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Just diagonalize: a curvelet-based approach to seismic amplitude recovery Herrmann, Felix J. 2008
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Title | Just diagonalize: a curvelet-based approach to seismic amplitude recovery |
Creator |
Herrmann, Felix J. Moghaddam, Peyman P. Stolk, Christiaan C. |
Publisher | European Association of Geoscientists & Engineers |
Date Created | 2008-03-05 |
Date Issued | 2007 |
Extent | 3716169 bytes |
Subject |
Curvelet Seismic Amplitude Transform Nonlinear Approximation Wave Propagation Hessian |
Genre |
Other |
Type |
Text |
File Format | application/pdf |
Language | Eng |
Collection |
Seismic Laboratory for Imaging and Modeling |
Date Available | 2008-03-05 |
DOI | 10.14288/1.0107393 |
Affiliation |
Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Herrman, Felix J., Mogahaddam, Peyman P., Stolk, Christiaan C. 2007. Just diagonalize: a curvelet-based approach to seismic amplitude recovery. Presentation at EAGE Curvelet Workshop. |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty Other |
URI | http://hdl.handle.net/2429/523 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/340/items/1.0107393/source |
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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 demigration- migration, 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 .... Sharp Model 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 2000 2500 3000 3500 4000 4500 5000 Curvelet propagation Sharp Model 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 2000 2500 3000 3500 4000 4500 5000 Curvelet propagation Major challenge. Limit ourselves to migration amplitude recovery! Sharp Model 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 2000 2500 3000 3500 4000 4500 5000 “Imaged” curvelet Hessian/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 preserved Corresponds to a spatially-varying dip filter after appropriate preconditioning (=> zero order). Ψ( Ψf ) (x) := ( KT Kf ) (x) = ∫ Rd e−ix·ξa(x, ξ)f̂(ξ)dξ • curvelets remain invariant • approximation improves for higher frequencies (a) (b) (c) (d) (e) (f) (g) (h) Figure 2: Invariance of curvelets under the discretized normal operator Ψ for a smoothly varying background model (a so-called lens model see Fig. 4(a)). Three coarse-scale curvelets in the physical domain before (a) and after application of the normal opera- tor (b) in the physical (a-b) and Fourier domain (e-f). The results for three fine-scale curvelets are plotted in (c-d) for the physical domain and in (g-h) for the Fourier domain. Remark: The curvelets remain close to invariant under the normal operator, a statement which becomes more accurate for finer scale which is consistent with Theorem 1. The ex- ample also shows that this statement only holds for curvelets that are in the support of the imaging operator excluding steeply dipping curvelets. 52 (a) (b) (c) (d) (e) (f) (g) (h) Figure 2: Invariance of curvelets under the discretized normal operator Ψ for a smoothly varying background model (a so-called lens model see Fig. 4(a)). Three coarse-scale curvelets in the physical domain before (a) and after application of the normal opera- tor (b) in the physical (a-b) and Fourier domain (e-f). The results for three fine-scale curvelets are plotted in (c-d) for the physical domain and in (g-h) for the Fourier domain. Remark: The curvelets remain close to invariant under the normal operator, a statement which becomes more accurate for finer scale which is consistent with Theorem 1. The ex- ample also shows that this statement only holds for curvelets that are in the support of the imaging operator excluding steeply dipping curvelets. 52 Invariance under Hessian matrix Diagonal approximation of the Hessian Existing scaling methods Methods 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 Ψ Ψ Ψ • Allows for the decomposition in Rd. Lemma 1. With C ′ some constant, the following holds ‖(Ψ(x,D)− a(xν , ξν))ϕν‖L2(Rn) ≤ C ′2−|ν|/2. (14) To approximate Ψ, we define the sequence u := (uµ)µ∈M = a(xµ, ξµ). Let DΨ be the diagonal matrix with entries given by u. Next we state our result on the approximation of Ψ by CTDΨC. Theorem 1. The following estimate for the error holds ‖(Ψ(x,D)− CTDΨC)ϕµ‖L2(Rn) ≤ C ′′2−|µ|/2, (15) where C ′′ is a constant depending on Ψ. This main result proved in Appendix A shows that the approximation error for the diagonal approximation goes to zero for increasingly finer scales. The approximation derives from the property that the symbol is slowly varying over the support of a curvelet, an approximation that becomes more accurate as the scale increases. Decomposition of the normal operator By virtue of Theorem 1, the normal operator can be factorized ( Ψϕµ ) (x) $ (CTDΨCϕµ)(x) (16) = ( AATϕµ ) (x) with A := √ DΨC and AT := CT √ DΨ. Because the seismic reflectivity can be written as a superposition of curvelets, we can replace ϕµ in the above equation with the model m. We 15 ( Ψϕµ ) (x) ! (CTDΨCϕµ)(x) = ( AATϕµ ) (x) with A := √ DΨC and AT := CT √ DΨ. Approximation • Wavelet-vagulette like [Donoho, Candes] • Amenable to nonlinear recovery y(x) = ( Ψm ) (x) + e(x) ! (AATm)(x) + e(x) = Ax0 + e, Approximation Estimation of the diagonal scaling (a) (b) (c) (d) Figure 5: Estimates for the diagonal ũ are plotted in (a-d) for increasing η = {0.01, 0.1, 1, 10}. The diagonal is estimated according the procedure outlined in Table 1 with the reference and ’data’ vectors, v and b, plotted in Fig. 4(b) and 4(c). As expected the diagonal becomes more positive for increasing η. Herrmann et.al. – 54 Diagonal estimation Seismic amplitude recovery • Final form • Solve y = Ax0 + ε Recovery with x0 = ΓCm and ! = Ae. P : minx J(x) subject to ‖y −Ax‖2 ≤ ! m̃ = (AH)†x̃ with J(x) = sparsity︷ ︸︸ ︷ α‖x‖1 +β ‖Λ1/2 ( AH )† x‖p︸ ︷︷ ︸ continuity . Gradient of the reference vector lateral (m) de pt h (m ) 2000 4000 6000 8000 10000 12000 14000 500 1000 1500 2000 2500 3000 3500 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 Seismic Laboratory for Imaging and Modeling Migrated data Amplitude-corrected & denoised migrated data Seismic Laboratory for Imaging and Modeling Noise-free data Noisy data (3 dB) 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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