140 ieee journal of oceanic engineering, vol. 34, no. 2 ... · index terms—array signal...

140 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 34, NO. 2, APRIL 2009

Shallow-Water Acoustic Tomography PerformedFrom a Double-Beamforming Algorithm:

Simulation ResultsIon Iturbe, Philippe Roux, Barbara Nicolas, Jean Virieux, and Jérôme I. Mars, Member, IEEE

Abstract—Recent shallow-water experiments in sea channelshave been performed using two vertical coplanar densely sampledsource and receive arrays. Applying a double-beamforming algo-rithm on the two arrays both on synthetic numerical simulationsand on experimental data sets, we extract efficiently source andreceive angles as well as travel times for a large number of acousticrays that propagate and bounce in the shallow-water waveguide.We then investigate how well sound-speed variations in the wave-guide are reconstructed using a ray time-delay tomography basedon a Bayesian inversion formulation. We introduce both dataand model covariance matrices and we discuss on the syntheticnumerical example how to choose the a priori information onthe sound-speed covariance matrix. We attribute the partialsound-speed reconstruction to the ray-based tomography andwe suggest that finite-frequency effects should be considered asthe vertical and horizontal size of the Fresnel zone significantlyspreads in the waveguide. Finally, the contribution of a differentset of ray angles for tomography goal is also presented.

Index Terms—Array signal processing, shallow-water tomog-raphy, underwater acoustics.

I. INTRODUCTION

O CEAN acoustic tomography was introduced by Munkand Wunsch [1] to provide large scale images of the

ocean sound-speed fluctuations using low-frequency acousticwaves. Typical parameters of these experiments are a range ofhundreds of kilometers, a water depth of a few kilometers, andfrequencies around 60 Hz [2]. The method consists in identifi-cation and tracking of the different ray arrivals. The variationof the arrival time of these rays is used to solve the inverseproblem and estimate sound-speed fluctuations. Results arestrongly dependent on: 1) the number of solved ray paths and2) the uniform spatial coverage of the oceanic waveguide by the

Manuscript received May 13, 2008; revised October 23, 2008; accepted Jan-uary 06, 2009. First published March 24, 2009; current version published May13, 2009.

Associate Editor: J. F. Lynch.I. Iturbe, B. Nicolas, and J. Mars are with GIPSA-Lab, Images Signaux,

Saint-Martin-d’Hères, Rhone Alpes 38402, France (e-mail: [email protected]).

P. Roux and J. Virieux are with the Laboratoire de Géophysique Interne etTectonophysique (LGIT), Centre National de la Recherche Scientifique (CNRS5559), Université Joseph Fourier, Grenoble 38041, France (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JOE.2009.2015166

ray paths. To identify and use a ray in the tomography process,its arrival time has to be measured accurately but also the rayhas to be assigned to a theoretical ray path. A good backgroundmodel of the environment is then necessary. In practical at-seaexperiments in deep or shallow water, an accurate knowledgeof the range- and depth-dependent fluctuating environmentis not available, and therefore, the number of solved (ex-tracted/separated and identified) ray arrivals is often too smallto provide satisfying tomography results. Furthermore, besidesexperimental limitations, Munk and Wunsch [3] theoreticallydiscussed the number of resolved eigenrays or eigenmodesfrom a point-to-point deep-water experimental configurationwhere travel-time separation between low grazing angle rayshas to face bandwidth issues. The same analysis was pursuedwith rays in shallow water showing similar physical limitationsof travel-time-based tomography [4]. Indeed, as the scale ratiosbetween propagation range, depth, and wavelength are com-parable to large scale experiments, refraction and reflection inshallow-water environment provide multipath propagation asin deep-water configuration. Basically, because of the limitednumber of resolved rays, shallow-water tomography does notgive satisfying results. Furthermore, moored thermistor stringsand conductivity–temperature–depth sensors (CTD) bearinggliders are technologies that provide higher resolution andlower variance estimates of the ocean sound speed. However,if all eigenrays between a set of source/receivers could beextracted and identified in shallow water on a kilometer scale,for example, using source and receive arrays, then the uniqueadvantage of the acoustic tomography would be to monitora slice of the ocean structure in real time providing a spa-tial–temporal measurement of the dynamics of mixing layersin shallow-water environment.

To avoid the technical limitations of large scale experiments,a series of small scale experiments (ranging from 1 to 8 km, ina 50–100-m-deep waveguide, with a source at 3 kHz) havebeen conducted in the last few years (2003 Focused AcousticField (FAF03) [5] and 2005 Focused Acoustic Field (FAF05)[6] experiments carried out around Elba Island, Italy).

