Cramer-Rao Bounds for Matched Field Tomography
and Ocean Acoustic Tomography
by
Peter M. Daly
B.S.E.E., University of Rhode Island (1993)
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
I
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
~··'
MvAR 0 6 1997
February 1997
@ Peter M. Daly, MCMXCVII. All rights reserved.
Ln
The author hereby grants to MIT permission to reproduce and distribute publicly
paper and electronic copies of this thesis document in whole or in part, and to grant
others the right to do so.
Author ............................ """ " .....................
... ...............
Department of Electrical Engineering an Computer Science
January 15, 1997
Certified by ................... ..
......
....
....... .B.
Baggeroer
Arthur B.Baggeroer
Ford Professor of Electrical and Ocean Engineering
Thesis Supervisor
Accepted by ................
........
.•........
..
, ..........
Arthur C. Smith
Chairman, Departmental Committee on Graduate Students
Cram'r-Rao Bounds for Matched Field Tomography
and Ocean Acoustic Tomography
by
Peter M. Daly
Submitted to the Department of Electrical Engineering and Computer Science
on January 15, 1997, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
This paper demonstrates a technique for solving the Cramer-Rao lower estimation bounds of environmental parameters applied to Matched Field Tomography (MFT) and Ocean Acoustic Tomography
(OAT). MFT is a parameter estimation method which processes narrowband signals, using the interference pattern generated between elements in a sonar array. OAT, another estimation technique,
relies on acoustic travel times between a source and receiver; consequently, wideband signals are
used to provide high time resolution. OAT exploits signal coherence over a selected bandwidth,
while MFT does not. With knowledge of the Crambr-Rao bounds, one can determine the minimum variance attainable for an estimator of any environmental parameter, as well as determining
the coupling between any set of parameters in the ocean environment. This information is useful
for evaluating present estimation techniques, determining the feasibility and expected performance
of new estimators, and finding how changes in one parameter can affect the estimation of other
parameters in the ocean.
Attention was focused on modeling a range independent shallow water environment with a sediment
layer and hard bottom. For a source and receiver spaced 15 km apart, four sound velocity profile
parameters were estimated. A comparison was made between the relative performance of MFT and
OAT. At low SNR levels, OAT has superior performance over MFT. Above certain SNR levels, similar
performance was observed for both MFT and OAT. Under constant energy conditions, minimum
standard deviations decrease as signal bandwidth increases. Coupling between parameters appears
to be independent of SNR and inversion method (OAT vs. MFT), and only slightly influenced by
signal bandwidth. Parameter selection is very important in determining the CRB; improper selection
leads to artificially high estimation bounds.
This work was supported by the United States Navy, Office of Naval Research, under contracts
N00014-93-1-0774 and N00014-90-J-1725.
Thesis Supervisor: Arthur B. Baggeroer
Title: Ford Professor of Electrical and Ocean Engineering
Acknowledgments
First of all, I would like to thank my parents for teaching me an excellent work ethic. I would
like to thank Prof. Arthur B. Baggeroer for giving me this research topic, laying the foundation for
the work described in this thesis, and donating his old 486 for use as an X-terminal on my desk. I
would also like to thank Prof. Henrik Schmidt, who was always able to answer my quick questions,
and pointed me toward SuperSNAP when KRAKEN failed.
I would like to thank the students and staff of MIT's Ocean Engineering Acoustics Group for all
their help. Thanks to Pierre Elisseeff for showing me his KRAKEN modefile reading code. Thanks
to Brian Sperry and Kathleen Wage for cleaning up PRUFER, and to Brian again for showing me
how to calculate the Af for wideband acoustic propagation. I would like to thank Dr. Joe Bondaryk
for his help with power spectral estimation, as well as his insights on parameter coupling and SNR
calculation.
Thanks to the Office of Naval Research for sponsoring this work, through the NDSEG fellowship
program (Contract N00014-93-1-0774), and the ATOC program (Contract N00014-90-J-1725).
Finally, and most importantly, thanks to those civilian engineers in the United States Navy who
allowed me to rip their computers apart and load Linux on them, so I could run batch jobs at
night. Without their array of computational power, this thesis would never have been completed
in a timely manner. Thanks to Ken for approving this, and to Andy for supplying the first test
machine. Thanks to both Jim and Diane for keeping me focused. Thanks to Brian for pointing me
toward Bucker's implementation of Green's function.
Contents
1 Introduction
1.1
Roadm ap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2
Background Information . ...........
...
10
.....
10
1.3
1.4
2
9
....
....
....
1.2.1
Underwater Acoustic Propagation ...................
1.2.2
Normal Mode Theory ...............................
12
1.2.3
Green's Function ..................................
15
1.2.4
Matched Field Processing .............................
16
1.2.5
Ocean Acoustic Tomography ...........................
17
1.2.6
Matched Field Tomography ............................
19
Previous Work ................
.......................
21
1.3.1
CRB derivation ............
.......
21
1.3.2
Applications of CRB to MFT and OAT ...................
Summ ary . .. .. .. ............
.................
. .. .. ...
.. .. .. . ...
..
23
.....
26
Theory and Formulation
27
2.1
Crambr-Rao Bound .....................................
28
2.2
Mechanics of the CRB ...................................
28
2.3 Application of the CRB to MFT and OAT
2.3.1
2.5
................
Summ ary ....................
Environment
...............
30
32
...............................
Results
3.1
.......
Matched Field Tomography ............................
2.4 Ocean Acoustic Tomography
3
. . ....
35
.......
41
42
.. . .....................................
42
3.1.1
Source and Receivers ................................
3.1.2
Signal to Noise Ratio .....
3.1.3
Parameters ..........
. . . . . . . . . 44
Minimum standard deviation . . . .
Correlation Coefficients
3.3.1
.. . . .. .
High Parameter Correlation E:nvironment
4 Conclusions
4.1
Summary ...............
4.2 Contributions . ............
4.3 Future Work
.............
A Computational Procedure
A.1 Perturbation Magnitude .......
A.1.1 Application to test case ...
A.2 Frequency Spacing ..........
A.2.1 Calculating the optimal Af .
A.3 Normal Mode, Green's Function, and CRB calculation
A.4 Simulation Hardware ............................
68
. . . .. . . . .. . . . . 69
. . . .. . . . .. . . . . 72
. . . .. . . . .. . . . . 75
. . . .. . . . . .. . . . 75
. . . . . . . . . . . . . . 78
. .. . . . . . 79
List of Figures
1-1 Simplified acoustic environment.
. ..................
1-2 Simplified acoustic waveguide ....................
..........
11
............
12
1-3 Normal modeshapes and eigenvalues for a fictitious shallow water environment. . . .
15
1-4 Sample MFP ambiguity function[l]..........
18
... . . . . .
..........
.
1-5 Sample tomographic configuration .............................
19
1-6 Tolstoy's MFT scenario, using air-dropped explosive charges. . . . . . . . . . . . . .
20
1-7 Convergence zone scenario. ................................
24
3-1 Pekeris shallow water environment.
42
...........................
3-2 Shallow water environment under study, with four environmental parameters shown.
43
3-3 Normalized source signal, with bandwidths of 10, 50, and 100 Hz . ..........
45
3-4 CRB results for SNR of-20 dB ...............................
57
3-5 CRB results for SNR of-10 dB ...........................
....
3-6 CRB results for SNR of +20 dB. .............................
58
59
3-7 Correlation coefficients for SNR of -20 dB. . ..................
......
60
3-8 Changes to environment for high parameter correlation. . ................
61
3-9 CRB results for SNR of +10 dB, uncorrelated parameters. . ...............
62
3-10 CRB results for SNR of +10 dB, correlated parameters. . ................
63
A-1 Flowchart of computational process ............................
69
A-2 Variability of Green's Function ...........................
....
A-3 Variability of Green's Function ...............................
A-4 Linearity test for Green's Function ...................
70
71
.........
72
A-5 Plot of perturbations for parameter 1: water column sound speed, evaluated at 200
H z.
.. . . .. . . .. . . . .. . .. . . .. . .. . . .. . . . .. . . .. . . . . .. .
73
A-6 Plot of perturbations for parameter 1: water column sound speed, evaluated at 200 Hz. 74
A-7 Plot of group velocities for modified Pekeris profile. . ...................
76
A-8 Plot of minimum and maximum group slowness 1/u, for modified Pekeris profile...
77
A-9 Plot of difference between minimum and maximum group slowness 1/un for modified
Pekeris profile. .......................................
.
78
List of Tables
1.1 Signal types considered by Baggeroer and Borodin. . ...................
22
3.1
SNR levels at each hydrophone, and their corresponding noise variance values.
3.2
CRB results for Parameter 1: water column reference speed. . ..............
48
3.3
CRB results for Parameter 2: water column speed gradient. . ..............
49
....
46
3.4 CRB results for Parameter 3: bottom reference speed. . .................
50
3.5
CRB results for Parameter 4: bottom speed gradient.
51
3.6
Mean correlation coefficients for modified shallow water case, +10 dB SNR. ......
. .................
55
3.7 MFT correlation coefficient results for CRB shallow water case .............
56
3.8
OAT correlation coefficient results for CRB shallow water case. . ............
56
A.1 Selected perturbations for environmental parameters under study. ............
75
Chapter 1
Introduction
Long range acoustic source localization has always been a topic of interest for oceanographers
and engineers. There are many applications; ranging from monitoring over-the-horizon shipping
movements to tracking underwater vehicles.
Matched Field Processing (MFP) was developed to aid in source localization. Both simulated
and experimental results of MFP have proven to be very accurate. Typically, one needs only a few
vertical line arrays to determine the range, bearing, and depth of a sound source. Accurate results
depend on comprehensive knowledge of the ocean environment through which sound propagates.
Because of this, recent emphasis has shifted from source to environment estimation. In order to
evaluate objectively any estimator which is presented, theoretical lower estimation bounds must be
obtained.
This thesis describes the steps needed to calculate the Cramer-Rao lower estimation Bounds
(CRB), as applied to Matched Field Tomography (MFT) and Ocean Acoustic Tomography (OAT).
The Cramer-Rao Bounds find the minimum variance attainable for any unbiased parameter estimator. The objective of this thesis was to calculate the CRB for four environmental parameters of a
shallow water ocean environment, and provide a specific example of their implementation.
The CRB were calculated for signal bandwidths from 0 to 200 Hz, with a center frequency of
300 Hz. Correlation between these parameters was determined, as well as the effect of differing
signal to noise ratios (SNR) on the CRB.
At low SNR levels (-20 dB), OAT has superior performance over MFT. At high SNR levels (over 0
dB), OAT and MFT have coincident CRB; the performance is the same for the selected parameters.
Also, high SNR is needed to construct estimators for the gradients in both the water column and
bottom, since lower SNR yields high CRB.
The correlation between parameters appears to be independent of SNR and inversion method
(OAT vs. MFT), and only slightly influenced by signal bandwidth. Parameter selection is very
important in determining the CRB; improper selection leads to artificially high estimation bounds.
1.1
Roadmap
This thesis begins with a brief review of underwater acoustics, starting from Helmholtz' equation, running through normal mode propagation theory, and ending at Green's Function. From
there, Matched Field Processing, Matched Field Tomography, and Ocean Acoustic Tomography are
explained.
Chapter 2, Theory and Formulation,provides an overview of the Cramer-Rao lower bound, and
its application to both MFT and OAT. Assumptions about the structure of the propagating signal
are given with explanations.
Chapter 3, Results, outlines a shallow water propagation environment. Four parameters are
selected for CRB calculation, using varying SNR levels and signal bandwidths. The CRB for both
MFT and OAT are calculated, and results explained.
Chapter 4, Conclusion, summarizes the results and offers suggestions for future work.
The Appendix expands on computational issues surrounding calculation of the CRB. It furnishes
the reader with information required to duplicate results shown here.
1.2
1.2.1
Background Information
Underwater Acoustic Propagation
Sound propagation through the ocean involves three items: an acoustic source, the propagation
medium, and an acoustic receiver (see Figure 1-1).
An acoustic source can be anything which injects acoustic energy into the water. This includes
naturally occurring sources, such as sea life feeding and communicating, and artificial sources, such
as explosions, surface and submersible ships, and active sonar systems. Sources outside the water
(magma displacements, earthquakes, aircraft, and heavy vehicles operating on shore) can also project
sound into the water.
Acoustic
Source
Propagation
Channel
Acoustic Receivers
Surface ship
with towed array
------ Vertical
Line Array
4
___________~
--------------------------
-
;--------------------4-4
-- - - - - - 4 -----...
.04A- -- -AL
Submarine
with towed array
Bottom mounted
hydrophone arrays
Figure 1-1: Simplified acoustic environment.
An underwater acoustic receiver is usually a type of underwater microphone known as a hydrophone. Hydrophones vary in their sensitivities and frequency responses. To increase the received
signal to noise ratio (SNR), and to safeguard against hardware failure, hydrophones are typically
deployed in a geometric configuration known as an array. There is no predefined spatial arrangement
for an array, but placement of hydrophones or transducers is governed by the type of data one wishes
to extract from a received signal and the cost of design, implementation, and deployment. A one
dimensional a vertical line array (VLA) can resolve only the elevation angle of an arriving signal; it
cannot determine the azimuth. A two dimensional array (for example, a flat, "billboard" array), can
resolve both elevation and azimuth of an arriving signal, but cannot determine if the signal came
from the front or the rear of the array. A three dimensional array (for example, a ship mounted
spherical array) can completely resolve the elevation and azimuth angles of an incoming signal.
The ocean medium is the most complicated part of acoustic propagation. Sound is affected by
the properties of sea water, as well as the physical characteristics of the ocean. Sea water also has
sodium chloride, and trace quantities of other elements. These compounds affect sound propagation
by absorption of acoustic energy. Physical characteristics of the ocean, including depth of the water,
type of bottom, and average wave height also affect sound propagation. Temperature, density,
and pressure affect the speed of sound. Sediment layers can convert the acoustic compressional
water-borne wave into a combination of compressional and shear waves in the sediment, which can
be re-radiated into the water. The effect of these different environmental parameters on sound
propagation is still an active area of research.
