Computer implementation

Already in applying the standard method of characteristics I have had to use Mathematica to reduce large determinants. The characteristic equation for the equations of motion, while it reduces to P and S wave eikonal equations for the isotropic case, is in the general case a two-page nonlinear first-order equation. I must solve this equation numerically, even if an error is found that when corrected reduces it to the equation one-fifth as long resulting from the principal-symbol variant method discussed earlier.

In the later application of the FIO method to the equations of motion, as in the method of characteristics the process may reach a simpler stage beyond which analytic solutions are not possible and numerical methods must be applied. Except for simple test examples and approximations, the finding of a fully analytic solution that can then be coded onto a computer may be unlikely. The two-stage process of some analytic reduction and some application of numerical methods should result in a much more tractable numerical problem than direct numerical solution of the equations of motion, however. For at least one test example, ideally for known geology above which real data has been collected and is available to me, I will also apply a full numerical method, such as finite element or finite difference, directly to the equations of motion. In this case care is necessary to avoid possible problems with such numerical methods. On the other hand, it might also be possible to demonstrate that a full numerical method has problems in some cases where the FIO method does not. One objective, therefore, is to learn relative advantages and disadvantages of different solution methods by testing.