).
The synthetic data were
generated using a physics based sonar model. Table 1 presents the entire dataset and the
considered subset of target parameters utilized in this thesis. The targets considered are
smooth, spherical shells that have a wall thickness that is 5% of their radius. The incident
pulse is a linear frequency modulated (LFM) pulse and the data is sampled at 60 kHz.
Three acoustic signature representations are considered: the synthetic aperture sonar
beamformed image, the acoustic color plot image, and the bispectrum image. The target,
at 3 separate ranges (25 meters is considered), is either composed of aluminum or
stainless steel with either a radius of 0.25 meters or 0.5 meters.
The last parameter that is considered is the background image composition and
there are several types: two different seafloor intensity images that have been collected
from an actual sonar system, 72 background intensity images that have been generated
using a correlated k-distribution [20], free field, and a smooth bottom.
30
Table 1: Spherical Data.
Source Signal
10-40 kHz 2ms LFM Pulse
Sampling Frequency
60 kHz
SAS Beamformed;
Signature Representation
Acoustic Color;
Bispectrum
Aluminum;
Material Type
Steel
0.25;
Target Radius (meters)
0.50
Target Thickness
5%(radius)
Target Range (meters)
15, 25*, 35
Bottom Types
2 Seafloor intensity images,
72 correlated k-distributed,
free-field, smooth bottom
* The range is kept at a constant of 25 meters for this
thesis
31
6.2. Signature Image Generation
The signature images are built from single target response to insure that the data
remains consistent and that only the representation of the data changes. The data begins
with a large SAS image of a target on a background. The size of the image can be seen in
Figure 8 where the red box outlines the target response. If the assumption is made that
the target has been located yet remains unclassified, we can crop this large image around
the target response. This cropping is performed by setting a fixed window size (constant
range and cross-range values) and applying this to every signature. The limitation to this
is that, in practice, the target location will not be precisely known and the target signature
can vary dramatically from case to case. Also, the target signatures might not align
correctly and then the comparison might not be between the targets considered.
However, when applied here, the cropping eliminates noise from the rest of the image
while reducing the amount of data to process. The result of the image cropping can be
seen in Figure 9.
32
Figure 8: Complete SAS beamformed image of an aluminum spherical shell
of radius 0.5 meters at a range of 25 meters.
33
Figure 9: Cropped SAS beamformed image of an aluminum spherical shell
of radius 0.5 meters at a range of 25 meters.
6.2.1. SAS Beamformed Images
The synthetic aperture sonar images are generated using the raw sonar time-series
data and implementing the Stolt mapping operation as described in section 3.4.1. The
result is a beamformed image containing the target response and it is here that the triple
structure found by [9,7] is observed; namely the specular response, followed by the
overlapping elastic response and multipath.
6.2.2. Acoustic Color Images
The acoustic color images are generated from the SAS beamformed image. The
SAS image is tightly cropped around the target response to isolate it and remove noise as
shown in Figure 9. Once this region has been isolated, the image is then put through the
34
process described in section 3.4.2, where the response is inverse beamformed (Figure 10)
and the spectrum is calculated along each ping. The signature in Figure 10 differs from
that in Figure 9 in that the latter is beamformed, the former is the raw sonar echo and the
parabolic nature of the response is due to the traveling of the sonar source along the
cross-range direction. The result is the acoustic color plot shown in Figure 11.
Figure 10: Sonar time-series echo of an aluminum spherical shell of radius
0.5 meters at a range of 25 meters.
35
Absolute value of the acoustic color plot (dB)
Figure 11: Acoustic Color plot of an aluminum spherical shell with a radius of
0.5 meters at a range of 25 meters.
6.2.3. Bispectrum Images
The inverse beamformed image is utilized so that the bispectrum can be implemented
on the time series data (Figure 10). The MATLAB HOSA toolbox [19] is used to
calculate the bispectrum using the direct fast Fourier transform (FFT) approach along the
zero aspect angle ping (Figure 7), as described in section 3.4.3. The resulting bispectrum
image can be seen in Figure 12.
36
Figure 12: Bispectrum of the time-series response at zero degrees aspect of
an aluminum spherical shell of radius 0.5 meters at a range of 25 meters.
6.3. Performance Evaluation
In order to quantify the effectiveness of these target signature representations in
differentiating between targets, a performance evaluation for each representation must be
conducted. This is done by using the relative Mean-Square Error (rMSE) performance
metric between target signatures of varying target parameters and the effective Signal-toNoise Ratio (SNReff) performance metric of a measured target vs. a referenced target.
The following sections describe the development of these performance metrics, as well as
37
the attempt to increase the reliability and effectiveness of the results using statistical
averaging and SNReff optimization via minimum thresholding.
The performance is evaluated by generating a matrix of values (look-up table) that
contain the results of the rMSE and SNReff. They are calculated as a measured signature
relative to a referenced signature of the same signature type.
This allows for the
observation of the performance metric values as each case goes from a match (measured
and reference signatures are of the same target) to a mismatch (measured and reference
signatures are of different targets).
6.3.1. Look-up Table
Let U(bU, sU, tU, mU, lU, aU, qU ) represent a vector of target signature attributes for
an unknown target and R(bR, sR, tR, mR, lR, aR, qR) represent a vector of target signature
features for a reference target. The image parameter indices are defined as follows: b is
the bottom specification index, s is the shape index, t is the thickness ratio index, m is the
material index, l is the range index, a is the signature index, and q is the bottom
realization index used for statistical averaging. The possible index values and their
corresponding target/signature parameters are listed in Table 2.
38
Table 2: Possible feature index values.
Index
Feature Index
Physical Target Parameter
Value
1
Bottom Type 1
2
Bottom Type 2
Bottom Specification
3
Free Field
Index (b)
4
Smooth Bottom
72 Averaged correlated
5
k-distributed
1
Sphere (0.5 m radius)
2
Sphere (0.25 m radius)
1
1 (solid)
Shape Index (s)
Thickness Ratio
0.05*radius
Index (t)
2
(thin shell)
1
Aluminum
2
Stainless Steel
1
15 m
2
20 m
3
25 m
4
30 m
5
35 m
6
40 m
Material Index (m)
Range Index (l)
39
Signature Index (a)
1
SAS Image
2
Acoustic Color Plot
3
Bispectrum
Bottom Realization
Correlated k-distributed
4-75
Index (q)
images
6.3.2. Relative Mean-Square Error
To quantify the difference between a match and a mismatch, a relative difference
between the two respective signatures (measured and referenced) must be calculated. To
calculate this, the Mean-Square Error relative to a referenced image is used and it is
presented in (6.1). This is a comparison of the image intensity on a pixel-by-pixel basis
whose locations are denoted by (n1,n2) and a denotes the signature representation type.
The rMSE is calculated for all possible combinations of targets; each target exemplar is
treated either as the unknown or as the reference target. The formulation of the rMSE, its
mean and its standard deviation are, respectively, denoted as
N1
EU2 , R ,qU ,qR 
N2

n1 1 n2 1
aU ,qU  n1 , n2   aR ,qR  n1 , n2 
N1
N2
  a n , n 
n1 1 n2 1
2
U ,R
E

QU
QR

qU 1 qR 1
40
R , qR
1
EU2 , R ,qU ,qR
Q
2
(6.1)
2
2
(6.2)
U ,R 
QU
QR

EU2 , R ,qU ,qR  EU2 , R
 Q  1
qU 1 qR 1
2
.
(6.3)
Here, the subscripts are defined in Table 2, n is the pixel index, N is the total
number of pixels, q is the realization index and Q is the total number of realizations. The
subscripts U and R denote the measured and referenced image, respectively. In the use of
statistical averaging (correlated k-distributed images, b = 5, see section 6.3.4) the mean
and the unbiased estimation of the standard deviation of the rMSE will be used, and they
are denoted, respectively in (6.2) and (6.3).
6.3.3. Effective Signal-to-Noise Ratio
As previously mentioned, the SNReff is inversely proportional to the rMSE. The
SNReff is utilized to represent the target discrimination in terms of relative acoustic
power. Two formulations of the SNReff have been developed; the first formulation is
used to observe the impact of the bottom on the signatures, and the second formulation is
used to observe the signatures’ ability to discriminate between targets on a target vs.
target case. The analysis in this thesis will focus on the second formulation.
1
SNReff

Target's acoustic power (rigid and elastic response, free-field)
,
Target's acoustic power (bottom scattering, multipath)
(6.4)
Measured target's acoustic power
.
Referenced target's acoustic power
(6.5)
 2
SNReff

41
In other words, and using the index notation introduced in Table 2, SNReff(1) holds all
measured and referenced target parameters identical except the measured image is in the
free field while the reference image is on a bottom type. This formulation is;
1
SNReff

1
 10 log10 
 E2
 U ,R

.

sR  sU ,t R tU , mR  mU ,lR lU , aR  aU ,bR 5,bU 3 
(6.6)
The flaw with this formulation is that both the measured and referenced images
are assumed to share the same physical target features. This does not provide any
indication of the signatures’ potential to discriminate between two targets under identical
environmental conditions. Therefore, the second SNReff formulation is used to resolve
this issue. The formulation observes the acoustic power of the measured signature
relative to the measured signature’s acoustic power. In other words, this formulation
keeps the target range, background type and signature type constant between images
while varying all other physical target parameters as expressed in
 2
SNReff

1
 10 log10 
 E2
 U ,R
sR  sU and/or t R  tU and/or mR  mU and lR lU ,aR  aU ,bR bU

.


