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General analysis for VTI media

The correction terms, i.e., those contained in the factor $\Delta$ in (35) for quasi-SV waves in anisotropic media, are proportional to the factor
{\cal A} \equiv (a-l)(c-l)-(f+l)^2 = 2c(c-l)(\epsilon-\delta),
 \end{eqnarray} (41)
which is sometimes called the anellipticity parameter. Similarly, we will call $\Delta$ the anellipticity correction. For the case of strong anisotropy that we are considering here, the presence of ${\cal A}/(c-l)$ 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 $\rho$. So it will be worth our time to study this factor in more detail. First note that, after rearrangement, we have the general identity
{\cal A} = (f+l)(a + c -2f -4l) + (a-f-2l)(c-f -2l),
 \end{eqnarray} (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 $\mu_1^*$ and $\mu_3^*$ associated with a and c, namely,
\mu_1^* \equiv a - m - f \qquad\hbox{and}\qquad 2\mu_3^* \equiv c - f.
 \end{eqnarray} (43)
Furthermore, it was shown that the combination defined by
G_{eff} = (\mu_1^* + 2\mu_3^*)/3
 \end{eqnarray} (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:
{\cal K} = f + l +
 \end{eqnarray} (45)
{\cal G} = \left[3G_{eff} + m - 4l\right]/3.
 \end{eqnarray} (46)
Then, (42) can be simply rewritten as
{\cal A} = 3{\cal K} {\cal G}.
 \end{eqnarray} (47)
This result is analogous to, but distinct from, a product formula relating the effective shear modulus Geff and the bulk modulus
K = f + \left[{{1}\over{a - m - f}} + {{1}\over{c -f}}\right]^{-1}
 \end{eqnarray} (48)
to the eigenvalues of the elastic matrix according to
\chi_+\chi_- = 6KG_{eff}.
 \end{eqnarray} (49)
Eq. (49) can be motivated by noting that, in the isotropic limit, the eigenvalues are 3K and $2\mu$.

[Side notes concerning layered materials: In the isotropic limit, when $\mu \to constant$,we have $K \to f + 2\mu/3$, while ${\cal K} \to f + \mu$. So these two parameters are not the same, but they do have strong similarities in their behavior. In contrast, $G_{eff} \to \mu$, while ${\cal G} \to 0$ in the same limit. It is also possible to show for layered materials that in general $l \le {\cal K} - f \le m$,with the lower limit being optimum, i.e., attainable.]

Also, since Thomsen's $\delta$ plays an important role in (39), it is helpful to note that (17) can also be rewritten as
c\delta = -(c-f-2l)
\left[1 - {{c - f -2l}\over{2(c-l)}}\right],
 \end{eqnarray} (50)
which shows that, at least for weakly anisotropic media (in which case the deviation from unity inside the brackets is neglected), $c\delta$ is very nearly a direct measure of the quantity c - f - 2l.

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