Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 319

+ (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 )
Equations 6.67 to 6.69 must be solved subject to the usual initial and boundary
conditions, which vary from problem to problem.

6.5.1.4 The seismic limit.
Despite the simplifications, Equations 6.67 to 6.69 are nonetheless
complicated, and therefore form the basis for ongoing research. However, they
can be simplified, using facts obtained from empirical observation.

6.5.1.5 Example 6-4: Simple rock formations.
The above transient equations apply to dissipative and dispersive waves in
one-dimensional homogeneous media. Again, in kinematic wave modeling, the
exact differential equations for the particular seismic motion need not be
available: the functions r (k) and i (k) may be determined empirically in the
laboratory if desired. Examples 6.1 to 6.3 show how damping affects phase and
amplitude, but they apply specifically to “ u/ t” dissipation, and then only to the
classical wave equation. This damping may or may not be relevant to all facets
of seismics. In general, different fluid-bearing rocks may damp differently,
depending on pore pressure, fluid type, rock matrix, and so on, and

i (k) must

be left free in a general formulation. The real frequency

r (k), likewise, need

not be simple.
We will explore one specific limit of the foregoing modulation equations.
We will assume that the waves in question are nondispersive , their phase and
group velocities being identical and independent of k. This assumption is non-
controversial. Thus, when “c” is a constant sound speed, we may write
r (k) = ck

(6.70)

so that

r k (k) = c (6.71)
r kk (k) = r kkk (k) = r kkkk (k) = ... = 0 (6.72)
We also invoke the attenuation model used previously, that is, take the
imaginary frequency in the form

i (k) = - c 2 k 2 /(4f 0 ) < 0 (6.73)
For this model, the following derivatives are found,
i k (k) = - c 2 k/(2f 0 ) < 0 (6.74)
i kk (k) = - c 2 /(2f 0 ) < 0 (6.75)