INTRODUCTION

Schoenberg and Muir (1989), present an efficient technique for finding the homogeneous bulk equivalent of a heterogeneous layered medium. This S&M technique is finding wider and wider use in predicting not only the bulk behavior of layered media, but the effects of adding sets of cracks to uncracked background media as well. Because this technique is so convenient, it is tempting to apply it without verifying that the underlying approximations are applicable. In a strict sense, the technique is on firm theoretical footing only when it is used to predict the behavior of a flat, infinite stack of layers (possibly with some of the layers representing ``cracks'') under a static traction.

In practice, it is known that if the waves propagating through the stack have sufficiently long wavelengths the zero-frequency approximation can be very accurate. The problem of what ``sufficiently long wavelength'' means in practice has been investigated by Nichols (1988). He concluded that if the medium were approximately uniform in the distribution of types of layers on the scale of a wavelength, the wave would not ``feel'' the layers. What about the ``flat infinite layer'' assumption? This assumption is commonly broken when the S&M method is used to model truncated dipping layers (such as those in Figure 1), cracks cross-cutting layers, or non-parallel sets of cracks.

The purpose of this paper is to test how well the method works when applied to truncated dipping layers. For our models we use an annulus of layered media embedded in its S&M homogeneous equivalent. The source is in the center of the annulus. We compare the wavefields after propagation through the annulus with the wavefields in the corresponding homogeneous model.

 

layers Figure 1 Schoenberg-Muir averaging for canted layers. S&M averaging assumes stationarity in the direction of the arrow, and infinite layers in the perpendicular direction. The truncated layers in the shaded block violate these assumptions.