= Height propagation of an angular-spectrum correlation
{title2=$m(\xi,z)=(2\pi)^{-1}\int S(\nu)e^{i\nu\xi-2\operatorname{Im}q(\nu)z}d\nu$}
Use $\widehat\psi(\nu)=\int\psi(x,0)e^{-i\nu x}dx$ and $q(\nu)=\sqrt{k^2-\nu^2}$ on the upward branch. If the uncentered field correlation is translationally homogeneous and $\langle\widehat\psi(\nu)\widehat\psi(\nu')^*\rangle=2\pi S(\nu)\delta(\nu-\nu')$, its correlation at height $z$ is the displayed integral. The propagating spectrum is unchanged; evanescent contributions decrease as $e^{-2\sqrt{\nu^2-k^2}z}$. A <spectral measure of a stationary random field> permits discrete coherent components. Without correlation homogeneity a double spectral covariance integral depends on both observation coordinates rather than only their separation.
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