(6.7)
6.3.4. Statistical Averaging
Statistical averaging must be performed to generate a statistically meaningful
quantification of the target discrimination in each signature representation and target
case. Additionally, a confidence interval is calculated. This confidence interval will
42
indicate the reliability of the SNReff calculations and its formulation is shown in (6.8) and
(6.9), where rMSE is the mean and rMSE is standard deviation (σ), respectively.
S  SNReff  S


(6.8)

10log10 rMSE  rMSE  SNReff  10log10 rMSE  rMSE

(6.9)
In the cases presented in this thesis, statistical averaging is performed by placing
the targets, within the image, on realizations, or variations, of the background images
(images of the seafloor). Every background image shares the same intensity distribution
but has a different profile. Figure 13 shows a side-by-side comparison of the result of
two background realization images sharing the same intensity distribution but having
completely different profiles. It should be noted that without any statistical averaging
(q=1) the SNReff reaches to infinity in the case of a perfect match, which is a meaningless
result. This is because the measured and reference target are identical, making the rMSE
go to zero.
43
Figure 13: Examples of the correlated k-distributed background realizations.
6.3.5. Target Discrimination
To more clearly and concisely observe the target discrimination capabilities of each
signature type, a minimum and maximum difference in calculated SNReff between a
match and all possible mismatches, within the data set, are calculated. This is used to
identify the range of SNReff values that can be expected for a mismatch. To determine
the discriminatory SNReff range, a comparison is performed in two ways. In the first
case, all possible measured image SNReff results are compared to the same reference
image SNReff result. In the second case, the measured image SNReff result is held
constant and is compared to all the SNReff results from every reference image
44
combination. In the first case equation (6.10) is used. In the second case, equation (6.11)
is used.
SNReff ,min  min  SNReff  bm , sm , tm , mm , lm   SNReff  bU , sU , tU , mU , lU  
SNReff ,max  max  SNReff  bm , sm , tm , mm , lm   SNReff  bU , sU , tU , mU , lU   .
(6.10)
bm  3,5; sm  1, 2; tm  2; mm  1, 2; lm  3; bU  3,5; sU  1, 2; tU  2; mU  1, 2; lU  3
SNReff ,min  min  SNReff  bm , sm , tm , mm , lm   SNReff  bR , sR , t R , mR , lR  
SNReff ,max  max  SNReff  bm , sm , tm , mm , lm   SNReff  bR , sR , t R , mR , lR   .
(6.11)
bm  3,5; sm  1, 2; tm  2; mm  1, 2; lm  3; bR  3,5; sR  1, 2; t R  2; mR  1, 2; lR  3
SNReff ,range  SNReff ,max  SNReff ,min .
(6.12)
Here, ΔSNReff,range denotes the difference range in SNReff between a match and
mismatch, where the subscripts min and max denote the minimum and maximum values
of the range, respectively. The index subscript m denotes a match (referenced and
measured image are identical) where bU = bR, sU = sR, tU = tR, mU = mR, lU = lR. The range
of values is chosen to correspond with the considered target and signature parameters
given in Table 2. The background index, b denotes if the background is either free field
(b=3) or if it is the averaged correlated k-distributed images, the shape index ranges from
a sphere of 0.5 meter radius (s=1) to a sphere of 0.25 meter radius (s=2), the material
index ranges from aluminum (m=1) to stainless steel (m=2), and the range index is the
same though out this thesis at 25 meters (l=3). This is so that all possible combinations
of target and signature parameters of the considered dataset are observed.
45
Under ideal conditions, the minimum and maximum mismatch differences are both
large in magnitude and there is little difference between the two. This means that the
signature has the ability to undoubtedly differentiate between a match and a mismatch
while having the ability to do so consistently with very little fluctuation or ambiguity.
6.4. Performance Optimization
Clearly, we want to keep the ΔSNReff,min as large as possible to avoid ambiguity
during target classification. In an attempted to increase the ΔSNReff,min even more, a
minimum threshold is applied to each image.
6.4.1. Minimum Threshold
In an initial attempt to increase the SNReff performance, a minimum value
threshold is applied to focus on the peaks within the images and isolate them from all the
intensity values that fall below the threshold value. This makes the rMSE presented in
(6.1) now a function of the minimum threshold T




a
 n , n  if aU ,qU ,T  n1 , n2   TU max aU ,qU ,T
aU ,qU ,T  n1 , n2    U ,qU ,T 1 2
0 elsewhere

a
 n , n  if aR ,qR ,T  n1 , n2   TR max aR ,qR ,T
aR ,qR ,T  n1 , n2    R ,qR ,T 1 2
,
0 elsewhere

(6.13)
0  TU  TR  1, 0  n1  N1  1, 0  n2  N 2  1
N1
EU2 , R ,qU ,qR ,T 
N2

n1 1 n2 1
aU ,qU ,T  n1 , n2   aR ,qR ,T  n1 , n2 
N1
N2
 a
n1 1 n2 1
46
R , qR ,T
 n1 , n2 
2
.
2
(6.14)
The minimum threshold is set as a percentage of the maximum values of the signature
and the same threshold value is applied to both the measured and referenced images,
across all exemplars.
By applying a minimum threshold, some of the target data contained within the
signature will be lost. Therefore, the criteria for the threshold values are that the values
sought should increase the relative difference between a match and a mismatch while
retaining as much target information from the original, unthresheld signature. To find a
compromise, an observation of the SNReff as a function of threshold between the
measured and referenced image is required. The impact of the threshold on the SNR eff
for an example signature can be seen in Figure 14. The case shown here is the SNReff(1)
formulation (referenced image is a correlated k-distributed background image while the
measured image is a free field image) and this is representative for each other case.
47
Figure 14: Acoustic Color SNReff as a function of measured and referenced
minimum threshold (The red dot is the chosen threshold value of 25%, see Section 7.2.1).
It is observed that, the SNReff is inversely proportional to the threshold value.
However, in order to make a determination on the threshold value to be chosen, a closer
look at the impact of the threshold on the percentage of retained information must be
taken. This is performed by comparing the energy contained within the thresheld image
to the energy contained within the original unthresheld image:
N1
% Energy Remaining 
N2
  a n , n 
n1 1 n2 1
N1 N 2
1
2
  a n , n 
n1 1 n2 1
48
T
1
2
.
(6.15)
The threshold value must be chosen very carefully otherwise some critical target
information could be lost.
49
7. RESULTS
To illustrate the discriminatory abilities of each signature type, a couple of example
target cases are chosen and their results for each signature type (not thresheld and
thresheld) are discussed. The measured targets chosen are an aluminum spherical shell of
radius 0.5 meters and a thickness of 5% of the radius and a stainless steel spherical shell
of the same dimensions. The respective index values for the SNReff(2) formulation are
U(5,1,2,1,3) and U(5,1,2,2,3) and they are compared to every referenced image.
Following this, the generalized results across a larger number of combinations are shown.
It is beneficial to discuss the physical referenced target parameters as they appear in
the tables in the following sections and in Appendices B through G. The first column
corresponds to an aluminum spherical shell of radius 0.5 meters, the second column
corresponds to an aluminum spherical shell of radius 0.25 meters, the third column
corresponds to a stainless steel spherical shell of radius 0.5 meters and, lastly, the fourth
column corresponds to a stainless steel spherical shell of radius 0.25 meters.
The
instances in which a match has occurred are distinguished by a highlighted cell.
For a complete compilation of results for each signature and each performance
metric, see Appendices B through G. The results in these Appendices include, for each
signature type; the SNReff, the minimum and maximum mismatch difference
50


