Stochastic Representations of Seismic Anisotropy: Locally Isotropic Transversely Isotropic Media

Xin Song, & Thomas H. Jordan

Submitted August 15, 2016, SCEC Contribution #6860, 2016 SCEC Annual Meeting Poster #236

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 [Jordan,2015]. Our model assumes the medium can be represented as a spatially homogeneous, transversely isotropic random field with a covariance tensor that can be factored into a one-point, tensor-valued variance and a two-point, scalar-valued correlation function. In the low-frequency limit, where the seismic wavenumbers are small compared to the characteristic wavenumbers of the heterogeneity, the locally isotropic stochastic medium can be replaced by a homogeneous “effective medium” with a transversely isotropic (TI) stiffness tensor that depends only the one-point covariance matrix of the two Lamé parameters and a dimensionless number η that measures the horizontal-to-vertical aspect ratio of heterogeneity. If η = 1, the heterogeneity is geometrically isotropic. The scattering caused by a ~ 10% variation in seismic velocities reduces the effective velocities by about 1%. As η --> 0, the medium is stretched into a vertical stochastic bundle, and as η --> ∞, the medium is flattened into a horizontal stochastic laminate. In the latter limit, the expressions for the anisotropic effective moduli reduce to Backus’s (1962) second-order expressions for a 1D stochastic laminate. Comparisons with the exact Backus theory show that the second-order approximation predicts the effective anisotropy for non-Gaussian media fairly well with an error less than 10% for media with velocity fluctuations less than about 15%, which should be adequate for most crustal studies. We also note that the 2nd-order theory is exact for heterogeneities that are gamma-distributed. We apply the theory to heterogeneities in the Los Angeles basin determined from well-log analysis [e.g. Plesch et al., 2014] and compare the predicted anisotropy with seismic constraints [Chen et al., 2007]. The observed shear wave splitting ratios overlaps the values calculate from effective medium theory with the relative Vp variations about 4-7% and η in the range of 5 to 200.

Song, X., & Jordan, T. H. (2016, 08). Stochastic Representations of Seismic Anisotropy: Locally Isotropic Transversely Isotropic Media. Poster Presentation at 2016 SCEC Annual Meeting.

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