UBC Faculty Research and Publications

Just diagonalize: a curvelet-based approach to seismic amplitude recovery Herrmann, Felix J.; Moghaddam, Peyman P.; Stolk, Christiaan C. 2007-12-31

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Just diagonalize: a  curvelet-based approach to seismic amplitude recoveryFelix J. Herrmann*,Peyman Moghaddam* & Chris Stolk (Universiteit Twente)*Seismic Laboratory for Imaging and Modeling slim.eos.ubc.caEAGE, London, June 11MotivationMigration generally does not correctly recover the amplitudes.Least-squares migration is computationally unfeasible.Amplitude recovery (e.g. AGC) lacks robustness w.r.t. noise.Existing diagonal amplitude-recovery methods? do not always correct for the order (1 - 2D) of the Hessian [see Symes ?07]? do not invert the scaling robustlyMoreover, these (scaling) methods assume that there? are no conflicting dips (conormal) in the model? is infinite aperture ? are infinitely-high frequencies? etc.Curvelets & seismologyWish listA transform that? detects the reflectors without prior information on the geologic dips? is sparse, i.e. the magnitude-sorted coefficients decay fast? is relative invariance under the demigration-migration, i.e. sparse on migrated imagesCurvelets ? were ?born? from studying high-frequency solution operators for wave propagation*? diagonalization of migration operators***See work by Stein, Smit, Donoho, Candes & Demanet** Main motivation for Douma & de Hoop and ChaurisNonlinear approximationMigrated mobil data setNonlinear approximationRecovery from largest 3 %Nonlinear approximationDifferenceNonlinear approximation ratesImaged Mobil data Reflectivity SEG AA? Curvelets & wave propagationTheoretical results that claim that curvelets near diagoanalize migration operators [Demanet et. al, de Hoop]Encouraging results for constant velocity media [Douma & de Hoop; Chauris]Challenge: discrete curvelets move off the grid? interpolation? definition of curvelet molecules [Demanet et. al, de Hoop]In not so smooth media curvelets spread significantly ....Curvelet propagationCurvelet propagationMajor challenge. Limit ourselves to migration amplitude recovery!?Imaged? curveletHessian/Normal operator[Stolk 2002, ten Kroode 1997, de Hoop 2000, 2003]Alternative to expensive least-squares migration.In high-frequency limit     is a PsDO? pseudolocal? singularities are preservedCorresponds to a spatially-varying dip filter after appropriate preconditioning (=> zero order).? curvelets remain invariant? approximation improves for higher frequenciesInvariance under Hessian matrixDiagonal approximation of the HessianExisting scaling methodsMethods are based on a  diagonal approximation of   .? Illumination-based normalization (Rickett ?02)? Amplitude preserved migration (Plessix & Mulder ?04)? Amplitude corrections (Guitton ?04)? Amplitude scaling (Symes ?07)We are interested in an ?Operator and image adaptive? scaling method which? estimates the action of    from a reference vector close to the actual image? assumes a smooth symbol of     in space and angle? does not require the reflectors to be conormal <=> allows for conflicting dips? stably inverts the diagonal ? Allows for the decompositionApproximation? Wavelet-vagulette like [Donoho, Candes]? Amenable to nonlinear recoveryApproximationEstimation of the diagonal scaling Diagonal estimationSeismic amplitude recovery? Final form? SolveRecoveryApplication to the SEG AA? modelExampleSEGAA? data:? ?broad-band? half-integrated wavelet [5-60 Hz]? 324 shots, 176 receivers, shot at 48 m? 5 s of dataModeling operator? Reverse-time migration with optimal check pointing (Symes ?07)? 8000 time steps? modeling 64, and migration 294 minutes on 68 CPU?sScaling requires 1 extra migration-demigrationSeismic Laboratory for Imaging and ModelingSeismic Laboratory for Imaging and ModelingSeismic Laboratory for Imaging and ModelingSeismic Laboratory for Imaging and ModelingMigrated data Amplitude-corrected & denoised migrated dataSeismic Laboratory for Imaging and ModelingNoise-free data Noisy data(3 dB)Data from migrated imageData from amplitude-corrected & denoised migrated imageNonlinear dataConclusionsCurvelet-domain scaling? handles conflicting dips (conormality assumption)? exploits invariance under the PsDO? robust w.r.t. noiseDiagonal approximation? exploits smoothness of the symbol? uses ?neighbor? structure of the curvelet transformResults on the SEG AA? show? recovery of amplitudes beneath the Salt? successful recovery of clutter? improvement of the continuity AcknowledgmentsThe authors of CurveLab (Demanet,Ying,  Donoho)Dr. or the 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), om the following organizations: oup, , vron,ExxonMobil and Shell.


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