Structured tensor missing-trace interpolation in the Hierarchical Tucker format Curt Da Silva and Felix J. Herrmann Sept. 26, 2013

Transcription

1 Structured tensor missing-trace interpolation in the Hierarchical Tucker format Curt Da Silva and Felix J. Herrmann Sept. 6, 13 SLIM University of British Columbia

2 Motivation 3D seismic experiments - 5D data expensive to acquire, store sample at sub- Nyquist rates Data exhibits low- rank structure exploit structure for interpolation Fully sampled data simultaneous sources in wave- equation based inversion mitigating multiples

3 [1] Kreimer, N, and Sacchi, M. D. "A tensor higher- order singular value decomposivon for prestack seismic data noise reducvon and interpolavon." (1) [] Gao, Jianjun, Vicente Oropeza, and Mauricio D. Sacchi. "EvaluaVon of a fast algorithm for the eigen- decomposivon of large block Toeplitz matrices with applicavon to 5D seismic data interpolavon." (11) Context Low- rank matrix/tensor completion via nuclear norm projection [1] Require SVDs on huge data matrices Not scalable to large problem sizes Data completion via Toeplitz embedding [] Problem size - (# data points)

4 Goals Generalization of Compressive Sensing to multiple dimensions what can we learn from 1D/D recovery? Randomized source/receiver acquisition reduce acquisition financial/time costs Efficient solver SVD- free, parallelizable # parameters << # data points

5 7.3 Hz - 75% missing receivers Common source gather x 1 6 x Receiver y 1 Receiver y Receiver x Receiver x True data Subsampled data

6 7.3 Hz - 75% missing receivers Common source gather x 1 6 x Receiver y 1 Receiver y Receiver x Receiver x 6 True data Recovered data - SNR 17. db

7 Compressive sensing with sparsity promotion Successful reconstruction scheme Signal structure sparsity Sampling subsampling decreases sparsity Optimization look for sparsest solution

8 Multidimensional interpolation with Hierarchical Tucker Successful reconstruction scheme Signal structure Hierarchical Tucker Sampling subsampling increases hierarchical rank Optimization fit data in the Hierarchical Tucker format

9 Matricization The matricization of a tensor X with dimensions 1,...,d along the dimensions t =(t 1,...,t r ) dimensions t along the rows and dimensions is the matrix formed by placing the t c along the columns Denoted X (t)

10 Example in Matlab n1 = ; n = ; n3 = ; n = ; % Tensor x = randn(n1,n,n3,n); X (1,) X (3,) X (1,3) % Matricization along dimensions 1 and x1 = reshape(x,n1 * n, n3 * n); % Matricization along dimensions 3 and y3 = permute(x,[3 1 ]); x3 = reshape(x, n3 * n, n1 * n); % Matricization along dimensions 1 and 3 y13 = permute(x,[1 3 ]); x13 = reshape(x,n1 * n3, n * n);

11 Hierarchical Tucker format X n 1 n n 3 n tensor B 13 k U T 3 3 n1n X (1,) = n1n U 1 n 3 n n 3 n k 1 SVD -like decomposition

12 Hierarchical Tucker format X n 1 n n 3 n tensor n1n U 1 n 1! U 1 n k 1 k 1

13 Hierarchical Tucker format X n 1 n n 3 n tensor B 1 n1n U 1 n 1! U 1 n 1! U 1 U T n k 1 k 1 n k k 1

14 Hierarchical Tucker format Intermediate matrices don t need to be stored U t,b t - small parameter matrices specify the tensor completely Separating groups of dimensions from each other dimension tree

15 A. Uschmajew, B. Vandereycken. The geometry of algorithms using hierarchical tensors. Linear algebra and its applications, 13. Example B 135 {1,, 3,, 5} = t r U 13 B 13 U 5 B 5 {1,, 3} {, 5} U 1 U 3 B 3 U U 5 {1} {, 3} = t {} {5} U U 3 {} = t 1 {3} = t

