Gaussian approximation for correlation function of apparent antenna pattern over beyond-the-horizon tropospheric path

It is well known that in free space the signal amplitude variation at the output of the receiving antenna, located in a far zone, is similar to the scanning transmitting antenna pattern (AP). In the case of wave scattering over a beyond-the-horizon path the amplitude becomes a random function and corresponds to the apparent antenna pattern (AAP). As a rule, the observed AAP is wider than the AP, so estimation of relative AP broadening has become a conventional method for its description. There have been some attempts to classify AAPs according to the character of the distortions. From the statistical point of view the correlation function contains more information. The objective of this paper is to test the possibility of using a bivariate Gaussian function to represent the limiting form of the correlation function for a narrowbeam antenna under the condition of multiple wave scattering. This gives a simple way to write down the Karhunen-Loe`ve expansion in terms of the set of the orthonormal Hermite functions and the set of eigenvalues