3D finite-difference method using discontinuous grids

We have formulated a 3D finite-difference method (FDM) using discontinuous grids, which is a kind of multigrid method. As long as uniform grids are used, the grid size is determined by the shortest wavelength to be calculated, and this constitutes a significant constraint on the introduction of low-velocity layers. We use staggered grids that consist of, on one hand, grids with fine spacing near the surface where the wave velocity is low, and on the other hand, grids whose spacing is three times coarser in the deeper region. In each region, we calculated the wavefield using a velocity-stress formulation of second-order accuracy and connected these two regions with linear interpolations. The second-order finite-difference (FD) approximation was used for updating. Since we did not use interpolations for updating, the time increments were the same in both regions. The use of discontinuous grids adapted to the velocity structure resulted in a significant reduction of computational requirements, which is model dependent but typically one-fifth to one-tenth, without a marked loss of accuracy.