When representing the ocean in an acoustic propagation problem, the physical characteristics
of the ocean are usually simplified. One creates a model to represent the major characteristics
of the ocean environment under study. If the model is overly simplistic, its validity comes under
doubt. Additional characteristics increase the robustness of the model, at the expense of increased
computation.
The first usual simplification is to assume horizontal stratification of the propagation medium.
One divides the water column into horizontal layers of varying thickness. Each layer has a discrete set of constant properties assigned to it, usually sound speed (both compressional and shear),
density, and attenuation. The next simplification is to assume the propagating medium is "range
independent;" it does not change between source and receiver. Unless one is modeling acoustic
propagation in a swimming pool, these two assumptions reduce the relevance of the acoustic model
significantly. Still, even these simplified models can be used to demonstrate characteristics of acoustic propagation. Specifically, these models show the waveguide nature of the medium and boundary
interactions between different layers.
1.2.2
Normal Mode Theory
Modeling the transmission of acoustic energy through the ocean can be accomplished by treating
the propagation medium as an acoustic waveguide [2]. Energy is transmitted through the medium
by means of a series of acoustic waves. One can visualize this by considering an ideal acoustic
two-dimensional (range and depth) waveguide, with a perfectly reflecting top and bottom and homogeneous interior.
Perfectly reflecting top boundary (e.g: vacuum or air)
Vertical
Line
Receiver
Array
Acoustic
Source
Perfectly reflecting bottom (e.g: basalt)
Figure 1-2: Simplified acoustic waveguide.
Energy transmitted through an acoustic waveguide can be modeled using normal modes, satisfying the Helmholtz equation[3], [4],
1
r
r Op(r, z)
r
Or
0
)+
1
Op(r, z)
p(z)
O
z (p z)
+
_2
c2(z) p(r, )
_
(r)(z
27rr
where r
=
range from source, in meters,
w
=
frequency (radians),
z
=
depth (meters),
z,
=
depth of source (meters),
c(z)
=
sound speed at depth z,
p(z)
=
density at depth z,
and p(r,z)
=
sound pressure at range r and depth z.
(1.1)
Equation 1.1 assumes a point source at depth z,, evaluated in polar coordinates, with a given
depth-dependent sound velocity profile c(z), and density profile p(z). This is a second order differential equation with a forcing term. Using the technique of separation of variables, a solution to
Equation 1.1 is
p(r,z) = 4(r)@(z).
(1.2)
Substituting this answer and re-arranging terms, one finds
-1 [r1dr(r• d(r)]
+ 1
4(r) [ dr dr /) + @(Z)
• (r)
d 1 dW(z) + c- 2 (z)
dz p(z) dzdz2 + C
(Z
[P(z)
P(Z)
= 0.
(1.3)
The first term in Equation 1.3 is a function of r, the second, a function of z. The only way for the
equation to be satisfied is for each term to be equal to a constant: kmI. Using this eigenvalue, the
terms can be rearranged to form
p(z)
[W)k 1 @m(Z)
c2 (Z)
1+
dz p(z) dz
= 0.
(1.4)
Boundary conditions must be specified to solve this differential equation. Each boundary condition has physical meaning. For instance, if one were to assume @(0) = 0, this would indicate
a perfectly reflecting boundary on the surface of the water column. Assuming the bottom of the
propagation medium to be at depth z = D, another boundary condition, d••()
dz
=D
= 0 would mean
a perfectly rigid bottom.
The form of Equation 1.4 satisfies a class of differential equations known as Strum-Liouville
(S-L) Equations[5]. Provided c(z) and p(z) are real functions, the equation reduces to a "proper"
(S-L) problem with homogeneous boundary conditions. Eigenvalue solutions are real, nonnegative
numbers, and eigenfunctions, or modeshapes, are real. There are an infinite number of eigenvalue/eigenfunction solutions. The norm of each solution, Cn, is a positive number and can be
calculated as
Cn =
/D42p(z)
(1.5)
dz.
Eigenfunctions can be scaled arbitrarily. If the eigenfunction, T(z) were scaled so the norm, Cn
is unity, then T(z) is normalized with respect to
.
Another feature of S-L problems is the
orthogonality characteristic of the eigenfunctions, with
D
o
qm(Z)
kp(z)
n(Z dz =
0
if
m : n.
(1.6)
The eigenfunction solutions form a complete and proper orthonormal set, provided '1(z) is normalized. Acoustic pressure at a point can be represented using a sum of weighted eigenfunctions,
p(r,z) = E
m=1
-Im(r)1@m(z).
(1.7)
The highest propagating mode has an eigenvalue equal to w/c. Higher order modes exist, but they
have imaginary eigenvalues. These represent non-propagating, or evanescent modes. If attenuation
were included in the analysis, the eigenvalues would be complex. The environmental model used
here neglects attenuation, and focuses only on propagating modes.
Normalized Modeshapes for Shallow Water Environment
0
. . .. ........
, .....
X
x
x
.
........
1.24 .................
10
20
1.22.
....
....
........
S..
30
40O
WaterEnvironment
Eigenvalues forShallow
1.26
1.2
. .I..
..-1..
.. .
-.
x
..
..
-
...
1.16...
..
1.14.
60
-..
70
ill
- i
80
I..li
i
x.......
1.12.
x
r__
i
1.1.
100
C0
0
2
4
6
8
12
10
Mode Number
14
16
2
4
6
20
18
8
10 12 14
ModeNumber
16
18 20
Figure 1-3: Normal modeshapes and eigenvalues for a fictitious shallow water environment. The
left plot shows normal modeshapes for a hypothetical 100 m deep environment, while the right plot
contains the eigenvalues associated with each normal mode. Note as mode number increases, the
eigenvalues decrease gradually, approaching w/c = 1.2566.
Figure 1-3 shows the first twenty modeshapes and eigenvalues for a shallow water acoustic waveguide, using a source frequency of 300 Hz.
1.2.3
Green's Function
Green's Function characterizes the effect of the ocean waveguide on the propagating signal. It
incorporates the position of the signal source and receiver along with the environmental characteristics of the propagating medium (speed of sound as a function of depth and range, as well as bottom
types and speed). Green's Function for normal mode propagation,
oo
G(r, zs, zr)
4zm(r, zs)Im(zr),
=
m=1
andia(r, z,)
=
kmr Trm(z,)
exp [j (kmr
(
+ 4)]
(1.8)
where
G(r, z,, zr)
km
Fm (z)
and
is Green's Function,
is the mth eigenvalue,
is the mth eigenfunction, evaluated at depth z,
zs
is the source depth, in meters,
zr
is the receiver depth, in meters,
r
is the distance from source to receiver, in meters,
exploits Equation 1.7 using the weighting function[6]. Green's Function acts as a transfer function,
describing the filtering effect of the propagating medium. The output of a convolution operation
using Green's Function and an acoustic source spectrum would yield the pressure field at the receiver.
With this Function, one can calculate the pressure field at any point in the water column, and also
determine the transfer function for the acoustic propagation channel.
1.2.4
Matched Field Processing
One item of interest in acoustic research is source localization. Given a signal received at a
hydrophone, one would like to be able to find where the signal came from. Applications include
Anti-Submarine Warfare (ASW) and acoustic oceanography.
Until recently, signal localization efforts were based on ray theory. One assumed sound traveled
from source to receiver through the water column, with its travel path influenced only by the speed
of sound in water. Multipath effects (sound bouncing from the top and bottom) interfered with
analysis and were deemed undesirable.
Instead of suppressing multipath, Matched Field Processing (MFP) exploits it. It uses the
information contained in all received acoustic energy in order to locate the sound source. MFP
starts by incorporating all known information about the propagating medium (sound velocity profile,
bottom type and depth, etc.) into an acoustic model. A narrowband, or CW source signal is
selected. Next, a grid of possible source locations is created. Using the acoustic model, Green's
Function is calculated for a narrowband signal propagating from each source to the receiver. The
simulated received signal is compared with the actual received signal in each case, with the resulting
correlation plotted as a function of simulated source position. This results in an ambiguity surface
(see Figure 1-4), with peaks representing areas of high correlation. By looking at the surface and
examining the highest peak, one would theoretically be able to localize the position of the signal
source.
MFP has shown great promise in simulation [7]. One of its largest drawbacks is its sensitivity
to the environmental parameters supplied to it. To simulate the propagation of sound accurately
one must have a complete knowledge of all aspects of the ocean environment. One must know the
speed of sound as a function of depth and range from source to receiver. Some parameters, such
as bathymetry and bottom type, are relatively constant through time, and can be cataloged for
simulation. Water temperature and currents, both of which affect sound propagation, change with
time.
Another problem lies with specifying the search grid for the source. The extremes of the grid can
be determined from a priori information, but the optimal spacing between sampling points must be
determined. If the spacing were too coarse, one could miss the actual location of the source (and its
correlation peak) entirely. Too fine and one wastes computational time.
Bucker's original MFP[8] used a conventional, or Bartlett beamformer on the received signal.
Conventional beamformers have wide main lobes; this provided a large peak which could be detected
by relatively coarse grid samplings. Unfortunately, the drawback to conventional beamformers is
their high (-13 dB) side lobe level. This resulted in several false peaks on the ambiguity surface.
Building on the work of T.C. Yang [9], Baggeroer et al. [1] solved this problem by replacing
the conventional beamformer with an adaptive, Maximum Likelihood Method (MLM) beamformer.
This reduced sidelobe levels considerably, eliminating spurious peaks in the ambiguity surface. The
narrow main lobe produced by the MLM beamformer was easy to miss during grid sampling.
Schmidt et al. [10] proposed a solution by changing the type of adaptive beamformer to a
Multiple Constraint Method, or MCM beamformer. This allowed the authors to specify the width
of the main lobe, while still keeping sidelobes down to a minimum. Using a wider mainlobe allowed
for coarser grid searches, thereby reducing the computational power needed to search for a peak on
the ambiguity surface.
1.2.5
Ocean Acoustic Tomography
Ocean Acoustic Tomography (OAT) is an older technique whose purpose is to acquire environmental information about the ocean. Before OAT, ocean parameters were measured by ocean-going
ships. These vessels crossed an area of interest and took measurements as they traveled through
pre-defined points in space. This technique had several drawbacks. First, ships were slow, and the
parameters which they measured changed with respect to time. Obtaining a "snapshot" of acoustic
conditions at one point in time was impossible. Next, ships were (and still are) expensive to operate;
AMBDR,ML
n
U-
o
will
20-
40106.0
105.0
60-
104.0
103.0
102.0
101.0
100.0
99.0
98.0
80-
97.0
100-
96.01
I
I
0
I
2
I
4
6
I
8
Range (km)
Limestone. H = 1 m. L = 10 m.
Figure 1-4: Sample MFP ambiguity function[1]. This shallow water example used an MLM beamformer to reduce erroneous sidelobes. The simulated source can be found as a peak at the center
of
the ambiguity function.
one could not expect to maintain a continuous presence to collect data.
Both the medical and geological communities have implemented methods of scanning areas which
are not easily accessible. Computerized Tomography (CT) employs a moving x-ray source
with
multiple receivers in order to reconstruct images of the internal structures of the body. Geologists
use sound waves and geophones to obtain information on the properties of the earth's crust.
Munk
and Wunsch [11] drew on tomographic applications to the medical and geological fields, resulting
in
OAT.
OAT is implemented by having an acoustic source ensonify a region of water. At the edge of the
region of interest are placed multiple acoustic receivers. Initially, researchers focused on recording
only the exact arrival time of acoustic energy at the receivers (see Figure 1-5). Measuring the travel
.71ý
Acoustic Source/Receiver
)
ARM
'JYK
Path between Source and
Receiver
~llh
Figure 1-5: Sample tomographic configuration.
time between source and receiver allowed them to map the sound speed as a function of position.
From there, information about the temperature of the water, or the presence of currents was derived.
In order to obtain precise temporal measurements, the locations of the source and receiver had
to be known to within an acoustic wavelength. To improve resolution in the time domain, wideband
signals were generated at the source. Typically, a deterministic FM chirp or M-sequence was used,
allowing easier detection at the receiver.
Initially, OAT focused on estimating sound speeds and current in large areas of the ocean. Munk
and Wunsch [12] established the temporal resolution needed to perform successful OAT, as well as
optimal center frequencies. Subsequent experiments (1981) focused on observing the "mesoscale"
eddy properties or the ocean, as well as large scale ocean circulation[13]. Later, the use of OAT for
measuring temperature changes in the ocean was proposed. [14] Measurement of ocean temperature
on a global scale required timing the arrival of sound from distant (thousands of kilometers away)
sources. A demonstration of the detection of acoustic signals over long ranges was conducted in
early 1991 [15], successfully showing global acoustic propagation was possible.
1.2.6
Matched Field Tomography
Matched Field Tomography (MFT) is similar to Matched Field Processing, except the input
and output are reversed. Instead of providing ocean environment information for acoustic source
localization, one uses a known source and receiver in order to derive environment information. As
with MFP, a range of possible parameter values is selected, and acoustic propagation through the
medium is simulated for each candidate parameter value. Correlation between simulated and actual
received signals results in an ambiguity surface, which theoretically give the values of each parameter
under study.
MFT has several advantages over OAT. The performance of OAT depends greatly on measuring
the travel time between signal sources and their receivers. The distance between source and receiver
must be known to within a fraction of an acoustic wavelength. Assuming a propagation speed of
1500 m/sec, and frequency of 300 Hz, an acoustic wavelength would be 5 meters. Even with the
aid of the Global Positioning System (GPS) and fixed moorings, establishing the exact distance and
assuring it remains constant would be extremely difficult.
