# American Institute of Mathematical Sciences

December  2020, 14(6): 985-1000. doi: 10.3934/ipi.2020052

## Two-dimensional seismic data reconstruction using patch tensor completion

 1 School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China 2 School of Computer, Central China Normal University, Wuhan 430079, China

* Corresponding author: Lihua Fu

Received  October 2019 Revised  May 2020 Published  December 2020 Early access  August 2020

Seismic data are often undersampled owing to physical or financial limitations. However, complete and regularly sampled data are becoming increasingly critical in seismic processing. In this paper, we present an efficient two-dimensional (2D) seismic data reconstruction method that works on texture-based patches. It performs completion on a patch tensor, which folds texture-based patches into a tensor. Reconstruction is performed by reducing the rank using tensor completion algorithms. This approach differs from past methods, which proceed by unfolding matrices into columns and then applying common matrix completion approaches to deal with 2D seismic data reconstruction. Here, we first re-arrange the seismic data matrix into a third-order patch tensor, by stacking texture-based patches that are divided from seismic data. Then, the seismic data reconstruction problem is formulated into a low-rank tensor completion problem. This formulation avoids destroying the spatial structure, and better extracts the underlying useful information. The proposed method is efficient and gives an improved performance compared with traditional approaches. The effectiveness of our patch tensor-based framework is validated using two classical tensor completion algorithms, low-rank tensor completion (LRTC), and the parallel matrix factorization algorithm (TMac), on both synthetic and field data experiments.

