Stable optimization of finite-difference operators for seismic wave modeling
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State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China ([email protected]) Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
Received: December 5, 2019; Revised: February 6, 2020; Accepted: April 22, 2020
ABSTRACT The finite difference scheme is now widely used in the reverse time migration and full waveform inversion. Their results are dependent on the accuracy of finite difference operators. In this paper, we combine the cosine function with the original window function to construct a new window function, in order to obtain higher precision finite difference operators. The absolute error curves of the optimized finite difference operators are close to zero for low wavenumbers. In other words, we do not observe an oscillating curve of absolute errors produced by other optimized methods. In order to overcome the limitations of a single graphics processing unit (GPU), we developed the multiple-GPU method for the elastic wave equation. Numerical experimental results show that our new window function can control the numerical dispersion better than the binomial window and scaled binomial window, and the multiple-GPU computation is very stable. K e y w o r d s : finite difference, window function, cosine function, elastic wave, multiple GPU
1. INTRODUCTION Seismic forward modeling based on the finite-difference (FD) scheme is now widely used in reverse time migration and full waveform inversion, which are included in several forward modeling schemes. The precision and speed of the forward modeling is important for imaging and inversion. Many scholars have performed extensive research on the finite difference schemes for the wave equation. In order to reduce the numerical dispersion, a variety of FD schemes have been developed, such as variable grids (Wang and Schuster, 1996; Hayashi and Burns, 1999), irregular grids (Opršal and Zahradník, 1999), staggered grids (e.g., Madariaga, 1976; Virieux, 1986; Graves, 1996; Robertsson, 1996; Bohlen and Saenger, 2006), and rotated staggered grids (Saenger et al., 2000; Krüger et al., 2005; Bansal and Sen, 2008).
Stud. Geophys. Geod., 64 (2020), DOI: 10.1007/s11200-019-0487-1, in print © 2020 Inst. Geophys. CAS, Prague
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J. Wang and L. Hong
The conventional classic coefficients of the high-order FD approximations for spatial derivatives are usually determined by a Taylor-series expansion of the spatial derivative term. Only changing the FD scheme does not minimize the numerical dispersion. If conventional FD coefficients are used for seismic wavefield forward modeling, strong numerical dispersion will be expected, especially in the high wavenumber range of the wave equation. In the past two decades, many optimization methods have been proposed, such as the Newton method (Kindelan et al., 1990), implicit scheme (Liu and Sen, 2009), time-space domain dispersion-relation-based method (Liu and Sen, 2011), simulated annealing algorithm (
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