MFT does not depend directly on ray arrival times from source to receiver; rather, it exploits the
constructive and destructive interference between the receiver array. In order to do this, multiple
element receiver arrays must be used, and the geometry of the array must be known. This brings
an added level of complexity over OAT, but eliminates the need to precisely measure the distance
between source and receiver.
Air-dropped explosive charges
Vertical Line Arrays
I
I
I
"'1Z
r
r
m J
J
1
p
~~I·
7
~--~-
Z
Figure 1-6: Tolstoy's MFT scenario, using air-dropped explosive charges.
To date, experiments demonstrating MFT have not been as numerous as those using MFP and
OAT. In 1991, A. Tolstoy proposed using air-dropped explosive sources for MFT (see Figure 1-6) [16]
[17]. Unlike OAT, MFT requires only approximate knowledge of source location; stationary, moored
sources are unnecessary. Further, deploying multiple air-dropped explosive sources is cheaper than
sending out a ship to deploy a fixed array, and using multiple sources provided better results than
a single fixed source. Using computer simulations, Tolstoy was able to estimate a three dimensional
sound speed profile, with errors of less than 0.2 meters/sec.
One of the main problems with MFT is the sheer number of environmental parameters which
exist in the ocean. For example, sound speed changes with respect to depth and range, and is
affected near the surface by currents, season, and time of day. It is difficult to determine what effect
any parameter in question will have on the received signal, if any.
One method of reducing the number of unknown parameters is to decompose a sound velocity profile into a set of eigenvalues and eigenfunctions. To do this, one chooses a baseline sound
velocity profile from past measurements or other historical data. Then, using a Karhunen-Lobve
expansion, one forms a set of empirical orthogonal basis functions (EOFs) which describe the sound
velocity profile [18] [19]. This reduces the number of unknown parameters to a set of EOF weighting
coefficients.
1.3
Previous Work
1.3.1
CRB derivation
Much has been published on estimation methods for OAT, but little has been done in deriving
the CRB for OAT. The bulk of the work has been accomplished by two authors: Arthur Baggeroer
of MIT, and V. V. Borodin of the Andreev Acoustics Institute of the Russian Academy of Sciences.
Both have focused on the CRB for sound velocity profile (SVP) estimation.
Borodin[20] started by referencing a baseline SVP derived from historical observations. The
difference between the baseline SVP and the "true" SVP was characterized by a set of eigenvalues
and eigenvectors, similar to the EOFs of a sound velocity profile, with
c(x) = co(Z) + 4(x),
where
co(z)
is the known reference profile of the sound velocity
(depth dependent only),
and
Z(x)
is an unknown perturbation,
(dependent on position and depth), and
(1.9)
4(x) =
CCoa (x),
where
C,
is an eigenvalue to be estimated,
and
yca
is an eigenfunction, supplied through normal propagating modes.
(1.10)
The eigenvectors were taken from the normal propagating modes. This reduced the estimation
problem to solving for the unknown eigenvalues. One needed only to find the CRB for these values.
Baggeroer[21] took a more general approach in deciding which environmental parameters needed
to be estimated. Rather than attempting to estimate all perturbations from a baseline SVP, he
selected a series of parameters and placed them in a vector of quantities to be estimated. These
parameters pointed to specific perturbations of the SVP.
Both authors did not assume the position of the source signal was known. Furthermore, both
assumed a vertical line array received the signal. The sound propagation between source and receiver
was described using Green's Function. In the frequency domain, Green's Function acted as a filter
between the signal source and the receiver. Both authors choose to model the received signal by
multiplying the source signal with Green's Function, summed with a noise term. Baggeroer's signal
model,
R(f) = b(f)Ss(f)G(f,a) + N(f,a),
where
a
G(f, a)
(1.11)
is a vector of the unknown parameters,
is a vector of Green's Function for propagation to the
receiver array,
S, (f)
is the Fourier transform of a coherent source signal,
b(f)
is a random process incorporating amplitude
and phase variability,
and
N(f, a)
is a stationary noise vector with spectral covariance
matrix K,(f, a),
followed this idea. Three different source signal types were considered by both authors (see Table 1.1).
The first type of signal: a known, deterministic type, was only considered by Borodin. This signal
Signal type
Known signal
Known magnitude, unknown phase
Baggeroer
Random signal
Borodin
/
/
/
/
Table 1.1: Signal types considered by Baggeroer and Borodin.
was not representative of true ocean conditions, but was used to demonstrate the mathematical
method. Had Baggeroer considered this type, his signal model would have been modified to set b(f)
to 1 and S,(f) to the source signal. The second type of signal (known magnitude, unknown phase)
was a better representation of the type of signal used with OAT. The phase was unknown due to
the difficulty of obtaining the exact propagation distance between the signal source and receiver.
Candidate signal types in this category included M-sequences and FM slides. Implementation of
this signal type was accomplished in Baggeroer's model by setting S,(f) to the signal, and b(f) to
an unknown scalar random variable.
The final class of problems focused on a source signal which was a random process. This type of
signal was typically found in applications of matched field tomography and source localization. In
order to apply Green's Function as a filter on the signal, one assumed the signal was a stationary
random process in the wide sense and used the Wiener-Khinchine theorem. [22] It was implemented
in the signal model by setting S,(f) to 1, and assumed b(f) was a random variable with a power
spectral density of Sb(f).
Borodin approached the CRB derivation problem by first obtaining a Maximum Likelihood (ML)
estimator. [23, 24] With enough data the ML estimator has been shown to be both unbiased and
efficient[19]. From the model of the received signal, one obtained a probability density function
(PDF) which had the unknown quantities to be estimated as nonrandom parameters. This PDF
was differentiated with respect to the unknown parameters, and set to zero. Then the unknown
parameters were solved for. The covariance matrix produced by the ML estimator was asymptotically
equal to the Fisher Information Matrix used to determine the CRB.
1.3.2
Applications of CRB to MFT and OAT
Borodin published a second paper which applied the CRB derivations to OAT [25], His objective
was to show what information was required in order to obtain varying levels of precision with
OAT. He first dealt with the case of estimating "global inhomogeneities," that is, reconstructing the
ocean environment with precision less than the typical convergence zone distance. A typical deep
ocean environment refracts downward propagating sound upward. After tens of kilometers, sound
converges again at the surface (hence the term, "convergence zone") is reflected, and propagates
again (see Figure 1-7). Borodin proved one needed to measure only the arrival times of normal
Deep water
sound velocity---------.................-----------------------.................----------..........
Convergence
profile
/
Zones
u-
Sound
Propagation
Paths
------------------------------------------Annn -
Acoustic
Source
/--- Bottom
1500
15401
1520
1560
(m/sec)
Figure 1-7: Convergence zone scenario. In deep water ocean environments, sound is refracted to the
surface, providing "convergence zones" of acoustic energy.
waves to successfully reconstruct the ocean environment in this manner. Only the travel times
for deterministic signals (or the time differences for stationary random signals) carried information
about the environment.
Next, Borodin solved for "small-scale inhomogeneities." By applying his previous derivation for
CRB to this problem, he was able to determine what data would be necessary in order to obtain
tomographic resolution greater than a typical convergence zone width. There it was shown the
ability to reconstruct the environment adequately depended on the location of the shadow zones.
Inside the shadow zone, reconstruction failed, while outside it succeeded. In order to succeed at
all depths and ranges, it was determined that several transmitters and receivers would be required,
placed at varying depths and orientations. This contrasted with the typical point source transmitter
and vertical line array (VLA) receiver.
Krolik and Narasimhan[26] used Baggeroer's CRB derivations to find the CRB for estimating a
depth dependent temperature profile in the Pacific ocean. The environment under consideration had
areas of "mesoscale variability," that is, the temperature profile differed on a scale of 100 km. The
objective of ATOC was to measure the temperature profile as a whole, so these areas of variability
had to be neglected. Krolik and Narasimhan decided to solve the CRB for cases where the receiving
vertical array did not span the entire water column. They concluded a priori knowledge of mesoscale
variations should reduce the CRB, as well as increase the number of sensors on the VLA.
Another paper submitted by Narasimhan and Krolik[27] derived the CRB for source range estimation in a coastal New England environment. Here, they decomposed the environment into a
set of EOFs, and calculated the CRB for varying SNR levels. They found the CRB diverged even
when the number of propagating modes was greater than the number of environmental unknowns,
unless one had a priori information on the statistics of the unknowns. Their results suggested source
localization could be improved significantly from contemporary methods.
Schmidt and Baggeroer[28] took the application of CRB one step further. They analyzed a
hypothetical shallow water environment, and solved for the CRB of several parameters. Schmidt
focused on the coupling between different parameters, and their effects on parameter estimation.
He concluded that some errors in adaptive parameter estimation algorithms can be attributed to
parameter coupling.
For example, rudimentary forms of MFT typically use a grid search algorithm to find the optimal
parameters for the supplied environment. One establishes a range in which to search, and the grid
resolution, or spacing, for each search. If only one parameter is desired, the grid is one-dimensional.
For two parameter estimation, the search grid is two dimensional, and so on. Each point on the
search grid corresponds to an ocean environment with "candidate" parameters. Acoustic propagation between the source and receiver is simulated using these "candidate" parameters. Correlation
between simulated and actual results is calculated, and plotted on an ambiguity function. The parameter set (grid point) with the highest degree of correlation is assumed to be the correct parameter
set.
Adaptive forms of MFT improve on the basic algorithm by limiting the search grid range, and
adjusting the grid resolution to recursively focus on solution points. Ideally, this allows one to
efficiently find the peak of an ambiguity function. Unfortunately, without appropriate a priori
information, an adaptive algorithm incorrectly limits the grid size, and converges on the wrong
peak. Schmidt suggested if consideration of parameter coupling were included in adaptive search
algorithms, the accuracy of peak localization would improve considerably.
1.4
Summary
This chapter has served as a review in basic underwater acoustics, normal mode propagation, and
Green's Function. Armed with this knowledge, the concepts of Matched Field Processing, Matched
Field Tomography, and Ocean Acoustic Tomography and their relationships to underwater acoustics
were explained. Additionally, a review of current literature in the application of the Cramer-Rao
lower bounds to these items was presented.
The next chapter shifts emphasis to the CramBr-Rao bounds themselves. It starts by describing
nonrandom parameter estimation, then explains the CRB applied to a scalar parameter. From
there, the application of the CRB to MFT and OAT is expressed in mathematical form. These two
chapters provide background for Chapter 3, which explains the simulations carried out in this thesis.
Chapter 2
Theory and Formulation
This document assumes the reader is familiar with the concepts of parameter estimation of nonrandom quantities. Kay[24] has written an excellent tutorial on estimation theory; this can be
used by the reader as a reference. Other supporting sources of information are Papoulis[23], and
van Trees[19]. These texts review the concepts of estimation, as well as the mechanics involved in
deriving and implementing nonrandom parameter estimators.
2.1
Cram6r-Rao Bound
The Cram6r-Rao lower Bound (CRB) finds the theoretical absolute lower bound on the covariance
of an unbiased estimator. It does not attempt to find the actual Minimum Variance, Unbiased (MVU)
estimator; but an estimator whose variance equates the CRB would be the MVU estimator. For a
scalar parameter, the CRB can be found by evaluating
Aa(a) >- 1
(2.1)
J- (a)
and
aJ4(a) = E (
where
lnp(Y;a))2
I
,
a
is the quantity to be estimated,
y
is observed data used as input to the estimator,
p,(Y; a)
(2.2)
is the probability density function of observed data y,
taking into account parameter a,
and
Jy (a)
is the Fisher Information quantity,
Aa (a)
is the variance of the estimated quantity, &.
Examination of equation 2.2 shows the derivative is taken with respect to the parameter, instead
of the observed quantity. Most probability density functions (PDF) are plotted with respect to an
observed quantity. It is intuitively easier to take the derivative with respect to the parameter if the
PDF were plotted with the parameter as the independent variable. The logarithm in equation 2.2
may not be possible to evaluate at all instances of a. If the argument of the logarithm were undefined
or zero, it would not be possible to calculate the CRB.
2.2
Mechanics of the CRB
Although equations 2.1 and 2.2 are adequate for calculating scalar CRB, extensions to the equation exist for calculating the CRB of several parameter at once, from sets of observed data. While
the details of deriving the CRB equation are beyond the scope of this thesis, the resulting formulas,
Aj,(a) ý [Jy'(a)] ,
(2.3)
and
(2.4)
[Jy(a)]ij = -E [02 lnpy(Y; a)]
where
a
is a column vector of parameters to be estimated:
a= [al a 2 ... a ],
Y
is a column vector of observed data for use as an
YN] T ,
input to the estimator: Y = [yl Y2 ...
py(Y; a)
is the probability density function of observed data Y,
taking into account parameter vector a,
and
Jy (a)
is the resulting Fisher Information Matrix,
A&(a)
is a matrix which describes the variance and covariance of the
estimated parameters,
are provided for reference. If one assumes the noise component the observed signal to be Gaussian
in nature,
(2.5)
y - N (p(a), K(a)) ,
then the properties of the distribution are exploited to better determine the CRB. Knowing a
Gaussian distribution can be completely characterized by its first two moments, the CRB expression,
[Jy(a)]i
(a)
t K-'(a)
+ 1 Tr K-'(a)
2 1
.
a
(2.6)
(
K-'(a)> K (a
i oaj
can be rewritten in a simpler form. Again, the details of deriving the CRB are excluded [24], but
equation 2.6 is provided for reference. This equation assumes a real mean and variance. Complex
parameters can be considered by removing the factor of 1/2 from the expression.
2.3
Application of the CRB to MFT and OAT
This thesis focuses on calculating the CRB for several parameters of the ocean environment.
Prof. A. B. Baggeroer derived the original equations for calculating the CRB applied to MFT and
OAT [29]. This derivation is summarized below.
