Broadband matched-field processing: Coherent and incoherent

Broadband matched-field processing: Coherent and incoherentapproaches

Cristiano Soaresa) and Sergio M. Jesusb)

SiPLAB–FCT, Universidade do Algarve, Campus de Gambelas, 8000-Faro, Portugal

~Received 2 January 2002; revised 7 October 2002; accepted 30 January 2003!

Matched-field based methods always involve the comparison of the output of a physical model andthe actual data. The method of comparison and the nature of the data varies according to the problemat hand, but the result becomes always largely conditioned by the accurateness of the physical modeland the amount of data available. The usage of broadband methods has become a widely usedapproach to increase the amount of data and to stabilize the estimation process. Due to thedifficulties to accurately predict the phase of the acoustic field the problem whether the informationshould be coherently or incoherently combined across frequency has been an open debate in the lastyears. This paper provides a data consistent model for the observed signal, formed by a deterministicchannel structure multiplied by a perturbation random factor plus noise. The cross-frequencychannel structure and the decorrelation of the perturbation random factor are shown to be the maincauses of processor performance degradation. Different Bartlett processors, such as the incoherentprocessor@Baggeroeret al., J. Acoust. Soc. Am.80, 571–587~1988!#, the coherent [email protected]. Michalopoulou, IEEE J. Ocean Eng.21, 384–392~1996!# and the matched-phaseprocessor@Orris et al., J. Acoust. Soc. Am.107, 2563–2375~2000!#, are reviewed and compared tothe proposed cross-frequency incoherent processor. It is analytically shown that the proposedprocessor has the same performance as the matched-phase processor at the maximum of theambiguity surface, without the need for estimating the phase terms and thus having an extremelylow computational cost. ©2003 Acoustical Society of America.@DOI: 10.1121/1.1564016#

PACS numbers: 43.30.Wi, 43.30.Pc, 43.60.Cg@DLB#

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I. INTRODUCTION

The introduction of physical models in underwatacoustic signal processing has been one of the most sigcant advances ever in this field.1–3 Defining a physical modefor a given practical scenario allows for a consistent incsion of a priori information on the signal estimation procesor. Thata priori information consists of the environmentcharacteristics of the propagation scenario which, by meof the solution of the wave equation on that scenario,stricts the received acoustic pressure to a well-defined cof expected signals. It is that reduction of the class ofpected signals that provides the highest performance gaterms of parameter estimation.

Since the definition of a physical model requires tknowledge~or the assumption! of a number of environmentally measurable quantities, the performance of the procebecomes dependent on those quantities. Conversely, ifemitted and received signals are known~or measurable! thenit is, in principle, possible to estimate the environmencharacteristics of the media of propagation—that is the bof the variousmatched-field~MF! based techniques beindeveloped in the last two decades: Matched-field proces~MFP! for source localization, matched-field tomograp~MFT! for ocean properties and matched-field invers~MFI! for geoacoustic parameter estimation.

There are at least two aspects that emerge by their

a!Electronic mail: [email protected]!Electronic mail: [email protected]

J. Acoust. Soc. Am. 113 (5), May 2003 0001-4966/2003/113(5)/2

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levance to the success of MF based techniques: one isability of a given MF processor to accurately pinpoint tsource location while rejecting sidelobes, and the other isimpact of erroneous or missing environmental informati~known as model mismatch! in the final parameter estimateThis study addresses the first aspect, regarding sidelobjection, while considering that the processor is working omismatch free situation. In that case, the capacity of deting the correct acoustic field among very close similar cdidates~the so-called discrimination! largely depends on thedegree of complexity of the received acoustic pressure fiAs an example, a single tone will have two discriminatiparameters: the amplitude and the phase. If a broadbandnal is transmitted, there are as many amplitudes and phas discrete frequencies, and the complexity of the receisignal is naturally increased leading to a higher MF discrimnation. This problem is similar—but not equal—to the detetion problem encountered in classical spectrum estimatio

There are a number of different ways to combine Minformation across frequency that can be classified in tbroad groups: the conventional incoherent methods, thatbased on the direct averaging of the autofrequency inproducts~average of real numbers! and the, say, less conventional methods, that perform a weighted average of the crofrequency inner products where the weights are thequency compensated phase shifts. The latter are genecalled coherent broadband methods since they combine cplex inner products.

Incoherent MF methods were first proposed by Bagroer et al.,4 where geometric averaging was found to be

2587587/12/$19.00 © 2003 Acoustical Society of America

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fective to reduce ambiguous Bartlett and minimum varian~MV ! MFP sidelobes in a shallow-water simulation studThe same principle was used in a countless number ofdies since then. More recently, the frequency domain coent approach was first suggested by Tolstoy.5 Michalopoulourecognized that incoherent processors discarded useful inmation contained in the off-diagonal terms of the crofrequency data covariance matrix.6 Coherent Bartlett andMV processors based on the formulation of ‘‘supervectocontaining field vectors of the frequencies to be proceswere proposed and successfully applied on tracking a sosource in the Hudson Canyon data set.7 Czenszak and Krolikproposed a coherent minimum variance beamformer withvironmental perturbation constrains~MV–EPC! designed fora short vertical array.8 Very recently Orriset al. proposed amatched-phase coherent processor that accounts for thetive phase relationships between frequencies.9 Those phasesare assumed to be unknown and are searched as free peters.

