# 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:

show all references

##### References:
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
 [1] Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems and Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601 [2] Yi Yang, Jianwei Ma, Stanley Osher. Seismic data reconstruction via matrix completion. Inverse Problems and Imaging, 2013, 7 (4) : 1379-1392. doi: 10.3934/ipi.2013.7.1379 [3] Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems and Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001 [4] Wei Wan, Weihong Guo, Jun Liu, Haiyang Huang. Non-local blind hyperspectral image super-resolution via 4d sparse tensor factorization and low-rank. Inverse Problems and Imaging, 2020, 14 (2) : 339-361. doi: 10.3934/ipi.2020015 [5] Juan Manuel Reyes, Alberto Ruiz. Reconstruction of the singularities of a potential from backscattering data in 2D and 3D. Inverse Problems and Imaging, 2012, 6 (2) : 321-355. doi: 10.3934/ipi.2012.6.321 [6] Ke Wei, Jian-Feng Cai, Tony F. Chan, Shingyu Leung. Guarantees of riemannian optimization for low rank matrix completion. Inverse Problems and Imaging, 2020, 14 (2) : 233-265. doi: 10.3934/ipi.2020011 [7] Joshua Hudson, Michael Jolly. Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations. Journal of Computational Dynamics, 2019, 6 (1) : 131-145. doi: 10.3934/jcd.2019006 [8] Thomas März, Andreas Weinmann. Model-based reconstruction for magnetic particle imaging in 2D and 3D. Inverse Problems and Imaging, 2016, 10 (4) : 1087-1110. doi: 10.3934/ipi.2016033 [9] Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001 [10] Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383 [11] Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669 [12] Simon Arridge, Pascal Fernsel, Andreas Hauptmann. Joint reconstruction and low-rank decomposition for dynamic inverse problems. Inverse Problems and Imaging, 2022, 16 (3) : 483-523. doi: 10.3934/ipi.2021059 [13] Yanfei Wang, Dmitry Lukyanenko, Anatoly Yagola. Magnetic parameters inversion method with full tensor gradient data. Inverse Problems and Imaging, 2019, 13 (4) : 745-754. doi: 10.3934/ipi.2019034 [14] Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas. Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations. Electronic Research Archive, 2021, 29 (3) : 2223-2247. doi: 10.3934/era.2020113 [15] Hiroshi Fujiwara, Kamran Sadiq, Alexandru Tamasan. Partial inversion of the 2D attenuated $X$-ray transform with data on an arc. Inverse Problems and Imaging, 2022, 16 (1) : 215-228. doi: 10.3934/ipi.2021047 [16] Adam Larios, Yuan Pei. Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data. Evolution Equations and Control Theory, 2020, 9 (3) : 733-751. doi: 10.3934/eect.2020031 [17] Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of borehole seismic data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022026 [18] Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems and Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052 [19] Shenglong Hu. A note on the solvability of a tensor equation. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021146 [20] Daniil Kazantsev, William M. Thompson, William R. B. Lionheart, Geert Van Eyndhoven, Anders P. Kaestner, Katherine J. Dobson, Philip J. Withers, Peter D. Lee. 4D-CT reconstruction with unified spatial-temporal patch-based regularization. Inverse Problems and Imaging, 2015, 9 (2) : 447-467. doi: 10.3934/ipi.2015.9.447

2020 Impact Factor: 1.639

## Metrics

• HTML views (251)
• Cited by (0)

• on AIMS