2‐D wave equation modeling and migration by a new finite difference scheme based on the Galerkin method

Abstract

2‐D wave equation modeling and migration using a new finite difference scheme based on the Galerkin method are presented. Since it involves the semi‐descretization of the finite element method (FEM), it is also called the finite element and finite difference method (FE‐FDM). For a 2‐D acoustic wave equation, by using the semi‐discretization technique of the finite element method (FEM) in the z direction with linear elements, the original problem can be written as a coupled system of lower dimensional partial differential equations (PDEs) that depend continuously upon time and space in the x direction. The fourth‐order finite difference method (FDM) is used to solve these PDEs. The concept and principle are introduced in this paper. Compared with the explicit finite‐difference method of the same accuracy, the stability condition becomes looser and shows an advantage over the conventional FDM. An absorbing boundary condition of fourth‐order accuracy is used to prevent boundary reflections. In numerical experiments, comparison is made between a FE‐FDM numerical solution and an analytic solution of the quarter‐plane. Here, FE‐FDM is shown to be accurate in numerical computation. In addition, a constant velocity model with two irregular interfaces is simulated to obtain a poststack seismic section, which is then successfully migrated. These examples show the potential of FE‐FDM in modeling and reverse‐time migration.