Analytic Model for Matched-filtered Scattered
Intensity of Volume Inhomogeneities in an Ocean
Waveguide Calibrated to Measured Seabed
Reverberation
MA
SSACHUSETTS INSTITUTE
OF TECHNOLOGY
by
JUL 2 9 2011
Anamaria Ignisca
LIBRARIES
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of ARCHVES
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2011
@ Massachusetts Institute of Technology 2011. All rights reserved.
.........................
Author ....................................
Department of Mechanical Engineering
May 20, 2011
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Certified by............
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Nicholas C. Makris
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by
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David E. Hardt
Graduate Officer, Department of Mechanical Engineering
2
Analytic Model for Matched-filtered Scattered Intensity of
Volume Inhomogeneities in an Ocean Waveguide Calibrated
to Measured Seabed Reverberation
by
Anamaria Ignisca
Submitted to the Department of Mechanical Engineering
on May 20, 2011, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
In this thesis, we derive full theoretical expressions for the moments of the matched
filtered scattered field due to volume inhomogeneities in an ocean waveguide and provide a computationally efficient time harmonic approximation to the matched filtered
model. Following the approach developed by Galinde et al 16], the expressions are derived from first principles, by applying Green's theorem and the Born approximation.
The scattered field and the total moment expressions are in terms of the fractional
changes in the bottom compressibility and density, as well as the waveguide Green
function and its gradients. The volume inhomogeneities are assumed to be statistically stationary, and assumed to be correlated in all three directions following a delta
correlation function. Sound propagation in the ocean is modeled using the parabolic
equation model and actual measurements of bathymetry and sound speed at the experimental locations. Monte Carlo simulations are used to account for the sound
speed variability in the ocean waveguide due to internal waves or other sources of
acoustic field randomization. The computationally efficient time-harmonic model is
shown to provide a good approximation to the full broadband matched filtered model
for a standard Pekeris waveguide. The time-harmonic model is then calibrated for
ocean bottom reverberation at several frequencies in the 415-1325 Hz band, with data
collected during the 2003 and 2006 ONR Geoclutter Experiments on the New Jersey
continental shelf and on the northern flank of Georges Bank in the Gulf of Maine,
respectively. The statistics for the inverted bottom parameters are summarized for
all frequencies and experimental locations considered. The acoustically determined
bottom parameters are shown to vary with approximately the wavelength cubed,
suggesting that, by different frequencies selecting the scale of the acoustic inhomogeneities, the acoustic effects dominate over the geophysical effects.
Thesis Supervisor: Nicholas C. Makris
Title: Professor of Mechanical Engineering
4
Acknowledgments
I would like to thank my thesis advisor, Professor Nicholas C. Makris, for the support
and guidance he offered on preparing this thesis and throughout my academic career at
MIT. I am grateful for the lessons Professor Makris taught me and my fellow students
not only in class, but also in the lab, through encouraging us to make progress one
step at a time and to learn through discovery.
I would also like to thank Dr. Mark Andrews, Prof. Purnima Ratilal, and Hadi
Tavakoli Nia for laying the ground work for the material in this thesis, and for the
guidance and direction they offered at the onset of this endeavor.
To Geoff Fox, Leslie Regan and Joan Kravit I thank for their help with all the
logistics related to preparing this thesis and completing my degree at MIT.
My colleagues and friends at MIT have also played a crucial role in the development of this work. Srinivasan Jagannathan, Ankita Deepak Jain, Dong Hoon Yi,
Wenjun Zhang, Kalina Gospodinova, Audrey Maertens and Derya Akkaynak Yellin
have always been there when I had questions related to acoustics or needed moral
support.
Lastly, I would like to thank my parents, brother and husband for their love,
support, and understanding.
6
Contents
1 Introduction
21
2 Analytic Formulation
23
.....................
2.1
Matched filtered scattered field ......
2.2
Full analytical expressions for the total moments of the matched filtered
scattered field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
26
Time harmonic approximation to the matched filtered scattered intensity 32
3 Model implementation for a Pekeris waveguide
35
3.1
Approximations to the matched filtered scattered intensity . . . . . .
36
3.2
Frequency dependence of simulated reverberation
. . . . . . . . . . .
44
4 Model calibration to bottom reverberation data
5
24
53
4.1
OAWRS bottom reverberation data . . . . . . . . . . . . . . . . . . .
53
4.2
Parameter inversion methodology . . . . . . . . . . . . . . . . . . . .
55
4.3
Model/data comparison using beam time data . . . . . . . . . . . . .
57
4.4
Model/data comparison using polar plots . . . . . . . . . . . . . . . .
62
4.5
Statistics and frequency dependence of inverted parameter . . . . . .
64
Conclusion
69
A Matched filtered total moment expressions based on conditional moments
B Second term of the time harmonic total variance
71
79
C Examples of calibration results for each experiment and frequency
C.1
85
New Jersey, 415 Hz . . . . . . . . . . . . . . . . . . .
86
C.2 New Jersey, 925 Hz . . . . . . . . . . . . . . . . . . .
91
C.3 New Jersey, 1325 Hz
. . . . . . . . . . . . . . . . . .
96
C.4 Georges Bank, 415 Hz
. . . . . . . . . . . . . . . . .
101
C.5 Georges Bank, 950 Hz
. . . . . . . . . . . . . . . . .
106
C.6 Georges Bank, 1125 Hz . . . . . .
111
List of Figures
3-1
Pekeris waveguide with sand bottom, where c,,,, p, and a,, are the
sound speed, density and attenuation of the water column, and cb, pb
and
ab
are those of the sea-bottom. The water column sound speed
profiles used in simulations are actual sound speed profiles measured
on the New Jersey continental shelf [16].
. . . . . . . . . . . . . . . .
36
3-2
The sound speed profiles measured on the New Jersey continental shelf. 37
3-3
Total moments of the full 390-440 Hz matched filtered scattered field
in a Pekeris sand waveguide with varying sound speed profiles, for 10
realizations of the ocean environment. The level of the mean squared
term is dependent on the number of realizations and on the range
increment dr, while the level of the variance term is stable. . . . . . .
3-4
38
Total moments of the full 390-440 Hz matched filtered scattered field
in a Pekeris sand waveguide with varying sound speed profiles, for a
single realization of the ocean environment. For the case of a single
realization, the total moments are equal to the moments conditional
on a set of deterministic Green functions. . . . . . . . . . . . . . . . .
3-5
Total 390-440 Hz matched filtered variance and its terms in a Pekeris
sand waveguide, for 10 realizations of the ocean environment . . . . .
3-6
39
40
Total 390-440 Hz matched filtered variance and its terms in a Pekeris
sand waveguide, for a single realization of the ocean environment. For
a single realization, the last two terms cancel each other and the total
variance is equal to the conditional variance .
. . . . . . . . . . . . .
41
3-7
Full 390-440 Hz matched filtered depth-integrated monostatic bottom
reverberation compared to time harmonic reverberation at center fre42
quency 415 Hz, for 10 realizations of the ocean environment. .....
3-8
Full 390-440 Hz matched filtered depth-integrated monostatic bottom
reverberation compared to time harmonic reverberation at center frequency 415 Hz, for a single realization of the ocean environment. . . .
3-9
42
Range-averaged full 390-440 Hz matched filter depth-integrated monostatic bottom reverberation compared to range-averaged time harmonic
reverberation at center frequency 415 Hz, for 10 realizations of the
ocean environment (range averaging over 500 m).
. . . . . . . . . . .
43
3-10 Range-averaged full 390-440 Hz matched filter depth-integrated monostatic bottom reverberation compared to range-averaged time harmonic
reverberation at center frequency 415 Hz, for a single realization of the
ocean environment (range averaging over 500 m).
. . . . . . . . . . .
43
3-11 Range-averaged full 390-440 Hz matched filter bottom reverberation
compared to its three terms: monopole, dipole and crossterm (range
averaging over 500 m). . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3-12 Range-averaged time harmonic bottom reverberation at center frequency 415 Hz compared to its three terms: monopole, dipole and
crossterm (range averaging over 500 m).
. . . . . . . . . . . . . . . .
45
3-13 Range averaged time-harmonic simulated monostatic reverberation level
in a sand Pekeris waveguide with isovelocity sound speed profile, at
16km from source/receiver (range averaging over 2000m). . . . . . . .
46
3-14 Range-averaged time-harmonic simulated monostatic reverberation level
in a sand Pekeris waveguide with isovelocity sound speed profile, at 415,
735, 950 and 1125 Hz, respectively (range averaging over 2000 m).
.
.
47
3-15 Range-averaged time-harmonic simulated monostatic reverberation level
in a sand Pekeris waveguide with isovelocity sound speed profile, for all
frequencies, at 4, 10 and 16 km in range, respectively (range averaging
over 2000 m ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3-16 Frequency dependence of time-harmonic simulated monostatic reverberation level at 16 km in range for a sand Pekeris waveguide with
isovelocity sound speed profile (range averaging over 2000 m), for bottom parameters V, Var(IT(rt)), Var(Fd(rt)) and Covar(F,,(rt)) constant for all frequencies.
. . . . . . . . . . . . . . . . . . . . . . . . .
49
3-17 Volume integral from Equation 3.3 plotted versus frequency for several
locations in range . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-18 Frequency dependence of volume integral from Equation 3.3. .....
49
50
3-19 Value of the integral in Equation 3.3 evaluated at several depths: the
integration is performed over a bottom layer that is 0.1 m thick, as
dz = 0.1 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3-20 Value of the integral in Equation 3.3 evaluated from the 100 m water/air interface down to several depths: the integration is performed
over all 0.1 m thick bottom layers located between 100 m and the
depths indicated in the legend . . . . . . . . . . . . . . . . . . . . . .
4-1
51
OAWRS image charting returns (dB re 1 pPa) received on the Northern
Flank of Georges Bank at 18:31 EDT, on 3 October 2006. The white
diamond indicates the source and the black line is the receiver track.
The white curves are bathymetric contours for the region. Clutter is
distinguished from seafloor returns by inspecting both the bathymetry
of the region and the stationarity of the returns over multiple pings.
Ship beams are also identified as radiating outward from the receiver,
symmetrically about the endfire of the receiver array. . . . . . . . . .
54
4-2 Measured normalized pressure level as beam time series for the New
Jersey continental shelf (left) and Georges Bank in the Gulf of Maine
(right). The black stars indicate the beams chosen for analysis. The
source is broadband for 415 Hz center frequency and 50 Hz bandwidth
and the source level is normalized to 0 dB re 1 pPa at 1m. . . . . . .
57
4-3
(Left) Bathymetric map of the New Jersey continental shelf indicating
the source and receiver positions at 10:31 EDT, on 13 May 2003. The
grey diamond indicates the source, located at 39.2312N, 72.8818W and
operating at 390-440 Hz. The black diamond indicates the receiver,
located at 39.2465N, 72.8626W, with heading 346*E. The black and
magenta stars indicate the beams chosen for analysis, corresponding to
the original and mirror beams, respectively. (Right) Bathymetric map
of the northern flank of Georges Bank in the Gulf of Maine indicating
source and receiver positions at 11:58 EDT, on 26 September 2006. The
grey diamond indicates the source, located at 41.8901N, 68.2134W and
operating at 390-440 Hz. The black diamond indicates the receiver,
located at 41.8212N, 68.3368W, with heading 137*E. The black and
magenta stars indicate the beams chosen for analysis, corresponding
to the original and mirror beams, respectively. . . . . . . . . . . . . .
58
4-4 Parameter estimates 0 (log scale) estimated from the data pings shown
in Figure 4-2 for the New Jersey (left) and Georges Bank (right) experiments, at 415 Hz. Each 0 corresponds to one beam and was estimated
using Equation 4.6, where the summation is performed over all n resolution cells Vs, along the beam. Typically, each beam extends for
25 to 35 km in range, which leads to values of n from 1667 to 2333,
respectively, given the sonar range resolution of 15 m. The mean of
the parameter estimates (0) is indicated by the red horizontal line and
is obtained using Equation 4.7, with p
=
1 and m representing the
number of beams in each ping. For this figure, m equals 31 and 36 for
the New Jersey and Georges Bank experiments, respectively. . . . . .
60
4-5 Model/data comparison along one beam for pings received at center frequency 415 Hz during the New Jersey (left) and Georges Bank (right)
experiments, respectively. The source level is normalized to 0 dB re 1
pPa at 1 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4-6
Model/data comparison along one beam for pings received at center frequency 415 Hz during the New Jersey (left) and Georges Bank (right)
experiments, respectively (range averaging over 2000 m). The source
level is normalized to 0 dB re 1 pPa at 1 m . . . . . . . . . . . . . . .
4-7
62
Polar plots of measured reverberation level (top), simulated reverberation level (middle) and their difference (bottom) for New Jersey (left)
and Georges Bank (right). The source level is normalized to 0 dB re 1
[tPaatlm.......
4-8
.................................
63
Histogram of the data-model differences as show in Figure 4-7 (bottom)
for New Jersey (left) and Georges Bank (right). For the New Jersey
errors (left), the mean is -1.1467 dB and the standard deviation 6.5095
dB. For the Georges Bank errors (right), the mean is -0.3526 dB and
the standard deviation 5.8003 dB. . . . . . . . . . . . . . . . . . . . .
4-9
64
Histograms of parameter values (log scale), for the New Jersey (left)
and Georges Bank (right) data. . . . . . . . . . . . . . . . . . . . . .
66
4-10 Mean (dots) and standard deviations (error bars) for the parameter
estimate 0, as given in Table 4.1, for the New Jersey and Georges
Bank data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4-11 Mean parameter estimates 0 (circles), as given in Table 4.1, for the
New Jersey and Georges Bank data, along with least squares fitted
lines for each experiment.
. . . . . . . . . . . . . . . . . . . . . . . .
B-1 Real part (left) and source spectrum
Q(f) 2
67
for 1 second long LFM
(top), CW (middle) and Gaussian (bottom) waveforms. . . . . . . . .
80
B-2 Simulated reverberation level using the first term of the matched filtered total variance for 10 Monte Carlo simulations, after normalization
by rB, for LFM, CW and Gaussian 1 second long broadband pulses
centered at 415 Hz and with 50 Hz bandwidth. . . . . . . . . . . . . .
81
B-3 Total matched filtered variance and its three terms, for 1 second LFM,
CW and Gaussian broadband pulses centered at 415 Hz with 50 Hz
bandwidth.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
B-4 Second term of total variance (Equation 2.18) for 1 Monte Carlo simulation, for CW pulses of 1 to 7 second duration, centered at 415 Hz
and with 50 Hz bandwidth.
. . . . . . . . . . . . . . . . . . . . . . .
83
C-1 Parameters obtained after fitting the model to the data for each beam,
in log scale. The red horizontal line indicates the mean of the parameter values (log scale). Only beams corresponding to relatively flat or
downward sloping bathymetry were included in the analysis. Endfire
beams as well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen for
calibration, as the model fails to predict well the data for small ranges.
C-2 list entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
87
C-3 Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original
beam (middle), range from receiver along the mirror beam (bottom),
for beams 30, 40 and 50, indicated by magenta stars in Figure C-2 (top
right). ........
...................................
88
C-4 For the same beams as in the previous figure C-3: comparison of measured and simulated reverberation time series after range/time averaging (top); comparison of the monopole term before adding the reverberation intensities for the original and mirror beams (bottom).
.
. .
89
C-5 Histogram of data-model differences for Figure C-6. . . . . . . . . . .
89
C-6 Beam time (left) and polar (right) plots of measured reverberation
level (top), simulated reverberation level (middle) and their difference
(bottom ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
C-7 Parameters obtained after fitting the model to the data for each beam,
in log scale. The red horizontal line indicates the mean of the parameter values (log scale). Only beams corresponding to relatively flat or
downward sloping bathymetry were included in the analysis. Endfire
beams as well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen for
calibration, as the model fails to predict well the data for small ranges.
C-8 list entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
92
C-9 Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original
beam (middle), range from receiver along the mirror beam (bottom),
for beams 30, 40 and 50, indicated by magenta stars in Figure C-8 (top
right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
C-10 For the same beams as in the previous figure C-9: comparison of measured and simulated reverberation time series after range/time averaging (top); comparison of the monopole term before adding the reverberation intensities for the original and mirror beams (bottom).
. .
C-11 Histogram of data-model differences for Figure C-12. .........
94
94
C-12 Beam time (left) and polar (right) plots of measured reverberation
level (top), simulated reverberation level (middle) and their difference
(bottom ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
C-13 Parameters obtained after fitting the model to the data for each beam,
in log scale. The red horizontal line indicates the mean of the parameter values (log scale). Only beams corresponding to relatively flat or
downward sloping bathymetry were included in the analysis. Endfire
beams as well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen for
calibration, as the model fails to predict well the data for small ranges.
C-14 list entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
97
C-15 Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original
beam (middle), range from receiver along the mirror beam (bottom),
for beams 30, 40 and 50, indicated by magenta stars in Figure C-14
(top right).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
C-16 For the same beams as in the previous figure C-15: comparison of
measured and simulated reverberation time series after range/time averaging (top); comparison of the monopole term before adding the
reverberation intensities for the original and mirror beams (bottom). .
99
. . . . . . . . .
99
C-17 Histogram of data-model differences for Figure C-18.
C-18 Beam time (left) and polar (right) plots of measured reverberation
level (top), simulated reverberation level (middle) and their difference
(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
C-19 Parameters obtained after fitting the model to the data for each beam,
in log scale. The red horizontal line indicates the mean of the parameter values (log scale). Only beams corresponding to relatively flat or
downward sloping bathymetry were included in the analysis. Endfire
beams as well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen for
calibration, as the model fails to predict well the data for small ranges. 101
C-20 list entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C-21 Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original
beam (middle), range from receiver along the mirror beam (bottom),
for beams 25, 37 and 42 indicated by magenta stars in Figure C-20
(top right).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C-22 For the same beams as in the previous figure C-21: comparison of
measured and simulated reverberation time series after range/time averaging (top); comparison of the monopole term before adding the
reverberation intensities for the original and mirror beams (bottom). . 104
C-23 Histogram of data-model differences for Figure C-24.
