Five-Dimensional Seismic Reconstruction Using Parallel Square Matrix Factorization

Seismic data acquired by geophones are processed to estimate images of the earth's interior that are used to explore, develop, and monitor resources and to study the shallow structure of the crust for geological, environmental, and geotechnical purposes. These multidimensional data sets are often irregularly sampled and incomplete in the so-called midpoint and offset acquisition coordinates. Multidimensional seismic data reconstruction can be viewed as a low-rank matrix or tensor completion problem. In this paper, we introduce a fast and efficient low-rank tensor completion algorithm named parallel square matrix factorization (PSMF) and adopt it to reconstruct seismic data in the typical seismic data processing coordinates: frequency, midpoint, and offset. For each frequency slice, we establish a tensor minimization model composed of a low-rank constrained term and a data misfit term. Then we adopt the PSMF algorithm for the recovery of the missing samples. In the PSMF method, we avoid using unbalanced "long strip" matrices that result from conventional tensor unfolding. Instead, the tensor is unfolded into almost square or square matrices that are low rank. We also compare the proposed PSMF method with other completion methods. Experiments via synthetic data and field data sets validate the effectiveness of the proposed algorithm.