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

Abstract

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.

High‐Resolution Mantle Transition Zone Imaging Using Multi‐Dimensional Reconstruction of SS Precursors

“…(2021) introduce the approximation of X using the robust damped rank‐reduction operator on the Hankel matrix P by $$$\widetilde{\mathbf{X}}[i]=\frac{1}{g}\sum\limits _{j=0}^{g-1}{Z}_{\tau }\left({U}_{1}^{\mathbf{P}}{{\Sigma }}_{1}^{\mathbf{P}}\beta {\left({V}_{1}^{\mathbf{P}}\right)}^{t}\right)\left[i-j\right],$$$ where the term $${U}_{1}^{\mathbf{P}}{{\Sigma }}_{1}^{\mathbf{P}}\beta {\left({V}_{1}^{\mathbf{P}}\right)}^{t}$$ corresponds to the approximation of X using the damped rank‐reduction operator (Chen, Zhang, et al., 2016). $${{\Sigma }}_{1}^{\mathbf{P}}$$ corresponds to the first diagonal matrix of the approximation signal using the traditional truncated singular value decomposition (Chen, Zhang, et al., 2016; Innocent Oboué et al., 2021; Oropeza & Sacchi, 2011). It contains the largest singular values of the block Hankle matrix P .…”

“…Symbol β indicates the damping operator with its mathematical formula given by β = I Σ P 1 ) d α d , where α represents the maximum element of the second diagonal matrix of the approximation signal using the traditional TSVD and d corresponds to the damping factor. The formula U P 1 Σ P 1 β V P 1 ) t can be generalized to deal with multi-dimensional (3D, 4D, and 5D) reconstruction problems using the robust damped rank-reduction method since P corresponds to the Hankel matrix with its structure determined by the size of the input data Innocent Oboué et al, 2021). Finally, the term 1 g ∑ moving-average (STMA) operator (Innocent Oboué et al, 2021), which combines the soft thresholding Z τ (⋅) (Donoho, 1995) and the moving-average & Oppenheim, 1989) operators to further improve the SNR.…”

“…The formula U P 1 Σ P 1 β V P 1 ) t can be generalized to deal with multi-dimensional (3D, 4D, and 5D) reconstruction problems using the robust damped rank-reduction method since P corresponds to the Hankel matrix with its structure determined by the size of the input data Innocent Oboué et al, 2021). Finally, the term 1 g ∑ moving-average (STMA) operator (Innocent Oboué et al, 2021), which combines the soft thresholding Z τ (⋅) (Donoho, 1995) and the moving-average & Oppenheim, 1989) operators to further improve the SNR. g denotes the window size, a key parameter used in the STMA function.…”

“…Compared to its traditional counterpart, the RDRR method is a more effective tool for interpolating noisecorrupted and highly decimated seismic data. More details of the RDRR method can be found in Innocent Oboué et al (2021).…”

Rank-constrained seismic data interpolation and denoising

“…Low-rank methods constitute another important category, which has been largely investigated in recent years (Trickett et al, 2010;Oropeza and Sacchi, 2011;Kreimer and Sacchi, 2012;Ely et al, 2015;Carozzi and Sacchi, 2021;Oboué et al, 2021;Cavalcante and Porsani, 2022). They assume that regular seismic data can be represented as low-rank matrices or tensors.…”