Robust source localization in an acoustic waveguide

Robust source localization in an acoustic waveguideBrian F. HarrisonNaval Undersea Warfare Center, Newport, Rhode Island 02841

Richard J. Vaccaro and Donald W. TuftsDepartment of Electrical Engineering, University of Rhode Island, Kingston, Rhode Island 02881

~Received 2 May 1995; accepted for publication 22 January 1996!

Matched-field source localization methods attempt to estimate the range and depth of a source in anacoustic waveguide. These methods give good results when the waveguide parameters are knownprecisely; however, matched-field methods have been shown to be very sensitive to model mismatchresulting from errors in the assumed environmental parameters. Described in this paper is anapproach which minimizes model mismatch, caused by uncertainty in the sound velocity profile, byjointly estimating the environmental parameters and source location. An efficient way to initializethe maximum-likelihood search by projecting the received data onto subspaces corresponding toregions in parameter space is proposed. Monte Carlo simulation results in a shallow-waterenvironment are reported. ©1996 Acoustical Society of America.

PACS numbers: 43.60.Gk, 43.20.Mv@JLK#

INTRODUCTION

Matched-field source localization methods attempt to es-timate the range and depth of a source in an acousticwaveguide.1,2 The measured sound field is compared, overthe set of possible source locations, to the sound field model~called the replica field in the matched-field literature! whichis based on the environmental parameters of the waveguide.The source location producing the best match between themeasured and modeled sound fields is assumed to be the truesource location. These methods have been shown to be verysensitive to model mismatch resulting from errors in the as-sumed environmental parameters.3,4 Small errors in the as-sumed environmental parameters can introduce large errorsin the source location estimates, especially in the presence ofnoise. This is a major difficulty which must be overcome inorder to apply matched-field methods to practical, real-worldproblems.

In this paper, we describe a two-step approach whichminimizes model mismatch by estimating the environmentalparameters and source location parameters together. In thefirst step of the procedure, the measured sound-field datavector is projected onto subspaces corresponding to regionsin parameter space. The region producing the largest projec-tion norm is assumed to be the most likely to contain the trueparameters. A maximum-likelihood parameter search is thenconducted over that region in step two. Initial environmentalparameter values are obtained from nominal,a priori infor-mation. Other approaches for dealing with environmental pa-rameter uncertainty have appeared in the literature.5–10

I. SIGNAL MODEL

The signal received by thel th sensor of a vertical linearray ofM omnidirectional sensors in an acoustic waveguidecan be modeled mathematically as11

yl~ t !5s~ t !* g~ t,zl uQ,C!1nl~ t !, ~1!

wheres(t) is the signal emitted by the source,g(t,zl uQ,C! isthe Green’s function of the medium,nl(t) is the additiveGaussian noise at thel th sensor, and* denotes convolution.The l th receiver depth is denotedzl , the vectorQ denotesthe unknown source location parameters, andC5[C1•••CP]

T is the vector of parameters used to describethe medium~e.g., parameters of the sound velocity profile,channel depth, sediment properties, etc.!. In this case, theparameters of interest are range and depth, therefore the es-timation problem is to determine

Q5@r ,zs#T.

The transfer function between the source and the re-ceiver located at depthzl can be obtained by Fourier trans-forming the Green’s function of the medium,

G~v,zl uQ,C!5F $g~ t,zl uQ,C!%. ~2!

At large horizontal distances between the source and receiverin an acoustic waveguide,G(v,zl uQ,C! can be approxi-mated by a normal mode expansion12

G~v,zl uQ,C!5 (m51

Q

am~v,Q,C!fm~v,zl ,C!, ~3!

where Q is the number of propagating modes andam~v,Q,C! is complex valued. The modal functionsfm(v,zl ,C! are found by solving the separated waveequation.13

If we sample the signals received at each sensor over aninterval of T seconds, we can obtain a frequency domainrepresentation of the output of each sensor by discrete Fou-rier transforming~DFT! the samples. WhenT is much largerthan the difference in propagation times of the fastest andslowest modes traversing the entire distance between thesource and array, a frequency domain approximation to~1!can be written14

yl~vn!5s~vn!G~vn ,zl uQ,C!1nl~vn!, ~4!

