Travel-time sensitivity kernels versus diffration patterns obtained through double beam-forming in shallow water Ion Iturbe a GIPSA Laboratory, INPG CNRS, 96 rue de la Houille Blanhe, Domaine Universitaire, BP 46, 3842 Saint Martin d Heres, Frane Philippe Roux and Jean Virieux LGIT, Universite Joseph Fourier CNRS, 38 rue de la Pisine, Saint Martin d Heres, 384 Frane Barbara Niolas GIPSA Laboratory, INPG CNRS, 96 rue de la Houille Blanhe, Domaine Universitaire, BP 46, 3842 Saint Martin d Heres, Frane Reeived 5 August 28; revised 22 May 29; aepted 27 May 29 In reent years, the use of sensitivity kernels for tomographi purposes has been frequently disussed in the literature. Sensitivity kernels of different observables e.g., amplitude, travel-time, and polarization for seismi waves have been proposed, and relationships between adjoint formulation, time-reversal theory, and sensitivity kernels have been developed. In the present study, travel-time sensitivity kernels TSKs are derived for two soure-reeiver arrays in an aousti waveguide. More preisely, the TSKs are ombined with a double time-delay beam-forming algorithm performed on two soure-reeiver arrays to isolate and identify eah eigenray of the multipath propagation between a soure-reeiver pair in the aousti waveguide. A relationship is then obtained between TSKs and diffration theory. It appears that the spatial shapes of TSKs are equivalent to the gradients of the ombined diretion patterns of the soure and reeiver arrays. In the finite-frequeny regimes, the ombination of TSKs and double beam-forming both simplifies the alulation of TSK and inreases the domain of validity for ray theory in shallow-water oean aousti tomography. 29 Aoustial Soiety of Ameria. DOI:.2/.358922 PACS number s : 43.6.Fg, 43.6.Rw WLS Pages: 73 72 I. INTRODUCTION The resolution limit of travel-time tomography has been studied from various aspets. 3 This investigation essentially relies on the speifi, maybe paradoxial, nature of travel-times, as extrated from time-series reordings. One piked, travel-times lose the frequeny information of the time series. For example, in seismology, hoosing times from high-frequeny impulsive seismograms or from broad-band low-frequeny seismograms will ertainly have an impat on the tomographi resolution. However, the frequeny information is not used in the travel-time tomography mahinery based on ray theory for example, see Ref. 4. Ad ho proedures for introduing frequeny information have been designed 5 with the so-alled fat-ray onept, based on reonstrution assumptions. The more physial onept of the wave path, as related to the wave propagation properties, was introdued by Woodward, 6 whih is losely related to Fresnel tomography in optis. 7 In reent years, this finitefrequeny influene has been systematially investigated for the different observables i.e., time, polarization, amplitude, and anisotropy in different studies, suggesting that higher resolution images an be obtained from this improved desription of wave propagation physis see Ref. 8 for a general review. Based on single-sattering effets, sensitivity kernels have been introdued and different omputational tehniques have been devised from ray theory as paraxial theory 9 or exat ray theory to numerial tools. Different studies 2 have questioned the differential tehniques used for the onstrution of these sensitivity kernels, with the emphasis on the so-alled banana-doughnut paradox: for travel-times, the sensitivity kernel is zero on the ray onneting the soure and the reeiver. Other studies 3 have shown that the traveltime tomographi problem with the speifi density of stations and soures enountered in seismology prevents an improvement of resolution. Overoming these limitations of data quality requires a denser deployment of soures and reeivers, whih an be expensive. Similar to seismi studies, onsideration of arrays of soures and reeivers is a lassial approah in underwater aoustis. 4 Wave-propagation problems an lead to similar features as in geophysis, and we would like to investigate the effets of the finite size of the soure and/or reeiver arrays in underwater tomographi reonstrutions. The onept of sensitivity kernels has been applied reently to this field, 5 and quite exiting theoretial and experimental investigations have led to fruitful ahievements with links to timereversal theory, 6,7 adjoint methods, and aousti and seismi imagings 8 or medial imaging. 9 2 In the present study, the relationships between traveltime sensitivity kernel TSK reonstrution and diffration theory in the ontext of shallow underwater aoustis is dea Author to whom orrespondene should be addressed. Eletroni mail: ion.iturbe@gipsa-lab.inpg.fr J. Aoust. So. Am. 26 2, August 29-4966/29/26 2 /73/8/$25. 29 Aoustial Soiety of Ameria 73