. . . . . . . . .1 104
C-24 Beam time (left) and polar (right) plots of measured reverberation
level (top), simulated reverberation level (middle) and their difference
(bottom).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
C-25 Parameters obtained after fitting the model to the data for each beam,
in log scale. The red horizontal line indicates the mean of the parameter values (log scale). Only beams corresponding to relatively flat or
downward sloping bathymetry were included in the analysis. Endfire
beams as well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen for
calibration, as the model fails to predict well the data for small ranges. 106
C-26 list entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
C-27 Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original
beam (middle), range from receiver along the mirror beam (bottom),
for beams 35, 45 and 50, indicated by magenta stars in Figure C-26
(top right).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
C-28 For the same beams as in the previous figure C-27: comparison of
measured and simulated reverberation time series after range/time averaging (top); comparison of the monopole term before adding the
reverberation intensities for the original and mirror beams (bottom). . 109
C-29 Beam time (left) and polar (right) plots of measured reverberation
level (top), simulated reverberation level (middle) and their difference
(bottom ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C-30 Parameters obtained after fitting the model to the data for each beam,
in log scale. The red horizontal line indicates the mean of the parameter values (log scale). Only beams corresponding to relatively flat or
downward sloping bathymetry were included in the analysis. Endfire
beams as well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen for
calibration, as the model fails to predict well the data for small ranges. 111
C-31 list entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
C-32 Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original
beam (middle), range from receiver along the mirror beam (bottom),
for beams 35 and 45, indicated by magenta stars in Figure C-31 (top
right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
C-33 For the same beams as in the previous figure C-32: comparison of
measured and simulated reverberation time series after range/time averaging (top); comparison of the monopole term before adding the
reverberation intensities for the original and mirror beams (bottom). .
114
C-34 Beam time (left) and polar (right) plots of measured reverberation
level (top), simulated reverberation level (middle) and their difference
(bottom).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
List of Tables
3.1
Statistical geologic properties of the New Jersey Strataform, as estimated by Galinde et al [6]. . . . . . . . . . . . . . . . . . . . . . . . .
4.1
37
Table showing the results of the parameter estimation. The first two
columns represent the corresponding center frequencies and experiment
locations, respectively. The number of pings, p, and total ping-beam
samples, mp, used in the analysis are given in the third and fourth
columns, respectively. The next six columns give the mean (level and
log scale), standard deviation (level and log scale), as well as the percentage of beam samples within one and two standard deviations, respectively. One 0 is estimated for each single beam in each single ping
using Equation 4.6, where the summation is performed over all n resolution cells Vs, along the particular beam. As mentioned in Figure 4-4,
we typically sum over 1667 to 2333 resolution footprints along each
beam. Then, the mean parameter estimate (6) is obtained by averaging the parameter estimates 0 over all mp ping-beam samples, as given
by Equation 4.7. (* the NJ and GB symbols stand for New Jersey and
Georges Bank, respectively)
. . . . . . . . . . . . . . . . . . . . . . .
65
20
Chapter 1
Introduction
The ability to accurately predict reverberation level in the ocean for different sediment
types and frequency ranges is critical for remote sensing and imaging systems. The
reverberation in shallow continental shelf environments for moderate sea surface wind
conditions, has been shown to be mainly due to scattering from the ocean seabed [22],
[19].
Several methods have been developed to model bottom scattering, including
interface roughness scattering models, based on the roughness at the water-bottom
interface [19], [21], [4], and volume scattering models, for scattering due to seabed
volume inhomogeneities [6], [19], [23], [11], [12], [20].
For areas of relatively flat
bathymetry and for the shallow grazing angles associated with low frequency longrange reverberation, it has been shown that volume scattering is the dominant factor
in total bottom reverberation [17], [20], [10].
Models for scattering from volume
inhomogeneities in the ocean seabed have been formulated in the literature, but have
generally not included the matched filter, which is applied in remote sensing systems
to "provide high-resolution imaging of the seafloor and submerged targets" [3]. The
matched filter "allows for scatterers extending over multiple range resolution cells of
the imaging system to be automatically localized in range, and does not require to
artificially break-up the scatterers to within each range resolution cell, as in the case
of time-harmonic or other broadband models" [1].
In this thesis, we derive full theoretical expressions for the moments of the matched
filtered scattered field due to volume inhomogeneities and provide a computationally
efficient time harmonic approximation to the matched filtered model. Following the
approach developed by Galinde et al [6], the expressions are derived from first principles, by applying Green's theorem and the Born approximation. The scattered field
and the total moment expressions are in terms of the fractional changes in the bottom
compressibility and density, as well as the waveguide Green function and its gradients.
The volume inhomogeneities are assumed to be statistically stationary, and assumed
to be correlated in all three directions following a delta correlation function. Sound
propagation in the ocean is modeled using the parabolic equation model RAM [5], and
actual measurements of bathymetry and sound speed at the experimental locations.
Monte Carlo simulations are used to account for the sound speed variability in the
ocean waveguide due to internal waves or other sources of acoustic field randomization. The computationally efficient time-harmonic model is shown to provide a good
approximation to the full broadband matched filtered model for a standard Pekeris
waveguide. The time-harmonic model is then calibrated for ocean bottom reverberation at several frequencies in the 415-1325 Hz band, with data collected during the
2003 and 2006 ONR Geoclutter Experiments on the New Jersey continental shelf and
on the northern flank of Georges Bank in the Gulf of Maine, respectively [16], [15].
The statistics for the inverted bottom parameters are summarized for all frequencies
and experimental locations considered.
This thesis is structured as follows: In Section 2, we present the analytic formulations for the moments of the matched filtered scattered field and a time harmonic
approximation. In Section 3, we implement the matched filtered and time harmonic
formulations to a standard Pekeris waveguide, using bottom parameters estimated by
Galinde et al [6] for the New Jersey continental shelf. Here, we show that the timeharmonic model provides a good approximation to the matched filtered model and
we investigate the frequency dependence of the time-harmonic model while keeping
the bottom parameters constant. In Section 4, we calibrate the computationally efficient time-harmonic model to ocean reverberation data and present statistics for the
inverted bottom parameters at several frequencies and at two different geographical
locations.
Chapter 2
Analytic Formulation
In this section, we derive analytic expressions for the total moments of the matched
filtered scattered field from random volume inhomogeneities extending over multiple
resolution cells in range. We start by deriving the scattered field in terms of the
single frequency
f.
We then apply the matched filter and use Fourier synthesis to
obtain the time dependent matched filtered scattered field.
Next, we derive full
formulations for the total scattered field moments, and indicate that the matched
filtered scattered intensity can be approximated by the incoherent term. We also
provide a computationally efficient time harmonic approximation to the matched
filtered scattered intensity.
We consider an ocean waveguide consisting of a water layer, located between an
air halfspace above, and a bottom halfspace below. We let (x, y, z) be the coordinates
of a cartesian coordinate system with origin at the air-water interface, and the z-axis
pointing down. We place the source at ro=(xo, yo, zo), the receiver at r=(x, y, z),
and the center of the inhomogeneity at rt=(xt, yt, Zt).
f
represents the frequency,
w = 27rf the angular frequency, c the sound speed, and k = w/c is the acoustic wave
number.
2.1
Matched filtered scattered field
To derive the matched filtered scattered field from random volume inhomogeneities,
we follow the approach developed by Morse and Ingard [18] and Galinde et al [6],
and start from the inhomogeneous Helmholtz equation for the single-frequency, timeindependent acoustic field in the presence of volume inhomogeneities:
V 2 4t(rt, f) + k2 5It(rt, f)
(2.1)
= -k 2 F(rt)4t(rt, f) - V - [Fd(rt)VDt(rt, f)].
Here, F,, is the fractional change in compressibility,
n,(rt) -
IF (rt)
_
,
(2.2)
and Fd is the fractional change in density,
]Fd(rt)
-
d(r) - d
d(rt)
(23)
where P. and d are the mean compressibility and density in the region, respectively
[18], [6]. The compressibility can be expressed in terms of the density and the sound
speed as , = 1/dc2 .
Applying Green's theorem [18] to Equation 2.1 above, we obtain the scattered field
4s(rsIr, ro, f) from inhomogeneities within Vs, a region which we choose to extend
along the entire positive z-axis in depth, over the sonar resolution angle in azimuth,
and over multiple sonar resolution cells in range. The region is centered at rs, and
the source and receiver are located at ro and r, respectively:
4
s(rslr, ro, f) =
J
vs [k2 ]p(rt),t(rt, f)G(rIrt, f)
(2.4)
+ Fd(rt)VDt(rt, f) - VG(rlrt, f)]dVt.
G(rlrt,
f) is the
Green function describing propagation from the location of the
inhomogeneity to the receiver, and 1t(rt,
f) is
the total field at the location of the
inhomogeneity. The total field can be expressed as the sum of the the incident and
scattered fields [18],
4t(rt, f) = eP(rtIro, f) + I1s(rt, f).
To set the source level to 0 dB re 1 pPa at 1m, we let 4 2 (rtlro,
f)
= (47r)G(rtero,
f).
For small fluctuations in density and compressibility, we can make the Born approximation, and approximate the total field at the inhomogeneity by the incident field
[18]. Then, we can express the scattered field at the receiver as
d s(rsIr, ro,
f) =(47r)
J
vs [k2 lT(r)G(rtjro, f)G(rlrt, f)
+ Pd(rt)VG(rtro,
(2.5)
f) - VG(rlrt, f)]dVt.
Next, we derive the matched filtered scattered field for a source that transmits a
broadband waveform q(t) with Fourier transform
center frequency
fc
Q(f)
and bandwidth B around the
by applying the matched filter to Equation 2.5. We express the
matched filter, "a normalized replica of the original transmitted waveform" [3], as
H(fItM)
Q*(f)ei 2 ftm
=
(2.6)
where tM represents the time delay of the matched filter. The matched filter reaches a
peak at t
=
tM. The source energy is Eo = f IQ(f)I 2df and equals 1 for a normalized
source. Then, using Fourier synthesis, we find the time-dependent matched filtered
scattered field to be
Ds(rslr,ro, t) = (47r)
f
/ff [k2]pK(r)G(rjro, f)G(rlrt,f)
vs
fc-B/2
+ F(rt)VG(rt~ro, f) -VG(rlrt, f)]
x
IQ(f)|2e-i 2 f(t-tM) dVt df.
(2.7)
2.2
Full analytical expressions for the total moments of the matched filtered scattered field
Here, we derive full analytical expressions for the total moments of the matched
filtered scattered field in terms of the statistical moments of fractional changes in
compressibility and density and in terms of average realizations of the Green functions
and gradients in a randomly fluctuating ocean waveguide. Implementation-friendly
expressions for the total moments of the scattered field based on conditional moments,
where we condition on a particular set of deterministic Green functions, are detailed
in Appendix A of this thesis.
We assume the bottom inhomogeneities to vary randomly in space, following a
stationary random process within the region Vs considered. Additionally, we assume
the fluctuations in bottom properties to be independent from the fluctuations in the
ocean waveguide. Thus, we can treat the random variables I, and Fd as independent
from the medium's Green function [6]. Then, the mean matched filtered scattered
field can be expressed as
(Cbs(rsIr, ro, t))
=
(47r)
fcfB/2
fc+B/2
fj
f))
[k2(IF.(r))(G(rtIro, f)G(rIrt,
+ (T'(rt))(VG(rtlro, f) - VG(rlrt, f))]
x
I MA
Qf)2 e-i27rt-tM) dVt df.
VIo
(2.8)
The second moment of the scattered field is
(1s(rsIr, ro, t) 2)
((4-r)
f
,:B2
fcB/ 2
[k 2]p"(rt)G(rtlro, f)G(rlrt, f)
s
+ ]7(r)VG(rtlrof) - VG(rlrt,f)] x
IQ(f)I e-i 2rf( t -tm)
2
dV df
12pp
(47r)jfc+B12
fc-B/2
(2.9)
ro,tM))
(,Ds(rsIr,ro, tm) *s(rslr,
(r'/)G*(r'l lro, f')G*(rlr', f')
[k'2
JJJVs
+ Fd(r')VG*(r'lro, f') - VG*(rlr', f')] x
I |Q(f')i2e-i27f'(t-tm) dV' df'),
which can be written as
(4,s(rsIr, ro, t)
(2.10)
2)
=(47r)2f fc+B/2
fcB/
c+B/2
f-/2
[k 2 k'/2 (.(rt)rr.(r'))(G(rt|ro,
1
f
Sf
f)G(rlrt, f)G*(r'|ro, f')G*(rlr',, f'))
+ (rd(rt)Fd(r))(VG(r|ro, f) - VG(rlrt, f) x VG*(r'|ro, f') -VG*(rlr', f'))
+ k 2 (F,(r)F(r'))(G(rtlro, f)G(rlrt, f)VG*(r'lro, f') - VG*(rlr', f))
± k' 2 (F,(r)Fd(rt))(G*(r'jro, f')G*(rjr', f')VG(rlro, f) - VG(rlrt, f))]
X1
IQ( f||1~)2
2w2(f -f')(t- t ) d V dVt' df df'
To model the statistics of the density and compressibility variations, we use a
delta correlation function and assume the parameters to be correlated in all three
dimensions. As we assume the parameters to follow a stationary random process, only
the second order statistics are needed. Moreover, as the density and compressibility
variations are assumed to be fully correlated [12], we may use only one coherence
volume, Vc, for both random variables. Following the approach developed by Galinde
et al [6], we let
(2.11)
(Fr.(rt)IF r.(r'))
=
V(rs, zt)[(I'(rt)) - |-
r') + (1](rt))(F-(r'+))
V(rs, zt)Var(I7.(rt)J(rt - r') + (F(rt))(FK(r')).
Similarly,
(2.12)
(Jd(rt)Fd(r')
=
V(rs, zt)Var(Fd(rt))6(rt
- r') + (Fd(rt))(Fd(r'))
(2.13)
(PK(rt)rd(r'))
V(rs,zt)Covar(F,(rt), Fd(rt))6(rt -
r',) +
(I'.(rt))(Fd(r'))
2.14)
(F((r'Fd(r)
= V(rs, zt)Covar(T,(rt), Fd(rt))6(rt
- r')
+ (F,(r')(Fd(rt)).
Then, Equation 2.10 becomes
(K<Ds(rslr, ro, t)12)
I fc+B/2
ffc+B/2 fff
-(4r)2
fcKB2
(2.15)
f
V
(rs, zt)
fc-B/2fc-/2Vs
x
[k2 k'2Var(ri)o(rt - r')(G(rro, f)G(rlrt, f)G*(r'ro, f')G*(rr', f))
f) - VG(rlrt, f) x VG*(rro, f') - VG*(rr, f'))
r',)(G(rtlro, f)G(rlrt, f)VG*(r'lro, f') - VG*(rlr', f'))
+ Var(Fd)&(rt - r')(VG(rt|ro,
+ k2 Covar('K, Fd)6(rt -
+ k'2 Covar(FK,
x
d) 6 (rt
-
rt)(G*(r' lro, f')G*(rlr', f')VG(rlro, f) - VG(rlrt, f))]
| Q(f }|21Qf 2e-is7ru-f'x,-tm) dV dVt' df df'+
ff
+fc+B/2 ffc+B/2
( fB/2
ff
Vs
Vs
Jf-B/2
x[k 2k'12 (r(rt))(rr.(r'))(G(rtlro,
f)G(rlrt, f)G*(r'l~ro, f')G*(rlr',,f'))
+ (Fd(rt))(Fd(r'))(VG(rtlro, f)
VG(rlrt, f) x VG*(r'lro, f') - VG*(r r',f'))
+ k2 (r.(rt))(Fd(r'))(G(rtlro, f)G(rlrt, f)VG*(r'ro, f) - VG*(rr', f'))
± k' 2 (F(r'))(rd(rt))(G*(r'jro, f')G*(rr', f')VG(rt|ro,
1
X
|IQ( f |2 Q{ gi 2e-i27r f-f')(t-t" ) dV dVt' df df'.
F0
f) - VG(rtrt, f))]
After canceling the delta function with one of the volume integrals in the first
term of the above equation, we have the full expression for the matched filtered total
second moment
(lkbs(rslr, ro, t)|2)
I fc+B/2
(2.16)
fc+B/2
(47r)2
f-B/2
f
/ gBg
Vc(rs, zt)
V
fc-B/2
x [k 2 k'2Var (r.)(G (r
Iro, f )G(r Irt, f )G* (rt Iro, f ') G*(r Irt, f'))
+ Var(Fd)(VG(rtlro, f) -VG(rjrt, f) x VG* (rt ro, f') -VG* (rirt, f'))
+ k2 Covar(IF, Fd)(G(rtIro, f)G(rlrt,
f)VG*(rtlro, f') - VG* (rirt, f'))
+ k'2 Covar(Fr, 1Fd)(G*(rtlro, f')G*(rlrt, f')VG(rtjro,
x
|IQ(f) 1 2jQ(fi)12e-i2(f-f')(t-tm)
+ +(47r)2fc+B2
IcB/
f
fc+B/2
fc-B/2 ]]Vs
1 1
f) - VG(rlrt, f))]
dV df df'+
Vs
x[k 2 k'2 (1F(rt))(Fn(r'))(G(rtiro,
f)G(rlrt, f)G*(r'|ro, f')G*(rlr',, f'))
+ (Fd(rt))(Fd(r'))(VG(r|ro,f) - VG(rlrt, f) x VG*(r'lro, f') - VG*(rlr', f'))
+ k2 (F.(rt))(Pd(r'))(G(rtIro, f)G(rIrt, f)VG*(r'Iro, f)-VG*(rIr', f'))
+ k' 2 (F.(r'))(Pd(rl)) (G*(r'ro, f')G*(rlr', f')VG(rro, f) - VG(rjrt, f))]
X
1o| Q(f}|21Q(fgi
2e--i27rtf-f'x(t-tm) dV dV' df df'.
