American Institute of Mathematical Sciences

November  2013, 7(4): 1379-1392. doi: 10.3934/ipi.2013.7.1379

Seismic data reconstruction via matrix completion

 1 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States 2 Department of Mathematics, Harbin Institute of Technology, Harbin, China 3 Department of Mathematics, University of California, Los Angeles, CA 90095

Received  February 2012 Revised  February 2013 Published  November 2013

In seismic processing, one goal is to recover missing traces when the data is sparsely and incompletely sampled. We present a method which treats this reconstruction problem from a novel perspective. By utilizing its connection with the general matrix completion (MC) problem, we build an approximately low-rank matrix, which can be reconstructed through solving a proper nuclear norm minimization problem. Two efficient algorithms, accelerated proximal gradient method (APG) and low-rank matrix fitting (LMaFit) are discussed in this paper. The seismic data can then be recovered by the conversion of the completed matrix into the original signal space. Numerical experiments show the efficiency and high performance of data recovery for our model compared with other models.
Citation: Yi Yang, Jianwei Ma, Stanley Osher. Seismic data reconstruction via matrix completion. Inverse Problems & Imaging, 2013, 7 (4) : 1379-1392. doi: 10.3934/ipi.2013.7.1379
References:
 [1] M. Sacchi, T. Ulrych and C. Walker, Interpolation and extrapolation using a high-resolution discrete fourier transform,, IEEE Transactions on Signal Processing, 46 (1998), 31. doi: 10.1109/78.651165. Google Scholar [2] A. Duijndam, M. Schonewille and C. Hindriks, Reconstruction of band-limited signals, irregularly sampled along one spatial direction,, Geophysics, 64 (1999), 524. doi: 10.1190/1.1444559. Google Scholar [3] B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records,, Geophysics, 69 (2004), 1560. doi: 10.1190/1.1836829. Google Scholar [4] S. Xu, Y. Zhang, D. L. Pham and G. Lambaré, Antileakage Fourier transform for seismic data regularization,, Geophysics, 70 (2005). doi: 10.1190/1.1993713. Google Scholar [5] F. J. Herrmann and G. Hennenfent, Non-parametric seismic data recovery with curvelet frames,, Geophysical Journal International, 173 (2008), 233. doi: 10.1111/j.1365-246X.2007.03698.x. Google Scholar [6] R. Shahidi, G. Tang, J. Ma and F. J. Herrmann, Application of randomized sampling schemes to curvelet-based sparsity-promoting seismic data recovery,, Geophysical Prospecting, (2013), 973. doi: 10.1111/1365-2478.12050. Google Scholar [7] M. Naghizadeh and M. Sacchi, Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data,, Geophysics, 75 (2010). doi: 10.1190/1.3509468. Google Scholar [8] S. Hauser and J. Ma, Seismic data reconstruction via directional weighted shearlet-regularized inpainting,, Preprint, (2012). Google Scholar [9] S. Spitz, Seismic trace interpolation in the f-x domain,, Geophysics, 56 (1991), 785. Google Scholar [10] S. Crawley, J. Claerbout and R. Clapp, Interpolation with smoothly nonstationary prediction-error filters,, in 69th Annual International