In that classification, time domain methods play a diffent role but can, to some extent, be included in the coheclass. Time domain methods were first suggested by Cl10

under the form of an optimum matched-filter for sourcecalization. The same technique was used by Liet al.11 inlaboratory experimental data. Also Frazeret al.12 testedClay’s technique with simulated data and a single hydphone. In 1992, Milleret al.13 showed, with computer simulations, that it is possible to localize short duration acoussignals in a range-dependent shallow water environment.same approach was followed by Knobleset al.14 with bottommoored sensors using a broadband coherent matchedprocessor proposed by Westwood.15 Time domain source lo-calization was actually achieved with real data by Brienet al.16 using data received on a vertical array in a deep waarea on the Monterrey fan. The technique used was a cbination of time domain filtering for each sensor~matched-filter! and then a space domain beamformer.

Despite the considerable amount of work on broadbmethods there is a lack of understanding on why and whecoherent method provides a better detection or localizaperformance than an incoherent method. This is the mtopic addressed in the present study, that starts by presea physical-based linear data model with suitable randomturbation terms as opposed to the traditional fully stochamodel. Under this model, it is shown that the advantageusing the cross-frequency terms resides in its ability to renoise, while its disadvantage is that the result is limitedthe correlation of the random phase terms together withdeterministic correlation of the channel response acrossquency. An efficient algorithm for combining crosfrequency information is derived that is shown to haveequivalent localization performance than that of tmatched-phase coherent processor with a much lower cputational burden. Then, the performance of the coherentincoherent processors are compared for different numbefrequencies using simulated data. Real data analysis issented to support the physical-based model as well asjustifying the distributions of the random perturbation termFinally, a real data example shows the effect of a wise se

2588 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 C

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tion of frequency bands on the final match of the modelthe data.

II. DATA MODEL

A. The physical data model

A widely used data model forM farfield point sourcesemitting narrowband signals received in a L-sensor arragiven by

y~ t !5A~w!s~ t !1u~ t !, ~1!

wherey(t) is the L-sensor array received acoustic pressuA(w) is the L3M steering matrix, which entries are thappropriate delays for each array sensor and each sourcem atbearing wm , s(t) is an M-dimensional vector with theMsource inputs at timet and u(t) is the observation additivenoise. A common assumption is to consider that the addinoise is white, Gaussian, zero-mean and uncorrelatedthe signalss(t), that themselves are zero-mean and uncolated stochastic processes. This model is useful for descing a field of dependent noise sources emitting througnondispersive unbounded media and received on a horizoarray. When dealing with shallow water dispersive scenardeterministic sources and nonhorizontal arrays this modeunable to account for the complexity of the received fielda mixture of correlated~partially! deterministic signal reflec-tions from sea bottom and sea surface.

An alternative approach is to start from the wave eqtion and directly calculate its solution with appropriaboundary conditions and environmental assumptions~e.g.,azimuth and range independent isotropic media, spatial psource, etc.!. In a cylindrical two-dimensional coordinatsystem, the acoustic pressure measured at receiver depzl

due to a point source at ranger and depthzs can be written17

p~zl ,t;r ,zs!52 i

2p E (j 51

JM

s~v!C j~zl !C j~zs!

Ark j

3e2 i ~kj r 2p/4!2g j reivt dv, ~2!

wheres(v) is the source spectrum,$C j ( ), j 51,...,JM% arethe waveguide mode functions, andkj andg j are the modehorizontal wave numbers and mode attenuation coefficierespectively;JM is the number of discrete modes supportby the waveguide.

Under the ray approximation the received acoustic prsure, using the same notation, can be written as

p~zl ,t;r ,zs!51

2p E (j 51

JR

ajs~v!e2 ivt jeiv~ t1up/2vu!dv, ~3!

where the number of eigenraysJR , the ray amplitudesaj andthe delayst j , fully characterize the propagation channel fthe specific source and receiver locations, (0,zs) and (r ,zl),respectively.

Assuming the propagation channel as a linear filter,lows for writing the received signal as the frequency prodbetween the source signals(v) and the channel transfefunction h(v), defined as the sum of modal terms~or rays!

. Soares and S. M. Jesus: MFP: coherent and incoherent approaches

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for a particular source–receiver location. Thus, a suitamodel for the array-received signal from an harmonic souat frequencyv would be

y~zl ,v;r ,zs!5h~zl ,v;r ,zs!s~v!1u~zl ,v!, ~4!

whereu(zl ,v) is a zero-mean stochastic process represing additive observation noise and whereh(zl ,v;r ,zs) canbe easily deduced either from~2! or from ~3! depending onwhich model—normal mode or ray model—is being usedis a common assumption to consider the observation noisbe wide-sense time stationary. Taking into account the Frier transform properties for sufficiently long observatitimes it can be considered that the frequency samplesuare asymptotically uncorrelated.

If the source inputs(t) is deterministic, signal detectiousing model~4! becomes a problem of detecting a determnistic signal in white noise, which optimal solutions are wknown.

In the past decade, with the development of methodsacoustic inversion using deterministic signals, it has bobserved that repeated emissions at very high SNR resuin successive receptions suffering rapid changes in shortintervals possibly caused by small scale environmentalturbations, source and/or receiver motion, and sea surand bottom roughness, which, partially or all together, ctribute to unmodeled fluctuations in the signal part of~4!.

Since such changes cannot be attributed to the noiseto the high SNR, a complex random factora5uauexp(jf)can be included such that the data model is written as

y~v,u0!5a~v!h~v,u0!s~v!1u~v!, ~5!

where a more compact notation has been adopted by inducing a vectorial notation for the L-sensor array asy5@y(z1),y(z2),...,y(zL)# t and similar definitions forh andu, the channel transfer function and the additive observanoise, respectively;s(v) is the source spectrum at frequenv andu0 is a vector with the relevant parameters undertimation. The noise processu is assumed to be uncorrelatefrom sensor to sensor and with random factora. Note thatrandom factora is space invariant but is assumed to be fquency dependent. For the design of optimal estimatorsuseful to consider thata is zero-mean and Gaussian distriuted. Whether that assumption is verified in practice issubject of the next section.