The total variance can be expressed in terms of the total second moment and the
squared of the mean field as
(Pes(rs r, ro, t)|2 ) - I(Ps(rsIr, ro, t))|2 ,
Var (s(rsIr, ro, t))
(2.17)
which can be further expanded as
Var(4s(rsIr,ro, t))
(2.18)
fcB/
Sfc+B/2
S(47r)2 V(rs, z,)
fc-B/2
f-B/2
x [k 2k'2Var (:P,)(G (rt Iro,
VS
f ) G(r Irt, f )G* (r Iro, f')G*(r Irt, f'))
+ Var(Jd)(VG(rtlro, f) - VG(rlrt, f) x VG*(rtlro, f') - VG*(rlrt, f'))
+ k 2 Covar(7,
IF)(G(rtlro, f)G(rlrt, f)VG*(rtlro, f') - VG*(rlrt, f'))
+ k' 2Covar(]F, Fd)(G*(rtro, f')G*(rrt, f')VG(rtlro, f) - VG(rlrt,
x
f))]
|IQ(f)21Q(fI)j2e-i2(f-f')(t-tm) dV df df'+
fc-B/2Ic-B/2LV
+ (47r)2Jfc+B2
f+B/2
x[k2 k'2(rr.(rt))
(rr (r')) (G(rt Iro, f )G(r Irt, f )G* (r'tro, f')G*(r Ir', f'))
+ (Fd(rt))(Fd(r'))(VG(rtlro, f) - VG(rlrt, f) x VG*(r'lro, f') - VG*(rlr', f'))
+ k 2 (F,(rt))(Fd(r'))(G(rtlro, f)G(rlrt, f)VG*(r'Iro, f') - VG*(rlr', f'))
+ k'2 (F.(r'))(Fd(rt))(G*(r'jro, f')G*(rr', f')VG(rtlro, f) -VG(rlrt, f))]
x
| Q( f}|21Qf 2e-i27rf-f'x(t-tm) dV dVt' df df'-
fcB/2
-
fc+B/2
(47)2 Jfc-B/2 jfe-B/2
VsJ
Vs
xk
{2 k'2(Fr.(rt))(Fr.(r'))(G(rtlro,
f)G(rlrt, f))(G*(r'lro, f')G*(rlr',f'))
± (Fd(rt))(Td(r'))(VG(rtlro, f) -VG(rlrt, f))(VG*(r'jro, f') VG*(rlr', f'))
+ k2 (]F(rt))(Fd(r'))(G(rtro, f)G(rlrt, f))(VG*(r' Iro, f') VG*(rIr', f'))
+ k' 2 (FK(r'))(Fd(rt))(G*(r'ro, f')G*(rlr', f'))(VG(rtlro, f) -VG(rlrt, f))]
X
1
_|Q(f||21Q(
Eo
i)12e-i27r
f -fl't-t" ) dV dVt df df'.
While the above equation represents the full expression for the matched filtered
total variance, we show in Section 3 that the last two terms of the variance Equation
2.18 are negligible, so that the variance can be approximated by the first term. Since
the first two terms in the variance Equation 2.18 are the same as the terms of the
second moment Equation 2.16, both the second moment, or matched filtered scattered
intensity, and the variance, can then be approximated by
Var(4s(rslr,ro, t))
(47r)2
(2.19)
Sfc+B/2
~fc+B/2
fI
fc-B/2
fc-B/2
I
Vc(rs, z,)
Vs
x [k2 k'2Var(1K)(G(rlro, f)G(rlrt, f)G*(rtlro, f')G*(rlrt, f'))
+ Var(Fd)(VG(rtjro, f) - VG(rlrt, f) x VG*(rtlro, f') - VG*(rlrt, f'))
+ k2 Covar(r, rd)(G(rtlro, f)G(rlrt, f)VG*(rtlro,
± k' 2 Covar(P, Fd)(G*(rlro,
X
1 |Q( f }|21Q(
Eo
)12e-i27rf-f'xt-tm)
f') - VG*(rjrt, f'))
f')G*(rlrt,f)VG(rtlro, f) - VG(rlrt, f))]
dV df df'.
The equations presented above are expressed analytically in terms of the moments
of the density and compressibility variations, as well as in terms of averages of Green
function products over multiple realizations of the ocean environment. Implementation of the full second moment and variance formulations as expressed in Equations
2.16 and 2.18, respectively, requires, however, double integration over the volume Vs.
In Appendix A we develop a more efficient method for implementing the full formulations for the total variance and second moment, based on moments conditional on
a set of deterministic Green functions.
Time harmonic approximation to the matched
2.3
filtered scattered intensity
In this section, we present a computationally efficient approximation to the matched
filtered scattered intensity derived in Section 2.2. Starting with Equation 2.5 for the
time harmonic scattered field, where Vs now represents the resolution footprint of
the imaging system, we following a procedure similar to the one used in the previous
section, and find that the time harmonic total variance is
Var(4s(rsIr,ro,
= (47r)2
f))
(2.20)
JJvs V(rs, zt)
x [k'Var(FK)(IG(rtlro, f)12 |G(rlrt, f)12 )
+ Var(Fd)(|VG(rtlro,
2
f) -VG(rlrt, f)1
)
± k 2 Covar(T, Fd)(2R{G(rtIro, f)G(rlrt, f)VG*(rtlro, f) - VG*(rlrt,
+ (47r)2
f)})] dV
Js j[k4(](r))(rr(r'))(G(rt|ro, f)G(rlrt, f)G*(r'lro, f)G*(rlr', f))
+ (Fd(rt))(Fd(r'))(VG(rtIro, f) - VG(rlrt, f) x VG*(r' Iro, f) VG*(rIr',f))
Sk 2 (F.(rt)) (Fd(r')) (G(rtIro, f)G(rlrt, f)VG*(r'lro, f) VG*(rIr', f))
+ k2 (F.(r')) (1d(rt)) (G*(r'\ro, f)G*(rr'7, f)VG(rtro, f) -VG(rlrt, f))] dV dV'
-
(47c)2
vJ
vJ[k4 (P(rt))(L'(r'))(G(rtjro, f)G(rlrt, f))(G*(r'jro, f)G*(rlr', f))
+ (Fd(rt)) (Fd(r')) (VG(rtlro, f) -VG(rlrt, f))(VG*(r'ro, f) -VG*(rlr', f))
+ k2 (F.(r))(r(r'))(G(rtIro, f)G(rIrt, f))(VG*(r'lro, f)
VG*(rIr, f))
+ k2 (F.(r'))(Fd(rt))(G*(r'lro, f)G*(rIr', f))(VG(rtIro, f) - VG(rIrt, f))] dV dV'.
In Section 3.1 we show, using Monte Carlo simulations in a standard Pekeris
waveguide, that the matched filtered scattered intensity derived in Equation 2.19 can
be approximated by the first term of the total time harmonic variance, also given by
Galinde et al [6] as:
Var(<Ds(rslr,ro,
=
(47r)2
f))
(2.21)
vl Vc(rs, zt)
x [k4Var(I)(IG(rtlro, f)I 2 |G(rlrt, f)12)
± Var(rd)(|VG(rtro,
f) - VG(rlrt, f)12)
± k2 Covar(d,
a)(2R{G(rtjro, f)G(rirt, f)VG*(rtlro, f) - VG*(rlrt,
f)})] dVt.
34
Chapter 3
Model implementation for a
Pekeris waveguide
In this section, we implement the theoretical formulations developed in Section 2 and
Appendix A for the total moments of the scattered field to a standard sand Pekeris
waveguide, using Monte Carlo simulations of the ocean environment. We show that
the following three approximations can be made not only for multiple realizations,
but also for a single realization of the ocean waveguide: (1) the total matched filtered
second moment (Equation 2.16) is dominated by and can be approximated by the
total matched filtered variance (Equation 2.18); (2) the total matched filtered variance
(Equation 2.18) is dominated by and can be approximated by its first term (Equation
2.19), while the last two terms are negligible; (3) the first term of the total matched
filtered variance (Equation 2.19) is dominated by and can be approximated by the
first term of the total time harmonic variance (Equation 2.21). Lastly, we investigate
the frequency dependence of the time harmonic model for the scattered intensity in
an isovelocity sand Pekeris waveguide, while keeping the bottom parameters constant
for all frequencies.
3.1
Approximations to the matched filtered scattered intensity
Here, we implement the formulations for the total moments of the scattered field
to a Pekeris waveguide with sand bottom such as the one illustrated in Figure 3-1,
and demonstrate that the three approximations stated above hold for any number of
realizations of the ocean environment.
water
depth
p = 1000 kg/m 3
Sand bottom
cb = 1700 m/s
Pb = 1900 kg/m 3
b =0.8 dB/A
al,= 6 x 10~' dB/A
Figure 3-1: Pekeris waveguide with sand bottom, where C., pw and a are the sound
speed, density and attenuation of the water column, and cb, Pb and ab are those of the
sea-bottom. The water column sound speed profiles used in simulations are actual
sound speed profiles measured on the New Jersey continental shelf [16].
We allow the sound speed profiles to vary in range, in order to model the random
fluctuations in the ocean environment caused by internal waves. From a set of sound
speed profiles measured on the New Jersey continental shelf [16], we pick one sound
speed profile every 500 m in range [2] and use the parabolic equation model RAM
[5] to compute the Green functions for each randomly selected set of sound speed
profiles. The set of measured sound speed profiles used for random selection is shown
in Figure 3-2. The geometry of the problem is monostatic, with the source and
receiver collocated at the depth of 50 m. The source and receiver comprise each of
a single element. The source level is normalized to 0 dB re 1 LPa at 1 m and the
waveform transmitted is a 1 second long linear frequency modulated (LFM) pulse
centered at 415 Hz, with a 50 Hz bandwidth. The corresponding range resolution,
Ar, is then 15 m, as Ar = c/2B, where c, the reference sound speed, is 1500 m/s,
and B, the bandwidth, is 50 Hz. The statistics of the fractional changes in density
and compressibility used in the simulation are those estimated by Galinde et al [6]
for the New Jersey continental shelf, summarized in Table 3.1. The region over which
we integrate, Vs, corresponds to a seafloor patch that extends over 10m in depth
(more than sufficient for convergence) and 3* in azimuth. In range, the patch extends
from 100 m to 5500 m relative to the source/receiver location. The depth and range
increments used for computing the Green functions and gradients are dz = 0.4 m and
dr = 3 m, respectively. In azimuth, we assume the Green functions are constant over
the integration angle.
New Jersey water column sound speed profiles
102030E4050-
60708090-
1460
1470
1480
1490
1500
1510
sound speed (m/s)
Figure 3-2: The sound speed profiles measured on the New Jersey continental shelf.
Table 3.1: Statistical geologic properties of the New Jersey Strataform, as estimated
by Galinde et al [6].
Statistical parameters
Vc
(Fk)
(Fd)
(Fi)
(Fj)
Var(Fk)
Var(Fd)
Covar(J'k, Fd)
Average values over the region
0.0030 (M3 )
0.0000
-0.0063
0.0083
0.0066
0.0083
0.0065
-0.0065
First, we show that the matched filtered scattered intensity is dominated by the incoherent intensity, or the variance. We implement the full expressions for the matched
filtered total second moment, total variance and the magnitude squared of the total
mean, as expressed in Equations A. 12, A. 14 and A.3, respectively, from Appendix A.
These equations represent formulations for the scattered field moments alternative
to those in Equations 2.16, 2.18 and 2.8 from Section 2: the former are formulations
expressed in terms of moments conditional on a set of deterministic Green functions,
while the latter are analytic formulations, expressed in terms of bottom parameter
statistics and averages of Green function products over multiple realizations of the
ocean environment. Equations A. 12, A. 14 and A.3, however, are more suitable for imTotal moments: 10 realizations
-80
second moment
---- variance
a
...............
=L
(D -1 0 0 -.....
C
S-140-
0
...
-180
-1
0
1
2
...
..
4
3
Range (km)
5
6
7
8
Figure 3-3: Total moments of the full 390-440 Hz matched filtered scattered field in
a Pekeris sand waveguide with varying sound speed profiles, for 10 realizations of the
ocean environment. The level of the mean squared term is dependent on the number
of realizations and on the range increment dr, while the level of the variance term is
stable.
plementation, since they don't require a double integration over the patch Vs. Figures
3-3 and 3-4 show the matched filtered moments for 10 and 1 realizations, respectively.
Both figures show that, except at the edges of the patch, where the coherent intensity is non-negligible, the second moment, or scattered intensity is dominated by the
variance, or the incoherent intensity.
We note that the level of the coherent term, the mean squared, is dependent on
the number of realizations and on the range increment dr, while the level of the
incoherent term, the variance, is stable. For the latter, this implies that increasing
the number of realizations renders a smoother curve, but one realization is sufficient
to determine the correct level. For the case of one realization, in fact, the Green
function is deterministic and the total moments are equal to the conditional moments
described in Equations A.11, A.9 and A.2 for the second moment, variance and mean,
respectively, which are functions of the statistics of the bottom parameters.
Total moments: 1 realization
-80
second moment
-- variance
100.
-100
square of mean
..-
-120 -
O
P -140
-160
O
-180
1
0
1
2
3
4
Range (km)
5
6
7
8
Figure 3-4: Total moments of the full 390-440 Hz matched filtered scattered field in
a Pekeris sand waveguide with varying sound speed profiles, for a single realization
of the ocean environment. For the case of a single realization, the total moments are
equal to the moments conditional on a set of deterministic Green functions.
Next, we verify that the matched filtered variance, as given in Equation A.14, is
dominated by the first term. In Figures 3-5 and 3-6 we plot the variance and its
three terms, for the cases of 10 and 1 realizations, respectively. For both cases, we
note that the first term dominates the total variance, while the last two terms are
negligible. For the case of 10 realizations, the third term is the least significant, as
it decreases both with an increase in the number of realizations, as well as with a
decrease in dr. The average level of the second term, however, is dependent only on
the range increment dr. For the case of a single realization, the second and third
terms cancel each other because of the common Green function product. Thus, for
one realization, the total variance is equal to the conditional variance.
Variance terms: 10 realizations
-80
termi1
---term 2
te rm 3
...................
variance
im
0
Ca
-~-140
N160-180
-1
*.
0
1
2
3
4
Range (kin)
5
6
7
8
Figure 3-5: Total 390-440 Hz matched filtered variance and its terms in a Pekeris
sand waveguide, for 10 realizations of the ocean environment.
Lastly, we demonstrate that we can further approximate the matched filtered
scattered intensity by the time harmonic expression in Equation 2.21, where VS is
now the resolution footprint of the sonar. Thus, for the time harmonic expression in
Equation 2.21, the integration region Vs corresponds to a seafloor patch that extends
over l1in in depth,
3'
in azimuth and 15m in range. We have shown that the matched
filtered scattered intensity can be approximated by the incoherent term, which, at its
turn, can be approximated by the first term of the total matched filtered variance, as
expressed in Equation 2.19. By implementing Equations 2.19 and 2.21 to the same
Pekeris sand waveguide, we show in Figures 3-7 and 3-8 that the full 390-440 Hz
matched filtered expression can be approximated by the time harmonic expression at
the 415 Hz center frequency, for both 10 and 1 realizations of the ocean environment,
respectively. Figures 3-9 and 3-10, which compare the range-averaged values of the
Variance terms: 1 realization
term
:L
o-100
-
C
............
-120 Q
-140
N -160
M
E
z
-180
-1
-
0
1
2
3
4
Range (km)
5
6
7
8
Figure 3-6: Total 390-440 Hz matched filtered variance and its terms in a Pekeris sand
waveguide, for a single realization of the ocean environment. For a single realization,
the last two terms cancel each other and the total variance is equal to the conditional
variance .
two series, show the results are within a few dB. For the last two figures, we have
found that if a patch at least 10 km long were considered, averaging over a 2000
m range reduces the difference between the two curves to as low as 1 to 2 dB. We
restricted the analysis to a shorter patch in order to reduce computation time.
Equation 2.21 has just been shown to provide a good approximation to Equation
2.19 and to the total scattered intensity, and is the only term assumed in heuristic
radiometry approaches. The total time harmonic variance expression of Equation
2.20, however, is not necessarily a good approximation. While the third term of
Equation 2.20 is negligible, the second term of Equation 2.20 may not be negligible
and is an artifact caused by misapplication of coherent single frequency theory, as
shown in Appendix B.
Monostatic reverberation in Pekeris waveguide: 10 realizations
-
0
1
2
3
4
Range (km)
5
6
7
8
Figure 3-7: Full 390-440 Hz matched filtered depth-integrated monostatic bottom
reverberation compared to time harmonic reverberation at center frequency 415 Hz,
for 10 realizations of the ocean environment.
Monostatic reverberation in Pekeris waveguide: 1 realization
0L1
time harmonic approximation
matched filter
-- time harmonic approximation - range averaged
-100
-120
-140
...
.. .
. . . .. .
. . . .. . .
-..
-160
-180
-1
0
1
2
3
4
Range (krn)
5
6
7
8
Figure 3-8: Full 390-440 Hz matched filtered depth-integrated monostatic bottom
reverberation compared to time harmonic reverberation at center frequency 415 Hz,
for a single realization of the ocean environment.
42
Monostatic reverberation in Pekeris waveguide: 10 realizations
-100
--- tie harmonic approximation
-mtched
110
filter
-12
-130
-140
i-150
E
O
-10
0
1
2
3
Range (km)
4
5
6
Figure 3-9: Range-averaged full 390-440 Hz matched filter depth-integrated monos-11
tatic bottom reverberation
compared to range-averaged time harmonic reverberation
at center frequency 415 Hz, for 10 realizations of the ocean environment (range averaging over 500 m).
-14
-16 0
.........
.....
........
........
Monostatic reverberation in Pekeris waveguide: 1 realization
~-150
-100
- tme harmonic approximation
-matched filter
-
00
0
1
2
3
Range (kin)
4
5
6
Figure 3-1: Range-averaged full 390-440 Hz matched filter depth-integrated monostatic bottom reverberation compared to range-averaged time harmonic reverberation
-10
at center frequency 415 Hz, for 10in realization of the ocean environment (ranger
aaging over 500 )
tatibotmrvreaincmprdt
-110gin
ag-vgdtime harmonic
approximation
-ve
mathe filte
3.2
Frequency dependence of simulated reverberation
In this subsection, we investigate the frequency dependence of the bottom reverberation using the time harmonic approximation given in Equation 2.21 and keeping the
bottom parameters V, Var(]F), Var(Fd) and Covar(',Fd) constant for all frequencies. The analysis is performed using a Pekeris waveguide similar to the one in the
previous subsection, where the actual sound speed profiles have been replaced with
an isovelocity profile, with the sound speed constant at 1500 m. The depth and range
increments are also modified from the previous section to dz = 0.1 m and dr = 15 m.
Because of the constant sound speed, we only implement the expression in Equation
2.21 for one realization, where the sound speed is constant.
