High order finite difference modeling and reverse time migration

Abstract

The treatment of the full acoustic wave equation by second order finite differences in space and time has been successfully used in exploration geophysics over the past one and a half decades for forward modeling. Migration, the inverse of modeling, was traditionally done by depth extrapolation of the upcoming wavefield. However, extrapolation in depth is not always a stable process. Furthermore, in treating only the upcoming energy, one must either sacrifice the generality of the full wave equation by using the 15 or 45 degree finite difference operator in space and time, or one must resort to Fourier methods such as the phase shift technique in which severe limitations are imposed on the velocity function. It is known that higher order finite difference methods significantly reduce the numerical dispersion associated with such techniques. However, efforts to apply this result to modeling problems, and not to mention inverse problems, have only recently been published. Here we investigate the application of fourth and higher order difference methods to the full acoustic wave equation for both modeling and reverse time migration. We also discuss the problems of model initialization, mathematical stability and dispersion, as well as the usefulness of the local character of these techniques for parallel computing as opposed to the global nature of Fourier techniques and the resulting implications for data motion, especially in 3D. Finally, we show an example of reverse time migration applied to a real data set from Western Australia.