(ΔSNReff), the SNReff  standard deviation SNReff , the rMSE and rMSE (mean and
standard deviation, respectively) for each case.
7.1. Effective SNR
The results presented in the following sections are broken down into two tables per
signature representation. The first table contains the results for the standard deviation
range of the two afore-mentioned targets. As was mentioned, the cases that indicate a
match are highlighted in yellow and the other cases indicate a mismatch. From this table,
a perspective of the SNReff(2) range of plus and minus one standard deviation can be seen
and this gives an insight into the discriminatory performance of the signature
representation (Section 6.3.3). The second table displays the minimum and maximum
mismatch difference for each signature representation (Section 6.3.5). This table is a
good indicator of the discrimination performance of the signature because it provides a
range of the performance consistency. For instance, if the range of values is high and
close together, this indicates good, consistent performance; however, if the values are low
and close together, this indicates consistently poor performance. If the values are widely
spread, this indicates that the signature’s performance can lead to ambiguous and
inconsistent results.
7.1.1. Synthetic Aperture Sonar
The performance of the SAS signature representation is very promising. There is
an indication of good target discrimination as seen when comparing the match and
mismatch cases in Table 3. Also, this good performance can be deemed consistent by
51
looking at the mismatch difference range. The minimum differences of 14.5802 dB and
15.5393 dB mean that even if the targets are similar they can be easily differentiated.
Table 3: SAS SNReff(2) standard deviation range results.
SAS SNReff(2) (dB)
Reference Image
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
15.1079
-4.3570
0.2160
-4.6222
(20.5566)
(-1.2701)
(3.2984)
(-1.5356)
0.8817
-2.7965
14.8185
-3.6185
(3.9635)
(0.2914)
(20.2488)
(-0.5306)
(5,1,2,1,3)
Measured
Image
(5,1,2,2,3)
Table 4: SAS SNReff(2) minimum and maximum mismatch difference.
SAS SNReff(2) (dB) Mismatch Difference
Minimum (Maximum)
15.5393
(5,1,2,1,3)
Measured
(20.3761)
Image
14.5802
(5,1,2,2,3)
(19.0784)
7.1.2. Acoustic Color plot
As was the case for the SAS signature results, the magnitude and range of the
SNReff(2) and the mismatch difference indicate that the acoustic color plot performed well.
Although the magnitude of the mismatch difference is not as high as it is for the SAS
signature, the minimum differences are relatively high at values of 10.3996 dB and
52
11.3997 dB. Table 5 demonstrates the signature’s ability to differentiate between targets
and the mismatch difference range is just as small as the SAS range at about 3 dB (Table
6). This indicates that the acoustic color plot representation performs just as constistently
as the SAS signature but the latter is more robust.
Table 5: Acoustic color plot SNReff(2) standard deviation range results.
ACP SNReff(2) (dB)
-σ(+σ)
Reference Image
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
22.1140
9.2242
12.1761
9.1569
Measured
(26.7952)
(12.3085)
(15.2624)
(12.2409)
Image
12.1529
9.3017
23.1070
9.1899
(15.2369)
(12.3856)
(27.7145)
(12.2746)
(5,1,2,1,3)
(5,1,2,2,3)
Table 6: Acoustic color plot SNReff(2) minimum and maximum mismatch difference.
ACP SNReff(2) (dB) Mismatch Difference
Minimum (Maximum)
10.3996
(5,1,2,1,3)
(13.4196)
Measured
Image
11.3997
(5,1,2,2,3)
(14.3624)
53
7.1.3. Bispectrum
The results for the bispectrum signature are ambiguous at best. As can be seen in
Table 7, the performance is good for some of the cases but bad for others. This means
that the performance is not consistent. Furthermore, the mismatch difference range is
very large, close to 20 dB for each case in fact. This means that the bispectrum is not a
robust signature for target discrimination.
Table 7: Bispectrum SNReff(2) standard deviation range results.
Bispectrum SNReff(2)
Reference Image
(dB)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
16.1147
-3.6424
14.8092
-4.2268
(24.7031)
(-0.3115)
(20.0338)
(-0.8825)
14.0100
-4.3270
16.9779
-4.9110
(19.9312)
(-0.9875)
(25.6272)
(-1.5694)
-σ(+σ)
(5,1,2,1,3)
Measured
Image
(5,1,2,2,3)
Table 8: Bispectrum SNReff(2) minimum and maximum mismatch difference.
SAS SNReff(2) (dB) Mismatch Difference
Minimum (Maximum)
1.8829
(5,1,2,1,3)
Measured
(21.4309)
Image
3.4015
(5,1,2,2,3)
(22.9866)
54
7.2. Thresheld Effective SNR
The results presented in the following sections are presented in the same manner as in
Section 7.1. These results display the impact of the performance optimization (Section
6.4) implemented as discussed in Section 7.2.1.
7.2.1. Minimum Threshold
The minimum threshold value chosen for this initial observation is a value of 25%
of the peak value within each signature image. This value was chosen as a compromise
to allow for the retention of some of the signal energy, or information, while removing a
portion of the lower intensity components which are believed to be mostly composed of
background noise. In other words, this can be looked at as an isolation of the image
peaks that occur above 25% of the maximum. Implementing (5.35) yields the resulting
plots shown in Figure 15 and Figure 16. These plots are representative for each signature
type and target combination. The impact on the SNReff can be seen in Figure 14 where
the red dot indicates the SNReff for a minimum threshold value of 25% applied equally to
the measured and referenced images.
55
Figure 15: Acoustic Color plot referenced image remaining signal energy as
a function of minimum threshold.
56
Figure 16: Acoustic Color plot measured image remaining signal energy as
a function of minimum threshold.
It can be seen, in Figure 15, that the remaining energy is approximately 25% of
the original referenced image and approximately 42% of the original measured image
(Figure 16). This high threshold is needed to focus on the peaks of the target’s signature,
where it is assumed that the significant target information lies. The complete results of
the SNReff after this minimum threshold is applied can be found in Appendices E through
G and an analysis of a few cases is presented in the following sections.
7.2.2. Synthetic Aperture Sonar
After the minimum threshold is applied to the SAS signatures, the result was
similar to before the minimum thresholding. The relatively large magnitude difference
between a match and a mismatch (Table 9) indicates good performance. However, the
57
reliability of the target discrimination drops.
The minimum thresholding causes an
increase in the maximum difference and a decrease in the minimum difference. This
increase in the mismatch difference range causes the performance to become less
consistent than they were without the minimum thresholding.
Table 9: SAS SNReff(2) thresheld standard deviation range results.
SAS SNReff(2) Thresheld
Reference Image
(dB)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
11.3067
-3.5548
-0.9635
-1.7231
(19.2167)
(-0.4198)
(2.1172)
(1.3579)
-9.8189
-9.4948
8.2605
-3.9937
(-6.6749)
(-6.1568)
(21.0688)
(-0.8642)
-σ(+σ)
(5,1,2,1,3)
Measured
Image
(5,1,2,2,3)
Table 10: Thresheld SAS SNReff(2) minimum and maximum mismatch difference.
Thresheld SAS SNReff(2) (dB) Mismatch Difference
Minimum (Maximum)
13.3564
(5,1,2,1,3)
Measured
(15.9299)
Image
13.7540
(5,1,2,2,3)
(19.5745)
58
7.2.3. Acoustic Color plot
The minimum thrsholding, when applied to the acoustic color plot signatures, had
much the same effect as it did on the SAS signatures (Table 11).
The minimum
mismatch difference decreased from 10.3996 dB and 11.3997 dB prior to the minimum
thresholding to 4.7454 dB and 5.7665 dB after the minimum thresholding. This indicates
a decrease in discrimination performance.
Furthermore, the minimum mismatch
difference decreased from about 10.5 dB prior to minimum thresholding to about 5 dB
after minimum thresholding. This decreased the mismatch difference range by about 5.5
dB.
This dramatically decreased the ambiguity (compared to the other signature
representations) and; therefore, improves the reliability of the acoustic color plot
signature as a target differentiator. This suggests that better discrimination performance
might be achieved with a tweaking of the minimum threshold value.
Table 11: Acoustic color plot SNReff(2) thresheld standard deviation range results.
ACP SNReff(2) Thresheld
Reference Image
(dB)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
4.1667
-0.0683
-0.6493
-0.0458
(9.1342)
(3.0227)
(2.5476)
(3.0454)
-0.8758
-0.0663
5.3029
-0.0415
(2.3635)
(3.0222)
(9.8257)
(3.0472)
-σ(+σ)
(5,1,2,1,3)
Measured
Image
(5,1,2,2,3)
59
Table 12: Thresheld Acoustic color plot SNReff(2) minimum and maximum
mismatch difference.
Thresheld acoustic color plot SNReff(2) (dB) Mismatch Difference
Minimum (Maximum)
4.7454
(5,1,2,1,3)
Measured
(5.3145)
Image
5.7665
(5,1,2,2,3)
(6.5518)
7.2.4. Bispectrum
The results of the bispectrum signature performance, post minimum thresholding,
are very interesting. The overall performance is still poor (Table 13); however, the
interesting factor lies in the resulting mismatch difference range. In this case the range
decreased by about 7 dB, which is more than the decrease observed in the acoustic color
signatures. This means that the minimum thresholding increase the reliability of the
bispectrum signature results, and this is what was originally expected of the minimum
thresholding process. This also suggests that better discrimination performance might be
achieved with a tweaking of the minimum threshold value.
Table 13: Bispectrum SNReff(2) thresheld standard deviation range results.
Bispectrum SNReff(2)
Thresheld (dB)
Reference Image
(5,1,2,1,3)
(5,2,2,1,3)
60
(5,1,2,2,3)
(5,2,2,2,3)
-σ(+σ)
3.9643
-6.6922
3.8768
-8.4471
(11.0567)
(-3.2755)
(9.0093)
(-4.8531)
3.0730
-7.2103
3.9676
-8.9638
(8.8916)
(-3.7194)
(10.2574)
(-5.3134)
(5,1,2,1,3)
Measured
Image
(5,1,2,2,3)
Table 14: Thresheld bispectrum SNReff(2) minimum and maximum
mismatch difference.
Thresheld Bspec SNReff(2) (dB) Mismatch Difference
