Full-Waveform Inversion with Gauss- Newton-Krylov Method

Transcription

1 Full-Waveform Inversion with Gauss- Newton-Krylov Method Yogi A. Erlangga and Felix J. Herrmann Seismic Laboratory for Imaging and Modeling The University of British Columbia (UBC), Vancouver The 79th SEG Meeting: SI3 Methods Houston, October 27, 29

2 Full-Waveform Inversion (FWI) Given experiment data P. With the misfit functional: E[m] = 1 2 P F[m] 2 2 Optimization Problem: Find ˆm = arg min m M E[m] subject to the (forward) modeling wavefields D receivers by. U F[m] = DU[m] restricted to the Lailly, 1983 Tarantola, 1984, 1986, 1987 Pratt and co-authors, 1996, 1998, 1999, 23

3 Frequency domain FWI Forward model: Helmholtz equation H[ω, m]u = Q, m = (m 1... m M ) T H : the Helmholtz matrix, function of angular freq ω Q = [q 1... q ns ] U = [u 1... u ns ] : the source matrix, with shots : the wavefield matrix n s

4 Impediments Fast, scalable solver for the forward and adjoint systems iterative method with Preconditioning with shifted Laplacian [E. et al. (26), Riyanti et al., (26)] Multilevel Krylov method [E. & Nabben (29), E. & Herrmann (28)] Multidimensional experiments (shots, frequencies): more data than model Data reduction via frequency subsampling [Sirgue & Pratt (24), Mulder & Plessix (24)] Compressive Sampling (CS) framework : data reduction via shot and frequency subsampling compressive wavefield computation [Lin, Herrmann (27), Herrmann, E. & Lin (29)] extension to compressive imaging Fast minimization solver (GN-type: Hessian) Gauss-Newton method with implicit computation of Hessian

5 Our solution Gauss-Newton with implicit Hessian (Gauss-Newton-Krylov, GNK) Dimensionality reduction [Herrmann, E. & Lin (29)] [Tim Lin: Compressive simultaneous full-waveform simulation, this meeting, SM1] Q = D HU = Q y = RMDU s }{{} single shots Q = D HU = Q y = DU RMs }{{} simul. shots FWI with CS

6 FWI with CS The misfit functional: E[m] = 1 2 RM(P F[m]) 2 2 with RM a CS-sampling matrix (reduces data size). Optimization Problem: Find ˆm = arg min m M E[m] subject to F[m] = DU[m]

7 Main contribution: [Hermann et al. (29), EAGE] See also: Krebs et al. (29), this meeting E[m] = 1 2 P F[m] 2 2 In line with this: Sampling of overdetermined systems [Drineas, Mahoney & Muthukhrisnan (26)] min E min E but is a bounded approximation.

8 Outline Newton method: Hessian Implicit computation of the GN Hessian Extension to CS framework Reduced numbers of shots and frequencies Examples related work: in time domain [Akcelic, Biros & Ghattas (22)] PDE-constrained optimization: KKT sytems, reduced systems, etc [Heinkenschloss (1991), Biros & Ghattas (25),...]

9 Newton Method E[m + δm] = E[m] + g T δm δmt Hδm m Initial model ; Update until convergence: δm = H 1 k 1 g k 1; m k = m k 1 + γ k 1 δm; with g k 1 g[m k 1 ] H k 1 H[m k 1 ] : the gradient, : the Hessian, γ k 1 : the step length.

10 H = [h i,j ] with ( ) E h i,j = m i m j ( 2 U = rowsum U m i m j m i Hessian: F m j ). Negative sign: not necessarily SP(S)D Fast/quadratic convergence only if close to the minimizer From the adjoint system: U m i V (back-propagated)

11 Gauss-Newton Method Simplify the Hessian by setting U m i nonlinear wave phenomena (e.g. multiples) F m j = Giving h GN i,j = rowsum ( 2 U m i m j ). This is associated with setting the back-propagated wavefield V = in the Hessian H GN = [h GN i,j ] is SP(S)D.