We begin by verifying the result in Galinde et al [6] that the monopole, dipole
and cross terms in Equation 2.21 are proportional to the full bottom reverberation.
Figures 3-11 and 3-12 show that this holds for both the matched filtered model and
the time harmonic approximation, respectively. The results have been computed for
an isovelocity Pekeris sand waveguide of Figure 3-1, and the levels have been averaged
in range over 500 m.
We further demonstrate that the proportionality holds for all frequencies. We
compute the simulated bottom reverberation for all frequencies by applying the time
harmonic model to the isovelocity Pekeris waveguide and keeping the bottom parameters constant. After range averaging over 2000 m, we plot, in Figure 3-13, the levels
of the full reverberation, monopole, dipole and crossterm at a single point in range. It
is clear from the figure that, at 16 km in range from the receiver, the three terms are
proportional to the full reverberation for all frequencies. The result was also found
to hold for any point in range.
After having shown that the full simulated reverberation level is proportional to
Matched filtered scattered intensity
3
Range (km)
Figure 3-11: Range-averaged full 390-440 Hz matched filter bottom reverberation
compared to its three terms: monopole, dipole and crossterm (range averaging over
500 m).
Time harmonic approximation
3
Range (km)
Figure 3-12: Range-averaged time harmonic bottom reverberation at center frequency
415 Hz compared to its three terms: monopole, dipole and crossterm (range averaging
over 500 m).
Reverberation level versus frequency at range = 16km
-120
Q
full
monople
-dipole
-130
crossterm
-
-140
0
-15
....
......
.....
.... ..-..
.
..
.....
..
..
.......
..
. ..
..
. ....
................
.......
..............
....................
..........
.............
.........
................
...
........
............
............
..
.......
...........
-0
_4
1)0
..........
.. .......
......
_Z
.......
....
.......... .......
0 470
0
z
-18u
400
800
900
700
Frequency (Hz)
600
500
1000
1100
1200
Figure 3-13: Range averaged time-harmonic simulated monostatic reverberation level
in a sand Pekeris waveguide with isovelocity sound speed profile, at 16km from
source/receiver (range averaging over 2000m).
the monopole term, given below as
(47r)2
fvs Vc(rs, zt)k 4Var(Fk(rt)) (IG(rtlro, f)1 |G(rlrt, f)
2
2
)dV,
(3.1)
we can use this term to investigate the frequency dependence of the model. Since
we keep the bottom parameters Vc and Var(Fk) constant for all frequencies, we can
further restrict our analysis to the frequency dependence of the following term:
k4
Jj
(IG(rtlro, f)12 |G(rlrt, f)12)dV.
(3.2)
For a monostatic scenario with the source and receiver collocated, this is equal to
k4
J j (IG(rtIro, f)1 )dVt.
4
(3.3)
Based on the equation above, for constant bottom parameters, we expect the
reverberation level to have a
f4
frequency dependence due to the k4 factor, and
a yet unknown frequency dependence due to the integral over the fourth power of
the Green function. Since penetration depth is known to decrease with increasing
frequency, however, we expect the volume integral in Equation 3.3 to be inversely
proportional to frequency.
To determine the actual frequency dependence of the simulated reverberation
numerically, we compute the time-harmonic simulated reverberation levels for a range
of frequencies using the standard Pekeris sand waveguide with isovelocity sound speed
profile. The bottom parameters are kept constant for all frequencies, and equal to
those previously summarized in Table 3.1.
In Figure 3-14 we plot the simulated
reverberation levels versus range, for four different frequencies. The figure clearly
shows an increase in reverberation level for higher frequencies.
Reverberation level versus range
-100
415Hz
---- 735Hz
950Hz
-110
12
-- I125Hz-
~
M~
~
20................
a-130
.
..-. .. ....
.....
............
H
.>
10
-140
.a
-150
10.15.20
.
..............
S-16 0 ..........
.
E -170
0
z
-180
0
II
5
10
15
20
Range (kin)
Figure 3-14: Range-averaged time-harmonic simulated monostatic reverberation level
in a sand Pekeris waveguide with isovelocity sound speed profile, at 415, 735, 950 and
1125 Hz, respectively (range averaging over 2000 m).
To test whether the increase in reverberation level is consistent across all frequencies, we plot the simulated reverberation level versus frequency for several points in
range, in Figure 3-15. We also plot the log-log of the simulated reverberation level
versus frequency for only the first range (4000 m) in Figure 3-16, and note that the
frequency dependence is approximately f4 .
Reverberation level versus frequency
-100
range =4km
-110
10km
.range=
range= 16km
mo-120....
130 -10
-180
400
500
600
700
800
900
1100
1000
1200
Frequency (Hz)
Figure 3-15: Range-averaged time-harmonic simulated monostatic reverberation level
in a sand Pekeris waveguide with isovelocity sound speed profile, for all frequencies,
at
4,
10
and
16
km
in
range,
respectively
(range
averaging
2000
over
m).
The fact that the reverberation level varies with approximately the fourth power
of frequency, and the presence of the k4 term in Equation 3.3 lead us to believe that
the volume integral in Equation 3.3 is nearly frequency independent. To verify this,
we plot the volume integral versus frequency in Figure 3-17 and versus the log of
frequency in Figure 3-18. The two figures confirm that the integral is indeed nearly
independent of frequency, as it varies approximately with f
--
2.
As penetration depth is known to decrease with an increase in frequency, which
makes the statement that the volume integral is nearly frequency independent counterintuitive, we provide further analysis to support this result. We plot the value of
the integral in Equation 3.3 at several depths, as indicated in Figure 3-19. As the
depth increment for this case is dz =-0.1 m, this represents an integral over a bottom
layer that is 0.1 m thick. Next, we plot the value of the integral from the 100 m
interface, down to several depths, as mentioned in Figure 3-20. Here, the integration
is performed over all 0.1 m thick bottom layers located above the depths indicated in
Frequency dependence of reverberation level at 16km
-145
Reverberation
*
f
4
-1501
0.25
-155
-;-------- -- -------
-160 -
-170 L
2.6
2.7
2.8
2.9
Frequency (dB)
3
3
Figure 3-16: Frequency dependence of time-harmonic simulated monostatic reverberation level at 16 km in range for a sand Pekeris waveguide with isovelocity sound
speed profile (range averaging over 2000 m), for bottom parameters V, Var(FK(rt)),
Var(Fd(rt)) and Covar(P,(rt)) constant for all frequencies.
IGreen
4
volume integral
-120
range= 4
- range= 10km
range= 16km
ai-130
-140
% -150
.........
... .....
.........
C
0
................ ...
..... ....
...
..
-160
.o
e -170
- -180
N
i..
E -190
0
z
-200
40 0
500
600
700
800
9(00
Frequency (Hz)
1000
1100
1200
Figure 3-17: Volume integral from Equation 3.3 plotted versus frequency for several
locations in range
49
-175 -
Freauency dependence of IGreen
Green 4 integral
4
integral at 16km
f
-180
f2
0.25
-
-185
-190
..
-195
-200
2.6
2.7
2.8
2.9
Frequency (dB)
3
3.1
Figure 3-18: Frequency dependence of volume integral from Equation 3.3.
the legend. The figures show that, although the intensity within the layers changes
for different frequencies and depths, causing the volume integral at low frequencies
to converge in depth slower than at high frequencies, the total volume integral remains relatively constant across frequencies. The most plausible explanation for this
phenomenon is that the top layers, with weak frequency dependence, are dominant.
The analysis presented in this section shows that the frequency dependence of
the simulated reverberation, with constant bottom parameters V, Var(1), Var(Fd)
and Covar(F],Fd), is approximately f4 . Literature detailing experimental results [8]
reveals, however, that measured ocean bottom reverberation has a low frequency dependence. This is consistent with our model if we let the coherence volume Vc vary
with frequency, as also suggested by Galinde et al [6], who state that the acoustically determined
, may not equal the geological coherence volume for the bottom
inhomogeneities, and it may vary with frequency. In the next section, we use the timeharmonic model for ocean bottom reverberation to invert for the bottom parameters
for several frequencies at two distinct geographical locations.
4
IGreen| at several depths, for range
-190
=16km
depth 1100. M
.dpt
-200
depth ==...
100.4m
100.4
-- depth =100.7m
depth =101.Om
.depth
101.3m
-210
depth= 101.6m
c-220
400
500
600
700
800
900
Frequency (Hz)
1000
1100
1200
Figure 3-19: Value of the integral in Equation 3.3 evaluated at several depths: the
integration is performed over a bottom layer that is 0.1 m thick, as dz = 0.1 m.
|Green
integrated to a certain depth, for range = 16km
-184
depth
depth
-- depth
depth
depth
depth
-186
-188
-190
-192
Va-194
-196
.....
...
...
...
..
..
..
...-..
...-.
.. . . .. .
. . ..
= 100.1m
= 100.4m
= 100.7m
= 101.Om
= 101.3m
=101.6m
......-....-.-..
...
...
...
..
...
...
..
.. .. . . .. . . .. . .
-198
-200
-9502
-204
400
500
600
700
800
900
Frequency (Hz)
1000
1100
1200
Figure 3-20: Value of the integral in Equation 3.3 evaluated from the 100 m water/air
interface down to several depths: the integration is performed over all 0.1 m thick
bottom layers located between 100 m and the depths indicated in the legend.
52
Chapter 4
Model calibration to bottom
reverberation data
In this section we compare simulated bistatic, range-dependent reverberation to reverberation measured during the course of two experiments at sea. We use the efficient time harmonic approximation of Equation 2.21 to the matched filtered model
and invert for the one parameter needed to scale the bottom parameters V(rs, zt),
Var(F,(rt)), Var(Fd(rt)) and Covar(]F,(rt)) given known ratios for the monopole,
dipole and cross terms. We use the least squares method to find the best fit parameter and provide statistics of the inverted parameter for each experiment, at three
different frequencies.
4.1
OAWRS bottom reverberation data
For model calibration, we use data acquired by the ocean acoustic waveguide remote
sensing (OAWRS) system during experiments conducted in 2003 on the New Jersey
continental shelf and 2006 on the northern flank of Georges Bank in the Gulf of Maine.
The 2003 experiment was conducted from April 27 to May 15, 2003 on the New
Jersey Strataform [2], [16], [13]. The bathymetry of the region is relatively flat, with
depths ranging from 65 to 80 m. The bottom sediment in the region consists mostly
of sand with a mean density, sound speed, and attenuation of approximately 1.9
g/cm 3 , 1700 m/s, and 0.8 dB/A, respectively [7], [6].
Over 100 sound speed pro-
files have been measured during the experiment to allow accurate characterization of
the randomly fluctuating continental shelf waveguide. The bistatic OAWRS system
consisted of a moored vertical array source centered at a depth of 47 m, and a horizontal receiving array towed at depths between 30 and 50 m. The source transmitted
Tukey-shaded linear frequency modulated (LFM) broadband pulses in three distinct
frequency ranges, 390 to 440 Hz, 875 to 975 Hz, and 1250 to 1400 Hz, each 1 second
in duration at every 50 second interval.
fora2006jd276t223115, 950Hz
-40
-30
-20
-10
0
(km)
10
20
30
40
Figure 4-1: OAWRS image charting returns (dB re 1 pPa) received on the Northern Flank of Georges Bank at 18:31 EDT, on 3 October 2006. The white diamond
indicates the source and the black line is the receiver track. The white curves are
bathymetric contours for the region. Clutter is distinguished from seafloor returns by
inspecting both the bathymetry of the region and the stationarity of the returns over
multiple pings. Ship beams are also identified as radiating outward from the receiver,
symmetrically about the endfire of the receiver array.
For the 2006 experiment, conducted from September 19 to October 6 [13], [8], [15]
on the Northern Flank of Georges Bank in the Gulf of Maine, the water depth ranged
from as shallow as 20 m to as deep as 400 m. The sediment at the experimental
site was found to consist of sand with sound speed of 1700 m/s, density of 1.9 g/cm3
and attenuation of 0.8 dB/A [8].
Roughly 200 sample sound speed profiles of the
water column have been taken during the experiment. The OAWRS imaging system
consisted of a moored vertical source array and a towed horizontal receiver array,
centered at 65 and 105 m depth, respectively. One second Tukey-windowed LFM
pulses were transmitted by the source in a 50 Hz frequency band around the center
frequencies of 415, 735, 950 and 1125 Hz, 75 seconds apart.
The pressure data collected from both experiments was beamformed and matched
filtered to determine the azimuth and range of arrivals and instantaneous OAWRS
images such as the one shown in Figure 4-1 have been created for every broadband
transmission [6], [8], [16], [15].
4.2
Parameter inversion methodology
We invert for only one parameter, as we have shown that the monopole, dipole and
cross terms are proportional to the total intensity for the 415 Hz center frequency (Figure 3-12), as well as for all frequencies in the 415-1125 Hz frequency range considered
(Figure 3-13). This proportionality allows us to express the total scattered intensity, or reverberation, as the product of one parameter and the monopole term. For
this, we assume that V(rs, zt) and the moments of the fractional changes in bottom
density and compressibility Var(PK(rt)), Var(Fd(rt)) and Covar(1F(rt)) are constant
across the seafloor as well as depth. Then, they can be taken outside the volume
integral in Equation 2.21 and the time harmonic reverberation can be expressed as:
Var(<bs(rsIr, ro,
f))
(4.1)
= (4,r) 2 VcVar(Fk)fJ
+ (47) 2 VcVar(Jd)
.(4w) 2 VcCov(Fk,
vs k'[(|G(rtjro, f)12 |G(rlrt, f)12 )]dV
Jfvs [(IVG(rtlro) -VG(rlrt) )]dVt
2
Jj
gId)
k2 [2R{G(rtjro, f)G(rlrt, f) x VG*(rtlro, f) - VG*(rlrt, f})D]dVt,
where the first integral is a monopole term, the second integral is a dipole term, and
the last integral is a crossterm. Using the fact that the three terms are proportional,
we can define Fd and Fc so that Fd represents the ratio between the dipole integral
and monopole integral, and Fe represents the ratio between the crossterm integral
and monopole integral. Then, we can further simplify Equation 4.1 as:
Var(4s(rsIr,ro,
=
f))
(4.2)
(47r) 2 Vc[Var(Fk) + FdVar(Ld) + FCov(Fk, Fd)]
x
vs k4 [(IG(rtlro, f)12 |G(rlrt, f)12 )]dVt.
Finally, the simplified one-parameter model we use for calibration is:
Var(4s(rslr,ro, f))
00 x
JJvs
k4[(IG(rtlro, f)12 |G(rlrt, f)| 2 )]dVt,
(4.3)
where the parameter we need to estimate is
S= (47r) 2 Vc[Var(Fk) + FdVar(Fd) + FcCov(Fk, Fd)].
(4.4)
To find an estimate 0 for the parameter 0 from the data, we use the method of
least squares. The equation used to find the estimate 0 from the data using the model
of Equation 4.3 is
min
n Q1 Ig(D~SiData(rs,Irro, f)12
12
) - 10log(VVar(4Is,(rsjIr~roj)))
p
r
10log(
ref
(4.5)
ref
where |4s Data(rs Ir, ro, f)12 represents the measured reverberation level at rsi. The
summation is over all n resolution footprints Vs, centered at rs, included in the
seafloor region we use for calibration. Using Equation 4.3, we find that the parameter
estimate 0 is given by
10log($)
nli 10log|I sData (rs,|r, ro, f)12
=
110log(fff,
k4[(|G(rtjrojf)|2|G (r Irt, f )|2)]dVt)'
(4.6)
We note that using least squares for logarithmic variables may lead to biases in 6 [14].
Using beam time received pressure intensity data for one ping at a time, we find
the best fit parameter for each beam in the respective ping. We do this by applying
Equation 4.6 and estimating one value of 0 for each beam within the particular ping.
To obtain a statistically significant set of inverted parameters, we analyze multiple
pings. We restrict our analysis to pings and beams that are not contaminated by
clutter or ship noise, and to regions of relatively flat or downward sloping bathymetry.
In the next section, we describe the beam time data used in the calibration and the
methodology for implementing the model along a beam. We then compare the model
simulation results with experimental data along beams. In the following section we
compare the model simulation results with data for wide area polar plots.
4.3
Model/data comparison using beam time data
In this subsection, we begin our model/data comparison by considering the receiver
reverberation intensity versus beam time and range.
In the next subsection, we
compare the model and data using wide area polar plots. All the analysis in the two
sections is performed for pings received at the center frequency 415 Hz.
Measured normalized pressure beam time series (dB re1pL
Pa)
-130
Measured normalized pressure beam time series (dB re 1 Pa)
--
-140
5
-140
10
10
-
0-150
-15
E15
20
-150
17
20a
17
-160
25
-160
25
30
-170
30
35
-1
130
-0.5
0
0.5
Sin of broadside angle
1
1-180
35-170
40
-1
-0.5
0
0.5
Sin of broadside angle
1
-180
Figure 4-2: Measured normalized pressure level as beam time series for the New
Jersey continental shelf (left) and Georges Bank in the Gulf of Maine (right). The
black stars indicate the beams chosen for analysis. The source is broadband for 415
Hz center frequency and 50 Hz bandwidth and the source level is normalized to 0 dB
re 1 p-Pa at 1m.
Each ping, for both the 2003 and 2006 ONR Geoclutter Experiments, consists of
65 or 66 beams of data covering an azimuthal sector of 1800, received continuously
over a 50 or 75 second interval, depending on the experiment. The directions of
the beams are uniformly spaced in sin(#), where
#
is the angle of the beam relative
to the broadside of the receiver. Figure 4-2 shows two examples of pings recorded
during the New Jersey and Georges Bank experiments, respectively. The pings shown
are partially contaminated by ship beams and clutter, which is often the case for
the data collected, raising a need to restrict our analysis to clean beams that are
stationary over multiple pings. Because of left-right ambiguity about the receiver
heading, the reverberation data for each beam corresponds to the sum of the returns
from a pair of two separate azimuthal directions, received along what we define as
original and mirror beams. The beam time reverberation data is thus a function of
Bathymetric map
Bathymetric map
0
30
40
0
30
-50
20-
-100
20
E*
50
10
0
-- 10
-250
8 -20
300
-30
-350
0
8
-150
-M ,-200
-30
-20
-10
0
10
Distance from receiver (km)
20
30
-40
-40
-20
0
20
Distance from receiver (km)
40
-400
Figure 4-3: (Left) Bathymetric map of the New Jersey continental shelf indicating
the source and receiver positions at 10:31 EDT, on 13 May 2003. The grey diamond indicates the source, located at 39.2312N, 72.8818W and operating at 390-440
Hz. The black diamond indicates the receiver, located at 39.2465N, 72.8626W, with
heading 346'E. The black and magenta stars indicate the beams chosen for analysis,
corresponding to the original and mirror beams, respectively. (Right) Bathymetric
map of the northern flank of Georges Bank in the Gulf of Maine indicating source and
receiver positions at 11:58 EDT, on 26 September 2006. The grey diamond indicates
the source, located at 41.8901N, 68.2134W and operating at 390-440 Hz. The black
diamond indicates the receiver, located at 41.8212N, 68.3368W, with heading 137'E.
