Curvelet imaging & processing: sparseness constrained least-squares migration

Transcription

1 Curvelet imaging & processing: sparseness constrained least-squares migration Felix J. Herrmann and Peyman P. Moghaddam (EOS-UBC) & thanks to: Gilles, Peyman and Candes, Sacchi, Rickett, WaveLab

2 Seismic imaging We are in the business of Improving the signal-to-noise ratio (SNR) Preserving edges on the model space Sparsifying (de)-migration & normal operators Coming up with ultimate preconditioners for out-of-the box imaging codes In the presence of noise... Lots of it SNR 0!

3 Seismic imaging Noisy image

4 Optimal imaging Denoised image

5 Basic idea Build on the premise that you stand a much better chance of solving an imaging problem when the model is represented optimally... local in space & angle sparse multi-scale and multi-directional Well behaved under migration!

6 Problem Seismic imaging is an inverse problem. Forward problem: data {}}{ d = K }{{} scat. oper. invoke curvelets to model/refl. {}}{ m + n }{{} noise estimate minimax. compress/precondition operators. invoke prior info (e.g. sparseness).

7 Inverse problem variational problem. Solve (Sacchi) ˆm : min m Solution Q: Can we use multiscale basis functions sparse, local & unconditional basis well behaved under imaging preserve the edges min. data mismatch {}}{ 1 2 d Km J(m) }{{} prior info

8 Simple (K=I) denoising with hard threshold: ˆm = composition {}}{ B Inspiration h Θ λγ }{{} thres./mute decomposition {}}{ B d with m Bm and d Bd solves ˆ m : min m 1 2 d m λ 2 m p

9 Inspiration Supplement constrained optimization (Candes 02): ˆm : with min m J(m) s.t. m ˆ m 0 µ e µ, µ ( d ) ˆ m 0 = Θ h λγ and ( d ) Θ h λ { d if d > λ 0 if d λ

10 Minimax estimation ˆm = B Θ t (Bd) approximates minimax, minimizes max. risk without prior info Bayes for least favorable prior preserves edges J(m) Lm 2 optimal/unconditional basis functions Mallat 97 Donoho 92

11 Wavelets Represent piece-wise smooth functions at no additional cost. Do not have to know where the singularities are. Only good for point-scatterers or horizon/vertically-aligned reflectors. Do NOT compress operators. Loose all beneficial properties.

12 courtesy WaveleLab + Beamlab 02 Directional wavelets

13 Directional wavelets

14 Directional wavelets

15 Directional wavelets

16 Seismic imaging Works so well because we exploit continuity along reflectors adaptive local smoothing Remaining challenges: deal with the operator/coloring compensate for the normal operator

17 Curvelet properties W j = {z, 2 j z 2 j+1, q q J p 2 j/2 } second dyadic partitioning Stein 93 Candes 02

18 Curvelet properties Curvelet in FK-dom@in

19 Imaging Q: Extend results to migration? deal with operators reformulate into denoising problem Curvelets compress FIO s (also ΨDO s) exploit compression with Lanczos methods (ultimate preconditioning) exploit Curvelet properties define new imaging schemes

20 Operators ˆm = ΨDO {}}{ ( ) 1 K T K K T }{{} FIO d YDO FIO d & m Wavelets Curvelets Theorem from Candes & Demanet 04: K T d K T trunc.d 2 C(# per col.) M for each M

21 Operators A Curvelet

22 Operators Migr@ted Curvelet

23 Operators Demigr@ted Curvelet

24 Operators Demigr@ted migr@ted Curvelet

25 Operators

26 Operators

27 Operators Curvelets remain curvelet-like 10 2 demigr@ted migr@ted norm@l 10 4 Compress the operators Almost diagonalize normal operator In particular BK T KB T diag ( diag ( )) BK T KB T Γ 2

28 Preconditioning Reformulate into preconditioned normal equations: with F T d = F T Fx + F T n F = KP, x = P T m and P = C T Γ 1 old yielding y = I {}}{ A x + n }{{} white

29 Preconditioning Lanczos

30 Estimation Ignore operator ( A I): equivalent to ˆx 0 = Θ λ (y) ˆm 0 = B ( Γ 2) (BK ) T ΘλΓ d approx. compensated for the normal operator minimax estimator brings us into convex

31 Estimation Impose prior info via constrained opt. ˆm : min m J(m) s.t. x ˆx 0 µ e µ µ with and e µ = { I µ if ˆx 0 µ λi µ λi µ if ˆx 0 µ < λi µ J(m) = m 1

32 Estimation ˆm : Include approx. normal operator min m J(m) s.t. A lanx ˆx 0 µ e µ µ with compressed operator A lan = QT k Q T k and T k = α 1 β β 1 α 2 β β k 1 0 β k 1 α k

33 Estimation Covariance operator: Cñ = E{ññ T } = BK T KB T with ñ BK T Monte-Carlo sample: Γ 2 ( ( )) 1 diag diag Cñ N N k=1 ñ 2 k

34 Examples Common-offset Kirchoff migration constant velocity model simple reflectivity Marmousi model complicated reflectivity Post-stack wave-equation migration

35 Examples Noisy Image

36 Examples Least-squares migrated Image

37 Examples Denoised after Thresholding

38 Examples Constrained Optimization

39 Observations Iterative non-regularized Leastsquares imaging fits the noise. Thresholding preserves the edges. Normal-operator correction restores the amplitudes. Constrained optimization removes the artifacts. Spikes remain due to L 1

40 Examples True model

41 Examples Noise-free image

42 Examples Preconditioned normal operator

43 Examples Noisy data SNR=0

44 Examples Inv. Curvelet Trans. Diag. Normal operator

45 Examples Thresholded

46 Examples Thresholded and corrected

47 Examples Optimized denoised

48 Conclusions Aimed at compression of operators & model: Optimal representation for m Ultimate preconditioner Thresholding brings us close to the solution Curvelets exploit smoothness along reflectors Constrained optimization is promising Finding appropriate norm is crucial & open Improved the SNR!

49 Acknowledgements Candes & Donoho for making their Curvelet code available. Sacchi and Rickett for making their migration codes available. Partially supported by a NSERC Grant.