12 Inverting the Hessian: Krylov H GN k 1δm = g k 1 SP(S)D Hessian: compute Four important steps in CG: δm with Conjugate Gradient (CG). compute: w := H GN k 1p solution update: residual update: search dir. update: δm δm + αp r r αw p r + βw α, β : CG step lengths, satisfying orthogonal projection p δ 2 m m : second variation (of the Lagrangian) of.

13 Second variation system of GN - derived from second variations of the Lagrange minimization functional - detailed treatment in weak (bilinear) formulation, see the abstract. Forward model: H[m]Ũ = ω2 diag(p)u Ũ : second variation of U Adjoint system/back propagation: H[m]Ṽ = D DŨ Ṽ : the second variation of V The action of Hessian on p H GN p := ω 2 rowsum(u Ṽ)

14 FWI: Examples Marmousi model: 742 x 298 m, 372 x 15 gridpoints, 37 shots. Freqs: 3, 5, 9 Hz. 1 CG iters for the Hessian Hard model Smooth model

15 First Update (in δm ) Gradient Method GNK Note: different scale (by 1^3)

16 Velocity after the first update Gradient Method GNK

17 After 5 iterations Hard model Inverted Result

18 FWI with compressive simultaneous source (CFWI) Minimization problem: ˆm R = arg min m M 1 2 RM(P F) 2 2 RM : CS-sampling matrix turns single shots into randomized simultaneous shots subsamples the shots (fewer shots) and frequencies Simultaneous shots: Beasley, Chambers & Jiang (1998), Beasley (28) Berkhout (28) Neelamani, Krohn, Krebs, Deffenbaugh & Romberg (28) Herrmann, E. & Lin (29)

19 Gradient method of CFWI Minimize functional: E = 1 2 RM(P DU) 2 2 Gradient update: = 1 2 (RM(P DU))T RM(P DU) ( ) g R = rowsum J T (RM(P DU)) J J(RMDU) ( ) g R = rowsum J (RM(P DU)) with the Jacobian.

20 Using wavefield-source equivalence [Herrmann, E., & Lin, 29] Gradient update with J J(DU) : the (compressed) Jacobian w.r.t. to the compressed simultaneous sources P δm = g = J T (P DU) : data obtained with simultaneous shot

21 Computing the Jacobian Compressed forward model: U m i = H 1 H m i U. δm = rowsum [ U T H T... HT m 1 m M HU = Q {}}{ H T (P DU) ] V V : backpropagated wavefield ass. with Q = RMQ. The GN Hessian can be derived in the similar way!

22 Complexity Analysis Gauss-Newton-Krylov (GNK) Gradient : forward + back-propagation n f n s n 2 log n Hessian : forward + back-propagation per CG iteration n CG n f n s n 2 log n Overall : Compressive FWI with GNK: n CG n f n s n 2 log n n CG n f n sn 2 log n n f n f, n s n s Construction of RM negligible compared to FWI

23 CFWI: Examples, GNK Iter #1 9% subsampled 37 randomized simul. shots 37 periodic shots Noisy image --> recover the image via sparsity promoting

24 CFWI: Examples, GNK Iter #5 9% subsampled 37 randomized simul. shots 37 periodic shots

25 CFWI: Examples, GNK Iter #1 99% subsampled 4 randomized simul. shots 4 periodic shots

26 CFWI: Examples, GNK Iter #5 99% subsampled 4 randomized simul. shots 4 periodic shots

27 Conclusion Viable inversion of GN Hessian with Krylov method Accuracy of the inversion of Hessian depends on the number of iterations --> better FWI result Faster convergence of CG by preconditioners Implicit BFGS-type preconditioner Curvelet-based preconditioner [Herrmann, Brown, E. & Moghaddam (29)] Memory-friendly algorithm (gradient and Hessian can be computed on the fly) With scalable implicit solver for forward and adjoint systems, matrixfree algorithm [E., Oosterlee & Vuik (26), E. & Nabben (29), E. & Herrmann (28)] Natural extension to compressive FWI Similar results but less computational work In the CS framework: l1 inversion

28 Acknowledgments 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 ( ) of Felix J. Herrmann. This research was carried out as part of the SINBAD II project with support from the following organizations: BG Group, BP, Petrobras, and Schlumberger. Further information: slim.eos.ubc.ca