Shear wave splitting

Definition

Let a homogeneous elastic medium with density $\rho$ and stiffness tensor $c_{ijkl}$ show seismic anisotropy. Fix a propagation direction $n$ and let $\Gamma_{ik} = c_{ijkl} n_j n_l$ be the Christoffel matrix, a positive definite symmetric matrix. Suppose two of its eigenvalues belong to quasi-shear waves, with speeds $v_{S1} > v_{S2}$ and orthonormal eigenvectors $e_1, e_2$, by the Christoffel equation. An incident S-wave travelling along $n$ with polarization $q$ in the plane of $e_1, e_2$ is decomposed as $q = (q \cdot e_1) e_1 + (q \cdot e_2) e_2$. After a path of length $L$ the two parts arrive separated by the delay $$\delta t = L\Big(\frac{1}{v_{S2}} - \frac{1}{v_{S1}}\Big) > 0.$$ This separation is shear wave splitting; its splitting parameters are the fast polarization direction $e_1$, usually given by an azimuth $\phi$, and the delay $\delta t$.