2-D wave modelling and reverse-time migration by a new finite difference scheme based on the Galerkin method

Xiang Du, John C. Bancroft

Full wave equation 2-D modeling and migration using a new finite difference scheme based on the Galerkin method are presented. Since it involves semi-discretization by the finite element method (FEM), it is also called finite element and finite difference method (FE-FDM). Using the semi-discretization technique of the finite element method (FEM) in the z direction with the linear element, the original problem can be written as a coupled system of lower dimensions partial differential equations (PDEs) that continuously depend upon time and space in the x direction. A 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-different method of the same accuracy, the stability condition is less constrained and shows its advantage over the conventional FDM. An absorbing boundary condition of fourth-order accuracy is used to prevent boundary reflections. In numerical experiments, a comparison is made between a FEFDM 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 get the poststack seismic section, which is then successfully migrated. These examples show the potential of FE-FDM in modeling and reverse-time migration.