Seismic anisotropy

Remarks

Example: a transversely isotropic medium with vertical symmetry axis has five independent stiffnesses $C_{11}, C_{33}, C_{44}, C_{66}, C_{13}$ in Voigt notation. Its P-speed is $\sqrt{C_{33}/\rho}$ along the axis and $\sqrt{C_{11}/\rho}$ across it, so it is anisotropic whenever $C_{11} \ne C_{33}$. Thomsen's parameters $\varepsilon = (C_{11} - C_{33})/(2C_{33})$, $\gamma = (C_{66} - C_{44})/(2C_{44})$ and $\delta = \big((C_{13} + C_{44})^2 - (C_{33} - C_{44})^2\big)/\big(2C_{33}(C_{33} - C_{44})\big)$ measure the anisotropy; all vanish for an isotropic medium. Non-example: an isotropic medium has $c_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk})$ with Lamé parameters $\lambda, \mu$. Then $\Gamma = (\lambda + \mu) n n^T + \mu I$, whose eigenvalues $\lambda + 2\mu$, $\mu$, $\mu$ do not depend on $n$, so it shows no anisotropy. Non-example: speed that changes from place to place, but not with direction, is heterogeneity, not anisotropy.