ADAPTIVE TRAVEL TIME TOMOGRAPHY WITH LOCAL SPARSITY. Michael Bianco and Peter Gerstoft

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1 ADAPTIVE TRAVEL TIME TOMOGRAPHY WITH LOCAL SPARSITY Mchael Banco and Peter Gerstoft UCSD Scrpps Insttuton of Oceanography La Jolla, CA ABSTRACT We develop a 2D travel tme tomography method whch regularzes the nverson by modelng sparsely patches of slowness pxels from dscrete slowness map, and adapts sparse dctonares to the slowness data. Ths locally-sparse travel tme tomography LST approach consders global and local behavor of slowness, whereas conventonal regularzaton methods consder only global covarance of pxels. We develop a maxmum a posteror formulaton of LST, and further explot the sparsty of patches usng dctonary learnng. We demonstrate the LST method on densely, but rregularly sampled synthetc slowness maps. Index Terms Sparse modelng, machne learnng, dctonary learnng, geophyscs, sesmcs. INTRODUCTION Travel tme tomography methods attempt to estmate complex Earth structure, whch contans smooth and dscontnuous features at multple spatal scales, usng sesmc and acoustc wave travel tmes between recordng statons [, 2]. The nverson of the travel tmes for a slowness model nverse of speed s ll-posed, wth often dense but rregular ray coverage of envronments. Conventonal tomography technques regularze the nverson by restrctng the models to be only smooth or dscontnuous, whch nclude st and 2nd order Tkhonov regularzaton [, 2]. Other regularzaton methods have been proposed whch employ of wavelet functons [3 6], total varaton TV [7, 8], or adaptve dscretzatons of slowness [9]. Recent works n acoustcs have utlzed sparse modelng and compressve sensng CS [0 2] to mprove performance n beamformng [3,4] and nverson for ocean acoustc propertes [5 8]. Smlarly, the more recent waveletbased methods n sesmc tomography, e.g. [4, 6, 9], assume sparse wavelet coeffcents. In sparse modelng and CS, nverse problems are regularzed by modelng sgnals as sparse combnatons of vectors or atoms from set or dctonary of atoms, whch can be prescrbed or learned [, 2, ]. Ths Ths work s supported by the Offce of Naval Research, Grant No. N paradgm s ubqutous n sgnal processng for mage denosng and npantng [, 2], and medcal magng [2, 22], to name a few examples. Learned dctonares can mprove reconstructon performance over prescrbed dctonares and recently, nverson methods wth dctonary learnng have been developed n the geoscences. Applcatons nclude denosng sesmc traces [23] and ocean acoustc recordngs [24], full waveform nverson [25], and estmaton of ocean sound speed profles [5, 6]. In ths paper, we develop a sparse and adaptve approach to 2D travel tme tomography, whch we refer to as locallysparse travel tme tomography LST. The LST sparsely models local behavors of overlappng groups of pxels from a dscrete slowness map, called patches. Large scale features n the slowness map are constraned usng least squares. Ths approach s smlar to works n mage denosng [26] and CS magnetc resonance magng MRI [22]. We develop a maxmum a posteror MAP formulaton to the problem and use the teratve thresholdng and sgned K-means ITKM dctonary learnng algorthm to mprove the slowness models over prescrbed dctonares. We demonstrate the performance of LST consderng 2D surface wave tomography wth synthetc slowness maps and travel tme data. The results are compared wth conventonal tomography. More detals of the approach and further expermental results are avalable n a forthcomng paper [27]. 2. OVERVIEW OF LST Gven travel tme perturbatons t R M from M ray paths through a dscrete slowness map see Fg. a, and tomography matrx A R M N, LST estmates the sparse slowness s s. We frst estmate the global slowness s g, and then obtan the patch slowness Dx for patch of s g. Here D R n Q s a dctonary of Q atoms, and x R n s the sparse coeffcents wth n the number of pxels n a patch. Fnally slownesses {D x } are averaged wth s g to obtan s s. 2.. Global slowness and travel tme We dscretze a 2D slowness map as a W W 2 pxel mage, shown n Fg. a, where each pxel has constant slowness. The slowness pxels are represented by the vector /8/$ IEEE 63 ICASSP 8