B. Random signal perturbation factor

This section deals with the distribution of the randosignal perturbation factora, introduced in the linear physicamodel ~5!. It is a common assumption to consider that radom factor to be complex zero-mean Gaussian distribut4

which implies that the module ofa follows a Rayleigh dis-tribution and that its phase is uniformly distributed in@2p,p#.18 In case that the real and/or imaginary parts ofacoustic pressure are not zero mean then the envelopelows a Rice distribution while the phase term does notpear to be uniform nor Gaussian distributed~see AppendixA!.

In order to obtain an empirical distribution of the signrandom perturbation, only possible using real data, one

J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 C. Soare

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to, first, assume that the signal-to-noise ratio~SNR! is suffi-ciently high, to be able to neglect the influence of the nou, and second, assume that the deterministic part of thenal, i.e.,h(v,u0)s(v) is time-stationary or slowly varyingUnder these two assumptions a possible estimator,a, of therandom factora at frequencyv is

an5yn

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whereyn , an , andfn are obtained for time snapshotn andfor an arbitrary frequency and receiver. This would impthat the distribution ofuau would be Rayleigh or Rice depending on whethera is zero-mean or not with, however,change on the amplitude axis due to the normalization cstantua0u. As an alternative and, if the stationarity assumtion for h(v,u0)s(v) is suspected not to hold, another esmator can be sought using a time sliding estimator as

an5yn

yn21'

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In this case the interpretation is a bit more elaborated sithe module ofa is the ratio of two Rayleigh~or Rice! ran-dom variables and the phase term is the difference betwtwo uniform variables ifa is zero mean. It is shown in Appendix B that the ratio of two independent Rayleigh distbuted random variables gives a nearly Cauchy distriburandom variable and that the difference of two uniformdistributed and independent random variables gives a pability density function~pdf! for the resulting random vari-able that is triangular in@22p, 2p#. Results obtained on readata using estimators~6! and ~7! are shown in Sec. VI.

III. SECOND ORDER STATISTICS AND BROADBANDMODEL FORMULATION

The correlation matrix can be directly written from~5!as

Cyy~v,u0!5E@y~v,u0!yH~v,u0!#

5E@ ua~v!u2#us~v!u2h~v,u0!hH~v,u0!

1su2~v!I , ~8!

where all terms have been previously defined and supersH denotes conjugate transpose. Equation~5! gives the essential description of the received data model in the narrowbacase. When a time-limited signal~impulse! is emitted by thesource, a significant band of frequencies of the acouchannel is excited giving rise to the need for a broadbaformulation. In order to introduce, as much as possiblecommon frame for the narrowband and broadband casesdefine an extended vector as

yO5@yT~v1!,yT~v2!,...,yT~vK!#T, ~9!

where superscriptT denotes matrix transpose andK is thetotal number of discrete frequency bins. In that case,broadband model can be written as

yO~u0!5H~u0!sO1uO , ~10!

2589s and S. M. Jesus: MFP: coherent and incoherent approaches

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wheresO is a K-dimensional random vector which entries as(vk)a(vk), i.e., the source spectrum multiplied by the radom perturbation factor at each frequencyvkP@v1 ,vK#; thematrix H(u0) is

H~u0!5F h~v1 ,u0! 0 ¯ 0

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where the noise extended vectoruO has an obvious notationsimilar to ~9!. It is interesting to write the correlation matrifor model~10!, which cross-frequency block matrix is giveby

Cyy~v i ,v j !

55us~v i !u2h~v i ,u0!h~v i ,u0!HE@ ua~v i !u2#1su

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i 5 j ,

s~v i !s* ~v j !h~v i ,u0!hH~v j ,u0!E@a~v i !a* ~v j !#,

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where the termE@a(v i)a* (v j )# denotes the correlation othe perturbation factor across frequency. Note that unlikeautofrequency entries (i 5 j ) the cross-frequency terms (iÞ j ) are noise free. This is due to the well-known propertythe Fourier transform for time-stationary processes that guncorrelated cross-frequency bins which might be also usif spatially correlated noise is present. In practice, with finobservation time, that property is only asymptotically vefied, which is often sufficient. In expression~12!, for iÞ j ,there are three contributions: the source cross-spectrums(v i)s* (v j ), the cross-frequency acoustic channel structterm h(v i ,u0)hH(v j ,u0) and the perturbation factor correlation E@a(v i)a* (v j )#. The first term is source dependeand will not be of concern here. The second term is chandependent and may significantly vary with environmenconditions, source position~range and depth! and receivingarray geometry. The third term on expression~12!, for iÞ j , concerns the correlation of the perturbation factor animpossible to obtain from simulations.

IV. BARTLETT MATCHED-FIELD PROCESSING

The Bartlett processor is possibly the most widely usestimator in MF parameter identification. The parametertimate u0 is given as the argument of the maximum of tfunctional

P~u!5E@wH~u!y~u0!yH~u0!w~u!#, ~13!

where the replica vector estimator is determined as the vew(u) that maximizes the mean quadratic power,

w~u!5arg maxw

E@wH~u!y~u0!yH~u0!w~u!#, ~14!

subject towH(u)w(u)51. In the narrowband case, usinmodel ~5! in ~14! gives the well-known nontrivial solution


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wNB~u!5h~u!

