Scattering of eletromagnetic waves from a random surface

Suppose $Z\left ( {x,y} \right )$ is a real random process, continuous in the mean over a finite region $D$, and with mean value zero and covariance function $r\left ( {x,y;x’,y’} \right )$. As such $Z\left ( {x,y} \right )$ defines a random surface (provided certain differentiability conditions are satisfied). Then, by Karhunen’s theorem on the representation of a random function, $Z\left ( {x,y} \right )$ has an expansion in terms of an orthogonal process and the eigenfunctions and eigenvalues of the covariance function. Introducing this expression into the far-zone form of the Stratton-Chu solution of the electromagnetic field equations then leads to an approximate expression for the radiation scattered from the random surface, from which mean and covariance of the scattered field can be determined.