Prestack seismic data reconstruction and denoising by orientation-dependent tensor decomposition

Multidimensional seismic data reconstruction and denoising can be achieved by assuming noiseless and complete data as low-rank matrices or tensors in the frequency-space domain. We have adopted a simple and effective approach to interpolate prestack seismic data that explores the low-rank property of multidimensional signals. The orientation-dependent tensor decomposition represents an alternative to multilinear algebraic schemes. Our method does not need to perform any explicit matricization, only requiring calculation of the so-called covariance matrix for one of the spatial dimensions. The elements of such a matrix are the inner products between the lower dimensional tensors in a convenient direction. The eigenvalue decomposition of the covariance matrix provides the eigenvectors for the reduced-rank approximation of the data tensor. This approximation is used for recovery and denoising, iteratively replacing the missing values. Synthetic and field data examples illustrate the method's effectiveness for denoising and interpolating 4D and 5D seismic data with randomly missing traces.