Solar Radiative Fluxes for Broken Cloud Fields above Reflecting Surfaces

A statistical bidirectional method for including the effects of underlying reflecting surfaces in Monte Carlo simulations of atmospheric photon transport is presented. It is illustrated for the idealized Lambertian surface and a general particulate surface. It is shown that the simple Lambertian surface should be an adequate approximation (to mildly anisotropic surfaces) for calculation of overall fluxes for broken cloud fields. The problem of incorporating multiple internal reflections of solar radiation between cloud and surface is reviewed, and cloud field reflectances to upwelling radiation rk as a function of the number of internal reflections k are calculated. Unlike plane-parallel and regular arrays of cloud, all rk are generally not equal for more realistic scaling cloud fields. Strictly speaking, this prohibits use of the familiar geometric sum formulas for flux calculation in a multiple reflecting system. Fortunately, though fortuitously, the geometric sum formulas can be used in such cases if all rk are assumed to equal approximately the spherical albedo of the cloud field.