Compressed Wavefield Extrapolation with Curvelets Tim T.Y. Lin and Felix J. Herrmann University of British Columbia SEG 2007 San Antonio, Sept 25 Introduction Concerned with explicit forms of wavefield propagator W of the linearized forward model P- = W- R+ W+ s+ ∆x3 x3 > 0 Would like to find explicit W suitable for waveequation migration: simultaneously operates on sets of traces fully incorporates velocity information of medium no parabolic approximations Introduction Compressed extrapolation with curvelets SM93 Goal: employ the complete 1-Way Helmholtz sparse reconstruction, Software: http://www.lx.it.pt/~mtf/GPSR/. REFERENCES operator for W Grimbergen, J., F. Dessing, and C. Wapenaar, 1998, Modal expansion of one- Bednar, and C. Shin, 2006, Two-way versus one-way: A comparison: 76th Annual International Meeting, SEG, cts, 2343–2347. ±by 82, Seismic migration. Imaging of acoustic energy olation: Elsevier. ompressive sensing: Presented at the Institute of Applied minars, University of British Columbia. L. Demanet, 2005, The curvelet representation of wave ptimally sparse: Communications on Pure and Applied 1472–1528. anet, D. Donoho, and L. Ying, 2006a, Fast discrete curveAM Multiscale Modeling and Simulation, 5, 861–899. Curvelets — A surprisingly effective L. Donoho, 2000a, esentation for objects with edges: in L. L. Schumaker et nd surfaces: Vanderbilt University Press. vering edges in ill-posed problems: Optimality of1 curves of Statistics, 30, 784–842. ight frames of curvelets and optimal representations of wise C2 singularities: Communications On Pure and Aps, 57, 219–266. Donoho, L. Demanet, and L. Ying, 2005, Fast discrete m: http://www.curvelet.org/papers/FDCT.pdf. 2Software: http://www.acm. . Romberg, 2005, ᐉ1-magic. gic/. berg, and T. Tao, 2006b, Stable signal recovery from inccurate measurements: Communications On Pure and atics, 59, 1207–1223. W way operator on laterally varying media: Geophysics, 63, 995–1005. Guitton, A., and D. J. Verschuur, 2004, Adaptive subtraction of multiples using the ᐉ1-norm: Geophysical Prospecting, 52, 27–27. Hale, D., N. R. Hill, and 1 J. Stefani, 1992, Imaging salt with turning seismic waves: Geophysics, 57, 1453–1462. 2 Discussion and reply1by authors in 1 GEO-58-8-1205-1206. He, C., M. Lu, and C. Sun, 2004, Accelerating seismic migration using FPGA-based coprocessor platform: 12th Annual Symposium on FieldProgrammable Custom Computing Machines, IEEE, 207–216. Hennenfent, G., and F. J. Herrmann, 2006a, Application of stable signal recovery to seismic interpolation: 76th Annual International Meeting, SEG, Expanded Abstracts, 2797–2801. ——–, 2006b, Seismic denoising with non-uniformly sampled curvelets: 2 in Science and Engineering, 8, 16–25. Computing Herrmann, F. J., U. Boeniger, and D. J. Verschuur, 2007, Nonlinear primarymultiple separation with directional curvelet frames: Geophysical Journal International, 17, 781–799. Koh, K., S. J. Kim, and S. Boyd, 2007, Simple matlab solver for 11-regularized least squares problems, Software: http://www-stat.stanford.edu/ ~tibs/lasso.html. Levy, S., D. Oldenburg, and J. Wang, 1988, Subsurface imaging using magnetotelluric data: Geophysics, 53, 104–117. Mulder, W., and1R. Plessix, 2004, How to choose a subset of frequencies in frequency-domain finite-difference migration: Geophysical Journal International, 158, 801–812. Oldenburg, D. W., S. Levy, and K. P. Whittall, 1981, Wavelet estimation and deconvolution: Geophysics, 46, 1528–1542. Paige, C. C., and M. A. Saunders, 1982, LSQR: An algorithm for sparse lin- =e ∓j∆xH H =H H Problem: computation & storage complexity creating and storing H is trivial however H is not trivial to compute and store H = H = Introduction In this case W decomposition is computed by eigenvalue H2 = LΛL = T L Λ L T W = ± L e−j √ Λ∆x3 LT requires, per frequency: 1 eigenvalue problem (O(n4)) 