Seismic denoising via truncated nuclear norm minimization

Because there are many similar geologic structures underground, seismic profiles have an abundance of self-repeating patterns. Thus, we can divide a seismic profile into groups of blocks with a similar seismic structure. The matrix formed by stacking together similar blocks in each group should be of low rank. Hence, we can transfer the seismic denoising problem to a series of low-rank matrix approximation (LRMA) problems. The LRMA-based model commonly adopts the nuclear norm as a convex substitute of the rank of a matrix. However, the nuclear norm minimization (NNM) shrinks the different rank components equally and may cause some biases in practice. The recently introduced truncated nuclear norm (TNN) has been proven to more accurately approximate the rank of a matrix, which is given by the sum of the set of the smallest singular values. Based on this, we have adopted a novel denoising method using truncated nuclear norm minimization (TNNM). The objective function of this method consists of two terms, the F-norm data fidelity and a TNN regularization. We develop an efficient two-step iterative algorithm to solve this objective function. Then, we apply the proposed TNNM algorithm to groups of blocks with similar seismic structure, and we aggregate all resulting denoised blocks to get the denoised seismic data. We update the denoised results during each iteration to gradually attenuate the heavy noise. Numerical experiments demonstrate that, compared with f-x deconvolution and the curvelet- and NNM-based methods, TNNM not only attenuates noise more effectively even when the signal-to-noise ratio is as low as -10 dB and seismic data have complex structures, but it also accurately preserves the seismic structures without inducing Gibbs artifacts.