Compressive sampling meets seismic imaging Felix J. Herrmann joint work with Tim Lin*, Peyman Moghaddam*, Gilles Hennenfent*, Deli Wang* & Chris Stolk (Universiteit Twente) *Seismic Laboratory for Imaging and Modeling slim.eos.ubc.ca PIMS-CINVESTAV 2007, Monterrey, October 19 Today’s challenges Aside from spurious local minima seismic waveform inversion is difficult because of lack of control on the image amplitudes missing data and noise computational cost to form the operators Today’s agenda is to leverage recent insights from applied harmonic analysis and information theory to restore amplitudes => affordable q-Newton updates stably reconstruct wavefields compress wavefield-extrapolation operators Motivation Exploit two aspects of curvelets, namely their parsimoniousness invariance under certain operators Formulate data-adaptive scaling algorithms non-adaptive wavefield reconstruction algorithms Applications nonlinear migration-amplitude recovery nonlinear sampling for wavefields nonlinear sampling for operators Today’s topics Sparsity-promoting seismic-image amplitude recovery curvelet-domain diagonal approximation of PsDO’s stable sparsity-promoting inversion Directional frame-based wavefield reconstruction by sparsity promotion curvelet parsimoniousness jitter sampling Compression of FIO’s through compressive sampling measurement basis diagonalizes operator The problem Minimization: c = arg min d − F [c] c 2 2 After linearization (Born app.) forward model with noise: d(xs , xr , t) = K[¯ c]m (xs , xr , t) + n(xs , xr , t) Conventional imaging: K d (x) = T y(x) = K Km (x) + K n (x) T T Ψm (x) + e(x) Ψ is prohibitively expensive to invert c] involves expensive wavefield evaluation of K[¯ extrapolators 2-D curvelets curvelets are of rapid decay in space x-t curvelets are strictly localized in frequency f-k Oscillatory in one direction and smooth in the others! Obey parabolic scaling relation length ≈ width2 Coefficients Amplitude Decay In Transform Domains Fourier Wavelets Curvelets Partial Reconstruction Fourier (1% largest coefficients) SNR = 2.1 dB Partial Reconstruction Curvelets (1% largest coefficients) SNR = 6.0 dB 3-D curvelets Curvelets are oscillatory in one direction and smooth in the others. Approximate linearized inversion by curvelet scaling & sparsity promotion Joint work with Chris Stolk* and Peyman Moghaddam Mathematics Department, Twente University, the Netherlands “Sparsity- and continuity-promoting seismic imaging with curvelet frames” to appear in ACHA Related work Wavelet-Vaguelette/Quasi-SVD methods based on homogeneous operators absorb “square-root” of the Gramm matrix in WVD’s Wavelets/curvelets near diagonalize the operator and are sparse on the model Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition (Donoho ‘95) Recovering Edges in Ill-posed Problems: Optimality of curvelet Frames (Candes & Donoho ‘00) Scaling methods based on a diagonal approximation of Ψ , assuming smoothness on the symbol and conormality reflectors Illumination-based normalization (Rickett ‘02) Amplitude preserved migration (Plessix & Mulder ‘04) Amplitude corrections (Guitton ‘04) Amplitude scaling (Symes ‘07) Hessian/Normal operator [Stolk 2002, ten Kroode 1997, de Hoop 2000, 2003] Alternative to expensive least-squares migration. In high-frequency limit Ψ is a pseudo-differential operator Ψf (x) := K T Kf (x) = R −ix·ξ ˆ(ξ)dξ e a(x, ξ) f d composition of two Fourier integral operators pseudolocal (near unitary) singularities are preserved symbol is smooth for smooth velocity models c¯ Corresponds to a spatially-varying dip filter after appropriate preconditioning (=> zero-order PsDO). To make this PsDO amenable to an approximation b Approximation stitutions are made for the scattering operator and the leading behavior for their composition, the normal operator Ψ, c leading behavior their composition, the normal operator Ψ, correspond e normal operator Ψ, corresponds to that for of an 1/2 α f )∧ (ξ) = |ξ|2α · fˆ(ξ). Alter m → (−∆) m with ((−∆) order-one invertible elliptic PsDO . of order 0, with homoSo let Ψ = Ψ(x, D) be a pseudodifferential operator order-one invertible elliptic PsDO . geneous principal symbol a(x, ξ). made zero-order byamenable composing thetodata sidebywith aby 1/2-o To this make this PsDO amenable an approximation curv To make PsDO to an approximation curvelets, the approximation by curvelets, the following sub- Substitutions: −1/2 operator stitutions are made for the scattering and theKmodel: −1/2 stitutions are made for the scattering operator thee.g. model: → K erator and the model: K K (−∆) and the→ time coordinate, i.e., K → ∂t Kand (see 3). After in Rd . 