## Finite difference - finite integral transform hybrid techniques: the coupled P-SV problem in a radially symmetric vertically inhomogeneous medium

### P. F. Daley

The hybrid finite difference finite integral transform method is developed for the P-SV wave propagation problem in a radially symmetric vertically inhomogeneous medium. Apart from the development of the finite difference analogues of the transformed equations of motion, a number of numerical considerations are addressed. As in most problems where numerical methods are employed in the solution, there are several areas that are given special attention to indicate how to improve run times and accuracy. Often this method of problem solving is referred to (erroneously) as the pseudo-spectral method. The solution approach described here is more general in that uniformly sampled grids of any spatial dimension are not required. It may be correct to say that the pseudo-spectral method is a subset of what is discussed here.

Presentation of the theory, with consideration given to finite Hankel transform theory, the development of finite difference analogues, stability analysis and numerical considerations to exploit the highly parallel nature of the problem are included. Numerical results for a range of geological models, for both amplitude versus offset (AVO) and vertical seismic profile (VSP) applications are presented.