Usually, small-scale tomographic experiments have beentested in the ocean between a single source and a receiver or anarray of receivers [7]. During FAF03 and FAF05, Roux et al.[5] used a time-reversal experiment data set, which providessignals between two source and receive arrays. They developeda double-beamforming algorithm (simultaneously on the sourcearray and on the receive array), allowing the identification of

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ITURBE et al.: SHALLOW-WATER ACOUSTIC TOMOGRAPHY PERFORMED FROM A DOUBLE-BEAMFORMING ALGORITHM 141

Fig. 1. Schematic of the two steps (a) and (b) of the double time-delay beam-forming process. (a) Beamforming is processed on the receive array, changing�� into �� . (b) Beamforming is processed on the source array,changing �� into �� . Thus, the double-beamforming processtransforms �� into �� .

each individual acoustic ray of the multipath signals by itslaunch angle, receive angle, and arrival time [6].

This double-beamforming approach between the source andreceive arrays tackles the problem of resolved/unresolved eigen-rays in a point-to-point or point-to-array configuration. Basi-cally, every acoustic ray is now isolated and identified giventhe diffraction limit of the system that depends on the size, thecentral frequency, and the bandwidth of the arrays. Moreover,double beamforming naturally increases signal-to-noise ratio(SNR) as an array gain is performed on both source and receivearrays. In their paper [6], Roux et al. discuss the suitability ofthe observed processed signals after double beamforming as es-timators of ray travel times.

Following this approach, we have two complementary goalsin this paper. First, in Section II, a complete ray solving algo-rithm based on double beamforming is presented. Its perfor-mance to extract a large number of ray arrivals between twosource and receive arrays is discussed, using at-sea data fromFAF03 experiment. Then, we investigate deeper our capacity ofsound-speed reconstruction on a synthetic data set starting froman experimental configuration equivalent to FAF03. We presentthe ray-based tomographic approach through a Bayesian formu-lation and we discuss the limitations of the ray-based inversionwith respect to a diffraction-based sensitivity kernel approach[8].

II. TRAVEL TIME TOMOGRAPHY USING

DOUBLE-BEAMFORMING ALGORITHM

In the FAF03 experiment, two vertical source and receivearrays are used in a shallow-water acoustic waveguide. A

broadband acoustic pulse is emitted for each source in a roundrobin sequence, and the acoustic response of the waveguide isrecorded on the receive array [5]. This procedure lasts a shorttime compared to the sound-speed temporal fluctuations in themedium. Thus, the propagation medium remains stationaryduring the acquisition and we consider the recorded data set asthe acoustic transfer matrix between all sources and receivers.The data matrix takes the form of a pressure fieldrecorded at a receiver depth for an emission at a source depth

.

A. Double Beamforming

Invoking spatial reciprocity principle, a time-delay beam-forming algorithm (similar to the Slant–Stack used in seismicexploration [9], [10]) is performed in two steps, both on thereceive array and the source array. In the first step, beam-forming is performed on the receive array for each source.The pressure field becomes , whereis the receive angle [Fig. 1(a)]. By reciprocity principle [11],[12], can be seen as the pressure field recorded atthe source array when directional emission at is producedby the receive array. As a result, the second step consists inapplying beamforming on the source array for each leadingto [Fig. 1(b)], where is the launch angle.

Equation (1) presents the mathematical formulation of thisprocessing called double beamforming, which is illustrated inFig. 1

(1)

where and are the receiver and source depths, respec-tively. corresponds to the time delay to be applied toone element of an array at depth to beamform in a direction .Coefficients are the amplitude weights of the array elementsto reduce secondary sidelobes of the beamforming process.

If sound speed is uniform along the array, plane-wave beamforming is obtained by

(2)

When the sound-speed profile significantly changes overdepth along the array, optimal time-delay beamforming is ob-tained by the turning-point filter [13]

(3)

In (2) and (3), is the chosen reference depth on the array. In(3), is the minimum sound speed along the array.

Consequently, the double beamforming provides the eigen-rays propagating between the reference source at and thereference receiver at , by their propagation time, receiveangle, and launch angle (in the domain). The goalof the double-beamforming process is then to extract from therecorded pressure field data matrix the acoustic contribution of



Fig. 2. Double beamforming is performed from several source/receive subar-rays to multiply the number of identified ray arrivals. Subarrays are chosen togather as many elements as possible around the subarray centers �� .

a given eigenray propagating between the reference source andthe reference receiver , and defined by its launch/re-ceive angles .

