Integration errors in a three-dimensional model for canopy albedos

Abstract

A three-dimensional model of the radiative transfer within a vegetation canopy gives albedo and bidirectional reflectance for climate and remote-sensing applications. Major sources of numerical error in this model are evaluated in the single-scattering limit by comparison with an analytic solution, and procedures for the amelioration of these errors are developed. The model consists of a three-dimensional array of cells and treats multiple scattering by individual cells through integration over discrete angular sections and iteration of the scattering. Two distinct sources of error are identified: (i) errors in bidirectional reflectance from the integration of absorption and emission along the path of the radiation, and (ii) errors in integrating the bidirectional reflectances over solid angle to synthesize the albedo. Path integration errors may result from neglect of self-absorption by individual cells upon emission, inclusion of insufficient number of cell interactions, and inclusion of too few cells, i.e., unless cells near the outside of the canopy have leaf area index LAI ⪡ 1, the implied gradients of light attenuation are too sharp to be resolved. These errors are minimized by including a self-absorption term, ensuring that the number of interactions are sufficient, selecting enough vertical layers, and providing most resolution in LAI near the outside of the canopy. The spherical integration error results from the coarseness and arrangement of angular sectors and from the procedure used to sum their contributions. Error may be reduced by adding more spherical resolution and by changing the angular integration to one with equal increments in the cosine of the zenith angle. Higher-order integration schemes may also be helpful.