Application of Effective Medium Theory to the Three-Dimensional Heterogeneity of Mantle Anisotropy

Abstract:

A self-consistent theory for the effective elastic parameters of stochastic media with small-scale 3D heterogeneities has been developed using a 2nd-order Born approximation to the scattered wavefield (T. H. Jordan, GJI, in press). Here we apply the theory to assess how small-scale variations in the local anisotropy of the upper mantle affect seismic wave propagation. We formulate a anisotropic model in which the local elastic properties are specified by a constant stiffness tensor with hexagonal symmetry of arbitrary orientation. This orientation is guided by a Gaussian random vector field with transversely isotropic (TI) statistics. If the outer scale of the statistical variability is small compared to a wavelength, then the effective seismic velocities are TI and depend on two parameters, a horizontal-to-vertical orientation ratio ξ and a horizontal-to-vertical aspect ratio, η. If ξ = 1, the symmetry axis is isotropically distributed; if ξ < 1, it is vertical biased (bipolar distribution), and if ξ > 1, it is horizontally biased (girdle distribution). If η = 1, the heterogeneity is geometrically isotropic; as η à∞, the medium becomes a horizontal stochastic laminate; as η à0, the medium becomes a vertical stochastic bundle. Using stiffness tensors constrained by laboratory measurements of mantle xenoliths, we explore the dependence of the effective P and S velocities on ξ and η. The effective velocities are strongly controlled by the orientation ratio ξ; e.g., if the hexagonal symmetry axis of the local anisotropy is the fast direction of propagation, then v_PH > v_PV and v_SH > v_SV for ξ > 1. A more surprising result is the 2nd-order insensitivity of the velocities to the heterogeneity aspect ratio η. Consequently, the geometrical anisotropy of upper-mantle heterogeneity significantly enhances seismic-wave anisotropy only through local variations in the Voigt-averaged velocities, which depend primarily on rock composition and not deformation history.