principles and applications of microearthquake networks

104 4. Seismic Ray Tracing for Minimum Time Puth
(4.108)
Therefore Eq. (4.105) becomes
yIA = Ojk = arcsin(vj/uk)
dx/ds = [(XB - XA)/h](uj/uk)
(4.109) dy/ds [(VB - ~4)/AI(uj/cJ
dz/ds = [l - (U,/Uk)2]”*
Using Eq. (4.66), the spatial derivatives of the travel time evaluated at the
source for this case are
aTjk/dXlA = -(-YE
- xA)/(A . uk)
(4.110) dTjJa~l.4 = -(.vB - v A)/(A uk)
aTjk/aziA = -b2 - UjL)”’/(Z’j.
and the take-off angle [according to Eq. (4.108)] is
(4.111) $. ,k = arcsin(vj/ck)
where the limits for j and k are given in Eq. (4.98).
For the direct wave with earthquake source in the first layer, the spatial
derivatives of the travel time and the take-off angle are the same as those
given in Section 4.4.2 for a constant velocity model.
For the direct wave with earthquake source in the jth layer (with
velocity vj), we find the direct ray path and its associated angle 4 by an
iterative procedure as described previously. The direction angle y at the
source is the take-off angle $ for the direct path. Since the take-off angle is
measured from the positive z direction whereas the angle is measured
from the negative z direction (see Fig. 24), we have
(4.112) yIA = $ = 180” - c#l
Thus, using Eqs. (4. IOS), (4.106), and (4.66), and noting that sin( 180” - 6)
= sin 4, and cos( 180” - 4) = -cos 4, the spatial derivatives of the travel
time evaluated at the source for this case are
dT/dxlA = -[(x,
Uk)
- xA)/(A * uj)] sin $I
(4.1 13) WdYIA = -“v, - Y A)/@ * 41 sin +
aT/azl, = cos 4/vj