B. Beamforming Resolution and Subarray Choice

Our goal is to perform a shallow-water tomography usingidentified ray paths between sources and receivers. As ex-plained above, double beamforming provides isolated rayarrivals between a reference source/receiver couple .To resolve more eigenrays, the propagation between sev-eral source/receiver references is analyzed. For each couple

, the subarrays on which the double beamforming willbe performed has to be chosen appropriately. In this paper,choice is conducted by two criteria, justified in this section:1) the number of sensors used in each subarray should be aslarge as possible; and 2) the subarrays must be centered on thereference source and the reference receiver (see Fig. 2).

As it is well known from array processing, with a uniformsound speed and -equally spaced elements without amplitudeshading , an analytic formulation of the single-beamforming pattern (SB) in the frequency domain [14]–[16] isgiven by

(4)

where is the spectrum of the signal, is the interelementdistance on the array, is the position of the reference point onthe array, and and are the actual arrival angle of the rayand the estimated one, respectively. In (4), the fraction and theexponential part express the amplitude gain of the beamformingfilter and its phase, respectively.

The choice of the largest subarray is due to the constraint ofan angle separation of different ray arrivals, and to the arraySNR gain provided by large arrays in the double-beamformingalgorithm. From (4), the angular resolution of the beamformingis given by

(5)

where is the wavelength, and is the lengthof the array. Because (5) is written in the frequency domain atthe central frequency, it remains a good approximation of thebroadband resolution. It follows that the angular resolution isdirectly related to the length of the array, implying the useof the largest array possible. In a depth-dependent ocean envi-ronment, subarray length is limited by refraction that makes thewavefront arrival incoherent over the whole array. This is notthe case in our simulated waveguide where acoustic propaga-tion is dominated by boundary reflections.

To justify that the subarray must be centered, the phase termof (4) has to be analyzed. The phase term at frequency is givenby

(6)

where is the depth of the first receiver. It corresponds to atime delay . If the array is centered on the refer-ence point , then and the time delay

is zero. But if this condition is not respected, and if the es-timation of the arrival angle is not exactly equal to the actualangle , a time-delayed version of the ray arrival is ex-tracted after beamforming. The consequence of this time-delayerror is important in differential time tomography, because theerror could be significant compared to the small arrival timevariations due to sound-speed fluctuations. To completely avoidthis problem, beamforming is always performed with a refer-ence point at the center of the subarray.

C. Interest of the Double Beamforming

Advantages of using beamforming on a receive array (alsocalled single beamforming in the following) have been summa-rized in [17]. In this section, the advantages of source/receivearray configuration are discussed. The analysis of the data in the

domain provided by double beamforming, comparedto an analysis in the initial domain or in the singlebeamforming domain, has three principal advantages:a better separation of different ray arrivals; a better identifica-tion of ray arrivals to theoretical ray paths; and finally, a higherSNR.

1) Ray Arrival Separation: The ability of double beam-forming to separate ray arrivals is analyzed in thedomain with and . In this 3-D domain,each ray arrival spans an elliptical volume of the space. Thethree diameters of this volume are given by , , and

, which are the time expansion of the signal (relatedto the bandwidth), the receive angle expansion of the rayarrival [ ]; and the launch angle expansionof the ray arrivals [ ], respectively. From ageometrical analysis, two ray arrivals defined byand are separated (they do not superpose at all) ifthe following condition is respected:

(7)



Fig. 3. (a) Separation ability of double beamforming is illustrated with two eigenrays extracted from the FAF03 experiment between a source at depth � �

87.3 m and a receiver at depth � � 39.3 m. (b) Normalized single-beamforming results performed on the receive array for a source depth � � 87.3 m and areceive reference � � 39.3 m. (c) Normalized double beamforming for � � 87.3 m and � � 39.3 m. Crosses indicate the ray prediction.

For the single beamforming, the equivalent condition is

(8)

and for the point-to-point configuration

(9)

Equations (7)–(9) show that the separation ability of the double-beamforming process is better than the single-beamforming andthe point-to-point one, because the sum of new positive terms onthe left-hand side of the inequality always helps in respecting it.