Minimum (Maximum)
0.4744
(5,1,2,1,3)
Measured
(13.2115)
Image
0.9883
(5,1,2,2,3)
(13.5727)
The minimum thresholding, in most cases, seems to do more harm than good (at least,
in the scope of this thesis) to the signatures’ discrimination performance. It introduces
ambiguity into the results by increasing the overall mismatch difference range
(ΔSNReff,range). The exact opposite was expected to happen, which suggests that the
subtle, less intense portions of the target response are influenced by the target’s physical
parameters. There is one exception to this observation; that is the bispectrum signature.
The thresholding of this signature reduces the overall mismatch difference; however, it
61
was not enough to redeem the bispectrum signature representation because the range is
still very large in comparison with the other two signature representations.
7.3. Overall Performance
To gain an overall view of the performance of each signature representation, the
mismatch difference is generalized to include all the ranges from each case and isolate
the absolute minimum and maximum ΔSNReff values. This gives a view of the average
performance of each signature as a whole and allows for a generalized conclusion to be
drawn as to which signature representation performs the best and most consistent target
discrimination.
Table 15 lists the target parameter and signature representation
combinations tested and the corresponding overall minimum and maximum difference
results are shown in Table 16.
62
Table 15: Target parameter combinations tested
Feature Index
Index Values
Bottom Specification Index
Physical Target Parameter
72 Averaged correlated k-
5
(b)
distributed backgrounds
1
Sphere (0.5 m radius)
2
Sphere (0.25 m radius)
Shape Index (s)
0.05*radius
Thickness Ratio Index (t)
2
(thin shell)
1
Aluminum
2
Stainless Steel
3
25 m
1
SAS Image
2
Acoustic Color Plot
3
Bispectrum
Material Index (m)
Range Index (l)
Signature Index (a)
63
Table 16: Overall target discrimination performances and ranges.
Thresheld
Mismatch
SNReff(2)
Mismatch
Difference
Signature
ΔSNReff,min
Type
(ΔSNReff,max)
SNReff(2)
Thresheld
Difference
Range
ΔSNReff,min
Range
ΔSNReff
(ΔSNReff,max)
[dB]
[dB]
[dB]
ΔSNReff
[dB]
14.5802
SAS
8.6896
5.8845
13.4214
(20.4647)
(22.1911)
Acoustic
9.1139
4.7454
color plot
(14.3624)
(21.7778)
1.8829
0.3360
5.2485
Bispectrum
17.0324
21.1037
(22.9866)
13.2367
(13.5727)
64
It can be seen from Table 16 that the SAS signature representation performs best
when compared to the acoustic color plot and bispectrum signature representations. This
is denoted by the magnitude of the overall minimum difference and by the relatively
small overall difference range.
A similar observation is made when the minimum
threshold is applied. The acoustic color signature representation is the second best
approach with an overall minimum difference of 9.1139 dB. The acoustic color plot
signature representation provides a very slightly more consistent performance than the
SAS signature representation as overall mismatch difference range is smaller.
The
implementation of the minimum threshold here both hinders and helps the acoustic color
plot signature’s performance by not only reducing the overall minimum mismatch
difference, but also by reducing the overall mismatch difference range. The bispectrum
performs the least favorably due to its low magnitude overall minimum difference; it also
performs inconsistently, as shown by its very large overall mismatch difference range.
65
8. CONCLUSION
The main goal of this thesis was the identification of a particular sonar signature
representation that allowed for a more robust and consistent differentiation between
targets of varying physical parameters. The sonar signature representations considered
were SAS, acoustic color plot and bispectrum. To quantify the signatures’ ability to
differentiate between targets, a series of performance metrics were developed and
implemented. A relative Mean-Square Error (rMSE) was developed as a means to
compare the sonar signatures from each target type. To turn this metric into a parameter
that is useful for comparing sonar signature energy, an effective Signal-to-Noise Ratio
(SNReff) metric was also developed and utilized. This allowed for a direct comparison in
dB of the signatures’ target discrimination capabilities. A minimum threshold value of
25% of the peak signature value was implemented and the results were tabulated
according to sonar signature.
An initial attempt to improve the signature performance in each representation, a
minimum threshold was applied to each signature. This technique was used to isolate the
peak values in each signature image since it is believed that the peaks are a product of the
variations in the target’s physical parameters.
The performance was tabulated and
compared to the results prior to the application of the minimum threshold.
It has been determined that the SAS signature representation provides the most
definitive and consistent performance in successfully differentiating between a matched
66
and mismatched target.
The SAS performance is shown to be accurate and least
ambiguous with its minimum and maximum mismatch difference of 14.5802 dB and
20.4647 dB, respectively.
The acoustic color plot signature performed well also;
however, with a minimum and maximum mismatch difference of 9.1139 dB and 14.3624
dB, respectively, the magnitude of the range is not as large as the SAS signature. This
makes it slightly less desirable as a means for target discrimination.
bispectrum signature led to very ambiguous results.
Finally, the
This is due to its very large
minimum and maximum difference range, 21.1037 dB.
The application of the minimum threshold proved to not be as beneficial as originally
believed. It caused an increase in the mismatch difference range for both the SAS and
the acoustic color plot signatures. Although it did not greatly degrade their ability to
differentiate between targets, it introduced a factor of ambiguity that, in turn, reduces the
reliability of their target discrimination performance. This being said, the application of
the minimum threshold on the bispectrum signature yielded interesting results. The
target discrimination ability remained poor; however, there was a relatively large
decrease in the mismatch difference range that indicates the potential for an increase in
performance with further analysis of the minimum threshold value.
Given additional resources and time, several areas would have been explored in
greater detail. These areas include analyzing the remaining simulated data that was not
included in the subset considered in this thesis, a further study into the minimum
threshold optimization, a close scrutiny of SNReff(1) to observe the impact of the
67
background on the target differentiation and then ultimately apply these techniques to the
experimental data collected during the PONDEX10 experiment.
68
9. APPENDIX
A. Nomenclature
ACP
Acoustic Color plot
Bspec
Bispectrum
FFT
Fast Fourier Transform
HOS
Higher-Order Statistics (spectrum)
HOSA
Higher-Order Spectral Analysis
LFM
Linear Frequency Modulated
MCM
Mine-Countermeasures
NSWC-PCD
Naval Surface Warfare Center – Panama City Division
PONDEX10
2010 NSWC – PCD Pond Experiment
rMSE
Relative Mean-Squared Error
RST
Resonant Scatter Theory
SAS
Synthetic Aperture Sonar
SNR
Signal-to-Noise Ratio
SNReff
Effective Signal-to-Noise Ratio
SNReff(1)
Free Field with Target vs. Background with Target SNReff
SNReff(2)
Target vs. Target SNReff
69
UXO
Unexploded Ordnance
ΔSNReff,range
Target Mismatch Difference Range
ΔSNReff,min
Minimum Target Mismatch Difference
ΔSNReff,max
Maximum Target Mismatch Difference
t
Convolution with respect to time
θ
Polar angle relative to target center
μ
Proportionality constant
ρ
Fluid density
ρB
Bottom density
τ
ct
a
φ
Azimuth angle relative to target center
ω
Angular frequency
ω0
Center frequency
PB
Bottom scatter pressure field
Mean value
Standard deviation value
P sc
Complex elastic scatter response (thin shell theory)
P s
Complex rigid scatter response
P se
Complex elastic scatter response (membrane approximation)
(bm, sm, tm, mm, lm, am, qm)
Matched image parameters (b: bottom index, s: shape index,
70
m: material index, l: range index, a: realization index, q:
signature representation index)
(bR, sR, tR, mR, lR, aR, qR)
Referenced image parameters (b: bottom index, s: shape index,
m: material index, l: range index, a: realization index, q:
signature representation index)
(bU, sU, tU, mU, lU, aU, qU)
Measured image parameters (b: bottom index, s: shape index,
m: material index, l: range index, a: realization index, q:
signature representation index)
(n1,n2)
Pixel coordinates
(u,t)
Receiver array coordinates
(x,y)
Image coordinates
*
Complex conjugate
′
Derivative
A
Spatial-temporal amplitude response
a
Target radius
A0
Anti-symmetric lamb wave
B
Bispectrum
c
Fluid sound speed
cB
Bottom sound speed
dA
Insonified bottom area
eem
Received echo
EEm
Received echo in the frequency-wave number domain
71
EL
Received acoustic color echo
f∞
Normalized form function of the steady state response of an
elastic, air-filled spherical shell
ff
Complex reflectivity distribution of the insonified area
Fo(O,X,Y,Z)
Fixed coordinate reference frame at the origin
FS(S,X,Y’,Z)
Moving coordinate reference frame at the source center
FT(T,X’’,Y’’,Z’’)
Fixed coordinate reference frame at the target center
G
Incident pulse spectrum
Gr
Gradient field
H
SAS image target height
HB
Bottom scatter directivity factor
hn
Hankle function of the 1st kind
HS
Source directivity factor
HT
Target scatter directivity factor
I
Acoustic intensity
I
Incident wave
Ii
Incident intensity
Ip
Pixel intensity
Is
Scatter intensity
jn
Bessel function of the 1st kind
k
Wave number
Kc
Chirp rate
72
kr
Radiated wave number
N
Total number of pixels
n
Index number
np
Pixel number
P
Acoustic pressure
P(t)
Received signal
P0
Incident pressure amplitude
pm
Complex LFM pulse
Pn
Legendre polynomial (m=0)
Psc
Transient backscatter pressure
Q
Total number of realizations
r
Distance from source
r0
Distance from source to target at 0o aspect angle
