Where do we go from here?

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.