1/2 ∧ (ξ) 2α · fˆ(ξ). Alternativel 1/2 αf ∧ (ξ) α 2α · = ˆ(ξ). m → (−∆) m with ((−∆) f ) |ξ| 2α · fˆ(ξ). Alternatively, these m → operators (−∆) mcan with ((−∆) ) = |ξ| f Alternatively, these op be or with operator Ψ becomes zero-order. Remark that these subsit 1. With C some constant, theby following made zero-order by holds composing with a 1/2-order made zero-order composing the datathe sidedata withside a 1/2-order fractional fra in deLemma with a 1/2-order fractional integration along tution made in the WVD−1/2 methods, −1/2 where vaguelettes are the time i.e., K → ∂t K → K∂ (see e.g. 3). After these substitutio thecoordinate, time coordinate, i.e., K (see e.g. 3). After these t e e.g. 3). After these substitutions, the normal −|ν|/2 (Ψ(x, D) − a(xν , ξν ))ϕν L2 (Rn ) ≤ C 2 . (14) operator Ψ the becomes zero-order. Remark that these subsitutions are simila Before detailing theRemark approximate operator Ψ becomes zero-order. that thesediagonalizat subsitutions hat these subsitutions aremappings. similar to substi- made the WVD where vaguelettes areDintroduced acco To approximate Ψ, tution weaccording define thein sequence u methods, := (uµ )µ∈M = a(x Let the introd tution made in the WVD methods, where µ , ξµ ). vaguelettes Ψ be are vaguelettes are introduced the same first discuss the properties of continuous curvelets under t mappings. Before detailing the approximate of the matrix with entries given by u. Next state our the result ondiagonalization the approximation of norm mappings. Before approximate diagonalization of atediagonal diagonalization of the normal operator, we wedetailing first discuss the properties of continuous curvelets under this operator. T D C. Ψ by C urvelets underΨthis operator. first discuss the properties of continuous curvelets under this ope APPROXIMATION OF THE NORM Tiling the ξ space ~2 j/2 !µ ~2 j In red, the essential frequency support of a curvelet φµ . 10 Scaling Ψ by C T DΨ C. Theorem 1. The following estimate for the error holds (Ψ(x, D) − C T DΨ C)ϕµ L2 (R ) n ≤ C 2−|µ|/2 , where C is a constant depending on Ψ. Allows for decomposition of the normal operator This main result proved in Appendix A shows that the approxim Ψϕµ (x) C DΨ Cϕµ (x) T diagonal approximation goes to zero for increasingly finer scales. The ap = AA ϕµ (x) T from the property that the symbol is slowly varying over the suppo with A := √ DΨ C and AT := C √ T DΨ . approximation that becomes more accurate as the scale increases. Matching procedure Compute reference vector <=> defines g migrate data apply spherical-divergence correction Create “data” <=> defines f demigrate migrate Estimate scaling by inversion procedure Define scaled curvelet transform Recover migration amplitudes by sparsity promotion. Key idea Estimation curvelet-domain scaling inversion of an underdetermined system over fitting positivity and reasonable scaling Solution: with use smoothness of the symbol formulate nonlinear estimation problem that minimizes 1 z 2 Jγ (z) = d − Fγ e 2 , 2 T z z gradJ(z) = diag{e } F Fe − d solve with l-BFGS [Noccedal, Symes ‘07] Key idea D1 x1 North quadrants Fine scales D2 x2 coarser scales 16 angles/ quad 8 angles/ quad West quadrants East quadrants Dθ θ South quadrants Key idea Impose smoothness via following system of equations f = C diag{Cg}w 0 = γLw T with L= T D1 T D2 T Dθ T first-order differences in space and angle directions for each scale. Equivalent to with 1 ˜ = arg min b − P[w] w w 2 2 2 + γ 2 Lw P = C