fined when the transfer funtion of the waveguide is reorded between soure and reeiver arrays. We show that the spatial shapes of TSKs are equivalent to the gradient of the ombined diffration pattern of the arrays. This relationship is exat when working with a point-to-point approah using only one soure and one reeiver, and it beomes more omplex and approximate when TSKs are alulated between two soure and reeiver arrays. In a shallow-water environment, array proessing using soure and/or reeiver arrays is neessary to improve the separation of the different ray paths. One standard array proessing method is time-delay beam-forming on the reeiver array, to separate the ray paths aording to their reeiver angles. 22 Reently, Roux et al. 23 proposed a more sophistiated time-delay double beam-forming DBF algorithm, based on spatial reiproity, whih takes advantage of both reeiver and soure arrays. The DBF algorithm an be applied when the entire transfer matrix is measured between eah pair of soure-reeiver transduers. DBF onsists of transforming the three-dimensional 3D data spae from soure depth, reeiver depth, and time into a new 3D spae that is related to ray propagation, desribed by the beamformed variables: soure angle, reeiver angle and time. As a onsequene, every eigenray of the multipath propagation for a soure-reeiver pair is identified and isolated through DBF aording to the reeiver and soure angles. In their very reent study, Roux et al. 23 went one step further. Every eigenray isolated through DBF proessing beame free from any interferene effets due to multipath propagation. Furthermore, DBF proessing provides array gain and robustness, sine every DBF eigenray arises from the summation of a large number of time-delayed soure-toreeiver signals. Thus, both the amplitude and phase of the DBF eigenray an be followed as a funtion of dynami oean flutuations, when, for example, internal waves loally perturb the sound-speed profile see Fig. 7 of Ref. 23. The stability and robustness of DBF proessing an lead to important onsequenes for future studies relating to oean aousti tomography. To date, travel-time tomography has mainly been performed from eho arrival peaking through point-to-point measurements. Indeed, only the envelope of the demodulated signal was a robust observable, regarding signal-to-noise ratio issues and rapidly hanging oean flutuations e.g., gravity waves at the oean surfae. The use of soure-reeiver arrays now allows the travel-time flutuations to be measured as phase hanges in the DBF signal, providing travel-time measurements with greater auray. Indeed, travel-time hange measured through the phase has an auray driven by the arrier frequeny F of the signal, while travel-time hanges measured through the envelope of the demodulated signal depend on the frequeny bandwidth f F. Note, however, that the travel-time hange measured through the phase of an eigenray is not a measurement of phase veloity, as lassially defined as the phase speed along the waveguide axis. For rays, the phase veloity along the waveguide axis is z /os, where z is the depthdependent sound-speed profile at the soure/reeiver depth z, and is the launh/reeiver eigenray angle. In water, the bulk modulus shows nearly no frequeny dependene, whih means that wave dispersion in the oean omes only from reverberation on interfaes and/or refration due to soundspeed gradients. As a onsequene, and with water being nondispersive, the group and phase veloities are the same along the ray path of an eigenray. The measurement of travel-time that hanges through the phase of the DBF eigenray is then just a more aurate observable for the measuring of hanges in the group veloity. In the ontext of DBF, the sensitivity kernel is no longer point-to-point but relies on all soure-reeiver time series. The kernels are omputed based on the onept that the proessed signal is a linear ombination of the time-delayed signals between all soures and reeivers. Throughout this study, we onentrate on the physis that onnets TSKs and Fresnel diffration in the ontext of soure-reeiver arrays and a multipath environment in whih DBF is performed to identify and isolate every eigenray. We show that TSKs assoiated with DBF result in inreased