principles and applications of microearthquake networks

100 4. Seismic Ray Tracing for Minimum Time Puth
jth layer, tfk is the corresponding critical distance, A is the epicentral
distance given by A = [(x, - xA)’ + (yB- Y ~)‘]]”~, the thickness 5 is given
by 5 = zA - (h, + hz + . . * + the symbol Oki = (ui- IJ?)~’~,
and
the symbol !& = ( ul - u?)112.
These equations allow us to calculate the travel times for various refracted
paths and the corresponding critical distances beyond which refracted
waves will exist. In many instances, one of the refracted paths is
the minimum time path sought. However, there are cases in which the
direct path is the minimum time path. This is considered next.
If the earthquake source is in the first layer (with velocity q), then the
travel time for the direct wave is the same as that in the constant velocity
model (see Section 4.4.2), i.e.,
(4.99) Td = [(X, - XA)~
4- (Ye - YA)? -k (ZB - Z~)~]*’*/2)1
However, if the earthquake source is in the second or deeper layer, there
is no explicit formula for computing the travel time of the direct path. In
this case, we must use an iterative procedure to find an angle 4 such that
its associated ray will reach the receiving station along a direct path. Let
us consider a model where the earthquake source is in the second layer, as
shown in Fig. 24. Let the first layer have a thickness hl, and let u1 and uz
be the velocity of the first and the second layer, respectively. In our model
the layer velocity increases with depth so that v2 > q. We choose a
coordinate system such that the z axis passes through the source at point
A, and the 7 axis passes through the station at point B. Let the coordinates
c
2
Fig. 24. Diagram to illustrate the computation of the travel path of a direct wave for a
layered velocity model.