## 1. Introduction

[2] Land surface models are used in atmospheric models to calculate fluxes of heat and moisture as lower boundary conditions for differential equations representing the dynamics and thermodynamics of the atmosphere. Surface state variables such as snow cover and soil moisture are also modelled, and land surface models are increasingly being used to calculate carbon fluxes and storage in coupled climate and carbon cycle models [*Cox et al.*, 2000]. The properties of the land surface and the meteorology of the lower levels of the atmosphere are invariably heterogeneous on the scales of the grids on which these models are applied—kilometers for mesoscale atmospheric models to hundreds of kilometers for global climate models. Several approaches have been adopted for dealing with landscape and meteorological heterogeneity. Distributed models with high-resolution grids (tens of meters) have been used extensively to model processes such as snowmelt [*Kirnbauer et al.*, 1994; *Marks et al.*, 1999], but they are prohibitively expensive for application over large areas. Models using effective parameters describing subgrid surfaces as a whole are much more economical, but due to nonlinearity in the processes involved, there are unlikely to be simple relationships between local and effective parameters [*Shuttleworth*, 1988]. Land surface heterogeneity has been represented by gathering distinct surface types within grid cells into homogeneous tiles for which calculations are performed separately [*Avissar and Pielke*, 1989; *Koster and Suarez*, 1992; *Essery et al.*, 2003], and an analogous approach has been used to account for variations in air temperature and precipitation due to subgrid orography by dividing grid cells into fixed elevation bands [*Arola and Lettenmaier*, 1996; *Essery*, 2003] or using a dynamic snowline [*Walland and Simmonds*, 1996; *Sloan et al.*, 2004]. The Météo France Safran-Crocus-Mepra system [*Durand et al.*, 1999] divides massifs into elevation and aspect bands to model snowpack conditions for avalanche forecasting, but this level of sophistication is unfeasible for large-scale models. *Avissar* [1992] suggested a statistical-dynamical approach to representing surface heterogeneity by integrating the energy balance over probability density functions for surface parameters and illustrated this by coupling a one-dimensional atmospheric model to a land surface model with a distribution of stomatal conductances. Probabilistic approaches have been more commonly used in hydrological models to represent subgrid variations in surface characteristics such as infiltration capacity [*Wood et al.*, 1992]. For meteorological variables, rainfall distributions are often used in models of infiltration and runoff [*Shuttleworth*, 1988], and *Bowling et al.* [2004] used a parametrization of windflow distributions over topography developed by *Essery* [2001] in a model of snow redistribution. An analogous approach might be used to parametrize the influences of spatial variations in meteorology on the surface energy balance.

[3] Solar radiation incident on the land surface under clear skies varies strongly with slope and aspect and on whether or not the surface lies in the shadow of remote topography; the relative importance of these factors in generating spatial variability in surface radiation has been investigated by *Oliphant et al.* [2003]. Solar radiation variations have strong influences on surface energy balances and have been the subject of long-standing interest [*Kondratyev and Manolova*, 1960; *Garnier and Ohmura*, 1968], but it was only with the availability of gridded elevation data sets, powerful computers, and efficient algorithms [*Dozier and Frew*, 1990] that simulations over large grids became possible. Simulations of variations in solar radiation over topography have been used in predicting things as diverse as snowmelt [*Blöschl et al.*, 1991; *Marks et al.*, 1999; *Lundquist and Flint*, 2006], soil temperatures [*Fu and Rich*, 2002], natural vegetation cover [*Franklin*, 1998], crop yields [*Reuter et al.*, 2005], distribution and energy balance of glaciers [*Arrell and Evans*, 2003; *Strasser et al.*, 2004], likely locations for ice deposits in permanent shadows on the moon [*Margot et al.*, 1999] and the influences of variations in illumination on remote sensing [*Baral and Gupta*, 1997; *Agassi and Ben Yosef*, 1997]. Climate models and numerical weather prediction models have, however, almost exclusively assumed the land surface to be flat on subgrid scales for the purpose of calculating the radiative components in their surface energy budgets although *Hauge and Hole* [2003] investigated the influence of allowing for sloping but plane grid cells in a high-resolution mesoscale model, and *Müller and Scherer* [2005] found that implementing a subgrid radiation parametrization in a mesoscale model gave better temperature forecasts. Radiation correction factors can be calculated explicitly in a preprocessing step and then read from lookup tables by an atmospheric model, giving no computational overhead, but this method requires detailed topographic information. This paper presents an alternative approach using functional parametrizations based on simple topographic statistics.

[4] Parametrizations of the spatial average and standard deviation of direct-beam solar radiation were developed by *Dubayah et al.* [1990] for topography with negligible shading. Statistical characterization of shading in mountainous topography was investigated by *Essery* [2004], and that work is extended here to derive statistical characterizations of solar radiation. Distributions of slope components calculated from digital elevation models (DEMs) are discussed in the next section. Parametrizations are then developed for the average and standard deviation of direct-beam solar radiation, shaded fractions, and sky view factors in section 3; results from the new parametrization and a modified form of the parametrization of *Dubayah et al.* [1990] are compared with gridded simulations of solar radiation for four sites of varying topography. Finally, the scaling of the necessary topographic parameters with the resolution of the DEM is discussed in section 4.