The Fn method for the one-angle radiative transfer equation applied to plant canopies

Abstract

Solutions of the radiative transfer equation describing photon interactions with vegetation canopies are important in remote sensing since they provide the canopy reflectance distribution required in the interpretation of satellite-acquired information. Although the most widely used transport models such as the moments formulations collapse the photon directional information into a limited number of directions (usually two or four), the more sophisticated methods such as discrete ordinates can, in principle, employ an unlimited number of discrete directions. These discrete methods, however, also contain spatial discretization error and lack testing against more accurate Numerical formulations for specific vegetation canopy scattering kernels. In this article, we consider a semianalytical approach to the solution of the one-angle radiative transfer equation in slab geometry called the F n method. This method has a its basis two integral equations specifying the intensities exiting the vegetation canopy boundaries. The solution is then obtained through an expansion in a set of basis functions with expansion coefficients to be determined. These coefficients are obtained from a collocation procedure resulting in a set of algebraic equations solved by matrix inversion. The advantage of this method is that only discretization in the angular variable is required, thus avoiding spatial truncation error entirely. The paper begins by considering a canopy where all the leaves are oriented at the same angle. Lambertian scattering will be assumed with unequal leaf reflectance and transmittance. This simple model contains all the difficulties of the more complex model incorporating a general leaf angle distribution which is to be considered in the latter part of the presentation. A sensitivity analysis is performed including variation of the numerical and model parameters. In addition, discrete ordinates calculations as well as field measurements are compared to the F n results.