RT
Distance from target
S
SAS image shadow zone length
s
Source signal
S{}
Stolt mapping operator
S0
Symmetric lamb wave
SR
Specular reflection wave
tB
Time delay
Tp
Pulse duration
TR
Referenced image minimum threshold
73
TU
Measured image minimum threshold
U
Measured image
w(t)
Tukey window
Xm
Fourier transform of the received echos
Zn
Shell acoustic impedance
zn
Radiation acoustic impedance
74
B. SAS Results
a) SAS SNReff (1) Results
Table 17: SAS mean SNReff(1) results.
Mismatch
SAS SNReff(1)
Reference Image
Difference
(dB)
(3,1,2,1,3) (3,2,2,1,3) (3,1,2,2,3) (3,2,2,2,3)
Min (Max)
13.4074
(5,1,2,1,3)
14.6692
-3.2472
1.2618
-3.5192
(18.1884)
13.1020
(5,2,2,1,3)
-0.8758
14.6033
-0.0037
1.4982
Measured
(15.4761)
Image
12.4652
(5,1,2,2,3)
1.8636
-1.7043
14.3288
-2.5188
(16.8476)
12.8694
(5,2,2,2,3)
-0.7988
1.7455
-0.4964
14.6150
(15.4137)
Overall: Min
Mismatch
Min
12.8056
12.8548
13.0671
13.1167
(Max)
Difference
(Max)
(15.5450)
(17.8475)
(14.8252)
(18.1342)
12.4652
(18.1884)
75
Table 18: Complete SAS SNReff(1) standard deviation range results.
SAS SNReff(1) (dB)
-σ(+σ)
Reference Image
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
13.2106
-4.1804
0.3310
-4.4522
(16.8813)
(-2.0570)
(2.4480)
(-2.3293)
-1.8067
13.0887
-0.9346
0.5674
(0.3107)
(16.9385)
(1.1827)
(2.6847)
0.9328
-2.6374
12.8804
-3.4521
(3.0499)
(-0.5143)
(16.5175)
(-1.3285)
-1.7297
0.8147
-1.4274
13.0801
(0.3878)
(2.9318)
(0.6902)
(17.0101)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
76
Table 19: SAS rMSE(1) results.
SAS rMSE(1)
Reference Image
Mean (Standard
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
0.0341
2.1121
0.7479
2.2486
(0.0136)
(0.5063)
(0.1788)
(0.5389)
1.2234
0.0347
1.0009
0.7082
(0.2925)
(0.0144)
(0.2392)
(0.1693)
0.6511
1.4806
0.0369
1.7860
(0.1556)
(0.3549)
(0.0146)
(0.4282)
1.2019
0.6690
1.1211
0.0346
(0.2874)
(0.1599)
(0.2680)
(0.0146)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
77
b) SAS SNReff(2) Results
Table 20: SAS mean SNReff(2) results.
Mismatch
SAS SNReff(2)
Reference Image
Difference
(dB)
(5,1,2,1,3) (5,2,2,1,3) (5,1,2,2,3) (5,2,2,2,3)
Min (Max)
15.5393
(5,1,2,1,3)
17.0285
-3.0823
1.4893
-3.3475
(20.3761)
15.4472
(5,2,2,1,3)
-0.5323
17.1743
0.3632
1.7271
Measured
(17.7065)
Image
14.5802
(5,1,2,2,3)
2.1548
-1.5214
16.7350
-2.3434
(19.0784)
15.0735
(5,2,2,2,3)
-0.4810
2.0437
-0.1423
17.1172
(17.5983)
Overall: Min
Mismatch
Min
14.8737
15.1305
15.2457
15.3901
(Max)
Difference
(Max)
(17.5608)
(20.2565)
(16.8773)
(20.4647)
14.5802
(20.4647)
78
Table 21: Complete SAS SNReff(2) standard deviation range results.
SAS SNReff(2) (dB)
-σ(+σ)
Reference Image
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
15.1079
-4.3570
0.2160
-4.6222
(20.5566)
(-1.2701)
(3.2984)
(-1.5356)
-1.8053
15.1891
-0.9100
0.4533
(1.2764)
(20.9366)
(2.1721)
(3.5371)
0.8817
-2.7965
14.8185
-3.6185
(3.9635)
(0.2914)
(20.2488)
(-0.5306)
-1.7541
0.7704
-1.4156
15.1185
(1.3275)
(3.8529)
(1.6669)
(20.9305)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
79
Table 22: SAS rMSE(2) results.
SAS rMSE(2)
Reference Image
Mean (Standard
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
0.0198
2.0334
0.7097
2.1615
(0.0110)
(0.6937)
(0.2418)
(0.7373)
1.1304
0.0192
0.9198
0.6719
(0.3850)
(0.0111)
(0.3133)
(0.2290)
0.0689
1.4195
0.0212
1.7153
(0.2074)
(0.4844)
(0.0118)
(0.5853)
1.1171
0.6246
1.0333
0.0194
(0.3805)
(0.2128)
(0.3521)
(0.0113)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
80
C. Acoustic Color plot Results
a) Acoustic Color plot SNReff(1) Results
Table 23: Acoustic color plot mean SNReff(1) results.
Mismatch
ACP SNReff(1)
Reference Image
Difference
(dB)
(3,1,2,1,3) (3,2,2,1,3) (3,1,2,2,3) (3,2,2,2,3)
Min (Max)
3.1823
(5,1,2,1,3)
15.1153
9.7182
11.9330
9.7025
(5.4128)
4.0609
(5,2,2,1,3)
10.4229
16.4908
10.3243
12.4299
Measured
(6.1666)
Image
4.5293
(5,1,2,2,3)
11.9277
9.6947
16.4570
9.6594
(6.7976)
4.6398
(5,2,2,2,3)
10.4233
13.1180
10.3311
17.7578
(7.4268)
Overall: Min
Mismatch
Min
3.1876
3.3728
4.5239
5.3279
(Max)
Difference
(Max)
(4.6924)
(6.7962)
(6.1327)
(8.0984)
3.1823
(8.0984)
81
Table 24: Complete Acoustic color plot SNReff(1) standard deviation range results.
ACP SNReff(1) (dB)
-σ(+σ)
Reference Image
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
14.1726
8.7846
10.9936
8.7694
(16.3211)
(10.9090)
(13.1333)
(10.8926)
9.4914
15.5027
9.3923
11.4907
(11.6103)
(17.7721)
(11.5124)
(13.6300)
10.9945
8.7621
15.4965
8.7272
(13.1179)
(10.8838)
(17.6920)
(10.8480)
9.4908
12.1808
9.3963
16.7558
(11.6124)
(14.3147)
(11.5238)
(19.0627)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
82
Table 25: Acoustic color plot rMSE(1) results.
ACP rMSE(1)
Reference Image
Mean (Standard
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
0.0308
0.1067
0.0641
0.1071
(0.0075)
(0.0256)
(0.0155)
(0.0257)
0.0907
0.0224
0.0928
0.0571
(0.0217)
(0.0057)
(0.0222)
(0.0138)
0.0642
0.1073
0.0226
0.1082
(0.0154)
(0.0257)
(0.0056)
(0.0259)
0.0907
0.0488
0.0927
0.0168
(0.0217)
(0.0117)
(0.0223)
(0.0043)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
83
b) Acoustic Color plot SNReff(2) Results
Table 26: Acoustic color plot mean SNReff(2) results.
Mismatch
ACP SNReff(2)
Reference Image
Difference
(dB)
(5,1,2,1,3) (5,2,2,1,3) (5,1,2,2,3) (5,2,2,2,3)
Min (Max)
10.3996
(5,1,2,1,3)
23.8503
10.4981
13.4507
10.4307
(13.4196)
8.8573
(5,2,2,1,3)
10.8749
23.0702
10.9757
14.2129
Measured
(12.1953)
Image
11.3997
(5,1,2,2,3)
13.4267
10.5755
24.8264
10.4640
(14.3624)
9.1139
(5,2,2,2,3)
10.8032
14.2091
10.8600
23.3229
(12.5198)
Overall: Min
Mismatch
Min
10.4236
8.8611
11.3757
9.1101
(Max)
Difference
(Max)
(13.0471)
(12.5721)
(13.9663)
(12.8923)
9.1139
(14.3624)
84
Table 27: Complete Acoustic color plot SNReff(2) standard deviation range results.
ACP SNReff(2) (dB)
-σ(+σ)
Reference Image
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
22.1140
9.2242
12.1761
9.1569
(26.7952)
(12.3085)
(15.2624)
(12.2409)
9.6013
21.5104
9.7002
12.9371
(12.6847)
(25.5277)
(12.7895)
(16.0270)
12.1529
9.3017
23.1070
9.1899
(15.2369)
(12.3856)
(27.7145)
(12.2746)
9.5284
12.9333
9.5834
21.7302
(12.6152)
(16.0232)
(12.6760)
(25.8645)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
85
Table 28: Acoustic color plot rMSE(2) results.
ACP rMSE(2)
Reference Image
Mean (Standard
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
0.0041
0.0892
0.0452
0.0906
(0.0020)
(0.0304)
(0.0154)
(0.0309)
0.0818
0.0049
0.0799
0.0379
(0.0279)
(0.0021)
(0.0273)
(0.0129)
0.0454
0.0876
0.0033
0.0899
(0.0155)
(0.0299)
(0.0016)
(0.0306)
0.0831
0.0379
0.0820
0.0047
(0.0284)
(0.0130)
(0.0280)
(0.0021)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
86
D. Bispectrum Results
a) Bispectrum SNReff(1) Results
Table 29: Bispectrum mean SNReff(1) results.
Mismatch
Bispectrum SNReff(1)
Reference Image
Difference
(dB)
(3,1,2,1,3) (3,2,2,1,3) (3,1,2,2,3) (3,2,2,2,3)
Min (Max)
0.6251
(5,1,2,1,3)
17.3199
-2.4395
16.6948
-2.9053
(20.2251)
14.4727
(5,2,2,1,3)
2.3121
16.7848
2.1979
1.5369
Measured
(15.2479)
Image
3.2702
(5,1,2,2,3)
15.0635
-3.3041
18.3337
-3.8210
(22.1547)
13.4339
(5,2,2,2,3)
2.8115
2.4008
2.7034
16.2455
(13.8447)
Overall: Min
Mismatch
Min
2.2564
14.3840
1.6389
14.7086
(Max)
Difference
(Max)
(15.0078)
(20.0889)
(16.1359)
(20.0664)
0.6251
(22.1547)
87
Table 30: Complete Bispectrum SNReff(1) standard deviation range results.
Bispectrum SNReff(1)
Reference Image
(dB)
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
15.4314
-3.4565
15.1372
-3.9174
(20.7370)
(-1.1091)
(19.1468)
(-1.5832)
1.3796
14.4502
1.2657
0.5756
(3.5011)
(22.1883)
(3.3863)
(2.7733)
13.5437
-4.3250
16.4361
-4.8337
(17.4217)
(-1.9671)
(21.7821)
(-2.4979)
1.8785
1.4524
1.7709
13.9804
(4.0015)
(3.6159)
(3.8923)
(21.2574)
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
88
Table 31: Bispectrum rMSE(1) results.
Bispectrum rMSE(1)
Reference Image
Mean (Standard
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
0.0185
1.7537
0.0214
1.9522
(0.0101)
(0.4627)
(0.0092)
(0.5124)
0.5872
0.0210
0.6029
0.7020
(0.1406)
(0.0149)
(0.1443)
(0.1739)
0.0312
2.1400
0.0147
2.4104
(0.0131)
(0.5671)
(0.0080)
(0.6330)
0.5234
0.5753
0.5366
0.0237
(0.1254)
(0.1404)
(0.1285)
(0.0163)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
89
b) Bispectrum SNReff(2) Results
Table 32: Bispectrum mean SNReff(2) results.
Mismatch
Bispectrum SNReff(2)
Reference Image
Difference
(dB)
(5,1,2,1,3) (5,2,2,1,3) (5,1,2,2,3) (5,2,2,2,3)
Min (Max)
1.8829
(5,1,2,1,3)
18.5620
-2.2888
16.6791
-2.8689
(21.4309)
14.5343
(5,2,2,1,3)
2.6400
17.1743
2.5274
1.4235
Measured
(15.7508)
Image
3.4015
(5,1,2,2,3)
16.0311
-2.9706
19.4326
-3.5540
(22.9866)
13.7488
(5,2,2,2,3)
3.1952
2.5898
3.0775
16.9400
(14.3502)
Overall: Min
Mismatch
Min
2.5309
14.5844
2.7535
15.5165
(Max)
Difference
(Max)
(15.9220)
(20.1449)
(16.9052)
(20.4940)
1.8829
(22.9866)
90
Table 33: Complete Bispectrum SNReff(2) standard deviation range results.
Bispectrum SNReff(2)
Reference Image
(dB)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
16.1147
-3.6424
14.8092
-4.2268
(24.7031)
(-0.3115)
(20.0338)
(-0.8825)
1.3609
14.2549
1.2497
0.0473
Measured
(4.4609)
(31.0007)
(4.3454)
(3.4498)
Image
14.0100
-4.3270
16.9779
-4.9110
(19.9312)
(-0.9875)
(25.6272)
(-1.5694)
1.9174
1.2992
1.8006
14.3603
(5.0136)
(4.4345)
(4.8941)
(24.1803)
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
91
Table 34: Bispectrum rMSE(2) results.
Bispectrum rMSE(2)
Reference Image
Mean (Standard
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
0.0139
1.6939
0.0215
1.9359
(0.0105)
(0.6195)
(0.0116)
(0.7106)
0.5445
0.00192
0.5588
0.7205
(0.1865)
(0.0184)
(0.1911)
(0.2687)
0.0249
1.9818
0.0114
2.2667
(0.0148)
(0.7265)
(0.0087)
(0.8314)
0.4792
0.5508
0.4923
0.0202
(0.1639)
(0.1906)
(0.1683)
(0.0164)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
92
E. SAS Thresheld Results
a) SAS SNReff (1) Thresheld Results
Table 35: SAS mean SNReff(1) thresheld results.
SAS SNReff(1)
Mismatch
Reference Image
Difference
Thresheld
(dB)
(3,1,2,1,3) (3,2,2,1,3) (3,1,2,2,3) (3,2,2,2,3)
Min (Max)
18.7251
(5,1,2,1,3)
18.2294
-3.8889
-0.4963
-2.9344
(22.1183)
14.8999
(5,2,2,1,3)
-1.8479
14.4088
-0.4911
-0.6452
Measured
(16.2567)
Image
13.8140
(5,1,2,2,3)
