principles and applications of microearthquake networks

4.4. Computing Travel Time and Derivatives 91
where I A denotes functional evaluation at point A. If we introduce slowness
u as defined by Eq. (4.46), then Eq. (4.26) which defines w becomes
(4.61) N' = u(X2 + ?',2 + Z2)112
By differentiating Eq. (4.61) with respect to X, y and z, respectively, and
using Eq. (4.31), we have
(4.62)
dw/dx = u d.r/ds. a~/aj = IA &/ds
awqai = u dz/ds
Using these relationships, Eq. (4.60) further reduces to
Noting that T is a function of the six end-point coordinates, the variation
6T can be written by the chain rule of differentiation as
(4.64) 6T = (dT/dxl) 6x, + (dT/ay,) 6y1 + (dT/dzl) 6z,
+ (dT/dx,) 6xz + (dT/ dy2) 6y2 + (aT/dz,) 6z2
Because the second end point B is fixed, 6x2 = 6y, = 8z2 = 0, and Eq.
(4.64) reduces to
(4.65) 6T = (dT/dx)I, 6x, + (aT/dy)lA 6y, + (dT/dz)l, 6z,
By comparing Eq. (4.63) with Eq. (4.65), we have
(4.66)
If we solve the ray equation in the form of Eq. (4.56), then Eq. (4.66)
becomes
(4.67)
aT/axl, = -u(o)w~(o),
aT/azl, = - u(O)o,(O)
dT/dyl, = -u(O)w,(O)
where w2, w4, and 06 are the second, fourth, and sixth components of the
solution vector o to Eq. (4.56). Since our normalized parameter t for Eq.
(4.56) is zero at point A, we write (0) to denote functional value at point A.
According to Eq. (4.57), wz(0), w4(0), and ~ ~(0) are the direction cosines of
the ray at the source.
The direction cosines of the seismic ray are the cosines of the direction
angles which are defined with respect to the positive direction of the
coordinate axes. By Eq. (4.34) these direction angles are