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The equations of motion and their solutions for seismic waves in
anisotropic media are well known, and have been derived in many places including
Berryman (1979) and Thomsen (1986). The dispersion relations for
phase velocities are
 
(29) 
for quasicompressional (+) waves and quasiSV () waves
(i.e., vertically polarized quasishear waves, by which we mean
the plane normal to the crossproduct of the polarization vector and
the propagation vector is vertical) and
 
(30) 
for horizontally polarized shear waves. In these equations,
is the overall
density (including fluids when present),
is the angular frequency, k_{1} and k_{3} are horizontal
and vertical wavenumbers (respectively), and the phase velocities are
determined simply by with .Elastically, the SH wave depends only on the two parameters
l and m, which are not dependent in any
way on layer Lamé parameter and, therefore, will play no role in the
poroelastic analysis. The densities of any fluids present
affect all three wave speeds equally, and cannot therefore
contribute to shear wave birefringence
by itself. Thus, we can safely ignore SH except when
we want to check for shear wave splitting  in which
case the SH results will be most useful as a baseline for such comparisons.
The dispersion relations for quasiP and quasiSVwaves can be rewritten in a
number of instructive ways. One of these that we will choose for
reasons that will become apparent shortly is
 

 (31) 
Written this way, it is obvious that the following two relations
hold:
 
(32) 
and
 
(33) 
either of which could have been obtained directly from (29)
without the intermediate step of (31).
We are motivated to write the equations in this way in order to try to
avoid evaluating the square root in (29) directly. Rather,
we would like to arrive at a natural approximation that is quite
accurate, but does not involve the square root operation. The desire
to do this is not new (Thomsen, 1986), but our goal is different since
we must necessarily treat strong anisotropy in this paper.
From a general understanding of the problem, it is clear that a reasonable
way of making use of (32) is to make the identifications
 
(34) 
and
 
(35) 
with still to be determined. Then, substituting these
expressions into (33), we find that
 
(36) 
Solving (36) for would just give the original
results back again. So the point of (36) is not to solve it
exactly, but rather to use it as the basis of an approximation scheme.
If is small, then we can presumably neglect it inside the
parenthesis on the left hand side of (36) 
or we could just keep a small number of terms in an expansion.
The leading term, and the only one we will consider here
(but see the Appendix for further discussion), is
 
(37) 
The numerator of this expression is known to be a positive quantity
for layers of isotropic materials (Postma, 1955; Berryman, 1979).
Furthermore, it can be rewritten (without approximation)
in terms of Thomsen's parameters as
 
(38) 
Using the first of the identities noted earlier in (5),
we can also rewrite the first elasticity factor in the denominator as
. Combining these results in the
limit of (for relatively small horizontal offset), we find that
 
(39) 
and
 
(40) 
with for very small angles
from the vertical.
These two equations may be recognized simply as small angle
approximations to the weakanisotropy equations of Thomsen (1986).
However, the main thrust of this paper (as we will soon see)
requires strong anisotropy and therefore also requires improved
approximations, which can be obtained to any desired order with
only a little more effort by using (36) instead of the first approximation derived here in (37).
Note that Eqs. (39) and (40) were derived without
assumptions about the smallness of or .
Although the approximations being discussed in this section are of
some practical interest in their own right, their elaboration at this
point would lead us away from the main theme of the paper. So,
to avoid further digression here from the issue of fluid effects on
shear modulus, we collect our remaining results concerning these
dispersion relation approximations in the Appendix.
Next: INTERPRETATION OF P AND
Up: Berryman: Seismic waves in
Previous: Approximate results if Thomsen
Stanford Exploration Project
5/23/2004