spatial diversity of the sensitivity kernels whih improves the range of validity of the ray theory for shallow-water aousti tomography in the low-frequeny regime. This report is divided into four setions. Following this introdution Se. I, in Se. II, the relationship between the TSK and the aousti diffration pattern is obtained for the point-to-point ase. In Se. III, the disussion is extended to soure-reeiver arrays through the DBF algorithm, whih provides identifiation of every eigenray in the waveguide. The disussion ontinues in Se. IV relating to the use of DBF in the ontext of shallow-water oean aousti tomography. II. TSKs VERSUS DIFFRACTION In this setion, we investigate the links between TSKs and aousti diffration patterns for point-to-point, sourereeiver onfigurations. Starting from the review of Skarsoulis et al., 5 who first introdued TSKs into oean aousti tomography, we show here that in the far-field approximation, the TSK is the gradient of the diffration pattern, orreted by a spatial fator. The TSK is a measure of the travel-time perturbation of an aousti path versus any spatial perturbation of the rangedependent and depth-dependent sound-speed profile. The pressure-field in the waveguide is expressed as the onvolution of the soure distribution over the soure volume and the Green s funtion G r,r s,. Under the first Born approximation, the Green s funtion perturbation G has a linear relationship with the perturbation of the sound-speed distribution,, aording to G r r,r s, = 2 2 G r,r s, G r r,r, r 3 r dv r. The temporal expression of the pressure-field p t is written as the inverse Fourier transform of the frequeny-domain pressure-field through 74 J. Aoust. So. Am., Vol. 26, No. 2, August 29 Iturbe et al.: Sensitivity kernels vs diffration
p t = 2 G r r,r s, P s e j t d, 2 where P s is the soure spetrum. Then, a variation in the pressure-field p has a linear relation with the Green s funtion perturbation: p t = 2 G r r,r s, P s e j t d. 3 Equations 3 and provide linear relationship between the pressure-field perturbation and the sound-speed perturbation. For estimation of the TSK, the perturbation of the traveltime related to the perturbation of the pressure-field needs to be onsidered. For aousti propagation, Skarsoulis et al. 5 proposed that the travel-time is defined as the peak of the envelope of the demodulated or analytial signal. In the ase of strong signal-to-noise ratios, as disussed above, the travel-time hange is performed diretly as the phase hange of the pressure-field. In theory, this phase hange an be measured at any time of the pressure-field. We hose to measure the travel-time hange at the yle peak of maximal amplitude i. In this ase, the relationship between the traveltime perturbation and the signal perturbation simplifies to i = ṗ i, 4 p i where p i is the seond-order derivative of the pressure-field p at time i, and ṗ i is the first-order derivative of the pressurefield perturbation p evaluated at time i Eq. 3. Note that as ompared to Skarsoulis et al., 5 where they dealt with analyti omplex signals, here the p and p defined in Eqs. 2 and 3 are real quantities. Combining Eqs. 4, the travel-time perturbation i is related to a perturbation of a sound-speed distribution through the integral: i = r K i r,r s,r r dv r, where the expression K i is the TSK. In Ref. 5, a general formulation of K i is given for the ase of analyti signals. Again, as we only deal with real signals, the TSK K i simplifies to K i r,r s,r r = Q r,r s,r r, e j id, 6 2 j p i where Q is given aording to 5 Q r,r s,r r, = G r,r s, G r r,r, 2 2 Ps 3. 7 r In Fig., the TSK results are illustrated for two different geometries. The first was obtained through numerial simulation. For a given ray path in the waveguide, the TSK is built by omputing Green s funtions using a paraboli equation ode 24 Fig. a. The seond geometry was obtained through analytial Green s funtions in a free-spae medium where the reeiver is the image of the atual reeiver in the waveguide with respet to the waveguide boundary onditions pressure release at the air-water interfae and rigid 5 2 3 4 5 2 4 6 8 2 4 5 (a) (b) 2 5 5 5 5 5 5 TSK (s 2 m 4 ) () 2 2 x 9 5 5 d(m) x 9 5 bottom, Fig. b. Consequently, the travel path in the freespae medium is the unfolded version of the atual travel path in the waveguide. The soure signal has a 2.5 khz entral frequeny and 4 khz bandwidth. Interestingly, even if the aousti fields are different in the two onfigurations, the two methods give similar