Citation: Qun Liu, Lihua Fu, Meng Zhang, Wanjuan Zhang. Two-dimensional seismic data reconstruction using patch tensor completion. Inverse Problems and Imaging, 2020, 14 (6) : 985-1000. doi: 10.3934/ipi.2020052
##### References:
 [1] D. Gemechu, H. Yuan and J. Ma, Random noise attenuation using an improved anisotropic total variation regularization, J. Appl. Geophys., 144 (2017), 173-187.  doi: 10.1016/j.jappgeo.2017.07.003. [2] Z. Hua, X. Chen, H. Li, G. Huang and C. Xiao, 3D seismic data de-noising approach based on Curvelet transform, Oil Geophysical Prospecting, 52 (2017), 226-232. [3] S. Jahanjooy, R. Nikrouz and N. Mohammed, A faster method to reconstruct seismic data using anti-leakage Fourier transform, J. Geophys. Eng., 13 (2016), 86-95.  doi: 10.1088/1742-2132/13/1/86. [4] L. Ji, P. Musialski, P. Wonka and J. Ye, Tensor completion for estimating missing values in visual data, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 208-220. [5] M. M. N. Kabir and D. J. Verschuur, Restoration of missing offsets by parabolic Radon transform, Geophysical Prospecting, 43 (1995), 347-368. [6] T. G. Kolda and B. W. Bader, Tensor decompositions and applications, Siam Review, 51 (2009), 455-500.  doi: 10.1137/07070111X. [7] D. Kong and Z. Peng, Seismic random noise attenuation using shearlet and total generalized variation, J. Geophys. Eng., 12 (2015), 1024-1035.  doi: 10.1088/1742-2132/12/6/1024. [8] N. Kreimer and M. D. Sacchi, A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation, Geophysics, 77 (2012), V113–V122. doi: 10.1190/geo2011-0399.1. [9] R. Kumar, C. D. Silva, O. Akalin, A. Y. Aravkin and F. J. Herrmann, Efficient matrix completion for seismic data reconstruction, Geophysics, 80 (2015), V97–V114. doi: 10.1190/geo2014-0369.1. [10] C. Li, G. Liu, Z. Hao, S. Zu, F. Mi and X. Chen, Multidimensional seismic data reconstruction using frequency-domain adaptive prediction-error filter, IEEE Transactions on Geoscience & Remote Sensing, 56 (2018), 2328-2336.  doi: 10.1109/TGRS.2017.2778196. [11] Z. Liu, J. Ma and X. Yong, Line survey joint denosing via low rank minimization, Geophysics, 84 (2018), 1-74. [12] C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin and S. Yan, Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization, in IEEE Conference on Computer Vision and Pattern Recognition, (2-17), 5249–5257. doi: 10.1109/CVPR.2016.567. [13] J. Ma, Three-dimensional irregular seismic data reconstruction via low-rank matrix completion, Geophysics, 78 (2013), V181–V192. doi: 10.1190/geo2012-0465.1. [14] M. Naghizadeh and K. A. Innanen, Seismic data interpolation using a fast generalized Fourier transform, Geophysics, 76 (2011), V1–V10. doi: 10.1190/1.3511525. [15] M. Naghizadeh and M. D. Sacchi, F–x adaptive seismic-trace interpolation, Geophysics, 74 (2009), V9–V16. doi: 10.1190/1.3008547. [16] M. Naghizadeh and M. D. Sacchi, Seismic data reconstruction using multidimensional prediction filters, Geophysical Prospecting, 58 (2010), 157-173.  doi: 10.3997/2214-4609.20148097. [17] V. Oropeza and M. Sacchi, Simultaneous seismic data denosing and reconstruction via multichannel singular spectrum analysis, Geophysics, 76 (2011), V25–V32. [18] M. D. Sacchi, Fx singular spectrum analysis, in Cspg Cseg Cwls Convention, (2009), 392–395. [19] 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, 61 (2013), 973-997.  doi: 10.1111/1365-2478.12050. [20] S. Spitz, Seismic trace interpolation in the F–X domain, Geophysics, 56 (1991), 785-794. [21] K. C. Toh and S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems, Pacific J. Optim., 6 (2010), 615-640. [22] D. Trad, T. Ulrych and M. Sacchi, Latest views of the sparse Radon transform, Geophysics, 68 (2003), 386-399.  doi: 10.1190/1.1543224. [23] S. Trickett, F–xy Cadzow noise suppression, Seg. Technical Program Expanded Abstracts, 27 (2008), 2586-2590.  doi: 10.1190/1.3063880. [24] S. Trickett, L. Burroughs, A. Milton, L. Walton and R. Dack, Rank-reduction-based trace interpolation, in Seg. Technical Program Expanded Abstracts, (2010), 1989–1992. doi: 10.1190/1.3513645. [25] S. Trickett, L. Burroughs and A. Milton, Interpolation using Hankel tensor completion, in Seg. Technical Program Expanded Abstracts, (2013), 3634–3638. doi: 10.1190/segam2013-0416.1. [26] L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, 31 (1966), 279-311.  doi: 10.1007/BF02289464. [27] J. Wang, J. Ma and B. Han, Seismic data reconstruction via weighted nuclear-norm minimization, Inverse Probl. Sci. Eng., 23 (2015), 277-291.  doi: 10.1080/17415977.2014.890616. [28] Z. Wen, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Math. Program. Comput., 4 (2012), 333-361.  doi: 10.1007/s12532-012-0044-1. [29] Y. Xu, R. Hao, W. Yin and Z. Su, Parallel matrix factorization for low-rank tensor completion, Inverse Probl. Imaging, 9 (2017), 601-624.  doi: 10.3934/ipi.2015.9.601. [30] Y. Yang, J. Ma and S. Osher, Seismic data reconstruction via matrix completion, Inverse Probl. Imaging, 7 (2013), 1379-1392.  doi: 10.3934/ipi.2013.7.1379. [31] Y. Q. Zhang and W. K. Lu, 2D and 3D prestack seismic data regularization using an accelerated sparse time-invariant Radon transform, Geophysics, 79 (2014), V165–V177. doi: 10.1190/geo2013-0286.1. [32] H. Zhang, H. Yang, H. Li, G. Huang and H. Yang, Random noise attenuation of non-uniformly sampled 3D seismic data along two spatial coordinates using non-equispaced curvelet transform, J. Appl. Geophys., 151 (2018), 221-233.  doi: 10.1016/j.jappgeo.2018.02.018. [33] P. Zhou and J. Feng, Outlier-robust tensor PCA, in IEEE Conference on Computer Vision and Pattern Recognition, (2017), 3938–3946. doi: 10.1109/CVPR.2017.419. [34] P. Zhou, C. Lu, Z. Lin and C. Zhang, Tensor factorization for low-rank tensor completion, IEEE Trans. Image Process., 27 (2017), 1152-1163.  doi: 10.1109/TIP.2017.2762595.