So, from the previous theoretical analysis, the separation ofthe independent ray arrivals is obtained thanks to the number ofdiscriminant parameters provided by double beamforming. Inthe domain, a ray arrival spreads over the whole watercolumn. Indeed, if two different ray paths at a given anddo not respect (9), the two ray arrivals produce interferences onthe recorded data. A way to partially avoid these interferencesis to compute beamforming on the receive array. Indeed, eachray arrival is then localized around its receive angle and itstravel time . If the two ray paths have the receive angles dif-ferent enough to respect (8), the two arrivals do not interfere.However, if the receive angles are close, the arrivals cannot beseparated (Fig. 3). For example, from the FAF03 data set [5], weextract two ray paths shown in Fig. 3(a) with close travel times

and receive angles. In Fig. 3, the black crosses indicate the theo-retically predicted travel time/receive angle positions of the tworay paths. The single beamforming gives only one spot, whichis actually an interfering spot between two ray arrivals. On theother hand, double beamforming yields separation as launch an-gles of the two ray paths are very different [Fig. 3(c)].

2) Identification: An obvious advantage of double beam-forming is that it provides a robust identification of a ray arrivalwith its theoretical ray path. In the point-to-point configuration,the only parameter that enables to associate each arrival to atheoretical ray path is its travel time . Single beamforming pro-vides an additional parameter to perform the identification.But ambiguity problem remains when two ray paths have thetravel time and the receive angles close to each other. Further-more, the ambiguity is reinforced, as shown by Roux et al. [6],by the fact that the two parameters may vary with sound-speedfluctuations. Double beamforming is then more robust for thisidentification task, since it provides one additional parameter

that most of the time solves the ambiguity. The interest ofdouble beamforming to identify ray arrival to its theoretical raypath is illustrated in [6].

To illustrate the resolving ability of double beamformingversus point-to-point configuration, we use synthetic numericalsimulations in shallow water that will be further describedin Section IV. Comparing point-to-point configuration todouble-beamforming process on the same source/receive ar-rays, the number of unresolved rays varies from 1493 to 156,respectively, for a total number of 6171 eigenrays.



3) SNR Gain: SNR gain obtained by double beamforming ishigher than the single-beamforming one. In the case of a uni-form weighting of all the sensors and of spatiallywhite noise, the SNR gain (in decibels) is as follows:

• for single beamforming;• for double beamforming;

where and are the number of sources and the number ofreceivers used in the double beamforming, respectively.

D. From Ray Extraction to Tomography

Time-delay tomography requires the association of ray ar-rival times extracted from the data to ray simulated by a prop-agation model [18]. To perform this association, for each refer-ence source and reference receiver , a ray simulation isfirst computed in a reference model. Approximate travel timesand source and receive angles are computed for all possible raypaths identified by a ray code [19]. For every ray, double beam-forming is performed on the recorded data in the neighborhood(in terms of ) of the simulated ray. We choose the an-gles if the double-beamforming algorithm exhibitsan intensity spot showing an intensity maximum in the pro-cessed signals. The time-domain pressure fieldis then associated with the selected ray path.

Obviously, an accurate shallow-water model yields a bettermatch of ray arrivals. However, both the double-beamformingalgorithm and the ray tracing code are robust to model mis-match. Furthermore, differential time-delay tomography takes abackground of the ocean state at time , and then creates imagesof the spatial sound-speed fluctuations after . Consequently,we do not need the absolute arrival times of the rays, but onlyvariations of these arrival times, which reduces even more theconstraint on the ocean model.

III. TRAVEL-TIME TOMOGRAPHIC INVERSION:BAYESIAN APPROACH

Travel-time tomographic inversion is a nonlinear problemwhen one wants to reconstruct absolute travel times. Up tonow, the only successful case is for the depth-dependentinvestigation [20]. As soon as one wants to consider lateralvariations of celerity, we must rely on the travel time delays asthe differences between the observed travel times and the traveltimes in an initial medium [21], which leads to a linear problem[17]. Indeed, considering a reference model for the ocean mayhelp the separation and identification procedure by allowing usa narrow investigation in the double beam space.

For the inversion problem, the depth and range-dependentsound speed is parametrized as the projection of the celerityvariations on a global basis of sinusoidal functions. As usu-ally done in ocean acoustic tomography (OAT) [17], the forwardproblem is expressed through the linear expression

(10)

where represents the time arrival fluctuations between twosuccessive waveguide transfer matrices and represents thecelerity variations related to these two waveguides. This systemof linear equations defined by the so-called sensitivity matrix

, also known as the Fréchet matrix, is built up for the max-imum number of well-identified rays.

