Scattering matrix coefficient

For anisotropic media, it is necessary to make the distinction between group and phase properties. The equations relating the several reflection and transmission coefficients with the elastic properties of the medium are derived for plane waves, which imply that the coefficients are associated to ratios between phase amplitudes.

One possible approach is to perform a local plane-wave decomposition, integrating the component of the vector-wavefield in the local polarization direction (${\bf j}(x,z,t)$) along different directions p in order to build a function A(p,x,z,t), which represents the phase amplitude for each point of the space-time as a function of the horizontal slowness, and a polarization vector function ${\bf j}(x,z,t)$. Time averaging should in this case provide three expectations:$\tilde{A}(x,z)$, $\tilde{p}(x,z)$, and $\tilde{j}(x,z)$. Next we need to use the Christoffel equation with the background elastic parameters used in the migration to find the wave type associated with each polarization vector. Then, an anisotropic inversion scheme can estimate the elastic parameters that best fit the scattering relation involving $\tilde{A}$ for the different modes and ${\bf j}$, as a function of the horizontal slowness $\tilde{p}$.