= P-SV characteristic reconstruction
{c}
{title2=$(Z_+-Z_-)^{-1}=D(2\rho\omega)^{-1}\operatorname{diag}(1/p,1/s)$}
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.
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