TSK patterns away from the waveguide boundaries, where different ehoes interfere Fig.. The Green s funtion we onsider in this waveguide is the sum of different eigenray ontributions that an be separated through the DBF analysis. Eah ray an have a omplex trajetory urved and/or broken lines, whih an inlude rebounds on the waveguide boundaries. As we have been able to separate eah ray from other neighboring rays, we an ompute the travel-time and amplitude onsidering refletions at boundaries and/or any variations in speed. For simpliity, we will only onsider here a uniform sound-speed and straight rays in the waveguide, although our study an also be extended to refrated rays. As stated above, the group and phase veloities along eah eigenray are idential in this shallow-water regime, whih means that the free-spae approah will provide similar results away from the waveguide interfaes. Therefore, we proeeded with the free-spae TSK for our analysis, and we an illustrate the results with the waveguide TSKs in some speifi ases. Obviously, the omputational osts in a free-spae medium, where a simple analyti expression for the pressure-field is available, are muh lower than in a waveguide where geometri dispersion has to be taken into aount through modes or rays. The standard well-known shape of the TSK is seen in Fig., for the ase of a uniform sound speed. We observe zero-sensitivity on the ray path that refers to the so-alled x 9 5 FIG.. a TSK s 2 m 4 in a waveguide for a ray path seleted between two soure-reeiver arrays. b TSK for the equivalent ray path in free spae note sale hange on vertial axis with respet to a. Cross-setion of the TSK perpendiular to the ray path along the lines in a and b at the middle of the soure-reeiver range. Solid-line orresponds to a, and irles orrespond to b. J. Aoust. So. Am., Vol. 26, No. 2, August 29 Iturbe et al.: Sensitivity kernels vs diffration 75
y D = j Re j u y. 2 FIG. 2. Shemati of the soure s and reeiver r onfiguration in free spae with polar oordinates r, and the definition of the angles RP and SP. banana-doughnut shape of the TSK. A negative sensitivity zone is then seen, whih is known as the first Fresnel zone. Higher-order Fresnel zones follow when moving away from the ray path. On the other hand, the diffration pattern between the soure and the reeiver has a maximum on the ray path. Its spatial derivative along a diretion perpendiular to the ray is therefore zero on the ray. One may wish to investigate the link between this derivative and the TSK. Following the Huygens Fresnel priniple and invoking reiproity, the aousti diffration pattern between a soure and a reeiver observed from any point of the medium r is omputed as a produt of both the Green s funtions at the reeiver r r and the soure r s : D r,r s,r r, = G r,r s, G r,r r,. Considering the Green s funtion in a homogeneous freespae medium: G r,r, = it follows that with and 4 d r,r e j d r,r /, D r,r s,r r, = R r,r s,r r e j r,r s,r r, R r,r s,r r = 4 2 d r,r s d r,r r, r,r s,r r = d r,r s + d r,r r. Any hange in r along the ray does not produe any phase hange in the diffration pattern D sine remains onstant, and only produes small and smooth amplitude variations in the far field. On the ontrary, a hange in r along a perpendiular diretion to the ray path will affet the phase, as investigated by Romanowiz and Snieder 25 and Snieder and Romanowiz 26 when onsidering seismi veloity perturbations. The gradient of the diffration D on the perpendiular diretion to the ray path is given by y D = D u y = R j R e j u y, 8 9 where the u y vetor is the unitary vetor along the y-diretion shown in Fig. 2. At large distane from the soure and reeiver, R an be ignored R R and Eq. redues to Inside Eq. 2, there are the propagation terms R and, whih are related to both of the soure/diffrator and reeiver/diffrator propagations that determine the phase and amplitude of D; 2 the oblique gradient as the u y term, related to the loal perturbations we are interested in. Considering the free-spae Green s funtion G, we an rewrite Eq. 2 as y D = j G r,r s, G r,r r, u y, 3 Comparing Eqs. 7 and 3, we an see that the sourereeiver diffration pattern is proportional to the temporal derivative of Q multiplied by a spatial fator, y, whih is analyzed below. Taking the referene point on the enter of the ray-path trajetory, is written in polar oordinates r, as see Fig. 2 r,r s,r r r 2 + d 2 2rd os + r 2 + d 2 2rd os =. 