384 384J. Acoust. Soc. Am. 100 (1), July 1996 0001-4966/96/100(1)/384/8/$6.00 © 1996 Acoustical Society of America

Downloaded 23 Oct 2012 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

whereyl(vn) is the nth DFT coefficient of thel th sensor,s(vn) is thenth DFT coefficient of the source, andnl(vn) isthenth DFT coefficient of the noise at thel th sensor.

Equation~4! can be written in matrix form as

y~vn!5a~vn ,Q,C!s~vn!1n~vn!, ~5!

where

y~vn!5@y1~vn!•••yM~vn!#T,

a~vn ,Q,C!5@G~vn ,z1uQ,C!•••G~vn ,zMuQ,C!#T,~6!

n~vn!5@n1~vn!•••nM~vn!#T.

Equation~5! can be generalized to include multiple sourcesas in Ref. 11.

The vector a~vn ,Q,C! can also be written as thematrix-vector product

a~vn ,Q,C!5V~vn ,z,C!m~vn ,Q,C!, ~7!

where z is the vector of receiver depths,z5[z1•••zM]T,

V~vn ,z,C! is aM3Q matrix of modal amplitudes~i.e., themodes sampled at the receiver depths!

V~vn ,z,C!

5F f1~vn ,z1 ,C! ••• fQ~vn ,z1 ,C!

A A

f1~vn ,zM ,C! ••• fQ~vn ,zM ,C!G , ~8!

andm~vn ,Q,C!5@a1~vn ,Q,C!•••aQ~v,Q,C!#T.

II. MAXIMUM-LIKELIHOOD ESTIMATOR

A. Known environment

Consider the case of a bandlimited signal that can berepresented byL DFT coefficients. Assuming the noise pro-cesses are uncorrelated across frequency, the likelihood func-tion for N independent observations of the sensor array isPn51

L p($yi(vn)% i51N uQ,$si(vn)% i51

N ) which equals

`~Q,si !5K expH 2 (n51

L

(i51

N

iyi~vn!

2a~vn ,Q,C!si~vn!iRn21~vn!

2 J , ~9!

whereK is a constant,Rn21(vn) is the inverse of the noise

cross-spectral matrix,Rn(vn)5E$n(vn)nH(vn)%, which is

assumed known, and the superscript ‘‘H ’’ denotes complex-conjugate transpose. Maximization of~9! is equivalent tominimizing the argument of its exponential. Therefore, themaximum-likelihood~ML ! estimates ofQ and si result byminimizing the log-likelihood function

Q,si~vn!5arg minQ,si

(n51

L

(i51

N

iyi~vn!

2a~vn ,Q,C!si~vn!iRn21~vn!

2. ~10!

Noting that for fixedQ, the optimal values ofsi(vn) are

@si~vn!#opt5a†~vn ,Q,C!yi~vn!, ~11!

wherea†5~aHRn21a!21aHRn

21. Substituting [si(vn)] opt into~10! results in a simpler optimization problem involving onlyQ,

Q5arg minQ

(n51

L

(i51

N

i~ I

2a~vn ,Q,C!a†~vn ,Q,C!…yi~vn!iRn21~vn!

2, ~12!

or equivalently,

Q5arg maxQ

(n51

L

(i51

N

ia~vn ,Q,C!a†~vn ,Q,C!

3yi~vn!iRn21~vn!

2

5arg maxQ

l ~Q!, ~13!

where l ~Q! is proportional to the log-likelihood function sowe will also refer to it as the log-likelihood function. There-fore,

arg maxQ

l ~Q!5arg maxQ

`~Q!,

where`~Q! results from substituting the optimal values ofsi(vn) into `~Q,si! as was done above.

