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 propose 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 to calculate 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. We present synthetic and field data examples to illustrate the method's effectiveness for denoising and interpolating 4D and 5D seismic data with randomly missing traces.
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.
Rank-constrained seismic data interpolation methods have been used to cope with spatial sampling deficiencies, but some fundamental aspects are often neglected. Understanding their underlying features is the first step for developing new solutions to overcome existing limitations. We intend to provide an intuitive description regarding low-rank strategies using the similarities between irregular samplings and noise in terms of their eigenimage representation. The interpretation of data recovery as iterative denoising helps to clarify how the traces are retrieved and the role of the rank. To emphasize either signal recovery or denoising along with the iterations, we explore non-linear versions of the decreasing weighting factor that drives the reinsertion of original samples. This type of weighting factor shows superior denoising results when raised to an integer power. Simple synthetic numerical examples illustrate the mechanics of low-rank procedures and their responses to different parameters. We also show 3D field data examples from a land survey to demonstrate the robustness of reduced-rank approaches.
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