Peter M. Daly. A uthor... """ "... Department of Electrical Engineering an Computer Science January 15, Certified by...b.

Transcription

1 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 MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1997 MvAR 0 6 Peter M. Daly, MCMXCVII. All rights reserved. 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. ~ ' I L n A uthor... """ " 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

2 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 N and N J Thesis Supervisor: Arthur B. Baggeroer Title: Ford Professor of Electrical and Ocean Engineering

3 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 N ), and the ATOC program (Contract N 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.

4 Contents 1 Introduction R oadm ap Background Information Underwater Acoustic Propagation Normal Mode Theory Green's Function Matched Field Processing Ocean Acoustic Tomography Matched Field Tomography Previous W ork CRB derivation Applications of CRB to MFT and OAT Sum m ary Theory and Formulation Crambr-Rao Bound Mechanics of the CRB Application of the CRB to MFT and OAT Matched Field Tomography Ocean Acoustic Tomography Sum m ary Results Environment

5 3.1.1 Source and Receivers Signal to Noise Ratio Parameters... Minimum standard deviation.... Correlation Coefficients High Parameter Correlation E:nvironment Conclusions 4.1 Summary Contributions 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 A.4 Simulation Hardware... calculation

6 List of Figures 1-1 Simplified acoustic environment Simplified acoustic waveguide Normal modeshapes and eigenvalues for a fictitious shallow water environment Sample MFP ambiguity function[l] Sample tomographic configuration Tolstoy's MFT scenario, using air-dropped explosive charges Convergence zone scenario Pekeris shallow water environment Shallow water environment under study, with four environmental parameters shown Normalized source signal, with bandwidths of 10, 50, and 100 Hz CRB results for SNR of-20 db CRB results for SNR of-10 db CRB results for SNR of +20 db Correlation coefficients for SNR of -20 db Changes to environment for high parameter correlation CRB results for SNR of +10 db, uncorrelated parameters CRB results for SNR of +10 db, correlated parameters A-1 Flowchart of computational process A-2 Variability of Green's Function A-3 Variability of Green's Function A-4 Linearity test for Green's Function A-5 Plot of perturbations for parameter 1: water column sound speed, evaluated at 200 H z

7 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 A-8 Plot of minimum and maximum group slowness 1/u, for modified Pekeris profile A-9 Plot of difference between minimum and maximum group slowness 1/un for modified Pekeris profile

8 List of Tables 1.1 Signal types considered by Baggeroer and Borodin SNR levels at each hydrophone, and their corresponding noise variance values CRB results for Parameter 1: water column reference speed CRB results for Parameter 2: water column speed gradient CRB results for Parameter 3: bottom reference speed CRB results for Parameter 4: bottom speed gradient Mean correlation coefficients for modified shallow water case, +10 db SNR MFT correlation coefficient results for CRB shallow water case OAT correlation coefficient results for CRB shallow water case A.1 Selected perturbations for environmental parameters under study

9 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

10 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 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.

11 Acoustic Propagation Acoustic Receivers Source Channel Surface ship with towed array ~ 4.04 A AL ; Vertical Line Array 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

12 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 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.

13 Perfectly reflecting top boundary (e.g: vacuum or air) Acoustic Source Vertical Line Receiver Array 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 Op(r, z) 0 1 Op(r, z) _2 _ r ) + p(z) + p(r, ) (r)(z (1.1) r Or O z ( p z) c2(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. 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)

14 Substituting this answer and re-arranging terms, one finds -1 1 d d(r)] + 1 d 1 dw(z) + 2 4(r) [r[ dr r (r dr P(Z) dz p(z) dz2 C (Z [P(z) dz + c- (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) 1+ [W)- k = 0. (1.4) dz p(z) dz c2 (Z) Boundary conditions must be specified to solve this differential equation. Each boundary condition has physical meaning. For instance, if one were to = 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 = 0 would mean =D 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 /D Cn = dz. (1.5) 42p(z) 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 qm(z) n(z dz = 0 if m : n. (1.6) o kp(z) 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 -Im(r)1@m(z). (1.7) m=1

