The Derivation of a Multivariate Probability Density Function Having an Exponential-Type Bivariate Marginal Density

摘要

For the calculation of electromagnetic scattering from rough terrain, relations have been established between the scattering cross section and the statistical characterization of the surface. These relations are based on the assignment of a bivariate probability density to the observed surface heights at any two points of a region. To implement a scattering calculation in a region, joint probability density functions that will reduce to appropriate bivariate marginal densities are associated with the distribution of heights in the area. In some instances, however, the required form of the multivariate probability density is not evident, since the original form is not necessarily preserved as successive marginal densities are formed. This report describes the procedures used to determine the multivariate probability density function that reduces to the particular bivariate exponential marginal density form having a known correspondence to electromagnetic scattering. Expressions are given both for the multivariate probability density and for the associated marginal densities of any order. (Author)