2 full matrix-vector for eigenspace transform (O(n2)) Introduction Band-diagonalization techniques like parabolic approximation trades for speed with approximations Is there another way? Our approach Consider a related, but simpler problem: shifting (or translating) signal −j ∆x 2π D operator is S = e D is differential operator D= Our approach Computation requires similar approach to W± D = LΛL = T L Λ L T S= L −j ∆x 2π Λ e LT However, for D, L = DFT , so computation trivial with FFT Our approach Suppose FFT does not exist yet ...... s(k) Fω=1,2,3,4,... s(x) Our approach suppose some nodes didn’t finish their jobs ...... s(k) Fω=1,4,... s(x) Our approach mathematically, the system is incomplete = Fω=1,4,... s(k) s(x) evidently some information of original invariably lost. Or is it? s(x) is Compressed Sensing SM78 Lin and Herrman states that given system of the form L ,L measured the fo y͉x=͉ s.t. Ax = y, ˜ x = argmin ʈxʈ = ͉x ͉ minʈxʈ1signal = ͑2͒ ͚ 1 i s i x i = 1accur x i = 1 linear model of restricted representation of on the ˜ sparse with the symbol hereby reserved for qua measurement process original data ereby reserved for (measurement quantities obtained by solvbasis) ing an optimization problem. The argmin cause x st problem. The argminx stands for the argument ofof the given the minimum, i.e., the value arg rows ue of the expression attains val e value of the given argument for which the val-its minimum previ cessful the measurement attains its minimum value. Thiswhen recovery is suc- and sparsity comp herent and when m is large enough compar asurement and sparsity representations are incoingreco n zero entries in x0. Because m N, this s large enough comparedsion to of theannumber of non-system. As long underdetermined N ͚ N H Compressed Sensing SM78 Lin and Herrman states that given system of the form Compressed extrapolation with curvelets L ,L REFERENCES measured the fo y͉x=͉ s.t. Ax = y, ˜ x = argmin ʈxʈ = ͉x ͉ minʈxʈ1signal = ͑2͒ ͚ 1 i s i x i = 1accur x i = 1 linear model of restricted representation of on the ˜ sparse with the symbol hereby reserved for qua measurement process original data ereby reserved for (measurement quantities obtained by solvbasis) ing an optimization problem. The argmin cause x st problem. The argminx stands for the argument ofof the given the minimum, i.e., the value arg rows SM78 Lin and Herrman ue of the expression attains its minimum val e value of the given argument for which the val can exactly “recover” x from y by solving L1 problem previ cessful the measurement N when attains its minimum value. This recovery is suc- and sparsity L H, L comp herent and when m is large enough compar the fo ˜ x = argmin ʈxʈ = ͉x ͉ s.t. Ax = y, ͑2͒ ͚ 1 i asurement and sparsity representations are incox ing n zeroi entries in x0. Because m N, this reco accur = 1 s large enough compared to the number of nonthe sion of an underdetermined system. As on long ˜ N ͚ Bednar, J. B., C. J. Bednar, and C. Shin, 2006, Two-way versus one-way: A case study style comparison: 76th Annual International Meeting, SEG, Expanded Abstracts, 2343–2347. Berkhout, A. J., 1982, Seismic migration. Imaging of acoustic energy by wave field extrapolation: Elsevier. Candès, E., 2007, Compressive sensing: Presented at the Institute of Applied Mathematics Seminars, University of British Columbia. Candès, E. J., and L. Demanet, 2005, The curvelet representation of wave propagators is optimally sparse: Communications on Pure and Applied Mathematics, 58, 1472–1528. Candès, E., L. Demanet, D. Donoho, and L. Ying, 2006a, Fast discrete curvelet transforms: SIAM Multiscale Modeling and Simulation, 5, 861–899. Candès, E. J., and D. L. Donoho, 2000a, Curvelets — A surprisingly effective nonadaptive representation for objects with edges: in L. L. Schumaker et al., eds., Curves and surfaces: Vanderbilt