principles and applications of microearthquake networks

4.3. Numerical Solutions qf the Ray Equation 83
4.3.1. Initial Value Formulation
The ray equation can be solved numerically as an initial value problem.
Eliseevnin (1965) presented an initial value formulation for the propagation
of acoustic waves in a heterogeneous medium. Since the ray equation
for seismic waves is the same as that for acoustic waves, Eliseevnin's
formulation can be applied directly to seismological problems. For the
two-dimensional case, Eliseevnin derived a set of three first-order differential
equations from the eikonal equation. For the three-dimensional
case, he gave a set of six first-order equations derived in a manner analogous
to the two-dimensional case. These equations are
dxldt = v cos a, dy/dt = 2: cos p. dz/dt = 2: cos y
da _-- av av av
- sin a -- cot a cos p -- cot (Y cos y
dt ax ay az
(4.33) dv t3V
dp=
av
dr ax ay dZ
- - cos a cot 0 + - sin p - - cot p cos y
av dV t3V
+ = -- cos a cot y -- dt t3X av
cos p cot y + - sin y
dZ
where a, p, and y are the instantaneous direction angles of the ray at point
(x, y , z) in a heterogeneous and isotropic medium in which the velocity is
specified by v = dx, y, z), and t is time. The direction cosines are defined
by
(4.34) cos IY = dx/ds, cos /3 = dy/ds, cos y = dz/ds
where ds is an element of the ray path. The first three members of Eq.
(4.33) specify the change of ray position with respect to time, and the last
three members specify the change of ray direction with respect to time.
The six initial conditions for Eq. (4.33) are provided by the three spatial
coordinates of the source and the three direction angles of the ray at the
source. If they are specified, then Eq. (4.33) can be solved numerically by
integration as described in many texts (e.g., Acton, 1970; Gear, 1971;
Shampine and Gordon, 1975).
Equation (4.33) is given in the Cartesian coordinate system. In microearthquake
studies, this coordinate system is appropriate because the
space under consideration is small. However, the spherical coordinate
system should be used for global problems. For example, Jacob (1970) and
Julian (1970) have derived equivalent sets of equations for this case.
The initial value formulation has been applied to a number of seismological
problems. For example, Jacob (1970) and Julian (1970) developed
numerical techniques to trace seismic rays in a heterogeneous medium in