15 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. 0 Normalized Modeshapes for Shallow Water Environment , X x x Eigenvalues for Shallow Water Environment S....I O x ill - i i r i I.. li x x C Mode Number Mode Number 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 = Figure 1-3 shows the first twenty modeshapes and eigenvalues for a shallow water acoustic waveguide, using a source frequency of 300 Hz 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, G(r, zs, zr) andia(r, z,) oo = 4zm(r, zs)im(zr), m=1 = Tr m(z,) exp [j (kmr + kmr ( 4)] (1.8)

16 where G(r, z,, zr) is Green's Function, km is the mth eigenvalue, Fm (z) is the mth eigenfunction, evaluated at depth z, zs is the source depth, in meters, zr is the receiver depth, in meters, and 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 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.

17 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 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;

18 n U- AMBDR,ML will o I I I I I 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

19 .71ý Acoustic Source/Receiver ) Path between Source and Receiver ARM 'JYK ~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 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

20 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 r r "'1Z m J J 1p ~~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]

21 [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 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), (1.9) 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

22 4(x) = CCoa (x), (1.10) where C, is an eigenvalue to be estimated, and yca is an eigenfunction, supplied through normal propagating modes. 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), (1.11) where a is a vector of the unknown parameters, G(f, a) 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

23 Signal type Baggeroer Borodin Known signal Known magnitude, unknown phase / / Random signal / / 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 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

24 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 Acoustic Source Annn - /--- Bottom (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.

25 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

26 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.

27 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.

28 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 ( lnp(y;a))2 I, (2.2) where a is the quantity to be estimated, y is observed data used as input to the estimator, p,(y; a) is the probability density function of observed data y, taking into account parameter a, Jy (a) is the Fisher Information quantity, and 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,

29 and Aj,(a) ý [Jy'(a)], [Jy(a)]ij = -E [02 lnpy(y; a)] (2.3) (2.4) where a Y is a column vector of parameters to be estimated: a= [al a 2... a ], is a column vector of observed data for use as an input to the estimator: Y = [yl Y2... YN] T, py(y; a) is the probability density function of observed data Y, taking into account parameter vector a, and Jy (a) A&(a) is the resulting Fisher Information Matrix, 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, y - N (p(a), K(a)), (2.5) 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) K-'(a)> K (a 2 1 a i oaj (2.6) 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.

30 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. (2.9) f - To/2

31 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, E[rn(t)r( 2 )] = Rr(t - t 2 ) = Sr(v)e-i 2 rv(t-t2)dv. (2.10) One incorporates Equation 2.10 to estimate the PSD, E [Rn(f 1 lto)r(f 2 To)] -= /2 E [r n(tj ) r f(t 2)]e-j 2 1(f i t -f2t2)dt l dt 2, n f0-/2 f- 0/2 +0 Sr(v) [ JToej2r(v-f / )tidt1 (2.11) S-f I To -To/2 ej2f(v-f2)t2dt 2 dv, S Sr(v) {sinc [27r(v - fl) 2 }sinc 2r(v - f2)j d. Next, one assumes Sr(v) is "smooth" in intervals of -L. 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, S()(2.13) (fl To)R~L (f To) ] = (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(f ITo)Rt(f ITo)] = (f) (2.14) results in a covariance matrix biased by To.

32 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 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), (2.16) where: 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, and 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.

33 By arranging the received signal vector, R (filt o ) R 2 (f To) R = RN(fi IT o ) RI (f21to) R 2 (f2t o ) (2.18) 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) = To Sb(fl)G(fl, a)gt(fl, a) Sb(f 2 )G(f 2,a)Gt(f 2,a) To 0 0. Sb(fM)G(fM,a)Gt(fM,a) S.(fl) o Sn(f 2 )... 0 (2.19) o 0.. 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, and [Jr(a)]ij = Tr [Kl '(fma)kr (fm (fm,a) OKr (fm, a) (2.20) m=1 Kr(fm,a) = To [Sb(fm)G(fm, a)gt(fm, a) + Sn(fm)], (2.21)

