principles and applications of microearthquake networks

90 4. Seismic Ray Trucing .for Minimum Time Path
sands of rays in a given problem, it is not economical at present to apply
these numerical ray tracing techniques in routine microearthquake studies.
Consequently, much simplified velocity models are used. Before we
discuss these models, we will derive formulas to compute travel time
derivatives and the take-off angle for the three-dimensional heterogeneous
case. These formulas are readily adaptable to the simplified models.
4.4.1. Heterogeneous Velocity Model
In the unified formulation (Section 4.3.3), the minimum travel time is
one of the variables to be solved from Eq. (4.56). In the solution of Eq.
(4.56), we also obtain the variables d.u/ds, dylds, and dzlds. These variables
are the direction cosines of the ray as given by Eq. (4.34). Using a
variational technique due to R. Comer (written communication, 1979), we
now derive formulas for the spatial derivatives of the travel time in terms
of the direction cosines of the ray at the source. The formula for the
take-off angle at the source follows readily from these derivatives.
Let us consider the minimum time path rl between a source at point A
and a station at point B. Let the spatial position of point A be rl = (x1, y,,
zl), and that for point B be r2 = (x2. y,, 2,). Let us also consider a neighboring
minimum time path r2 between points C and B, where C is a
neighboring point of A and its spatial position is given by rl + 6r,. Let the
position along each path be parameterized by an arbitrary parameter q as
discussed in Section 4.2.2, such that q = q1 at point A and point C, and q
= q2 at point B.
The variation of travel time Ton passing from rl to r2 is given by Eq.
(4.28). Because F1 and T2 are minimum time paths, they satisfy Euler’s
equation as given by Eq. (4.29). Thus, the three integrals on the right-hand
side of Eq. (4.28) vanish, and Eq. (4.28) reduces to
(4.59)
Because the second end point B is the same for rl and T2, the right-hand
side of Eq. (4.59) evaluated at q = q2 is zero. Therefore, the variation of
travel time is