TY - JOUR

T1 - Fixed-point proximity algorithms solving an incomplete Fourier transform model for seismic wavefield modeling

AU - Wu, Tingting

AU - Shen, Lixin

AU - Xu, Yuesheng

N1 - Funding Information:
T. Wu was supported in part by the Natural Science Foundation of China under grants 11971276 and 11771257 , and Shandong Province Higher Educational Science and Technology Program under grant J18KA221 . L. Shen was supported in part by the National Science Foundation, USA under grant DMS-1913039 . Y. Xu was supported by the National Science Foundation, USA under grant DMS-1912958 and by the Natural Science Foundation of China under grant 11771464 .

PY - 2021/3/15

Y1 - 2021/3/15

N2 - 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 ℓ1 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 ℓ1 norm, we characterize solutions of the two ℓ1 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.

AB - 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 ℓ1 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 ℓ1 norm, we characterize solutions of the two ℓ1 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.

KW - Compressed sensing

KW - Incomplete Fourier transform

KW - Proximity algorithms

KW - Seismic wavefield modeling

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U2 - 10.1016/j.cam.2020.113208

DO - 10.1016/j.cam.2020.113208

M3 - Article

AN - SCOPUS:85091967920

VL - 385

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

M1 - 113208

ER -