Inverting Incomplete Fourier Transforms by a Sparse Regularization Model and Applications in Seismic Wavefield Modeling

Abstract

We propose a sparse regularization model for inversion of incomplete Fourier transforms and apply it to seismic wavefield modeling. The objective function of the proposed model employs the Moreau envelope of the \(\ell _0\) norm under a tight framelet system as a regularization to promote sparsity. This model leads to a non-smooth, non-convex optimization problem for which traditional iteration schemes are inefficient or even divergent. By exploiting special structures of the \(\ell _0\) norm, we identify a local minimizer of the proposed non-convex optimization problem with a global minimizer of a convex optimization problem, which provides us insights for the development of efficient and convergence guaranteed algorithms to solve it. We characterize the solution of the regularization model in terms of a fixed-point of a map defined by the proximity operator of the \(\ell _0\) norm and develop a fixed-point iteration algorithm to solve it. By connecting the map with an \(\alpha \)-averaged nonexpansive operator, we prove that the sequence generated by the proposed fixed-point proximity algorithm converges to a local minimizer of the proposed model. Our numerical examples confirm that the proposed model outperforms significantly the existing model based on the \(\ell _1\)-norm. The seismic wavefield modeling in the frequency domain requires solving a series of the Helmholtz equation with large wave numbers, which is a computationally intensive task. Applying the proposed sparse regularization model to the seismic wavefield modeling requires data of only a few low frequencies, avoiding solving the Helmholtz equation with large wave numbers. This makes the proposed model particularly suitable for the seismic wavefield (SW) modeling. Numerical results show that the proposed method performs better than the existing method based on the \(\ell _1\) norm in terms of the SNR values and visual quality of the restored synthetic seismograms.