Fixed-point proximity algorithms solving an incomplete Fourier transform model for seismic wavefield modeling

Seismic wavefield modeling is an important tool for the seismic interpretation. We consider modeling the wavefield in the frequency domain. This requires to solve a sequence of Helmholtz equations of wave numbers governed by the Nyquist sampling theorem. Inevitably, we have to solve Helmholtz equations of large wave numbers, which is a challenging task numerically. To address this issue, we develop two methods for modeling the wavefield in the frequency domain to obtain an alias-free result using lower frequencies of a number fewer than typically required by the Nyquist sampling theorem. Specifically, we introduce two l(1) regularization models to deal with incomplete Fourier transforms, which arise from seismic wavefield modeling in the frequency domain, and propose a new sampling technique to avoid solving the Helmholtz equations of large wave numbers. In terms of the fixed-point equation via the proximity operator of the l(1) norm, we characterize solutions of the two l(1) regularization models and develop fixed-point algorithms to solve these two models. Numerical experiments are conducted on seismic data to test the approximation accuracy and the computational efficiency of the proposed methods. Numerical results show that the proposed methods are accurate, robust and efficient in modeling seismic wavefield in the frequency domain with only a few low frequencies. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

This study proposes two methods for modeling seismic wavefield in the frequency domain to obtain an alias-free result with fewer frequencies than typically required by the Nyquist sampling theorem. The methods address the numerical challenge of solving Helmholtz equations of large wave numbers by introducing l(1) regularization models and a new sampling technique. Numerical experiments demonstrate that the proposed methods are accurate, robust, and efficient in modeling seismic wavefield with only a few low frequencies.