-2.9360
-4.9295
10.8780
-3.2772
(13.3828)
11.4572
(5,2,2,2,3)
-2.0269
-1.6324
-0.1014
11.3559
(13.3828)
Overall: Min
Mismatch
Min
20.0773
16.0412
10.9793
12.0010
(Max)
Difference
(Max)
(21.1654)
(19.3383)
(11.3743)
(14.6331)
10.9793
(22.1183)
93
Table 36: Complete SAS SNReff(1) standard deviation range thresheld results.
SAS SNReff(1) Thresheld
Reference Image
(dB)
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
16.7583
-4.8231
-1.4301
-3.8673
(20.4709)
(-2.6972)
(0.6949)
(-1.7446)
-2.7785
12.1009
-1.4263
-1.5766
(-0.6620)
(19.6570)
(0.7022)
(0.5421)
-3.8668
-5.8675
8.8816
-4.2143
(-1.7499)
(-3.7315)
(14.6825)
(-2.0807)
-2.9576
-2.5668
-1.0346
9.7630
(-0.8410)
(-0.4402)
(1.0889)
(13.8980)
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
94
Table 37: SAS rMSE(1) thresheld results.
SAS rMSE(1) Thresheld
Reference Image
Mean (Standard
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
0.0150
2.4485
1.1211
1.9653
(0.0061)
(0.5876)
(0.2689)
(0.4710)
1.5303
0.0362
1.1197
1.1602
(0.3657)
(0.0254)
(0.2690)
(0.2775)
1.9661
3.1114
0.0817
2.1268
(0.4599)
(0.7501)
(0.0477)
(0.5127)
1.5948
1.4563
1.0236
0.0732
(0.3811)
(0.3496)
(0.2454)
(0.0324)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
95
b) SAS SNReff(2) Thresheld Results
Table 38: SAS mean SNReff(2) thresheld results.
SAS SNReff(2)
Mismatch
Reference Image
Difference
Thresheld
(dB)
(5,1,2,1,3) (5,2,2,1,3) (5,1,2,2,3) (5,2,2,2,3)
Min (Max)
13.3564
(5,1,2,1,3)
13.6656
-2.2642
0.3092
-0.4503
(15.9299)
8.6896
(5,2,2,1,3)
-2.8595
8.9007
0.1302
0.2112
Measured
(11.7602)
Image
13.7540
(5,1,2,2,3)
-8.5254
-8.1389
11.0491
-2.7049
(19.5745)
12.5244
(5,2,2,2,3)
-6.3987
-5.1710
0.1811
12.7055
(19.1042)
Overall: Min
Mismatch
Min
16.5251
11.1650
10.7398
12.4943
(Max)
Difference
(Max)
(22.1911)
(17.0396)
(10.9189)
(15.4104)
8.6896
(22.1911)
96
Table 39: Complete SAS SNReff(1) standard deviation range
thresheld results.
SAS SNReff(2) Thresheld
Reference Image
(dB)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
11.3067
-3.5548
-0.9635
-1.7231
(19.2167)
(-0.4198)
(2.1172)
(1.3579)
-4.1562
6.1535
-1.1427
-1.0618
Measured
(-1.0023)
(18.1992)
(1.9385)
(2.0196)
Image
-9.8189
-9.4948
8.2605
-3.9937
(-6.6749)
(-6.1568)
(21.0688)
(-0.8642)
-7.6789
-6.5124
-1.0943
10.5122
(-4.5756)
(-3.2199)
(1.9945)
(17.3529)
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
97
Table 40: SAS rMSE(2) thresheld results.
SAS rMSE(2) Thresheld
Reference Image
Mean (Standard
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
0.0430
1.6843
0.9313
1.1092
(0.0310)
(0.5828)
(0.3171)
(0.3778)
1.9317
0.1288
0.9705
0.9525
(0.6721)
(0.1137)
(0.3305)
(0.3244)
7.1210
6.5146
0.0785
1.8642
(2.4706)
(2.3872)
(0.0707)
(0.6440)
4.3639
3.2893
0.9592
0.0536
(1.4960)
(1.1904)
(0.3274)
(0.0352)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
98
F. Acoustic Color plot Thresheld Results
a) Acoustic Color plot SNReff (1) Thresheld Results
Table 41: Acoustic color plot mean SNReff(1) thresheld results.
ACP SNReff(1)
Mismatch
Reference Image
Difference
Thresheld
(dB)
(3,1,2,1,3) (3,2,2,1,3) (3,1,2,2,3) (3,2,2,2,3)
Min (Max)
0.2867
(5,1,2,1,3)
1.7999
1.4676
-0.4929
1.5132
(2.2928)
2.3439
(5,2,2,1,3)
-4.6346
10.5994
-4.4283
8.2555
Measured
(15.2340)
Image
3.1645
(5,1,2,2,3)
-0.3218
1.1064
4.3180
1.1534
(4.6397)
2.1077
(5,2,2,2,3)
-4.6140
7.8738
-4.4088
9.9814
(14.5954)
Overall: Min
Mismatch
Min
2.1217
2.7256
4.8109
1.7259
(Max)
Difference
(Max)
(6.4345)
(9.4930)
(8.7462)
(8.8280)
0.2867
(14.5954)
99
Table 42: Complete Acoustic color plot SNReff(1) standard deviation range
thresheld results.
ACP SNReff(1) Thresheld
Reference Image
(dB)
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
0.4479
0.5366
-1.5124
0.5822
(3.7735)
(2.6542)
(0.8418)
(2.6999)
-5.7646
9.5928
-5.4732
7.3185
(-3.1031)
(11.9121)
(-3.0496)
(9.4520)
-1.3402
0.1755
3.1240
0.2226
(1.0111)
(2.2928)
(5.9701)
(2.3398)
-5.7453
6.9318
-5.4548
8.9432
(-3.0800)
(9.0783)
(-3.0283)
(11.3485)
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
100
Table 43: Acoustic color plot rMSE(1) thresheld results.
ACP rMSE(1) Thresheld
Reference Image
Mean (Standard
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
0.6007
0.7132
1.1202
0.7058
(0.2413)
(0.1705)
(0.2964)
(0.1687)
2.9071
0.0871
2.7722
0.1494
(0.8639)
(0.0227)
(0.7541)
(0.0360)
1.0769
0.7751
0.3700
0.7668
(0.2846)
(0.1853)
(0.1171)
(0.1833)
2.8933
0.1632
2.7598
0.1004
(0.8610)
(0.0395)
(0.7516)
(0.0271)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
101
b) Acoustic Color plot SNReff(2) Thresheld Results
Table 44: Acoustic color plot mean SNReff(2) thresheld results.
ACP SNReff(2)
Mismatch
Reference Image
Difference
Thresheld
(dB)
(5,1,2,1,3) (5,2,2,1,3) (5,1,2,2,3) (5,2,2,2,3)
Min (Max)
4.7454
(5,1,2,1,3)
5.9759
1.2078
0.6614
1.2304
(5.3145)
6.3032
(5,2,2,1,3)
-4.8110
16.9668
-4.5792
10.6636
Measured
(21.7778)
Image
5.7665
(5,1,2,2,3)
0.4486
1.2090
7.0004
1.2338
(6.5518)
5.7276
(5,2,2,2,3)
-4.7448
10.7148
-4.5107
16.4424
(21.1872)
Overall: Min
Mismatch
Min
5.5272
6.2520
6.3390
5.7788
(Max)
Difference
(Max)
(10.7869)
(15.7589)
(11.5796)
(15.2120)
4.7454
(21.7778)
102
Table 45: Complete Acoustic color plot SNReff(2) standard deviation range
thresheld results.
ACP SNReff(2) Thresheld
Reference Image
(dB)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
4.1667
-0.0683
-0.6493
-0.0458
(9.1342)
(3.0227)
(2.5476)
(3.0454)
-6.2197
14.8979
-5.9287
9.2874
Measured
(-2.7129)
(21.0586)
(-2.6110)
(12.6898)
Image
-0.8758
-0.0663
5.3029
-0.0415
(2.3635)
(3.0222)
(9.8257)
(3.0472)
-6.1553
9.3600
-5.8618
14.1669
(-2.6427)
(12.6944)
(-2.5390)
(21.5101)
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
103
Table 46: Acoustic color plot rMSE(2) thresheld results.
ACP rMSE(2) Thresheld
Reference Image
Mean (Standard
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
0.2526
0.7572
0.8587
0.7533
(0.1305)
(0.2586)
(0.3025)
(0.2573)
3.0276
0.0201
2.8703
0.0858
(0.1600)
(0.0123)
(1.0460)
(0.0320)
0.9019
0.7570
0.1995
0.7527
(0.3216)
(0.2584)
(0.0954)
(0.2569)
2.9818
0.0848
2.8254
0.0227
(1.1442)
(0.0311)
(1.0310)
(0.0156)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
104
G. Bispectrum Thresheld Results
a) Bispectrum SNReff (1) Thresheld Results
Table 47: Bispectrum mean SNReff(1) thresheld results.
Bispectrum SNReff(1)
Mismatch
Reference Image
Difference
Thresheld
(dB)
(3,1,2,1,3) (3,2,2,1,3) (3,1,2,2,3) (3,2,2,2,3)
Min (Max)
0.0634
(5,1,2,1,3)
12.6121
-5.0437
12.5487
-5.2458
(17.8579)
10.7209
(5,2,2,1,3)
0.3159
11.0406
0.3197
-1.1578
Measured
(12.1984)
Image
1.6947
(5,1,2,2,3)
11.3298
-5.7035
13.0245
-5.8941
(18.9186)
8.3743
(5,2,2,2,3)
0.8850
-0.1512
0.8974
9.2718
(9.4230)
Overall: Min
Mismatch
Min
1.2822
11.1918
0.4758
10.4295
(Max)
Difference
(Max)
(12.2962)
(16.7441)
(12.7048)
(15.1659)
0.0634
(18.9186)
105
Table 48: Complete Bispectrum SNReff(1) standard deviation range thresheld results.
Bispectrum SNReff(1)
Reference Image
Thresheld
(dB)
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
10.9342
-6.0404
11.1124
-6.2626
(15.3825)
(-3.7478)
(14.7095)
(-3.9158)
-0.6183
9.2880
-0.6139
-2.1086
Measured
(1.5076)
(14.0262)
(1.5105)
(0.0614)
Image
9.8924
-6.7019
11.3449
-6.9105
(13.4933)
(-4.4050)
(15.7997)
(-4.5648)
-0.0466
-1.1133
-0.0339
7.6433
(2.0728)
(1.0864)
(2.0847)
(11.9074)
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
106
Table 49: Bispectrum rMSE(1) thresheld results.
Bispectrum rMSE(1)
Reference Image
Thresheld
Mean (Standard
(3,1,2,1,3)
(3,2,2,1,3)
(3,1,2,2,3)
(3,2,2,2,3)
0.0548
3.1942
0.0556
3.3464
(0.0258)
(0.8241)
(0.0218)
(0.8828)
0.9299
0.0787
0.9290
1.3055
(0.2231)
(0.0391)
(0.2228)
(0.3195)
0.0736
3.7184
0.0498
3.8852
(0.0289)
(0.9610)
(0.0235)
(1.0244)
0.8156
1.0354
0.8133
0.1183
(0.1952)
(0.2568)
(0.1945)
(0.0538)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
107
b) Bispectrum SNReff(2) Thresheld Results
Table 50: Bispectrum mean SNReff(2) thresheld results.
Bispectrum SNReff(2)
Mismatch
Reference Image
Difference
Thresheld
(dB)
(5,1,2,1,3) (5,2,2,1,3) (5,1,2,2,3) (5,2,2,2,3)
Min (Max)
0.4744
(5,1,2,1,3)
6.1997
-5.3116
5.7253
-7.0118
(13.2115)
7.5066
(5,2,2,1,3)
-0.8061
6.8245
-0.6821
-3.5166
Measured
(10.3411)
Image
0.9883
(5,1,2,2,3)
5.0730
-5.8066
6.0613
-7.5114
(13.5727)
6.0475
(5,2,2,2,3)
-0.3900
-1.4011
-0.2750
5.7725
(7.1735)
Overall: Min
Mismatch
Min
1.1267
8.2256
0.3360
9.2891
(Max)
Difference
(Max)
(7.0058)
(12.6311)
(6.7434)
(13.2838)
0.3360
(13.5727)
108
Table 51: Complete Bispectrum SNReff(2) standard deviation range thresheld
results.
Bispectrum SNReff(2)
Reference Image
Thresheld (dB)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
3.9643
-6.6922
3.8768
-8.4471
(11.0567)
(-3.2755)
(9.0093)
(-4.8531)
-2.1044
4.3999
-1.9742
-4.9120
Measured
(1.0543)
(12.8053)
(1.1655)
(-1.4480)
Image
3.0730
-7.2103
3.9676
-8.9638
(8.8916)
(-3.7194)
(10.2574)
(-5.3134)
-1.6738
-2.7238
-1.5577
3.2273
(1.4405)
(0.5103)
(1.5533)
(12.6944)
-σ(+σ)
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
109
Table 52: Bispectrum rMSE(2) thresheld results.
Bispectrum rMSE(2)
Reference Image
Thresheld
Mean (Standard
(5,1,2,1,3)
(5,2,2,1,3)
(5,1,2,2,3)
(5,2,2,2,3)
0.2399
3.3975
0.2676
5.0255
(0.1615)
(1.2715)
(0.1420)
(1.9684)
1.2040
0.2078
1.1701
2.2473
(0.4195)
(0.1553)
(0.4054)
(0.8516)
0.3110
3.8076
0.2477
5.6382
(0.1819)
(1.4529)
(0.1534)
(2.2392)
1.0940
1.3807
1.0654
0.2647
(0.3762)
(0.4916)
(0.3661)
(0.2109)
Deviation)
(5,1,2,1,3)
(5,2,2,1,3)
Measured
Image
(5,1,2,2,3)
(5,2,2,2,3)
110
H. Rigid Spherical Scatterer
Figure 17: Rigid spherical shell diagram.
In order to discuss the elastic scatter from a spherical target, it is helpful to begin
with a rigid scatterer in terms of surface acceleration. The resultant pressure, recorded by
the receiver is a function of both incident pressure, P0 and the scattered pressure, Ps .
The subscript  denotes the consideration of a target with infinite impedance making it a
completely rigid target.
Consider a sphere of radius a and a source at such a distance away so that plane
wave approximation holds. The surface acceleration of the sphere is expressed as a
function of both polar angle  , azimuthal angle  , and in a double series in Legendre
functions [2]:

