Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 312

or, upon division by the length { } 1/2 ,

{ } -1/2 d/d [ c -1 { } -1/2 dx/d ] + c -2 c x = 0 (6.28)

Now, since Equation 6.17 implies that
(dx/d ) 2 + (dy/d ) 2 +( dz / d ) 2 } 1/2 d = ds (6.29)
Equation 6.28 simplifies to the requirement
d/ds {c -1 dx/ds} + c -2 c x = 0 (6.30)
for the actual path between two points, taken by the signal which renders the
time of travel stationary (or less correctly, the ray representing the path which
requires the least time for a signal to travel from the first point to the second).

6.2.3 Eikonal solution satisfies least time condition.
So far, we have only asked for the constraints needed to minimize the
integral or definition in Equation 6.19, and the result is the requirement defined
by Equation 6.30. How can this requirement be achieved in reality? The
dynamics of the physical problem have not yet been invoked in deriving
Equation 6.30. At this point , we consider Equation 6.15a, which states that (1/c)
dx/ds = / x. Together with Equation 6.15d, this implies that d{(1/c)dx/ds}/ds
= - c -2 c x (x,y,z), which is identical to variational condition required by Equation
6.30 ! Similar arguments apply to the y and z components of the wave flow. But
the bottom line is this: solutions of the eikonal equation in Equation 6.5, which
arise from Equation 6.1, satisfy Fermat's “Principle of Least Time.” Equation
6.1 with a variable c(x,y,z), again, assumes isotropic, nondissipative media, and
it is this specialized wave model that is consistent with Fermat' s principle.

6.3 Fermat's Principle Revisited Via Kinematic Wave Theory
In the previous section, we found using the calculus of variations that the
integral in Equation 6.19 can be minimized provided Equation 6.30 is satisfied.
Then, we showed that solutions to the eikonal equation based on the classical
undamped wave model in Equation 6.1 satisfied this requirement. Here, we will
rederive Fermat's Principle starting with the kinematic wave theory formalism
of Chapter 2. The three-dimensional KWT ray equations are given in Equations
2.147 to 2.152; for simplicity, we discuss the x-component of wave propagation
only. Equations 2.147 and 2.150 are
dk x /dt = -

r x

(6.31)

dx/dt =

r kx

(6.32)

Consistently with Equation 6.3, which is strictly applicable to Equation 6.1, we
assume that
r = c(x,y,z){k x 2 + k y 2 + k z 2 } 1/2

(6.33)