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# SCHOENBERG-MUIR THEORY

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 qS 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.

carlos
Figure 2
Left: Wave propagation in a layered medium. There are 512 layers from top to bottom, chosen randomly from a set of 11. Right: The homogeneous S&M medium equivalent to the stack of layers on the left. For both models, the source is a vertical force at the center. The X component of displacement is shown. The wavelength of the qS waves is about 75 gridpoints, which is not quite enough for a good average if the waves are traveling nearly parallel to the layers.

Next: Running rings around Schoenberg-Muir Up: Dellinger, Muir & Etgen: Previous: INTRODUCTION
Stanford Exploration Project
1/13/1998