P-SV characteristic reconstruction ($(Z_{+}-Z_{-}{)}^{-1}=D(2\rho \omega {)}^{-1}\mathrm{diag}(1/p,1/s)$)

For a mixture of forward and backward elastic waves, $\mathbf{v}={\mathbf{v}}^{+}+{\mathbf{v}}^{-}$ and $\mathbf{t}=-Z_{+}{\mathbf{v}}^{+}-Z_{-}{\mathbf{v}}^{-}$. The diagonal inverse displayed above gives ${\mathbf{v}}^{+}=-(Z_{+}-Z_{-}{)}^{-1}(\mathbf{t}+Z_{-}\mathbf{v})$ and ${\mathbf{v}}^{-}=(Z_{+}-Z_{-}{)}^{-1}(\mathbf{t}+Z_{+}\mathbf{v})$. It is well-defined when both modes propagate strictly away from grazing. At a vanishing vertical wavenumber its singularity signals a failure of this strict directional split.