2 necessary to adequately approxmate R s s. Atoms can be prescrbed functons, e.g. wavelets or the dscrete cosne transform DCT, or learned from the data see Sec DERIVATION OF LST MAP OBJECTIVE Startng wth Bayes rule, we derve the LST MAP objectve for s s, ncorporatng both local sparse pror and global constrants. For the dervaton, we assume the dctonary D and sensng matrx A known. In Sec. 3.2, dctonary learnng s ncluded n the algorthm. The posteror densty s formulated as p s g, s s, X t p t s g, s s, X p s g s s, Xp s s Xp X, 2 Fg. : a 2D slowness patches and slowness map parameters, wth b example patch dstrbuton. Synthetc slowness s for c checkerboard map and d smooth-dscontnuous map W = W 2 = 00 pxels km. e 6 straght ray paths surface wave from 64 sesmc statons red X s. s = s g + s 0 R N, where s 0 s reference slownesses and s g s perturbatons from the reference, wth N = W W 2. We assume travel tme observatons t = t + t 0 from M straght ray paths, where t 0 and t are the reference travel tme and perturbatons. Snce s 0 and t 0 = As 0 are known, we estmate the perturbatons t = As g + ɛ, where ɛ R M s Gaussan nose N 0, σ 2 ɛ I, wth mean 0 and covarance σ 2 ɛ I, n the travel tme observatons. We call the global model, as t captures the large-scale features that span the dscrete map and generates t. We assume dense ray coverage, and do not explctly account for varyng ray densty see Sec Local sparse model Each patch s a n n group of pxels from s s, see Fg. a. The patches are selected from s s by the bnary matrx R {0, } n N. Hence the slownesses n patch are R s s. Each patch s ndexed by the row w and column w 2 of ts top-left pxel n the 2D mage as w,, w 2,. We consder all overlappng patches, wth w, and w 2, dfferng from ther neghbor by ± strde of one. Thus, for a W W 2 pxel mage, the number of patches s I = W n + W 2 n +. R s s s approxmated by sparse combnatons atoms from D. The coeffcents x are estmated usng the l 0 pseudonorm see 7, whch penalzes the number of non-zero coeffcents [2]. We call 7 the local model, as t captures the smaller scale, localzed features contaned by patches. The atoms n D are consdered elemental patches, where only a small number of atoms from Q I are where X = [x,..., x I ] R Q I are the coeffcents descrbng all patches. If s g s s s known, so s t s g, whereby p s g, s s, X t p t sg p sg ss p s s Xp X. 3 We assume p t s g, p sg s s, and p ss X are Gaussan, whch results n a smple LST objectve. Hence, for the global model, p t sg = N Asg, Σ ɛ and p s g ss = N ss, Σ g where Σ ɛ R K K s the covarance of ɛ and Σ g R N N s the covarance of s g. For the local model, the patch slownesses {R s s } are consdered ndependent, gvng the local lkelhood p s s X p s s X = p R s x s = N Dx, Σ p, where Σ p, R n n s the covarance of the patch slownesses for each patch. Assumng the coeffcents x ndependent and sparse, ln p X = ln p x, wth ln p x x 0. We further assume the number of non-zero coeffcents T s the same for every patch for whch the l 0 -norm penalty s well suted, and errors d wth Σ ɛ = σɛ 2 I, Σ g = σgi, 2 and Σ p, = σp, 2 I, where I s the dentty matrx. Hence the MAP estmate { ŝ g, ŝ s, X } s from 3 {ŝg, ŝ s, X} = arg mn s g, s s, X + σp, 2 Dx R s s Solvng for the MAP estmate 4 { σɛ 2 t As g σs 2 s g s s 2 2 } 5 subject to x 0 = T. We fnd the MAP estmates { ŝ g, ŝ s, X } solvng 5 va blockcoordnate mnmzaton, smlar to [22, 26]. The global objectve s wrtten from 5 ŝ g = arg mn s g t As g λ s g s s 2 2, 6 where λ = σ ɛ /σ g 2 s a regularzaton parameter. The local objectve from 5 for each patch s solved wth s s = ŝ g decouplng the local and global objectves, gvng x = arg mn Dx R ŝ g 2 2 subject to x 0 = T. 7 x 64

