Multi-Gaussian Representation of the Cox–Munk Distribution for Slopes of Wind-Driven Waves

The Cox–Munk probability density function (PDF) for slopes (γx, γy) of wind-driven ocean waves was obtained about 50 years ago and until now has remained the most complete result. This PDF is widely used in different applications. With respect to the problem of wave scattering from a sea surface, this PDF completely determines the scattering lidar cross sections. For Ku and C bands this PDF is important for radars and thermal radiation at small grazing angles. Unfortunately, the analytical representation of the Cox–Munk PDF has an imperfection because in certain regions of slopes it leads to negative probabilities. This feature is a consequence of the general theorem proven by Marcinkiewicz that any truncated cumulant expansion (except in the case of Gaussian distribution) cannot represent a true PDF. An alternative representation of a non-Gaussian multivariate PDF is used. Any non-Gaussian PDF is presented as a linear combination of Gaussian functions having different weights, mean values γx, γy, and covariance matrixes {〈γ2x〉, 〈γ2y〉, 〈γxγy〉}. To find these parameters, the Cox–Munk PDF is approximated in a region where it is positive as a result of such a linear combination. The results are for a wind speed range between 5 and 15 m s−1. The method used is rather general and can be employed in the wide class of statistical problems dealing with non-Gaussian multivariate distributions. The representation of the PDF presented in this paper allows the calculation in a closed analytical form of all mean values appearing in the theory of electromagnetic wave scattering from the sea surface.