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

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Just diagonalize: a curvelet-based approach to seismic amplitude recovery Felix J. Herrmann*, Peyman Moghaddam* & Chris Stolk (Universiteit Twente) *Seismic Laboratory for Imaging and Modeling slim.eos.ubc.ca EAGE, London, June 11  Motivation Migration 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 robustly  Moreover, these (scaling) methods assume that there  are no conflicting dips (conormal) in the model     is infinite aperture are infinitely-high frequencies etc.  Curvelets & seismology  Wish list A 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 demigrationmigration, i.e. sparse on migrated images  Curvelets    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 Chauris  Nonlinear approximation Migrated mobil data set  Nonlinear approximation Recovery from largest 3 %  Nonlinear approximation Difference  Nonlinear approximation rates Imaged Mobil data  Reflectivity SEG AA’  Curvelets & wave propagation Theoretical 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 propagation Sharp Model  5000  50 100  4500  150 4000 200 250  3500  300 3000 350 2500  400 450  2000  500 50  100  150  200  250  300  350  400  450  500  Curvelet propagation Sharp Model  5000  50 100  4500  150 4000 200 250  3500  300 3000 350 2500  400 450  2000  500 50  100  150  200  250  300  350  400  450  500  Major challenge. Limit ourselves to migration amplitude recovery!  “Imaged” curvelet Sharp Model  5000  50 100  4500  150 4000 200 250  3500  300 3000 350 2500  400 450  2000  500 50  100  150  200  250  300  350  400  450  500  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 PsDO  Ψf (x) := K Kf (x) = T     R  −ix·ξ  e d  a(x, ξ)fˆ(ξ)dξ  pseudolocal singularities are preserved  Corresponds to a spatially-varying dip filter after appropriate preconditioning (=> zero order).  Invariance under Hessian matrix  (a)  (b)  (c)  (d)  (a)  (b)  (c)  (d)  curvelets remain invariant (e) (f)  (g)  (h)  (e)  (g)  (h)  • •  (f)  approximation improves for higher frequencies  Diagonal approximation of the Hessian  Existing scaling methods Methods are based on a diagonal approximation of Ψ.      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 Ψ from a reference vector close to the actual image assumes a smooth symbol of Ψ in space and angle does not require the reflectors to be conormal <=> allows for conflicting dips stably inverts the diagonal  Ψ by C T DΨ C.  Approximation  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 the decomposition  This main result proved in Appendix A shows that the approxima  Ψϕµ (x)  C DΨ Cϕµ (x) T  diagonal approximation goes to zero for increasingly finer scales. The app T  =  AA ϕµ (x)  from the property that the symbol is slowly varying over the support  with A :=  √  DΨ C and AT := C  √ T  DΨ .  approximation that becomes more accurate as the scale increases.  Approximation y(x)  =  Ψm (x) + e(x) AA m (x) + e(x) T  = Ax0 + e,  • •  Wavelet-vagulette like [Donoho, Candes] Amenable to nonlinear recovery  Estimation of the diagonal scaling  Diagonal estimation  (a)  (b)  Seismic amplitude recovery  Recovery  Final form  y = Ax0 + ε with x0 = ΓCm and Solve P: with  = Ae.    minx J(x) subject to  y − Ax  2    H †˜ ˜ m = (A ) x sparsity  J(x) = α x  1  +β Λ  1/2  A  H  †  continuity  x  p  .  ≤  Gradient of the reference vector  500  1000  depth (m)  1500  2000  2500  3000  3500 2000  4000  6000  8000 lateral (m)  10000  12000  14000  Application to the SEG AA’ model  Example SEGAA’ data:     “broad-band” half-integrated wavelet [5-60 Hz] 324 shots, 176 receivers, shot at 48 m 5 s of data  Modeling operator     Reverse-time migration with optimal check pointing (Symes ‘07) 8000 time steps modeling 64, and migration 294 minutes on 68 CPU’s  Scaling requires 1 extra migration-demigration  Seismic Laboratory for Imaging and Modeling  Seismic Laboratory for Imaging and Modeling  Seismic Laboratory for Imaging and Modeling  Migrated data  Seismic Laboratory for Imaging and Modeling  Amplitude-corrected & denoised migrated data  Noise-free data  Noisy data (3 dB)  Seismic Laboratory for Imaging and Modeling  Data from migrated image  Data from amplitude-corrected & denoised migrated image  Nonlinear data  Conclusions Curvelet-domain scaling     handles conflicting dips (conormality assumption) exploits invariance under the PsDO robust w.r.t. noise  Diagonal approximation    exploits smoothness of the symbol uses “neighbor” structure of the curvelet transform  Results on the SEG AA’ show     recovery of amplitudes beneath the Salt successful recovery of clutter improvement of the continuity  Acknowledgments The authors of CurveLab (Demanet,Ying, Candes, Donoho) Dr. Symes for 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), from the following organizations: BG Group, BP, Chevron,ExxonMobil and Shell.  

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