principles and applications of microearthquake networks

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84 4. Seismic Ray Tracing for Minimum 7ime Path
order to study the propagation of seismic waves in island arc structures.
Engdahl (1973) and Engdahl and Lee (1976) applied the ray tracing technique
as developed by Julian (1970) to relocate earthquakes in the central
Aleutians and in central California, respectively.
4.3.2. Boundary Value Formulation
In many seismological applications, tracing seismic rays between two
end points is required. It is therefore natural to seek a solution of the ray
equation with the spatial coordinates of the two end points as boundary
conditions. For example, Wesson (1971) presented a boundary value formulation
for two-point seismic ray tracing in a heterogeneous and isotropic
medium. He noted that the three second-order members of the ray
equation [Eq. (4.32)] are not independent, because the direction cosines
[Eq. (4.34)] are related by Eq. (4.16). If the ray path is parameterized by
(4.33 y = y(x,, z = z(x)
and the coordinate system is transformed such that the two end points lie
in the xz plane, then Eq. (4.32) reduces to two second-order equations
”[
3 +
Yf + y12 + Z ‘ y - au =o
dx v(l + y” + z‘*)”~ V2 dY
(4.36)
”[
I +
Zf
(1 + yf‘ + 2’2)’’‘
-=o av
dx v(l + yf2 + 2”)”’ V2 aZ
where y’ = dy/dx, and zr = dz/dx.
By using central finite differences to approximate the derivatives in Eq.
(4.36), Wesson (1971) derived a set of nonlinear algebraic equations which
he solved by an iterative procedure. For two selected areas in California,
he was able to construct theoretical velocity models that are consistent
with travel time data from explosions and with the known geological
structure.
Chander (1975) used a method due originally to L. Euler that applied a
sum to approximate the integral for the travel time and solved for the
minimum time path directly. Yang and Lee (1976) showed that this formulation
was equivalent to the central finite-difference approximation used
by Wesson (1971).
Yang and Lee (1976) introduced a different technique for solving the
two-point seismic ray tracing problem. In order to reduce the ray equation
to a set of first-order equations they recast Eq. (4.36) into
(4.37)
y” = F,(y’, z’, z”, v, v,, v,, v,)
z” = F&’, Yl, y”, v, v,, u,, v,)