principles and applications of microearthquake networks

4.4. Computing Travel Time and Derivatives 99
arbitrary depth in a model consisting of N multiple horizontal layers over
a half-space as shown in Fig. 23. Let the earthquake source be at point A
with coordinates (xA, y,, z,), and the station be at point B with coordinates
(xB, yB, zB). We construct our model such that the station is at the top of the
first layer, and we use ui and hi to denote the velocity and the thickness of
the ith layer, respectively. In Fig. 23, we let the source be inside thejth
layer at depth 5 from the top of thejth layer. The difference between this
case and the surface-source case is that the down-going travel path of the
refracted wave starts at depth zA instead of at the surface. This means that
the down-going seismic wave has the following less layers to travel: the
first ( j - 1) layers from the surface, and an imaginary layer within thejth
layer of thickness 5. We can therefore write down a set of equations for
this case in a manner similar to Eq. (4.97)
(4.98)
forj = 1,2, . . . , N - 1 andk = 2, 3, . . . , N, where Tjk is the travel time
for the ray that is refracted along the top ofthekth layer from a source at the
I
I
4
Fig. 23.
jth layer.
nth layer
Diagram of the refracted path along the kth layer for a source located in the