principles and applications of microearthquake networks

4. Seismic Ray Tracing for
Minimum Time Path
The earth is continuously being deformed by internal and external
stresses. If the stresses are not too large, elastic and/or plastic deformation
will occur. But if the stresses accumulate over a period of time,
fracture will eventually take place. Fracture involves a sudden release of
stress and generates elastic waves that travel through the earth. A seismic
network at the earth’s surface may record the ground motion caused by
the passage of elastic waves. The major problem is to deduce the earthquake
source parameters and seismic properties of the earth from a set of
surface observations. Before we attempt to solve this inverse problem, we
need to understand the forward problem, namely, how elastic waves
travel in the earth. We will now present an outline of the forward problem
in order to provide a background for seismological applications.
4. I Elastic Wave Propagation in Homogeneous and Heterogeneous Media
Karal and Keller (1959) presented a modern treatment of elastic wave
propagation in homogeneous and heterogeneous media, and our outline
here follows their approach. The equation of motion for a homogeneous,
isotropic, and initially unstressed elastic body may be obtained using the
conservation principles of continuum mechanics (e.g., Fung, 1969, p. 261)
as
(4.1)
where 8 = & auj/axj is the dilatation, p is the density, Ui is theith component
of the displacement vector u, t is the time, xi is the ith component of
the coordinate system, and h and p are elastic constants. Equation (4.1)
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