University Press. ——–, 2000b, Recovering edges in ill-posed problems: Optimality of curvelet frames: Annals of Statistics, 30, 784–842. ——–, 2004, New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities: Communications On Pure and Applied Mathematics, 57, 219–266. Candès, E. J., D. L. Donoho, L. Demanet, and L. Ying, 2005, Fast discrete curvelet transform: http://www.curvelet.org/papers/FDCT.pdf. Candès, E. J., and J. Romberg, 2005, ᐉ1-magic. Software: http://www.acm. caltech.edu/limagic/. Candès, E., J. Romberg, and T. Tao, 2006b, Stable signal recovery from incomplete and inaccurate measurements: Communications On Pure and Applied Mathematics, 59, 1207–1223. with the symbol N H sparse reconstruction, Software: http://www.lx.it.pt/~mtf/GP Grimbergen, J., F. Dessing, and C. Wapenaar, 1998, Modal expa way operator on laterally varying media: Geophysics, 63, 995 Guitton, A., and D. J. Verschuur, 2004, Adaptive subtraction of ing the ᐉ1-norm: Geophysical Prospecting, 52, 27–27. Hale, D., N. R. Hill, and J. Stefani, 1992, Imaging salt with tur waves: Geophysics, 57, 1453–1462. Discussion and reply b GEO-58-8-1205-1206. He, C., M. Lu, and C. Sun, 2004, Accelerating seismic mig FPGA-based coprocessor platform: 12th Annual Symposiu Programmable Custom Computing Machines, IEEE, 207–21 Hennenfent, G., and F. J. Herrmann, 2006a, Application of stab covery to seismic interpolation: 76th Annual International M Expanded Abstracts, 2797–2801. ——–, 2006b, Seismic denoising with non-uniformly sample Computing in Science and Engineering, 8, 16–25. Herrmann, F. J., U. Boeniger, and D. J. Verschuur, 2007, Nonlin multiple separation with directional curvelet frames: Geophy International, 17, 781–799. Koh, K., S. J. Kim, and S. Boyd, 2007, Simple matlab solver fo ized least squares problems, Software: http://www-stat.s ~tibs/lasso.html. Levy, S., D. Oldenburg, and J. Wang, 1988, Subsurface imagin netotelluric data: Geophysics, 53, 104–117. Mulder, W., and R. Plessix, 2004, How to choose a subset of fr frequency-domain finite-difference migration: Geophysical J national, 158, 801–812. Oldenburg, D. W., S. Levy, and K. P. Whittall, 1981, Wavelet es deconvolution: Geophysics, 46, 1528–1542. Paige, C. C., and M. A. Saunders, 1982, LSQR: An algorithm f hereby reserved for quantities obtained by solv- cause signal in time domain r F signal in time domain signal in Fourier domain r F signal in time domain signal in Fourier domain R restricted signal in Fourier domain (real) r F signal in time domain signal in Fourier domain L1 R recovered signal in time domain restricted signal in Fourier domain (real) r Compressed Sensing x has to be sparse A has to be Fourier transform Compressed sensing theory gives us strict bounds on regions of recoverability Enables deliberate incomplete computations Compressed Sensing “Computation” −j ∆x 2π Λ if we “shift” s(k) with e we recover s(x) using s’(k)? = s (k) , what happens when = −j ∆x 2π Λ e s(k) Fω=1,4,... s(k) s(x) Compressed Sensing “Computation” −j ∆x 2π Λ if we “shift” s(k) with e we recover s(x) using s’(k)? = s (k) , what happens when = −j ∆x 2π Λ e s(k) Fω=1,4,... s(k) s(x) Answer: we recover a shifted s(x)! Compressed Sensing L1 F R incomplete signal in Fourier domain signal in space domain signal in space domain Compressed Processing FR signal in space domain L1 ∆x −j −j∆x Λ 2π 2π Λ ee incomplete and shifted signal in Fourier domain shifted signal in space domain Straightforward Computation F F −j ∆x 2π Λ e shifted signal in Fourier domain signal in space domain shifted signal in space domain Compressed Processing FR signal in space domain L1 −j ∆x 2π Λ e incomplete and shifted signal in Fourier domain shifted signal in space domain ear equationsdata: and deconvolution: Geophysics, 46, 1528–1542. netotelluric tions for the curvelet design: 76th Annual International Meeting, SEG, Excurvelet transform: http://www.curvelet.org/papers/FDCT.pdf. Software, 8,and 43–71 Paige, C. C., and M. A. Saunders, 1982, LSQR: An algorithm for sparse linMulder, W., R. panded Abstracts, 2406–2410. d its implicasparse reconstruc -magic. Software: http://www.acm. Candès, E. J., and J. 