Stochastic Representations of Seismic Anisotropy: Transversely Isotropic Effective Media Models

Xin Song, & Thomas H. Jordan

Published March 20, 2017, SCEC Contribution #7167

We apply Jordan’s (2015) self-consistent, second-order Born theory to compute the effective stiffness tensor for spatially stationary, stochastic models of 3-D elastic heterogeneity. The effects of local anisotropy can be separated from spatially extended geometric anisotropy by factoring the covariance of the moduli into a one-point variance tensor and a two-point correlation function. The latter is incorporated into the rescaled Kneer tensor, which is contracted against the one-point variance tensor to yield a second-order perturbation to the Voigt average. The theory can handle heterogeneity with orthotropic stochastic symmetry, but the calculations presented here are restricted to media with transversely isotropic (TI) statistics. We thoroughly investigate TI stochastic media that are locally isotropic. If the heterogeneity aspect ratio η is unity, the effective medium is isotropic, and the main effect of the scattering is to reduce the moduli. The two limiting regimes are a 2-D vertical stochastic bundle (η→0), where the P and S splitting ratios are negative, and a 1-D horizontal stochastic laminate (η→∞), where they are positive. The effective-medium equations for the latter yield the second-order approximation to Backus’s (1962) exact solution, demonstrating the connection between Backus theory and self-consistent effective-media theory. Comparisons of the exact and second-order results for non-Gaussian laminates indicate that the approximation should be adequate for moduli heterogeneities less than about 30% and thus valid for most seismological purposes. We apply the locally isotropic theory to data from the Los Angeles Basin to illustrate how it can be used to explain shallow seismic anisotropy. To assess the relative contributions of geometric and local anisotropy to the effective anisotropy, we consider a rotational model for stochastic anisotropic variability proposed by Jordan (2015). In this model, the axis of a hexagonally symmetric stiffness tensor is locally rotated to align it with a 3-D stochastic Gaussian vector field that has TI statistics and depends on a horizontal-to-vertical orientation ratio ξ. If ξ=1, the Voigt average and one-point variance tensor are isotropic, and the effective anisotropy depends only on η. In the vertical-alignment regime (ξ→0), the medium is homogeneous. In the horizontal-alignment regime (ξ→∞), the hexagonal symmetry axes are uniformly dispersed in the horizontal plane, which maximizes the scattering. In the rotational model, the effects of the local anisotropic alignment is at least an order of magnitude larger than the effects of geometric anisotropy. We apply the rotational model to Earth’s inner core. Normal-mode and P-wave estimates of the average anisotropy in the outer half of the inner core can be well fit by a rotational model in which the local anisotropy is specified by ab initio estimate of hexagonal close-packed iron by Martorell et al. (2013). The best match to the normal-mode average is given by an orientation ratio ξ=0.52 with a conditional 95% confidence interval of 0.32-0.85, corresponding to a moderate alignment of the hcp-Fe crystals along Earth’s rotation axis.

Song, X., & Jordan, T. H. (2017). Stochastic Representations of Seismic Anisotropy: Transversely Isotropic Effective Media Models. Geophysical Journal International, 209(3), 1831-1850. doi: 10.1093/gji/ggx112.