Introduction

Wave propagation in the Earth is usually described in terms of the space-time partial differential wave equation (acoustic or elastic), which governs the mechanical behavior of continuous media. The basic requirement for the applicability of the wave equation is that the displacements and its first order spatial derivatives be continuous; in other words, the medium must be continuous. Evidently, this assumption has proved to be a reasonably good approximation for the behavior of Earth within the wavelength range of typical reflection seismology experiments. It is possible, however, that in some cases a discrete system might better approximate the real earth than a continuous system would. Furthermore, any practical solution of the wave equation involves its discretization in both time and space, which brings some degree of inaccuracy to the solution. Although several high-order methods have been developed to overcome these numerical artifacts (see Etgen 1989 for a general discussion), it is worthwhile to consider the possibility that some of these so called "artifacts" have a physical existence and should be present in a more realistic simulation of wave propagation in the earth.

A spatially discrete system can be solved exactly, without the need of any numerical approximation. In this paper I compare the exact solution of a discrete system with the exact solution of the equivalent continuous system for some very simple 1-D models. Next, I derive the equations of motion for a 2-D discrete transverse isotropic medium and describe two methods of solution; one exact, using eigenvalue decomposition, and the other an approximation which is based in a recursive time relation.