The black and magenta stars indicate the beams chosen for analysis, corresponding
to the original and mirror beams, respectively.
sin($) and beam time. This data can be charted to range and converted into polar
plots by multiplying the beam time with the sound propagation speed in the ocean
and accounting for the bistatic geometry i.e. the distance traveled from source to each
scatterer location in range along the beam and from the scatterer to the receiver.
To accurately model the reverberation received along each beam for a certain ping,
we first account for all the variables of the problem, such as the actual bistatic geometry of the respective ping, bathymetry, sound speed profiles, seafloor and ocean properties and source and receiver array types. Each beam data time series corresponds
to two patches in the seafloor, placed symmetrically about the receiver heading, along
the original and mirror directions. The two patches extend over an azimuthal angle
equal to the sonar angular resolution, over the top 10 m deep bottom layer and over
25 to 35 km in range from the receiver location, depending on the length of the data
time series we use for comparison or the extent of the region for which we have bathymetric data. Since we use the time harmonic model for computational efficiency, we
only integrate over volumes Vs that are 15 m long in range, corresponding to the
sonar range resolution, at each point in range along the two patches. Examples of
bistatic geometries from the New Jersey continental shelf and the northern flank of
Georges Bank in the Gulf of Maine are depicted in Figure 4-3, showing the source and
receiver (as the grey and black diamonds, respectively), along with black and magenta
stars, corresponding to the original and mirror patches selected for analysis in this
subsection. The background of the maps represents the relatively flat bathymetry in
the region chosen for analysis, respectively.
Next, we compute the Green functions using the parabolic equation model RAM.
For both experiments, we use sand as the bottom sediment type, with sound speed of
1700 m/s, density of 1.9 g/cm 3 and attenuation of 0.8 dB/A, as mentioned in Section
4.1. We use a range increment dr = 15 in, a depth increment dz = 0.4 m, and we
assume the Green function is uniform in azimuth over the angular resolution. Since
both the original and mirror patches radiate outward from the receiver location,
computing the Green function from the receiver to each range increment along the
patch requires running RAM only once. However, for the bistatic case, computing
the Green function from the source to each range increment along the patch requires
running RAM a great number of times. For computational efficiency, we run RAM
from the source to only several points in range along each patch (sufficient for convergence) and interpolate the points to find the source Green function along the entire
patch. The interpolation is performed in order to obtain a value for both the source
and receiver Green functions at each range increment along the patch and thus increase the accuracy in fitting the model to the data. The Green functions are then
used to compute the volume integral for the monopole term in Equation 4.3 over
all resolution cells Vs along both the original and mirror patches. Since the time
harmonic result is a function of range, due to the implementation at each resolution
cell Vs, we convert range to time before adding the contributions from the original
and mirror patches. Lastly, we use the least squares method to find the parameter
that provides the best fit of the simulated reverberation to the beam time data along
each beam considered.
50
45
_4
0
Parameter estimates 0
Parameter estimates 0
50
-
45
4.
O ........
..... ....
.... .......
< 35
30
20
.......
...
<:35
30
40
Beam number
50
60
30'
20
40
30
Beam number
50
Figure 4-4: Parameter estimates 0 (log scale) estimated from the data pings shown in
Figure 4-2 for the New Jersey (left) and Georges Bank (right) experiments, at 415 Hz.
Each 0 corresponds to one beam and was estimated using Equation 4.6, where the
summation is performed over all n resolution cells Vs, along the beam. Typically, each
beam extends for 25 to 35 km in range, which leads to values of n from 1667 to 2333,
respectively, given the sonar range resolution of 15 m. The mean of the parameter
estimates (0) is indicated by the horizontal line and is obtained using Equation 4.7,
with p = 1 and m representing the number of beams in each ping. For this figure, m
equals 31 and 36 for the New Jersey and Georges Bank experiments, respectively.
The procedure described above can be repeated for all clutter-free beams within a
certain ping, as well as for multiple pings. Typically, we average 0 over all calibrated
certain ping, as well as for multiple pings. Typically, we average 0 over all calibrated
beams within a ping and over multiple pings, and then verify the fit of the model
using the mean parameter estimate (0). For the case of p pings each with m beams,
(0) is obtained by averaging over a total of mp ping-beam samples, and is given by
(0)
A
j-
Pj=1
h
(47)
jh-
h=1
In this subsection, however, we restrict our analysis to only one ping for each of
the two experiments, corresponding to the beam time series data shown in Figure
4-2. Thus, for this special case, p = 1. The parameters 0 (in log) estimated for each
of the m beams of the two pings are shown in Figure 4-4. A particular trend in 0
for consecutive beams within a ping could be an indication of a geographical feature
or of gradual ship beam contamination. The red line indicates the mean parameter
estimate (0) as given by Equation 4.7 with p = 1 and m representing the number of
beams in each ping. In Figure 4-5, we compare the measured reverberation level to
the reverberation level simulated using Equation 4.3 and (0) of Figure 4-4, along one
of the beams for each experiment. For the range averaged levels presented in Figure
4-6 for the same beams, the errors are within 5 to 6 dB.
Measured vs simulated reverberation level for beam 25
-100
Measured vs simulated reverberation level for beam 25
-100
110
=L~~
-Measured
Simulated
~~
-Measured
Simulated
..........
10 .....
-120-..
....
-130~
-130
-150t-150
-1
5
10
15
20
25
Beam time (s)
30
35
-10
15
20
25
Beam time (s)
30
35
Figure 4-5: Model/data comparison along one beam for pings received at center
frequency 415 Hz during the New Jersey (left) and Georges Bank (right) experiments,
respectively. The source level is normalized to 0 dB re 1 [pPa at 1 m .
Measured vs simulated reverberation level for beam 25
Measured vs simulated reverberation level for beam 25
Measured: range-averaged
.Simulated: range-averaged
110
.........
......
....
.....
.......
-140
-120
-130
0
-- 140
-0
-Measured: range-averaged
Sim ulated. range-averaged
-120
....
-.
....
. ..
..
....
..-.. .....
.
....
~.
.
~.
~~~
~.
..
~.
.....
-..
.....
........... .....
............ ....
0
.......-..
. ..
C>-130
. ......... .
....
. .......
........
...
..-
-140
,-150
cc:
-160
-170
S-170
10
5
10
15
20
Beam time (s)
25
30
35
15
10
20
25
30
35
Beam time
(s)
Figure 4-6: Model/data comparison along one beam for pings received at center
frequency 415 Hz during the New Jersey (left) and Georges Bank (right) experiments,
respectively (range averaging over 2000 m). The source level is normalized to 0 dB
re 1 pPa at 1 m.
4.4
Model/data comparison using polar plots
Implementing the methodology described in the previous subsection and calibrating
the model to data for consecutive beams within a ping enables us to create wide area
polar plots of the simulated reverberation, and to compare them with polar plots of
the data. Figure 4-7 shows the polar plots for the data (top), model (middle) and
data-model differences (bottom) for the New Jersey (left) and Georges Bank (right)
experiments. We can note from the figure that for the New Jersey ping, the beam
time series was converted to an approximately symmetric range on both sides about
the receiver heading, while for the Georges bank ping, a shorter range resulted on
one side than on the other. This is because the separation between the source and
receiver is larger for the Georges Bank ping, as we have seen in Figure 4-3. The
patches of uniform intensity located at the sharp angle of the polar plots for Georges
Bank are also due to the range stretching. Overall, the results are satisfactory, as the
data-model differences plots show most of the errors fall within 5 dB. The data-model
error at the ith resolution footprint Vs5 along beam
|<s.n( (rse3 Ir, ro,
100g(
"
ref
f)12
fff,
j
is given by
k4[(|G(rtlro, f)12|G(rlrt, f)}2 )dVt
(4.8)
- (8 "p)
ref
Polar plot of measured reverberation levels (dB re 1p Pa)
Polar plot of measured reverberation levels (dB re 1pgPa)
-130
-130
-140
-140
-150
-150
*
E
0
-30
-20
-10
0
10
Distance from receiver (km)
20
30
-160
-160
-170
-170
-180
-180
-_tJ
-1U
u
10
Zti
;d
Distance from receiver (km)
Polar plot of simulated reverberation levels (dB re 1IpPa)
Polar plot of simulated reverberation levels (dB re Ip P a)
-130
-130
-140
-140
-150
-150
E
0
-20
10
-10
0
Distance from receiver (km)
20
30
-160
-160
-170
-170
-180
-20
20
0
10
-10
Distance from receiver (km)
30
Polar plot of data-model differences (dB re 1 Pa)
Polar plot of data-model differences (dB re 1g Pa)
20
I20
-180
15
15
10
10
-5
00
E
-5
-10
-15
-30
-30
-20
-10
0
10
Distance from receiver (km)
20
30
-20
-20
20
10
0
-10
Distance from receiver (km)
30
Figure 4-7: Polar plots of measured reverberation level (top), simulated reverberation
level (middle) and their difference (bottom) for New Jersey (left) and Georges Bank
(right). The source level is normalized to 0 dB re 1 pPa at 1m
63
with
ks'j
Data
(rs Ir, ro,
f)I
being the measured reverberation level at rsej.
Again,
since the analysis in this section is restricted to one ping, the errors in Figure 4-3 are
computed using (0) for p = 1. Figure 4-8 shows the distribution of these errors for
the entire polar area analyzed.
Histogram of errors
Histogram of errors
0.05
0.05
0.04 -
0.04 --
0.03-
0.03 -
0.02 -
0.02-
0.01-
0.01 --
-20
-15
-10
-5
0
dB
5
10
15
20
-20
i
-15
-10
-5
0
dB
5
10
15
20
Figure 4-8: Histogram of the data-model differences as show in Figure 4-7 (bottom)
for New Jersey (left) and Georges Bank (right). For the New Jersey errors (left),
the mean is -1.1467 dB and the standard deviation 6.5095 dB. For the Georges Bank
errors (right), the mean is -0.3526 dB and the standard deviation 5.8003 dB.
Appendix C presents calibration results for other pings at several frequencies.
As the Appendix shows, for some of the pings recorded during the Georges Bank
experiment at the center frequencies 950 and 1125 Hz, the model was found to underestimate the data for short ranges. Potential explanations may include returns
that are not accounted for by the current approach, such as the use of too narrow
of a vertical angle in the parabolic equation, fathometer returns, or returns due to
multiple scattering.
4.5
Statistics and frequency dependence of inverted
parameter
A systematic calibration of the model to data for multiple pings and frequencies
for both the New Jersey and Georges Bank experiments allows us to determine the
second order statistics of the inverted parameter 0, which we tabulate in Table 4.1.
The parameter 0, as given in Equation 4.4, is a function of the coherence volume Vc
and the second order statistics of the bottom properties Var(F,(r,)), Var(rd(rt)) and
Covar(Fs(rt)), which we assume constant in depth and across the entire experimental
site. Since 4.1 is an equation with four unknowns, the values for each of the bottom
parameters cannot be estimated from 0. What we are most interested in, however, is
the mean and variance of the parameter 0 across the seafloor, for different frequencies
and experimental locations.
f
(Hz)
415
415
925
950
1325
1125
Experiment*
NJ
GB
NJ
GB
NJ
GB
Data analyzed
mp (#
p (#
ping-beam
pings)
samples)
5
180
4
85
4
144
4
99
4
144
4
114
Parameter statistics
(9)
0.000117
0.000134
0.0000032
0.0000059
0.0000019
0.0000059
(9) (dB)
-39.32
-38.73
-54.95
-52.29
-57.21
-52.29
Std (9)
0.0000411
0.0000461
0.0000019
0.0000027
0.0000005
0.0000022
Std(6)
(dB)
1.63
1.47
2.56
2.55
1.21
1.73
%
1 std
69.32
74.70
72.53
69.07
67.61
66.36
%
2std
94.87
95.18
97.18
96.91
95.07
96.36
Table 4.1: Table showing the results of the parameter estimation. The first two
columns represent the corresponding center frequencies and experiment locations,
respectively. The number of pings, p, and total ping-beam samples, mp, used in the
analysis are given in the third and fourth columns, respectively. The next six columns
give the mean (level and log scale), standard deviation (level and log scale), as well as
the percentage of beam samples within one and two standard deviations, respectively.
One b is estimated for each single beam in each single ping using Equation 4.6, where
the summation is performed over all n resolution cells Vs, along the particular beam.
As mentioned in Figure 4-4, we typically sum over 1667 to 2333 resolution footprints
along each beam. Then, the mean parameter estimate (0) is obtained by averaging
the parameter estimates 0 over all mp ping-beam samples, as given by Equation 4.7.
(* the NJ and GB symbols stand for New Jersey and Georges Bank, respectively)
The inversion results summarized in Table 4.1 confirm that, for a certain frequency
and for similar bottom sediments (in this case sand), the parameter estimates are
consistent. For example, at 415 Hz, for both New Jersey and Georges Bank we found
a parameter equal to about 0.0001, or negative 40 dB. This is also consistent with
the parameter estimate b we would obtain if we used the bottom parameter statistics
estimated by Galinde et al [6] for 415 Hz in New Jersey, summarized in Table 3.1.
Another observation is the fact that the value of the parameter decreases with
New Jersey parameter histograms
Georges Bank parameter histograms
415 Hz
925 Hz
1325 Hz
0415 Hz
M950 Hz
1125 Hz
.0.1
2.051
a.
0
-65
-60
-55
-50
-45
-40
Parameter value (dB)
0
-35
-30
-65
-60
-55
-50
-45
-40
Parameter value (dB)
-35
-30
Figure 4-9: Histograms of parameter values (log scale), for the New Jersey (left) and
Georges Bank (right) data.
frequency, except for Georges Bank, where the parameter 0 remains stable from 950
to 1125 Hz. To clearly illustrate this result, we plot in Figure 4-9 the distributions
of the inverted parameter estimates 6 using New Jersey and Georges Bank data at
three frequencies for each case. In Figure 4-10 we also plot the mean and standard
deviations of the estimated parameter 6 for both the New Jersey and Georges Bank
data. Since the statistics of the bottom properties should be independent of frequency,
the result supports our hypothesis from Section 3.2 that the acoustically determined
coherence volume Vc varies inversely with frequency, to compensate for the frequency
dependence of the k4 factor in Equation 3.3. After fitting linear curves in log-log
domain (Figure 4-11) through the data points for the mean estimates from figure 410, we find that the frequency dependence in the New Jersey case is
Georges Bank case
f-3.3.
f--7 , and in the
This result is consistent with the wavelength dependence of
scattered intensity found in speckle interferometry [9] and scintillation theory [2], [14].
For the case of volume scattering, when frequency dependence of scattered intensity
is greatest [9], the inhomogeneities are selected over the wavelength scale in all three
directions, leading to a dependence of approximately the wavelength cubed.
Mean and standard deviation of parameter estimates
-35
* New Jersey
-.-.-.-.-.-.--.-.--.
.......
-p
--.
-40
r-45
-.
4)
F- -50
-...--..--...-.
.....
.
.-
---------...
.. ......
-........
-..
- ...
.
..
..-.
...
-60
400
600
.-
.-.
.-.
.-
.......
-.
---.-.-.-.---.---.-.-.-.-.-.-.-
-55
c
Georges Bank -
800
1000
Frequency (Hz)
1200
-.
1400
Figure 4-10: Mean (dots) and standard deviations (error bars) for the parameter
estimate 0, as given in Table 4.1, for the New Jersey and Georges Bank data.
-40
0 New Jersey
4 Georges Bank
--...-..
---. ------ .
--..--..
-.-..-
3.7.
f-3.3
CO,
-.45 - -
- --
--
-
- - -
%hNN
~50 - --
--
- -.. - ---
-
--
60
2.8
2.7
28
2.9
3
3.1
3.2
Log of frequency (dB)
Figure 4-11: Mean parameter estimates 0 (circles), as given in Table 4.1, for the
New Jersey and Georges Bank data, along with least squares fitted lines for each
experiment.
67
68
Chapter 5
Conclusion
In this thesis, we derived full expressions for the total moments of the matched filtered scattered field, using the Green theorem application to the inhomogeneous wave
equation and the Born single scatter approximation. We provided approximate expressions for the total scattered intensity, including a time harmonic approximation
implemented in terms of the resolution footprint of the sonar. We showed, using
Monte Carlo simulations of the Green function using actual sound speed profiles and
a Pekeris waveguide, that the following are true: (1) the total matched filtered second
moment can be approximated by the total matched filtered variance; (2) the total
matched filtered variance can be approximated by its first term; (3) the first term of
the total matched filtered variance can be approximated by the first term of the total
time harmonic variance. Using the least squares method, we calibrated the computationally efficient time harmonic model to reverberation data collected during two
experiments at sea, for several frequency ranges. We provided the statistics of the
estimated parameter and discussed the frequency dependence of the model and parameters. We found that the acoustic signal selects seafloor scatterers of wavelength
scale.
70
Appendix A
Matched filtered total moment
expressions based on conditional
moments
Instead of implementing the analytical expressions for the total moments of the
matched filtered scattered field described in Section 2, we found it more efficient
to implement the total moment expressions derived from conditional moment expressions as described in this section.
To derive the expressions for the total moments using conditional moments, we
begin with Equation 2.7, for the matched filtered scattered field, which we repeat
below as
4bs(rsIr, ro, tM)
=
(47)f
If
k2r (rt)G(rtlro, f)G(rjrt, f)
(A.1)
Vs
fc-B/2
+ Fd(rt)VG(rtro, f) - VG(rlrt, f)]
1
x
IQ(f)I2e-i2
f(t-tm)
dV df.
