principles and applications of microearthquake networks

102 4. Seismic Ray Tracing for Minimum Time Path
The preceding procedure for a source in the second layer can be generalized
to a source in a deeper layer, say, the jth layer. Once a trial angle
& is chosen, we may use Snell’s law to compute successive incident
angles to each overlying layer until the trial ray reaches the surface at
point A, with the r) coordinate given by
(4.102)
& = q+ +
1
i=J-1
hi tan 8i
where 8, = 4*. The incident angles are related by
(4.103)
sin Oi
--
- sin
for 1 5 i < j
Vi
Vi+l
where Bi is the incident angle for the ith layer with velocity vi and thickness
hi. With the proper substitutions, Eq. (4.102) can also be used to
calculate Al and A2.
In this iterative procedure, the trial angle $* converges rapidly to the
angle 4 whose associated ray path reaches the station within the error
limit E. Since this ray path consists ofj segments of a straight line in each
of the j layers, we can sum up the travel time in each layer to obtain the
travel time from the source to the station.
Knowing how to compute travel time for both the direct and the refracted
paths, we can then select the minimum travel time path. The
spatial derivatives and the take-off angle can be computed from the direction
cosines using results from Section 4.4.1 as follows.
Let us consider an earthquake source at point A with spatial coordinates
(xA, y,, zA), and a station with spatial coordinates (xB, yB, zB). Let us
choose a coordinate system such that points A and B lie on the qz plane,
and A’ is the projection of A on z = zE. In Fig. 25, let us consider an
element of ray path ds with direction angles a, p, and y, and components
dx, dy, and dz (with respect to the x, y, and z axes, respectively). In Fig.
25a, the projection of ds on the r) axis is sin y ds. In Fig. 25b, this projected
element can be related to the components dx and dy of ds by
(4.104)
where a’ and p’ are the angles between the r) axis and the -x and y axes,
respectively. Consequently, from Eq. (4.104) and the definition of direction
cosine for y, we have
(4.105)
dx = cos a’ sin y ds,
dx/ds = cos a’ sin y,
dz/ds = cos y
dy = cos p’ sin y ds
dy/ds = cos p‘ sin y
The angles a’ and p‘ can be determined from Fig. 25b as