3 Gven: t R M, A R M N, s 0 s = 0 R N, D 0 = Haar, DCT or nose N 0, R n Q, λ, λ 2, T, and j = Repeat untl convergence:. Global estmate: solve 6 usng LSQR [28], ŝ j g = arg mn s j g As j g t λ s j g s j s Local estmate a: Settng s j s = ŝ j g, center patches {R ŝ j g } and. Dctonary learnng Fnd D j usng ITKM [].. Prescrbed dctonary Set D j = D 0.. Solve 7 usng OMP, x j = arg mn D j x j R ŝ j g 2 2 subject to x j 0 = T. x j b: Obtan ŝ j s by 0 as j = j + ŝ j s,n = λ 2ŝ j g,n+b n s j p,n λ 2 +b n Table : Sparse travel tme tomography LST algorthm wth fxed or adaptve dctonares Wth X = [ x,..., x I ] from 7 and ŝ g from 6, we fnd ŝ s from 5, assumng σ 2 p, = σ2 p ŝ s = arg mn λ 2 ŝ g s s s s D x R s s 2 2, 8 where λ 2 = σ p /σ g 2 s a regularzaton parameter. The estmate ŝ s s obtaned analytcally from 8 by ŝ s = λ 2 I + R T R λ 2 ŝ g + R T D x, 9 whch averages the pxels from the patch estmates {D x }, wth weght gven to ŝ g by λ 2. From 9, the average patch slowness s s p = RT R RT D x, wth b = dag RT R Z N the number of patches per pxel. Hence, 9 s expressed as an operaton at pxel n by ŝ s,n = λ 2ŝ g,n + b n s p,n λ 2 + b n LST algorthm wth dctonary learnng The results 6, 7, and 9 gve the LST algorthm for estmatng s s, shown n Table, as a MAP estmate wth local sparse prors usng a prescrbed dctonary D. Dctonary learnng va the ITKM [] s added to the LST n the soluton to the local objectve 7. The global objectve 6 s solved usng the sparse least squares program LSQR [28]. The local objectve 7 s solved usng OMP after the slowness patches {R ŝ g } are centered []. The complexty of each LST teraton s determned prmarly by LSQR computaton n the global estmate, O2MN, and by ITKM OknQI and OMP OT nqi n the local estmate, where k s the ITKM teratons see Table. For large slowness maps, we expect the LST complexty to be domnated by LSQR. In our smulatons we obtan reasonable run tmes see Sec Conventonal tomography We llustrate conventonal tomography wth a Bayesan approach [29], whch enforces smoothness regularzaton wth a global non-dagonal covarance. Consderng the measurements, the MAP estmate of the slowness s ŝ g = A T A + ησ A T L t, where η = σ ɛ /σ c 2 s a regularzaton parameter, σ c s the conventonal slowness varance, and smoothness Σ L, j = exp D,j /L. Here, D,j s the dstance between cells and j, and L s the length scale [29, 30]. 4. SIMULATION We demonstrate the performance of LST Sec. 3, Table relatve to a conventonal tomography Sec Experments are conducted usng smulated travel tmes from two synthetc 2D slowness maps Fg. c,d wth dmensons W = W 2 = 00 pxels km. The checkerboard pattern Fg. c contans only dscontnuous slowness whereas the smooth-dscontnuous map Fg. d contans a fault-lke dscontnuty n a smooth map. The slowness estmates from LST are plotted as ŝ s = ŝ s + s 0 R N, and for conventonal ŝ g = ŝ g + s 0 R N. The slownesses are sampled by M = 6 straght-rays between 64 sesmc statons see Fg. e. The travel tme vector t s found by ntegratng along these ray paths. We consder only straght ray propagaton, to focus on our proposed nverson approach. The reference slowness s calculated from the mean travel tme usng the the tomography matrx A. The LST nverson vald-regon s obtaned wth a dlaton operaton wth a patch template along the outermost ray paths. The conventonal vald regon s the outermost pxels along the ray paths. The conventonal vald regon s used for error calculatons for both methods. We consder the nose-free case σ ɛ = 0. The results of the LST and conventonal tomography are shown n Fgs. 2 and 3. RMSE s/km of the estmates ŝ s and ŝ s relatve to the true slowness s s prnted on the 2D estmates. We nvert usng LST wth and wthout dctonary learnng. We consder two prescrbed dctonares D, overcomplete Haar wavelet and DCT dctonares both Q = 69, n = 64, snce Haar wavelet dmensons power of Inverson parameters and results The regularzaton parameter values for LST and conventonal tomography were selected to mnmze RMSE s/km. For LST, the best parameters were: for both prescrbed dctonary 65

