Comparison of sparsity-constrained regularization methods for denoising and interpolation

Missing trace reconstruction is an ongoing challenge in seismic processing due to incomplete acquisition schemes and irregular grids. Noise is also a concern because it is naturally present in acquired seismic data through several mechanisms such as natural noise and equipment noise. Both problems need to be adequately addressed, especially because they negatively affect several important processing steps such as migration. While there are many approaches to solve these problems, most of the recent research on the subject focuses on transform domain approaches. These approaches commonly use sparsity-constrained inversion, i.e., one assumes that the signal to be estimated is sparse in a transform domain, in order to obtain a reasonable solution. Several formulations for denoising and missing trace reconstruction have been proposed based on the compressive sensing (CS) framework, which states that sparse signals can be recovered from a highly incomplete set of measurements. In particular, a specific kind of constraint, called synthesis approach, has been widely used in geophysical problems. The analysis approach, which can be considered as the synthesis’ dual problem, is an alternative for sparsity-constrained inversion. Although less popular than the synthesis problem, the analysis approach is more effective in several problems, such as denoising of natural images. In this paper, we compare the analysis and synthesis approaches as sparsity constraints for the denoising of seismic images and missing trace reconstruction. Our experiments show that the analysis approach can yield more accurate results than the synthesis approach for both problems, which makes it a viable approach for sparsity-constrained inversion for geophysical problems and should be considered along with the synthesis approach.