4 Taking into aount that os = os and sin =sin, the gradient of in polar oordinates beomes r,r s,r r = r + d os r 2 + d 2 +2rd os + d sin r 2 + d 2 +2rd os + r d os r 2 + d 2 2rd os d sin r 2 + d 2 2rd os. 5 Calulating the diretional gradient in the y-diretion, we obtain r sin y = u y = r 2 + d 2 +2rd os + r sin r 2 + d 2 2rd os = sin SP + sin RP, 6 where SP and RP are the soure and reeiver angles shown in Fig. 2. Finally, substituting Eq. 6 into Eq. 3 for the diretional gradient of the diffration pattern, and omputing the inverse Fourier transform, we obtain the omplete expression of the gradient of the diffration pattern in the time domain as y D r,r s,r r = 2 j R r,r s,r r e j r,r s,r r whih turns out to be sin SP + sin RP P s e j id, 7 76 J. Aoust. So. Am., Vol. 26, No. 2, August 29 Iturbe et al.: Sensitivity kernels vs diffration
.5.5 5 5 (a) 5 5 FIG. 3. a and b Point-to-point TSK solid-line versus diffration pattern gradient dashed and spatially orreted diffration pattern gradient irle line. The three plots have been normalized aording to their maxima. The soure-reeiver range is.5 km. The TSK was alulated at 75 m for a 7.6 aousti ray for a 2.5 khz entral frequeny pulse, and a 25Hzand b 4 Hz frequeny bandwidths. y D r,r s,r r = sin SP + sin RP 2 j G r,r s, G r,r r, P s e j id. 8 Similarly, ombining Eqs. 6 and 7, we an write the TSK as K i r,r s,r r = 2 j G r,r s, G r r,r, 2 2 P s p i 3 e j id. r 9 Equations 8 and 9 are similar but present two major differenes. The first of these onerns the geometrial influene of the soure/reeiver position. Figure 3 shows the TSK solid-line, the gradient of the diffration pattern dashed line, and the spatially orreted gradient of the diffration pattern y D / sin SP +sin RP, irle line for a 2.5 khz entral frequeny signal with.25 and 4 khz bandwidths, respetively. Figure 3 is obtained at a 75 m range position for a.5 km soure-reeiver range and 7.6 ray path. We an see in Fig. 3 a that the use of the spatial orreting fator sin SP +sin RP / allows a perfet fit between the TSK and the gradient of the diffration pattern far from the waveguide interfaes. Note that this orretion fator resembles the obliquity fator used in Fourier optis to aount for diffration effets from extended apertures. 27 The seond differene is more diffiult to assess. The frequeny ontents of Eqs. 8 and 9 appear to be different, although Fig. 3 a shows us that this is not the ase. The frequeny-dependent term 2 2 Ps / p i 3 r orresponds to the square of the aousti wave number / 2 ounterbalaned by the aeleration of the pressure-field. The aeleration of the pressure-field also justifies the missing minus term, as aeleration has an opposite phase with respet to pressure. As an illustration, Fig. 3 b shows the same omparison performed over a larger bandwidth. We an see that the fit between Eqs. 8 and 9 is no longer perfet, sine the 2 present in the TSK beomes signifiant when the bandwidth is large ompared to the entral frequeny. We also note that the use of a wider bandwidth eliminates the.5.5 (b) side lobes of the TSK, putting sensitivity in the first Fresnel zone only. We will see below that the DBF proessing leads to the same interesting phenomenon. The similarity between Eqs. 8 and 9 is quite general and stays valid in more ompliated waveguides with range-dependent and depth-dependent sound-speed patterns, as mentioned earlier. As long as we onsider that the Green s funtion is a ombination of well-identified separated eigenray ontributions, we an isolate travel-time and amplitude for eah eigenray onneting the soure and the reeiver following Eqs. 9 3, regardless of the path this ray has taken. In onlusion, the point-to-point TSK is nothing else but the spatial derivative of the diffration pattern that an be expeted from the single-sattering Born approximation. III. TSK VERSUS DIFFRACTION WITH DBF When using DBF proesses to extrat eigenrays between two soure-reeiver arrays, the TSK reonstrution must take into aount the geometry of the soure and reeiver arrays. More speifially, when using array proessing, the measured travel-time does not orrespond to the travel-time of a single arrival at one reeiver p t but orresponds to the travel-time of a linear ombination of the properly timedelayed pressure-fields reorded between the soure and reeiver arrays. When performing the DBF analysis, the beamformed pressure-field is given by N r N s p BF t, r, s = ap rs t T r r,z r T s s,z s, 2 r= s= where p rs t is the pressure-field between soure s and reeiver r, and a rs is an amplitude shading window. r and s are the observed reeiver and launh angles, respetively, and T r and T s are the delay orretions to the beam-form on r and s. If the sound speed is uniform along the arrays, the time-delay beam-forming is sin T,z = z z, 2 where z z is the distane between an array element at depth z and the enter of the array at depth z. The expression of an optimal delay orretion for soure-reeive arrays in a depth-dependent sound-speed profile is given by the turning-point filter approah. 