principles and applications of microearthquake networks

4.4. Computing Travel Time and Derivatives 89
lar problem at hand, and thus it produced more detailed computations
where it was necessary, as in the case of highly curved seismic rays.
Lee et ul. (1982b) applied the numerical technique developed by
Pereyra ef al. (1980) to trace seismic rays in several two- and threedimensional
velocity models. Three different types of models were considered:
continuous and piecewise continuous velocity models given in
analytic form, and general models specified by velocity values at grid
points of a nonuniform mesh. Results of their numerical solutions were in
excellent agreement with several known analytic solutions.
Using an initial value approach for tracing seismic rays, Luk et ul.
(1982) noted tliat solving Eq. (4.53) was computationally more efficient
than solving Eq. (4.33) because trigonometric functions were not involved.
They solved Eq. (4.53) by numerical integration using the DE-
ROOT subroutine described by Shampine and Gordon (1975). Thus starting
from a given seismic source, a set of initial direction angles may be
used to trace out a corresponding set of ray paths. This is very useful in
studying the behavior of seismic rays in a heterogeneous medium.
For two-point seismic ray tracing, one may solve a series of initial value
problems starting from one end point and design a scheme so that the ray
converges to the other end point. Luk et al. (1982) solved Eq. (4.53) by
shooting a ray from the source and adjusting the ray to hit the receiver
using a nonlinear equation solver. This technique was found to be just as
efficient as the method of Pereyra et ul. (1980) which used a boundary
value approach.
4.4. Computing Travel Time, Derivatives, and Take-Off Angle
In many seismological problems, such as earthquake location, we need
to compute travel times between a source and a set of stations, and the
corresponding spatial derivatives evaluated at the source. Take-off angles
of the seismic rays at the source to a set of stations are also needed to
determine the fault-plane solution. For a heterogeneous earth model,
travel times, derivatives, and take-off angles can be computed by solving
the ray equation as discussed in the previous section. But in practice, we
may not have enough information to construct a realistic threedimensional
model for the velocity structure of the crust and upper mantle
beneath a microearthquake network. Furthermore, it is expensive to solve
the three-dimensional ray equation numerically. For instance, about 0.3
sec CPU time in a large computer (e.g., IBM 3701168) is required to find
the minimum time path between a source and a station in a typical threedimensional
heterogeneous model. Because we may need to trace thou-