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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.

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Stanford Exploration Project

1/13/1998