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where the denominator is a normalization scalar and themerator contains the signal structure as ‘‘seen’’ at the recing array. This is simply the classical matched filter for tparticular parameter locationu. Substituting ~15! in ~13!gives the well-known generalized conventional narrow babeamformer for parameteru. If the search is made overu andthe maximum is selected, then an optimum mean lesquares estimateu0 of u0 is obtained.

In the broadband case, the estimator of the replica veis given in terms of frequency extended vectors using mo~10!, thus

wO BB~u!5arg maxwO

$wO H~u!H~u0!E@sOsOH#HH~u0!wO ~u!%,~16!

where the expectation of the signal matrixsOsOH relates to thecorrelation of the perturbation factora across frequencyweighted by the source power cross spectrums* (v i)s(v j ).No closed form forwO BB(u) can be given in this case withouexplicit knowledge of that signal matrix. There are a numbof possible implementations that represent suboptimal vsions of~16! with different assumptions for the structurethe perturbation correlation and signal weighting matrix.few cases are reviewed in the next section and a new cputational effective alternative to existing techniques is aproposed.

A. Broadband incoherent processor

The so-called incoherent broadband Bartlett procesoriginally proposed in Ref. 4, implicitly assumes that trandom factor is simplyE@a(v i)a* (v j )#5sa

2d i j , i.e., thatthe random perturbations are uncorrelated across frequand have a constant power. Using that expression of therelation ofa in ~12!, plugged in~16! and solved forw gives

wO inc~u!5H~u!sO

iH~u!sOi , ~17!

wheresO is a K-dimensional vector which entries ares(vk).Thus, by replacement into~13!, allows to obtain the processor expression

Pinc~u!5sa

2(k51K us~vk!u2hH~vk ,u!Cyy~vk ,vk!h~vk ,u!

iH~u!sOi2

~18!

which is nothing more than a source power weighted sumthe diagonal matched-filtered autofrequency block matriof the extended correlation matrixCyy . Notice that ifsa hadbeen assumed to be frequency dependent, a factorsa(vk)would appear as weighting the terms in the summation~18!. In the case of a flat source power spectrum, Eq.~18!reduces to a simple summation of the quadratic terms acthe discrete band of frequencies. When the source pospectrum is unknown but not flat, an unweighted incoherprocessor is generally used which leads to the suboptiincoherent broadband conventional estimator proposedRef. 4.


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B. Broadband coherent processor

Although there is good evidence that for many of treal underwater propagation channels most of the energconcentrated along the main diagonal of the cross-speccorrelation matrix~the autofrequency terms! it is also clearthat the same autofrequency terms would carry the npower as it can be seen in expression~12!. One of the moti-vations when performing coherent processing is to takevantage of the noiseless cross-frequency terms of~12!. Thesecross-frequency terms have no noise but the signal infortion they contain may also be reduced, according to bothchannel cross-frequency structure and the cross-frequcorrelation of the random perturbation factor, as explainedthe preceding section. This explains why in most studconcerned with coherent processing, only the crofrequency off-diagonal terms were used, excluding the dional autofrequency information.9 There are actually severabroadband coherent processors depending on the asstions made for approximating the cross-frequency perturtion terms of the signal matrixE@ sOsOH# of ~16!.

1. Coherent normalized processor

The coherent normalized processor~COH–N! has beenproposed by Michalopoulou7,19 and attempts to eliminate thsource spectrum–perturbation weighting across frequeAt each frequencyv i , a normalized model vector is defineas

nx~v i ,u0!5x~v i ,u0!

xl~v i ,u0!, ~19!

where xl(v i ,u0) is the signal received at sensorl. Thechoice ofl depends on the actual signal-to-noise ratio~SNR!at that particular sensor. In a high SNR situation, if the nocontribution at sensorl is neglected, the normalized damodel becomes

nx~v i ,u!'nh~v i ,u0!1u~v i !

hl~v i ,u0!s~v i !a~v i !. ~20!

Matching this model with an extended normalized replvector yields a perfect match for the signal and a stroncorrelated structure for the noise field due to the noise tin ~20!. In that case thecoherent-normalizedreplica vector iswritten as

wcoh-n~v i ,u!5nh~v i ,u!5h~v i ,u!

hl~v i ,u!, ~21!

and using that expression in the Bartlett processor gives

Pcoh-n~u!

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(j 51

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H~v j ,u0!nh~v j ,u!

1nhH~v i ,u!Cnunu

~v i ,v j !nh~v j ,u!, ~22!

whereCnunu(v i ,v j ) is the cross-frequency correlation m

trix of the normalized additive noise vectornu defined in thesecond term of~20!. Expression~22! shows a perfectly co-herent match for the signal model part whenu5u0 , and anoise term residual which is a constant wheni 5 j , due to the


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white noise assumption, and has a correlation structureiÞ j that is highly dependent on the cross-frequency corretion of the perturbationa~v!.

2. Matched-phase coherent processor

Another approximation to the broadband coherent pcessor has been recently proposed by Orris9 where the cor-relation terms are explicitly included in the replica vectorunknowns and have therefore to be estimated. A new repvector is defined as

wO coh-mp~u!5@hT~v1 ,u!ej fh~v1!,...,hT~vK ,u!ej fh~vK!#T,~23!

where the phase terms@fh(vk);k51,...,K# are the estimatesthat maximize the output power upon summation over senand frequency. Taking into account that, when carryingthat summation, each term has its complex conjugate,energy contained in the imaginary part is lost. The unknophase termsfh are estimated in such a way as to minimithat loss which, ideally, requires the unknown phase termbe symmetric to the phase of the signal matrix terms in~12!.If that is achieved all terms turn into real numbers andsum is carried out in phase. In that case, and for a flat sptrum source, this processor is optimum. Replacing~23! in theBartlett processor expression gives

Pcoh-mp~u!5(i 51

K

(l 51

K

hH~v i ,u!Cyy~v i ,v l !

