General analysis for VTI media

The correction terms for SV waves in weakly anisotropic media are proportional to the factor

$$\begin{eqnarray} {\cal A} \equiv (a-l)(c-l)-(f+l)^2 = 2c(c-l)(\epsilon-\delta), \end{eqnarray}$$ (47)

which is sometimes called the anellipticity parameter. For the case of weak anisotropy that we are considering here, the presence of this term in (46) just introduces ellipticity into the move out, but the higher order corrections that we neglected can introduce deviations from ellipticity, hence anellipticity.

Clearly, from (46) for quasi-SV-waves [and in layered media at this order of approximation], the anellipticity parameter holds all the imformation about 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

$$\begin{eqnarray} {\cal A} = (f+l)(a + c -2f -4l) + (a-f-2l)(c-f -2l), \end{eqnarray}$$ (48)

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 effective shear moduli $\mu_1^*$ and $\mu_3^*$ associated with a and c, namely,

$$\begin{eqnarray} \mu_1^* \equiv a - m - f \qquad\hbox{and}\qquad 2\mu_3^* \equiv c - f. \end{eqnarray}$$ (49)

Furthermore, it was shown that the combination defined by

$$\begin{eqnarray} G_{eff} = (\mu_1^* + 2\mu_3^*)/3 \end{eqnarray}$$ (50)

plays a special role in the theory, as it is the only effective shear modulus for the anisotropic system that may also contain information about fluid content. It turns out that (48) can be rewritten in terms of this effective shear modulus if we first introduce two more parameters:

$$\begin{eqnarray} {\cal K} = f + l + \left[{{1}\over{a-f-2l}}+{{1}\over{c-f-2l}}\right]^{-1} \end{eqnarray}$$ (51)

and

$$\begin{eqnarray} {\cal G} = \left[3G_{eff} + m - 4l\right]/3. \end{eqnarray}$$ (52)

Then, (48) can be simply rewritten as

$$\begin{eqnarray} {\cal A} = 3{\cal K} {\cal G}. \end{eqnarray}$$ (53)

This result is analogous to, but distinct from, a product formula relating the effective shear modulus G_eff and the bulk modulus

$$\begin{eqnarray} K = f + \left[{{1}\over{a - m - f}} + {{1}\over{c -f}}\right]^{-1} \end{eqnarray}$$ (54)

to the eigenvalues of the elastic matrix according to

$$\begin{eqnarray} \chi_+\chi_- = 6KG_{eff}. \end{eqnarray}$$ (55)

In the isotropic limit for layered materials, 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.

Also, since Thomsen's $\delta$ plays an important role in (45), it is helpful to note that (25) can also be rewritten as

$$\begin{eqnarray} c\delta = -(c-f-2l) \left[1 - {{c - f -2l}\over{2(c-l)}}\right], \end{eqnarray}$$ (56)

which shows that, at least for weakly anisotropic media, $c\delta$ is very nearly a direct measure of the quantity c - f - 2l.