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Seismic data processing with curvelets: a multiscale and nonlinear approach Herrmann, Felix J. 2007

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Seismic data processing with curvelets: a multiscale and nonlinear approachFelix J. Herrmannjoint work with Deli Wang, Gilles Hennenfent and Peyman MoghaddamMotivationExploit two aspects of curvelets, namely their? parsimoniousness? invariance under certain operatorsFormulate? non-adaptive wavefield reconstruction algorithms? data-adaptive matching algorithms Applications? nonlinear sampling theory for wavefields? nonlinear migration-amplitude recovery? nonlinear primary-multiple separationApproachEmploy parsimoniousness by sparsity promotion.Exploit behavior of certain operators in phase space? diagonalization <=> curvelet domain scaling? smoothness <=> structure of phase spaceCombine parsimoniousness with structure in phase space? diagonal approximation operators? stable amplitude recovery? improved adaptive separationMigration-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 promotionThe curvelet transform2-D curveletscurvelets are of rapid decay in spacecurvelets are strictly localized in frequencyx-t f-kOscillatory in one direction and smooth in the others!Obey parabolic scaling relationCoefficients Amplitude Decay In Transform DomainsFourierWaveletsCurveletsPartial ReconstructionFourier (1% largest coefficients)SNR = 2.1 dBPartial ReconstructionCurvelets (1% largest coefficients)SNR = 6.0 dBNon-adaptive curvelet-domain sparsity promotionLinear quadratic (lsqr):? model GaussianNon-linear:? model Cauchy (sparse)Problem:? data does not contain point scatterers? not sparseOur contributionModel as superposition of little plane waves.Compound modeling operator with curvelet synthesis:Exploit parsimoniousness of curvelets on seismic data & images ...Sparsity-promoting programProblems boil down to solving forwith ? exploit sparsity in the curvelet domain as a prior? find the sparsest set of curvelet coefficients that match the data, i.e.,? invert an underdetermined systemsignal = + noisecurvelet representation of ideal dataSeismic wavefield reconstruction with CRSISparsity-promoting inversion*Reformulation of the problemCurvelet Reconstruction with Sparsity-promoting Inversion (CRSI)? look for the sparsest/most compressible,physical solution KEY POINT OF THE RECOVERY* inspired by Stable Signal Recovery (SSR) theory by E. Cand?s, J. Romberg, T. Tao, Compressed sensing by D. Donoho & Fourier Reconstruction with Sparse Inversion (FRSI) by P. Zwartjessignal = + noisecurvelet representation of ideal dataOriginal data80 % missingCRSI recovery with 3-D curveletsAdaptive curvelet-domain matched filteringForward modelLinear model for amplitude mismatch:? spatially-varying dip filter? zero-order PseudoAfter discretization? linear operator? f and g known? matrix B is full and not known ....Forward modelDiagonal approximation in the curvelet domain:? curvelet domain scaling? opens the way to an estimation of wExamples:B f gmigration migrated ?image? ?reflectivity?multiple removal obliquity factor total data predicted multiplesKey ideaProblems with estimating w? inversion of an underdetermined system? over fitting? positivity and reasonable scaling by wSolution:? use smoothness of the symbol ? formulate nonlinear estimation problem that minimizeswith? solve with l-BFGSKey ideaEast quadrantsWest quadrantsNorth quadrantsSouth quadrants16 angles/quad8 angles/quadFine scalescoarserscalesKey ideaImpose smoothness via following system of equationswithfirst-order differences in space and angle directions for each scale. Equivalent towithSmoothness penaltyincreasing smoothness? reduces overfitting? scaling is positive and reasonableSmoothness penaltySmoothness penaltySmoothness penaltySeismic amplitude recoveryMatching procedureCompute reference vector <=> defines g? migrate data ? apply spherical-divergence correctionCreate ?data? <=> defines f? demigrate? migrateEstimate scaling by inversion procedureDefine scaled curvelet transformRecover migration amplitudes by sparsity promotion.Primary-multiple separationMatching procedurePredict 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 mismatchEstimate scaling by an inversion procedure.Define scaled curvelet-domain threshold.Separate primaries & multiples by sparsity promotion.Problem formulationSignal model for total dataMultiple prediction by e.g. SRME may contain amplitude errors, i.e.,Solvewith s the total data. Use z to correct the predicted multiples, i.e.,or correct the thresholdingSynthetic exampleTotal data SRME predicted multiplesSynthetic exampleCurvelet estimated primariesSynthetic exampleCorrected multiplesSynthetic exampleScaled thresholded primariesSynthetic exampleCurvelet estimated primariesScaled thresholded primariesReal exampleSRME predicted primariesReal exampleThresholded primaries Scaled thresholded primariesConclusionsCombining the parsimonious curvelet transform with phase-space structure allows us to  control diagonal estimation <=> over fitting  handle data with conflicting dips  stably recover & separateApplication  improved migration-amplitude recovery  improved primary-multiple separationsFuture  3-D  non-smooth symbolsAcknowledgmentsThe authors of CurveLab (Demanet,Ying,  Donoho)Christiaan C. or his contribution to phase-space smoothness.The SLIM team Sean Ross Ross, own and Henryk Modzeleweski for developing SLIMPy: verloading in pythonThese 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), om the following organizations: oup, , vron,ExxonMobil and 

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