Analysis of Thomsen parameters for finely layered VTI media

Since the work of Postma and Backus, much has been learned about elastic co nstants 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 t hat 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 flu ctuations in the Lame parameter lambda in the component layers. We show tha t delta can also be either positive or negative, and that for constant dens ity 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 rando m media, positive and negative delta are equally likely in finely layered m edia. (Of course, the delta s associated with real earth materials may span some smaller subset of those that are theoretically possible, but answerin g this important question is beyond our present scope.) Layered media havin g large fluctuations in lambda are those most likely to have positive delta . This is somewhat surprising since epsilon is often negative or a small po sitive number for such media, and we have the general constraint that epsil on - delta > 0 for layered VTI media. Since Gassmann's results for fluid-sa turated porous media show that the mechanical effects of fluids influence o nly the Lame parameter lambda, not the shear modulus mu, these results sugg est that small positive delta occurring together with small positive epsilo n (but somewhat larger than delta) may be indicative of changing fluid cont ent in a layered earth.