Exact results in terms of layer elasticity parameters

Combining results from Eqs. (18)-(15), we find after some work on rearranging the terms that

$$\begin{eqnarray} {{a+b-c}\over{f}} = \left<{{\lambda}\over{\lambda + 2\mu}}\righ... ... - \left<{{\mu}\over{\lambda + 2\mu}}\right\gt^2\right\}\right], \end{eqnarray}$$ (33)

where the correction involving $m - \mu$ in the numerator is the difference of the shear modulus from the layer-averaged shear modulus m, and will be the dominant correction when fluctuations in $\mu$ are small. The fact that $\left<(m - \mu)/\mu\right\gt = \left<\mu\right\gt\left<1/\mu\right\gt -1 \ge 0$, suggests that this dominant correction to unity (since the leading term is exactly unity) for this expression will be positive if $\lambda$ and $\mu$ are positively correlated throughout all the layers, but the correction could be negative in cases where there is a strong negative correlation between $\lambda$ and $\mu$. On the other hand, the term in curly brackets in (33) is again in Cauchy-Schwartz form (i.e., $\left<\alpha^2\right\gt\left<\beta^2\right\gt - \left<\alpha\beta\right\gt^2 \ge 0$), and therefore is always non-negative. But, since it is multiplied by -1, the contribution to this expression is non-positive. This term is also quadratic in the deviations of $\mu$from its layer average, and thus is of higher order than the term explicitly involving $m - \mu$. 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 layering are (as expected) the main source of deviations of (33) 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.