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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.

** Next:** 1-D discrete systems
** Up:** Cunha: Modeling a discrete
** Previous:** Cunha: Modeling a discrete
Stanford Exploration Project

12/18/1997