34 can be re-written in a more computationally tractable form. Using Woodbury's identity, both the receiver covariance matrix inverse, K '(fm,a) = {S _(fm ) - Snl(fm)G(fm,a) To x [Gt(fm,a)Sn'(fm)G()G(fm,a)+ Sb(fm)-1-1 G t (fm,a)s n (fm) (2.22) ( 1 S() (fm)g(fm, a)gt(fm, a) S(fm) To Gt(fm, a)s '(fm)g(fm,a) + Sb(fm)- 1 and its derivative, Krf, a) ToSb(fm) G (,a)gtfma) + G(fm, a) G (fm,a) aa IJ~i a (2.23) can be written. To aid in term simplification, four quadratic identities, d 2 (fm,a) = Gt(fm,a)Snl(fm)G(fm,a), S(fm, a) 1 + Sb(fm)d 2 (fm,a))' (2.24) (2.25) li(fm, a) ct(fma)s;l(fm) 'ai ' (2.26) and li,j(fm,a) Bai aaj (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, y(fm, a) is close to 2, but at higher SNRs, the Sb (fm)d 2 (fm, a) term becomes large, lowering -(fm, a). With this information, the Fisher Information Matrix (FIM) equation can be expanded, [Jr(a)]i,j = T r S; (sm) - +T] ) M TS ) S[(fm)G(fm,a)Gt(fm,a)Sn'(fm) E d 2 (fm,a) + Sb(fm) X Sb(fm) [G(fm, a) O(fm,a) G(fm,a)(fm, a)] (2.28) d2(m, a ) + I abaj iaj asb-j(m)

35 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, [Jr(a)]i,j -Tf, S (f )y(f,a) {Re [d2(a)ij(f d ra)(f,, - 1(f, a) (2.30) - t(f, a)re [l1(f, a)] Re [Ij (f, a)]} df, with d 2 (fm,a) = G t (fm,a)s (fm)g(fm, a), 2 (f, a) = Sb(fm)d 2 (fm, a))' li(fm,a) = Gt(fm,a)S; (fm) OG(fm,a) Bai and ij(fma) = c9gt(fm, a) )G(fm, a) Bai aand (fa) l(f9m) a simplified equation for the CRB applied to Matched Field Tomography results. Baj a 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)

36 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) and E[R(fi To,a)Rt(f2ITo, a)] = arsh(fl, f2) +T o Sn(fi, f 2 ) (2.33) 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, SH(fl, fl) SH(fi, 2).. SH(fi, fm) Kr(a) = abt To SH(f 2, fl) SH( 2, f2) SH(f 2, fm) (2.35) SH(fM, f)i) SH(fM, f2) SH(fM, fm) Sn(fi, fl) 0... Sn(f2, 2) "' ,. Sn (fm, fm) which is not block-diagonal. The next step is solving for the matrix inverse of K,(a) and derivative

37 with respect to a 2. 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(f 1 )G(fi, z 1, a) S,(fl)G(f 1, z 2, a) H(a) = Ss(fl)G(f,, zn, a) S (f 2 )G(f 2, zl, a) (2.36) 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, Kr1(a) 1 S -'H(a) 1 S [Ht(a)S'IH(a) +o~'] Ht(a)S.'} (2.38) = S-1 TO I n 1 Ht(a)S,1H(a)Ht H bt ()+ 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, H(a)Ht(a)S '1 2 Ht(a)Sn H(a) + Ub- [S - 1 H(a) H t n (a) S 1 ] n Ht(a)Sn-H(a) + - [H(a) oht ( 0aj 8Ht(a) + H(a) +ai + ai + H(a Bay Ht (a)] It(a)] (2.40) Attention should be focused on one of the the quadratic terms in the denominator of Equation 2.40.

38 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) S j(fi,zi,zi) 0 S Sn (fl, z 2, Z 2 ) S0 0 "" o o Snl(fl,zN, ZN) " 0 S (f2, Z, Z1) ".. 0 o 0 "o' 0 0 '" S I(fM,ZN,ZN) H(fl,zi,a) H(fl, z 2, a) X H(f, ZN, a) H(f 2, zi, a) (2.41) 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). m=1 n=1 (2.42) Retaining the vector notation for the spatial dimension, one would arrive at an expression H t (a)sn 1 H(a) = M Ht(fm,a)S-l(fm)H(fm,a), m=1 (2.43) similar to that found in the MFT derivation.