n
 ( ,  )   Wnm Pnm cos( ) cos(m ).
n 0 m0
111
(10.1)
,
Where Wmn are the coefficients, Pnm are the associated Legendre functions. The Legendre
polynomials can be determined by setting m  0 .
Pn0 ( )  Pn ( ),
  cos( ).
(10.2)
.
For the cases when m  1 , the Legendre functions become:
m
Pnm ( )  (1   2 ) 2
d m Pn ( )
.
d m
(10.3)
However, when the lane wave approximation is utilized, only the Legendre polynomials
remain m  0 . [2]
Recall the conditions of the considered sphere; it has a radius a and is insonified
by a plane wave approaching from the    direction. This approaching plane wave can
be described in spherical coordinates as:

P0 ( RT , )  P0eikRT cos  P0  (2n  1)i n Pn (cos  ) J n (kR).
n 0
(10.4)
,
Where Pi is the pressure amplitude of the approaching wave, RT is an arbitrary point
radially away from the center of the sphere, k is the wave number, and J n is a Bessels
function of the first kind.
The corresponding rigid surface acceleration distribution
becomes:
s ( ) 
kPi

 (2n  1)i
n 0
112
n
Pn (cos  ) jn (kRT ).
(10.5)
.
The rigid scattered pressure becomes:
ps ( RT , )   Pi  (2n  1)i n Pn (cos  )
n 0
jn (kRT )
hn (kRT ).
hn (ka)
(10.6)
Where hn denotes the spherical Hankle function of the first kind. However, in the far
field, where kRT  n2  1, the scattered pressure becomes
ps ( RT , ) 
ikRT
iPe
i
kRT
jn (kRT )
 (2n  1)i P (cos ) h (ka ) .
n
n
n 0
n
(10.7)
T
In the backscatter case; that is to say when    and Pn  (1)n , the scattered pressure
can be written as
ps ( RT ,  ) 
ikRT
iPe
i
kRT
 (2n  1)(1)
n 0
n
jn (kRT )
.
hn (ka)
(10.8)
Finally, to describe the surface pressure at any angle  , we set RT  a and obtain

