2-D Seismic Data Reconstruction via Truncated Nuclear Norm Regularization

Abstract

Rank-reduction (RR) methods have been widely applied for reconstructing seismic data. The popular convex relaxation formulation of RR is nuclear norm minimization (NNM), which is capitalized on its convexity. Consequently, global optimization can be effectively achieved with NNM method. However, NNM minimizes the summation of all singular values and is therefore not equivalent to the minimum of the rank function; in fact, the rank of the matrix is equal to the number of nonzero singular values. Thus, NNM is not a good approximation to the original RR problem, which affects the reconstruction performance. The truncated nuclear norm regularization (TNNR) method only minimizes the smallest min(m, n) - r singular values (m, n is the size of matrix and r is the rank), yielding more accurate approximations to the rank function in real applications. In this article, we propose a new method for seismic reconstruction via TNNR based on the texture patch matrix, and further provide a strategy to estimate r, which can efficiently reduce the cost of computation. Numerical experiments performed on both synthetic and real data indicate that the reconstruction quality based on our proposed algorithm is better than that based on the singular value thresholding algorithm and the low-rank factorization model (low-rank matrix fitting).