3h~v l ,u!e2 j @fh~v i !2fh~v l !#. ~24!

In practice, the problem associated to the matched-phasecessor, according to Orris,9 is the computation load necessato obtain the estimatesfh of the phase shiftsfh , for anexhaustive search over a realistic parameter space. Thatputation load is of the order ofo5JK3M3N, whereJ is thenumber of samples for the phase in@0, 2p#, K is the numberof frequencies andM3N is theu parameter search grid~e.g.,range versus depth!. In practice, and as mentioned by Orris9

if the source location and relative phases have to be exhtively searched, computation complexity limits the numbof frequencies toK53 while for a larger number of frequencies efficient search algorithms~e.g., simulated annealing!were proposed.

C. The cross-frequency incoherent processor

The cross-frequency incoherent processor is proposethis paper and represents an alternative to overcome the cputational burden of the matched-phase processor wkeeping the same performance. This processor stemsthe simple idea that the phase corrections for the surfmaximum (u5u0) are

fh~v i !2fh~v j !5/s~v i !s* ~v j !E@a~v i !a* ~v j !#, ~25!

for all i , j 51,...,K which can be seen by direct inspection~12! and where/ means ‘‘phase of.’’ When these correctionare correctly set the value of the maximum is just the suma series of real numbers, which are the modules of the qdratic terms across frequency, i.e.,


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FIG. 1. Ambiguity surfaces computed with synthetic data generated without perturbation factor for the ADVENT’99 scenario at frequencies 300,00,and 600 Hz, at SNR528 dB and for the following processors:~a! incoherent conventional,~b! coherent normalized,~c! matched-phase coherent, and~d!incoherent cross frequency.
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uhH~v i ,u!Cyy~v i ,v j !h~v j ,u!u. ~26!

The value of the maximum of the ambiguity surface obtainwith ~26! is exactly the same as that obtained with~24! withabsolutely no phase parameter search. Therefore, thewould have the same height and the same location, howthe aspect of the resulting surface would be much differbetween the cross-frequency and the matched-phase prsors: the former would have a smooth appearance, muchthe incoherent processor, and the latter would have extremnarrow peaks distributed along the surface with, howeveroverall envelop that is very similar to that of the crosfrequency incoherent processor. Examples on simulatedare given in the next section.

V. SIMULATION RESULTS

This section shows a few simulated data examples ofapplication of broadband MF processors to source localtion. The data was simulated using the C-SNAP model20 in a80 m deep range-independent shallow water scenario simto that of the ADVENT’99 experiment.21 The acoustic sourceis placed at 76 m depth and at 5 km range from a 32-senvertical array. The source is emitting a series of multitonbetween 300 and 600 Hz with 100 Hz increment. The signwere generated in the frequency domain using~4! with anSNR of 28 dB and the correlation matrix was estimatusing 32 snapshots. The noise level was set accordinglthe following SNR definition:

SNRdB510 log(k51

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where

ss2~vk!5

E@ ih~zl ,vk ,r ,zs!s~vk!i2#

L~28!

and

su2~vk!5

E@ iu~vk!i2#

L. ~29!

Figure 1 shows the range-depth ambiguity surfacestained for the above referred broadband Bartlett processPinc ~a!, Pcoh-n ~b!, Pcoh-mp ~c!, andPinc-xf ~d!. In cases~b!,~c!, and ~d! only the cross-frequency terms were used.expected, the incoherent processors~a! and~d!, gave similarsmooth surfaces with a lower sidelobe structure for the crofrequency processor. The coherent processors~b! and~c! alsogave similar responses with a large number of very narrpeaks~up to only 1 m wide in range! that are due to a perfecalignment of the surfaces for all grid points. By formulatinthe matched-phase and the incoherent cross-frequencycessors in terms of normal modes, it can be shown thatcorresponding ambiguity surfaces are oscillating functionsthe distance modulated by an amplitude factor that issame in both processors. The peaky structure shown bycoherent processors results from a periodic phase alignmof the correlation terms at each pair of frequencies. At lSNR the coherent normalized processor~b!, rapidly degradesdue to the SNR limitation pointed out in~20!. As explainedabove the matched-phase and the cross-frequency incohprocessors have analytically the same source detectionformance with comparable peak-to-sidelobe ratios of 2.5and 2.0 dB, respectively. Note that for the coherent procsors a subsampling of the ambiguity surface in rangehide the sidelobe structure. The detection performance of


FIG. 2. Probability of correct source localization obtained on ADVENT’99 conditions simulated data for the broadband incoherent processor~dashed! andbroadband cross-frequency processor~continuous! in the frequency band@500, 600# Hz for the following number of frequencies: 4~a!, 7~b!, 16~c! and also 16but in the band@400, 700# Hz.