For the case of spatially white, uncorrelated sensornoise, the ML estimator of~13! assumes the form of the wellknown ‘‘Bartlett’’ processor

Q5arg maxQ

(n51

L

(i51

N

uaH~vn ,Q,C!yi~vn!u2 ~14!

with unit norm replica vectors,ia~vn ,Q,C!i51.

B. Uncertain environment

The ML estimator of~13! assumes precise knowledge ofthe environmental parameters of the channel. Errors in theassumed environmental parameters will cause the maximumof ~13! to occur at a value ofQ away from the true sourcelocation. In this case, the uncertain environmental parametersneed to be estimated also and the ML estimator for thesource location and the environmental parameters is

F5arg maxF

(n51

L

(i51

N

ia~vn ,F!a†~vn ,F!yi~vn!iRn21~vn!

2,

~15!

where the vectorF5@QTCT] T is a concatenation of all theparameters to be estimated.

Consider an environment in which the sound velocityprofile ~SVP! is uncertain. The SVP can be modeled as theweighted sum of empirical orthogonal functions~EOF!,w l(z),

15 as

c~z!5(l51

J

glw l~z!1c0~z!, ~16!

wherec0(z) is the mean sound speed and21<gl<1. TheEOF’s can be computed from a historical database of mea-sured SVP files by performing a singular value decomposi-

385 385J. Acoust. Soc. Am., Vol. 100, No. 1, July 1996 Harrison et al.: Robust source localization


tion of the estimated SVP covariance matrix.16 The environ-mental parameter vectorC contains the EOF coefficientsglin ~16!.

III. ML ESTIMATOR IMPLEMENTATION

A. Conventional implementation

The implementation of the ML estimator in~15! requiresa nonlinear multivariate search over the source location anduncertain environmental parameters. An efficient means ofsearching the log-likelihood surface for its peak is requiredfor the estimator to be useful in practical applications. Gen-erally, the search is conducted in two steps. First, a coarsegrid search over the parameters to be estimated is performedon the surface. This coarse estimate is then used as the initialestimate for a gradient-based search in step two. Unfortu-nately, poor performance can result if the grid spacing isselected too coarsely and the peak is narrow or if the log-likelihood surface is very oscillatory and the initial estimateis not near the main lobe of the peak.

As an example, we will look at the log-likelihood sur-face @the norm-squared in~15!# for a typical acoustic wave-guide scenario. The waveguide is the same as that used inRef. 7 with the summer sound velocity profile. It consists ofa 120-m deep, range-invariant water layer above a medium-to-hard bottom. The SVP is modeled using a single EOF in~16!, i.e.,J51, with g50.4. Figure 1 shows the mean soundspeed and the EOF. This EOF is representative of the surfacewarming/cooling effects experienced in the summer. The re-ceive array consists of eleven evenly spaced sensors at aseparation of 5 m with the shallowest sensor at a depth of 25m. We consider the case of a single source of frequency 200Hz located at 22 000 m in range and 40 m in depth. In thiscase, the norm in~15! is the norm of the data vector pro-jected onto a replica vector which is a function of sourcelocation and environmental parameters. The noise cross-spectral matrix is assumed to be the identity.

Figure 2 shows a densely sampled cross section through

the log-likelihood surface parallel to the depth axis withgand r fixed at their correct values.~This figure and Fig. 3also show cross sections of the smoothed log-likelihood sur-face, which is defined in the next subsection.! A denselysampled cross section through the log-likelihood surfacewith g andzs fixed at their correct values is shown in Fig. 3.Notice the oscillatory nature of the cross sections and thesharpness of the peaks. At the cost of increased computation,more grid points could be computed to decrease the possibil-ity of missing the peak in the initial grid search. The main-lobe widths and sidelobe heights of the cross sections varysignificantly with frequency, source location, and environ-ment. Therefore, without completea priori knowledge of theenvironment and the source and its location, selection of themost efficient grid spacing is difficult.

