principles and applications of microearthquake networks

88 4. Seismic Ray Tracirig for Minimum Time Path
Thus the final set of eight first-order differential equations to be solved is
0; = 0 gw29 0; = 0 806
Wk = 08u(u, - GWZ),
0; = o~z'(u, - Gos)
(4.56) 6.J; = 0 804. 0; = 08u
W& = w ~z)(u~ - Gw~), w; = 0
t E [O, 11
The variables corresponding to the solution of these equations are
(4.57)
0 1 = x, o2 = dx/ds, w3 = y, 04 = dy/ds
o5 = z, wg = dz/ds, 07 = T, 0 8 = s
The boundary conditions are
0,(0) = xA, 03(0) = )'A, w5(0) = ZA
(4.58) ol(1) = Xg, Wg(1) = YB, Wg(1) = ZB
w,(O) = 0, w;(o) + wZ(0) + &O) = I
where the coordinates for pcint A are (xA, yA, zA) and those for point B are
(xg, yB. z~). The boundary condition for w7 is determined by the fact that
the partial travel time at the initial point A is zero. The last boundary
condition in Eq. (4.58) simply states that the sum of squares of direction
cosines is unity, as given by Eq. (4.16).
Equation (4.53) is a set of six first-order differential equations which are
simple and exhibit a high degree of symmetry. This set of equations is
equivalent to either Eq. (4.33) or Eq. (4.44) and is easier to solve using
recently developed numerical methods. Equation (4.56) is an extension of
Eq. (4.53) so that the travel time and the ray path can be solved together.
Otherwise, the travel time would have to be computed by numerial integration
after the ray path is solved.
Pereyraet a!. (1980) described in detail how to solve a first-order system
of nonlinear ordinary differential equations subject to general nonlinear
two-point boundary conditions, such as Eqs. (4.56)-(4.58). In their firstorder
system solver, high accuracy and efficiency were achieved by using
variable order finite-difference methods on nonuniform meshes which
were selected automatically by the program as the computation proceeded.
The method required the solution of large, sparse systems of
nonlinear equations, for which a fairly elaborate iterative procedure was
designed. This in turn required a special linear equation solver that took
into account the structure of the resulting matrix of coefficients. The
variable mesh technique allowed the program to adapt itself to the particu-