Seismic Data Reconstruction With Low-rank Tensor Optimization




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Seismic data recorded in the field often has gaps due to missing or failed receivers or aperture restrictions and may be contaminated by external noise. Reconstruction is the process of completing missing data and removing noise. Multi-dimensional seismic data, for example in 3, 4, or 5D, can be efficiently stored in a tensor or multi-dimensional array. Low-rank tensor optimization is a model used to reconstruct tensor data under the assumption that the underlying data has low rank. Data has low rank when it has redundant rows or columns, causing the singular values to decay at a rapid rate. Because minimizing rank is NP-hard, a relaxation of rank can be used such as the tensor nuclear norm (TNN), derived from the tensor singular value decomposition (tSVD). The alternating direction method of multipliers (ADMM) effectively solves the TNN model in which the sum of singular values is minimized. ADMM splits the minimization problem into smaller subproblems. The combination of the ADMM method and TNN model is referred to as TNN-ADMM and is useful for reconstruc- tion of missing data. Exploiting the conjugate symmetry of the multi-dimensional Fourier transform (the most expensive part of the tSVD algorithm) reduces the runtime of the tSVD algorithm for real- valued order-p tensors by approximately 50%. The relation between the tSVD of a tensor and the SVD of a corresponding block-diagonal matrix reveals how the singular values of the tensor and matrix change as the orientation of the tensor changes and provides evidence for the success of the most-square orientation when used for low rank data reconstruction. For seismic data, the most-square orientation has frontal faces formed over the spatial dimen- sions, so the tensor contains more redundancies than pairing a spatial dimension with time. On real data reconstruction examples, TNN-ADMM outperforms two other data completion methods, projection onto convex sets and multi-channel singular spectrum analysis (MSSA), with less error and 10-1000× faster runtime. In exploration seismology, an initial baseline survey informs decisions to produce a region, and monitor surveys conducted during production provide updated subsurface information. The time-lapse difference between the baseline and monitor surveys reveals changes in the Earth due to production. Non-repeatability issues, such as inconsistent receiver locations, negatively impact one’s ability to accurately identify time-lapse changes. If receiver locations are regularized onto a common grid, then the baseline and monitor surveys can be compared. The resulting tensors are incomplete due to the potential for grid blocks to not contain receivers, so we apply TNN-ADMM to each data tensor to successfully fill in missing data, and a time-lapse difference can be computed between the reconstructed tensors. The unconstrained formulation of TNN-ADMM simultaneously completes and denoises data. The convergence analysis of this method proves that the iterative solution converges to a local minimum, provided that the step size parameter is greater than one. The convergence proof requires new properties of the Frobenius norm for tensors, as well as Hadamard (entry- wise) product properties. On a synthetic problem unconstrained TNN-ADMM outperforms MSSA with 18%-27% less error and 10× faster runtime.



Mathematics, Geophysics