FIG. 1. Mean sound velocity profile and empirical orthogonal function usedin examples. FIG. 2. Cross sections of exact and smoothed log-likelihood surfaces par-

allel to the depth axis.

FIG. 3. Cross sections of exact and smoothed log-likelihood surfaces par-allel to the range axis.



Most of the matched-field literature does not adequatelyaddress the problem of grid spacing selection. The publishedresults of matched-field algorithms generally use an unreal-istically coarse grid but space the grid such that the truesource location is positioned exactly on a grid point. Rangeand depth resolution, which are functions of the waveguidephysics, can be used as guides to grid spacing selection. Fora discussion of range and depth resolution see Refs. 17–19.

B. Replica-subspace weighted projection

Since finding the global peak of the log-likelihood sur-face is a difficult problem, we would like to smooth thelog-likelihood surfacel (F! for the initial search. Smoothingwill allow us to more quickly identify the region most likelyto contain the global peak. A fine search of that region canthen be conducted to yield the ML estimate.

Assuming spatially white, uncorrelated sensor noise, theBartlett processor for the source location and environmentalparameters is given by

F5arg maxF

(n51

L

(i51

N

uaH~vn ,F!yi~vn!u2. ~17!

One method for smoothing the log-likelihood surface is topartition the surface into equally spaced regions in range, indepth, and in environmental parameters and sum all the pro-jection norms within each region.

Define the parameter vector at the center of each regionas

FC53r c5

r e1r b2

zcs5

zes1zb

s

2

C1,c5C1,e1C1,b

2A

CP,c5CP,e1CP,b

2

4 , ~18!

where the subscriptse andb denote the ending and begin-ning values of the parameter in the region. We can thensample each of the parameters about the center of the regionto form the sample-point sets,

$r b ,...,r c22]r ,r c2]r ,r c ,r c1]r ,r c12]r ,...,r e%,

$zbs ,...,zc

s22]zs,zcs2]zs,zc

s ,zcs1]zs,zc

s12]zs,...,zes%,

$Cp,b ,...,Cp,c22]Cp ,Cp,c2]Cp ,Cp,c ,

Cp,c1]Cp ,Cp,c12]Cp ,...,Cp,e%,

where]r denotes the sampling interval in range,]zs denotesthe sampling interval in depth, and]Cp denotes the sam-pling interval in thepth environmental parameter. DefinePas a matrix whose columns are all of the possible combina-tions of the parameter samplesP5@F1,...,FC ,...,FV# andAas aM3V matrix whose columns are the replica vectorscorresponding to each of the columns ofP,A~vn ,P!5@a~vn ,F1!•••a~vn ,FV!#. Thus the columns ofAare replica vectors computed on a grid of points over theregion$r b2r e ,zb

s2zes% in range/depth space at each of theG

combinations of the samples of theP environmental param-eters, i.e.,

~19!

In the known environment situation,Fv can be formed with-out $C1 ,...,CP% to provide robustness to range/depth gridspacing alone.

If we identify each region by its center, smoothing of thelog-likelihood surface can be expressed as

l S~FC!5 (FiPP

l ~Fi ! ~20!

and the smoothed Bartlett-processor becomes

F5arg maxP

(n51

L

(i51

N

iAH~vn ,P!yi~vn!i2, ~21!

whereF is defined to be the parameter values at the center ofthe region, i.e.,F5FC . Although F can only be estimatedto within a region in parameter space rather than a singlepoint, resolution performance is not compromised since thisprocedure will be followed by a fine search.

