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At first sight the notion of an equivalent model that is exact only at
zero frequency is not very satisfying. Our branch of Seismology deals
with dynamic and kinematic problems, not with static ones. Fortunately
it appears that S&M is both statically and kinematically correct. As
far as we know this is not proven, but there are reasonable hand-waving
arguments in its behalf. By ``kinematically correct'' we mean that in the
far field, the center (defined in some reasonable way) of the wavelet
should travel with
the velocity predicted by S&M. A suggested line of argument is via one
of the Abel/Tauber theorems that tie behavior at infinity in the Time
Domain to behavior around the origin in the Frequency Domain (and vice
versa).
The extension of S&M to Dynamic correctness looks difficult but
doable. There are two ideas worth pursuing. The first is to recognize
that in a randomly layered medium, energy is continuously converted
from coherent to scattered. Since we are interested in the behavior of
the coherent energy, this suggests equivalencing the heterogeneous,
scattering medium with a homogeneous, visco-elastic one. In both
cases energy is lost out of the coherent system, and the analogy is
sufficient for many of our purposes. The second idea is to employ an
ergodic principle, and instead of averaging over layers as the S&M
paper did, average over realizations as we did in Figure 6.

** Next:** APPENDIX
** Up:** DISCUSSION
** Previous:** A possible modeling application
Stanford Exploration Project

1/13/1998