In order to derive an equation for the Cram6r-Rao bounds (CRB) applied to both MFT and
OAT, one must start by making assumptions about the characteristics of the received signal. First,
and most importantly, one limits the analysis to zero mean signals which are embedded in Gaussian
white noise. Next, one allows for complex statistics in the Gaussian distribution. This allows one to
solve for elements of the Fisher Information Matrix (FIM) using
[Jr(a)]ij =
Tr [Kr- (a)0 r(a
Kr-'(a)
aj
(2.7)
a simplified version of the CRB equation. The input of Equation 2.7, K,(a) is a standard covariance
matrix formed by taking the expected value of the observed input, R, multiplied by its complex
conjugate,
Kr(a) = E [RR t ].
(2.8)
One solves the CRB for a specific signal bandwidth. This requires one to incorporate a frequency
dependence into the CRB equation. This is handled by the R vector, which contains the received
signal for each sensor and frequency of interest.
The signal at the receiver array is acquired initially in the time domain. In order to convert it
to the frequency domain, a periodogram type of spectral estimation is assumed. This is performed
on the time-domain signal at each receiver. The result is a spectral estimation of the received signal
at each sensor.
One starts the spectral estimation process by taking the zero mean received signal at sensor n,
rn(t), and calculating its short time Fourier Transform,
Rn(f ITo)
+/2rn(t)e-j2-7ftdt.
f - To/2
(2.9)
Next, one estimates the received signal Power Spectral Density (PSD) using the periodogram
method. The result is an estimate of the power spectral density of the received signal for sensor n,
during observation time T,.
Recall the correlation function for a time domain signal,
Sr(v)e-i 2 rv(t-t2)dv.
E[rn(t)r( 2)] = Rr(t - t 2 ) =
(2.10)
One incorporates Equation 2.10 to estimate the PSD,
E [Rn(f1lTo)R(f
2
To)]
-=
n f0-/2 f+0 Sr(v) [
S-f-To/2
ITo
S
/2E [r n(tj) r
2)]e-j
i t -f2t2)dt l dt ,
1(f f(t
2
2
0/2
JToej2r(v-f )tIdt1
/
ej2f(v-f2)t2dt 2
Sr(v) {sinc [27r(v -
Next, one assumes Sr(v) is "smooth" in intervals of
-L.
fl)
(2.11)
dv,
2
}sinc
2r(v - f2)J
d.
If Sr(v) were relatively constant, it would
be taken outside the dv integral,
E[R(fi|To)R(f ITo)]
2
= Sr(fi)
sinc [2r(v
-
fT)
sinc [2r(v - f2)] dv, (2.12)
SSr(f) sinc [2(fi - f2) •
If fi were equal to f2, then Parseval's theorem would be used to reduce the expression,
E [R, (fl To)R~L(fTo)]= S()(2.13)
(2.13)
to a single term. This formulation can be extended to the vector case, where one has a set of N
sensors receiving a signal, R. The power spectral density estimate, Sr, would be a matrix instead
of a scalar. It includes cross terms describing the frequency correlation between receiving sensors.
The vector form,
E [R(fITo)Rt(f ITo)] =
results in a covariance matrix biased by To.
(f)
(2.14)
At this point, a notational change must be made to conform with Prof. Baggeroer's original
derivation. In the equations above, it is obvious a bias of 1/To was introduced into the spectral
density estimate. In order to obtain the "true" estimate, the bias must be multiplied out. Prof.
Baggeroer's formulation assumed the bias to be incorporated into Sr(f), eliminating the need to
show a division by To. However, he factored the bias out by multiplying his representation of S,(f)
by To,
Kr(f) = ToSr(f).
(2.15)
The Kr(f) matrix represents the spectral covariance matrix of the received signal. It is this expression
which is used in the Gaussian CRB equation listed above.
2.3.1
Matched Field Tomography
The next step is to examine the structure of the received signal. For MFT, the received signal
R(f, a) = b(f)Ss(f)G(f, a) + N(f, a),
where:
and
(2.16)
R(f, a)
=
received signal vector, evaluated at frequency f,
b(f)
=
random process with power spectral density equal to the source, Sb(f)
S,(f)
=
assumed to be 1 for MFT case,
G(f, a)
=
Green's function,
N(f, a)
=
noise vector,
is broken down as a combination of signal and noise. Taking the expected value of the received
signal squared, and adjusting for the spectral density estimate bias results in
E[R(f To)Rt(f To)] = ToSb(f)G(f, a)Gt(f,a) + ToSn(f),
(2.17)
the receiver covariance matrix. Note the formation of the spectral covariance matrix is for a single
frequency only. In order to calculate the CRB, all frequencies in the selected bandwidth of interest
would be needed. However, MFT assumes the received signal is uncorrelated across frequency. This
allows one to process the result on a frequency by frequency basis.
By arranging the received signal vector,
R (filT o)
R2(fTo)
R =
RN(fi
ITo)
(2.18)
RI (f21To)
R2 (f2T o)
RN(fMITo)
in such a manner that it spans M frequencies and N receiver sensors, one can process all frequencies
and receivers of interest simultaneously. Solving for the spectral covariance matrix,
K,(a) =
Sb(fl)G(fl, a)Gt(fl, a)
0
...
0
0
Sb(f 2 )G(f 2 ,a)Gt(f 2 ,a)
...
0
0
0
.
Sb(fM)G(fM,a)Gt(fM,a)
To
...
0
Sn(f 2 ) ...
0
0
S.(fl)
o
+ To
o
0
..
(2.19)
Sn(fM)
results in a block diagonal form.
Referring back to the Gaussian CRB equation, both the inverse and derivative of K,(a) are block
diagonal matrices.
Multiplication of four block diagonal matrices together yields another block
diagonal matrix, and the trace of a block diagonal matrix is equivalent to the sum of the trace of
the submatrices. With these assumptions, the CRB equation,
[Jr(a)]ij
=
m=1
Tr [Kl '(fma)Kr (fm
(fm,a) OKr (fm, a)
(2.20)
and
Kr(fm,a) = To [Sb(fm)G(fm, a)Gt(fm, a) + Sn(fm)] ,
(2.21)
can be re-written in a more computationally tractable form. Using Woodbury's identity, both the
receiver covariance matrix inverse,
K '(fm,a)
=
To
x
{S _(fm) - Snl(fm)G(fm,a)
[Gt(fm,a)Sn'(fm)G()G(fm,a)+ Sb(fm)-1
-
1
G t
(fm,a)Sn (fm)
(2.22)
(
S() (fm)G(fm, a)Gt
(fm, a)S(fm)
1
To
Gt(fm, a)S '(fm)G(fm,a) + Sb(fm)- 1
and its derivative,
Krf, a)
aa
ToSb(fm)
G (
IJ~i
,a)Gtfma)
+ G(fm, a) G (fm,a)
a
(2.23)
can be written. To aid in term simplification, four quadratic identities,
d2 (fm,a)
= Gt(fm,a)Snl(fm)G(fm,a),
S(fm, a)
and
(2.25)
1 + Sb(fm)d 2(fm,a))'
li(fm, a)
ct(fma)S;l(fm)
li,j(fm,a)
Bai
(2.24)
'ai '
aaj
(2.26)
(2.27)
are introduced. Each quadratic term has significance in the equation. The Sb(fm)d 2 (fm, a) term
is the Signal to Noise Ratio (SNR) for Green's function in additive noise. li(fm, a) is a measure
of the mean of the parameter sensitivity under additive noise conditions, while li,j (fm, a) measures
the convexity of parameter sensitivity. At low SNR,
Sb (fm)d 2 (fm, a) term becomes large, lowering
y(fm, a) is close to 2, but at higher SNRs, the
-(fm, a).
With this information, the Fisher Information Matrix (FIM) equation can be expanded,
M
[Jr(a)]i,j
=
TS
)
Tr S; (sm)--
S[(fm)G(fm,a)Gt(fm,a)Sn'(fm)
+T])
d2 (fm,a) + Sb(fm)
E
X Sb(fm) [G(fm, a) O(fm,a)
G(fm,a)(fm, a)]
d2(m,a ) +
I
aBaj
aSb-j(m)
iaj
(2.28)
as the observation time, To, cancels out. After some algebra, the quantity inside the summation can
be expressed with 16 individual terms, which combine to form
[Jr(a)]i,j
=
ES(fm)7(fm, a) Re [d2(f,,a)li,j(fm,a) - li(fm,a)t(fm,a)]
(2.29)
- y(fm,a)Re[li(fm,a)] Re [lj(fm,a)]}.
Finally, the summation can be converted to an integral. One must multiply and divide by Afs,
the inter-frequency spacing. The inverse of Af, is equal to the sampling time, T,. Provided Af, is
small,
-Tf,
[Jr(a)]i,j
S (f )y(f,a) {Re [d2(a)ij(f
dra)(f,
, a) - 1(f,
a)
(2.30)
- t(f, a)Re [l1(f, a)] Re [Ij (f, a)]} df,
with
d2(fm,a) = Gt (fm,a)S (fm)G(fm, a),
2
(f, a) = 11+
+ Sb(fm)d2 2 (fm,a))'
li(fm,a)
and
=
ij(fma) =
Gt(fm,a)S; (fm) OG(fm,a)
Bai
c9Gt(fm, a)
)G(fm,
aand (fa)l(f9m)
Bai
a
a)
Baj
a simplified equation for the CRB applied to Matched Field Tomography results.
2.4
Ocean Acoustic Tomography
Ocean Acoustic Tomography (OAT) differs from MFT in its treatment of the received signal.
While MFT operates on incoherent signals, OAT is better suited for processing broadband signals
which are coherent across a given frequency range. The received signal
R(f,a) = b(f)Ss(f)G(f,a) +N(f,a),
(2.31)
assumes both a signal and noise component. b(f) is a scalar random variable with variance ab.
Both exact distance and attenuation over the propagation channel cannot be known exactly; b(f)
allows for random phase and amplitude at the receiver. The source signal, S,(f) is deterministic,
as is Green's function G(f, a). The noise vector, N(f, a), is Gaussian in nature, and is uncorrelated
across both space and frequency.
Taking the expected value of the received signal squared, and factoring out the spectral estimation
bias,
SH(fl, f 2) = S (fl)S;(f 2)G(fi,a)Gt(f 2,a)
(2.32)
E[R(fi|To,a)Rt(f2ITo,a)] = arSH(fl, f2) +T o Sn(fi, f 2 )
(2.33)
and
results in the OAT receiver covariance matrix.
Estimation of Ub does not rely on an observation time; ab is assumed to be a known quantity.
Matrix computation is simplified if To is present in both terms of Equation 2.33, so a new variable,
a2, = a2/T, is introduced into the receiver covariance matrix,
E[R(fiITo, a)Rt(f
2
To, a)] = C
TToSH(fl, f2) + ToSn(fl,
f 2 ).
(2.34)
OAT assumes coherency across a frequency band. The resulting power spectral covariance matrix
is not block diagonal (as in the MFT case). Because of this, the problem cannot be broken down
and solved for single frequency increments; the entire frequency range of interest must be considered
at the same time.
Assuming a received signal vector arrangement as in Equation 2.18, one arrives at a spectral
covariance matrix,
Kr(a)
=
abT To
SH(fl, fl)
SH(fi, 2)
..
SH(fi, fM)
SH(f 2 , fl)
SH(
•
SH(f
SH(fM,
f)i)
Sn(fi, fl)
2 ,f2)
SH(fM, f2)
0
...
Sn(f2, 2) "'
0
2 , fM)
(2.35)
SH(fM, fM)
0
0
.,. Sn (fM,fM)
which is not block-diagonal. The next step is solving for the matrix inverse of K,(a) and derivative
with respect to a2 . Again, since K,(a) is not block diagonal, the problem needs to be reformulated
slightly.
Consider combining the source signal and Green's function into one vector,
Ss(f1 )G(fi, z1,a)
S,(fl)G(f1 , z2,a)
H(a) =
(2.36)
Ss(fl)G(f,,zn, a)
S (f 2 )G(f 2 , zl, a)
S,(fM)G(fM,zN,a))
Then, rewrite the received spectral covariance matrix,
K,(a) = To [S, + H(a)aoTHt(a)],
(2.37)
into a simplified form. Using Woodbury's identity, the inverse of the receiver covariance matrix,
1S
1 S --'H(a)
Kr1(a)
=
S-1
TO I n
[Ht(a)S'IH(a) +o~'] Ht(a)S.'}
(2.38)
H•bT
Ht(a)S,1H(a)Ht ()+
1
is found. Its derivative is
oK,(a)
oai
= aTTo [H(a)a
( •) + H(a)Ht (
(2.39)
Substituting these expressions into Equation 2.7 yields another multiple term equation, with the
observation time, To, canceling out,
Jij (a)=
{
'bTTr [
[S -n1
S, H(a)Ht(a)S '1
2
Ht(a)Sn H(a) + Ub-
8Ht(a) + H(a)
+ai
+
ai
It(a)]
(2.40)
Sn H(a) H t n(a) S 1 ] [H(a) oHt (
+H(a
Bay Ht (a)]
Ht(a)Sn-H(a)
+ 0aj
Attention should be focused on one of the the quadratic terms in the denominator of Equation 2.40.
Recall the vector H(a) and matrix S-1 span all receivers and frequencies of interest. The quadratic
term,
Ht(a)S
1
H(a) =
H*(fl,zi,a)
H*(fl,z 2 ,a)
*
H*(fl, ZN,a)
H*(f2,zi,a)
...
H*(fM, ZN, a)
Sj(fi,zi,zi)
0
S0
0
...
0
0
Sn (fl, z 2 ,Z2)
S0
0
""
0
0
0
0
...
0
o
o
"..
0
o
0
SSnl(fl,zN, ZN)
• "
0
"o'
0
S
(f2,Z , Z1)
0
'"
S
I(fM,ZN,ZN)
H(fl,zi,a)
H(fl, z 2 , a)
X
H(f, ZN, a)
H(f
2
(2.41)
, zi, a)
H(fM, ZN,a)
can be expanded into matrix form and re-written. Recall the assumptions made for the noise matrix
S,: the noise was Gaussian, with no correlation across space or frequency. This made Sn a diagonal
matrix, which simplifies calculation of the quadratic term,
MN
Ht(a)S- 1 H(a) =
H*(fm,zn,a)Sn1(fm, Z,, Zn)H(fm, Zn, a).
