Enhancing 3-D Seismic Data Using the t-SVD and Optimal Shrinkage of Singular Value

We consider the three-dimensional (3-D) seismic data as a tensor data of size n ₁ × n ₂ × n ₃ contaminated with the white Gaussian noise. A new version of the tensor robust principal component analysis (TRPCA) is employed for denoising the 3-D seismic data. In the new TRPCA, the singular values are extracted using the optimal shrinkage method. We recover the low-rank matrices from noisy data by shrinkage of the singular values, in which the singular value thresholding in the Fourier domain is exploited to extract the low-rank component of the tensor. The algorithm is as follows. First, by assuming the incoherency conditions the whole tensor is modeled as a combination of a low-rank component and a sparse component. Second, the Fourier transform of the tensor is computed along the third dimension of the tensor, then the singular value decomposition (SVD) is computed in the Fourier domain, then the low-rank component is extracted by shrinkage of the singular values. Finally, the steps mentioned above are repeated until the Frobenius norm of the error matrix reaches the desired value. We evaluate the performance of the proposed method, which is called tensor optimal shrinkage of SVD based on the qualitative and quantitative measurements, and compare it with state-of-the-art methods such as iterative tensor singular value thresholding and 4-D block matching using different types of synthetic and real seismic data.