Time domain sparsity promoting LSRTM with source estimation

Transcription

1 Time domain sparsity promoting LSRTM with source estimation Mengmeng Yang, Philipp Witte, Zhilong Fang & Felix J. Herrmann SLIM University of British Columbia

2 Motivation Features of RTM: pros - no dip limitation - strong lateral velocity variations cons - inaccurate amplitudes & low resolution Problems of LS- RTM: iterations that touch all shots are too expensive data can be overfitted 2

3 RTM w/ correct wavelet Depth (km) Distance (km)

4 Sparsity promoting LS-RTM w/ correct wavelet Depth (km) Distance (km) -.3

5 Sparsity promoting LS-RTM w/ wrong wavelet Depth (km) Distance (km) -.3

6 LS-RTM min m P ns i=1 kj i[m, q i ] m b i k 2 m J i m : background model : Born modelling operator for shot : model perturbation i th q i b i : source wavelet for shot i th : vectorized reflections for shot i th 6

7 Herrmann F J, Li X. Efficient least-squares imaging with sparsity promotion and compressive sensing[j]. Geophysical prospecting, 212, 6(4): Sparsity promoting inversion min x kxk 1 s.t. ns X i=1 k J i [m, q i ]C {z } Ĵ x b i {z} b k 2 apple C : the transpose of Curvelet transform x : Curvelet coefficients : tolerance for noise or modelling error 7

8 Felix J. Herrmann,Ning Tu and Ernie Esser, Fast online migration with Compressive Sensing, EAGE Annual Conference Proceeding, 215, vol. 6, p , 212 Lorenz, Dirk A.; Wenger, Stephan; A sparse Kaczmarz solver and a linearized Bregman method for online compressed sensing. arxiv: Randomized subsampling Ĵ x = b Ĵ r(k) x = b r(k) n s n s n s 8

9 W, Yin. Analysis and generalizations of the linearized Bregman method. SIAM J. Imaging Sci., 3(4): , 21. Herrmann F J, Tu N, Esser E. Fast online migration with Compressive Sensing[J]. Solvers for sparsity promoting inversion Many solvers for sparse. inversion: Iterative soft thresholding (simple, but slow convergence, cooling of threshold...) Spectral projected gradients w/ L1 constraint SPGL1 (expensive, difficult to implement, slow convergence) Linearized Bregman (LB) (easy to implement, proven convergence w/ subsampling) 9

10 Sparsity promoting LS-RTM w/ correct wavelet & SPGL Depth (km) Distance (km) -.3

11 Sparsity promoting LS-RTM w/ correct wavelet & LB Depth (km) Distance (km) -.3

12 W, Yin. Analysis and generalizations of the linearized Bregman method. SIAM J. Imaging Sci., 3(4): , 21. Herrmann F J, Tu N, Esser E. Fast online migration with Compressive Sensing[J]. Modification minimize x kxk kxk2 s.t. kĵx bk 2 apple strongly convex objective function because of additional 2- norm term for big enough solves BP problem 12

13 Workflow for LB minimize x kxk kxk2 s.t. kĵx bk 2 apple 1. Initialize x =, z =, q,, batchsize n s n s 2. for k =, 1, Randomly choose shot subsets I 2 [1 n s ], I = n s Ĵ k = {J i (m,q i )C } i2i 5. b k = {b i } i2i 6. z k+1 = z k t k Ĵ T k P (Ĵ k x k 7. x k+1 = S (z k+1 ) b k ) 8. end note: S (z k+1 )=sign(z k+1 ) max{, z k+1 } P (Ĵ k x k b k ) = max{, 1 kĵ k x k b k k k x k b k ) 13

14 Toy example Sparsity recovery with tall ill- conditioned matrix A: 2 X 1, with Rank 5 x: 1 X 1, with 2 non- zeros 14

15 15 SPGL1 vs LB no subsampling

16 16 SPGL1 vs LB 5% subsampling

17 17 SPGL1 vs LB 8% subsampling

18 18 SPGL1 vs LB 9% subsampling

19 An analysis of seismic wavelet estimation. Ayon Kumar Dey, 1999, University of Calgary, PhD thesis What if source signature is unknown? Estimate source by solving least- square problem: min w P Ntr j kw b j b j k 2 s.t.kwk 2 =1 where Suppose that, and q is the same for all shots q = w q q = w q is the initial guess of q 19

20 Initial wavelet setting.3.2 signal in time domain initial q true q Amp approximate duration time (s) frequency spectrum Energy initial q true q frequency bandwidth wider due to factorization frequency (HZ) 2

21 Combine the image inversion & source estimation start w/ sufficiently small threshold to allow main reflectors to enter into solution update m start with initial wavelet update wavelet 21

22 Workflow for sparsity-promoting LS-RTM w/ source estimation 1. Initialize x =, z =, q,, 2, batchsize n s n s, weights r 2. for k =, 1, 3. Randomly choose shot subsets I 2 [1 n s ], I = n s 4. Ĵ k = {J i (m,q )C } i2i 5. b k = {b i } i2i 6. bk = Ĵ k x k 7. w k = arg min w PI kw b k b k k 2 + kr(w q )k kw q k 2 8. z k+1 = z k t k Ĵ w k k?p (w k b k b k ) 9. x k+1 = S (z k+1 ) 1. end 22

23 Experiments Data: 295 shots with shot interval 15m 295 receivers with receiver interval 15m 4s record, 15Hz peak frequency designed wavelet synthetic linearized data Experiments: one pass through the data with batch sizes 2.5% data randomized subset of shots normalized true source wavelet & initial guessed wavelet 23

24 background model and model perturbation.5 s 2 /km 2.4 Depth (km) Distance (km) 24

25 Sparsity promoting LS-RTM w/ correct wavelet & LB Depth (km) Distance (km) -.3

26 Sparsity promoting LS-RTM w/ wrong wavelet & LB Depth (km) Distance (km) -.3

27 Sparsity promoting LS-RTM w/ source estimation w/ LB Depth (km) Distance (km) -.3

28 Sparsity promoting LS-RTM w/ source estimation, via LB Depth (km) Distance (km)

29 Sparsity promoting LS-RTM w/ source estimation.3.2 signal in time domain estimated initial q true q Amp time (s) frequency spectrum 4 estimated initial q true q Energy frequency (HZ) 29

30 Residual & model error Residuals Relative model error 3

31 31 Robustness of source estimation starting w/ zero-phase wavelet

32 32 Sparsity promoting LS-RTM w/ correct wavelet & LB

33 33 Sparsity promoting LS-RTM w/ source estimation & LB

34 Conclusions LB with correct source signature gives image with sharp interfaces w/ correct amplitudes Computational complexity is controlled to ~1 RTM w/ randomized source subsampling LB improves inversion results compared to other one- norm solvers LB can be combined w/ on- the- fly source estimation w/o a large computational overhead 34

35 Acknowledgements This research was carried out as part of the SINBAD project with the support of the member organizagons of the SINBAD Consorgum. 35