Analysis of Thomsen parameters for finely layered VTI media

Since the work of Postma and Backus, 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 some uncertainty about the possible range of Thomsen's anisotropy parameters epsilon and delta for such media. We use both Monte Carlo studies and detailed analysis of Backus' equations for both two- and three-component layered media to establish the results presented. We show that epsilon lies in the range -3/8 less than or equal to epsilon less than or equal to 1/2[< v(p)(2)>< v(p)(-2)>-1], for finely layered media; smaller positive and all negative values of epsilon occur for media with large fluctuations in the Lame parameter lambda in the component layers. We show that delta can also be either positive or negative, and that for constant density media, sign (delta) = sign (< v(p)(-2)> - < v(s)(-2)>< v(s)(2)/v(p)(2)>). Monte Carlo simulations show that among all theoretically possible random media, positive and negative delta are equally likely in finely layered media. (Of course, the delta s associated with real earth materials may span some smaller subset of those that are theoretically possible, but answering this important question is beyond our present scope.) Layered media having large fluctuations in lambda are those most likely to have positive delta. This is somewhat surprising since epsilon is often negative or a small positive number for such media, and we have the general constraint that epsilon - delta > 0 for layered VTI media. Since Gassmann's results for fluid-saturated porous media show that the mechanical effects of fluids influence only the Lame parameter lambda, not the shear modulus mu, these results suggest that small positive delta occurring together with small positive epsilon (but somewhat larger than delta) may be indicative of changing fluid content in a layered earth.