Low‐rank seismic data reconstruction and denoising by CUR matrix decompositions

ABSTRACT Low‐rank reconstruction methods assume that noiseless and complete seismic data can be represented as low‐rank matrices or tensors. Therefore, denoising and recovery of missing traces require a reduced‐rank approximation of the data matrix/tensor. To calculate such approximation, we explore the CUR matrix decompositions, which use actual columns and rows of the data matrix, instead of the costly singular vectors derived from singular value decomposition. By allowing oversampling columns and rows, CUR decompositions obviate the need for the exact rank. We evaluate three different procedures for randomly selecting columns and rows to obtain the CUR. Once the low‐rank approximation is estimated, data reconstruction is achieved by an iterative optimization scheme. To demonstrate the effectiveness of CUR matrix decompositions for multidimensional seismic data recovery, we present examples of 3D and 4D synthetic and field data. Results derived by CUR compare well to conventional eigenimage‐family methods.