39 Expanding the H(f,n, a) vector into its components, Hi (a)s 'H(a) M = Ht(fm,a)S;l(fm)H(fm,a) m=1 M Ss(fm)Gt(fm,a)Sn'(fm)G(frm,a)Ss(fm) m-1 M = I(fm)l 3 2 Gt(fm,a)Sx'(fm)G(fm,a) m-1 (2.44) shows the quadratic term in its simplified form. Note the quadratic expression inside the summation is the d 2 (fm, a) term found in the MFT derivation. Substituting this expression in, and converting the summation to an integral, Ht(a)SnlH(a) M I S (fm)12 Gt (fm,a)s;n'(fm)g(fm,a) -A f m-1 I S9(fm)1 2 d 2 (fm,a)afs ST f, IS(f)12 d 2 (fa)df = d2 (a), (2.45) 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 -yt(a) lit(a) a2tt(a ) ' 0 H ( ) " Htb(aa)S, 1 = Ht(a)S;1O = T, IS,(ff) 2 l(f,a)df, (2.46) (2.47) and lijt(a) =S; ' = T], ISS((f)1 2 I4,,(f, a)dfl. (2.48)

40 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 Ht(a----) H- a)ht(a)] OHt(a) &H(a) + Ht(a)] -yt(a),h(a)ht(a)s-] 2 [H(a) Oa- + H a), (2.49) expanded to sixteen terms, and simplified again. The final expression, Ji,j(a) = b(toy(a) Re [d2(a)li,j(a)- li(a)l (a)] + y(a)re [li(a)] Re [j(a)]), (2.50) with d 2 (a) -(a) 1i(a) and l4,i (a) = f IS 8 (f)1 d 2 (f, a)df, 1±+ T, d2(a)' = aw SS(f)1 2 1i(fa)df, = faw ISs(f) 2 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)]} (2.51) with d 2 (a) -y(a) 1 i(a) and li4j (a) Aw IS,(f) 2 d 2 (f, a)df, d 2 (a)' = WISS(f)1 2 i(f, a)df, = law ISs(f)1 2 1 i,j(f,a)df, would reduce to a simpler form.

41 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.

42 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) c = 1500 m/sec 1 p = 1000 kg/m 3 Water Layer c = 1800 m/sec p = 1800kg/m Sediment Layer 140 Bedrock I I I I I I 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.

43 Originally, the Pekeris model provided for a bottom composed of a fluid infinite halfspace. However, this model was not considered very realistic, so a subbottom was added. The final model had a water column of 100 m, with a sound speed of 1500 m/sec and a density of 1000 kg/m 3. Below was a 50 m sediment layer, with a sound speed of 1800 m/sec, density of 1800 kg/m 3, and compressional attenuation of 0.15 db/ap. Finally, a rigid subbottom was chosen. This final layer was 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. 0 Distance between source and receivers (km) 15 I I 0 - Signal Source 20 - at 20m %I p = 1000 kg/m 3 Water Layer 19 receiver elements evenly spaced from 5 to 95 m depth Sediment Layer Z' Z ' " Bedrock c,,f=5250rr'1ec %= 0.1 db/ X / c,= 2500 m/sec /a,=0.2 db/;x, I I I I I I Sound Speed (m/sec) Figure 3-2: Shallow water environment under study, with four environmental parameters shown. Again, the depth depended sound velocity profile is superimposed on the diagram. Dashed lines indicate the effect of the four parameters under study on the sound velocity profile.

44 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) = (3.1) where: f = frequency (Hz), S, (f) = source signal, evaluated at frequency f, fc = center frequency of signal (Hz), and bw = bandwidth of signal (Hz). 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: 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

45 Source spectra for MFT and OAT problems n n Frequency (centered at 300 Hz) )0 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, (3.3) where and SNR SL TL NL is the Signal to Noise Level, in db, is the Source Level, in db, is the Transmission Loss, in db, is the Noise Level, in db.

