[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

Range of the P-wave anisotropy parameter for finely layered VTI media

James G. Berryman

berryman@sep.stanford.edu

ABSTRACT

Since the work of Postma (1955) and Backus (1962), much is known about elastic constants in vertical transversely isotropic (VTI) media when the anisotropy is due to fine layering of isotropic elastic materials. I review earlier work and then show that the P-wave anisotropy parameter c₁₁/c₃₃ lies in the range ${1\over4} \le c_{11}/c_{33} \le \left<\lambda+2\mu\right\gt\left<1/(\lambda+2\mu)\right\gt$,when the layers are themselves composed of isotropic elastic materials with Lamé constants $\lambda$ and $\mu$ and the vertical average of the layers is symbolized by $\left<\right\gt$. The upper bound is true for all finely layered media of this type, while the lower bound is the best possible for fine layering with two constituents. This lower bound corrects an error in Postma's (1955) paper on two-component VTI media. The method used here shows in general terms that the P-wave anisotropy parameter is smallest when the variation in the layer $\lambda$ Lamé parameter is large, independent of the variation in the shear modulus $\mu$. This result is therefore important for applications to porous layers where the effects of fluids influence only the $\lambda$ Lamé constants, not $\mu$.Thus, the P-wave anisotropy parameter may be a useful hydrocarbon indicator in some situations.