Direct computation of~21! requires the evaluation ofmany log-likelihood points since the parameter space issampled densely. A reduction in computation can be ob-tained using the singular value decomposition~SVD! of A,

A~vn ,P!5USVH5@U1U2#FS1 0

0 S2GFV1

H

V2HG , ~22!

whereU isM3M and unitary,V isV3V and unitary, andS



isM3V and ‘‘diagonal’’ in that^S& i j50 unlessi5 j .20 ThesubmatrixU1 hasr columns corresponding to the ‘‘signifi-cant’’ singular values ofA. After setting the small singularvalues in~22! to zero, we substitute~22! into ~21! to obtain

F5arg maxP

(n51

L

(i51

N

iS1U1Hyi~vn!i2 ~23!

sinceV is unitary. Since the matrixV is not required in thecomputation of~23!, a computational savings can be gainedby calculatingS and U from an eigendecomposition ofA~vn ,P!AH(vn ,P!, i.e.,

A~vn ,P!AH~vn ,P!5US2UH, ~24!

whereS2 is a diagonal matrix of the squared singular valuesof A~vn ,P!.

The magnitude-squared term in~17! is the norm squaredof the data vector projected onto a replica vector; a singlepoint in parameter space. Therefore, the norm-squared termin ~23! can be interpreted as the norm squared of the datavector projected onto a replicasubspace. Since the columnsof U1 are weighted by the singular valuesS1, ~23! will becalledreplica-subspace weighted projectionsand the surfaceobtained by weighted projection onto replica subspaces willbe called thesmoothed log-likelihood surface. For the caseof colored noise,~23! becomes

F5arg maxP

(n51

L

(i51

N

iS1U1Hyi~vn!iR

n21~vn!

2. ~25!

The computational advantage to using~23! over ~21! is thatthere are much fewer columns inU1 than inA ~i.e., r!V!and therefore much fewer vectors to store and inner productsto compute. From~22! we know that any linear combinationof the columns ofA can be approximated by a linear com-bination of the columns ofU1. Additionally, since we aresearching for theregion of maximum ‘‘energy’’ instead ofthe point, the replica vectors inA do not have to be sampledas finely as the vectors in an equivalent replica vector search.Assume the subspace regions are selected to be just largeenough to encompass the main lobe. Then if a subspace re-gion contains the true peak, projecting the data vector onto itwill produce a large norm-squared value because we aresumming values sampled on the mainlobe. With a replicavector search, we have to sample finely enough to ensure thatwe sample the mainlobe at a point higher than the highestsidelobe is sampled.

Every replica vector for a given environment is thecomplex-weighted sum of the columns of the modal ampli-tude matrix for that environment~see Ref. 21!. Therefore, allof the replica vectors lie in the subspace spanned by thecolumns of the modal amplitude matrix. Thus the maximumpossible rank of any subspace is equal to the rank of theaugmented modal amplitude matrix defined by

VA~vn ,z,C!5@V~vn ,z,C1!u•••uV~vn ,z,CG!#, ~26!

whereCi are vectors of possible combinations of the envi-ronmental parameters$C1 ,...,CP%.

Using the same scenario as before, cross sectionsthrough the smoothed log-likelihood surface using the rep-

lica subspace weighted projections were calculated. In Fig. 2,a cross section is shown using subspaces withg fixed at 0.4,a width of 200 m in range over fixed range values of 22000–22200 m, 5 m width in depth, and overlapped the adjacentsubspace by 2.5 m in depth. The values are plotted at thecenter of each subspace. Figure 3 shows a cross section usingsubspaces withg fixed at 0.4, a width of 5 m over fixeddepth values of 35–40 m, 200 m width in range, and over-lapped the adjacent subspace by 160 m in range. Large over-lap of the subspaces was used to depict fine detail in thecross sections.

An alternative to the approach described in this sectionwas proposed in Ref. 5. This alternate approach is essentiallyreplica vector grid searches over source location averagedover all possible environments. Therefore, it appears to besusceptible to the grid spacing difficulties discussed earlierand is computationally intensive. A related approach usingthe SVD to compute a low-rank approximation to a con-straint matrix based on sound-speed perturbations is given inRef. 6. This approach seeks to reduce the sensitivity of theminimum variance beamformer to SVP variations by usingmultiple constraints designed to maintain the beamformergain over the range of wavefront perturbations caused bythese variations. The sector-focused stability method21 formscorrelation matrices of replica vectors for sectors in range/depth space. However, the objective of this method is tostabilize the processing of the minimum variance beam-former in the presence of modal noise. In this case, the cor-relation matrices of replica vectors are used to transform thedata cross-spectral matrix to a lower-dimensional subspace.Simulated annealing and genetic algorithms have also beenproposed as nonexhaustive methods of searching for sourcelocation and environmental parameters in uncertainenvironments.8,9

