Seismic data restoration via data-driven tight frame

Restoration/interpolation of missing traces plays a crucial role in the seismic data processing pipeline. Efficient restoration methods have been proposed based on sparse signal representation in a transform domain such as Fourier, wavelet, curvelet, and shearlet transforms. Most existing methods are based on transforms with a fixed basis. We considered an adaptive sparse transform for restoration of data with complex structures. In particular, we evaluated a data-driven tight-frame-based sparse regularization method for seismic data restoration. The main idea of the data-driven tight frame (TF) is to adaptively learn a set of framelet filters from the currently interpolated data, under which the data can be more sparsely represented; hence, the sparsity-promoting l(1)-norm (SPL1) minimization methods can produce better restoration results by using the learned filters. A split inexact Uzawa algorithm, which can be viewed as a generalization of the alternating direction of multiplier method (ADMM), was applied to solve the presented SPL1 model. Numerical tests were performed on synthetic and real seismic data for restoration of randomly missing traces over a regular data grid. Our experiments showed that our proposed method obtains the state-of-the-art restoration results in comparison with the traditional Fourier-based projection onto convex sets, the tight-frame-based method, and the recent shearlet regularization ADMM method.