Seismic data processing with curvelets: a multiscale and nonlinear approach Felix J. Herrmann joint work with Deli Wang, Gilles Hennenfent and Peyman Moghaddam Motivation Exploit two aspects of curvelets, namely their parsimoniousness invariance under certain operators Formulate non-adaptive wavefield reconstruction algorithms data-adaptive matching algorithms Applications nonlinear sampling theory for wavefields nonlinear migration-amplitude recovery nonlinear primary-multiple separation Approach Employ parsimoniousness by sparsity promotion. Exploit behavior of certain operators in phase space diagonalization <=> curvelet domain scaling smoothness <=> structure of phase space Combine parsimoniousness with structure in phase space diagonal approximation operators stable amplitude recovery improved adaptive separation Migration-amplitude recovery methods are based on diagonal approximation of Pseudo’s estimate scaling from a reference vector and demigrated-migrated reference vector Illumination-based normalization (Rickett ‘02) Amplitude corrections (Guitton ‘04) Amplitude scaling (Symes ‘07) Primary-multiple separation methods are based on diagonal approximation in the Fourier domain estimate scaling from mismatch pred. multiples & data adaptive subtraction (Verschuur and Berkhout ‘97) We are interested in a formulation that estimates the scaling with smoothness control prevents overfitting allows for conflicting dips incorporates curvelet-domain sparsity promotion The curvelet transform 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! 2 Obey parabolic scaling relation length ≈ width 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 Non-adaptive curveletdomain sparsity promotion Inline 0 Linear quadratic (lsqr): • 2 s.t. Ax − y 2 ≤ model Gaussian 500 300 400 500 Inline 1 s.t. Ax − y 2 ≤ 0.5 Time (s) model Cauchy (sparse) Problem: • 200 400 1.5 2.0 0 • 100 300 1.0 Non-linear: ˜ = arg min x x x 200 0.5 Time (s) ˜ = arg min x x x 100 1.0 data does not contain point scatterers 1.5 not sparse 2.0 Our contribution Inline 0 Model as superposition of little plane waves. Time (s) Compound modeling operator with curvelet synthesis: 0.5 1.0 1.5 K m0 ˜ m T → KC = ˜ C x → x0 T Exploit parsimoniousness of curvelets on seismic data & images ... 2.0 100 200 300 400 500 Sparsity-promoting program Problems boil down to solving for x0 signal y = A + n noise x0 curvelet representation of ideal data with P : ˜ = arg minx x x ˜ = CT x ˜ m 1 s.t. Ax − y 2 ≤ exploit sparsity in the curvelet domain as a prior find the sparsest set of curvelet coefficients that T ˜ match the data, i.e., y ≈ KC x invert an underdetermined system Seismic wavefield reconstruction with CRSI Sparsity-promoting inversion* Reformulation of the problem signal y = H RC + n noise x0 curvelet representation of ideal data Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI) P : look for the sparsest/most compressible, physical solution KEY POINT OF THE RECOVERY data misfit sparsity constraint sparsity constraint = arg ˜ H H − PC minmin x 0x 0 s.t. s.t. y −yPC x 2x≤2!≤ ! x =x˜arg ˜0 )= arg minxx Wx x s.t. Ax − y 2 ≤ (P0 )(Px 1 T ˜ ˜H f = C x H ˜f = ˜fC= xC ˜ x˜ * inspired by Stable Signal Recovery (SSR) theory by E. Candès, J. Romberg, T. Tao, Compressed sensing by D. Donoho & Fourier Reconstruction with Sparse Inversion (FRSI) by P. Zwartjes Original data 80 % missing CRSI recovery with 3-D curvelets Adaptive curveletdomain matched filtering Forward model Linear model for amplitude mismatch: Bf (x) = B x∈R e d b(x, k)fˆ(k)dk = Pseudodifferential operator b(x, k) = jk·x the symbol spatially-varying dip filter zero-order Pseudo After discretization f = Bg linear operator f and g known matrix B is full and not known .... Forward model Diagonal approximation in the curvelet domain: f = Bg ≈ C diag{w}Cg T curvelet domain scaling opens the way to an estimation of w Examples: B migration multiple removal T K K obliquity factor f migrated “image” total data g “reflectivity” predicted multiples Key idea Problems with estimating w inversion of an underdetermined system over fitting positivity and reasonable scaling by w 