DISCUSSION

Currently it is not clear what this result means. Symes (1990) has suggested that fixed-stencil methods like the one used here (in contrast to upwind methods like the one in Van Trier and Symes) cannot track first arrivals across velocity discontinuities properly. Perhaps that is what is happening here, but Symes suggests that such methods should also necessarily be unstable. This method does appear to be stable, it just doesn't always follow the first arrival. If Symes is right, switching to upwind finite-differences will probably resolve the problem evidenced in Figure 3.

Referring back to the bottom plot in Figure 2, we see the traveltime contours also did not follow the leading branch of the shear wave triplication in the homogeneous case. Would upwind finite-differences have behaved the same way? It isn't clear. Triplicating shear waves don't even really have a well-defined ``first arrival''-- precursive energy reaches out all the way to the P wave.

In any case, it appears that calculating anisotropic finite-difference traveltimes is possible, if not quite as straightforward to implement as one might hope.