show all references

##### References:
 [1] D. Gemechu, H. Yuan and J. Ma, Random noise attenuation using an improved anisotropic total variation regularization, J. Appl. Geophys., 144 (2017), 173-187.  doi: 10.1016/j.jappgeo.2017.07.003. [2] Z. Hua, X. Chen, H. Li, G. Huang and C. Xiao, 3D seismic data de-noising approach based on Curvelet transform, Oil Geophysical Prospecting, 52 (2017), 226-232. [3] S. Jahanjooy, R. Nikrouz and N. Mohammed, A faster method to reconstruct seismic data using anti-leakage Fourier transform, J. Geophys. Eng., 13 (2016), 86-95.  doi: 10.1088/1742-2132/13/1/86. [4] L. Ji, P. Musialski, P. Wonka and J. Ye, Tensor completion for estimating missing values in visual data, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 208-220. [5] M. M. N. Kabir and D. J. Verschuur, Restoration of missing offsets by parabolic Radon transform, Geophysical Prospecting, 43 (1995), 347-368. [6] T. G. Kolda and B. W. Bader, Tensor decompositions and applications, Siam Review, 51 (2009), 455-500.  doi: 10.1137/07070111X. [7] D. Kong and Z. Peng, Seismic random noise attenuation using shearlet and total generalized variation, J. Geophys. Eng., 12 (2015), 1024-1035.  doi: 10.1088/1742-2132/12/6/1024. [8] N. Kreimer and M. D. Sacchi, A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation, Geophysics, 77 (2012), V113–V122. doi: 10.1190/geo2011-0399.1. [9] R. Kumar, C. D. Silva, O. Akalin, A. Y. Aravkin and F. J. Herrmann, Efficient matrix completion for seismic data reconstruction, Geophysics, 80 (2015), V97–V114. doi: 10.1190/geo2014-0369.1. [10] C. Li, G. Liu, Z. Hao, S. Zu, F. Mi and X. Chen, Multidimensional seismic data reconstruction using frequency-domain adaptive prediction-error filter, IEEE Transactions on Geoscience & Remote Sensing, 56 (2018), 2328-2336.  doi: 10.1109/TGRS.2017.2778196. [11] Z. Liu, J. Ma and X. Yong, Line survey joint denosing via low rank minimization, Geophysics, 84 (2018), 1-74. [12] C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin and S. Yan, Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization, in IEEE Conference on Computer Vision and Pattern Recognition, (2-17), 5249–5257. doi: 10.1109/CVPR.2016.567. [13] J. Ma, Three-dimensional irregular seismic data reconstruction via low-rank matrix completion, Geophysics, 78 (2013), V181–V192. doi: 10.1190/geo2012-0465.1. [14] M. Naghizadeh and K. A. Innanen, Seismic data interpolation using a fast generalized Fourier transform, Geophysics, 76 (2011), V1–V10. doi: 10.1190/1.3511525. [15] M. Naghizadeh and M. D. Sacchi, F–x adaptive seismic-trace interpolation, Geophysics, 74 (2009), V9–V16. doi: 10.1190/1.3008547. [16] M. Naghizadeh and M. D. Sacchi, Seismic data reconstruction using multidimensional prediction filters, Geophysical Prospecting, 58 (2010), 157-173.  doi: 10.3997/2214-4609.20148097. [17] V. Oropeza and M. Sacchi, Simultaneous seismic data denosing and reconstruction via multichannel singular spectrum analysis, Geophysics, 76 (2011), V25–V32. [18] M. D. Sacchi, Fx singular spectrum analysis, in Cspg Cseg Cwls Convention, (2009), 392–395. [19] 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, 61 (2013), 973-997.  doi: 10.1111/1365-2478.12050. [20] S. Spitz, Seismic trace interpolation in the F–X domain, Geophysics, 56 (1991), 785-794. [21] K. C. Toh and S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems, Pacific J. Optim., 6 (2010), 615-640. [22] D. Trad, T. Ulrych and M. Sacchi, Latest views of the sparse Radon transform, Geophysics, 68 (2003), 386-399.  doi: 10.1190/1.1543224. [23] S. Trickett, F–xy Cadzow noise suppression, Seg. Technical Program Expanded Abstracts, 27 (2008), 2586-2590.  doi: 10.1190/1.3063880. [24] S. Trickett, L. Burroughs, A. Milton, L. Walton and R. Dack, Rank-reduction-based trace interpolation, in Seg. Technical Program Expanded Abstracts, (2010), 1989–1992. doi: 10.1190/1.3513645. [25] S. Trickett, L. Burroughs and A. Milton, Interpolation using Hankel tensor completion, in Seg. Technical Program Expanded Abstracts, (2013), 3634–3638. doi: 10.1190/segam2013-0416.1. [26] L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, 31 (1966), 279-311.  doi: 10.1007/BF02289464. [27] J. Wang, J. Ma and B. Han, Seismic data reconstruction via weighted nuclear-norm minimization, Inverse Probl. Sci. Eng., 23 (2015), 277-291.  