46 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 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 Noise Variance, a n (db re 1 ppa at I m) (db re 1 ppa at 1 m) x x x x x 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.

47 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. a 2 : Gradient of sound velocity in water column, 3. a 3 : Bottom sound velocity, and 4. a 4 : 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) = [ ai] + [a 2 * (z )] 0 < z < 100m, and (3.5) c(z) = [ a3] + [a4 * (z )] 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.

48 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( aw) (3.7) where and & (AW) a,b is a log-linear approximation to CRB curve, is the bandwidth of source signal, are coefficients. Tables 3.2 through 3.5 show the curve fit results for each parameter. SNRII (db) II a b db / decade o at 1 Hz (m/sec) MFT x x x x x x10-1 OAT x10 - ' x10 - b x x x x10 - 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 db per decade, while 3 has a slope of 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

49 SNRI a b db / a at 1 Hz (db) decade (1/sec) MFT x x x x x x10-5 OAT x x x x x 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 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 m/sec and 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

50 SNR I a b db / a at 1 Hz (db) decade (m/sec) MFT x x x x x x10-7 OAT x x x x x 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.

51 SNR (db) a b db / decade a at 1 Hz (1/sec) MFT x x x x x x 10 - OAT x x x x x x10-4 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 = EMFT = EOAT (3.8) = T f Sb(f)df = (f)i 2 df J W J W where E (AW) Sb(f) s (f) To and ab 2 is the energy contained in the source signal, is the bandwidth of source signal, is the power spectral density of the source signal for MFT, is the deterministic source signal for OAT, is the observation time, in seconds, 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

52 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 ( db/decade for Parameter 2, and db/decade for Parameter 4), showing nearly level MSD as bandwidth is increased. Also, the OAT bounds have a negative slope ( db/decade for Parameter 2; 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: x 10-2 sec - 1 and 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 sec - 1, and 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 db/decade drop to 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 db/decade at -20 db SNR, to 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 x 10-5 m/sec at -20 db SNR, to x 10-5 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 - 1 at -20 db SNR, to 9.60 x 10-3 sec - 1 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

53 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 x 10-7 m/sec; for Parameter 2, the MSD dropped to x 10-4 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 db/decade, while the OAT curve had a slope of 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.

54 Calculation of the correlation coefficient provided a good measurement of the coupling, with PxY - xy (3.9) where PXy is the correlation coefficient for random variables x and y, Ax is the covariance of two random variables, x and y, and ax,,a 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 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.

55 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 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 3-8). 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 MFT OAT Coefficient (original) (correlated) (original) (correlated) P P P P p P 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 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

56 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 x 10-5 m/sec, from its original 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. SNR P12 P13 P14 P23 P24 P34 (db) I Table 3.7: MFT correlation coefficient results for CRB shallow water case. SNR (db) P12 P13 P14 P23 P24 P Table 3.8: OAT correlation coefficient results for CRB shallow water case.

57 L -3 IU Cramer-Rao bounds for reference sound speeds in water and bottom E 10-4a C 0 " , Cf) 4 A MFT OAT 1 - -MFT3 OAT Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -20 db Cramer-Rao bounds for sounds speed gradients in water and bottom ".ti C 0 co cc S10 ca Wn - MFT2 -- OAT MFT4... OAT 4 S Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -20 db Figure 3-4: CRB results for SNR of -20 db.

58 ^-4 10 Cramer-Rao bounds for reference sound speeds in water and bottom (D E 0 u -5., S10 (D Cz ca.) -n MFT 1 OAT MFT3... OAT Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -10 db 10-1 Cramer-Rao bounds for sounds speed gradients in water and bottom oa) 0 " -2 >10 a) Cz ca U) MFT 2 - OAT2 - -MFT4 OAT I......, I......~ Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -10 db Figure 3-5: CRB results for SNR of -10 db.

59 L -5 Iv Cramer-Rao bounds for reference sound speeds in water and bottom 1-6 E 10 C a, -o ", 10-7 "4" MFT 1 - -OAT MFT3... OAT Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 20 db -2 1U Cramer-Rao bounds for sounds speed gradients in water and bottom a, O 0 a) CU o 10 ca MFT 2 -- MFT4 OAT4 I -5 1n Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 20 db Figure 3-6: CRB results for SNR of +20 db.