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Deman ͑MCA͒: Applied and Computational Harmonic Analysis, 19, 340–358. 1 eeting, SEG, Compressed Sensing “Computation” Compressed Wavefield Extrapolation Recall the similarity between W± and W = ± S H2 = L √ −j Λ∆x3 e LT D= S= F −j ∆x 2π Λ e F T Compressed Wavefield Extrapolation H2 = LΛL T Structure of H1 analytically H2 = H1 H1 H2 = k 2 (x) + ∂µ ∂µ discretely H2 = C + D2 H2 = ω c1 2 0 0 .. . ω c2 0 0 .. . 2 ··· 0 ··· .. . 0 .. . ··· ω cn1 + D2 2 H1 = LΛ 1/2 T L Compressed Wavefield extrapolation eigenfunctions of H2 at 30 Hz for constant velocity medium 0 5 10 15 20 25 Eigenvalue Index Asymptotically identical to the Cosine transform Compressed Wavefield extrapolation eigenfunctions of H2 at 30 Hz for Marmousi velocity medium 4400 4200 4000 Wave Velocivty (m/s) 3800 3600 3400 3200 3000 2800 2600 2400 0 50 100 150 Position Index 200 250 Compressed Wavefield extrapolation eigenfunctions of H2 at 30 Hz for Marmousi velocity medium 0 5 10 15 20 25 Eigenvalue Index fairly close to the Cosine transform Straightforward 1-Way inverse Wavefield Extrapolation LT e−j √ L Λ∆x3 back-extrapolated to impulse source in space-time domain back-extrapolated wavefield in H2 domain wavefield in space-time domain Compressed 1-Way Wavefield Extrapolation LT √ −j Λ∆x3 Re wavefield in space-time domain incomplete backextrapolated wavefield in H2 domain L1 back-extrapolated to impulse source in space-time domain Compressed wavefield extrapolation simple 1-D space/time propagation example with point scatters 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4000 3200 2400 4 6 8 10 Compressed wavefield extrapolation simple 1-D space/time propagation example with point scatters 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 0 2 4 6 propagated 1.5km down 8 10 1 0 2 4 6 8 recovered though L1 inverson Restricted L transform to ~0.01 of original coefficients 10 Compressed extrapola Sparsity through curvelets REFERENCES for extrapolation to reflectivity, we first transform Bednar, J.into B., C.aJ. sparsifies Bednar, and C.reflectivity Shin, 2006, Two-way versus one-way: A signal case study style comparison: 76th Annual International Meeting, SEG, Expanded Abstracts, 2343–2347. Berkhout, A. J., 1982, Seismic migration. Imaging of acoustic energy by we know reflectivityElsevier. are sparse in curvelets wave field extrapolation: Candès, E., 2007, Compressive sensing: Presented at the Institute of Applied Mathematics Seminars, University of British Columbia. Candès, E. J., and L. Demanet, 2005, The curvelet representation of wave propagators is optimally sparse: Communications on Pure and Applied Mathematics, 58, 1472–1528. Candès, E., L. Demanet, D. Donoho, and L. Ying, 2006a, Fast discrete curvelet transforms: SIAM Multiscale Modeling and Simulation, 5, 861–899. Candès, E. J., and D. L. Donoho, 2000a, Curvelets — A surprisingly effective nonadaptive representation for objects with edges: in L. L. Schumaker et al., eds., Curves and surfaces: Vanderbilt University Press. ——–, 2000b, Recovering edges in ill-posed problems: Optimality of curvelet frames: Annals of Statistics, 30, 784–842. ——–, 2004, New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities: Communications On Pure and Ap- Example (Canadian overthrust) original reflectivity downward extrapolated 50m inverse extrapolated explicitly Example (Canadian overthrust) inverse extrapolated explicitly inverse extrapolated with compressed computation ~15% coefficients used Discussions Bottom line: synthesis, operation, and storage