The influence of topography on the radiation balance in complex terrain has so far been investigated either with very simple or very sophisticated approaches that are limited, respectively, by an uncontrolled spatial representation of radiative fluxes or heavy computational efforts. To bridge this gap in complexity, this paper proposes the radiosity approach, well known in computer graphics, to study anisotropic reflections of radiation in complex terrain. To this end the radiosity equation is rederived in the context of three-dimensional radiative transfer. The discretized equation is solved by means of an adapted version of progressive refinement iteration. To systematically study terrain effects, the geometrical disorder provided by the topography is considered in its simplest approximation by Gaussian random fields. These model topographies capture the most important length scales of complex terrain, namely a typical elevation and a typical valley width via the variance and the correlation length of the field, respectively. The mean reflected radiation is computed as a function of these length scales and sun elevation, thereby explicitly addressing finite system sizes and grid resolutions. A comparison with an isotropic parameterization of terrain reflections reveals that mean values are similar whereas spatial distributions vary remarkably. It is also shown that the mean reflected radiation in real topography is reasonably well characterized by the Gaussian approximation. As a final application of the method, the effective albedo of a topography is shown to vary with sun elevation and domain-averaged albedo, leading to albedo differences up to 0.025.

Fractional snow-covered area (SCA) is a key parameter in large-scale hydrological, meteorological and regional climate models. Since SCA affects albedos and surface energy balance fluxes, it is especially of interest over mountainous terrain where generally a reduced SCA is observed in large grid cells. Temporal and spatial snow distributions are, however, difficult to measure over complex topography. We therefore present a parameterization of SCA based on a new subgrid parameterization for the standard deviation of snow depth over complex topography. Highly resolved snow depth data at the peak of winter were used from two distinct climatic regions, in eastern Switzerland and in the Spanish Pyrenees. Topographic scaling parameters are derived assuming Gaussian slope characteristics. We use computationally cheap terrain parameters, namely, the correlation length of subgrid topographic features and the mean squared slope. A scale dependent analysis was performed by randomly aggregating the alpine catchments in domain sizes ranging from 50 m to 3 km. For the larger domain sizes, snow depth was predominantly normally distributed. Trends between terrain parameters and standard deviation of snow depth were similar for both climatic regions, allowing one to parameterize the standard deviation of snow depth based on terrain parameters. To make the parameterization widely applicable, we introduced the mean snow depth as a climate indicator. Assuming a normal snow distribution and spatially homogeneous melt, snow-cover depletion (SCD) curves were derived for a broad range of coefficients of variations. The most accurate closed form fit resembled an existing fractional SCA parameterization. By including the subgrid parameterization for the standard deviation of snow depth, we extended the fractional SCA parameterization for topographic influences. For all domain sizes we obtained errors lower than 10% between measured and parameterized SCA.

The spatial distribution of snow in the mountains is significantly influenced through interactions of topography with wind, precipitation, shortwave and longwave radiation, and avalanches that may relocate the accumulated snow. One of the most crucial model parameters for various applications such as weather forecasts, climate predictions and hydrological modeling is the fraction of the ground surface that is covered by snow, also called fractional snow-covered area (fSCA). While previous subgrid parameterizations for the spatial snow depth distribution and fSCA work well, performances were scale-dependent. Here, we were able to confirm a previously established empirical relationship of peak of winter parameterization for the standard deviation of snow depth σ_HS by evaluating it with 11 spatial snow depth data sets from 7 different geographic regions and snow climates with resolutions ranging from 0.1 to 3 m. An enhanced performance (mean percentage errors, MPE, decreased by 25 %) across all spatial scales ≥ 200 m was achieved by recalibrating and introducing a scale-dependency in the dominant scaling variables. Scale-dependent MPEs vary between −7 % and 3 % for σ_HS and between 0 % and 1 % for fSCA. We performed a scale- and region-dependent evaluation of the parameterizations to assess the potential performances with independent data sets. This evaluation revealed that for the majority of the regions, the MPEs mostly lie between ±10 % for σ_HS and between −1 % and 1.5 % for fSCA. This suggests that the new parameterizations perform similarly well in most geographical regions.