Robust damped rank-reduction method for simultaneous denoising and reconstruction of 5-D seismic data

We have developed a new method for simultaneous denoising and reconstruction of 5-D seismic data corrupted by random noise and missing traces. Several algorithms have been proposed for seismic data restoration based on rank-reduction methods. More recently, a damping operator has been introduced into the conventional truncated singular value decomposition (TSVD) formula to further remove residual noise, the presence of which disturbs the quality of the seismic results. Despite the success of the damped rank-reduction (DRR) method when the observed data have an extremely low signal-to-noise ratio (SNR), random noise is still a limiting factor for obtaining perfect quality of the result. Therefore, how to accurately solve the simultaneous denoising and reconstruction problem with high fidelity is still challenging. We assume that introducing only the damping operator into the TSVD formula is not enough to remove the random noise and restore the useful signal well. Here, by combining the soft thresholding operator and the moving-average filter, we first develop a new operator, which we call soft thresholding moving-average (STMA) operator. Then, by introducing the STMA operator into the DRR framework, we develop a new algorithm known as the robust damped rank-reduction (RDRR) method, which aims at mixing the advantages of the STMA operator and the damping operator. The STMA operator is applied to the Hankel matrix after damped truncated singular value decomposition (DTSVD) to better remove the residual noise. Examples of the proposed approach on synthetic and field 5-D seismic data demonstrate the better performance in terms of the visual examination and numerical test compared with the DRR approach. The proposed method aims at producing an effective low-rank filter and, thus, can perfectly enhance the SNR of the simultaneously denoised and reconstructed results with higher accuracy.