Robust Seismic Image Interpolation with Mathematical Morphological Constraint.

Seismic image interpolation is a currently popular research subject in modern reflection seismology. The interpolation problem is generally treated as a process of inversion. Under the compressed sensing framework, various sparse transformations and low-rank constraints based methods have great performances in recovering irregularly missing traces. However, in the case of regularly missing traces, their applications are limited because of the strong spatial aliasing energies. In addition, the erratic noise always poses a serious impact on the interpolation results obtained by the sparse transformations and low-rank constraints-based methods. This is because the erratic noise is far from satisfying the statistical assumption behind these methods. In this study, we propose a mathematical morphology-based interpolation technique, which constrains the morphological scale of the model in the inversion process. The inversion problem is solved by the shaping regularization approach. The mathematical morphological constraint (MMC)-based interpolation technique has a satisfactory robustness to the spatial aliasing and erratic energies. We provide a detailed algorithmic framework and discuss the extension from 2D to higher dimensional version and the back operator in the shaping inversion. A group of numerical examples demonstrates the successful performance of the proposed technique.