principles and applications of microearthquake networks

92 4. Seismic Ruy Tracing for Minimum Time Path
(4.68)
a = arccos(dx/ds).
y = arccos(dz/ds)
p = arccos(dy/ds)
In the usual Cartesian coordinate system, the positive z axis is pointing
upward. However, in seismological applications, it is more convenient to
let the positive z axis point downward. The take-off angle I#J at the source
(with respect to the downward vertical) is just the direction angle y at the
source, i.e., + = yIA, or
(4.69) $ = arccos(dz/ds)/A
If we solve the ray equation in the form of Eq. (4.56), then ws(0) is the
value of dz/ds at the source. Therefore, Eq. (4.69) may be written as
(4.70) $ = arccos[w6(0)]
Equation (4.66) is very important because it relates the spatial derivatives
of the travel time to the direction cosines of the seismic ray and the
slowness (Le., the reciprocal of the velocity) of the medium at the earthquake
source. We will use this equation frequently in the following
subsections.
4.4.2. Constant Velocity Model
This is the simplest velocity model; the velocity v is a constant and is
independent of the spatial coordinates. The minimum time path between
any two points A and B is a straight line (i.e., AT). If the earthquake
source is at point A with coordinates (xA,
zA), and the receiving station
is at point B with coordinates (xs, Ve, zB), then the path length S between A
and B is [(xB - xA)2 + (ys - yA)2 + (za - zA)*]”*. The travel time T is
simply given by
(4.71) T = S /E
The direction cosines for the ray are constant and are given by (xE-
xA)/S, (Ys - yA)/S, and (zB- zA)/S. Using Eq. (4.661, the spatial derivatives
of the travel time at the source for this case are
aT/ax/A = -(xB - xA)/(ns), aT/ay(A = -(YE - 4)A)/(uS)
(4.72)
aT/azl, = -(zB - zA)/(us)
Using Eq. (4.69), the take-off angle at the source is
(4.73) 4 = arccos[(zB - zA)/S]
In this simple model, we do not really need to apply the formulas for the
general three-dimensional case. Eq. (4.72) for the spatial derivatives can