principles and applications of microearthquake networks

4.3. Numericul Solutions of the Ray Equatiotz 87
Let us denote du/ds by G, then according to the chain rule of differentiation
(4.50)
du(r) -
Ge----
du dx du
+--
dy
+--
au dz
ds ax ds ay ds az ds
= u> + u,y + u,i
where the dot symbol denotes differentiation with respect to s, and u, =
au/ax, u, = au/ay, and u, = au/az. Thus Eq. (4.49) may be written as
three second-order differential equations
(4.51) i = U(U, - GX), j; = U(U, - Gj), Z U(U, - GZ)
If we denote the variables to be solved by a vector o whose components
are
(4.52)
then Eq. (4.51) is equivalent to the following six first-order differential
equations :
where G as given by Eq. (4.50) is
(4.54) G = t!d,O2 + 1I,WI 4- 14@.)6
Since in many seismological applications, the travel time between point A
and point B, T = s: u ds, is of utmost interest, we introduce an additional
variable w7 to represent the partial travel time 7, along a segment of the ray
path from point A. The corresponding differential equation is simply Cj7 =
u. With this addendum, the total travel time is given by T = T (at point B)
= w7(S), where S is the total path length, and is computed to the same
precision as the coordinates describing the ray path. To determine S
(which is a constant for a given ray path), we introduce one more variable,
wg = S, and its corresponding differential equation, Cj8 = 0. It is also
computationally convenient to scale the arc length s such that its value is
between 0 and 1. We therefore introduce a new variable t for s such that
(4.55) t = s/s
and we use the prime ('1 symbol to denote differentiation with respect to t.