22 Physially speaking, the DBF onsists of phasing a olletion of signals aording to given launh and reeive angles; these are then averaged. After summation, the wave front assoiated with the launh and reeiver angles is preserved, sine it is oherently averaged, while the other field omponents are inoherently averaged and disappear. In this part, we investigate the spatial shape of the TSK when using DBF. As for the ase of point-to-point TSK, two linearized expressions are required between p BF and, and between BFi and p BF, to define the DBF-TSK. As the DBF is a linear ombination of pressure-fields between different soure/reeivers, we have the following expression: J. Aoust. So. Am., Vol. 26, No. 2, August 29 Iturbe et al.: Sensitivity kernels vs diffration 77
p BF t, r, s = r= N r s= N s a rs p rs t T r r,z r T s s,z s. 22 The linear relationship between p and see Eqs. 3 and implies a linear relationship between p BF and. Similarly, the relation between BFi and p BF is given by Eq. 4, replaing p by p BF. Thus, using Eqs., 3, 4, and 22, we define the linear relationship between the hange in travel-time BFi and the loal perturbations of the sound-speed profile: BFi = r K BFi r,r s,r r dv r, 23.5.5 5 5 (a).5.5.5.5 5 5 (b).5.5 where the TSK is now: K BFi r,r s,r r = 2 j p BFi Q BF r,r s,r r, e j id, 24 with Q BF as a linear ombination of the point-to-point Q rs expressed as Q BF r,r s,r r, = r= N r s= N s a rsq rs r,r s,r r, e j T r r +T s s. 25 When using DBF, the diffration pattern between the two arrays is given by D BF r,r s,r r, = r= N r s= N s a rsd r,r s,r r, e j T r r +T s s. 26 If we take for D the expression in Eq., and then ompute the diretional gradient of D BF perpendiular to the ray path in the far-field approximation, we obtain the expression N r N s y D BF = j a rsg r,r s, G r,r r, r= s= e j T r r +T s s r,r s,r r. u y 27 The spatial fator r,r s,r r u y annot be pulled out of the sum beause of its speifi dependene on eah soure and eah reeiver. So we annot dedue the TSK from the spatial derivative of the diffration pattern by a spatial orretion fator, as we did previously in the point-to-point ase. However, we an approximate this orretion fator by the onstant r,r s,r r u y at the entral positions r and s of the soure-reeiver arrays. Atually, r,r s,r r u y also orresponds to the mean value of the diretional gradients. Finally, we have the approximate relationship K BFi r,r s,r r yd BF r,r s,r r, P s e j id. sin S P + sin R P 28 Figure 4 shows a omparison between K BFi and the righthand term in Eq. 28. As expeted, the omparison degrades when the size of the soure-reeiver arrays is inreasing. 5 5 () However, we an see that the orretion fator sin SP +sin RP / helps with a better approximation of the TSK from y D BF, even in the ase of the 9 length array. In partiular, we note that the diretional gradient of the diffration pattern has zero-sensitivity on the ray path, and that the spatial orretion fator removes this zero-sensitivity when using arrays. IV. ANALYSIS OF TSK USING DBF 5 5 FIG. 4. a and b TSK versus the diffration pattern gradient and spatially orreted diffration pattern gradient, using the DBF with different array lengths. The three plots have been normalized aording to their maxima. The soure-reeiver range is.5 km. TSK is alulated at 75 m for a 7.6 aousti ray for a 2.5 khz entral frequeny pulse and a 25 Hz bandwidth. From a to d, the soure-reeiver arrays are point to point, 3 long, 6 long, and 9 long, respetively. The line odes are the same as in Fig. 3. The relationship between the TSK and the diffration pattern of the soure-reeiver arrays is interesting sine it an provide a simpler and faster way to alulate the TSK in shallow-water waveguides. Indeed, alulating TSK after DBF is a very time-onsuming task, sine it requires the omputing of an ensemble of TSKs between eah sourereeiver pair, taken among the soure-reeiver arrays as required by Eqs. 24 and 25. Equation 28 shows that the TSK an be approximated from the produt of the soure and reeiver array diffration patterns only. For example, in