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 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 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=5 200 0 Seismic amplitude recovery 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. Primary-multiple separation Matching procedure Predict multiples <=> defines g apply conventional Fourier matched filtering Consider total data as “true” multiples <=> defines f do not know true multiples use total data instead minimize energy mismatch Estimate scaling by an inversion procedure. Define scaled curvelet-domain threshold. Separate primaries & multiples by sparsity promotion. Problem formulation Signal model for total data s = s1 + s2 Multiple prediction by e.g. SRME may contain amplitude errors, i.e., Solve s2 = B˘s2 s2 ≈ C diag{w}C˘s2 T 1 s − Fγ ez 22 , Jγ (z) = 2 with s the total data. Use z to correct the predicted multiples, i.e., ˜ z ˜ ˜ =e ˘s2 → C diag{w}C˘ s2 with w T or correct the thresholding ˜ t = diag{w}|C˘ s2 | Synthetic example Total data s SRME predicted multiples ˘s2 Synthetic example SRME predicted primaries ˘s1 Curvelet estimated primaries ˜s1 = CT Tt Cp t = C˘s2 Synthetic example Corrected multiples T ˘scorr. diag{w}C˘s2 for γ = 0 = C 2 ˘scorr. 2 Corrected multiples = CT diag{w}C˘s2 for γ = 0.5 Synthetic example Scaled thresholded primaries T ˜s1 t = C Tt Cp = diag{w}|C˜s2 | Scaled thresholded primaries T ˜s1 t = C Tt Cp = diag{w}|C˜s2 | Synthetic example Scaled thresholded primaries ˜s1 t = CT Tt Cp = diag{w}|C˜s2 | Curvelet estimated primaries ˜s1 = CT Tt Cp t = C˘s2 Real example SRME predicted multiples ˘s2 SRME predicted primaries ˘s1 Real example Thresholded primaries ˜s1 = CT Tt Cp t = C˘s2 Scaled thresholded primaries ˜s1 t = CT Tt Cp = diag{w}|C˜s2 | Conclusions Combining the parsimonious curvelet transform with phase-space structure allows us to control diagonal estimation <=> over fitting handle data with conflicting dips stably recover & separate Application improved migration-amplitude recovery improved primary-multiple separations Future 3-D non-smooth symbols Acknowledgments The authors of CurveLab (Demanet,Ying, Candes, Donoho) Christiaan C. Stolk for his contribution to phase-space smoothness. The SLIM team Sean Ross Ross, Cody Brown and Henryk Modzeleweski for developing SLIMPy: operator overloading in python These results were created with Madagascar developed by Sergey Fomel. 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
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Seismic data processing with curvelets: a multiscale and nonlinear approach Herrmann, Felix J. 2007-12-31
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Title | Seismic data processing with curvelets: a multiscale and nonlinear approach |
Creator |
Herrmann, Felix J. |
Publisher | Society of Exploration Geophysicists |
Date Issued | 2007 |
Description | In this abstract, we present a nonlinear curvelet-based sparsity-promoting formulation of a seismic processing flow, consisting of the following steps: seismic data regularization and the restoration of migration amplitudes. We show that the curvelet's wavefront detection capability and invariance under the migration-demigration operator lead to a formulation that is stable under noise and missing data. |
Extent | 5716560 bytes |
Subject |
migration demigration curvelet seismic data regularization restoration migration amplitudes nonlinear curvelet domain sparsity smoothness overfitting 2D 2-D |
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File Format | application/pdf |
Language | eng |
Date Available | 2008-03-20 |
Provider | Vancouver : University of British Columbia Library |
DOI | 10.14288/1.0107416 |
URI | http://hdl.handle.net/2429/600 |
Affiliation |
Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Herrmann, Felix J. 2007. Seismic data processing with curvelets: a multiscale and nonlinear approach. SEG 77th Annual Meeting and Exposition. |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty |
Copyright Holder | Herrmann, Felix J. |
Aggregated Source Repository | DSpace |
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