(2.42)
m=1 n=1
Retaining the vector notation for the spatial dimension, one would arrive at an expression
M
H t (a)Sn 1 H(a) =
Ht(fm,a)S-l(fm)H(fm,a),
m=1
similar to that found in the MFT derivation.
(2.43)
Expanding the H(f,n, a) vector into its components,
M
=
Hi (a)S•'H(a)
Ht(fm,a)S;l(fm)H(fm,a)
m=1
M
Ss(fm)Gt(fm,a)Sn'(fm)G(frm,a)Ss(fm)
(2.44)
m-1
M
2 Gt(fm,a)Sx'(fm)G(fm,a)
I(fm)l
3
=
m-1
shows the quadratic term in its simplified form. Note the quadratic expression inside the summation
is the d2 (fm, a) term found in the MFT derivation. Substituting this expression in, and converting
the summation to an integral,
M
IS (fm)12 Gt (fm,a)S;n'(fm)G(fm,a)
Ht(a)SnlH(a)
-
IS9(fm)12 d2 (fm,a)Afs
Af
(2.45)
m-1
ST f, IS(f)12 d2 (fa)df
=d2 (a),
finishes the quadratic expression. Note the multiplication of T,, the sampling time, to counter the
Af, used in summation over frequency. Similarly, the quadratic terms obtained in the MFT case
have equivalent OAT representations, with
2
-yT(a)
'
1 + a2TT(a
0 H) (
Htb(aa)S,
liT(a)
and lijT(a)
=
Ht(a)S;1O
=S;
'•
(2.46)
) "
1
= T,
=
IS,(ff)|2 l(f,a)df,
T], ISS((f)1 2 I4,,(f, a)dfl.
(2.47)
(2.48)
Using these expressions, Equation 2.40 can be simplified,
Jij (a)=
aSTTr
x [Isn
S-1 'T(a)2 .TSiH(a)Ht(a)S-I] [H(a) O
OHt(a)
-yT(a) •,H(a)Ht(a)S-]
2
[H(a)
Ht(a----)
H- a)Ht(a)]
&H(a)
+ Ht(a)]
Oa- +
H a)
(2.49)
,
expanded to sixteen terms, and simplified again. The final expression,
Ji,j(a) =
b(To y(a) Re [d2(a)li,j(a)- li(a)l (a)] + y(a)Re [li(a)] Re [j(a)]) ,
(2.50)
with
d2 (a) =
-(a)
1i(a)
and l4,i(a)
IS8 (f)1 d2 (f, a)df,
f
1±+T,
d2(a)'
=
aw SS(f)12 1i(fa)df,
=
2
faw ISs(f) lij(f,a)df,
shows the CRB applied to Ocean Acoustic Tomography. The observation time, To, should always
be equal to or larger than the sampling time, T,. If To were equal to Ts, the OAT FIM equation,
Ji,j(a) = 4'y(a) {Re [d2(a)li,j(a) - li(a)l (a)] + y(a)Re [1i(a)] Re [lj(a)]}
with
d2 (a)
Aw IS,(f)
-y(a)
1i(a)
1+
=
and li4j (a) =
would reduce to a simpler form.
2 d2(f,
a)df,
2d2 (a)'
WISS(f)12 i(f, a)df,
lAw ISs(f)12 1i,j(f,a)df,
(2.51)
2.5
Summary
This chapter has reviewed the Cramnr-Rao lower estimation bounds, and its application to
signals embedded in Gaussian noise. By making assumptions about the signal structure in both
Matched Field Tomography and Ocean Acoustic Tomography, simplified expressions for the CRB
have been derived. In the next chapter, these expressions will be used to calculate the CRB for four
environmental parameters in a simplified shallow water environment.
Chapter 3
Results
3.1
Environment
A range independent shallow water environment was selected for testing. The initial setting was
derived from the Pekeris model (Figure 3-1).
0
20
40
c 1= 1500 m/sec
p = 1000 kg/m3
c = 1800 m/sec
p = 1800kg/m
Water Layer
60
80
100
120
Sediment Layer
140
Bedrock
I
I I
0
I
1000
500
I
I
2000
1500
2500
Sound Speed (m/sec)
Figure 3-1: Pekeris shallow water environment. A simplified two-dimensional model is used to
describe the medium. To aid in comprehension, the depth dependent sound speed is superimposed
on the diagram.
Originally, the Pekeris model provided for a bottom composed of a fluid infinite halfspace. However,
had a
this model was not considered very realistic, so a subbottom was added. The final model
3
Below
water column of 100 m, with a sound speed of 1500 m/sec and a density of 1000 kg/m .
3
was a 50 m sediment layer, with a sound speed of 1800 m/sec, density of 1800 kg/m , and com-
layer was
pressional attenuation of 0.15 dB/Ap. Finally, a rigid subbottom was chosen. This final
modeled as basalt, with a compressional sound speed of 5250 m/sec and shear speed of 2500 m/sec.
Compressional and shear attenuation in this layer were 0.1 dB/Ap, and 0.2 dB/A,, respectively.
In Figure 3-2, the baseline case was annotated with a zero (0). Parameters under study were
shown with their respective numerals (1, 2, 3, and 4).
Due to limitations of some propagation models, only compressional sound waves were considered;
shear waves were neglected. Although this detracts from the realism of the ocean model, higher
priority was given to maintaining a consistent environment across all propagation models.
15
Distance between source and receivers (km)
0
I
I
0
Signal
Source
at 20m
-
20 40 -
%
Water Layer
I
19 receiver elements
evenly spaced from
5 to 95 m depth
p = 1000 kg/m3
60 80 100 -
Sediment Layer
120 140 -
/
I
I
I
I
500
"
'•
Bedrock
%=0.1 dB/X
2500 m/sec /a,=0.2
c,=
dB/;X,
I
2000
1000
0
I
Z' Z
c,,F=5250rr'1ec
1500
2500
Sound Speed (m/sec)
shown.
Figure 3-2: Shallow water environment under study, with four environmental parameters
lines
Dashed
Again, the depth depended sound velocity profile is superimposed on the diagram.
indicate the effect of the four parameters under study on the sound velocity profile.
3.1.1
Source and Receivers
A signal source was placed at 20 m depth. A vertical array of receivers was located 15 km away,
spanning the entire water column. The first receiver was situated at a depth of 5 m, while the last
was at 95 m. Nineteen receivers were deployed, each at 5 m intervals.
The deterministic OAT signal source had a raised cosine spectrum, with
cos [
=
ss(fo)
where: f
=
frequency (Hz),
S, (f)
=
source signal, evaluated at frequency
fc
=
center frequency of signal (Hz),
and bw
=
bandwidth of signal (Hz).
(3.1)
f,
This spectrum was normalized such that the energy was constant regardless of the bandwidth selected, with
E=
S,(f)|2 df = 1.
(3.2)
--OO
In MFT, the signal source was a stationary random process, with power spectral density Sb(f).
The spectral distribution, while not Gaussian, was assumed to be similar to that generated by
pseudo-random noise: a raised cosine. The same frequency representation of the source signal used
for OAT was employed in MFT.
Noise was assumed to have a Gaussian distribution, and was independent of any parameter a.
Throughout this document, the noise covariance matrix was assumed to be a diagonal matrix:
3.1.2
unI.
Signal to Noise Ratio
Determining the Signal to Noise Ratio (SNR) at the receiver requires knowledge of the source
strength, transmission loss, and ambient noise level in the ocean. According to Urick[30], a source
which radiates 1 Watt of acoustic power has a source level of approximately 171 dB re 1 pPa at
1 meter. For this simulation, a slightly weaker source was assumed, with a power level of 150 dB.
Calculation of Green's function across the frequency range of interest yielded a mean transmission
loss of approximately 60 dB (see Figure A-2). The noise level for representative shallow water
Source spectra for MFT and OAT problems
n
00
220
240
260
280
300
320
340
360
380
400
200
220
240
260
280
300
320
340
360
380
400
n
)0
Frequency (centered at 300 Hz)
Figure 3-3: Normalized source signal, with bandwidths of 10, 50, and 100 Hz
environments ranged from 70 to 80 dB[30], but these were considered to be optimistic by the author.
With these three quantities, the SNR at each hydrophone could be calculated on a per-Hertz basis,
with
SNR = SL - TL - NL,
where
and
SNR
is the Signal to Noise Level, in dB,
SL
is the Source Level, in dB,
TL
is the Transmission Loss, in dB,
NL
is the Noise Level, in dB.
(3.3)
One should realize these numbers are averaged over the frequency range, and are to be taken as
approximate values. Substituting the numbers cited above (with NL equal to 80 dB), one obtains
an SNR of +10 dB. Additional processing on the received signal (taking advantage of array gain,
for instance) would improve this number. The SNR can be adjusted by changing any of the three
input parameters: source level, transmission loss, and noise level.
The CRB equations incorporate SNR through the diagonal term of the noise covariance matrix,
S,. This term represents the noise level of the environment (NL). Equation 3.3 is used to calculate
NL, with
NL = SL - TL - SNR.
(3.4)
Recall from Equation 3.2 the simulated power level is actually 0 dB. The simulated transmission
loss is 60 dB, with the desired SNR at the receiver of +10 dB. This gives a simulated noise level of
-70 dB, or 0.0000001. For the CRB calculations, the diagonal term of S, was set to this number.
Variations of hydrophone SNR levels led to changes in the diagonal term. Actual diagonal terms
used for simulation are listed in Table 3.1.
SNR at hydrophone
(dB re 1 pPa at I m)
-20
-10
0
10
20
30
Noise Variance, a n
(dB re 1 pPa at 1 m)
1 x 101 x 101 x 10- 6
1 x 10- 7
1 x 101 x 10- 9
Table 3.1: SNR levels at each hydrophone, and their corresponding noise variance values. The
diagonal value of the noise covariance matrix incorporated the SNR at each receiver element.
3.1.3
Parameters
Four parameters were chosen for solving the CRB. Recall the objective was to find the lowest
variance which an estimator could obtain when trying to find the parameters in question. The four
parameters were
1. al: Water column sound velocity,
2. a2: Gradient of sound velocity in water column,
3. a3: Bottom sound velocity, and
4. a4 : Gradient of sound velocity in bottom.
In order to calculate the CRB, the baseline environment was modified.
For each parameter, a
quantity in the environment was changed. For example, solving the CRB for the first parameter
involved simulating the ocean environment for both the baseline environment and an environment
with a slightly different water column sound velocity. Solving the CRB for the second parameter
required simulating the original baseline environment, as well as an environment with a tilted water
column sound speed. Similar environmental perturbations were required for the third and fourth
parameters.
The water column and bottom sound speeds were
c(z) = [1500.0 + ai] + [a2 * (z - 50.0)]
c(z) = [1800.0 + a3] + [a4 * (z - 125.0)]
0 < z < 100m, and
(3.5)
100 < z < 150m.
(3.6)
Calculating the CRB for all four parameters simultaneously helped one gain insight on the minimum
covariance of an estimator involving two parameters. This allowed one to gauge the dependence of
one parameter on another, quantitatively establishing the relationship between different parameters.
3.2
Minimum standard deviation
Figure 3-4 shows the minimum standard deviation for each parameter plotted as a function of
bandwidth, at a SNR of -20 dB. The upper plot shows curves for the water and bottom reference
sound speeds, while the lower plot shows results for the sound speed gradients.
When plotted on a logarithmic scale, the curves appear linear. To quantitatively compare results
for different SNR levels, the CRB results were fit to a power curve using a least squares algorithm,
& = 10b(
where
&
is a log-linear approximation to CRB curve,
is the bandwidth of source signal,
(AW)
and
(3.7)
aw)
are coefficients.
a,b
Tables 3.2 through 3.5 show the curve fit results for each parameter.
SNRII
(dB)
MFT
OAT
II
a
b
dB /
decade
o at 1Hz
(m/sec)
x10 x 10x10 x10 x10 -
-20
-10
0
10
20
0.00805
-0.06863
-0.17408
-0.21001
-0.21552
-4.1760
-4.9609
-5.4924
-5.9788
-6.4759
0.0805
-0.6863
-1.7408
-2.1001
-2.1552
7.6453
1.3561
3.7245
1.1549
3.6442
30
-0.21616
-6.9755
-2.1616
1.1521 x10 - 1
-20
-10
0
10
20
-0.24668
-0.24125
-0.24067
-0.24061
-0.24061
-4.4396
-4.9524
-5.4537
-5.9539
-6.4539
-2.4668
-2.4125
-2.4067
-2.4061
-2.4061
3.7889
1.1482
3.6149
1.1426
3.6131
30
-0.24061
-6.9539
-2.4061
1.1426 x10-
x10 - '
x10- b
x 10x10 x 10-
Table 3.2: CRB results for Parameter 1: water column reference speed.