60 E E L CRB: Shallow water case > p23 p14 _ '~"--- _ - p24 p _1 111 I 1 I 111 I I11111,,. 1 ) Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: -20 db Figure 3-7: Correlation coefficients for SNR of -20 db.

61 Original low correlation environment U Modified high correlation environment Water column Bottom 1I ~V sound speed (m/sec) sound speed (m/sec) Figure 3-8: Changes to environment for high parameter correlation.

62 [ 10-5 Cramer-Rao bounds for reference sound speeds in water and bottom a.) U) oc,) cn E C 0 (U O > 10 a) -6 e 0 co CD... OAT 3 4 A-7 I ' 1 ' I I 1 I 02 I I ' Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 10 db I IU - 2 Cramer-Rao bounds for sounds speed gradients in water and bottom 3. ~ 1... OAT 4 1 n I.. I..... I Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 10 db Figure 3-9: CRB results for SNR of +10 db, uncorrelated parameters.

63 S-3 Iv Cramer-Rao bounds for reference sound speeds in water and bottom toa) t- 0) MFT 1 -- OAT MFT3 OAT I I,.I ' I Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 10 db 4-2 IU Cramer-Rao bounds for sounds speed gradients in water and bottom -r UAIn Bandwidth centered at 300Hz, obs. time of 50 sec, SNR: 10 db Figure 3-10: CRB results for SNR of +10 db, correlated parameters.

64 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 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 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 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 The highest coupling was witnessed between the bottom sound speed and gradient, with a correlation coefficient of

65 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

66 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 Orthogonal Functions (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.

67 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.

68 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.

69 START Choose which parameters for CRB calculation Determine frequency spacing for broadband modeling Calculate normal modes Determine magnitude of perturbation for each parameter Choose frequency range for modeling Calculate Green's function Solve for Fisher Information Matrix FINISH A.1 Perturbation Magnitude Figure A-1: Flowchart of computational process 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.

70 Magnitude of Greens Function E Z g I Frequency (Hz) 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).

71 Unwrapped phase corrected Greens function E a o0 [ UU U3 JU4 Frequency, (Hz) 0 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(-j21r fr/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-

72 C1 G(f,a) G(f,a).6 G 2 2G, -- 2 G Coa, o a) U G I I I I a I -OI G1 G 2 I I I I I I I I I I I I I I I 0 a, 2a, 0 a, 2a 1 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. A.1.1 Application to test case Prior to evaluation of Green's Function, the perturbation magnitude was determined for all four parameters under study. A baseline magnitude of 1 x 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

73 Perturbation 1 Frequency 200 Hz Initial Magnitude le-10 Perturbation 1 Frequency 200 Hz Initial Magnitude le-10 I U q 1U o Perturbation magnitude (2An) Phase U L IU Ie io Perturbation magnitude (2An) Magnitude 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 I cni ln G(f, 2ai)] (A.4) where: i is the parameter number being studied, n is the magnitude of perturbation: 2 n, and G(f, aj) 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.

74 Perturbation 1 Frequency 200 Hz Initial Magnitude 1 e i40 "D D cc Perturbation magnitude (2An) 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.

75 Evaluation of all the plots resulted in the selection of environmental perturbations for each parameter. The perturbations are listed in Table A.1. Parameter Description Column Perturbation on plot Magnitude Selected 1 Sound speed in water column x 10 - m/sec 2 Slope of sound speed in water column x 10-6 sec - 3 Sound speed in bottom x 10 - ' m/sec 4 Slope of sound speed in bottom 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

76 mode across all frequencies. Using the eigenvalue associated with each mode, the group velocity was approximated using the following equation: (f df dkrn - 2r(f + Af) - 2-rf ki,(27r(f + Af)) - krn(27rf) (A.5) where: f is the frequency (Hz), krn(f) 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 0 CL 0a, 0 2(9 (C, V 0 Frequency (Hz) Figure A-7: Plot of group velocities for modified Pekeris profile. Each velocity of one propagating normal mode. line represents the group