Orthogonal dictionary learning based on l 4-Norm maximisation for seismic data interpolation

Due to geological conditions, acquisition environment, and economic restrictions, acquired seismic data are often incomplete and irregularly distributed, and this affects subsequent migration imaging and inversion. Sparse constraint-based methods are widely used for seismic data interpolation, including fixed-base transform and dictionary learning. Fixed-base transform methods are fast and simple to implement, but the basis function needs to be pre-selected. The dictionary learning method is more adaptive, and provides a means of learning the sparse representation from corrupted data. K-singular value decomposition (K-SVD) is a classical dictionary learning method that combines sparse coding and dictionary updating iteratively. However, the dictionary atoms are updated column-by-column, leading to high computational complexity due to long SVD calculation times. In this study, we evaluated the dictionary learning method via l 4-norm maximisation using an orthogonal dictionary, which is different from the traditional l 0-norm or l 1-norm minimisation, and interpolated the missing traces in the projection onto convex sets (POCS) framework. The optimal objection function is convex, but can be solved using a simple and efficient Matching, Stretching and Projection (MSP) algorithm, which greatly reduces the dictionary learning time. Numerical experiments using synthetic and field data demonstrate the effectiveness of the proposed method.