doi: 10.1080/17415977.2014.890616. [28] Z. Wen, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Math. Program. Comput., 4 (2012), 333-361.  doi: 10.1007/s12532-012-0044-1. [29] Y. Xu, R. Hao, W. Yin and Z. Su, Parallel matrix factorization for low-rank tensor completion, Inverse Probl. Imaging, 9 (2017), 601-624.  doi: 10.3934/ipi.2015.9.601. [30] Y. Yang, J. Ma and S. Osher, Seismic data reconstruction via matrix completion, Inverse Probl. Imaging, 7 (2013), 1379-1392.  doi: 10.3934/ipi.2013.7.1379. [31] Y. Q. Zhang and W. K. Lu, 2D and 3D prestack seismic data regularization using an accelerated sparse time-invariant Radon transform, Geophysics, 79 (2014), V165–V177. doi: 10.1190/geo2013-0286.1. [32] H. Zhang, H. Yang, H. Li, G. Huang and H. Yang, Random noise attenuation of non-uniformly sampled 3D seismic data along two spatial coordinates using non-equispaced curvelet transform, J. Appl. Geophys., 151 (2018), 221-233.  doi: 10.1016/j.jappgeo.2018.02.018. [33] P. Zhou and J. Feng, Outlier-robust tensor PCA, in IEEE Conference on Computer Vision and Pattern Recognition, (2017), 3938–3946. doi: 10.1109/CVPR.2017.419. [34] P. Zhou, C. Lu, Z. Lin and C. Zhang, Tensor factorization for low-rank tensor completion, IEEE Trans. Image Process., 27 (2017), 1152-1163.  doi: 10.1109/TIP.2017.2762595.
Illustration of texture-patch pre-transformation and patch tensor pre-transformation
Singular value plots for the mode-1, mode-2, and mode-3 unfolding matrices of patch tensor of the field data. (a) field data. (b) the singular value plots for the mode-1 unfolding matrix of the patch tensor of field data (blue) and field data with missing columns (red star). (c) the singular value plots for the mode-2 unfolding matrix of the patch tensor of field data (blue) and field data with missing columns (red star). (d) the singular value plots for the mode-3 unfolding matrix of the patch tensor of field data (blue) and field data with missing columns (red star)
Illustration of unfolding a 3D tensor into a matrix and inversely folding a matrix into a 3D tensor
Synthetic seismic data recovery. (a) Original seismic data. (b) Corrupted data with 50% randomly missing traces. (c) Recovered signal via APG. (d) Recovered signal via LMaFit. (e)Recovered signal via LRTC. (f) Recovered signal via TMac
Comparison of the 9th trace taken from original synthetic data and reconstructed data processed by different methods: (a) APG, (b) LMaFit, (c) LRTC, and (d) TMac
Comparison of f-k spectra of synthetic data. (a) Original data. (b) Corrupted data. (c) Recovered data by APG. (d) Recovered data using the LMaFit algorithm. (e) Recovered data using the LRTC algorithm. (f) Recovered data using the TMac algorithm
Reconstruction results for the APG, LMaFit, LRTC, and TMac algorithms. (a) Signal-to-noise-ratio (SNR) versus the sampling ratio. (b) Computational time versus the sampling ratio
Reconstruction of post-stack seismic data. (a) Original data. (b) Corrupted data with 50% randomly missing traces
Comparison of the 58th trace taken from the original post-stack data and reconstructed data and the differences between them. Panels (a), (c), (e), and (g) show this reconstructed single trace using the APG, LMaFit, LRTC, and TMac algorithms, respectively. Panels (b), (d), (f), and (h) show the differences between the original single trace and the reconstructed single traces when using the APG, LMaFit, LRTC, and TMac algorithms, respectively
Comparison of f-k spectra for post-stack seismic data. (a) Original data. (b) Corrupted data. (c) Data recovered using the APG algorithm. (d) Data recovered using the LMaFit algorithm. (e) Data recovered using the LRTC algorithm. (f) Data recovered using the TMac algorithm
Reconstruction results of the post-stack seismic data. (a) The SNR versus the sampling ratio for the APG, LMaFit, LRTC, and TMac algorithms. (b) Amplitude spectrum comparison of the 58th single trace. The spectrums of the original data, data from the LMaFit and data from TMac are displayed in black, red, and blue lines, respectively
The reconstructed SNR for post-stack seismic data with the increase of iteration numbers in different patch size by LRTC
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