principles and applications of microearthquake networks

4.3. Numerical Solutions of the Ray Equation 85
where
(4.38)
Fl = (1 + z”)-’{(~”/v,[uzy’ - CV( 1 + 2’2) + u,z’y’] + z’”’y’}
F2 = (I + y’*)-l{(p/v)[czz’ - t’*( 1 + y ’2) + v,y’z’] + y’”’z’}
(4.39)
6 = (1 + .s’2 + p ) ’ / 2
(4.40) u, = ar/i3x, t‘, = w ay, v, = au/az
They noted that second- or higher-order systems of ordinary differential
equations could be reduced to first-order systems by introducing auxiliary
variables. They defined
(4.41) y1 = 4‘, y, 3 y; = y‘; z, = z, z, = z; = Z ’
as new variables and reduced Eq. (4.37) to a system of four first-order
equations:
(4.42) y; = Y2, y; = F1. Z; = 22, Z; = F2
This system of equations was solved by using an adaptive finite-difference
program later published by Lentini and Pereyra (1977). For twodimensional
linear velocity models with known analytic solutions for the
minimum time paths, Yang and Lee (1976) showed that their approach
was more accurate and used less computer time than the central finitedifference
technique used by Wesson ( 1971).
Although the parameterization method adopted by Wesson (1971) and
Yang and Lee (1976) had two instead of three second-order equations to
be solved, Julian and Gubbins (1977) pointed out its inherent difficulty. If
the ray path is defined by functions y(s) and z(x) as in Eq. (4.35)’ these
functions can be multiple valued and can have infinite derivatives at turning
points. Therefore, it is better to solve the ray equations parameterized
in arc length such as that given in Eq. (4.32). Starting from Euler’s equation
[Eq. (4.29)], Julian and Gubbins (1977) chose the parameter q to be:
(4.43) q = s/s
and derived the following set of second-order equations:
-Uz(j2 + Z2) + U&;it + u$Z + ux = 0
(4.44)
-14,(X2 + Z2) + u,ty + u,$Z + uj = 0
+ y j + ii’ = 0
where s is an element of arc length, S is the total length of the ray path
between the two ecd points, u is the slowness or the reciprocal of the
velocity, the dot symbol indicates differentiation with respect to q, and u,.