First, focus on the MFT results. Here, both Parameters 1 and 3 show increased Minimum
Standard Deviation (MSD) as bandwidth is increased. Parameter 1 has a slope of 0.0805 dB per
decade, while 3 has a slope of 0.1622 dB/decade. As signal bandwidth is increased, it becomes more
difficult to estimate the reference sound speeds in both the water column and bottom. This is a
counterintuitive result; one would expect the additional information furnished by a higher bandwidth
SNRI
(dB)
MFT
OAT
-20
-10
0
10
20
30
-20
-10
0
10
20
30
a
b
-0.01309
-0.09055
-0.19611
-0.23105
-0.23625
-0.23684
-0.24393
-0.23853
-0.23796
-0.23790
-0.23789
-0.23789
-1.2964
-2.0811
-2.6150
-3.1041
-3.6018
-4.1015
-1.5948
-2.1075
-2.6089
-3.1090
-3.6090
-4.1090
dB /
decade
-0.1309
-0.9055
-1.9611
-2.3105
-2.3625
-2.3684
-2.4393
-2.3853
-2.3796
-2.3790
-2.3789
-2.3789
a at 1 Hz
(1/sec)
5.3898
9.5998
2.6433
8.2009
2.5878
8.1814
2.6554
8.0471
2.5334
8.0076
2.5321
8.0072
x10 - 2
x10 x10 x10 -4
x10 - 4
x10 - 5
x10 - 2
x 10 x10 x10 - 4
x10 - 4
x107 -
Table 3.3: CRB results for Parameter 2: water column speed gradient.
signal would aid in parameter estimation. One should remember, however, that MFT is inherently
a narrowband process. It assumes a wideband signal is incoherent across frequency. This suppresses
a substantial amount of information in a wideband signal, and this information is needed in order to
lower the CRB. Additionally, the upward sloping CRB shows at this low SNR, there is insufficient
information in the incoherent signal to lower the CRB; rather, the additional noise included in the
wider signal bandwidths increases the CRB.
Next, observe the relative positions of the MFT lines in Figure 3-4. Parameter 1 has a MSD of
7.65 x 10- 5 m/sec at 1 Hz bandwidth, while Parameter 3 is 1.143 x 10- 4 m/sec. For a signal of 1
Hz bandwidth, using MFT analysis, one could at best build an estimator with a minimum standard
deviation of 0.0000765 m/sec and 0.0001143 m/sec, respectively. For MFT, this is an extremely
small MSD, and shows promise for parameter estimation at this signal level. It is interesting to note
the MSD of Parameter 3 is consistently higher than Parameter 1. This indicates, given the current
scenario, that it is easier to estimate the reference speed in the water column than the bottom. This
is an intuitive result, given both the source and receivers are in the water column. Information about
the bottom is obtained indirectly; one would speculate the MSD of the bottom reference sound speed
to be lower if receivers were placed in the bottom.
Focus now on the OAT results for Parameters 1 and 3. These are substantially different than
their MFT counterparts. First, the slope of the CRB lines is downward as bandwidth increases.
For Parameter 1, the slope is down 2.47 dB/decade, and for Parameter 3, 2.19 dB/decade. In this
MFT
OAT
SNR I
a
(dB)
0.01622
-20
-0.06224
-10
0 -0.16975
10 -0.20663
20 -0.21238
30 -0.21305
-20
-0.21888
-10
-0.21348
0 -0.21291
10 1 -0.21285
20 -0.21285
30 -0.21285
b
-3.9920
-4.7730
-5.2999
-5.7847
-6.2813
-6.7809
-4.2808
-4.7935
-5.2948
-5.7949
-6.2949
-6.7949
dB /
decade
0.1622
-0.6224
-1.6975
-2.0663
-2.1238
-2.1305
-2.1888
-2.1348
-2.1291
-2.1285
-2.1285
-2.1285
a at 1 Hz
(m/sec)
1.1433 x10 - 4
2.0387 x10 - 5
5.6173 x10 - 1
1.7430 x10 5.5002 x10 1.7389 x10 - 7
5.6052 x10 1.6988 x10 - 5
5.3483 x 101.6095 x10- 6
5.3457 x10- 7
1.6904 x10-
Table 3.4: CRB results for Parameter 3: bottom reference speed.
scenario, as one increases the bandwidth of the signal, the MSD declines substantially, making it
easier to estimate the parameter. The decline can be attributed to how OAT processes information.
Unlike MFT, OAT assumes signal coherency across frequency. Expanding signal bandwidth provides
additional information not only in the form of the added frequencies, but also in the interaction
between different frequencies. This is exploited by the CRB equations, resulting in a lower MSD.
Examination of the relative positions of the OAT lines shows, at 1 Hz bandwidth, the MSD for
Parameter 1 is 3.79 x 10- 5 m/sec, and for Parameter 3 is 5.61 x 10-
5
m/sec. As in the MFT case,
this shows the bottom reference sound speed to be more difficult to estimate than the water column
reference speed.
SNR
a
(dB)
MFT
OAT
b
dB /
decade
a at 1Hz
(1/sec)
x10 - 1
x10 - -
-20
-10
0
10
20
30
-20
-10
0
10
-0.00202
-0.07703
-0.17805
-0.21247
-0.21792
-0.21856
-0.22623
-0.22085
-0.22028
-0.22023
-0.7164
-1.5040
-2.0419
-2.5302
-3.0272
-3.5269
-1.0221
-1.5348
-2.0361
-2.5362
-0.0202
-0.7703
-1.7805
-2.1247
-2.1792
-2.1856
-2.2623
-2.2085
-2.2028
-2.2023
2.2485
3.9902
1.0953
3.3951
1.0712
3.3866
1.1081
3.3600
1.0579
3.3438
20
-0.22022
-3.0363
-2.2022
1.0574 x10- 3
30
-0.22022
-3.5363
-2.2022
3.3437 x10 - 4
x10 - 2
x 10x10 - 3
x 10x10 x10-
x10 x10-
Table 3.5: CRB results for Parameter 4: bottom speed gradient.
In order to make objective comparisons between MFT and OAT, similar source signal spectra
were chosen. Care was taken to ensure the amount of received energy was the same for each method,
E
=
=
where
and
E
(3.8)
EMFT = EOAT
T
fJ WSb(f)df
=
W
J W
2df
(f)I
is the energy contained in the source signal,
(AW)
is the bandwidth of source signal,
Sb(f)
is the power spectral density of the source signal for MFT,
s (f)
is the deterministic source signal for OAT,
To
is the observation time, in seconds,
2
ab
variance of the travel time uncertainty for OAT.
Similar signals were chosen for both MFT and OAT. This made the quantities inside the integral
in equation 3.8 equal. The source spectra were normalized to ensure the integrated quantities were
unity, regardless of the bandwidth selected. These were multiplied by the MFT sampling time, and
the OAT arrival time variance, a . To maintain equality between MFT and OAT, a0 was set equal
to Ts, at 50 seconds.
Setting the received signal energy for both MFT and OAT resulted in equal CRB as bandwidths
approached zero Hz. This result was expected; OAT could not benefit from signal coherence across
different frequencies, so its performance was reduced to that of the narrowband MFT process.
Turning one's attention to the lower plot in Figure 3-4, one sees the CRB lines for the sound
speed gradients plotted. Here, the MFT bounds have a slightly negative slope (-0.1309 dB/decade
for Parameter 2, and -0.0202 dB/decade for Parameter 4), showing nearly level MSD as bandwidth
is increased. Also, the OAT bounds have a negative slope (-2.439 dB/decade for Parameter 2; -2.262
dB/decade for Parameter 4). These results support the conclusions put forth for the upper plot in
Figure 3-4.
Observe the 1 Hz MSD for Parameters 2 and 4: 5.390 x 10-2 sec -
1
and 2.249 x 10- 1 sec - 1,
respectively. For this scenario of -20 dB SNR (per Hz, at a single hydrophone, before any processing)
and 1 Hz bandwidth signal, the best estimator one could construct for water sound speed slope
would have a minimum standard deviation of 0.054 sec - 1 , and 0.225 sec - 1 for bottom speed slope.
Although these quantities are low, slope estimation is more useful if the MSD is on the order of 10-
3
or lower. Thus, for a -20 dB SNR signal, slope estimation is not ideal.
Figure 3-5 shows the CRB for a SNR (per Hertz, at a single hydrophone, before processing) of
-10 dB. At first glance, this plot is remarkably similar to Figure 3-4. However, several items have
changed. As the SNR has increased from its initial -20 dB level, the slope of the MFT lines has
decreased. Parameter 1 has seen a 0 dB slope of 0.0805 dB/decade drop to -0.6863 dB/decade.
Similar drops have occurred for each parameter (see Tables 3.2 through 3.5). As SNR rises, the
amount of coherent signal information to reach the receiver also rises, which reduces the MSD.
Conversely, the slope of the OAT CRB for Parameter 1 rose, from -2.4668 dB/decade at -20 dB
SNR, to -2.4125 dB/decade at -10 dB SNR. This reduction in slope is difficult to explain. One would
expect better performance as the SNR increases; but instead, the rate of improvement as bandwidth
increased has decreased. However, the overall downward slope is consistent with the conclusions
reached for OAT in the -20 dB case.
The 1 Hz bandwidth CRB for Parameter 1 decreased from 7.645 x 10- 5
to 1.3561 x 10
5
m/sec at -20 dB SNR,
m/sec at -10 dB SNR. This decrease allows one to determine the reference sound
speed with more precision than the -20 dB case. This reduction in MSD is seen for the bottom
reference sound speed, as well.
For the sound speed gradient in water, the MSD dropped from 5.39 x 10- 2 sec - 3
SNR, to 9.60 x 10
- 1
sec
1
at -20 dB
at -10 dB SNR. Again, increased SNR has led to a decrease in the MSD.
This MSD is would be appropriate for gradient estimation in this environment; with an optimal
estimator, the sound speed uncertainty at either end of the water column would be on the order of
0.1 m/sec.
Performance improved as the SNR was raised to +20 dB/Hz/phone (see Figure 3-6).
This
extremely high SNR had lower MSD values for each of the four parameters. For example, the MFT
MSD of Parameter 1 was reduced to 3.644 x 102.588 x 10-
4
7
m/sec; for Parameter 2, the MSD dropped to
sec - 1 . This lower value for the sound speed gradient MSD is encouraging; it is low
enough to make estimation of the gradient worthwhile.
One interesting observation is the overlap of the MFT and OAT MSD curves. As the SNR
increased, the slope of the MFT curves declined, while the OAT curves ascended. By 0 dB SNR,
the curves have met to overlap. For Parameter 1 at 30 dB SNR, the MFT curve had a slope of -2.16
dB/decade, while the OAT curve had a slope of -2.41 dB/decade. Similar overlaps were observed
for all four parameters.
This overlapping shows that for high SNR in this environment, there is no difference in the
performance between MFT and OAT for parameter estimation. The strength of the signal obliterates
any advantage which coherence across frequency would give. Additional bandwidth alone gives
sufficient information to reduce the minimum standard deviation.
The similarity between MFT and OAT MSD curves should have some influence in planning atsea experiments. When performance between two methods has been judged to be equivalent, other
factors, such as cost or feasibility, decide which method is used. MFT requires a geometric array
of receiver hydrophones in order to be effective. Hydrophones cost money, and maintaining their
geometry for the duration of an experiment is difficult. For example, a vertical line array (VLA) can
be distorted from its straight line shape by subsurface currents. On the other hand, OAT requires
only one receiver at each location, but the position of the receiver must be known to within one
acoustic wavelength.
3.3
Correlation Coefficients
Solving for the CRB resulted in a Fisher Information Matrix (FIM), J (see equations 2.31
and 2.51). The inverse of this square matrix produced minimum variance (MV) values for each
parameter. If a diagonal term were selected, the MV for that particular parameter was obtained.
Off-diagonal terms produced Minimum Covariance (MCV) values. One FIM was generated for each
source bandwidth under consideration.
Comparison of the MCV values exhibited information on the coupling between each parameter.
Calculation of the correlation coefficient provided a good measurement of the coupling, with
PxY
where
PXy
Ax
and
ax, ,a
-
xy
(3.9)
is the correlation coefficient for random variables x and y,
is the covariance of two random variables, x and y,
are standard deviations of random variables x and y.
The correlation coefficient normalizes the covariance to the standard deviations between two
random variables. It has a range between -1 and + 1. A value of zero indicates the two random
variables are completely uncorrelated, while a value of ±1.0 indicates perfect correlation between
the two random variables.
Figure 3-7 shows the correlation coefficients for the shallow water test case at -20 dB SNR. Apart
from some initial narrowband data, the correlation coefficients are somewhat linear with respect to
bandwidth. Also, differences between OAT and MFT correlation coefficients are small (see Tables 3.7
and 3.8).
The relative positions of the correlation coefficient lines show the degree of coupling for each
parameter. l12 has the lowest correlation, with an average of 0.04. This shows the gradient and
reference sound speeds in the water were nearly completely uncorrelated.
Low correlation does not imply independence; one cannot use these statistics to assert that the
reference sound speed is independent of the sound speed gradient. However, one can conclude that
the set of information which is used to estimate the reference sound speed cannot be effectively used
to estimate the gradient, and vice-versa. Also, the CRB of the reference sound speed has no effect
on the CRB of the gradient (and vice-versa).
The correlation coefficient for water column reference sound speed and bottom sound speed
gradient (P14) hovers around 0.20, regardless of SNR, for both MFT and OAT. Also, the coefficient
for water column gradient and bottom gradient (P24) stays near -0.19, again, for both MFT and
OAT, for all SNR tested. A coefficient value of ±0.19 shows low amounts of correlation, but one
cannot say these parameters are completely uncorrelated.
The strongest level of correlation exists between the bottom reference and bottom gradient sound
speeds (P34). Here, the mean correlation coefficient is -0.52, for both MFT and OAT, and for all
SNR levels tested. This moderate level of correlation indicates that a change in the bottom reference
sound speed would affect an estimate of the bottom gradient sound speed.
Calculation of the correlation coefficient gives an extremely good insight on the coupling between
different parameters. Each coefficient describes the dependence of one parameter on another. Higher
coefficient magnitudes indicate changes in one parameter will have effects on the performance of
estimators for other parameters. Here, it has been demonstrated the coupling between parameters
is nearly independent of SNR used, or of processing method (MFT vs. OAT). Rather, the coupling
appears to be a function of the chosen environment.
