Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 325

containing reflections, but this subject is beyond the scope of this chapter.
These methods have, in addition to computation intensive requirements,
significant disadvantages. Shooting methods often experience difficulty finding
correct rays in shadow zones, while bending methods do give answers there.
However, in both methods, it is possible that the solution is only a local
minimum, with the global minimum travel time and corresponding ray path
remaining unknown.
Other methods have also appeared in the literature. Like the bending
method, a recent study by Moser (1991) also takes advantage of Fermat's
Principle of Least Time. However, the “network theory based” implementation
is quite novel. The author views the seismic ray trace problem as analogous to
the well known “traveling salesman problem,” where a hypothetical salesman
(hopefully) selects the shortest possible routes from city to city in order to
minimize total travel time. With this analogy in mind, the author also chooses
methods traditionally used to solve the traveling salesman problem, namely,
methods from “network theory.” Refer to the original paper for details.
6.7.2 Applications to surface seismics.
Travel-time calculations play important roles in seismic processing. For
example, Kirchhoff methods of migrating and modeling seismic data require the
calculation of Green' s functions, which in turn depend on travel-times between
surface survey points and depth points in the assumed velocity model. Ray
tracing is often used, but the extensive interpolations required to extrapolate data
to points on regular grids are impractical. For complicated velocity models, rays
may cross each other, or they may not penetrate shadow zones.
6.7.3 Finite difference calculation of travel times.
Rather than directly integrating the ray equations, using shooting or
bending methods, or working with network theory approaches, several authors
recently suggested a direct finite difference attack on the eikonal equation itself,
solving the nonlinear equation
( / x) 2 + ( / y) 2 + ( / z) 2 = 1/c(x,y,z) 2 (6.99)
which we may recognize as Equation 6.5. The reader should refer to Chapter 4
for a review of low-order finite difference methods - in many seismic
applications, higher-order methods (e.g., fourth-order accurate) are used and are
definitely recommended for those in research or software development. In this
approach, wavefronts, rather than traditional rays, are tracked ; head waves are
properly treated, according to early authors, and shadow zones are filled by
appropriate diffractions. We will refer the details of the implementation to the
original papers, e.g., Vidale (1988), Vidale and Houston (1990), Qin et al
(1992), and Coultrip (1993), since they do not fall within the objectives of the
present chapter.