The *q*P wavelength in this example is about 65 gridpoints.
By Nichols' rules of thumb (1988),
this would seem to be more than enough of a distance to average over,
since there are only
11 kinds of layers, each one gridpoint thick.
Probably this difference in results is because we used
a much more extreme variation between layers in our examples.

Figure 5

We know the S&M approximations break down when the layers are too thick
compared to the wavelengths of the waves passing through,
as in Figure 5.
Muir has conjectured that perhaps S&M gives an answer that is
correct in expectation in such cases.
Figure 6 tests this hypothesis. To make it, we ran 20 models
similar to that
in Figure 5, but with independently chosen sets of random layers.
Because of the symmetry of the problem, we averaged in each result
and each result with the Z axis reversed, to effectively double the
number of models at no additional cost.
The results are much cleaner, but there is still a slight phase
shift that did not average out. This is especially noticeable for the
higher-frequency *q*S wave. (This suggests that it might be useful
to work on extending S&M theory to non-zero frequencies.)

Figure 6

1/13/1998