Bayesian approach for tomography has been largely dis-cussed in seismology [22]–[27] as well as in underwateracoustics [28]–[33]. A reference work on the general inverseproblem has been published by Tarantola [34]. In Bayesianapproach, the general inverse problem is solved by findingthe maximum of the a posteriori probability density. Theresult is called the maximum a posteriori (MAP) solution.If uncertainties have Gaussian distributions, the optimizationproblem is completely defined by second-order statistics: themaximization of the a posteriori probability density is equiva-lent to the minimization of the least squares objective function

expressed as

(11)

where the matrix transpose operation is denoted by the super-script . The covariance matrix for the data space will beinferred as uncertainties on differential travel time measure-ments as well as uncertainties related to forward modeling ap-proximations. The covariance matrix for the model spaceshould be defined as the uncertainties on the a priori informationthat we consider for the tomographic reconstruction .Note that the objective function (11) is an adimensional functionthanks to the introduction of covariance matrices. The a prioridifferential model will be set to zero as the backgroundoceanic state is expected to be the most probable model. Otherchoices are possible as an expected low speed zone in a specificarea coming from a systematic warming effect for example. Thecovariance matrix will be taken as a diagonal matrixwith a uniform uncertainty for each independent measurement.

We assume that the a priori model covariance matrix isa diagonal matrix with Gaussian decrease in spatial frequency.Physically, the decorrelation of different frequency componentscorresponds to the case where spatial correlation lengths and

are stationary in the whole medium. The Gaussian decreasecorresponds to a Gaussian correlation in the space domain. In-deed, for a global uncertainty , the diagonal components of

are defined for each of the spatial frequency com-ponents as

(12)

where and are the entire range and the entire depthchosen for the construction of the discrete sine and cosine basis.This frequential weighting allows us to stabilize the result ofthe inversion, diminishing the degrees of freedom of the result(the higher and are, the smoother the final result is). Onthe other hand, this decomposition of the medium of this globalbase will carry less resolution.

Finally, for the forward modeling through a linear (10), theminimum of the objective function can be obtained analytically[34] by the inverse operator

(13)

as long as covariance matrices can be inverted. The MAP solu-tion is given by

(14)



The correlation lengths and could be provided throughthe complementary measurements (such as CTD chain measure-ments) in underwater acoustics as well as the expected uncer-tainty on model parameters. They can also be estimated aspart of the inversion problem as done in a seismological ap-proach [35]. We have selected the second approach here. Wecompute the inversion several times, using at each time differentcorrelation lengths and and different global uncertainties

. At each inversion, the normalized objective function of (11)is minimized and values of the minimum given by

(15)

are recorded. In (15), the celerity perturbation obtained bythe inverse procedure using , , and is denoted by

, the associated value of the objective function by. Quantities and are the contri-

butions of the data space and of the model space to the totalobjective function, respectively. After performing the inversionfor various , , and , the correlation lengths and globaluncertainty that gives the smallest objective function value

are selected and the corresponding result ischosen as a solution for the inverse problem.

IV. SYNTHETIC DATA EXPERIMENT

To assess the different performances of both double beam-forming and travel time tomography, we design a numericalexperiment where environmental parameters are close to ex-perimental parameters encountered during the FAF03 experi-ment. We introduce sound-speed fluctuation between two suc-cessive acquisitions of the waveguide transfer matrix. Then, weanalyze the quality of the estimated sound-speed fluctuation.The goal of this tomographic simulation is to check for themethod performance in an ocean environment similar to shallowwater—high-frequency ocean experiments.

The range between the two arrays is 1.5 km, the water depthis 50 m, and the source signal has a central frequency of 2.5 kHz,with a 1-kHz frequency bandwidth. Sound speed in the wave-guide is uniform (1500 m/s), and the source and receive arraysare made of 32 equally spaced transducers spanning 46.5-mdepth in the water column (between 1.5 and 48 m). The dis-tance between the two elements of an array is 1.5 m to becompared to a 0.6-m central wavelength . Not respecting thespatial Shanon condition of leads to aliasing the prob-lems that we take into account in the inversion part by givinga smaller reliability to eigenrays subject to aliasing errors. Thebottom is made reflective (bottom sound speed 2500 m/s) toobtain the largest number of ray arrivals with significant ampli-tude, and we do not introduce noise in the simulation.

We perform two simulations that will be used as “real data.”The first one is performed with a uniform sound-speed pro-file that defines a reference waveguide. Then, we introduce twolocal sound-speed variations in the second one with a nega-tive 0.3-m/s fluctuation and a positive 0.25-m/s fluctuation

Fig. 4. Spatial–temporal representation of the normalized pressure field inten-sity on the receive array for a 24-m source depth. The legend bar is in decibels.