16 Hierarchical Tucker format Storage apple dnk +(d )K 3 + K Compare to N d storage for the full tensor Effectively breaking the curse of dimensionality when K N d Low frequency data compresses in HT

17 Hierarchical Tucker example For a cube with max rank N = 1,d=,K = Full storage: N d = 1 8 values HTucker storage: values Compression of a factor of 99.97%

18 Seismic Hierarchical Tucker We consider a 3D seismic survey with coordinates (src x, src y, rec x, rec y, time) We take a Fourier transform in time and restrict ourselves to a single frequency slice

19 Seismic Hierarchical Tucker For a frequency slice with coordinates (src x, src y, rec x, rec y), there are essentially two choices of dimension splitting (by reciprocity) {src x, srcy, rec x, rec y} {src x, rec x, src y, rec y} {src x, src y} {rec x, rec y} {src x, rec x} {src y, rec y} {src x} {src y} {rec x} {rec y} {src x} {rec x} {src y} {rec y} Canonical Decomposition Non- canonical Decomposition

20 Matricizations 1 x rec, y rec Full data Normalized singular value x src, y src 1 6 (Rec x, Rec y) matricization - Canonical ordering

21 Matricizations 1 x src,x rec (Rx Ry) matricization (Sx Rx) matricization Normalized singular value y src,y rec 1 6 (Src x, Rec x) matricization - Noncanonical ordering

22 Multidimensional interpolation with Hierarchical Tucker Successful reconstruction scheme Signal structure Hierarchical Tucker Sampling subsampling increases hierarchical rank Optimization fit data in the Hierarchical Tucker format

23 Matrix Completion X Normalized singular value

24 Matrix Completion A(X) Normalized singular value

25 Tensor Completion Structure - recover a tensor Well represented in HT X which has low hierarchical rank Sampling - random removal of points increases rank Poorly represented in HT Idealized sampling

26 Idealized recovery 75% random entries removed Common receiver gather x 1 3 x Source x 5 Source x Source y Source y 1 True data Subsampled data

27 Idealized recovery 75% random entries removed Common receiver gather x x Source x 5 Source x Source y Source y True data Subsampled data

28 Idealized recovery 75% random entries removed Common receiver gather x x Source x 5 Source x Source y Source y True data Recovered data - SNR 19.3 db

29 Sampling Sampling (x src,y src,x rec,y rec ) points from the data idealized recovery impossible to physically implement Sampling (x rec,y rec ) points from the data less idealized possible to acquire data - e.g. ocean bottom nodes

30 Realistic recovery 5% random receivers removed x rec, y rec No subsampling 5% missing sources Normalized singular value x src, y src 1 6 (Rec x, Rec y) matricization - Canonical ordering

31 Realistic recovery 5% random receivers removed x src,x rec No subsampling 5% missing sources y,y src rec Normalized singular value (Src x, Rec x) matricization - Noncanonical ordering

32 Data organization In summary: (rec x, rec y) organization High rank Missing sources operator - removes columns Poor recovery scenario (src x, rec x) organization Low rank Missing sources operator - removes blocks Closer to ideal recovery scenario

33 Multidimensional interpolation with Hierarchical Tucker Successful reconstruction scheme Signal structure Hierarchical Tucker Sampling subsampling increases hierarchical rank Optimization fit data in the Hierarchical Tucker format

34 Optimization Given data b with missing sources and/or receivers, subsampling operator A, full tensor expansion operator :(U t,b t )! C n 1...n d solve min x=(u t,b t ) 1 ka (x) bk

35 A. Uschmajew, B. Vandereycken. The geometry of algorithms using hierarchical tensors. Linear algebra and its applications, 13 C. Da Silva and F. J. Herrmann, OpFmizaFon on the Hierarchical Tucker manifold - applicafons to tensor complefon, 13 Differential geometry HT tensors parametrize a submanifold of full tensor space C n 1...n d Nonlinear, nonconvex space Generalization of curved surfaces Steepest Descent, Conjugate gradient, Gauss- Newton without SVDs in the full tensor space