Next, using the assumption that the bottom parameters are uncorrelated with
the propagation Green functions, we condition on a deterministic set of Green functions (G(rtlro, f), G(rlrt, f)), and define the conditional mean for the matched filtered
scattered field as
(A.2)
(4s(rslr, ro, tM)|G(rtro,f), G(rlrt,f))
-(47)
f
fc-B/2
Jjj
[k2 (7(rt))G(rtjro, f)G(rlrt, f)
Vs
+ (1F(r))VG(rtlro, f) - VG(rlrt, f)]
x
I |Q(f)|2 e-i 2 rf(t-tM) d
df.
The expression above is a function of the bottom parameters statistics (17,) and
(Pd) and a deterministic set of Green functions (G(rtlro,
f), G(rlrt, f)),
which rep-
resent one realization of the ocean environment, where at every 500m in range we
randomly selected a new sound speed profile from a large set of measured profiles.
The total matched filtered mean can then be expressed in terms of the conditional
mean as
=(Ds(rsIr, ro, tM)|IG (r Iro, f ), G (r Irt, f).
(4)s(rsIr, ro,7 tM)
(A.3)
The expression above is equivalent to the analytic expression from Equation 2.8
and does not facilitate implementation. We are more interested in the formulations
for the second moment and variance, which we derive in the remainder of this section.
The conditional second moment is now
(l4)s(rslr, ro, t M) 12|G(rtiro, f), G(rlrt,f))
2 r(rt)G(rtlro,
((47r) f f[Ik
fc- B/2
(A.4)
f)G(rlrt, f)
Vs
+ Fd(rt)VG(rtlro, f) - VG(rlrt, f)] x
(47r)
f
cB2
fc-B/2
1
IQ(f)I2 e-i2f(t-tM) dVt df
[k'r, (r')G*(r'l ro, f')G*(r Ir', f')
JJJVs
+ Fd(r')VG*(r'ro, f') - VG*(rlr', f')] x
IQ(f')|2e-i24 '(t -tm) dV' df'),
and can be written as
(|GDs(rsjr,ro, tm)|2|G(rtjro,f), G(rlrt, f))
=
(47r)2 jfc+B2 jfc+B/2
(A.5)
HA Il
[k 2k'(IF,(rt)IF,(r'))G(rt Iro, f ) G(r Irt, f )G* (r'l ro, f')G*(r Ir', f'
± (Fd(rt)Fd(r'))VG(rtjro, f) - VG(rlrt, f) x VG*(r'lro, f') - VG*(rrf')
± k2 (F(rt)F(r'))G(rtro, f)G(rIrt, f) VG*(r'Iro, f') -VG*(rIr', f')
+ k' 2 (F,(r')1Fd(rt))G*(r'lIro, f')G*(rIr, f')VG(rtIro, f) VG(rIrt, f)]
x
F0
| Q( f||21Q(fi)1so-i27r(f-f'xt-tm) dV dVt df df'.
Note that there are now no expectations around the Green function products, as the
Green function and its gradients are assumed deterministic. We can further simplify
this expression, using Equations 2.11 and 2.12, to obtain
(l4s (rs Ir, ro, tM) 12|G(rtlro, f), G(rlrt, f))
=(47r) 2 jf+1
( 2f
f+1
fc+B/2
ff
f
j
(A.6)
V
Vc(rs, zt)
x [k2 k'2 Var(IF,) J (rt - r')G(rt Iro, f )G(rlrt, f )G*(r' Iro, f')G*(rIr', f')
+ Var(L'd)6(rt - r')VG(rtIro,
f) - VG(rlrt, f) x
VG*(r'lro,
+ k2 Covar(FK, Fd)6(rt - r')G(rtIro, f)G(rlrt, f)VG*(r'lro,
f') - VG*(rlr', f')
f') -VG*(rlr', f')
+ k'2 Covar(FP, ld) 6 (rt - r')G*(r'jro, f')G*(rlr', f')VG(rt ro, f) - VG(rlrt,
X 1o| Q( f |21Q{gi)12e-127(f-f'x(t-tm) dV dVt' df df'+
f)]
fc+B/2
fc+-VB/2
J~v
+ (47r)2
fc-B/2 Jfc-B/2
x[k2 k'2
Vs
(r.(rt)) (r,(r'))G(rt Iro, f )G(r Irt, f )G* (r' Ilro, f')G*(r Ir', f')
± (Fd(rt))( d(r'))VG(rt|ro, f) - VG(rlrt, f) x VG*(r'lro, f') - VG*(rr',f')
+ k2 (F(rt)) (Fd(r'))G(rtIro, f)G(rlrt, f)VG*(r'l ro, f') - VG*(rr', f')
+ k' 2 (F.(r'))(F(r))G*(r'|ro, f)G*(rlr', f')VG(rtjro,f) - VG(rlrt, f)]
X
Eo
|IQ( f||21Q(fi 2e-'27r f -f'xt-tm) dV dVt' df df'.
By canceling the delta function with one of the volume integrals in the first term,
we have
(I<bs (rsIr, ro, tM) 12|G(rtlro, f), G(rlrt,f))
Ifc+B/
=
(47r)2 f
fc,-B/2
(A.7)
f
fc+B/2
fI fV(rs,
zt)
Vs
fc-B/2
x [k2 k'2 Var(1)G(rlro, f)G(rlrt, f)G*(rtlro, f')G*(rlrt,
+ Var(Fd)VG(rtlro,
f')
f) - VG(rlrt, f) x VG*(rt|ro, f') - VG*(rjrt, f')
+ k2 Covar(r,F)G(rtro, f)G(rlrt, f)VG*(rtlro, f') - VG*(rlrt, f')
+ k' 2 Covar(F, Fd)G*(rtlro, f')G*(rlrt,
x
+
f')VG(rtjro, f) - VG(rlrt, f)]
I|Q(f)|21Qgfr)12e-i27r(f-f"'x,-t) dV df df'+
J47)2
fc+B/2
fc-B/2
fc+B/2.
Vs
fc-B/2
x[k 2 k'12(Ts(rt))(r,,(r'))G(rt|ro,
+ (Fd(rt))(]P(r'))VG(rtIro, f)
V
f)G(rlrt, f)G*(r'l|ro, f')G*(rlr',,f')
VG(rIrt, f) x VG*(r'4ro, f') -VG*(rIr',f')
+ k2 (F(rt))(F(r'))G(rtlro, f)G(rlrt, f)VG*(r'lro, f') - VG*(rlr', f')
+ k' 2(P(r'))(F(r))G*(r'|ro, f)G*(rr', f')VG(rt|ro, f) - VG(rlrt, f)]
x In
Q(pfe|2 owQ
2e-2r
tf-f'wnt-Mt
t lat tr
i
at
s
By inspection, however, we notice that the last term in the above equation is the
magnitude squared of the conditional mean expression from Equation A.2:
|(4s(rslr, ro, tM)|G(rtlro,f), G(rjrt, f))|2
-(4)2
Jf2
jfc+B/2
fc -B/2
fc:-B/2
IJi~iL
Vsf
x[k 2k'2(rn(rt))(r,,(r'))G(rtlro,
(A.8)
Vs
f)G(rlrt, f)G*(r'lro, f')G*(rlr',f')
+ (Fd(rt))(Fd(r'))VG(rtjro, f) - VG(rlrt, f) x VG*(r'ro, f') - VG*(rr', f')
+ k2 (r.(rt))(Fd(r'))G(rtro, f)G(rlrt, f)VG*(r'lro, f') - VG*(rlr', f')
+ k' 2 (r.(r'))(rd(rt))G*(r'ro, f')G*(rlr', f')VG(rtjro, f) - VG(rr,, f)]
| Q(f }|21Qf 2e-i27r(f-f')xt-tm) dt dVt df df'.
x
We can also note that the first term of Equation A.7 is the conditional variance
Var ( s (rsIr, roI tmM)|G(rt Iro, f ), C(rIrt, f ))
= (l4,s(rsjr,rotM)
2
(A.9)
G(rtlro, f), G(rlrt, f))
- |(4s(rsIr,ro, tM)IG(rtIro, f), G(rIrt, f))12
(47r)2
1fc+B/2 ~fc+B/2
If
fc-B/2
fc-B/2
f
IB
Vc(rs, zt)
Vs
x [k2 k'2Var(r,)G(rtIro, f)G(rIrt, f)G*(rtro, f')G*(rIrt, f')
± Var(Ld)VG(rtjro,
f) - VG(rlrt, f) x VG*(rtlro, f') - VG*(rlrt, f')
+ k2 Covar(FP, Pd)G(rtjro, f)G(rlrt, f)VG*(rtro, f') - VG*(rlrt, f')
+ k'2 Covar(IF, Fd)G*(rtlro, f')G*(rlrt, f')VG(rtlro,
x
1|Q(f)|21
2e-i27(f-f'Xt-t()
f) - VG(rlrt, f)]
d% df df'.
(A. 10)
Then, the conditional second moment can be expressed as
(l45s(rsjr, ro, tm)|2|G(rtjro,f), G(rlrt, f))
(A. 11)
= Var(4s(rsIr,ro, tm)|G(rtIro,f), G(rIrt, f))
+ I(Qs(rslr, ro, tm)|G(rtjro,f), G(rlrt, f))| 2,
and the total second moment as
(Ds(rs Ir, ro, tM ) 12)
=
=(Is
(rsIr, r0 , tM) 12|G(rtlro, f), G(rlrt, f)))
(A. 12)
(Var(Ds(rsIr,ro, tm)IG(rIro,f), G(rIrt, f)))
+ (I(Ds(rsIr,ro, tM)IG(rtIro,f), G(rrt, f))12 ),
where the conditional variance Var(@s(rslr,ro,tM)|G(rt~rof),G(rlrt,f)) and the
conditional mean ( 4 s(rslr, ro,tM)IG(rtlro, f), G(rjrt, f)) are as given in Equations
A.9 and A.2 respectively. This expression is easier to implement than Equation 2.16
in Section 2.2, which has a double integral over volume.
To derive the total variance based on the conditional expressions derived above,
we use the fact that the total variance is the difference between the total second
moment and the magnitude squared of the total mean
Var(Ps(rsIr,ro, tM))
(A.13)
= (IPs(rsIr,ro, tM)12 ) - I(Ds(rsIr, ro tM)) 12
By applying Equations A.12 and A.3, we find that the total matched filtered
variance is
Var(4s(rsIr,ro, ItM))
= (Var(4)s(rsjr, ro, tm)|G(rtjro, f), G(rlrt, f)))
+ (I(Ds(rsIr,ro, tM)IG(rtIro, f), G(rrt,f))|2 )
- I(((Ds(rsIr, ro, tm)IG(rtIro, f), G(rIrt,f)))| 2.
(A.14)
Equation A.14 is also easier to implement than the full expression for the total
matched filtered variance given in Equation 2.18, due to the fact none of its terms
have a double integral over the volume Vs. To note is also that the first term in
Equation A.14 is equivalent to the expression given in Equation 2.19:
(Var(4s(rslr, ro, tM)IG(rtIro,f), G(rlrt, f)))
(47r)2j
_fcB/
J
fc+B2
JJL
(A.15)
Vc(rs, zt)
x [k2 k' 2 Var(F.) (G(rt Iro, f)G(rIrt, f)G*(rtlro, f')G*(rIrt,
± Var(Fd)(VG(rtlro,
f'))
f) - VG(rlrt, f) x VG*(rtlro, f') -VG*(rlrt, f'))
+ k 2 Covar(]F, 1Fd)(G(rtlro, f)G(rlrt,
f)VG*(rtlro, f') - VG*(rlrt, f'))
+ k'2 Covar(1F, Pd)(G*(rtlro, f')G*(rlrt, f')VG(rtro, f) - VG(rlrt,
X
1 IQ(f)|used2
Eo
2ap-xr(i-a't
-t)
f))]
dVt df df',
and can be used to approximate the total matched filtered scattered intensity.
78
Appendix B
Second term of the time harmonic
total variance
In section 3.1 we showed that the time harmonic expression in Equation 2.21 provides a good approximation to the matched filtered scattered intensity. Equation
2.21, however, corresponds to only the first term of the total time harmonic variance,
given by Equation 2.20. Numerically, we haven't been able to show that the first
term of Equation 2.20 dominates the total time harmonic variance. While the third
term of Equation 2.20 becomes negligible for a fine range increment and for several
realizations of the ocean environment, the second term is independent of the number
of realizations, and even for a fine range increment, dominates or is comparable to
the first term. This was not the case for the second term of the total matched filtered
variance given in Equation 2.18, where both the second and the third terms have
been shown to be negligible (Figures 3-5 and 3-6). We believe that the dominance of
the second term in Equation 2.20 is an artifact of the time harmonic approximation,
which cannot approximate a finite duration pulse and so cannot resolve a scattering
patch. Implementing the time harmonic approximation is effectively equivalent to
using a continuous wave CW waveform of infinite duration. We show in this Appendix that while a 1 second long CW waveform renders results similar to the linear
frequency modulated (LFM), increasing the duration of the CW signal reduces the
range resolution and consequently overestimates the second term in Equation 2.18 for
the matched filtered variance.
The simulation in this appendix was performed for an actual New Jersey continental shelf environment approximately 80 m deep. Monte Carlo simulations using
actual sound speed profiles have been run to randomize the waveguide and to allow computation of the total moments. The depth and range increments used were
dz = 0.4 m and dr = 3 m, respectively. The single element source and receiver were
collocated and situated at 50 m in depth. The integration patch Vs extended over
the top 10 m layers in depth, from 100 m to 5,500 m in range and over a 3* angle in
azimuth. The center frequency and bandwidth were 415 Hz and 50 Hz, respectively,
for all waveforms considered.
LFM signal
LFM spectrum
0.04
2
0.02
0
0.5
1
1.5
Time (s)
CW signal
1
0
390
2
430
440
410 420 430
Frequency (Hz)
Gaussian spectrum
440
410 420 430
Frequency (Hz)
440
400
410
420
Frequency (Hz)
CW spectrum
I
0.5
1
1.5
Time (s)
Gaussian signal
2
0
390
400
1
1.5
Time (s)
2
390
400
0.5
-20
0
0.5
Figure B-1: Real part (left) and source spectrum Q(f) 2 for 1 second long LFM (top),
CW (middle) and Gaussian (bottom) waveforms.
We first wish to study the effect that the choice of waveform has on the simulated reverberation and in particular to determine whether the first term of the total
matched filtered variance dominates for all three waveforms. We consider the LFM
waveform typically used in experiments, along with the CW and Gaussian waveforms.
We keep the signal duration r constant to 1 second for all three waveforms. The real
80
part of the time domain signal and the corresponding normalized spectra
Q(f) 2
for
each waveform are depicted in Figure B-1. As expected, the CW and Gaussian waveforms have a much narrower bandwidth than the LFM. To obtain consistent results
for the simulated reverberation using all three waveforms, we normalize by the timebandwidth product TB. Figure B-2 shows that, indeed, the simulated reverberation
level using the first term of the matched filtered variance given in Equation 2.19, is
consistent for the three waveforms.
Simulated reverberation level: 10 realizations
-80
-LFM
-- CW
Gaussian
-100
.--
-120
-
-160-
-180
-1
0
1
2
3
4
Range (km)
5
6
7
8
Figure B-2: Simulated reverberation level using the first term of the matched filtered
total variance for 10 Monte Carlo simulations, after normalization by rB, for LFM,
CW and Gaussian 1 second long broadband pulses centered at 415 Hz and with 50
Hz bandwidth.
While we confirmed that the three waveforms give consistent results, we also wish
to verify whether the second and third terms of the full matched filtered variance
given in Equation 2.18 are negligible compared to the first term. Figure B-3 shows
the total matched filtered variance and its three terms, for 10 Monte Carlo realizations,
for the LFM, CW and Gaussian waveforms. For all three waveforms, the first term
dominates, while the second and third terms are negligible. As the range resolution
Ar =
,
where c is the sound speed and B is the bandwidth, the CW and Gaussian
Total variance terms: LFM
I
2
I
I
4
3
Range (km)
I--
5
Total variance terms: CW
..
2
!.. ... -.
..
4
3
Range (km)
5
Total variance terms: Gaussian
2
3
4
Range(km)
5
6
Figure B-3: Total matched filtered variance and its three terms, for 1 second LFM,
CW and Gaussian broadband pulses centered at 415 Hz with 50 Hz bandwidth.
pulses, with much narrower bandwidth than the LFM, offer a worse range resolution.
This explains why the plots of the CW and Gaussian variance terms in Figure B-3 are
smoother than the corresponding LFM plots. As the signal duration T increases, the
range resolution of the CW waveform decreases even further. The two peaks in the
second term found at the end points of the patch due to discontinuity of the medium
could, then, overlap for a certain value of
T,
causing the second term to be at the
level of the two peaks. The next part of the Appendix studies the behavior of the
second term as the duration T of the CW signal increases.
Second variance term: 1 realization
-80
Is CW
2s CW
-- 3s CW
-- 4s CW
5s CW
CW
7s CW
-100
-1206s
-140
-180
....
...
-160
0
2
4
Range (km)
6
8
10
Figure B-4: Second term of total variance (Equation 2.18) for 1 Monte Carlo simulation, for CW pulses of 1 to 7 second duration, centered at 415 Hz and with 50 Hz
bandwidth.
To study the behavior of the second term as the duration r of the CW signal
increases, we only need to consider one realization of the ocean waveguide, as the
average level of the second term has been found to remain constant across realizations.
We consider CW pulses of durations r = 1 to 7 s, and plot in Figure B-4 the second
term of the matched filtered variance (Equation 2.18) for each r, for one realization
of the ocean waveguide. The figure shows that, as the signal time duration increases,
the peaks at the edges of the patch become wider, and begin to overlap.
For a
hypothetical CW pulse of infinite time duration, even if wrap-around would not be
problem and the time window for the fourier transform was large, the two peaks
would still overlap and form a monotonous curve, causing an apparent dominance of
the second term.
Appendix C
Examples of calibration results for
each experiment and frequency
C.1
New Jersey, 415 Hz
Implementation to New Jersey data, 415Hz, ping fora2003jd133t143135.
