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Multiple prediction from incomplete data with the focused curvelet transform Herrmann, Felix J. 2007-03-20

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Multiple prediction from incomplete data with the focused curvelet transformFelix J. Herrmannjoint work with Deli Wang and Gilles Hennenfent.The problemTotal data 85 % traces missingThe problem cont?dSRME from complete data SRME from missing dataOur solutionSRME from recovered data SRME from original dataMotivationData-driven (SRME) multiple prediction requires fully sampled data.The Focal transform (Berkhout & Verschuur ?06) allows for? mapping of multiples => primaries? incorporation of prior information in the recoveryPresent a curvelet-based scheme for sparsity-promoting? recovery of missing data? prediction of primaries from multiples? data inverse ...The curvelet transformProperties curvelet transform:? multiscale: tiling of the FK domain into dyadic coronae? multi-directional: coronae sub-partitioned into angular wedges, # of angle doubles every other scale? anisotropic: parabolic scaling principle? Rapid decay space? Strictly localized in Fourier? Frame with moderate redundancy (8 X in 2-D and 24 X in 3-D)Transform Underlying assumptionFK plane waveslinear/parabolic Radon transform linear/parabolic eventswavelet transform point-like events (1D singularities)curvelet transform curve-like events (2D singularities)Representations for seismic datafine scale data coarse scale data 2-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 relationCurvelets are oscillatory in one direction and smooth in the others.3-D curveletsCurvelet sparsity promotionSparsity-promoting programSolve forwith ? exploit sparsity in the curvelet domain as a prior.? find the sparsest set of curvelet coefficients that match the data.? invert an underdetermined system.signal = + noiserestricted compounded curvelet representation of ideal dataFocused wavefield reconstruction with curveletsFocused recoveryNon-data-adaptive Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI) derives from curvelet-sparsity of seismic data.Berkhout and Verschuur?s data-adaptive Focal transform derives from focusing of seismic data by the major primaries.Both approaches entail the inversion of a linear operator.Combination of the two yields? improved focusing => more sparsity? curvelet sparsity   => better focusingPrimary operatorReceiversShotsShotsReceiversFrequency Frequency dominantprimaries.Primary operatorPrimary operatorPrimaries to first-order multiples: First-order multiples into primaries:with the acquisition matrix?inverting? for source and receiver wavelet wavelets geometry and surface reflectivity.Curvelet-based Focal transformSolve with 3-D curvelet transformTotal dataSRME estimate for the primariesFocused with the primariesDifferenceSolveRecovery with focussingCRSIfCRSIMultiple prediction with fCRSIincomplete dataCRSISRMEfCRSIrecovered dataSRME primary operatorWavefield reconstruction with fCRSIOriginal data80 % missingCurvelet recoveryOriginal dataFocused curvelet recoveryOriginal dataMultiple predictionReference predicted multiplesOriginal predicted multiplesPredicted multiples from missing dataPredicted multiples from CRSI recoveryPredicted multiples from focussed CRSI recoveryReference predicted multiplesOriginal predicted multiplesPrimary prediction with fCRSIincomplete dataCRSISRMEfCRSIrecovered dataCurvelet-based Focal transformSolveFocal transform from complete data80 % missingFocal transform from missing dataAn encore ...preliminary results for the data inverseCurvelet-based seismic data inversewithis the data to be invertedCurvelet-sparsity regularized data inverse computed for the whole data volume ......Curvelet-based seismic data inverseCurvelet-based seismic data inverseCurvelet-based seismic data inverseConclusionsCRSI? recovers data by curvelet sparsity promotion? uses sparsity as a priorFocused CRSI? incorporates additional prior information? strips interaction with the surface <=> more sparsity? improves the recovery and hence predicted multiples? precursor of migration-based CRSIResults of curvelet-based computation of the data inverse are encouraging.AcknowledgmentsThe authors of CurveLab (Demanet,Ying,  Donoho)Dr. Verschuur for his synthetic data and the estimates for the primaries.The SLIM team Sean Ross Ross, own and Henryk Modzeleweski for developing SLIMPy: verloading in pythonThese results were created with Madagascar developed by Dr.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, ed through ITF (the Industry Technology Facilitator), om the following organizations: oup, , vron,  

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