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Combining results from Eqs. (8)(12), we find after
some work on rearranging the terms that
 

 (27) 
where the correction involving in the numerator
is the difference of the shear modulus from the layeraveraged
shear modulus m, and will be the dominant correction when
fluctuations in are small. The fact that
, suggests that this dominant correction to unity (since the
leading term is exactly unity)
for this expression will be positive if and are
positively correlated throughout all the layers, but the correction
could be negative in cases where there is a strong negative
correlation between and .
On the other hand, the term in curly brackets in (27) is
again in CauchySchwartz form (i.e.,
),
and therefore is always nonnegative. But, since it is multiplied by
1, the contribution to this expression is nonpositive.
This term is also quadratic in the deviations of from its layer average, and thus is of higher order than the
term explicitly involving . So, if the fluctuations in
shear modulus are very large throughout the layered medium,
the quadratic terms can dominate  in which case the overall
result could be less than unity. Numerical examples developed by
applying a code of V. Grechka [used previously in a similar context
by Berryman et al. (1999)] confirm these analytical results.
Our main conclusion is that the shear modulus fluctuations giving rise
to the anisotropy due to stacks of thin isotropic layers
are (as expected) the main source of
deviations of (27) from unity. But now we can say more,
since positive deviations of this parameter from unity are generally
associated with smaller magnitude fluctuations of the layer shear
modulus, whereas negative deviations from unity must be due to large
magnitude fluctuations in these shear moduli.
Next: Approximate results if Thomsen
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Stanford Exploration Project
5/23/2004