2D wave-equation migration by joint finite element method and finite difference method

A new method of migration using the finite element method (FEM) and the finite difference method (FDM) is jointly used in the spatial domain. It has been applied to solve a time relay 2D wave equation. By using the semi-discretization technique of FEM in the spatial domain, the origin problem can be written as a coupled system of lower dimensions partial differential equations (PDEs) that continuously depend upon time and space. FDM is used to solve these PDEs. The concept and theory of this method are also discussed in this paper. Two numerical examples of 2D wave-equation migration show the successful result and its potential application. INTRODUCTION The finite element–finite difference method (FE–FDM) is one of the numerical methods using FEM and FDM in the spatial domain to solve partial differential equations. FE-FDM uses FEM in some dimensions and FDM in the remaining dimensions and in the time domain. The FE-FDM has strong resemblance to a number of numerical methods such as the finite difference method and the finite element method. A brief emphasis on the basic differences between FE-FDM and the above mentioned methods is as follows: FEM fully discretizes a static problem into a system of algebraic equations with discrete nodal values as the basic unknowns. For the time relay problem, FEM fully discretizes it in spatial domain into ODEs and solves them with the FD method (Hughes, 1987), whereas the FE-FDM semi-discretizes the PDE using FEM in the spatial domain into a coupled system of PDEs. These PDEs still continuously depend upon both time and space (although not all the space dimension), and are solved with FD method. Thus, the strengths of FEM, the adaptation to arbitrary domain, boundary, material and loading are retained. The shortcomings of FEM, such as large demand on computer memory and high computation costs are reduced because of the semi-discretization. Compared with the FD method, the computation precision is increased by FEM semi-discretization. The technique of FD for solving PDEs in lower dimensions can decrease frequency dispersion in space and has looser conditions of stability for explicit FD schemes. In this paper, the basic concept and theory of the finite element–finite difference method are described through the 2D wave equation. Two numerical examples of 2D wave equation migration are given as well to demonstrate the tremendous performance of this method. * Tsinghua University Du, Dong, and Bancroft 2 CREWES Research Report — Volume 15 (2003) PRINCIPLE Consider the hyperbolic model problem, with the 2D scalar wave equation