IV. A COMPLETE ALGORITHM USING REPLICA-SUBSPACE WEIGHTED PROJECTIONS

A. Algorithm

The replica-subspace weighted projection~RSWP!method provides a robust initialization step to a completeestimation algorithm for source location and environmentalparameters. Generally, the range of environmental param-eters to search over is small compared to the range of hypo-thetical source locations. RSWP is an environmentally robustway to focus in on the region of the range/depth plane whichcontains the true peak. Once that region is identified, a finesearch of it over the environmental parameters can be con-ducted to yield accurate source location and environmentalparameter estimates. The environmental parameter estimatescan then be used in known environment processing of sub-sequent source location estimates.

If the subspaces for a given search region are formedsuch that the true source location’s mainlobe is divided be-tween subspaces, then the maximum energy region will alsobe spread across multiple subspaces. Therefore, in the pres-ence of noise the subspace region which produces the maxi-mum norm-squared projection may not always be the trueregion. To compensate for this, a threshold can be set and



any subspace region whose norm-squared projection exceedsit will be included in the fine search. Overlapping the sub-spaces may also be beneficial. The steps of the completealgorithm are then as follows.

~1! Select subspace-defining values~width in range,width in depth, sampling interval in range and in depth!.

~2! Perform RSWP method over the range/depth andenvironmental parameter search space.

~3! Perform fine grid searches using~17! on all sub-space regions whose norm-squared projection is greater thanthe threshold.

~4! The global maximum over source location and en-vironmental parameters is assumed to be the true solution.

In step 1, range width and range sampling interval canbe selected based on the width of the mainlobe in range. Thiscan be calculated for a given environment and frequency as

Dr54p

KMDmin2KMDmax, ~27!

whereKMDmin andKMDmax are the horizontal wave numbersof the lowest- and highest-order modes, respectively.22 Thesubspace range-width should then be selected to be on theorder ofDr . This would allow all of the energy in the main-lobe to be included in a single subspace. In order to ensurethat there is at least one range sample on the mainlobe, therange sampling interval can be selected asDr /3. Similarly,the mainlobe width in depth approximately given by

Dz52zmaxQ

, ~28!

wherezmax is the depth of the waveguide andQ is the num-ber of modes,17 can be used to select the depth width anddepth sampling interval. It has been our experience that thenumber of subspace regions in step 3 is much smaller thanthe total number of regions in step 2 even at low signal-to-noise ratios.

B. Simulation results in a shallow-water environment

In order to demonstrate the performance of the RSWPalgorithm, Monte Carlo simulations were conducted usingthe acoustic waveguide and the array described in the previ-ous section. The source signal was assumed to have a uni-form band from 200 to 210 Hz and was located at a range of22000 m and at a depth of 40 m. The frequency resolution inthe Fourier decomposition was 2 Hz. Although the band-width of the signal in this simulation is only 10 Hz, theRSWP algorithm can process signals of any bandwidth. Asbefore, the SVP was modeled using a single EOF withg50.4. The range/depth search space was 20 000 to 24 000m in range and 10 to 90 m in depth and the assumed range ofSVP uncertainty in terms of the environmental parametergwas 0<g<0.5. All acoustic model calculations were per-formed using a MATLAB version of the KRAKEN normalmode program.23

To illustrate the effects of environmental mismatch onsource localization, two exact log-likelihood surfaces usingthe signal described above were generated. There was nonoise added to the signal. Figure 4 shows a contour plot of

the surface generated with no environmental mismatch; theenvironmental parameters are known exactly. The highestsidelobe is approximately 2 dB below the peak value. Figure5 shows a contour plot of the surface where the environmentwas modeled as the nominal~i.e., g50!. This amounts to amismatch of 2 m/s of the SVP in the isovelocity layer. Seri-ous degradation has occurred due to this mismatch. The peakis no longer at the correct source location and the sidelobelevel has increased significantly. Therefore, if the source lo-cation estimates were based on the peak of this log-likelihood surface, consistently incorrect estimates would beobtained over a set of Monte Carlo runs.

