[1] Proof:  

$$\begin{eqnarray} { \enq{\end{eqnarray}$$ (1)

x M A C D E F G K L P R S T V W Int Bdy

Analysis of Thomsen parameters for finely-layered VTI media

James G. Berryman, Vladimir Grechka, and Patricia A. Berge

berryman@sep.stanford.edu, vgrechka@dix.mines.edu, berge@s44.es.llnl.gov

ABSTRACT

Since the work of Postma (1955) and Backus (1962), much has been learned about elastic constants in vertical transversely isotropic (VTI) media when the anisotropy is due to fine layering of isotropic elastic materials. Nevertheless, there has continued to be a degree of uncertainty about the possible range of Thomsen's anisotropy parameters $\epsilon$ and $\delta$ for such media. We show that $\epsilon$ lies in the range $-3/8 \le \epsilon \le {1\over2}\left[\left<v_p^2\right\gt \left<v_p^{-2}\right\gt - 1\right]$, for finely layered media having constant density; smaller positive and all negative values of $\epsilon$occur for media with large fluctuations in the Lamé parameter $\lambda$.We show that $\delta$ can also be either positive or negative, and that for constant density media ${\rm sign}(\delta) = {\rm sign}\left(\left<v_p^{-2}\right\gt - \left<v_s^{-2}\right\gt \left<v_s^2/v_p^2\right\gt\right)$.Among all theoretically possible random media, positive and negative $\delta$ are equally likely in finely layered media limited to two types of constituent layers. Layered media having large fluctuations in Lamé $\lambda$are the ones most likely to have positive $\delta$.Since Gassmann's results for fluid-saturated porous media show that the effects of fluids influence only the $\lambda$ Lamé constant, not the shear modulus $\mu$, these results suggest that positive $\delta$ occurring together with positive but small $\epsilon$ may be indicative of changing fluid content in layered earth.