4 Range km a 00 Slowness s/km e b True Estmated f c g d h Range km 00 Slowness s/km a d 0.02 Slowness s/km b e True Estmated Range km 00 c f Range km Slowness s/km 0.02 Fg. 2: LST and conventonal tomography for checkerboard map Fg. c. 2D and D from black lne n 2D slowness estmates aganst true slowness for: a,b conventonal ŝ g; LST ŝ s wth c,d Haar dctonary and e,f wth DCT dctonary D; and g,h wth dctonary learnng. RMSE s/km s prnted on each 2D mage. g j h k l and dctonary learnng, λ = 0 km 2 n 6 and λ 2 = 0 n 8; for prescrbed dctonares T = 2 non-zero coeffcents n 7; for dctonary learnng, T =, n = 00, and for the checkerboard smooth-dscontnuous Q = 66 Q = 268 atoms. Snce for the nose free case σ ɛ = 0, we expect λ = σ ɛ /σ g 2 = 0 km 2 to be best. We assume the slowness patches are well approxmated by the sparse model 7, and expect σ p σ g. Hence, we expect the best value of λ 2 = σ p /σ g 2 n 8 to be small. For conventonal tomography Sec. 3.3, the best parameters were L = 0 km and η = 0. km 2 n for the both the checkerboard and smooth-dscontnuous maps, whch devates from the expected value of η = σ ɛ /σ c 2 = 0. Whle the dscontnuous shapes n the Haar dctonary are smlar to the dscontnuous content of the checkerboard mage, the local features n the hgher order Haar wavelets overft the ray samplng where samplng s poor near the edges of the nverson, Fg. 2c,d. The performance of the Haar wavelets s better for the smooth-dscontnuous slowness map Fg. 3d f than for the checkerboard. As shown n Fg. 3d f, the Haar wavelets add false hgh frequency structure to the slowness reconstructon but the trends n the smoothdscontnuous features are well preserved. The nverson performance of the DCT transform Fg. 2e,f and Fg. 3g,j s better than the Haar wavelets for both cases, but matches less closely the dscontnuous slowness features, as the DCT atoms are smooth. The smoothness of the DCT atoms better preserve the smooth slowness structure. The LST wth dctonary learnng Fg. 2g,h and Fg. 3j l acheves the best RMSE relatve to the true slowness s. As n the other cases, the performance degrades near the edges of the ray samplng, where the ray coverage s poor, but hgh resoluton s mantaned across a large part of the samplng regon. The RMSE of the Haar wavelet nverson for the checkerboard s greater than for the conventonal method, Range km 00 Range km Slowness s/km Fg. 3: LST and conventonal tomography for smooth-dscontnuous map Fg. d. 2D and D horzontal and vertcal, from black lnes n 2D slowness estmates aganst true slowness for: a c conventonal ŝ g; LST ŝ s wth d f Haar dctonary and g wth DCT dctonary D; and j l wth dctonary learnng. RMSE s/km s prnted on each 2D mage. although resoluton s lost n the conventonal MAP nverson near the more densely sampled regon of the wave speed maps. The RMSE for the DCT s less than the that of the Haar wavelets and also the conventonal MAP nverson. A better qualtatve ft to the true slowness s also observed. The LST algorthm Table used 00 teratons for all cases and the ITKM used 50 teratons. In Matlab, the nverson wth dctonary learnng took 5 mn on a Macbook Pro 2.5 GHz Intel Core CONCLUSIONS We derved a travel tme tomography method whch ncorporates a sparse pror on patches of the slowness mage, whch we refer to as the LST algorthm. The LST uses prescrbed or learned dctonares, though the learned dctonares mprove performance. The local sparse pror and dctonary learnng provde an mproved slowness model, whch s capable of modelng smultaneously smooth dscontnuous features. We consdered 2D surface wave tomography, and for densely sampled slowness maps obtaned superor results from the LST over conventonal tomography. The LST s relevant to other tomography scenaros where slowness structure s rregularly sampled, n for nstance ocean [6] and terrestral [3] acoustcs. 66

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