36 Optimization program (x) Full- tensor space Parameter space

37 Optimization program

38 Derivatives Derivatives of a particular node with respect to its children can be computed efficiently, i.e. via (I U tl U H t l )hu T t r Z, B t i (,3),(,3) (I U tr U H t r )hu T t l 1 Z, B t i (1,3),(1,3) X P i Q multiplies Q by P in dimension i, hx, Y i (1,3),(1,3) = X i1,:,i Y 3 i1,:,i 3 i 1,i 3 The chain rule gives the gradient of the function

39 Derivatives Only involves matrix- matrix multiplications of small matrices compared to the full- tensor space Parallelizable - multilinear product can be done in parallel SVD- free - no large- scale SVDs, unlike nuclear norm- based methods

40 Results

41 Synthetic BG Group data Unknown model 68 x 68 sources with 1 x 1 receivers, data at.68hz, 7.3 Hz Receivers subsampled to 1 x 1 Recovered with Gauss- Newton

42 .68 Hz - 75% missing receivers Common source gather x 1 x Receiver y 1 Receiver y Receiver x Receiver x True data Subsampled data

43 .68 Hz - 75% missing receivers Common source gather x 1 x Receiver y 1 Receiver y Receiver x Receiver x True data Recovered data - SNR 19.7 db

44 .68 Hz - 75% missing receivers Common source gather x 1 x Receiver y 1 Receiver y Receiver x Receiver x True data Difference

45 .68 Hz - 75% missing receivers Common receiver gather - no data initially x 1 8 x Source y 3 Source y Source x Source x 8 True data Recovered data - SNR 3 db

46 .68 Hz - 75% missing receivers Common receiver gather - no data initially x 1 8 x Source y 3 Source y Source x Source x 8 True data Difference

47 7.3 Hz - 75% missing receivers Common source gather x 1 x Receiver y 1 Receiver y Receiver x Receiver x True data Subsampled data

48 7.3 Hz - 75% missing receivers Common source gather x 1 x Receiver y 1 Receiver y Receiver x Receiver x True data Recovered data - SNR 17.6 db

49 7.3 Hz - 75% missing receivers Common source gather x 1 x Receiver y 1 Receiver y Receiver x Receiver x True data Difference

50 7.3 Hz - 75% missing receivers Common receiver gather - no data initially x 1 x Source y 3 1 Source y Source x Source x True data Recovered data - SNR 16. db

51 7.3 Hz - 75% missing receivers Common receiver gather - no data initially x 1 x Source y 3 Source y Source x Source x True data Difference

52 7.3 Hz - simultaneous receivers - 9% data reduction Common source gather x 1 x Receiver y Receiver y Receiver x Receiver x True data Input data - A T Ab A - subsampling operator b - full data

53 7.3 Hz - simultaneous receivers - 9% data reduction Common source gather x 1 x Receiver y Receiver y Receiver x Receiver x True data Recovered data - SNR 16. db

54 7.3 Hz - simultaneous receivers - 9% data reduction Common source gather x 1 x Receiver y Receiver y Receiver x Receiver x True data Difference

55 Conclusion 3D seismic data has an underlying structure that we can exploit for interpolation (Hierarchical Tucker format) Different schemes for organizing data - important for recovery

56 Conclusion We can interpolate HT tensors with missing entries using the Riemannian manifold structure of the HT format Achieve good results from largely subsampled data (75% missing receivers) Can use this method to create full volumes from subsampled data Migration, multiple removal, etc.

57 Acknowledgements Thank you for your attention This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant (R815) and the Collaborative Research and Development Grant DNOISE II (3751-8). This research was carried out as part of the SINBAD II project with support from the following organizations: BG Group, BGP, BP, CGG, Chevron, ConocoPhillips, ION, Petrobras, PGS, Total SA, WesternGeco, Statoil, and Woodside.