Suppose we have a flat stack of infinite elastic layers, and apply some static traction across the top and bottom surfaces. Schoenberg and Muir's layer averaging method tells us how to find the equivalent homogeneous medium that exactly mimics our original stack. (This is really a glorified elastic version of the standard physics problem of finding out the effective spring constant when two different springs are connected end-to-end.) Although the Schoenberg-Muir method is exact for static tractions, it should also approximately apply to elastic waves if their wavelength is sufficiently large. There are two reasons why long wavelengths are required. First, the time scale of the wave passing by must be sufficiently slow that the medium has time to adjust in a quasi-static way. Second, each wavelength must ``feel'' a good representative sample of the set of layers.

As an example of S&M layer-averaging,
Figure 2 shows two snapshots
of *q*S waves propagating away from a point source. The model on the left
is *not* anisotropic, despite the prominent shear-wave triplication visible.
It consists of 512 layers; each individual layer
is chosen randomly from the set of 11 possible isotropic media listed in
the table in Appendix 1.
(Note that this is rather an extreme test; the P velocity, S velocity,
and density all vary by a factor of 2 between the extremal layers.
We used 11 layers so that there was little chance the same elastic constants
would be picked several times in a row, resulting in a ``thick'' layer.)
The model on the right uses the homogeneous S&M
medium equivalent to the stack of layers on the left.
The shear waves are high enough in frequency to ``feel''
the layers in the left model to some extent (especially the occasional
double-thick ones),
so there is some scattered energy, but otherwise the wavefields in the
layered and homogeneous models are nearly identical. The two results
cannot be subtracted because they show *displacement*
instead of *energy*, which depends strongly on the elastic
constants of the medium.

Why did we choose this particular model? The S&M calculus is quite general, and we could have chosen layers with a high degree of anisotropy. Instead we elected to use isotropic layers. There are many people who believe that anisotropy is a consequence of heterogeneity, without any intrinsic quality. We wished to demonstrate that triplication is a normal, geometric consequence of layering, and not a special artifact of a homogeneous anisotropic medium.

Figure 2

1/13/1998