Tutorial: Hybrid finite difference - finite transform methods for hyperbolic partial differential equations
P. F. Daley
The use of integral transforms has long been a solution method for hyperbolic equations or systems. Finite integral transforms, which are discussed here, became popular with the advent of digital computers, with the first references in the literature appearing in the early 1960's. Combining these transforms with finite difference methods has been shown to provide highly accurate numerical solutions to problems of wave propagation in elastic media, as the reduced spatial dimensionality employed in the finite difference part of the numerical solution results in a reduction of grid dispersion. Also, for problems where elastic parameters are dependent upon only one spatial coordinate, a considerable saving of the space required is achieved.
In this report, the simple case of the SH potential equation in a medium whose elastic parameters are only dependent upon depth is considered. This is a fairly simplistic case but does demonstrate, for tutorial purposes, many of the concepts required for the practical, numerical computation of synthetic traces using this method. Apart from a fairly standard treatment of the finite-difference aspect of the problem, the subject of replacing the infinite series in the inverse transform by a truncated finite series is explored for a band limited source wavelet.