Discontinuous Galerkin method for solving viscoacoustic wave equations with amplitude dissipation and phase dispersion separation in isotropic and anisotropic media

Abstract

SUMMARY The standard-linear-solid (SLS) theory works well for viscoelasticity. However, the coupling of amplitude dissipation and phase dispersion makes it impossible to investigate their effects separately by the discontinuous Galerkin method (DGM). In this paper, we have derived newly viscoacoustic wave equations with amplitude dissipation and phase dispersion separation in isotropic and anisotropic media, based on a Fourier method, which is suitable for using a time–space-domain DGM on unstructured meshes. The basic framework of DGM is constructed and the amplitude-dissipation effect and the phase-dispersion effect in viscoacoustic wave equations are investigated. The original equation is first transformed into the frequency–wavenumber domain, where the amplitude dissipation and phase dispersion are separated effectively, and then the decoupled formulation is converted back to the time–space domain. The new equivalent and approximate equations can be obtained. Compared with the original equation, the newly approximated equation enables us to separate the amplitude-loss and phase-delay terms, respectively, and experiences four kinds of effects, namely acoustic effect, only amplitude-dissipation effect, only phase-dispersion effect and both amplitude-dissipation and phase-dispersion effect. Moreover, the stability condition and numerical dispersion for using DGM to solve the new and old equations are presented. Several numerical examples are used to verify the correctness and effectiveness of the modified approximate equations in viscoacoustic isotropic and anisotropic media. The numerical results in a cave and SEG/EAGE salt models demonstrate that the new equations combined with DGM have performances on viscoacoustic media with complex geological structures.