Quasi‐analytical treatment of spatially averaged radiation transfer in complex terrain

We provide a new quasi‐analytical method to compute the subgrid topographic influences on the shortwave radiation fluxes and the effective albedo in complex terrain as required for large‐scale meteorological, land surface, or climate models. We investigate radiative transfer in complex terrain via the radiosity equation on isotropic Gaussian random fields. Under controlled approximations we derive expressions for domain‐averaged fluxes of direct, diffuse, and terrain radiation and the sky view factor. Domain‐averaged quantities can be related to a type of level‐crossing probability of the random field, which is approximated by long‐standing results developed for acoustic scattering at ocean boundaries. This allows us to express all nonlocal horizon effects in terms of a local terrain parameter, namely, the mean‐square slope. Emerging integrals are computed numerically, and fit formulas are given for practical purposes. As an implication of our approach, we provide an expression for the effective albedo of complex terrain in terms of the Sun elevation angle, mean‐square slope, the area‐averaged surface albedo, and the ratio of atmospheric direct beam to diffuse radiation. For demonstration we compute the decrease of the effective albedo relative to the area‐averaged albedo in Switzerland for idealized snow‐covered and clear‐sky conditions at noon in winter. We find an average decrease of 5.8% and spatial patterns which originate from characteristics of the underlying relief. Limitations and possible generalizations of the method are discussed.