Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 324

6.7 Travel-Time Modeling
While the subject of high-order derivative terms is in itself of fundamental
interest, and perhaps insofar as accurate ray tracing in the presence of
dissipation and dispersion is concerned, it is its role in explaining some
controversial results encountered by recent authors that is by far the most
fascinating.

6.7.1 Applications to crosswell tomography.
In crosswell tomography, an array of acoustic transmitters (or, a single
transmitter moved among several different physical locations) located within a
wellbore, transmits signals that are received at nearby wells instrumented along
their lengths with receiver arrays. “First arrival time data” is used to provide
“velocity estimates” for the formation “velocity model” between the transmitters
and the receivers. In most applications, the velocity function c(x,y,z) is assumed
by trial and error, and the ray equations

dx/dt = c(x,y,z) (6.96)
dy/dt = c(x,y,z) (6.97)
dz/dt = c(x,y,z) (6.98)
are numerically integrated in time. If the computed first arrival times
corresponding to the guess for c(x,y,z) are correct for all transmitter and receiver
points, the conjecture for c(x,y,z) is assumed to be representative of the actual
velocity distribution in the underground formation. Of course, the solution may
be non-unique, but that represents a separate issue. Additional constraints,
beyond those called for in first arrival time matching, can be invoked. One
possibility involves the matching of “relative received energies” detected at the
receivers. These can be calculated using the energy and momentum equations
obtained in Chapter 2, in particular,

E/ t + (

r k (k,x,t)E)/ x = E{2 i + r t (k,x,t)/ r }

(2.104)

M/ t + ( r k (k,x,t)M)/ x = M{2 i - r x (k,x,t)/k} (2.105)
There are two methods for accomplishing the required ray integration
discussed earlier. In “shooting methods” for ray tracing, a “fan” of rays is shot
from one point in the general direction of the other. The denser the fan, the
greater the accuracy, and the higher the probability of successful integration
(there is no guarantee that any particular ray will reach the desired receiver). In
“bending methods,” an (incorrect) guess for the ray path provides an initial
guess. This conjectured path is bent by perturbation methods until it satisfies a
minimum travel-time criterion; thus, the method, following our earlier
arguments, implicitly assumes a nondissipative system. Additional information
about the formation can be obtained from the more complete time trace