The first-order scattering approximation: A closed-form extension to Beer's law, accurate for weakly scattering media. (March 2021) - Ramírez-Cabrera, M.A.
Highlights: An asymptotic expansion is proposed to solve the Radiative Transfer Equation. Beer's Law is extended, accounting for single scattering events in phase-space. The approximation is accurate for albedo < 0.36 or optical thickness < 0.85 . A novel numerical method is proposed for the weakly scattering range. The method is as versatile and accurate as MC method and as efficient as P1 method. Abstract: This article applies perturbation theory to obtain an asymptotic expansion that approximates the Radiative Transfer Equation (RTE) Green's function. The integral (source function) formulation of the RTE yields a recursive operator; applying the operator once to a point source collimated beam yields a closed-form extension to Beer's Law. In the weakly scattering range (i.e., albedo below 0.36 or domain optical thickness below 0.85), the approximation correctly describes the energy transported in phase space for any phase function (mean squared errors below 10%). Besides the theoretical methodology's value, the closed-form approximation allows developing a deterministic numerical method, free of directional meshes, that computes the global irradiance field in three-dimensional enclosures under different internal and boundary conditions. For the cases tested, its accuracy and versatility are comparable to a Monte Carlo method in the weakly scattering range, while its computational efficiency resembles that of the P 1 approximation.