3.3.1
High Parameter Correlation Environment
The four parameters investigated so far have relatively low levels of correlation. To illustrate an
example of high parameter correlation, the shallow water environment was modified (see Figure 38). In the original environment, the water sound speed gradient (Parameter 2) had an intersection
point at the center of the water column. Similarly, the bottom sound speed gradient (Parameter 4)
intersected the reference sound speed at the center. These two parameters were modified to intersect
at the top of the water and bottom layers, respectively. While this appears to be a relatively minor
change, the effects on parameter coupling were significant.
Correlation
Coefficient
P12
P13
P14
P23
p24
P34
(original)
0.0449
-0.2829
0.1951
0.4440
-0.1946
-0.5244
MFT
(correlated)
-0.9990
-0.2193
0.2054
0.2103
-0.1967
-0.9994
(original)
0.0431
-0.2863
0.2013
0.4438
-0.1932
-0.5219
OAT
(correlated)
-0.9991
-0.2165
0.2026
0.2074
-0.1939
-0.9994
Table 3.6: Mean correlation coefficients for modified shallow water case, +10 dB SNR.
Table 3.6 shows the updated correlation coefficients. The most obvious changes are
P12
and P34;
these have moved to 0.999. This indicates nearly perfect correlation between the two parameters.
Estimation of both parameters is made more difficult, since any effect on one parameter will have
an impact on the other. For example, a change in the sound speed gradient would negatively impact
the ability to estimate the reference sound speed.
Figure 3-10 shows the MSD for the updated environment. When compared with Figure 3-9, the
effect of the parameter coupling becomes evident. The lower plot shows no difference between the
two figures. One can assume the change in environment had no effect in the ability to estimate the
slope of the sound speed in the water column and the bottom. On the other hand, the CRB for
the reference sound speeds is considerably higher. For Parameter 1, MFT, at 1 Hz bandwidth, the
MSD increased to 2.6736 x 10- 5 m/sec, from its original 1.1549 x 10-6 m/sec. One can conclude the
coupling between parameters has significantly reduced the ability to estimate the reference sound
speed in both layers.
The importance of proper parameter selection cannot be overstated. Uncorrelated parameters
yield lower CRB curves, which allow for better estimation. Solving for the correlation coefficients
enables one to determine which parameters are orthogonal and which are biased. This enables one
to modify the parameters until an orthogonal set is found.
P14
P23
-20 0.0447 -0.2812
I
0.1958 0.4461
-0.1918 -0.5196
-10
0
10
20
30
0.1950
0.1948
0.1951
0.1952
0.1952
0.4455
0.4457
0.4440
0.4438
0.4438
-0.1926
-0.1943
-0.1946
-0.1946
-0.1946
SNR
(dB)
P12
0.0453
0.0451
0.0445
0.0443
0.0443
P13
-0.2819
-0.2824
-0.2828
-0.2829
-0.2829
P24
P34
-0.5203
-0.5230
-0.5244
-0.5245
-0.5245
Table 3.7: MFT correlation coefficient results for CRB shallow water case.
SNR
P12
P13
P14
P23
P24
P34
-20
-10
0
10
20
30
0.0431
0.0431
0.0431
0.0431
0.0431
0.0431
-0.2863
-0.2863
-0.2863
-0.2863
-0.2863
-0.2863
0.2013
0.2013
0.2013
0.2013
0.2013
0.2013
0.4438
0.4438
0.4438
0.4438
0.4438
0.4437
-0.1932
-0.1932
-0.1932
-0.1932
-0.1932
-0.1932
-0.5219
-0.5219
-0.5219
-0.5219
-0.5219
-0.5219
(dB)
Table 3.8: OAT correlation coefficient results for CRB shallow water case.
Cramer-Rao bounds for reference sound speeds inwater and bottom
-3
IU
..
- ....
".
4
E 10--4a
C
0
MFT1
.,10-5
--
03
-
OAT 1
- -MFT3
Cf)
OAT 3
101...
.. 100..
.. 101..
10- 1
100
101
102....
4 A-6
L
10-2
102
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -20 dB
Cramer-Rao bounds for sounds speed gradients inwater and bottom
"
.ti
C
0
co
-
cc
S10
ca
MFT2
OAT 2
--
Wn
----
- - MFT4
.....
OAT 4
S-3
-2
10
10
-1
100
101
102
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -20 dB
Figure 3-4: CRB results for SNR of -20 dB.
Cramer-Rao bounds for reference sound speeds inwater and bottom
^-4
10
(D
E
0
-5
u
(DS10
.,
-
MFT 1
OAT 1
Cz
ca
- -
.)
MFT3
..... OAT 3
110101010
-n
6
10
- 2
-1
102
101
100
10
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -10 dB
Cramer-Rao bounds for sounds speed gradients inwater and bottom
-1
10
o
a)
0
"
>10
a)
-2
MFT 2
- OAT2
Cz
ca
- -MFT4
U)
OAT 4
1
-3
.
10
-2
-1
10
. ..
.
I.
.
. .
,
100
.
.
. .
.
. .I
.
.
. .
101
.
.~ .
102
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -10 dB
Figure 3-5: CRB results for SNR of -10 dB.
. .
. . .
Cramer-Rao bounds for reference sound speeds in water and bottom
-5
Iv
-6
E110
C
a,
-o
",10-7
MFT 1
-
- - OAT 1
-.- MFT3
"4"-
.... OAT 3
.
1
.
.
100.
101
10
100
101
102
- 8
-2
10
10
-1
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 20 dB
Cramer-Rao bounds for sounds speed gradients in water and bottom
-2
1U
a,
O
0
a)
CU
o
MFT 2
10
ca
-- MFT4
OAT4
I
1n
-5
L
10 - 2
10
-1
100
101
102
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 20 dB
Figure 3-6: CRB results for SNR of +20 dB.
CRB: Shallow water case
0.5
12
p23
1>
p14
_12
E
p24
p131
..· ·-- ·- '~"--- _ -0.5
E
_1
111
I
1
I111
I
111
1..
I11111
,,
.
1
)111
L
10-
-1
102
10
100
101
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -20 dB
Figure 3-7: Correlation coefficients for SNR of -20 dB.
10
Original low correlation environment
Modified high correlation environment
U
Water column
1 0
-
Bottom
1I ~V
1400 1600 1800
1500 1700 1900
sound speed (m/sec)
1400 1600 1800
1500
1700
1900
sound speed (m/sec)
Figure 3-8: Changes to environment for high parameter correlation.
Cramer-Rao bounds for reference sound speeds in water and bottom
-5
10
a.)
U)
oC,)
cn
E
C
0
(U
O
-6
> 10-6
a)
e
0
co
CD
....
OAT 3
4A-7
.
-2
-2
I10
10
.101
100
10- 1
100
1
'
101
'
I I1I02
I
I'
I
102
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 10 dB
Cramer-Rao bounds for sounds speed gradients in water and bottom
- 2
IU
· 3.
1
.....
1n
- 4
I .
·~
OAT 4
.
I. . . .. I
[
-2
10
10- 1
100
101
102
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 10 dB
Figure 3-9: CRB results for SNR of +10 dB, uncorrelated parameters.
Cramer-Rao bounds for reference sound speeds inwater and bottom
S-3
Iv
04
-to 10
- 4
a)
MFT 1
-- OAT 1
t-
0)
- - MFT3
OAT 3
10
-6
.
'
.
.-
I,.I
.
.
.
. . .
......
I
|
-2
-1
10
10
100
101
I
102
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 10 dB
4
Cramer-Rao bounds for sounds speed gradients inwater and bottom
-2
IU
-r
UAIn4
1•
-4
-2
10
10- 1
100
101
102
Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 10 dB
Figure 3-10: CRB results for SNR of +10 dB, correlated parameters.
Chapter 4
Conclusions
4.1
Summary
In this thesis, the Cramer-Rao bounds for four parameters of a range independent shallow water
ocean environment were determined. Attention was focused on the sound speed and its gradient, in
both the water column and bottom layer.
Matched Field Tomography (MFT) and Ocean Acoustic Tomography (OAT) processing techniques were compared using a broadband source centered at 300 Hz. At low SNR (-20 dB), the
Minimum Standard Deviation (MSD) given by the CRB using MFT for all selected parameters
increased an average of 0.023 dB/decade, but for OAT, the MSD declined an average rate of 2.34
dB/decade. At high SNR (+30 dB), the MSD for both MFT and OAT decreased an average rate of
-2.24 dB/decade.
As SNR increased, the MSD for each parameter dropped, regardless of signal bandwidth or
tomographic method. For example, the MFT MSD for the reference water column sound speed
at -20 dB was 7.65 x 10-
5
m/sec; but this dropped down to 1.15 x 10 - 7 m/sec at +30 dB. Until
approximately +10 dB SNR, the MSD for both water and bottom sound speed gradients was too
high (above 0.001 sec - 1) for practical use.
The correlation between the four parameters was determined to be consistent regardless of SNR,
tomographic method, or signal bandwidth. Coupling was lowest between the reference and gradient
sound speeds in the water column, with a correlation coefficient of 0.04. The highest coupling was
witnessed between the bottom sound speed and gradient, with a correlation coefficient of -0.52.
4.2
Contributions
Several concepts have been demonstrated which may have an impact on future ocean acoustic
tomographical work.
* High SNR performance for both MFT and OAT: At low (-20 dB) SNR, the performance of
MFT and OAT differed with respect to signal bandwidth. However, at high SNR levels, the
minimum standard deviation for all parameters under consideration was nearly equivalent for
both MFT and OAT. This may have a significant impact in future at sea experiments; if both
tomographical methods have similar performance, then the decision to use on or the other will
be based on cost and ease of deployment alone.
* Gradient estimation in shallow water: The reference sound speeds in both water and bottom
had low MSD levels, even at low SNR. However, the gradients in both the water and bottom
had high MSD levels, making slope estimation impractical. Only when the SNR was raised
beyond +10 dB, did the MSD drop to a practical level. For detailed, second order knowledge
of the ocean environment, high SNR levels will be needed for ensonification.
* Parameter Correlation:Use of the off-diagonal terms of the inverted Fisher Information Matrix
yields the minimum covariance between two parameters. Now, the effects of one parameter
on the estimation of another can be conclusively determined. This will allow engineers to
verify their estimation parameters are uncorrelated before wasting computational time on
simulations, or expensive time at sea.
4.3
Future Work
Opportunities to extend the work described in this thesis abound. Two areas can be emphasized:
simulation and real-world acoustics. This document dealt entirely with a fictitious simulated ocean.
Areas which can be improved are:
* Use different propagation models: Only one acoustic propagation program, SuperSNAP, was
used to obtain Green's function. Other models, including KRAKEN and PRUFER, were found
to be ill suited for the task. SuperSNAP was modified to compute and store quantities in double
precision. KRAKEN[31] was too large to convert to double precision, and PRUFER[32] was
not suitable for Pekeris-type environments. Other models, such as SAFARI/OASES[33] and
RAM[34], were not considered due to time constraints. Inclusion of these models in future
work would increase the credibility of the results.
* More realistic environment: The ocean environment studied in this document can only be
duplicated in a very large and deep swimming pool. A real shallow water environment has
additional properties not considered here, such as bottom and surface attenuation, and rangeindependent features. Assuming a Pekeris-type sound velocity profile in shallow water is far
too simplistic for practical use; the environment should be derived from measured data for a
particular geographical region.
* Empirical OrthogonalFunctions (EOF):The use of EOFs has not been considered in this thesis,
but can be used to represent a relatively complicated sound velocity profile. By decomposing
a SVP into a set of weighted orthogonal functions, and using the weighting coefficients as
uncoupled parameters, the estimation problem can be simplified.
* Tabulate correlation: A larger project would be to collect data on each measurable parameter
in either a deep ocean or shallow water environment, and determine the coupling between
all parameters. This would give insight into the effects of all parameters on basic acoustic
propagation, perhaps uncovering new acoustic properties which have not reached the open
literature. Tabulation of these parameters and their effect on coupling would be a great help
to the underwater acoustics community.
However, no simulation is credible unless it has been verified in a real ocean environment. Several
opportunities exist for applying the information in this thesis to current propagation problems:
* Verification: One could conduct a shallow water experiment using the shallow water environment, in an attempt to verify the calculated Cramer-Rao estimation bounds. Locations on
the United States continental shelf would be appropriate for at sea experiments. This would
provide hard data on the feasibility of parameter estimation.
* Source Localization: Another idea for an at-sea test would be to use the information gathered
from MFT or OAT to improve the performance of MFP source localization. Initially, one would
estimate the ocean environment using uncoupled parameters with low MSD. This derived
environment would be fed into a MFP localization algorithm to identify acoustic sources.
Acoustic parameter estimation will always require a method to objectively evaluate the quality
of the results. Solving for the Cramer-Rao lower estimation bounds provides an easy way of doing
this for any selected parameter.
Appendix A
Computational Procedure
The main body of this thesis covers the background, theory, and application of Cramer-Rao
bounds (CRB) for both Matched Field Tomography (MFT) and Ocean Acoustic Tomography (OAT).
In order to emphasize the results and their impact on ocean acoustics, much of the computational
detail was left out. The purpose of this appendix is to give enough information to the reader to
duplicate the results given in Chapter 3.
Figure A-1 gives a road map of the procedure used to calculate the CRB. One starts by choosing
the parameters which parameters to focus on. In this thesis, this was the reference and gradient
sound speeds in the water column and bottom layers. Referring back to Equations 2.31 and 2.51,
one must take the derivative of Green's Function with respect to the candidate parameter. This is
done by taking a numerical derivative, requiring calculation of Green's Function twice: once for a
"baseline" case, and again for a "perturbed" case. The magnitude of the perturbed Green's Function
is selected in such a manner so it retains the linearity of Green's Function. Next, a relevant frequency
range must be selected. In order to simulate broadband propagation using narrow band simulation
code, an optimal inter-frequency spacing must be selected. From there, normal modes are calculated
for each frequency and perturbed parameter. Green's Function is calculated, and the results are
plugged into Equations 2.31 and 2.51 to solve for the Fisher Information Matrices.