j (kR )h (ka) 
p(a, )  ps  P0  P0  (2n  1)i n Pn (cos  )  jn (ka)  n T n
, (10.9)
hn (ka)
n 0


which can be reduced so that the numerator in the latter term takes the form of the
Wronskian;
iP0
p ( a,  ) 
(ka) 2
(2n  1)i n Pn (cos  )
 Pi 1  (i23 )ka cos  .

hn (ka)
n 0
for (ka) 2  1
113
(10.10)
The backscatter response for ka ~ 1.5 , is denoted as the resonance region and leads to
creeping waves, which introduces the elastic acoustic response of the target.
114
I. Elastic Spherical Scatterer (Using Membrane Approximation)
It helps to discuss the elastic response of a spherical target in terms of surface
velocity and acceleration since these parameters are in direct relation with the resonant
scattering of the target. Consider a spherical shell with a thin wall and an evacuated
center. This target is able to react as a rigid target or react elastically. It is assumed that
the shell wall is thin and the frequency is low so that flexural stresses can be ignored.
The shell’s response to a surface excitation is denoted as
p(a, )   pn Pn (cos  ),
(10.11)
n 0
and when expressed in terms of modal velocity amplitudes as well as shell impedance Zn
and radiation impedance zn ; the surface velocity becomes [2]
 ( )  
n 0
pn Pn (cos  )
Z n  zn ,
(10.12)
zn  rn  i mn
rn   c

2
 for ka  n  1,
mn   a(ka) 
.
(10.13)
2
zn is a function of both a resistance component and an inertial component; where rn is the
resistance and mn is the modal accession to inertia per unit area. It is noted that,
Pn 
i n 1 (2n  1) P0
,
(ka) 2 hn (ka)
and we obtain the shell’s rigid scattered pressure response
115
(10.14)
 ( )  
 n 1
P0
i (2n  1) Pn (cos  )

.

2
(ka)
( Z n  zn )hn (ka)
n 0
(10.15)
The far field radiated pressure is formed when kRT  n2  1,
i  cP0eikR
pr ( RT , ) 
kRT

(2n  1)i n Pn (cos  )
 (Z
n 0
 zn )  kahn (ka)
2
n
.
(10.16)
The total scattered pressure is formed when combining equation (10.16) and the rigid
scatter repose given in (10.10)
pse ( RT , ) 

i  cP0eikR  (2n  1)i n Pn (cos  ) 
c
jn (ka) 


.
2
kRT n 0
hn (ka)
(ka) hn (ka)( Z n  zn )  (10.17)

for kRT  n 2  1
It should be noted that, as the frequency sweeps through resonance there is an abrupt
change in sign in the radiated pressure, Pr if pr ~ ps . It is observed that the pressure
radiated by the resonant mode takes the form
iP0 (2n  1) Pn (cos  )ei ( kRT  2 n )
prn ( RT ,  ) 
kRT
 y (ka) 
 n  tan  n
 = arg{hn (ka)}.
 jn (ka) 
(10.18)
1
Where ka  (n j )
cp
c
( j)
; s  n ,  n is the natural frequency, n is the radiation loss
factor associated with the radiation of acoustic power by the nth structural mode,  s is the
structural loss factor, c p is the phase velocity, and yn is a spherical Bessels function of the
116
second kind. This derivation is useful for observing the scattered pressure’s dependence
on the mode in which the target is vibrating.
Until now the assumption that the radiated impedance is much greater than the
structural impedance; that is to say that the shell in question is considered to be lossless.
In the case in which this is not so, the impedance term, (Zn  zn ) can be replaced by a
structural resistance term rs . With this considered, the radiated pressure by the resonant
mode is defined as
prn 
iP0 (2n  1) Pn (cos  )  ceikRT
kRT (kahn (ka)) 2 rs
iPi (2n  1) Pn (cos  )rn ei ( kRT  2 n )

.
kRT rs
Where rs  rn , ka  (n j )
cp
c
(10.19)
, rn is the modal resistance and rs is the specific acoustic
resistance associated with ms and the structural resistance of a shell associated with the
structural loss factor,  s . ms is the accession to inertia per unit area on a plane boundary,
for a standing wave field of infinite extent, of structural wave number, k s . Again, the
modal resistance is given by
rn   c(ka hn (ka) ) 2

 c(ka)2 n  2
(n  1)(1 3...(2n  1))
  c, ka  n 2  1
117
2
, (ka) 2  2(2n  1)
(10.20)
J. Higher-Order Spectral Analysis
Higher-order spectra analysis is an up and coming technique for signal
processing. Statistics is a very common way to extract features from non-image based
sonar analysis. Common low-order statistical analysis parameters include the mean, autocorrelation, cross correlation, power spectral density, variance, and various other lower
ordered moment quantities. Higher-order spectra are exactly as the name suggests.
Higher ordered moments and cumulants are used in order to extract information from a
signal that ordinarily goes unnoticed when using lower ordered statistics.
The
information that can be extracted includes: phase information, deviations from
Gaussianity, and the presence of nonlinearities in the signal. [18]
Consider a set of n real, random variables {x1, x 2,..., x n }. Their joint moments of
order rk1 k2 ...kn are defined as


 r (1 , 2 ,..., n )
Mom[ x , x ,..., x ] E{x , x ,..., x }  (1)
1k1 2k2 ...nkn   ...
k1
1
k2
2
kn
n
k1
1
k2
2
kn
n
n
1
2
(10.21)
n
where E denotes the mathematical expectation and
(1 , 2 ,..., n )  E{exp(i(1 x1  2 x2  ...  n xn ))}
(10.22)
It follows that, given a set of two random variables [x1, x2 ] ,
Mom[ x1 , x2 ]  E{x1 , x2 }
(10.23)
Considering the same random variables in (10.23), the joint cumulants of order r are
defined as follows:
118
Cum[ x1k1 , x2k2 ,..., xnkn ] (1) r
 r  (1 , 2 ,..., n )
1k1 2k2 ...nkn  
1
 (1 , 2 ,..., n )
2 ...n
(10.24)
ln  (1 , 2 ,..., n ) 
It follows that the moments from order one (m1) to three (m3) of random variables can be
defined as:
m1  Mom[ x1 ]  E{x1}
m2  Mom[ x1 , x1 ]  E{x12 }
(10.25)
m3  Mom[ x1 , x1 , x1 ]  E{x13}
These values can be related to their corresponding ordered cumulants;
c1  Cum[ x1 ]  m1
c2  Cum[ x1 , x1 ]  m2  m12
(10.26)
c3  Cum[ x1 , x1 , x1 ]  m3  3m2 m1  2m
2
1
However, if the process is of zero mean, the various ordered cumulants reduce to
c2  m2
(10.27)
c3  m3
Here, the main focus of the use of higher-order statistics is the bispectrum.
The
bispectrum is defined as the Fourier transform of the third-order moment of the sequence
x(k), c3x , and can be written discretely as
C3x (1 , 2 ) 


   c ( ,

1  2 
x
3
1
2
for 1 , 2   ; 1  2  
119
)ei (11 2 2 )
(10.28)
It is important to note that the values given by the bispectrum are rotation, scale,
amplification and translation invariant.
The other parameter that is noteworthy is the skewness (  3x ); expressed in terms of
cumulant spectra.
These parameters are calculated by taking the inverse Fourier
transform of the third order cumulant, respectively


1
C (1 ,..., 2 ) 
...  cnx (1 ,...n1 )ei (11 ...n1 n1 ) d1...dn 1 (10.29)
n 1 
(2 )  
x
n
By setting  1  0 and choosing n  3 we obtain the equation to calculate the skewness
C (0, 0) 
x
3
 
1
2
2
  c ( ,  )d d
x
3
 
120
1
2
1
2
(10.30)
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125