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The correction terms, *i.e.*, those contained in the factor
in (35) for quasi-SV waves in anisotropic media,
are proportional to the factor

| |
(41) |

which is sometimes called the *anellipticity parameter*. Similarly,
we will call the *anellipticity correction*. For the case
of strong anisotropy that we are considering here, the presence of
in (40) just introduces
ellipticity into the move out, but the higher order corrections
that we neglected can introduce deviations from ellipticity --
hence anellipticity.
Clearly, from (40) for quasi-SV-waves [and in layered media
at this order of approximation], the anellipticity parameter
holds all the information about the presence or absence of fluids that
is not already contained in the density factor . So it will be
worth our time to study this factor in more detail. First note that,
after rearrangement, we have the general identity

| |
(42) |

which is true for all transversely isotropic media.
In some earlier work (Berryman, 2003), the author has shown that
it is convenient to introduce two special-purpose effective shear moduli
and associated with *a* and *c*, namely,

| |
(43) |

Furthermore, it was shown that the combination defined by
| |
(44) |

plays a particular role in the theory, as it is only this
effective shear modulus for the anisotropic system
that can also contain information about fluid content.
It turns out that (42) can be rewritten in terms of
this effective shear modulus if we first introduce two more parameters:
| |
(45) |

and
| |
(46) |

Then, (42) can be simply rewritten as
| |
(47) |

This result is analogous to, but distinct from, a product formula
relating
the effective shear modulus *G*_{eff} and the bulk modulus
| |
(48) |

to the eigenvalues of the elastic matrix according to
| |
(49) |

Eq. (49) can be motivated by noting that, in the isotropic
limit, the eigenvalues are 3*K* and .
[Side notes concerning layered
materials: In the isotropic limit, when ,we have , while . So these two
parameters are not the same, but they do have strong similarities in
their behavior. In contrast, , while
in the same limit. It is also possible to show
for layered materials that in general ,with the lower limit being optimum, *i.e.*, attainable.]

Also, since Thomsen's plays an important role in (39),
it is helpful to note that (17) can also be rewritten as

| |
(50) |

which shows that, at least for weakly anisotropic media
(in which case the deviation from unity inside the brackets is neglected),
is very nearly a direct measure of the quantity *c* - *f* - 2*l*.

** Next:** Analysis for isotropic layers
** Up:** INTERPRETATION OF P AND
** Previous:** INTERPRETATION OF P AND
Stanford Exploration Project

5/23/2004