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cross-frequency incoherent processor is shown in Fig. 2the model without perturbation. This performance was msured in terms of probability of correct source localizationdetermining how often the peak appeared at the correccation in 50 realizations. The environment is always thatthe ADVENT’99 experiment. The effect of increasing thnumber of frequencies within a relatively small frequenband of 100 Hz around 550 Hz is shown in plots~a! to ~c! ofFig. 2, where the number of frequencies is 4, 7, andrespectively. It can be noticed that the performance ofcross-frequency incoherent processor is always superiothat of the conventional~autofrequency! incoherent processor due to the higher number of frequencies involved andthe noise immunity, despite the inevitable decrease in chnel structure power transmission at certain cross frequenoff the main diagonal. The number of ambiguity surfacincreases asK for the incoherent processor and asK3(K21)/2 for the cross-frequency incoherent processor. Tfact results in a steady increase of the difference in permance with the number of discrete frequencies from 4 tand then to 16. With 16 frequencies there is a gain in detion performance estimated to approximately 4 dB at usedetection probabilities. The result shown in plot~d! was ob-tained for a number of frequencies equal to 16@the same asin plot ~c!# but within a frequency band enlarged to 300 Halways centered at 550 Hz. The result is that there is a sldecrease of the performance of both processors, whiledecrease is stronger for the incoherent cross frequency,there is a net loss of performance of the cross-frequeprocessor relative to the incoherent autofrequency proceOther tests performed for relatively small number of closspaced discrete frequencies clustered around center freqcies along the whole band gives better results than unifordistributed frequencies in the same band.

In practice, with real data, these performance predictiobtained in simulation have to be balanced by the correla


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of the perturbation factor across-frequency contributing tnet decrease of performance when enlarging the bandwaround a given center frequency. That fact clearly favorssolution of using the proposed cross-frequency incoheoperating in closely spaced frequencies clustered at varfrequencies in the useful band.

VI. A REAL DATA EXAMPLE

The ADVENT’99 data set was used as to provide a rworld example for the assertions made in the precedingtions. The ADVENT’99 experiment took place during thmonth of May of 1999, in a nearly range independent areathe Strait of Sicily, Italy. The approximate depth of the arwas 80 m and the acoustic signals were transmitted frobottom mounted sound source and received on ahydrophone vertical array. Various signal sequences at difent frequencies and repetition rates were used. Also, thetical array was successively located at rangesapproximately 2, 5, and 10 km. A complete description of texperimental setup can be found in Sideriuset al.21

A. The perturbation factor

In order to justify the perturbation factor distribution,signal tone at 200 Hz was extracted from the time serecorded during 18 hours on a mid-water-depth hydrophat 10 km range from the signal source. The signal-to-noratio is expected to be.20 dB at that frequency, and thadditive noise is assumed negligible compared to the sigterm. Figure 3 shows the estimated pdf’s based on the higrams obtained for the module—~a! and ~c!—and for thephase—~b! and ~d!. In plots ~a! and ~b!, module and phasepdf’s are estimated using the normalization proposed in~6!.It can be seen that the module is approximately Rayledistributed, with parameterl51 due to the normalization byy0 , while the phase is noncentered~also due to the normal


tio
FIG. 3. Estimated probability density functions of signal perturbation factor from the ADVENT’99 data set at 200 Hz: using first element normalizan forthe module~a! and phase~b!; using a sliding window along time for module~c! and phase~d!.
orth,

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ph

x-

t

a

ing.

ighs-n

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e-

thsr tlow

the

alisticariotes thethe

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ization!, almost uniform in@2p,p# with an outstanding peakat f520.5 rad of unknown nature. Instead, the sliding nmalization of~7!, applied in the same data set, providesresults shown in plots~c! and ~d! for the module and phaserespectively. The module—plot~c!—is in this case approxi-mately distributed according to the approximate Caucgiven by ~B2!, while that of the phase—plot~d!—does notresembles to a triangular function as a result from the difence of two uniformly distributed random variables. The dtribution is approximately symmetric in@22p, 2p#, but has amuch narrower peak than expected for a triangular shapdf. Due to the complicated form of the expression of tphase pdf in the noncentered case@Appendix A, Eq.~A14!#,it is difficult to theoretically predict what could be the epected pdf for the phase random variablefn2fn21 . Somenumerical simulations using expression~A14! and realisticvalues fors suggest that a bell-shaped centered pdf as thaFig. 3~d! can most likely be obtained forma.0 and mb

'0. @Note that the empirical distribution of Fig. 3~d! is, ac-cording to the theory, the autocorrelation of two identicpdf’s as that obtained in~A14!#. A similar behavior was veri-fied on the ADVENT’99 data set at various frequenciesthe interval @200,1500# Hz, with however, an increasinbroadening of the peak of the phase pdf with frequencybroader pdf means a larger value fors which is a wellknown effect leading to highly variable phase shifts at hfrequency making it difficult to accurately predict. This dicussion brings a key question for broadband applicatiothat is to determine which is the degree of correlation ofsignal across frequency.

Using the same ADVENT’99 data set along a wide frquency band@200,1600# Hz, the correlation of the perturbation factor using the normalization~6! was estimated. Theresult is shown in Fig. 4 where a broad diagonal alongwhole frequency band can be observed. Additional effectfrequency bandpass of the two transducers used to covewideband of frequencies can be seen on the artificially


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levels of energy in the diagonal at about 800 Hz, which isoverlap transit in frequency band.

In order to obtain a complete view of the received signcorrelation along frequency one has to add the determineffect of the channel correlation. As an example, a scensimilar to that of the ADVENT’99 was simulated to computhe cross-array coherence of the acoustic channel acrosfrequency band of interest. Figure 5 shows the result ofexpression

Ch~v i ,v j !5hH~v i !h~v j !, ~30!

for v i ,v jP@200,1600# Hz. It can be easily noticed fromthat figure that the energy is not concentrated on a sindiagonal but on a band of frequencies around that diagoThe bandwidth varies with frequency and with sourcreceiver geometry~not shown!, e.g., it tends to be larger alonger ranges due to stronger multipath. There is also a

FIG. 4. Estimated correlation of normalized signal perturbation factor othe band 200–1600 Hz using the LFM data of the ADVENT’99 experimat 5 km source–receiver range.


