## 1. Introduction

[2] The importance of including terrain effects into the shortwave radiation balance in complex terrain has been widely known for a long time [*Dozier and Outcalt*, 1979]. Local incoming fluxes might be strongly reduced at locations which are shadowed by remote terrain or might be significantly enhanced at locations which receive additional reflected radiation from adjacent terrain. The latter effect is particularly important for snow-covered areas where surface albedos are high. As a result, terrain effects generally increase spatial heterogeneities of local incoming fluxes when compared to flat surfaces. But also spatially averaged values of incoming and reflected fluxes change due to the presence of terrain. As addressed in*Weihs et al.* [2000]for UV radiation, the so-called effective albedo of a large mountainous domain is lower than a simple area average of the surface albedo. Similar terrain effects on the albedo must be taken into account for remote sensing application [*Wen et al.*, 2009]. The notion of an effective albedo has important consequences for coarse-resolution meteorological, land surface, or climate models which do not fully resolve the topography and resort to so-called subgrid parametrizations to include terrain effects. The impact of resolved topography on radiation transfer in numerical weather prediction has been recently addressed by*Manners et al.* [2012]for the Met Office Unified Model. In particular for snow cover processes in mountainous terrain, as a sensitive indicator for climate change, subgrid topography increases the model efficiency of large-scale models [*Parajka et al.*, 2010].

[3] The impact of terrain effects on the radiation balance is certainly best investigated by photon tracing simulations [*Chen et al.*, 2006; *Liou et al.*, 2007]. A recent application of the Monte Carlo approach specifically highlighted its relevance for applications in climate modeling [*Lee et al.*, 2011]. However, the computational complexity of these methods still prevents their direct incorporation into large-scale models [*Lee et al.*, 2011] and the analysis of Monte Carlo simulations must eventually resort to empirical regressions to relate topographic parameters to simulated fluxes. To connect sophisticated simulations to simple subgrid parametrizations it would be desirable to aim at simplified model systems of radiation transfer in complex terrain, yielding simple, analytical parametrization formulas which guide the development of parametrization schemes.

[4] On a semiempirical level a large number of studies has been hitherto devoted to the parametrization of terrain effects in complex topography [*Dozier and Frew*, 1990; *Dubayah et al.*, 1990; *Olyphant*, 1986; *Müller and Scherer*, 2005; *Essery and Marks*, 2007]. Most of them include shadowing and limited sky view as the most important geometric influences of the topography. However, shadowing and limited sky view must be computed from the horizon line, an inherent nonlocal quantity. Since horizons cannot be computed from nearest neighbor heights of the underlying digital height model (DHM), the incorporation of shadowing is less straightforward. For some parametrizations the degree of simplification remains unclear and their relation to Monte Carlo approaches can barely be put on firm theoretical grounds. Toward a remedy *Helbig et al.* [2009] have derived the radiosity approach under controlled simplifications from generic radiative transfer in complex terrain. It can be shown that the radiosity approach compares reasonably well to Monte Carlo simulations for point measurements [*Helbig et al.*, 2010] for clear sky days. On the other hand domain averages within the radiosity approach compare well to a parametrization developed in *Helbig and Löwe* [2012] which is based on the sky view factor and the parametrization by *Dubayah et al.* [1990] for the direct flux. The validation has been carried out for Gaussian random fields as model topographies which could be shown to capture relevant geometrical aspects of realistic complex terrain [*Helbig and Löwe*, 2012]. However, the parametrization of [*Dubayah et al.*, 1990] and likewise [*Helbig and Löwe*, 2012] do not include partial shading of the terrain by remote topography for low Sun elevations. In addition the sky view factor is explicitly contained as a parameter which must be determined in advance.

[5] In this paper we present a new, quasi-analytical method to derive parametrizations for all radiation components and the sky view factor in complex terrain from the radiosity equation on Gaussian random fields. By accepting the underlying simplifications in the first place we can make significant progress from a well-defined mathematical framework. To this end we show how the effective albedo, as required for coarse resolution models, originates from geometrical properties of the topography in a high-resolution model. Thereby domain-averaged radiation fluxes can be solely expressed in terms of slope characteristics, i.e., local quantities. This essential step is accomplished by relating domain-averaged fluxes in complex terrain to a type of level-crossing probability of the topographic surface. Thereby nonlocal horizon effects, namely, sky view factor and shadowing are treated implicitly and related to integrals over the level-crossing probability. This constitutes a main difference to [*Essery and Marks*, 2007] where the horizon must be computed explicitly from the DHM. By using long-standing results put forward for acoustic scattering from sea surfaces we are able to compute the integrals numerically. This enables us to derive practical formulas for the radiation components and the effective albedo solely in terms of the mean-square slope. These parametrizations include partial shading of the terrain for low Sun elevations close to sunset. Our results enable a straightforward application in large-scale models by efficient DHM preprocessing without prior computation of sky view factors. This is demonstrated by computing the effective albedo for the entire Swiss Alps.

[6] Our method requires that the grid size of the coarse model is sufficiently large compared to the correlation length of the subgrid topography. Additionally, by using the radiosity approach as presently formulated in *Helbig et al.* [2009] we neglect atmospheric effects and thus focus on the influence of topography under clear sky conditions. Similar to [*Lee et al.*, 2011] we assume however that our results provide a reasonable first-order estimate for the shortwave fluxes. Some limitations of our approach can however be overcome with existing generalizations of the present level-crossing framework.