cost savings versus L1-solver cost require good sparsity-promoting basis (ie Curvelets) potential to apply same technique to a variety of different operators Conclusions 1) Take linear operator with suitable structure for compressed sensing, having a diagonalizing basis which is incoherent with the signal basis 2) Compressed sensing theory tells us how much computation we can throw away while still recovering full signal with L1 solver 3) Then we can take advantage of results in compressed sampling for compressed computation Take home point: Exploit compressed sensing theory for gains in scientific computation Still awake? Check-out the full paper at: Lin, T.T.Y. and F. Herrmann, 2007, Compressed wavefield extrapolation: Geophysics, 72, SM77-SM93 Compressed wavefield extrapolation y ˜ x ˜ u √ Λ∆x3 LT u = Re = arg minx x ˜ =x −jω 1 s.t. RL x = y T Randomly subsample in the Modal domain Recover by norm-one minimization Capitalize on the incoherence between modal functions and impulse sources reduced explicit matrix size Compressed wavefield extrapolation with curvelets y ˜ x ˜ u √ Λ∆x3 LT CT u = Re = arg minx x ˜ =x −jω 1 s.t. RL C x = y T T Original and reconstructed signals remain in the curvelet domain Original curvelet transform must be done outside of the algorithm
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Compressed wavefield extrapolation with curvelets Lin, Tim T. Y.; Herrmann, Felix J. 2007-12-31
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Title | Compressed wavefield extrapolation with curvelets |
Creator |
Lin, Tim T. Y. Herrmann, Felix J. |
Publisher | Society of Exploration Geophysicists |
Date Issued | 2007 |
Description | An \emph {explicit} algorithm for the extrapolation of one-way wavefields is proposed which combines recent developments in information theory and theoretical signal processing with the physics of wave propagation. Because of excessive memory requirements, explicit formulations for wave propagation have proven to be a challenge in {3-D}. By using ideas from ``\emph{compressed sensing}'', we are able to formulate the (inverse) wavefield extrapolation problem on small subsets of the data volume{,} thereby reducing the size of the operators. According {to} compressed sensing theory, signals can successfully be recovered from an imcomplete set of measurements when the measurement basis is \emph{incoherent} with the representation in which the wavefield is sparse. In this new approach, the eigenfunctions of the Helmholtz operator are recognized as a basis that is incoherent with curvelets that are known to compress seismic wavefields. By casting the wavefield extrapolation problem in this framework, wavefields can successfully be extrapolated in the modal domain via a computationally cheaper operatoion. A proof of principle for the ``compressed sensing'' method is given for wavefield extrapolation in {2-D}. The results show that our method is stable and produces identical results compared to the direct application of the full extrapolation operator. |
Extent | 2684048 bytes |
Subject |
Helmholtz operator compressed sensing wavefield extrapolation eigenfunctions curvelets incoherent compressed processing compressed wavefield extrapolation |
Genre |
Conference Paper |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-03-20 |
Provider | Vancouver : University of British Columbia Library |
DOI | 10.14288/1.0107427 |
URI | http://hdl.handle.net/2429/607 |
Affiliation |
Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Herrmann, Felix J., Lin, Tim T.Y. 2007. Compressed wavefield extrapolation with curvelets. SEG 77th Annual Meeting and Exposition. |
Peer Review Status | Unreviewed |
Scholarly Level | Graduate Faculty |
Copyright Holder | Herrmann, Felix J. |
Aggregated Source Repository | DSpace |
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