diag{Cg} T 2 2 Smoothness penalty increasing smoothness reduces overfitting scaling is positive and reasonable Smoothness penalty 1 0.9 20 0.8 40 0.7 60 0.6 80 0.5 100 0.4 120 0.3 140 0.2 160 0.1 50 100 150 γ=0 200 0 Smoothness penalty 1 0.9 20 0.8 40 0.7 60 0.6 80 0.5 100 0.4 120 0.3 140 0.2 160 0.1 50 100 150 γ = 1/2 200 0 Our approach “Forward” model: y ≈ Ax0 + ε with y = A := C Γ T AA r K = K Km + ε T migrated data T T ≈ K Kr = the demigration operator = migrated noise. diagonal approximation of the demigration-migration operator costs one demigration-migration to estimate the diagonal weighting Solution Solve P: with minx J(x) subject to y − Ax 2 ˜ = (AH )† x ˜ m sparsity J(x) = α x 1 +β Λ 1/2 A H † continuity x p . ≤ Imaging example Migrated data Amplitude-corrected & denoised migrated data two-way reverse time wave-equation migration with checkpointing [Symes ‘07] adjoint state method with 8000 time steps T evaluation K takes 6 h on 60 CPU’s Observations Curvelet-domain scaling handles conflicting dips (conormality assumption) exploits invariance under the PsDO Diagonal approximation exploits smoothness of the symbol uses “neighbor” structure of curvelets Results on the SEG AA’ show recovery of amplitudes beneath the Salt successful recovery from clutter improvement of the continuity robust w.r.t. noise Curvelet-domain matched filter ... A primer on compressive sampling Compressive sensing [Candes, Romberg & Tao, Donoho, many others] Three key ingredients existence of a sparsifying transform existence of a sub-Nyquist sampling strategy that reduces coherent aliases handle wavefronts & reflectors with conflicting dips incoherence random sampling scheme existence of a large-scale (norm-one) solver sparsity promotion by iterative thresholding and cooling Simple example Fourier few significant coefficients transform 3-fold under-sampling Fourier transform Fourier transform Seismic Laboratory for Imaging and Modeling ✓ significant coefficients detected ✗ ambiguity Forward problem restriction operator with = y A := RFH A Fourier transform signal x0 Fourier coefficients (sparse) Seismic Laboratory for Imaging and Modeling Naive sparsity-promoting recovery detection + Fourier data-consistent transform amplitude recovery Ar data-consistent amplitude recovery = inverse Fourier transform detection y = A†r y = A AH Seismic Laboratory for Imaging and Modeling y x0 Undersampling “noise” “noise” – due to AHA ≠ I – defined by AHAx0-αx0 = AHy-αx0 1 out of 2 1 out of 4 1 out of 6 1 out of 8 less acquired data 3 detectable Fourier modes Seismic Laboratory for Imaging and Modeling 2 detectable Fourier modes Sparsity-promoting wavefield reconstruction restriction operator with = y A := RSH A acquired data sparsifying transform for seismic data x0 complete wavefield (transform domain) H ˜ ˜ with f = S x Interpolated data given by ˜ = arg min ||x||1 x x Seismic Laboratory for Imaging and Modeling s.t. y = Ax [Sacchi et al ‘98] [Xu et al ‘05] [Zwartjes and Sacchi ‘07] [Herrmann and Hennenfent ‘07] Observations bla bla generalized to A=RMS^H depends on solver, sampling strategy and sparsity transform Seismic Laboratory for Imaging and Modeling Compressive sampling of wavefields joint work with Deli Wang (visitor from Jilin university) and Gilles Hennenfent “Curvelet-based seismic data processing: a multiscale and nonlinear approach” & to appear in Geophysics, “Non-parametric seismic data recovery with curvelet frames” and “Simply denoise: wavefield reconstruction via jittered undersampling” General form compressive sampling Solution of P : ˜ = arg minx x x ˜f = ST x ˜ 1 s.t. with A = RMS T R = restriction matrix M = measurement matrix T = sparsity synthesis matrix y = RMf S recovers the function f. Ax − y 2 ≤ The problem Total data 85 % traces missing Requirements Sparsifying transform (S) curvelet focussed curvelets Sampling scheme (RM) random sampling random jittered sampling => control largest gaps Sparsity promoting solver (P) Iterative thresholding (Landweber + soft threshold) Discrete random jittered undersampling random