the Fraunhofer approximation, the angle-dependent diffration pattern of an N-element linear array around angle at angular frequeny is given by B,, = sin Na sin sin /2 sin a sin sin /2, 29 where a is the array pith. In the far-field approximation, the diffration pattern D BF an be approximated by B s, s, B r, r,, where B s and B r are the diffration patterns of the soure and reeiver arrays, respetively. Calulating the TSK K BFi onsists then in simply omputing the diretional gradient of the D BF perpendiular to the ray path, weighted by the appropriate spatial orretion fator, as shown in Eq. 28. (d) 78 J. Aoust. So. Am., Vol. 26, No. 2, August 29 Iturbe et al.: Sensitivity kernels vs diffration
2 3 4 5 5 5 (a) 2 3 4 5 5 5 (b) () 5 2 25 3 7 75 8 x 8.5.5 x 8 Furthermore, Eq. 28 provides some physial insight in the shape evolution of the TSK with respet to the sourereeiver array size. Two effets are revealed in Fig. 4. First, the side lobes beome smaller when using extended antennas. This is an effet similar to the use of a larger frequeny bandwidth see the similarity between the solid-line in Fig. 3 b and the solid-line in Fig. 4 b. Indeed, when integrating Eq. 29 over a frequeny bandwidth f, it appears that an inrease in f or array size Na leads to a similar derease in seondary side lobes in the diffration pattern. The same effet was seen by Raghukumar et al. 28 when analyzing the sensitivity kernel of a time-reversal mirror: they observed that the derease in the sensitivity of the time-reversal fous to a sound-speed perturbation is due to the interferene between more aousti paths provided by the use of a larger array. The seond effet is that the sensitivity on the ray path beomes nonzero and even beomes maximal with large aperture arrays Fig. 4, solid-line. This result is non-intuitive, sine it ontradits the well-known banana-doughnut shape lassially observed with point-to-point TSK on the ray path. Indeed, as shown in Fig. 5 a, the point-to-point TSK remains zero on the ray path even at the high-frequeny, large bandwidth limit. It is only the ombination of TSK with DBF from soure-reeiver arrays that provides a non-zerosensitivity of travel-times on the ray path. Indeed, the pointto-point zero-sensitivity on the ray path is a onsequene of the stationary phase theorem that is assoiated to the Fermat priniple that is no longer valid when the beam-forming proess is performed on the soure-reeiver arrays. In oean aousti tomography, suh behavior for TSK is expeted, assuming that the 3D TSK should be spatially averaged on two-dimensional 2D sound-speed flutuations in 5 2 25 3 (d) 7 75 8.5.5.5 x 8.5 x 8 FIG. 5. TSKs s 2 m 4 at 2 khz entral frequeny with 3 khz bandwidth. a point to point; b with DBF over 2 length arrays. and d Zoom in on the entral part of the ray, as defined by the blak square in a and b : point to point; d with 2 length arrays..5.5 the oean. In this ase, TSK sensitivity on the ray path is obtained assuming a 2D TSK alulation for the sound-speed perturbation in the oean. In our ase, the maximal traveltime sensitivity on the ray path is due to the use of arrays on both sides of the waveguide and does not depend of assumptions on the shape of the oean spatial flutuations. Of note, the high sensitivity seen after the DBF on the ray path is interesting from the point of view of oean aousti tomography. Indeed, in the limit of high-frequeny, large bandwidth, and large arrays, the TSK in an aousti waveguide shows a uniform kernel, with a sensitivity that is nearly limited to the ray path Fig. 5 b. This further validates the use of ray theory and soure-reeiver arrays for shallow-water oean aousti tomography. This last aspet of DBF and TSK leads to two interpretations. It ould be said that if only a few rays are available in the waveguide, we may have more diffiulties in loating anomalies, as the ambiguity of the perturbation position will now be on a broader zone. On the other hand, the uniform pattern of TSK after DBF in the Fresnel zone appears to have an advantage with respet to robustness. Indeed, sound-speed mismath is always an issue with real data and will have a signifiant effet on the Fresnel zone osillations seen on the point-to-point TSK shown in Fig. 5, while being negligible on the smoothly varying TSK measured after DBF Fig. 5 d. V. CONCLUSIONS In the present study, we have analyzed geometrially the mapping between TSKs and diffration patterns. More preisely, we have shown that the point-to-point TSK is equivalent to the gradient of the diffration pattern orreted by a spatial fator. When the pressure-field is double beamformed on soure-reeiver arrays, we obtain an approximate relationship between the TSK and the diffration pattern between the two arrays. Finally, the DBF proess signifiantly modifies the spatial struture of TSK, in suh a way that oean aousti tomography ould improve the robustness of its performane when onsidering ray theory approximation. ACKNOWLEDGMENTS The authors would like to thank Brue Cornuelle and Shane Walker for helpful omments on this study. P. Williamson, A guide to the limits of resolution imposed by sattering in ray tomography, Geophysis 56, 22 27 99. 