principles and applications of microearthquake networks

80 4. Seismic Ruy Trucing for Minimum Time Path
following set of equations from Eq. (4.19):
These are three members of the ray equation in which the index of refraction
n characterizes the medium.
4.2.2. Derivation from Fermat’s Principle
The ray equation may also be derived from Fermat’s principle. This
approach is most appealing when one investigates the ray path and travel
time between two end points as required in several seismological applications.
The derivation is well known in optics (e.g., Synge, 1937, pp. 99-
107) and is shown in detail by Yang and Lee (1976, pp. 6-15). Our treatment
here is brief.
One basic assumption in the mathematical derivation concerns the functional
properties of the velocity c’ in a heterogeneous and isotropic medium.
We assume that the velocity (1) is a function only of the spatial
coordinates, and (2) is continuous and has continuous first partial derivatives.
The time T required to travel from point A to point B is given by
(4.22)
where ds is an element along a ray path. For a minimum time path,
Fermat’s principle states that the time T has a stationary value.
Let us introduce a new parameter, q, to characterize a ray path, and
write
(4.23) x = x(q), y = y(q). z = z(q)
An element of the ray path may then be written as
(4.24)
where the dot symbol indicates differentiation with respect to the parameter
q. Equation (4.22) now becomes
(4.25)
where
(4.26)
and q = q1 at A, andq = q2 at B.
dLC. = (-i2 + +2 + Z?)1,’2
dq