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# Two-dimensional seismic data reconstruction using patch tensor completion

• * Corresponding author: Lihua Fu
• 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.

Mathematics Subject Classification: 15A29, 15A83.

 Citation: • • Figure 1.  Illustration of texture-patch pre-transformation and patch tensor pre-transformation

Figure 2.  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)

Figure 3.  Illustration of unfolding a 3D tensor into a matrix and inversely folding a matrix into a 3D tensor

Figure 4.  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

Figure 5.  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

Figure 6.  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

Figure 7.  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

Figure 8.  Reconstruction of post-stack seismic data. (a) Original data. (b) Corrupted data with 50% randomly missing traces

Figure 9.  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

Figure 10.  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

Figure 11.  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

Figure 12.  The reconstructed SNR for post-stack seismic data with the increase of iteration numbers in different patch size by LRTC

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