Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 318

6.5.1.1 The low-order limit.
Let us explore the consequences of Equations 6.62 to 6.64 in the low-order
limit, and understand exactly how Fermat's Principle arises as a consequence.
Following Chapter 2, Equation 6.64 is expanded in the form k/ t + r k k/ x =
- r x using the chain rule. Then, since the total differential dk = k/ t dt + k/ x
dx, we have dk/dt = k/ t + dx/dt k/ x. Direct comparison leads to dk x /dt = -
r x and dx/dt = r kx ... that is, Equations 6.31 and 6.32, which led to Fermat' s
Principle of Least Time.

6.5.1.2 Extended eikonal equations.
Now that we understand how the principle of least time arises as a low-
order consequence of KWT , we are in a position to extend the overall model so
that problems with both dissipation and dispersion can be simulated . To do
this, we understand that the frequency in Equation 6.64 must allow an
amplitude dependence which is important over large scales, in addition to the
usual dependence on k and possibly x, as suggested in Equation 6.62. Thus, we
need to write k/ t + / x = 0 as
k/ t + (k,a,x,t)/ x = 0 (6.65)
k/ t + k k/ x + a a/ x + x = 0 (6.66)
When Equation 6.62, modified to include any heterogeneities that may exist in a
particular application, is available explicitly, Equation 6.66 and Equation 6.63
(similarly altered to include the low order radiation stress terms implied in E/ t
+ ( r k (k,x,t)E)/ x = E{2 i + r t (k,x,t)/ r }) provide the required “extended
eikonal equations.” At this order, an additional coupling with wave amplitude
exists, so that the motion in not entirely kinematic or amplitude-independent.

6.5.1.3 Extended eikonal equation in homogeneous medium.
To illustrate the basic ideas with a concrete example, we consider large
space-time wave propagation in a uniform homogeneous medium. In this limit,
the explicit dependence on x disappears, that is,

(k,a,x,t) becomes (k,a), and

x vanishes. The governing equations for k(x,t) and a(x,t) reduce to
k/ t + k/ x + a/ x = 0 (6.67)

k a

where

r (k)

a x

i k (k)/a + 1/2 k x i kk - 1/2 a xx r kk /a

(6.68)

- 1/2 a x k x

r kkk /a - 1/8 k x 2 r kkkk - 1/6 k xx r kkk

a 2 / t + (

r k a 2 )/ x = 2 i (k)a 2

(6.69)

i kk + aa x k x i kkk

- (aa xx

+ 1/4 a 2 k x 2

i kkkk + 1/3 a 2 k xx i kkk )