optimally jittered poorly jittered regular Type Sampling scheme Typical spatial convolution kernel (amplitudes) Averaged spatial convolution kernel (amplitudes) receiver positions PDF receiver positions PDF receiver positions PDF receiver positions Seismic Laboratory for Imaging and Modeling [Hennenfent and Herrmann ‘07] Curvelet-based recovery Solution of P : ˜ = arg minx x x ˜f = ST x ˜ 1 with A = RIC T R = jitter sampling I = Dirac basis T = curvelet synthesis y = Rf C recovers the wavefield f. s.t. Ax − y 2 ≤ Model Seismic Laboratory for Imaging and Modeling Regular 3-fold undersampling Seismic Laboratory for Imaging and Modeling CRSI from regular 3-fold undersampling SNR = 6.92 dB Seismic Laboratory for Imaging and Modeling SNR = 20 × log10 model 2 reconstruction error 2 Random 3-fold undersampling Seismic Laboratory for Imaging and Modeling CRSI from random 3-fold undersampling SNR = 9.72 dB Seismic Laboratory for Imaging and Modeling SNR = 20 × log10 model 2 reconstruction error 2 Optimally-jittered 3-fold undersampling Seismic Laboratory for Imaging and Modeling CRSI from opt.-jittered 3-fold undersampling SNR = 10.42 dB Seismic Laboratory for Imaging and Modeling Model Seismic Laboratory for Imaging and Modeling Regular 3-fold undersampling SNR = 12.98 dB Seismic Laboratory for Imaging and Modeling Regular 3-fold undersampling SNR = 12.98 dB Seismic Laboratory for Imaging and Modeling Optimally-jittered 3-fold undersampling SNR = 15.22 dB Seismic Laboratory for Imaging and Modeling Optimally-jittered 3-fold undersampling SNR = 15.22 dB Seismic Laboratory for Imaging and Modeling Focussed recovery Solution of P : ˜ = arg minx x x ˜f = ∆PCT x ˜ with A ∆P y = R∆PC T = main primaries = Rf recovers the wavefield f. 1 s.t. Ax − y 2 ≤ 80 % missing data off one interaction with the surface, focusing primaries to (directional) sources, which leads to a sparser Focused curvelet urvelet representation. recovery By compounding the non-adaptive curvelet transform with the data-adaptive focal transform, i.e., A := R∆PCT , he recovery can be improved by solving P . The solution of P now entails the inversion of ∆P, yielding the sparsst set of curvelet coefficients that matches the incomplete data when ’convolved’ with the primaries. Applying the nverse curvelet transform, followed by ’convolution’ with ∆P yields the interpolation, i.e. ST := ∆PCT . Comparng the curvelet recovery with the focused curvelet recovry (Fig ?? and ??) shows an overall improvement in the ecovered details. SEISMIC SIGNAL SEPARATION Predictive multiple suppression involves two steps, namely multiple prediction and the primary-multiple separation. n practice, the second step appears difficult and adap- operation by filling in the zero traces. Since seismic the missing data can be Curvelet recovered by compounding recovery modeling operator, i.e., A := RCT . With this definit P corresponds to seeking the sparsest curvelet vec followed by the picking, matches the data at the n transform (with S := C in P ) gives the interpolated An example of curvelet based recovery is presente data volume is recovered from data with 80 % traces traces are selected at random according to a discrete d Original data Observations Regular subsampling is unfavorable random sampling favorable but suffers from gaps jitter sampling favorable and controls gaps Focal transform is reminiscent of an imaging operator improves recovery <=> additional compression Solver solves norm one problem for 200-300 matrix-vector multiplications for 230 unknowns ... Outlook Migration-based wavefield reconstruction sparsity on the image focussing of the image (extra constraint) or a more “blue sky” approach of compressive oneway wavefield extrapolation Compressed wavefield extrapolation joint work with Tim Lin “Compressed wavefield extrapolation” in Geophysics Motivation Synthesis of the discretized operators form bottle neck of imaging Operators have to be applied to multiple right-hand sides Explicit operators are feasible in 2-D and lead to an order-of-magnitude performance increase