principles and applications of microearthquake networks

82 4. Seismic Ray Tracing for Minimum Time Pucth
”[
x
dq U(X2 + y 2 + Z2)”* 1-
(S2 + Y’ + 22)”’ _d (1) z 0
ax
The parameter q along the minimum time path r is still arbitrary. If we
choose q = s, the arc length along r, then in order to satisfy Eq. (4.24), we
must have
(4.3 1) (X2 + j2 + i 2)”2 = 1
along r. Hence, Eq. (4.30) reduces to
(4.32)
When multiplied by a reference velocity uo, Eq. (4.32) is exactly the same
as the ray equation given by Eq. (4.21).
4.3. Numerical Solutions of the Ray Equation
In order to study the heterogeneous structure of the earth, several
different techniques to trace seismic rays in three dimensions have been
developed; for example, see Jackson (1970), Jacob (1970), Julian (1970),
Wesson (1971), Yang and Lee (1976), Julian and Gubbins (1977), Pereyra
et al. (1980), Lee et al. (1982b), and Luk et al. (1982). Some authors
formulated seismic ray tracing as an initial value problem, whereas others
formulated it as a two-point boundary value problem. These two approaches
and a unified formulation are described in the following subsections.
As pointed out by Wesson (1971), tracing seismic rays between two end
points is required in many seismological applications, such as earthquake
location (e.g., Engdahl, 1973; Engdahl and Lee, 1976) and determining
three-dimensional velocity structure under a seismic array (e.g., Aki and
Lee, 1976; Lee et al., 1982a). However, computer programs to trace seismic
rays are complex and time consuming, so they have not been widely
applied in microearthquake studies.