Seislet-based morphological component analysis using scale-dependent exponential shrinkage

Conclusion and discussion

We have developed a seislet-based MCA method for seismic data processing. PWD filter can be utilized to estimate the slopes of seismic data. An exponential shrinkage function is introduced to diversify the capability of sparsity-promoting shaping operator. The proposed seislet-based MCA using scaled-dependent shaping regularization is promising in the application to seismic trace interpolation, and multiple removal. The numerical results reveal that the exponential shrinkage operator in sparsity-promoting shaping regularization plays an extremely important role in successful seislet-based MCA separation, superior to many existing thresholding operators. The additional parameter $p$ provides us more flexibility for approximating many existing shrinkage operators to achieve better separation performance. Meanwhile, it is free of non-differential singularity and unifies many existing shrinkage operators.

Seislet-MCA using PWD-based dip estimation is of special physical meaning for geophysical data in seismic processing, while the sparse dictionaries reported in Starck et al. (2004) are useful in image processing but lacking seismic attributes. However, a computational expensive optimization problem using least-squares minimization, which is not involved in the method of Starck et al. (2004), has to be solved to estimate the slope fields before applying our seislet-based method. Besides the computational expensive slope estimation, the proposed method is very efficient for interpolation and separation. The method fails to interpolate the missing traces when the random decimating rate is larger than 70% for 2D seismic data, which honors the necessity of high dimensional data reconstruction using 3D seislet transform. Although numerically working well, up to now we have no theoretical convergence proof of the nonlinear shaping algorithm, and it remains an open problem for future works.

Seislet-based morphological component analysis using scale-dependent exponential shrinkage