Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 317

6.5 Ray Tracing Over Large Space-Time Scales
The low-order kinematic wave theory described by Equations 2.41 to 2.70
is consistent with the fact that wave dissipation does not affect phase at early
times. Over large space and time scales, the examples of Chapter 1 show that
ray trajectories are affected, as are the corresponding amplitudes calculated
along wave fronts. In surface seismics , typical propagation distances are on the
order of tens of feet, and phase changes are not important. In crosswell
seismics , however, ray paths extend hundreds and thousands of feet, so that
nondissipative interpretation models, e.g., ( / x) 2 + ( / y) 2 + ( / z) 2 =
1/c(x,y,z) 2 , can be inherently incorrect whenever attenuation arising from oil
and gas-bearing formations exists. Higher order accuracy is required. In
tomographic analysis, care must be taken to account for the cumulative effects
of locally small dissipative terms, a consideration emphasized in many areas of
continuum mechanics (Whitham, 1974). For simplicity, we will restrict
ourselves to one-dimensional applications; extensions to three dimensions,
implicit in Equations 2.130 to 2.158, will be left as exercises for reader.

6.5.1 High-order modulation equations.
In simple one-dimensional problems, the uniform plane wave is in general
characterized by a complex dispersion relation of the form
(k) = r (k) + i i (k) (6.61)
We showed in Chapter 2 that the slowly varying wavetrain (for example, the
large-time asymptotic solution for an exploding point source) satisfies the
coupled phase and amplitude laws given by Equations 2.94 and 2.95. Again, the
phase relation and amplitude law shown, that is,

r (k)

a x

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

(6.62)

- 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 = i (k)a 2

( 2 (6.63)

- (aa xx

i kk + aa x k x i kkk

+ 1/4 a 2 k x 2

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

+ (1/3 aa xxx

r kkk + 1/2 aa xx k x r kkkk

+ 1/3 aa x k xx

r kkkk

+ 1/12 a 2 k xxx

r kkkk

+ 1/6 a 2 k x k xx

r kkkkk + 1/4 aa x k x 2 r kkkkk

+ 1/24 a 2 k x 3 r kkkkkk )
are coupled through the “wave crest conservation” law
k/ t + / x = 0 (6.64)