Extension towards 3-D problematic storage of the explicit operators convergence of implicit time-harmonic approaches First go at the problem using CS techniques to compress the operator ... Related work Curvelet-domain diagonalization of FIO’s The Curvelet Representation of Wave Propagators is Optimally Sparse (Candes & Demanet ‘05) Seismic imaging in the curvelet domain and its implications for the curvelet design (Chauris ‘06) Leading-order seismic imaging using curvelets (Douma & de Hoop ‘06) Explicit time harmonic methods Modal expansion of one-way operators in laterally varying media (Grimbergen et. al. ‘98) A new iterative solver for the time-harmonic wave equation (Riyanti ‘06) Fourier restriction How to choose a subset of frequencies in frequency-domain finitedifference migration (Mulder & Plessix ‘04) 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 Inspiration Suppose we want to shift a sparse spike train, i.e., u = Tτ v −τ D = e v −jτ Ω = Le H L v where D = LΩL H L = The Fourier Transform Eigen modes <=> Fourier transform. Can this operation be compressed by compressive sampling? Operators on spikes [Candes et. al, Donoho] Calculate instead y A ˜ u j Ωτ = Re Fv = RF = arg minu u s.t. Au = y Take compressed measurements in Fourier space. Recover with sparsity promotion Shift operator is compressed by the restriction R∈R 1 m×N with m N yielding compressed rectangular operators. Extend this idea to wavefield extrapolation? Representation for seismic data [Berkhout] p- s+ WW+ R+ Different representations diagonalization operator parsimony wavefield SVD/Lanczos/ modal ✓ ✕ curvelets ✕ ✓ Different representations diagonalization operator parsimony wavefield SVD/Lanczos/ modal ✓ ✕ curvelets ✕ ✓ If incoherent this may actually work .... Sparsity promoting formulation Buys us stability w.r.t. missing data provided measurement and sparsity representations are mutually incoherent sufficient mixing <=> random restriction Different strategy: Let the physics define the measurement basis Use the modal domain (domain of eigenfunctions) to define the measurement basis See what you can recover Study eigenfunctions: mutual coherence with sparsity representation modal spectrum on the to-be-extrapolated wavefield One-Way Wave Operator Structure of A confounds the meaning of its exponentiation, due to it being an operator (Simon & Reed; Dessing ‘97; Grimbergen ‘98) A = Two-way Wave Operator H2 contains 0 ωρ 1 −1/2 ρ ) 0 (H 2 1/2 ωρ H2 = k (x, ω) + ∂µ ∂µ 2 information about medium velocity One-Way Wave Operator Solution of the one-way wave equation W(x3 ; x3 ) = exp(−j(x3 − x3 )H1 ) After discretization solve eigenproblem on H2 H2 = ω c¯1 2 0 0 .. . ω c¯2 0 0 .. . 2 ··· 0 ··· .. . 0 .. . ··· ω c¯n1 + D2 2 Helmholtz operator is Hermitian (Claerbout, 1971; Wapenaar and Berkhout, 1989) monochromatic ¯ varies laterally velocity c Modal transform Solve eigenproblem & take square root H1 = LΛ 1/2 H L L is orthonormal & defines the modal transform that diagonalizes one-way wavefield extrapolation Eigenvalues play role of vertical wavenumbers Extrapolation operator is diagonalized −j Λ W = F Le H 1/2 (x3 −x3 ) H L F Compressed wavefield extrapolation Forward model −j Λ u = Le Original events 1/2 ∆x3 H L v Recorded Data Reconstruct point scatterers from recorded data .... Compressed wavefield extrapolation y A ˜ x v ˜ = RL u 1/2 jΛ ∆x3 H = Re L H = arg minx x ˜ =x 1 s.t. Ax = y Randomly subsample & phase rotate in Modal domain Recover by norm-one minimization Capitalize on the incoherence modal functions and point scatterers reduced explicit matrix size constant velocity <=> Fourier recovery Compressed wavefield extrapolation Reconstruction Recorded Data Reconstructed events Only 1 % of original modes were used ... Observations Despite the existence of evanescent (exponentially decaying) waves modes recovery is successful If you are looking for pointscatterers, we have a proof of concept that is fast (a) (b) Earth is more complex ... (c) Compressed wavefield extrapolation Extend to general wavefields Use curvelets as the sparsity