Fig. 5. (a) Range and depth sound-speed fluctuations introduced in the numer-ical forward model on top of an unperturbed waveguide with range-independentuniform sound speed. Forward propagation is calculated using a wide angle par-abolic equation code. (b) Travel time tomography result performed after extrac-tion of 6015 rays. The color bar has units of meters per second.

[Fig. 5(a)]. A wide angle parabolic equations algorithm is usedto provide the real data in the time domain [36]. Our travel-timetomography goal is to estimate the sound-speed variations be-tween the two experiments, using double beamforming to solvethe ray arrivals. A spatial–temporal representation of the fieldreceived on the receive array is shown in Fig. 4, for a sourcedepth 24 m.

Double beamforming is performed on 11 different source andreceive subarrays (so reference source/receivepairs), placed at and

); 6015 ray arrivals are resolved, which corresponds to97% of all theoretical ray paths. It provides a complete sensingof the entire waveguide. However, note that for thereference pairs that are close to the waveguide boundaries, thesubarray lengths become smaller, and the angular resolution be-comes smaller as well. Indeed, the first ray arrivals with closetravel time, launch angle, and receive angle cannot be separated,which explains the 3% of unresolved rays.

A. Bayesian Inversion

The generalized inverse matrix is obtained through (13) whilethe solution is deduced from (14) using ray theory for the esti-mation of the sensitivity matrix [17]. Forty horizontal fre-quential components and 20 vertical frequential components areused to model the celerity variations. As sound-speed fluctu-ations are modeled on sine and cosine bases, four parametersare used for each frequency ( , , , and

, where subindices and represent the “vertical” and



Fig. 6. Normalized objective function �� obtained for different hori-zontal and vertical correlation lengths � and � .

“horizontal” components, respectively). Indeed, 3200 parame-ters (40 20 4) are inverted. Uncertainties on travel times aretaken as 2 10 s.

The inversion is performed for several , , and valuesand the is recorded. Thanks to the linearized forwardproblem, the inversion could be performed efficiently as raysare traced only once. We have found that the uncertainty0.5 m/s gives the lowest value. Similar values for andshow that both data and a priori information have contributionsin our reconstruction. A cross section is shown in Fig. 6for our optimal global uncertainty 0.5 m/s. An increasingvalue of is observed with increasing correlation lengths. In-creasing correlation lengths make the model smoother as we re-move the short wavelength components of the medium. Withthese constraints, the data are less fitted and, therefore, the firstpart of the objective function increases. On the other hand,the cost function also increases when correlation lengths be-come smaller. As we increase the number of degrees of freedom,

includes the values with shorter wavelengths, which leadto an increase of second terms of the objective function. Min-imal value of the objective function is obtained for 37 mand 6 m. The ratio of the vertical correlation length overthe entire depth is equal to 10% while the ratio of the horizontalcorrelation length over the entire range is equal to 2.5%. Thus,geometry of acquisition provides a better horizontal resolutionthan a vertical one.

For the minimum of the objective function (Fig. 6), thecelerity variation estimation is shown in Fig. 5(b).The standard deviation of the data is 1.6 10 s, and thestandard deviation between the data and reconstructed traveltime variations is 1.4 10s, leading to a reduction down to 8.75% of the standard devia-tion of the data. The two sound-speed fluctuations (one positiveand one negative) are well localized with a better lateral reso-lution than a vertical one. However, the recovered fluctuationsdo not reach the true amplitudes used for computing syntheticdata. In fact, we only recover half of the true amplitude in thecenter of both fluctuations because the reconstructed surface iswider and ghost anomalies are present due to the ray samplingas intrinsic limitations of ray approach [37]. As shown by [38],this partial reconstruction comes from the use of ray theory inthe inversion procedure and the fact that the wavefront healing

Fig. 7. Vertical cross sections of the 3-D TSKs (s /m ) for travel timemeasured after double beamforming performed on different subarray length:(a) point-to-point; (b) 9-m source/receive subarrays; (c) 21-m source/receivesubarrays; and (d) 30-m source/receive subarrays.

and other diffraction effects are not included in the forwardproblem.

B. Finite-Frequency Investigation

The simulated data set has been computed without noise, butthe parabolic equation code includes diffraction effects. Doublebeamforming has been performed with a perfect knowledge ofthe sound-speed profile leading to optimal travel time measure-ments. The tomography result [Fig. 5(b)] does not reach the ex-pected sound-speed fluctuations [Fig. 5(a)]: various diffractioneffects are not included in the estimation of travel times throughray tracing [38]. Indeed, waves emitted at 2.5 kHz with 1-kHzbandwidth in the 60-m-deep waveguide exhibit features far fromthose expected by ray theory according to the ratio between thewaveguide dimensions and the acoustic wavelength.