In the simulations, each subspace region had a rangewidth of 200 m and a depth width of 5 m with a 33% overlapof the regions in range resulting in a total of 480 regions. Theapproximate mainlobe widths from~27! and ~28! for thisenvironment wereDr5210 m andDz512 m. There were

FIG. 4. Exact log-likelihood surface with no mismatch. Contours depictlevels from21.5 to 0 dB.

FIG. 5. Exact log-likelihood surface with SVP mismatch of 2 m/s. Contoursdepict levels from21.5 to 0 dB.



five samples across range and two samples across depth ineach subspace, i.e.,

$r b ,r c250,r c ,r c150,r e% $zbs ,ze

s%.

The environmental parameter was sampled at an interval of0.1 from 0 to 0.5. This results in subspace spanning sets,A,of 60 vectors each. However, after computing the SVD ofeachA it was found that each could be efficiently repre-sented by 7 or 8 vectors.

The array snapshots for each run were computed usingthe model of~5! where s(vn) and n(vn) are independentcomplex Gaussian random vectors of zero mean. The aver-age signal-to-noise ratio ~SNR! is defined by1/L(n51

L (ss2ia~vn ,Q,C!i2/Msw

2 ), whereM511 is the num-ber of array sensors,L56 is the number of DFT coefficients,ss2 is the signal amplitude variance, andsw

2 is the noisevariance.

The results of Monte Carlo runs from26 to 15 dB SNRwith 100 runs per SNR are presented in Fig. 6. There werefive array snapshots per run~i.e., N55!. To compare theeffects of using different threshold values, results using threedifferent threshold levels are shown. The levels were set cor-responding to a percentage of the peak of the smoothed log-likelihood surface.Ad hocthreshold levels of 98%, 95%, and90% of the smoothed log-likelihood surface maximum wereused. Unique realizations ofs(vn) and n(vn) were gener-ated for each snapshot. A run was considered successful ifthe source location estimates were within a neighborhood of625 m in range and62 m in depth of the true source loca-tion and the environmental parameter estimate was equal tothe true value,C50.4. Figure 7 shows the average numberof regions included in the final search at each SNR for thethree threshold levels. These results clearly demonstrate theeffectiveness of using RSWP in an uncertain environment.

V. CONCLUSIONS

This paper presented a maximum-likelihood estimatorfor source location and environmental parameter estimationin an uncertain sound speed, acoustic waveguide. A environ-mentally robust method using replica-subspace weightedprojections~RSWP! to search the log-likelihood surface wasdescribed. It was seen that a thorough search of the surfacecould be performed with much fewer function evaluations.The RSWP method was seen as an efficient way of identify-ing regions in parameter space which were most likely tocontain the true parameters. A fine search of these regionscould then be conducted to yield accurate estimates of theparameters. Monte Carlo simulation results were shown toillustrate the effectiveness of the RSWP method in an uncer-tain sound velocity profile, shallow water environment.

Currently under investigation are extensions of theRSWP method to the case of many uncertain environmentalparameters. In this case, a stochastic technique for samplingthe environmental parameters given theira priori probabilitydistributions, similar to the technique used in Ref. 10, is usedto form the replica subspaces. Those results will be the topicof a future paper.

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FIG. 6. Number of successful runs versus average SNR for the completesource location, environmental parameter estimation algorithm at three dif-ferent threshold levels.

FIG. 7. Average number of regions included in final search versus averageSNR for three different threshold levels.



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Robust source localization in an acoustic waveguide