representation Use the full & compressed forward operator operator Compressively extrapolate back 600m to the source (a) Restriction & sparsity strategies Forward extrapolation: W1 : 1/2 jΛ ∆x3 H y = Re L A := RLH F CT ˜ = arg minx x 1 s.t. Ax = y x u ˜ = CT x ˜, Inverse extrapolation: F1 : H y = RL F u 1/2 jΛ ∆x3 H H A = Re L C ˜ x = arg min x s.t. A x = y 1 x T v ˜=C x ˜. Forward Extrapolation (a) (b) (c) (d) (a) is Full extrapolation (b)-(d) is compressed extrapolation, (b) p = 0.04, (c) Figurep11:=Compressed forward 0.16, (d) p extrapolation = 0.24 according to W1 (cf. Eq. (42)) for different Inverse Extrapolation (a) (c) (b) (a) p = 0.04 (d) (b) p = 0.16, (c) pf=0.4, px=0.4 Evanescent Recovery (a) (b) (c) (a) is downward extrapolated wavefield (b) is matched filter Figure 14: Inversion of the evanescent wavemodes according vm = WT u or v (c) is “compressed” inverse extrapolation −1 = F1 [u] Velocity model Compressed inverse extrapolation Overthrust exploding reflector Matched filter Full forward extrapolation Recovered from p=0.25 Multiscale and angular compressed wavefield extrapolation Propose a scheme motivated by extensions of CS j F1 : yj = Rj Mj u A := Rj M CT j j j ˜ j = arg minx xj x j T v ˜j, ˜ = j Cj x with j = {j, l} the scale and angle. (Tsaig and Donoho ‘06) 1 s.t. Aj xj = yj adapt discretization & restriction parallel implementation Conclusions Curvelets sparsity on the model and near diagonalization yields stable inversion Gramm matrix Jittered sampling and focussing in combination with curvelets leads to wavefield recovery Compressed wavefield extrapolation reduction in synthesis cost inverse extrapolation works well when focussed mutual coherence curvelets and modes performance of norm-one solver keep the constants under control ... Double-role CS matrix is cool ... upscaling to “reallife” is a challenge .... Open problems What deeper insights can CS give? inversion near unitary operators coherency generalized to frames to study cols modeling operator <=> curvelets radiation vs guided modes <=> curvelets Norm-one solver for reduced system as fast a LSQR on the full system Fast random eigenvalue solver does not exist yet ... Extension of CS to waveform inversion & to compressed computations ... Many more ... Acknowledgments The audience for listening and the organizers for putting this great workshop together .... The authors of CurveLab (Demanet,Ying, Candes, Donoho) Dr. W. W. Symes for his reverse-time migration code This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant (22R81254) and the Collaborative Research and Development Grant DNOISE (334810-05) of F.J.H. This research was carried out as part of the SINBAD project with support, secured through ITF (the Industry Technology Facilitator), from the following organizations: BG Group, BP, Chevron,ExxonMobil and Shell.
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Compressive sampling meets seismic imaging Herrmann, Felix J. 2007
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Title | Compressive sampling meets seismic imaging |
Creator |
Herrmann, Felix J. |
Contributor | University of British Columbia. Seismic Laboratory for Imaging and Modeling |
Date Issued | 2007 |
Extent | 10592568 bytes |
Subject |
q Newton wavefield extrapolation compression wavefield reconstruction Gramm matrix Compressed wavefield extrapolation |
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Conference Paper |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-03-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | All rights reserved |
IsShownAt | 10.14288/1.0102592 |
URI | http://hdl.handle.net/2429/602 |
Affiliation |
Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Herrman, Felix J. 2007. Compressive sampling meets seismic imaging. PIMS-CINVESTAV Mathematics of Oil Exploration Workshop, Annual meeting of the Mexican Mathematical Society. |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty |
Copyright Holder | Herrmann, Felix J. |
AggregatedSourceRepository | DSpace |
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