Inverted parameters for ping fora2003jd133t143135 at 415Hz
0
10
20
30
40
Beam number
50
60
70
Figure C-1: Parameters obtained after fitting the model to
the data for each beam, in log scale. The red horizontal line
indicates the mean of the parameter values (log scale). Only
beams corresponding to relatively flat or downward sloping
bathymetry were included in the analysis. Endfire beams as
well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen
for calibration, as the model fails to predict well the data for
small ranges.
Measured normalized pressure level beam time series
Polar plot of measured normalized pressure level
-130
-130
-5
0
-140
-140
5
10
-150
-150
15
E
-160
20
-160
25
-170
-170
30
35
10
0
-10
Distance from receiver (km)
-180
20
-0.5
0
Sin of broadside angle
-180
0.b
Polar plot of measured normalized pressure level
Bathymetric map
0
-130
30
20
-50
-140
E
10
D
-100
1-150
E 0
-160
S-10
-150
-30
-20
10
0
-10
Distance from receiver (km)
20
3U
0
-170
-30
-20
10
0
-10
Distance from receiver (km)
20
30
-180
Figure C-2: New Jersey continental shelf: (top left) Polar plot of received normalized
pressure level by OAWRS at 14:31 EDT, on 13 May 2003. The grey diamond indicates
the source, located at 39.2312N, 72.8818W and operating at 390-440 Hz. The black
diamond indicates the receiver, the coordinate origin, located at 39.2465N, 72.8626W,
with heading 346'E. The black and magenta stars indicate the beams chosen for
analysis, corresponding to the original and mirror beams, respectively. (top right)
Received normalized pressure level as beam time series. The black stars indicate
the beams chosen for analysis. The three magenta stars correspond to the beams
analyzed in Figures C-3 and C-4. (bottom left) Bathymetric map of the region with
depth contours at 60, 80 and 100m. The grey and white diamonds correspond to the
source and receiver, respectively. The black and magenta stars indicate the beams
chosen for analysis. (bottom right) Polar plot of received normalized pressure with
overlain bathymetric contours.
Measured
vs simulatedreverberationlevelfor beam25
Measuredvs simulatedreverberationlevelfor beam 35
Measured vs simulatedreverberation level for beam 45
Measuredvssimulatedreverberation
levellor beam25
5
10
15
20
25
Rangealongoriginalbeam(km)
Measuredvssimulatedreverberation
levelfor beam35
Measured
vs simulatedreverberationlevelfor beam45
Measured
Measuredvs simulatedreverberationlevelfor beam25
S100
Bea limeure
Figure1
C-30 Comp
Measuredvs simulatedreverberation
levelfor beam35
1223
uris
of measurated
120
1-Simulated
ca-120
1-130
-
140-
-
120
13013
140
5-150
Measuredvssimulatedreverberation
levelfor beam45
12t5
- MeasuredBMesurreds
-
10-1-150
S-160
-160
-170
-
-180
--
c 160
-170
1-70
-
-
1SO-1-
5
1 2 15 25
Beam time(s)
3
355
10
15
20
25
Beamtime (S)
30
35
5
10
15
20
25
Beam time(s)
30
35
Figure C-3: Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original beam (middle), range from receiver along the mirror beam (bottom), for beams 30, 40 and 50,
indicated by magenta stars in Figure C-2 (top right).
88
-100
-100
-
-110
range-averaged
Measured:
range-averaged
Simulated:
levelfor beam45
Measuredvs simulatedreverberation
levelfor beam3!
Measured
vs simulatedreverberation
Measuredvs simulatedreverberationlevelfor beam25
-100
-
Measured:
range-avera
Simulated:range-averal
a- 110
c -120
C -120-
120
130
5 -130-
-130
.140
-140
-150
m150
150
S-160
-160
160
-170-
-170
too
-140 \
-170-
180
5
10
15
25
20
Beamtime(s)
30
35
5
10
15
10
5
35
30
20
25
Beamtime(s)
25
20
15
Beamtime (s)
30
Monopoletermsbeforeadditionfor beam45
Monopoletermsbeforeadditionfor beam35
Monopoletermsbeforeadditionforbeam25
range-averaged
Measured:
Simulated:
range-averaged
-0
aOriginal
0--70-
F
beam
Mirrorbeam
-80a-90
100
-120130E 140
5
10
15
20
25
Beam time (s)
30
30
20
25
15
Beamtime(a)
10
5
35
150
35
5
10
15
25
20
Beamtime(s)
30
35
Figure C-4: For the same beams as in the previous figure C-3: comparison of measured
and simulated reverberation time series after range/time averaging (top); comparison
of the monopole term before adding the reverberation intensities for the original and
mirror beams (bottom).
Histogram of errors
0.045
0.040.0350.030.0250.02
0,0150.010.0050"
-20
-1:
-10
0
dB
-5
5
10
15
Figure C-5: Histogram of data-model differences for Figure C-6.
89
Polar plot of measured reverberation levels
Bean time series of measured reverberation
-140
15
-150
20
0
-160
cO -
C-.
5
-0.4
-0.2
0
0.2
0.4
Sin of broadside angle
-30
-30
0.6
20
30
Polar plot of simulated reverberation levels
Beam time series of simulated reverberation
.15h
Ef
10
0
-10
Distance from receiver (km)
-20
-150
20-
E
0
-160
a
R
0
-0.6
-0.4
0
0.2
0.4
-0.2
Sin of broadside angle
0.6
0.8
10
0
-10
Distance from receiver (km)
-20
20
30
20
30
Polar plot of data-model differences
Beam time series of data-model differences
10
10
5
Z
0
0
-5
-10
|
-10
5
-20
-15
-0.6
-0.4
-0.2
0
0.2
0.4
Sin of broadside angle
0.6
0.8
-20
-30
-30
-20
-10
0
10
Distance from receiver (km)
Figure C-6: Beam time (left) and polar (right) plots of measured reverberation level
(top), simulated reverberation level (middle) and their difference (bottom).
C.2
New Jersey, 925 Hz
Implementation to New Jersey data, 925Hz, ping fora2003jd124t155725.
Inverted parameters for ping fora2003jd1 24t1 55725 at 925 Hz
70
60
minRIIMWMM 91
50
E
ca 400
a)
2 30
CU
20
10
U
20
30
50
40
Beam number
60
70
Figure C-7: Parameters obtained after fitting the model to
the data for each beam, in log scale. The red horizontal line
indicates the mean of the parameter values (log scale). Only
beams corresponding to relatively flat or downward sloping
bathymetry were included in the analysis. Endfire beams as
well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen
for calibration, as the model fails to predict well the data for
small ranges.
Measured normalized pressure level beam time series
Polar plot of measured normalized pressure level
0-130
00
-105
1
E
-150
E15
-30
20
11
-20
-10
0
10
Ee
Distance from receiver (km)
20
30
-18
-10-10
20
-5
1
30
-160
-18
-130
-1170
-170
-140
1-85
-200
-180
Sin of broadside angle
Bathymetric map
Polar plot of
0
o20
20
rceived normalized
-3
2
30L-5
10
-50
35
1
-305
-20
0-
-10
0
10
20
30
- 20
-0
0
1 0 . 20
3
1
3
100
-1016
lm
0 -20C
b150
-20
-0
a20
0
10
B
20
30
-200
0
2
the source, located at 39.2694N, 72.8630W and operating at 875-975 Hz. The black
diamond indicates the receiver, the coordinate origin, located at 39.3108N, 72.9116W,
with heading 120'E. The black and magenta stars indicate the beams chosen for
analysis, corresponding to the original and mirror beams, respectively. (top right)
Received normalized pressure level as beam time series. The black stars indicate
the beams chosen for analysis. The three magenta stars correspond to the beams
analyzed in Figures C-9 and C-10. (bottom left) Bathymetric map of the region with
depth contours at 60, 80 and 100m. The grey and white diamonds correspond to the
source and receiver, respectively. The black and magenta stars indicate the beams
chosen for analysis. (bottom right) Polar plot of received normalized pressure with
overlain bathymetric contours.
50
-100Measuredvs simulatedreverberatonlevelfor beam
Measuredvs simulatedreverberationlevel for beam 40
Measured vs simulated reverberationlevelfor beam 30
110
I
Simulated
130
-140
150
160
-17010
5
15
20
10
Range along originalbeam kmun
-
levelfor beam40
Measuredvs sinulatedreverberation
levelfor beam30
Measuredvs simulatedreverberation
Rangealongmirrorbeam(km)
levelfor beam50
Measuredvssimulatedreverberation
Neasured vs simulated reverberationlevel for beam 40
levelfor beam30
Measuredvssimulatedreverberation
100--
110
Smltd
c 120130140'
-150160170r
5
10
15
20
25
Beamtime(s)
30
35
-105
10
15
20
Beam lime(5)
25
30
35
5
10
15
25
20
Beamtime(s)
30
35
Figure C-9: Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original beam (middle), range from receiver along the mirror beam (bottom), for beams 30, 40 and 50,
indicated by magenta stars in Figure C-8 (top right).
Measured
vs simulatedreverberation
levelfor beam30
Measured
vs simulated
reverberation
levelfor beam40
Measuredvssimulated reverberation
levelfor beam50
-100
Measured:
range-averaged
-Simulated: range-averaged
~
Il0
100
Measured:
range-averaged
Simulated range-averaged
S110
S120
-120
S1301
140
O130
130
140
-140
-150
-150
-
-160-
-120
150
I -160
-160
-70r
-
-180
10
15
20
Beamtime(s)
25
30
170
-170--
180,
180-
10
Monopoleterms before additionfor bean 30
15
20
25
Beamtime(s)
30
Monopoletermsbeforeadditionfor beam40
-0
Originalbeam
Mirrorbeam
L 70
Measured:range-averaged
-Simulated: range-averaged
110-
--10
15
20
Beamtime(s)
25
30
Monopoletermsbeforeadditionfor beam50
-70
100
-20
-100
-10
-110
4-120
io
4120
-130
130
1_e1401
-150
-140
5
10
15
20
25
Beam time(s)
-
30
35
-150
10
5
15
20
25
Beam time(S)
30
-150-5
35
10
15
20
25
Beamtime(s)
30
Figure C-10: For the same beams as in the previous figure C-9: comparison of measured and simulated reverberation time series after range/time averaging (top); comparison of the monopole term before adding the reverberation intensities for the original and mirror beams (bottom).
Histogram of errors
0.045
0.040.0350.03
0.025
0.02
0.0150.010.005 0-20
-15
-10
-5
0
dB
5
10
15
20
Figure C-11: Histogram of data-model differences for Figure C-12.
Beam time series of measured reverberation
Polar plot of measured reverberation levels
-130
-130
3
-140
-140
-150
-150
2
E~
20
E
0
-160
-160
25
C-1
9
-170
30
-180
35
-0.2
0
0.4
0.2
Sin of broadside angle
0.6
-30M
-30
0.8
-180
-10
-20
u
10
20
Distance from receiver (km)
Polar plot of simulated reverberation levels
Beam time series of simulated reverberation
-130
5
-130
10
-140
-140
15
$?
-150
-150
0
20
E
E
20
25
-160
-160
0
C
-170
30
35
-30
-30
-180
-0.2
Sin of broadside angle
10
0
-10
Distance from receiver (km)
Beam time series of data-model differences
Polar plot of data-model differences
0
0.2
U.4
Ulb
-20
20
30
-180
20
5
15
10
10
10
15
5
0
E
0
-=20
E
-5
-5
~-10
-10
r5
-15
-0.2
0
0.4
0.2
Sin of broadside angle
0.6
0.8
-30 m
-30
-20
10
0
-10
Distance from receiver (km)
20
30
-20
Figure C-12: Beam time (left) and polar (right) plots of measured reverberation level
(top), simulated reverberation level (middle) and their difference (bottom).
C.3
New Jersey, 1325 Hz
Implementation to New Jersey data, 1325Hz, ping fora2003jd124t155135.
Inverted parameters for ping fora2003jd124t155135 at 1325Hz
70
60
5 50
E
40
CD0
.230020
10-
20
30
40
50
Beam number
60
70
Figure C-13: Parameters obtained after fitting the model to
the data for each beam, in log scale. The red horizontal line
indicates the mean of the parameter values (log scale). Only
beams corresponding to relatively flat or downward sloping
bathymetry were included in the analysis. Endfire beams as
well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen
for calibration, as the model fails to predict well the data for
small ranges.
Measured normalized pressure level beam time series
Polar plot of measured normalized pressure level
-130
-140
-
1(
1150
E
0
-150
E
-160
160
-170
-2(
-3(
-20
20
10
0
-10
Distance from receiver (km)
-0.5
-1
30
0.5
0
Sin of broadside angle
Polar plot of measured normalized pressure level
Bathymetric map
-180
-130
21
-140
-50
2(
-150
a
-100
S
-160
-20
10
0
-10
Distance from receiver (km)
20
30
-150
-2
-200
-
-170
-20
-10
0
10
Distance from receiver (km)
-180
Figure C-14: New Jersey continental shelf: (top left) Polar plot of received normalized
pressure level by OAWRS at 15:51 EDT, on 4 May 2003. The grey diamond indicates
the source, located at 39.2694N, 72.8630W and operating at 1250-1400 Hz. The black
diamond indicates the receiver, the coordinate origin, located at 39.3140N, 72.9185W,
with heading 117*E. The black and magenta stars indicate the beams chosen for
analysis, corresponding to the original and mirror beams, respectively. (top right)
Received normalized pressure level as beam time series. The black stars indicate the
beams chosen for analysis. The three magenta stars correspond to the beams analyzed
in Figures C-15 and C-16. (bottom left) Bathymetric map of the region with depth
contours at 60, 80 and 100m. The grey and white diamonds correspond to the source
and receiver, respectively. The black and magenta stars indicate the beams chosen
for analysis. (bottom right) Polar plot of received normalized pressure with overlain
bathymetric contours.
5
Measuredvssimulatedreverberationlevelfor beam30
Measuredvs simulatedreverberationlevelfor beam 40
Measuredvssimulatedreverberation levelfor beam30
Measuredvssimulatedreverberation
levelfor beam40
Measuredvs simulatedreverberatlon
levelfor beam30
Measuredvssimulatedreverberation
levelforbeam40
10
15
20
25
Beamtime(s)
30
35
5
10
15
20
25
Beamtime(s)
30
Measured vs simulatedreverberation level for beam 50
Measured
vs simulatedreverberation
levelfor beam50
35
5
10
15
20
25
Beamtime(s)
30
35
Figure C-15: Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original beam (middle), range from receiver along the mirror beam (bottom), for beams 30, 40 and 50,
indicated by magenta stars in Figure C-14 (top right).
Measuredvs simulatedreverberatonlevelfor beam40
levelfor beam30
Measuredvs simulatedreverberation
Measured:
range-averaged
Stirmulated:
range-averaged
110
1
levelfor beam50
Measuredvssimulatedreverberation
Measured:
range-averaged
Simulated:range-averaged
-110
Measured:range-averaged
-Simulated: range-averaged
110
-120
120
-130
-130
-140
-140
-140
- -160
7@-160
-170
- 170
1-170
10
-180
120
130
-150
-150
3-160
15
10
20
Beamtime(s)
25
30
Monopoletermsbeforeaddition
for beam30
10
15
20
25
Beamdme (s)
-o180
30
10
15
20
Beamtime(s)
30
25
Monopoletermsbeforeaddidon
for beam40
-70
Mrorba
-80is-90
L
100-tl'
ki
-1101i
1 -120
z
5
10
15
20
25
Beamtime (s)
30
35
- 130o-140-150
5
10
15
20
25
Bearntime(s)
30
5
35
10
15
20
25
Beamtime(s)
30
35
Figure C-16: For the same beams as in the previous figure C-15: comparison of
measured and simulated reverberation time series after range/time averaging (top);
comparison of the monopole term before adding the reverberation intensities for the
original and mirror beams (bottom).
Histogram of errors
0.05
0.04
0.03
0.02
0.01
0
-20
-15
-10
-5
0
dB
5
10
15
20
Figure C-17: Histogram of data-model differences for Figure C-18.
Beam time series of measured reverberation
Polar plot of measured reverberation levels
5
-130
-130
30
-140
-140
15r
10
150-150
E
-160
25-
-160
-10
-170
30
-0.2
0
0.2
0.4
Sin of broadside angle
0.6
0.8
-170
-20
-180
-30
-30
-130
30-130
Beam time series of simulated reverberation
-20
-10
0
10
Distance from receiver (km)
20
30
-180
Polar plot of simulated reverberation levels
10
5-150
m-1-160
E
200
~
-zt
M
-160
25C
0
-6
1
30
35
-0.2
0
0.2
0,4
Sin of broadside angle
0.6
0.8
-170
-20
-180
-30
-30
-170
-20
Beam time series of data-model differences
5
-10
0
10
Distance from receiver (km)
20
30
Polar plot of data-model differences
20
30
-20
15
10
15
20
10
10-15
7;
10
5
-20
E
ca
-10
--
0
-5
-10
30r-
-0.2
0
-5
r
E
35
5
0
0
25S
0.2
0.4
Sin of broadside angle
0.6
-180
15
0.8
-20
-10
20
-20
-15
-30
-30
-20
-10
0
10
Distance from receiver (km)
20
30
-20
Figure C-18: Beam time (left) and polar (right) plots of measured reverberation level
(top), simulated reverberation level (middle) and their difference (bottom).
100
C.4
Georges Bank, 415 Hz
Implementation to Georges Bank data, 415Hz, ping fora2006jd269t165140.
Inverted parameters for ping fora2006jd269t1 65140 at 415Hz
70
60
50E
.
- ...
.40
0
10
Ca
0
20
25
30
35
Beam number
40
45
50
Figure C-19: Parameters obtained after fitting the model to
the data for each beam, in log scale. The red horizontal line
indicates the mean of the parameter values (log scale). Only
beams corresponding to relatively flat or downward sloping
bathymetry were included in the analysis. Endfire beams as
well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen
for calibration, as the model fails to predict well the data for
small ranges.