2 P. R. Williamson and M. H. Worthington, Resolution limits in ray tomography due to wave behaviour: Numerial experiments, Geophysis 58, 727 735 993. 3 F. Dahlen, Resolution limit of travel-time tomography, Geophys. J. Int. 57, 35 33 24. 4 J. Virieux and G. Lambaré, Theory and Observation-Body Waves: Ray Methods and Finite Frequeny Effets Elsevier, Amsterdam, 27, pp. 27 55. 5 S. Husen and E. Kissling, Loal earthquake tomography between rays and waves: Fat ray tomography, Phys. Earth Planet. Inter. 23, 27 47 2. 6 M. Woodward, Wave-equation tomography, Geophysis 57, 5 26 992. 7 M. Born and E. Wolf, Priniples of Optis, 6th ed. Pergamon, New York, 98. 8 G. Nolet, A Breviary of Seismi Tomography Cambridge University Press, J. Aoust. So. Am., Vol. 26, No. 2, August 29 Iturbe et al.: Sensitivity kernels vs diffration 79
Cambridge, England, 28. 9 F. A. Dahlen, S. H. Hung, and G. Nolet, Fréhet kernels for finitedifferene travel-times I. theory, Geophys. J. Int. 4, 57 74 2. S. Gautier, G. Nolet, and J. Virieux, Finite-frequeny tomography in a rustal environment: Appliation to the western part of the Gulf of Corinth, Geophys. Prospet. 56, 493 53 28. J. Tromp, C. Tape, and Q. Liu, Seismi tomography, adjoint methods, time reversal, and banana-donut kernels, Geophys. J. Int. 6, 95 26 25. 2 M. V. de Hoop and R. D. van der Hilst, On sensitivity kernels for waveequation transmission tomography, Geophys. J. Int. 6, 62 633 25. 3 A. Sieminski, J.-J. Lvque, and E. Debayle, Can finite-frequeny effets be aounted for in ray theory surfae wave tomography?, Geophys. Res. Lett. 3, L2464. L2464.4 24. 4 P. Roux, W. Kuperman, W. Hodgkiss, H. C. Song, T. Akal, and M. Stevenson, A non reiproal implementation of time reversal in the oean, J. Aoust. So. Am. 6, 9 5 24. 5 E. K. Skarsoulis and B. D. Cornuelle, Travel-time sensitivity kernels in oean aousti tomography, J. Aoust. So. Am. 6, 227 238 24. 6 C. Tape, Q. Liu, and J. Tromp, Finite frequeny tomography using adjoint methods Methodology and examples using membrane surfae waves, Geophys. J. Int. 68, 5 29 27. 7 M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J. Thomas, and F. Wu, Time-reversed aoustis, Rep. Prog. Phys. 63, 933 995 2. 8 A. Tarantola, Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation Elsevier, Amsterdam, 987. 9 G. Herman, Image Reonstrution From Projetions: The Fundamentals of Computerized Tomography Aademi, New York, 98. 2 A. J. Devaney, Inverse-sattering theory within the Rytov approximation, Opt. Lett. 6, 374 376 98. 2 A. J. Devaney, A filtered bak-propagation algorithm for diffration tomography, Ultrason. Imaging 4, 336 35 982. 22 M. Dzieiuh, P. Worester, and W. Munk, Turning point filters: Analysis of sound propagation on a gyre sale, J. Aoust. So. Am., 35 49 2. 23 P. Roux, B. D. Cornuelle, W. A. Kuperman, and W. S. Hodgkiss, The struture of raylike arrivals in a shallow-water waveguide, J. Aoust. So. Am. 24, 343 3439 28. 24 M. Collins and E. Westwood, A higher-order energy-onserving paraboli equation for range-dependent oean depth, sound speed and density, J. Aoust. So. Am. 89, 68 75 99. 25 B. Romanowiz and R. Snieder, A new formalism for the effet of lateral heterogeneity on normal modes and surfae waves: II. General anisotropi perturbation, Geophys. J. Int. 93, 9 99 998. 26 R. Snieder and B. Romanowiz, A new formalism for the effet of lateral heterogeneity on normal modes and surfae waves: I. Isotropi perturbations, perturbations of interfaes and gravitational perturbations, Geophys. J. Int. 92, 27 222 998. 27 J. Goodman, Introdution to Fourier Optis MGraw-Hill, San Franiso, 968. 28 K. Raghukumar, B. D. Cornuelle, W. S. Hodgkiss, and W. A. Kuperman, Pressure sensitivity kernels applied to time-reversal aoustis, J. Aoust. So. Am. 22, 323 27. 72 J. Aoust. So. Am., Vol. 26, No. 2, August 29 Iturbe et al.: Sensitivity kernels vs diffration