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nificant amount of energy well apart from the diagonal duemode interference.

These two last Figs. 4 and 5 can be compared by meof a third figure that is the frequency correlation matrixthe signals received at 5 km range during the ADVENT’sea trial~Fig. 6!. The first comment is that the resemblanbetween this figure and that obtained with simulated datstriking. It appears that the cross-frequency energy sprout of the diagonal is largely attenuated when compared wthe synthetic data example, that is particularly true in the lfrequency range but is also evident at high frequencies whthe main diagonal lobe is narrower. An estimation of teffective 23 dB bandwidth shows that at least 100 Hz aavailable throughout the analyzed frequency range betw200 and 1600 Hz. Comparison of Figs. 5 and 6 shoulddone under the assumption that the latter contains infortion on the source spectrum level that might alter the resNote that the values plotted in the last three figures wnormalized, so there is no information on the relative levof each term on the final observed signal.

FIG. 5. Channel coherence of simulated acoustic field in the band 200–Hz in the ADVENT’99 conditions with a source–receiver range of 5 km

FIG. 6. Estimated correlation of received signal over the band 200–160using the LFM data of the ADVENT’99 experiment at 5 km source–receirange.


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B. Broadband MFP

The results shown in the preceding section suggestdue to the limited correlation of the perturbation factocross-frequency broadband processors should preferablyerate on relative narrow bands of 100 or 200 Hz than on wfrequency bands. In order to illustrate that point with a redata example, a series of vertical array observations at 5range and in the band 400–1000 Hz was drawn fromADVENT’99 data set and processed with the proposcross-frequency incoherent Bartlett processor for randepth source localization purpose. Figure 7 shows the Btlett power results obtained during approximately five cosecutive hours of data for two processing schemes: the stones at 100 Hz spacing between 400 and 1000 Hz wprocessed in a single frame~* ! and the same tones werprocessed in three groups of three frequencies each~s!—groups ~400,500,600!, ~600,700,800!, and ~800,900,1000!.The number of cross-frequency terms is 21 in the fischeme and nine in the second scheme. Despite that dence the Bartlett power, i.e., the value of the normalized pin the final ambiguity surface at the correct source locatiis always higher for the processing in the clustered band tin the wide band. The range-depth source localization permance was the same for both processors. The groupinfrequencies in a limited band acts as an automatic schemexclude the correlation terms that yield worst SNR atprocessor’s output caused by low cross correlation ofsignal components.

VII. CONCLUSION

For many years, underwater acoustic signal processwas devoted to the detection and/or localization of narroband or broadband random sources using a multisensorzontal array. Localization here meant bearing estimatiwhich was the main scope of a wide, yet powerful, suitetechniques. That situation has dramatically changed withwide spread of physical model codes being able to prethe acoustic channel propagation characteristics at var

00

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FIG. 7. Bartlett power for source localization using the cross-frequeincoherent processor in the 5 km range ADVENT’99 data set: 200 Hzquency band clustered processing~s! and 600 Hz wide band processing~* !.


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ranges and depths and in different environmental conditiwith practical relevance. These are the generically camatched-field~MF! techniques, that are used not only fdetecting and localizing submerged targets but also,more importantly, to probe the ocean~ocean tomography!and the seafloor~geoacoustic inversion!. From a purely sig-nal processing point of view, the problem has lost most ofinterest since the knowledge of an image of the receisignal limits the range of~optimum! methods to the well-known matched filter. However, numerous tests with rdata have shown that physical models, at least in thpresent form, can not account for acoustic channel fluctions between the source and the receiver~s!.

This paper approaches the problem of modeling theceived signal as a mixture of a deterministic structure, tcan be predicted by a suitable acoustic model, and a ranperturbation factor that is supposed to be space invar~within the physical sensor array limits! and time variant.Estimation of that perturbation factor on the ADVENT’9data set has shown that its amplitude was approximaRayleigh distributed but its phase did not follow a unifordistribution as it is assumed in many texts. Those distritions were apparently frequency invariant with, howeverconsistent variance increase for the random perturbaphase term. It was also shown, based on the same realset, that a band of frequencies extending to 100 Hz cansafely assumed to contain a significant channel and ranperturbation cross-frequency signal correlation.

Making use of that data model allowed for derivationsoptimum broadband MF processors, according to the varassumptions on the signal and perturbation factor correlaacross frequency. The uncorrelated perturbation assumpled to the well-known incoherent broadband processor.often used unweighted processor was shown to be optimonly on the flat source spectrum case. Other coherent brband processors proposed in the literature are shown tovide either suboptimum performance in real noisy situatioor to have serious limitations in terms of the number of fquencies processed in a reasonable computation timealternative incoherent algorithm is proposed that is shownhave the same detection performance as the matched-pcoherent processor. That processor—the incoherent crfrequency processor—is able to process any number ofquencies with only a slightly larger computation time ththat of the incoherent processor with however, the advanof using the asymptotically noise-free cross-frequency teand without making any use of the source spectrum. Insense the proposed incoherent cross-frequency processobe compared with that developed by Westwood,15 since nei-ther used the source spectrum knowledge with, however,main difference that is that the former uses cross-frequeterms while the latter only used autofrequencies. Finallysimple simulated test on realistic conditions, illustrateddetection performance of the proposed cross-frequency iherent processor when compared with the autofrequencycoherent processor for a well chosen frequency band relato the band of coherence of the underwater channel. Itconcluded that the cross-frequency processor always ouformed the autofrequency processor clearly showing tha


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was advantageous to chose clustered sets of closely spdiscrete frequencies instead of an equivalent number offormly distributed frequencies along the whole band.