Meeting, (1999), 1913. doi: 10.1190/1.1820707. Google Scholar [11] M. Porsani, Seismic trace interpolation using half-step prediction filters,, Geophysics, 64 (1999), 1461. doi: 10.1190/1.1444650. Google Scholar [12] Y. Liu and S. Fomel, Seismic data interpolation beyond aliasing using regularized nonstationary autoegression,, Geophysics, 76 (2011). doi: 10.1190/geo2010-0231.1. Google Scholar [13] M. Naghizadeh and M. Sacchi, Seismic data reconstruction using multidimensional prediction filters,, Geophysical Prospecting, 58 (2010), 157. doi: 10.1111/j.1365-2478.2009.00805.x. Google Scholar [14] S. Trickett, L. Burroughs, A. Milton, L. Walton and R. Dack, Rank-reduction-based trace interpolation,, 80th Annual meeting, (2010). doi: 10.1190/1.3513645. Google Scholar [15] V. Oropeza and M. Sacchi, Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis,, Geophysics, 76 (2011). doi: 10.1190/1.3552706. Google Scholar [16] R. Vautard and M. Ghil, Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series,, Physica D, 35 (1989), 395. doi: 10.1016/0167-2789(89)90077-8. Google Scholar [17] E. J. Candes and B. Recht, Exact matrix completion via convex optimization,, Foundations of Computational Mathematics, 9 (2009), 717. doi: 10.1007/s10208-009-9045-5. Google Scholar [18] H. Ji, C. Liu, Z. Shen and Y. Xu, Robust video denoising using low rank matrix completion,, in Computer Vision and Pattern Recognition (CVPR), (2010). doi: 10.1109/CVPR.2010.5539849. Google Scholar [19] J.-F. Cai, E. J. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion,, SIAM Journal on Optimization, 20 (2008), 1956. doi: 10.1137/080738970. Google Scholar [20] S. Ma, D. Goldfarb and L. Chen, Fixed point and bregman iterative methods for matrix rank minimization,, Mathematical Programming: Series A and B, 128 (2011), 321. doi: 10.1007/s10107-009-0306-5. Google Scholar [21] J. Yang and X. Yuan, An inexact alternating direction method for trace norm regularized least squares problem,, Optimization Online, (2010). Google Scholar [22] R. Keshavan, A. Montanari and S.Oh, Matrix completion from a few entries,, IEEE Transactions on Information Theory, 56 (2010), 2980. doi: 10.1109/TIT.2010.2046205. Google Scholar [23] Y. Nesterov, A method of solving a convex programming problem with convergence rate o($1/k^2$),, Soviet Mathematics Doklady, 27 (1983), 372. Google Scholar [24] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM Journal on Imaging Sciences, (2009), 183. doi: 10.1137/080716542. Google Scholar [25] K.-C. Toh and S. Yun, An advanced proximal gradient algorithm for nuclear norm regularized linear least squares problems,, Pac. J. Optim., 6 (2010), 615. Google Scholar [26] Z. Wen, W. Yin and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm,, Mathematical Programming Computation, 4 (2012), 333. doi: 10.1007/s12532-012-0044-1. Google Scholar [27] H. Schaeffer and S. Osher, A low patch-rank interpretation of texture,, SIAM J. Imaging Sci., 6 (2013), 226. doi: 10.1137/110854989. Google Scholar [28] Y. Hua, Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,, IEEE Transactions on Signal Processing, 40 (1992), 2267. Google Scholar

