The Interpolation of Sparse Geophysical Data

Geophysical data interpolation has attracted much attention in the past decades. While a variety of methods are well established for either regularly sampled or irregularly sampled multi-channel data, an effective method for interpolating extremely sparse data samples is still highly demanded. In this paper, we first review the state-of-the-art models for geophysical data interpolation, focusing specifically on the three main types of geophysical interpolation problems, i.e., for irregularly sampled data, regularly sampled data, and sparse geophysical data. We also review the theoretical implications for different interpolation models, i.e., the sparsity-based and the rank-based regularized interpolation approaches. Then, we address the challenge for interpolating highly incomplete low-dimensional data by developing a novel shaping regularization-based inversion algorithm. The interpolation can be formulated as an inverse problem. Due to the ill-posedness of the inversion problem, an effective regularization approach is very necessary. We develop a structural smoothness constraint for regularizing the inverse problem based on the shaping regularization framework. The shaping regularization framework offers a flexible way for constraining the model behavior. The proposed method can be easily applied to interpolate incomplete reflection seismic data, ground penetrating radar data, and earthquake data with large gaps and also to interpolate sparse well-log data for preparing high-fidelity initial model for subsequent full-waveform inversion.