Analytic Framework for Calculating Brdfs of Randomly Rough Surfaces

This report presents an analytic framework for calculating BRDFs (bi-directional reflectance distribution function) of randomly rough surfaces. The fundamental assumptions are the tangent plane approximation and statistical approach sufficiency. By treating a local surface area as its tangent plane, the tangent plane approximation validates the description of waves using geometric rays, and the statistical approach sufficiency validates the calculation of reflection using statistical concepts such as surface height probability and correlation function. Based on these assumptions, we derive a generic equation for reflection due to single scattering at a rough surface. As an example, the generic equation is applied to an isotropic surface with Gaussian statistics to obtain a specific analytic form of BRDF. An interesting result we have obtained is that as the surface smoothness varies, the derived BRDF shows four distinct reflection regimes, namely, mirror reflection, grazing reflection, retro-reflection, and normal reflection. In this report, we have also derived an explicit form of self-shadowing for a surface with Gaussian statistics, and the result agrees well with computer simulation. Our analytic solution for single scattering is useful to describe the entire reflection of an opaque surface that is highly or moderately smooth. It also offers a basis for calculating reflection that involves multiple scattering. While this study focuses on optical reflection, the method and solution also apply to reflections of other forms of waves.