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References:
 [1] M. Sacchi, T. Ulrych and C. Walker, Interpolation and extrapolation using a high-resolution discrete fourier transform,, IEEE Transactions on Signal Processing, 46 (1998), 31. doi: 10.1109/78.651165. Google Scholar [2] A. Duijndam, M. Schonewille and C. Hindriks, Reconstruction of band-limited signals, irregularly sampled along one spatial direction,, Geophysics, 64 (1999), 524. doi: 10.1190/1.1444559. Google Scholar [3] B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records,, Geophysics, 69 (2004), 1560. doi: 10.1190/1.1836829. Google Scholar [4] S. Xu, Y. Zhang, D. L. Pham and G. Lambaré, Antileakage Fourier transform for seismic data regularization,, Geophysics, 70 (2005). doi: 10.1190/1.1993713. Google Scholar [5] F. J. Herrmann and G. Hennenfent, Non-parametric seismic data recovery with curvelet frames,, Geophysical Journal International, 173 (2008), 233. doi: 10.1111/j.1365-246X.2007.03698.x. Google Scholar [6] R. Shahidi, G. Tang, J. Ma and F. J. Herrmann, Application of randomized sampling schemes to curvelet-based sparsity-promoting seismic data recovery,, Geophysical Prospecting, (2013), 973. doi: 10.1111/1365-2478.12050. Google Scholar [7] M. Naghizadeh and M. Sacchi, Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data,, Geophysics, 75 (2010). doi: 10.1190/1.3509468. Google Scholar [8] S. Hauser and J. Ma, Seismic data reconstruction via directional weighted shearlet-regularized inpainting,, Preprint, (2012). Google Scholar [9] S. Spitz, Seismic trace interpolation in the f-x domain,, Geophysics, 56 (1991), 785. Google Scholar [10] S. Crawley, J. Claerbout and R. Clapp, Interpolation with smoothly nonstationary prediction-error filters,, in 69th Annual International Meeting, (1999), 1913. doi: 10.1190/1.1820707. Google Scholar [11] M. Porsani, Seismic trace interpolation using half-step prediction filters,, Geophysics, 64 (1999), 1461. doi: 10.1190/1.1444650. Google Scholar [12] Y. Liu and S. Fomel, Seismic data interpolation beyond aliasing using regularized nonstationary autoegression,, Geophysics, 76 (2011). doi: 10.1190/geo2010-0231.1. Google Scholar [13] M. Naghizadeh and M. Sacchi, Seismic data reconstruction using multidimensional prediction filters,, Geophysical Prospecting, 58 (2010), 157. doi: 10.1111/j.1365-2478.2009.00805.x. Google Scholar [14] S. Trickett, L. Burroughs, A. Milton, L. Walton and R. Dack, Rank-reduction-based trace interpolation,, 80th Annual meeting, (2010). doi: 10.1190/1.3513645. Google Scholar [15] V. Oropeza and M. Sacchi, Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis,, Geophysics, 76 (2011). doi: 10.1190/1.3552706. Google Scholar [16] R. Vautard and M. Ghil, Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series,, Physica D, 35 (1989), 395. doi: 10.1016/0167-2789(89)90077-8. Google Scholar [17] E. J. Candes and B. Recht, Exact matrix completion via convex optimization,, Foundations of Computational Mathematics, 9 (2009), 717. doi: 10.1007/s10208-009-9045-5. Google Scholar [18] H. Ji, C. Liu, Z. Shen and Y. Xu, Robust video denoising using low rank matrix completion,, in Computer Vision and Pattern Recognition (CVPR), (2010). doi: 10.1109/CVPR.2010.5539849. Google Scholar [19] J.-F. Cai, E. J. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion,, SIAM Journal on Optimization, 20 (2008), 1956. doi: 10.1137/080738970. Google Scholar [20] S. Ma, D. Goldfarb and L. Chen, Fixed point and bregman iterative methods for matrix rank minimization,, Mathematical Programming: Series A and B, 128 (2011), 321. doi: 10.1007/s10107-009-0306-5. Google Scholar [21] J. Yang and X. Yuan, An inexact alternating direction method for trace norm regularized least squares problem,, Optimization Online, (2010). Google Scholar [22] R. Keshavan, A. Montanari and S.Oh, Matrix completion from a few entries,, IEEE Transactions on Information Theory, 56 (2010), 2980. doi: 10.1109/TIT.2010.2046205. Google Scholar [23] Y. Nesterov, A method of solving a convex programming problem with convergence rate o($1/k^2$),, Soviet Mathematics Doklady, 27 (1983), 372. Google Scholar [24] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM Journal on Imaging Sciences, (2009), 183. doi: 10.1137/080716542. Google Scholar [25] K.-C. Toh and S. Yun, An advanced proximal gradient algorithm for nuclear norm regularized linear least squares problems,, Pac. J. Optim., 6 (2010), 615. Google Scholar [26] Z. Wen, W. Yin and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm,, Mathematical Programming Computation, 4 (2012), 333. doi: 10.1007/s12532-012-0044-1. Google Scholar [27] H. Schaeffer and S. Osher, A low patch-rank interpretation of texture,, SIAM J. Imaging Sci., 6 (2013), 226. doi: 10.1137/110854989. Google Scholar [28] Y. Hua, Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,, IEEE Transactions on Signal Processing, 40 (1992), 2267. Google Scholar
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