101
Polar plot of measured normalized pressure level
Measured normalized pressure level beam time series
-130
-130
300
-14
-140
-140
-8
-150
1
0-150
-
20
10-
2
0-160
20
25
010
-160
30
-170
10
2
0
-20
-10
0
10
20
Distance from receiver (km)
30
Bathymetrdc map
-180
-1
level bO
-0.5
S
0
0.5
Sin of broadside angle
t 1:
ed on
1
-180
26 Seem
3013
16
-104
6
-130
Q
-250
015
10-100
3
T
0
-10
-140
1
4
-200
0
B
-10
0
0
170
3
-35
-20
beas,
esectvey.
to riht Reeiedoor
map of0th region30
-400
Distance from receiver (km)
alie ploro
essure nomlee presur beame
-20 -10
0 a0
20
30
Distance from receiver (km)
im
-180
Figure C-20: Northern flank of Georges Bank in the Gulf of Maine: (top left) Polar
plot of received normalized pressure level by OAWRS at 11:58 EDT, on 26 September
2006. The grey diamond indicates the source, located at 41.8901N, 68.2134W and
operating at 390-440 Hz. The black diamond indicates the receiver, the coordinate
origin, located at 41.8212N, 68.3368W, with heading 137'E. The black and magenta
stars indicate the beams chosen for analysis, corresponding to the original and mirror
beams, respectively. (top right) Received normalized pressure level as beam time
series. The black stars indicate the beams chosen for analysis. The three magenta
stars correspond to the beams analyzed in Figures C-21 and C-22. (bottom left)
Bathymetric map of the region with depth contours at 100, 150 and 200m. The grey
and white diamonds correspond to the source and receiver, respectively. The black
and magenta stars indicate the beams chosen for analysis. (bottom right) Polar plot
of received normalized pressure with overlain bathymetric contours.
102
for beam 25
level
level for beam42
Measuredvssimulatedreverberation
Measuredvs simulatedreverberationlevel for beam 37
Measured vs simulated reverberation
Measured
-Simulated
iii
vi~
20
25
15
Rangealong originalbeam(km)
10
25
15
20
Rangealongoriginalbeam(km)
10
30
5
30
25
10
15
20
Range alongoriginalbeam(km)
30
Measured
vs simulatedreverberationlevelfor beam42
vs simulatedreverberation
levelfor beam37
Measured
levelfor beam25
Measuredvssimulatedreverberation
Measured
SImulated
-110
o -120
-130
-140
-150
-160
-170
z
5
-100
10
20
15
Rangealongmirrorbeam(km)
Measuredvs simulatedreverberationlevelfor beam42
Measuredvs simulatedreverberation
levelfor beam37
Measuredvs simulatedreverberationlevelfor beam 25
Measured
Simulated
110
-10
-
1
E:Simulated
-110
o -120
M -120
-130
-130
140
-140Li
-150
-150
10
S-160
-170
z
-10
10
15
25
20
Beamtime(s)
30
35
0
15
20
25
Beam time(s)
30
35
10
15
20
25
Beamtime(s)
30
35
Figure C-21: Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original beam (middle),
range from receiver along the mirror beam (bottom), for beams 25, 37 and 42 indicated
by magenta stars in Figure C-20 (top right).
103
Measuredvssimulatedreverberationlevelfor beam25
Measured
vs simulatedreverberation
levelfor beam37
100 - Measured:
range-averaged
110
-Simulated: range-averaged
100
SMeasured:range-averaged
Simulated:
range-averaged
110
-120-
.120
-130
-130-
-130
-140
140-
-140
-150
150
-160
160
-160-
70-
15
20
25
30
35
10
Beamtime(s)
-
-170
z
10
Measured:
range-averaged
Simulated:
range-averaged
IL -110
-120,
-170
Measuredvs simulatedreverberationlevelfor beam42
100
15
20
25
30
Beamtime(s)
Monopoletermsbeforeaddition
for beam25
10a'
35
10
Monopoletermsbeforeadditionfor beam37
15
20
25
Beamtime (s)
30
35
Monopoletermsbeforeadditionfor beam42
Beam60melt
-70
-70
-
Mirrorbeam
B
-80
-4-90
-40
-100
-
100
-110
110-
-120,
-120
-130
8-140
- 150
10
0140
15
20
25
Beam time(5)
30
35
10
20
25
Beamtime(s)
15
30
35
10
15
20
25
Beam time(a)
30
35
Figure C-22: For the same beams as in the previous figure C-21: comparison of
measured and simulated reverberation time series after range/time averaging (top);
comparison of the monopole term before adding the reverberation intensities for the
original and mirror beams (bottom).
Histogram of errors
0.03
0.02
0.01
0
-20
M
-15
-10
-5
0
dB
5
10
15
20
Figure C-23: Histogram of data-model differences for Figure C-24.
104
Beam time series of measured reverberation
Polar plot of measured reverberation levels
-140
-140
E
20
-150
-150
-2
E
0
E 25
-160
-160
9~
0-
-170
-170
-0.3
-0.2
0
0.1
0.2
-0.1
Sin of broadside angle
0.3
0.4
-180
-20
10
20
-10
0
Distance from receiver (km)
30
-180
Polar plot of simulated reverberation levels
Beam time series of simulated reverberation
10
15-
-130
-130
-140
-140
-150
-150
*
E
-160
-160
C
i-
-170
-170
-0.3
-0.2
0.1
0.2
-0.1
0
Sin of broadside angle
-180
0.3
-20
-10
u
10
-180
2u
Distance from receiver (km)
Polar plot of data-model differences
Beam time series of data-model differences
20
20
15
15
10
10
5
5
.
0
0
2
-5
*
-5
ir5
-10
C
-10
-15
-15
-0.3
-0.2
-0.1
0
0.1
U.2
0.3
0.4
-20
Sin of broadside angle
-20
20
10
0
-10
Distance from receiver (km)
30
-20
Figure C-24: Beam time (left) and polar (right) plots of measured reverberation level
(top), simulated reverberation level (middle) and their difference (bottom).
105
C.5
Georges Bank, 950 Hz
Implementation to Georges Bank data, 950Hz, ping fora2006jd278t025345.
Inverted parameters for ping fora2006jd278t025345 at 950Hz
70
60-a
550
E
E,
40
Q
0
.2 30-
-
10-
0
A
20
25
30
35
40
45
Beam number
50
55
60
Figure C-25: Parameters obtained after fitting the model to
the data for each beam, in log scale. The red horizontal line
indicates the mean of the parameter values (log scale). Only
beams corresponding to relatively flat or downward sloping
bathymetry were included in the analysis. Endfire beams as
well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen
for calibration, as the model fails to predict well the data for
small ranges.
106
......
..........
Measured normalized reverberation level beam time series
Polar plot of measured normalized reverberation level
-140
140
5-150
20
10
-170
-170
0
-40
010
-20
20
4-170
-10
60
Polar plot of measured normalized reverberation level
plot of received
-514
-40
-160
40
40#
06
-50
30
30
-140
-50
0.20
105
Distance
from receiver (kmn)
10
Ditn ofbromareier kn)
-106
-200
j5
-0-
-00
-16
0-30
-50
-50
-40
-20
-400 20 20
Distance from receiver (km)
-170
-40
-40--350
400
1400-40
-
4
20
-0-20
00
0
Distance from receiver (km)
40118
40
Figure C-26: Northern flank of Georges Bank in the Gulf of Maine: (top left) Polar
plot of received normalized pressure level by OAWRS at 22:53 EDT, on 4 October
2006. The grey diamond indicates the source, located at 41.9696N, 68.3507W and
operating at 925-975 Hz. The black diamond indicates the receiver, the coordinate
origin, located at 42.0776N, 68.2707W, with heading 153'E. The black and magenta
stars indicate the beams chosen for analysis, corresponding to the original and mirror
beams, respectively. (top right) Received normalized pressure level as beam time
series. The black stars indicate the beams chosen for analysis. The three magenta
stars correspond to the beams analyzed in Figures C-27 and C-28. (bottom left)
Bathymetric map of the region with depth contours at 100, 200 and 300m. The grey
and white diamonds correspond to the source and receiver, respectively. The black
and magenta stars indicate the beams chosen for analysis. (bottom right) Polar plot
of received normalized pressure with overlain bathymetric contours.
107
Measuredvs simulatedreverberationlevel for beam 30
M
Measured vs simulatedreverberaton level for beam 40
e vMeasured
v-
110-
Simulated
e
-10-
I
tt20
-120
Measured vs simulated reverberationlevel for beam 50
Measured
Simulated
3
-130
140
150
-
-160
-60
160
-170
-170-
-100
110 -27:
5
10
15
20
25
Rangealongoriginalbeam(km)
30
5
-130
130
-40-
30
-100
--
mMeasured
Simulated
r 110 a
~2
G 120-
Measured
Simulated
-130
40140
50
ri-150
-1616
Is
z
-170
~z
30
35
40
Range
along
mirror
beam
(km
Measured
vs
simulated
reverberation
level
or
beam
30
100
5
10
15
20
25
=110
Smlae
-150
-
1-160
-170
-10
-1705
10
15
20
25
30
35
40
Range
along
mirror
beam
(km)
Measured
vs
simulated
reverberation
level
for
beam
40
100
110
o 120
-Smltd
1SO
110
o 120
-130
-130
-140-
A40
e-150
-10
.2-160
-170-
10
15
20
25
30
Beam time(s
35
40
45
50
-150
-160-
-160
-170
-70
1,80
'-180
10
15
Smlae
CD-120-
130
m
5
10
15
20
25
30
35
Range
along
mirror
beam
(km)
Measured
vs
simulated
reverberation
level
for
beam
50
-100
-140
-100
10
15
20
25
Rangealongorigina beam(km)
Measuredvssimulatedreverberationlevelfor beam50
-2205
Somua
do 10
120
-180
-
Measured
vs simulatedreverberation
levelfor beam40
120
M
I
Simulated
150
-
-170
5
10
15
20
25
30
Rangealongoriginalbeam(km
Measuredvssimulatedreverberation
levelfor beam30
0
Mmasurmds
I
130
-140
S-150
-
120-
30
-140
Measured
.11G
20
25
30
35
Beam time(S)
40
45
50
-
10
-
15
20
25
30
35
40
45
50
Beam time(s)
Figure C-27: Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original beam (middle), range from receiver along the mirror beam (bottom), for beams 35, 45 and 50,
indicated by magenta stars in Figure C-26 (top right).
108
Measuredvssimulatedreverberation
levelfor beam50
Measured
vs simulatedreverberationlevelfor beam40
levelfor beam30
Measuredvssimulatedreverberation
-100
-100
Measured:
range-averaged
---Simulated:
range-averaged
Measured:
range-averaged
SImulated:range-averaged
110-
Measured:
range-averaged
Simulated:
range-averaged
110
-120-
e120
-130-
-130
-140
-140
i 50-
-150
-160
-160
-170
z
z
35
25
30
Beamtime(s)
20
15
15
46
40
20
30
35
25
Beamtime Is)
45
40
-170
-180
15
20
35
30
25
Beamtime(a)
45
40
Monopoletermsbeforeadditionfor beam50
Monopole
termsbeforeadditionfor beam40
for beam30
Monopoletermsbeforeadditilon
Originalbeam
a-70
-Mirror beam
D
7
irrba
-o
- 90
1
1-90
10-mMirrorbea
1ea
_14-40z
z
10
15
20
25 1 30
320 35
35
1025
Beam time(S)
40
40
45
4
50
50
1
10
15 1
20
25
35
30
Beam time(s)
40
45
50
10
20
30
40
Beamtime(s)
50
Figure C-28: For the same beams as in the previous figure C-27: comparison of
measured and simulated reverberation time series after range/time averaging (top);
comparison of the monopole term before adding the reverberation intensities for the
original and mirror beams (bottom).
109
Beam time series of measured reverberation
Polar plot of measured reverberation levels
-130
40
-130
30
-140
-140
20
-150
10
-150
-160
-160 -10
-160
EU0
-170
5-20
-170
-30
-0.2
0
0.2
0.4
Sin of broadside angle
0.6
-180
-40
-40
Beam time series of simulated reverberation
-20
0
20
Distance from receiver (km)
40
-180
Polar plot of simulated reverberation levels
-130
-140
20
-150
-
10
-150
-10
-160
E
0
-160
5 -20
-170
-170
-30
-0.2
0
-180
0.2
0.4
Sin of broadside angle
-401
-40
Beam time series of data-model differences
-20
0
20
Distance from receiver (km)
40
-180
Polar plot of data-model differences
20
15
10
V
-5
-
-10
-0.2
0
0.2
0.4
Sin of broadside angle
E
0.6
5
-40
-20
0
20
Distance from receiver (km)
Figure C-29: Beam time (left) and polar (right) plots of measured reverberation level
(top), simulated reverberation level (middle) and their difference (bottom).
110
....
.. ....
..
C.6
Georges Bank, 1125 Hz
Implementation to Georges Bank data, 1125Hz, ping fora2006jd276t223615.
Inverted parameters for ping fora2006jd276t223615 at 1125Hz
70
60 p 50E
CZ
'- 40
0.
0
300
10u
25
30
35
40
45
Beam number
50
55
Figure C-30: Parameters obtained after fitting the model to
the data for each beam, in log scale. The red horizontal line
indicates the mean of the parameter values (log scale). Only
beams corresponding to relatively flat or downward sloping
bathymetry were included in the analysis. Endfire beams as
well as beams contaminated by clutter were also excluded.
The second half of the data and model time series were chosen
for calibration, as the model fails to predict well the data for
small ranges.
111
Polar plot of measured normalized pressure level
Measured normalized pressure level beam time series
-130
bu
-130
-140
0-140
20
CD
150
-150
- 30
E
0
2
160
-160
c40
-170
-170
60
50
-5
-20
0
20
Distance from receiver (km)
Bathymetric map
-40
40
0 50.
-180
-1
-0.5
0
0.5
Sin of broadside angle
1
Polar plot of measured normalized pressure level
-50
-130
1-180
-140
-150
-150
0
--200
0
0
-250
C-D
-160
-170
-350
-40
-20
-50
0
20
Distance from receiver (km)
40
-00
-0
-40
-20
111110-180
0
20
Distance from receiver (km)
40
Figure C-31: Northern flank of Georges Bank in the Gulf of Maine: (top left) Polar
plot of received normalized pressure level by OAWRS at 18:36 EDT, on 3 October
2006. The grey diamond indicates the source, located at 42.2088N, 67.6892W and
operating at 1100-1150 Hz. The black diamond indicates the receiver, the coordinate
origin, located at 42.2305N, 67.8553W, with heading 177'E. The black and magenta
stars indicate the beams chosen for analysis, corresponding to the original and mirror
beams, respectively. (top right) Received normalized pressure level as beam time
series. The black stars indicate the beams chosen for analysis. The two magenta
stars correspond to the beams analyzed in Figures C-32 and C-33. (bottom left)
Bathymetric map of the region with depth contours at 100, 200 and 300m. The grey
and white diamonds correspond to the source and receiver, respectively. The black
and magenta stars indicate the beams chosen for analysis. (bottom right) Polar plot
of received normalized pressure with overlain bathymetric contours.
112
...
..
...
....
Measured vs simulated reverberation level for beam 35
Measured vs simulated reverberation level for beam 45
measurea
0
-
i
-110Simulated
~
~ CL-110
measurea
Simulated
imlae
-120-
-120-
-130
-130
-140
CD-140
t5 -150
CD-150
3 -160
3 -160
E -170
8
z
z
-100 ,
5
'
'-180
15
20
25
30
Range along original beam (km)
10
-170
'
5
35
30
10
15
20
25
Range along original beam (km)
35
Measured vs simulated reverberation level for beam 45
Measured vs simulated reverberation level for beam 35
-100
-100
-Measured
Measured
-110
-Simulated
- 1
Simulated
0
=L0
-120-
-o
c -120
-130
-130
a
Q
-140
-140
-150-
-D
8-150
0
0
3-160
3-160
e
-E -170L
-180
5
O
15
20
10
Range along mirror beam (km)
-180
25
- -Measured
-1101
Simulated
20
10
15
Range along mirror beam (km)
25
Measured vs simulated reverberation level for beam 45
100-
Maue
Simulated
-110
-120
-120
a.
a.
-130
20
130
-140
. 140
-150-
150Q
3 160-.
78 -160
-170 -180
5
0
Measured vs simulated reverberation level for beam 35
-100
S
-170
10
15
-170 -
..
-...
20
25
30
Beam time (s)
35
-10
40
10
15
20
25
30
Beam time (s)
35
40
Figure C-32: Comparison of measured and simulated reverberation levels plotted
against: beam time (top), range from the receiver along the original beam (middle),
range from receiver along the mirror beam (bottom), for beams 35 and 45, indicated
by magenta stars in Figure C-31 (top right).
113
Measured vs simulated reverberation level for beam 35
inn
-100
0-
Measured: range-averaged
Simulated: range-averaged
-110
Measured: range-averaged
Simulated: range-averaged
-L110
m -120
m-120
0
$
Measured vs simulated reverberation level for beam 45
-130
S-130
-140
-140
CD
Z
-150
( -150
cc
3 -160
-8 -160-
5 -170
9 -170
0
0
z
z
-180
15
20
25
30
Beam time (s)
35
40
15
Monopole terms before addition for beam 35
10
15
20
25
30
35
20
25
30
Beam time (s)
35
Monopole terms before addition for beam 45
40
10
Beam time (s)
15
20
25
30
35
40
45
Beam time (s)
Figure C-33: For the same beams as in the previous figure C-32: comparison of
measured and simulated reverberation time series after range/time averaging (top);
comparison of the monopole term before adding the reverberation intensities for the
original and mirror beams (bottom).
114
Beam time series of measured reverberation
Polar plot of measured reverberation levels
-130
-140
-140
20
10
-150
_
-160
-150
0
-160
-170
5-20
-170
-30
-0.1
0
0.1
0.2
0.3
Sin of broadside angle
0.4
0.5
-180
-401
-40
Beam time series of simulated reverberation
-20
0
20
Distance from receiver (km)
40
-
-180
Polar plot of simulated reverberation levels
-130
-140
-150
2:
-150
E
0
-160
-160
A0
-170
-0.1
0
0.1
0.2
0.3
Sin of broadside angle
-170
-180
0.4
-20
0
20
Distance from receiver (km)
Beam time series of data-model differences
Polar plot of data-model differences
20
20
15
15
10
10
5
.5
.(
0
-0.1
0
0.1
0.2
0.3
Sin of broadside angle
0.4
-180
0.5
0
-5
*-
-5
-10
5-
-10
-15
-
-15
-20
--
-20
0
20
Distance from receiver (km)
-20
Figure C-34: Beam time (left) and polar (right) plots of measured reverberation level
(top), simulated reverberation level (middle) and their difference (bottom).
115
116
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