This appendix assumes the relevant parameters have already been selected. Each step of the
flowchart is explained, followed by a description of the hardware used to calculate the results.
Determine frequency
spacing for broadband
modeling
START
Choose which parameters
Calculate normal modes
for CRB calculation
Calculate Green's function
Determine magnitude
of perturbation for each
parameter
Solve for Fisher Information
Matrix
Choose frequency range
for modeling
FINISH
Figure A-1: Flowchart of computational process
A.1
Perturbation Magnitude
Equation 2.31 and its OAT counterpart both have one term which included a derivative of Green's
Function,
dG(f, a)
(A.1)
The optimal approach would take an analytical derivate of Green's Function, but this is not practical,
given the potentially complicated nature of the propagation environment (see Figures A-2 and A-3).
Instead, the derivative is approximated using a finite difference (see Equation A.2).
dG(f, a)
dai
G(f, al)- G(f, O)
ai
(A.2)
It is desirable to approximate the derivative of Green's Function when it is linear with respect to
the perturbation. For example, if the perturbation were doubled, the difference between the baseline
(G(f, 0)) and perturbed (G(f, ai)) Green's Function should also double.
Magnitude of Greens Function
-50
-60
-70
I
E
gZ
-80
-90
-100
-110
295
296
297
298
299 300 301
Frequency (Hz)
302
303
304
Figure A-2: Variability of Green's Function. This plot shows the magnitude of Green's Function
over a 10 Hertz interval. The units of the colorbar are in dB re 1 pPa at 1 m. Note the rapid
changes with frequency and depth. This plot is also useful in determining the Transmission Loss
(TL) between source and receiver.
Figure A-4 illustrates this point. Green's Function is evaluated for both the baseline G(f, 0), as
well as the perturbed case G(f, ai). Optimally, Green's Function is linear between the two points.
If it is, then the numerical derivative is accurate (left plot). If not (right plot), then the numerical
derivative is inaccurate.
The magnitude of the perturbation must be chosen with care. One expects perturbation of
the input sound velocity profile (or other environmental setting) would change the output Green's
Function by an appreciable amount. If the magnitude of the perturbation were too small, the
difference between the baseline and perturbed Green's Function would have been below the level of
precision available to the computer. There would have been no difference between Green's Function
in the baseline case and the perturbed case. If the perturbation were too large, the derivative would
not be linear (left plot, Figure A-4).
Unwrapped phase corrected Greens function
300
250
[
1200
E
150
a
o0
100
50
0
295
296
297
298
299
30UU
31
Frequency, (Hz)
302
3U3
JU4
Figure A-3: Variability of Green's Function. This plot shows the unwrapped phase of Green's
Function over a 10 Hertz interval. The units of the colorbar are in radians. The phase has been
multiplied by exp(-j21rfR/co), with f as frequency, R as range, and co as the baseline sound
speed, 1500 m/sec. This removed range and frequency dependence from the phase. Even with this
processing, the phase shows high variability.
Green's Function is treated as linear system. As the SVP perturbation is doubled, the resulting
change between the perturbed Green's Function and the baseline function also doubles:
G(a + 6) - G(a) oc 6
(A.3)
However, if the magnitude of the perturbation becomes too large, the proportional relationship
between the baseline and perturbed Green's Functions ceases to exist. To ensure linearity, the phase
difference between the baseline and perturbed Green's Functions should be limited to 30 degrees.
The range of linearity in Green's Function often changes with respect to frequency. The CRB
equations limit one to choosing a single perturbation value throughout the frequency range in ques-
G(f,a)
G(f,a)
C1
.6 G 2
-------
2G,
2
Co-
U
G
--
Go
-OI
I
a,
G1 G2
I
a)
II
0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
2a,
0
a,
2a1
II
a,
a
Perturbation
Perturbation
Green's function in a linear range
Green's function in a nonlinear range
Figure A-4: Linearity test for Green's Function.
tion. One must carefully select a perturbation magnitude which is within the linear range of Green's
Function at all frequencies of interest.
Application to test case
A.1.1
Prior to evaluation of Green's Function, the perturbation magnitude was determined for all four
parameters under study. A baseline magnitude of 1 x 10-10 m/sec (for parameters 1 and 3), or
1/sec-
1
(for parameters 2 and 4) was selected. Green's Function was calculated every 50 Hz from
200 to 600 Hz, for the baseline case and each individual perturbation. The perturbation was doubled,
and Green's Function recalculated for baseline and perturbed cases. This procedure continued until
each perturbation had been doubled at least twenty (20) times.
In Figure A-5, the plot on the left shows differences in phase, while the plot on the right shows
magnitude differences. The areas in light blue indicate where Green's Function was linear. Each plot
shows the impact of perturbation variation at a different frequency. The magnitude of the selected
perturbation was doubled for each column.
Each perturbed Green's Function was subtracted from the unperturbed case. The magnitude of
the difference between Green's Function was separated from the phase difference, and a logarithm
taken of both results. Then, a numerical derivative was taken with respect to the power of the
perturbation magnitude, and was in turn raised to an exponential. The final result was limited
between zero and 5.0. Since the perturbation was doubled each time, Green's Function was linear
Magnitude le-10
Perturbation 1 Frequency 200 Hz Initial
Perturbation 1 Frequency 200 Hz Initial Magnitude le-10
I
U
q
•
1U
1
magnitude (2An)
Perturbation
14
10
10o
U
L
9
0
0
IU
Ie
19
10
io
Perturbation
magnitude
(2An)
Magnitude
Phase
Figure A-5: Plot of perturbations for parameter 1: water column sound speed, evaluated at 200 Hz.
in the ranges where the final result was 2.0.
Li = exp
where:
and
IcnIln G(f, 2ai)]
i
is the parameter number being studied,
n
is the magnitude of perturbation: 2n ,
G(f, aj)
(A.4)
is Green's Function evaluated at frequency f,
taking into account parameter ai.
An additional constraint is the magnitude of the difference in Green's Function. One must not
exceed a phase difference of 30 degrees between the perturbed and baseline results. To enforce this,
candidate perturbations were fed into Green's Function, and the magnitude of the phase differences
from baseline plotted. The phase differences were limited to a maximum of 50 degrees. Figure A-6
demonstrates this; the colorbar indicates the degree differences between each perturbation iteration.
Perturbation 1 Frequency 200 Hz Initial Magnitude 1e-10
50
10
45
20
40
30
35
130
i40
"D
D50
25
60
20
70
15
cc
10
80
90
0
2
4
6
8
10
12
14
Perturbation magnitude (2An)
16
18
0
Figure A-6: Plot of perturbations for parameter 1: water column sound speed, evaluated at 200 Hz.
Colorbar units are in degrees. It is desirable to keep the phase difference below 30 degrees.
Evaluation of all the plots resulted in the selection of environmental perturbations for each
parameter. The perturbations are listed in Table A.1.
Parameter
1
2
3
4
Description
Sound speed in water column
Slope of sound speed in water column
Sound speed in bottom
Slope of sound speed in bottom
Column
on plot
4
15
4
15
Perturbation
Magnitude
Selected
1.6 x 10- m/sec
3.2768 x 10-6 sec 1.6 x 10- ' m/sec
3.2768 x 10-6 sec -
Table A.1: Selected perturbations for environmental parameters under study.
A.2
Frequency Spacing
Another objective is to determine how signal bandwidth affects the CRB for a given parameter.
SuperSNAP[35], the selected normal mode propagation model, solves for modes and modeshapes
only for individual frequencies. A method was needed to extend SNAP for use with wide band signal
modeling.
Fortunately, the CRB equations call for Green's Function evaluated at a single frequency, then
numerical integration of Green's Function through the chosen bandwidth. One need only to determine how finely to sample the bandwidth in order to obtain an accurate integration result.
A.2.1
Calculating the optimal Af
Assuming normal mode propagation, the optimum inter-frequency spacing (A f) can be calculated. Given the distance between source and receiver, the difference in time between the first and
the last mode arrival at the receiver is solved over the entire frequency range. The inverse of this
quantity, oversampled by a factor between two and three yields the desired Af.
A frequency range of interest was established first. For the modified Pekeris test case, this was
from 200 to 400 Hz. Then, a coarse inter-frequency spacing was chosen, and normal mode eigenvalues
were calculated through the frequency range. Inter-frequency spacing needed only be fine enough
to generate an approximation of group velocity. For the shallow water environment in this thesis,
an initial spacing of 5 Hz was selected.
Solving for the mode arrival times required calculating the group velocity for each propagating
mode across all frequencies. Using the eigenvalue associated with each mode, the group velocity was
approximated using the following equation:
2r(f + Af) - 2-rf
(f
df
dkrn - kI,(27r(f + Af)) - krn(27rf)
where:
f
krn(f)
(A.5)
is the frequency (Hz),
is the eigenvalue associated with nth normal mode,
evaluated at frequency f,
and
u (f)
is the modal group velocity associated with nth normal mode
evaluated at frequency f,
Plot of modal group velocity vs frequency
E
o
o0
CL
0a,
0
2(C,
(9
V
0
Frequency (Hz)
Figure A-7: Plot of group velocities for modified Pekeris profile. Each line represents the group
velocity of one propagating normal mode.
The time difference between first and last mode arrival over all modes and frequencies is calculated
by
At = R
where:
R
1
-
(A.6)
i
is the range between source and receiver (m),
Umax
is the maximum group velocity over all modes in
the frequency range of interest,
and
Umin
is the minimum group velocity over all modes in
frequency range of interest.
Plot of minimum and maximum group slowness
x 10-3
E
C)
CO
(n
CO
(.D
0
0
0
0
10
20
30
40
50
60
Mode number
70
80
90
100
Figure A-8: Plot of minimum and maximum group slowness 1/un for modified Pekeris profile.
For the modified-Pekeris test case, the difference between the maximum and minimum group
slowness cases (see Figure A-9) peaked at approximately 1.5 x 10-
3
sec/m.
Multiplied by the
distance between source and receiver (15000 m), this resulted in a At of 22.5 seconds. The inverse
x 10
Plot of difference between max and min group slowness
-3
E
a)
C)
C,
0(U)
(D
o
(9
0
0
10
20
30
40
50
60
Mode number
70
80
90
100
Figure A-9: Plot of difference between minimum and maximum group slowness 1/un for modified
Pekeris profile.
of this quantity was approximately 0.044 Hz. Oversampling this number by a factor of 2 produced
an inter-frequency spacing (Af) of 0.02 Hz. This spacing was used throughout the simulation.
A.3
Normal Mode, Green's Function, and CRB calculation
After the perturbation magnitude, frequency range and spacing are determined, the next step
was to solve for the modeshapes and eigenvalues across all frequencies, for both the baseline and
perturbed ocean environments.
For every frequency and environment under consideration, a separate SuperSNAP job was run.
Ocean environment files were assembled automatically, and each job produced a single output file
consisting of modes and modeshapes for a given frequency and environment. The process repeated
until all frequencies and environments had been simulated.
A.4
Simulation Hardware
This thesis was done entirely using IBM-PC compatible computers running the Linux operating
system. Linux is a free, POSIX-compliant OS for computers using the Intel 80386 (and descendants), Alpha, and SPARC microprocessors.
More information on Linux can be obtained from
http://www.linux.org/.
The bulk of the computational work on this thesis was performed between June, 1995 and
December, 1996. During that time, several different versions of Linux were used, starting with
Linux Slackware 2.3, and ending with Linux Slackware 3.1. For more information on the Slackware
distribution of linux, see http: //www. cdrom. com/.
Several different versions of ocean acoustic propagation codes were employed throughout this
thesis, with differing levels of success. The KRAKEN[31] Normal Mode Program, by Mike Porter
of NRL, was initially used. The sources were obtained from ftp://oalib.nj it. edu/pub/
AcousticsToolbox/atsgi.tar.Z, and compiled using gcc 2.6.3 and f2c. Another normal mode
code, PRUFER[32], developed by Prof. A. B. Baggeroer of MIT, was also utilized. Finally, SuperSNAP [35], written by Finn Jensen of SACLANT Undersea Research Centre
http://www. saclantc.nato. int/models/snap. html was selected as the propagation model of
choice for this thesis.
Every attempt was made to treat these models as "black boxes," using them without modification
as they appeared on the net. In the end, minor modifications had to be made to allow the programs
to run effectively under the Linux operating system.
In order to compile and run KRAKEN, a number of C functions were written, implementing
some of the more obscure FORTRAN functions. More modifications were needed for SuperSNAP;
neither f2c+gcc nor g77 (two FORTRAN compilers shipped with Linux) were able to compile SuperSNAP to the point where it generated correct results. Instead, a commercial FORTRAN compiler
was purchased and used. Information on the Absoft Fortran 77 compiler can be obtained from
http://www. absoft. com/f77_linux.html. SuperSNAP was also modified to perform all computation using double precision arithmetic, and all storage in double precision variables.
Internal
computational mesh sizes were increased in order to accurately generate normal modeshapes.
Substantial amounts of code were written to take these results and calculate normal mode shapes,
group speeds, and the Crambr-Rao bounds.
All code was initially written in ANSI C, then ported over to MATLAB. MATLAB is a technical
computing environment for numerical computation and visualization, marketed by The Math Works
(http://www.mathworks. com/). Although MATLAB provided an excellent development environment, the execution speed of the scripts was suboptimal. The final code was re-ported back over to
C, with only the graphing functions handled by MATLAB.
Most of the code development was performed on the author's Gateway 2000 P5-75 microcomputer. This machine had a 75 MHz Pentium microprocessor with 40 MB of RAM and 5.5 GB of disk
space. The bulk of the normal mode propagation code was executed on six US Navy owned computers, located at several Navy R&D centers spread around the country. Processing power ranged
from a single 100 MHz 80486 up to 133 MHz Pentium computers. All machines had at least 32 MB
of RAM, and were linked together and to the author's machine via NFS and TCP/IP. None of the
Pentium-based machines contained microprocessors with the Pentium FDIV bug.
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