Complex-valued adaptive-coefficient finite-difference frequency-domain method for wavefield modeling based on the diffusive-viscous wave equation

Abstract

The diffusive-viscous wave (DVW) equation is an effective model for analyzing seismic low-frequency anomalies and attenuation in porous media. To effectively simulate DVW wavefields, the finite-difference or finite-element method in the time domain is favored, but the time-domain approach proves less efficient with multiple shots or a few frequency components. The finite-difference frequency-domain (FDFD) method featuring optimal or adaptive coefficients is favored in seismic simulations due to its high efficiency. Initially, we develop a real-valued adaptive-coefficient (RVAC) FDFD method for the DVW equation, which ignores the numerical attenuation error and is a generalization of the acoustic adaptive-coefficient FDFD method. To reduce the numerical attenuation error of the RVAC FDFD method, we introduce a complex-valued adaptive-coefficient (CVAC) FDFD method for the DVW equation. The CVAC FDFD method is constructed by incorporating correction terms into the conventional second-order FDFD method. The adaptive coefficients are related to the spatial sampling ratio, number of spatial grid points per wavelength, and diffusive and viscous attenuation coefficients in the DVW equation. Numerical dispersion and attenuation analysis confirm that, with a maximum dispersion error of 1% and a maximum attenuation error of 10%, the CVAC FDFD method only necessitates 2.5 spatial grid points per wavelength. Compared with the RVAC FDFD method, the CVAC FDFD method exhibits enhanced capability in suppressing the numerical attenuation during anelastic wavefield modeling. To validate the accuracy of our method, we develop an analytical solution for the DVW equation in a homogeneous medium. Three numerical examples substantiate the high accuracy of the CVAC FDFD method when using a small number of spatial grid points per wavelength, and this method demands computational time and computer memory similar to those required by the conventional second-order FDFD method. A fluid-saturated model featuring various layer thicknesses is used to characterize the propagation characteristics of DVW.