These effects can be included in the forward modelingthrough travel time sensitivity kernels (TSKs) developed by[39] and [40] in seismology and by [8] in underwater acoustics.Very little has been said about the shape of these TSKs whenconsidering double beamforming. In Fig. 7, the TSKs arecomputed in the waveguide for a 15 ray path when increasingthe size of the subarray lengths ranging from point-to-pointsource/receiver pair [Fig. 7(a)] to a 30-m-long source/re-ceive arrays [Fig. 7(d)] on each side of the waveguide. Thepoint-to-point TSKs clearly show the limit of ray theory inthe waveguide since the Fresnel zone associated with TSKsstrongly spreads in the water column. As expected, we alsonotice the travel time zero sensitivity along the ray path thatcorresponds to the ”banana-doughnut” shape of the TSKs. Onthe other hand, the TSKs of the double-beamforming algorithmshow an interesting pattern when the size of the source/receivesubarrays increases. Two effects are clearly observed. First,the use of large arrays cancels the TSK sidelobes on each



Fig. 8. Tomography results (in meters per second) using ray paths (a) withangles smaller than 16 , (b) with angles between 16 and 29 , and (c) withangles larger than 29 .

side of the ray path. Second, the TSKs do not exhibit zerosensitivity on the ray path: the classical ”banana-doughnut”shape of the Fresnel zone is changed into a ”banana-only”shape when double beamforming is performed on finite-lengtharrays at the source/receiver locations. As such, the TSKs ofthe double-beamforming algorithm will allow similar inversionprocedure as the one based on the ray theory, albeit diffractioneffects will be included.

A good illustration of the Fresnel zone influence on the to-mography inversion can be seen with rays with different launchangles. In Fig. 8, tomography inversion is performed with inputsconsisting of different fans of rays. Fig. 8(a) is the sound-speedfluctuation reconstructed with travel times corresponding to rayangles smaller than 16 only. Similarly, Fig. 8(b) shows the to-mography results for ray angles in the [16 , 29 ] interval whileFig. 8(c) is obtained with ray angles larger than 29 . The numberof ray paths used for each inversion is similar ( 2000) and theonly difference comes from the different angle intervals linkedto the length of each ray and, therefore, to the size of the Fresnelzone.

A good estimation of the sound-speed fluctuations is obtainedusing low-angle rays only, but ghosts are now more importantin Fig. 8(a) when compared to Fig. 5(b). Celerity perturbationamplitudes are partly reconstructed. When using only higherangle ray paths [Fig. 8(b) and (c)], the tomography result is de-graded both in shape and amplitude. It appears then that, usingray theory as a forward model, the main part of travel time in-formation is carried out by low angles, while higher angles onlyhelp improve the result in amplitude and spatial resolution. Thetomography resolution is directly related to the length of thefirst Fresnel zone [37] equal to [37] where is the raypath length. In our case, high angles correspond to long raypaths while small angles correspond to short ray paths. Then,

Fig. 9. Singular values of the SVD decompositions of the sensitive matrix� ,for the three angle sets shown in Fig. 8. The solid, dashed, and dashed–dottedlines correspond to small, intermediate, and high launch angles, respectively.

small angles lead to a better resolution than high angles. Thefact that low angles provide a good tomography result is alsogood news for shallow-water oceanic environments where a 16critical angle corresponds to a realistic 1560-m/s bottom soundspeed.

A singular value decomposition (SVD) analysis of the sen-sitive matrix highlights the influence of launch angles onthe tomography reconstruction. In Fig. 9, the first 500 singularvalues of the SVD decompositions of the three angle sets areshown. The SVD results for small angles show much more sig-nificant singular values than the SVD for high angles. As ex-pected, it confirms the fact that the number of “independent”rays is larger when considering small angles than when consid-ering high angles.

V. CONCLUSION

Performance of shallow-water acoustic tomography has beenanalyzed using a ray identification algorithm based on a double-beamforming processing. This array processing helps in the sep-aration and the identification of the maximum possible numberof acoustic rays, thanks to their source/receive angles and traveltimes between the source array and the receive array in a wave-guide. Performing double beamforming on several subarrayspermits the extraction of a large number of ray-like arrivals andtheir travel time measurements. From these ray paths, a uni-form coverage of the waveguide by the ray paths is obtained,from which sound-speed fluctuations are reconstructed using aray-based direct tomography model.

