Iterative Double Laplacian-Scaled Low-Rank Optimization for Under-Sampled and Noisy Signal Recovery

Abstract

Recovering signal from under-sampled and erratic noise-corrupted seismic data is indeed a challenging task because of its difficulty in simultaneous modeling of erratic noise and missing signal. Assuming that the recorded data are the superposition of low-rank and sparse components, many related works have been reported using a hybrid rank-sparsity constraint. Those published works typically detect the rank and erratic noise using empirical and global thresholds, which often fail to well characterize the varying sparsity and easily cause biased estimation in case of their nonstationary distribution. We propose an iterative double Laplacian-scaled low-rank optimization to adaptively select the sparsity and rank regularizer parameters for robust signal recovery. Comparing with the published approaches with global threshold, Laplacian-scaled mixture, which is obtained by multiplying Laplacian variable with a Gamma variable, is used to locally model the sparsity of erratic noise and the low-rank feature of signal. Then, the expectation–maximization (EM) algorithm is used to transform the Laplacian-scaled mixture problem into a localized reweighted $\ell _{1}$ minimization scheme. The weighted coefficient appearing in its EM solver provides a variable constraint to locally address the rank and erratic noise, and hence, their regularizer parameters can dynamically reflect the different importance of those coefficients. We tested the effectiveness of the proposed method using under-sampled synthetic and field data that are corrupted by erratic noise and used other state-of-the-art methods as comparisons. The results showed that more exact estimations of the signal and erratic noise can be obtained using the proposed method.