principles and applications of microearthquake networks

78 4. Seismic Ray Tracing for Minimum Time Path
4.2. Derivation of the Ray Equation
The seismic ray approach is particularly relevant to the problem of
inverting arrival times observed in a microearthquake network. To carry
out the inversion, the travel time and ray path between the source and the
receiving stations must be known. This information may be obtained by
solving the ray equation between two end points for a given earth model.
Two independent derivations of the ray equation are given next in order to
provide some insight into its solution and applications.
4.2.1.
The wave equation may be transformed to a first-order partial differential
equation known as the eikonal equation whose solutions can be interpreted
in terms of wave fronts and rays (e.g., Officer, 1958, pp. 3642). In
brief, the general three-dimensional wave equation [Eq. (4.5)] has associated
with it the equation of characteristics (Officer, 1958, p. 37) given by
(4.9)
Derivation from the Wave Equation
(d+/dx)Z + (d+/dy)Z + (a$/az)z = (l/V”(ag/at)2
A particularly simple solution for the wave potential a,b in Eq. (4.5) or Eq.
(4.9) is
(4.10) I,IJ = $(UX + by + cz - ut)
where a, b, and c are the three direction cosines. A more general solution
takes the form
(4.11) rcr = rL[WX, Y, 4 - uotl
where W describes the wave front and uo is a constant reference velocity.
If we substitute the solution for t,!~ [Eq. (4.1 I)] into the equation of characteristics
[Eq. (4.9)], we obtain the time-independent equation known as
the eikonal equation
(4.12) (dW/a~)~
+ (dW/dy)* + (aW/dz)’ = vf/v* = n*
where we have defined the index of refraction n = uo/u. The eikonal
equation leads directly to the concept of rays. It is especially useful in the
solution of problems in a heterogeneous medium where the velocity is a
function of the spatial coordinates.
The eikonal equation [Eq. (4.12)] is a first-order partial differential equation,
and its solutions,
(4.13) W(x, y, z) = const