Modeling and migration by a new finite difference scheme based on the Galerkin method for irregular grids

Full wave equation 2D modeling and migration using a new finite difference scheme based on the Galerkin method (FDGM) for irregular grids are presented. Since these involve semi-discretization by the finite element method (FEM) in the depth direction with the linear element, spatially irregular grids can be used to compute the wavefield in modeling and reverse-time migration. The mesh can be made locally thin to better represent structural complexity and lower velocity zones, which are treated by a fine grid, while the remaining parts of the models are represented by a coarse grid with equal accuracy. No interpolation is needed between the fine and coarse parts due to the rectangular grid cells. The accuracy of the proposed technique has been tested with a comparison to an analytical solution. The effectiveness of the method is verified by its application to a thin-layer model. At the same time, its efficiency is shown through an impulse and an oblique interface with a variable velocity media. INTRODUCTION The design of finite-difference (FD) schemes to handle nonuniform grids is an important topic in seismic modeling and migration. It offers the possibility of a more rational discretization in which the mesh can be made locally thin to better represent structural complexities and low velocity zones, avoiding oversampling in the rest of the model. A simple kind of irregular rectangular mesh may be obtained by variation of the sample interval along the xand z-axes. Mufti et al. (1996) demonstrated that a variable vertical grid step which adapts to the changes in the velocity with depth can greatly increase the efficiency of the acoustic-wave reversed time migration. In the elastic case, Oprsal and Zahradnik (1999) developed a FD scheme to solve the wave equation for displacements in irregular rectangular grids. Sergio (2003) extended the work to apply a fourth-order FD approximation to the complete acoustic wave equation and handle irregular rectangular grids. Pitarka (1999) put forward the velocity-stress formulation for nonuniform rectangular staggered grids. This paper presents a new approach to build irregular grids based on the Galerkin method. Considering that the field seismic data are uniformly recorded along the survey line, and velocities of layers change with the depth, a FD scheme with an irregular grid along the depth direction and a regular grid along the survey line is treated in this work. Following the Galerkin method, we used finite element discretization along the depth direction and the FD method in the spatial domain to solve partial differential equations, which has been clearly addressed by Du and Bancroft (2004). In previous work, we tested the effect of the new method on a regular mesh. Since the FEM discretization along the depth direction is applied, the irregular grid computation along the depth direction will be adopted. As a result, oversamping of large high-velocity areas, typical for fine regular grids, is avoided in this way; it also offers the