In the shallow-water configuration, ray-based travel time to-mography only reconstructs half of the amplitude as diffrac-tions effects (wavefront healings, for example) are not included.Diffraction effects computed through the shape of the TSKs inthe waveguide confirm the limitation of ray-based tomographyinversion. This influence of the Fresnel zone could also be ob-served through a selection of launch angles. We have shown thatlow angles are sufficient to provide satisfactory tomography re-sults while high angles provide locally better range/depth reso-lution.

From these benchmark results, future work in shallow-watertomography will follow two principal directions. The first taskwill be to couple double beamforming and a direct model thatwill use appropriate sensitivity kernels and diffraction-basedforward modeling to go beyond the limits of ray approximationand the first Fresnel zone influence. The second goal will be



to perform double beamforming and tomography inversion onthe experimental data, which will introduce several challengingdifficulties such as complexity of depth-dependent sound-speedprofiles and scattering phenomena produced by surface waves.

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Ion Iturbe received the M.S. degree in signal pro-cessing from the Grenoble-INP, Grenoble, France, in2006. Currently, he is working towards the Ph.D. de-gree in signal processing at GIPSA-Lab, Images Sig-naux, Saint-Martin-d’Hères, Rhone Alpes, France.

His research deals with signal processing and un-derwater acoustic tomography.

Philippe Roux received the Ph.D. degree in the ap-plication of time reversal to ultrasounds from the Uni-versity of Paris, Paris, France, in 1997.

He is a physicist with a strong background inultrasonics and underwater acoustics. Between2002 and 2005, he was a Research Associate at theMarine Physical Laboratory, Scripps Institution ofOceanography, San Diego, CA. He is now a full-timeResearcher at the Laboratoire de Géophysique In-terne et Tectonophysique (LGIT), Centre National dela Recherche Scientifique (CNRS 5559), Université

Joseph Fourier, Grenoble, France, where he develops small scale laboratoryexperiments in geophysics.

Dr. Roux is a Fellow of the Acoustical Society of America.



Barbara Nicolas graduated from the Ecole Na-tionale Supérieure des Ingénieurs Electriciens deGrenoble, France, in 2001. She received the M.S.and Ph.D. degrees in signal processing from theGrenoble-INP, France, respectively, in 2001 and2004.

In 2005, she held a postdoctoral position atthe French Atomic Energy Commission (C.E.A.)in medical imaging. Since 2006, she has beenCNRS researcher at GIPSA-Lab, Images Signaux,Saint-Martin-d’Hères, Rhone Alpes, France. Her

research in signal processing and underwater acoustics include geoacousticinversion, acoustic tomography, underwater source detection and localization,time-frequency representations, and beamforming.

Jean Virieux received the degree in physics from theEcole Normale Supérieure, Paris, France, in 1978.He defended a University Thesis on ”Earthquakes;rupture and waves” under the direction of Prof. R.Madariaga, in 1986.

Currently, he is a Professor at the Laboratoire deGéophysique Interne et Tectonophysique (LGIT),Centre National de la Recherche Scientifique (CNRS5559), Université Joseph Fourier, Grenoble, France.His main scientific interest is related to wave prop-agation for various purposes as 1) understanding by

modeling these waves in complex media, 2) imaging rupture processes at theearthquake source, and 3) imaging structures inside which waves propagate. Fordoing so, he has developed numerical techniques for full waveform modelingand for asymptotic waveform modeling and the associated nonlinear techniquesfor recovering parameters of various complex media.

Jerome Mars (M’08) received the M.S. degreein geophysics from Joseph Fourier University,Grenoble, France, in 1986 and the Ph.D. degreein signal processing from the Institut NationalPolytechnique, Grenoble, France, in 1988.

From 1989 to 1992, he was a PostdoctoralResearcher at the Centre des Phénomènes Aléatoireset Geophysiques, Grenoble, France. From 1992 to1995, he was a Visiting Lecturer and Scientist atthe Materials Sciences and Mineral EngineeringDepartment, University of California, Berkeley. He

is currently a Professor in Signal Processing at the Department Image andSignal, GIPSA-Lab, Images Signaux, Saint-Martin-d’Hères, Rhone Alpes,France. He is the Leader of ”Signal-Image-Physics” team. His research interestsinclude seismic and acoustic signal processing, wavefield separation methods,time-frequency time-scale characterization, and applied geophysics.

Dr. Mars is a member of the Society of Exploration Geophysicists (SEG) andthe European Association of Geoscientists and Engineers (EAGE).


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