ACKNOWLEDGMENTS

The authors would like to thank Ju¨rgen Sellschopp, Mar-tin Siderius and Peter Nielsen responsible for the AVENT’99 experiment design and data collection, and tSACLANT Undersea Research Center for providing the dof the ADVENT’99 experiment. This work was partially financed by Fundac¸ao para a Cieˆncia e a Tecnologia–FCTPortugal, under Contract No. ATOMS,PDCTM/P/MAR15296/1999.

APPENDIX A: ENVELOPE AND PHASEDISTRIBUTIONS

Let a5a1 jb be a random variable such that

a:N~0,s2! and b:N~0,s2!, ~A1!

where a and b are uncorrelated, in which case it is weknown that the polar notationa5uauexp(jf) implies that

uau:RFsAp

2,s2S 22

p

2 D G ,F:U2p,pS 0,

p

)D ,

where R and U designate Rayleigh and Uniform distributions, respectively. The question is to determine the distrition of V5uau andF whena andb are not zero mean. So, leus assume that

a:N~ma ,s2!, b:N~mb ,s2!,

with joint probability density function~pdf!

pA,B~a,b!51

2ps2 expF2~a2ma!21~b2mb!2

2s2 G . ~A2!

It is known that the square moduleY5A21B2 follows anoncentral chi-square distributionx2(s) with the noncentral-ity parameters25ma

21mb2. The pdf ofY is given by

pY~y!51

2s2 expS 2y1s2

2s2 D I 0SAys

s2 D , y>0 ~A3!

whereI 0 is the zeroth-order modified Bessel function of firkind. Thus a simple change of variableR5AY gives us thepdf of R as

pR~r !5pY~r 2!uJu, ~A4!

whereJ52r is the Jacobian of the transformation giving

pR~r !5r

s2 expS 2r 21s2

2s2 D I 0S rs

s2D , r>0, ~A5!

which represents a Rice distribution with parameters25a2

1b2.For the phasef the calculation is more elaborated an

the result is not easy to interpret. Let us first make the traformation

HV25A21B2

F5arctan~B/A!⇔ HA5V cosF,

B5V sinF, ~A6!


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fi-ble

nse

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-

blee

s

n to

r-

w

with the Jacobian,uJu5v, thus the joint pdf of the new variables (V,F) is

pV,F~v,f!

5pA,B~a,b!uJu

5v

2ps2 expF2~v cosf2ma!21~v sinf2mb!2

2s2 G .~A7!

The marginal distribution can be obtained as

pF~f!5E0

`

pV,F~v,f!dv. ~A8!

The first step to solve the integral obtained by replacing~A7!in ~A8! is to develop the sum of squares in the exponenget ~only for the exponent!

21

2s2 @v222v~ma cosf1mb sinf!1ma21mb

2#, ~A9!

which can be made a square of the sum, by subtractingadding the term (ma sinf2mb cosf)2 which gives for thepdf,

pF~f!51

2ps2 e2~ma sin f1mb cosf!2/2s2

3E0

`

v expH 2@v2~ma cosf1mb sinf!#2

2s2 J dv.

~A10!

Performing a change of variablez5v2(ma cosf1mb sinf) reduces to

pF~f!51


3E2~ma cosf1mb sin f!

`

ze2z2/2s2dz1¯

1ma cosf1mb sinf


3E2~ma cosf1mb sin f!

`

e2z2/2s2dz. ~A11!

The first integral equates to

expF2~ma cosf1mb sinf!2

2s2 G , ~A12!

that gives by replacement in~A11! and by knowing that thesecond integral is even, allows for changing the sign ofintegration bounds


o

nd

e

pF~f!5

expS 2ma

21mb2

2s2 D2ps2 1¯1

ma cosf1mb sinf

2ps2

3e2~ma sin f2mb cosf!2/2s2

3E2`

ma cosf1mb sin f

e2z2/2s2dz. ~A13!

Now, a small change of variablel5z/s allows to view thislast integral as the distribution function of a standard nmally distributed random variable as

pF~f!5

expS 2ma

21mb2

2s2 D2ps2 1¯1

ma cosf1mb sinf

2ps

3e2~ma sin f2mb cosf!2/2s2

3E2`

~ma cosf1mb sin f!/se2l2/2 dl. ~A14!

It is not possible to continue any further knowing the difculties to calculate the integral in the second term. Availaapproximate expressions exist for largema cosf1mb sinf/s but that assumption does not makes much sefor the problem at hand.

APPENDIX B: ESTIMATING THE RANDOMPERTURBATION FACTOR DISTRIBUTION

Let an and an21 be two independent Rayleigh distributed random variables with pdf’s,

pa5a

le2a2/2l2

, a>0. ~B1!

The random variableZ defined asZ5an /an21 can beshown to follow a pdf as

pZ~z!52ln

2

ln212

z

~z21ln2/ln21

2 !2 , z>0. ~B2!

Separately, ifFn and Fn21 are two independent Uniformly distributed random variables in@2p, p#, then it canbe easily demonstrated that the pdf of the random variaDF5Fn2Fn21 is given by the correlation between thpdf’s of the two random variablesFn andFn21 , i.e.,

pDF~Df!5E2`

`

pFn~Df1t!pFn21

~t!dt, ~B3!

which, can be easily evaluated for Uniform distributions a

pDF~Df!5H 1

8p2 ~2p2Df!, 22p<Df<2p,

0, otherwise.

~B4!

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Broadband matched-field processing: Coherent and incoherent

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