**2. Methods and Data**

Most GIS-based solar radiation models provide estimates of solar radiation over large areas using digital terrain models (DTMs) and selected atmospheric and land cover parameters derived from ground-based or satellite-based data [19–22]. These topographic solar radiation models can only be used for 2D surfaces, such as land surfaces or rooftops. There are 2 models implemented in GRASS GIS that are based on the same fundamental basis of solar radiation, but they work differently with the geometric representation of the Earth's surface. The r.sun model is for 2D surfaces, such as terrain or roofs, represented by rasters, while v.sun is for 3D city models represented by 3D vectors [5].

The solar radiation methodology used in the r.sun and v.sun models is based on the European Solar Radiation Atlas (ESRA) methodology [23,24] and described in [8] and [5]. The calculation of the direct (beam) component of solar radiation on surfaces for clear-sky atmospheric conditions *B* (W/m2) is quite straightforward:

$$B = G\_0 \exp\left\{-0.8662 T\_{LK} m \delta\_R(m)\right\} \sin \delta\_{exp} \tag{1}$$

where *G*<sup>0</sup> is the normal extra-terrestrial irradiance outside the atmosphere (W/m2), *TLK* is the air mass 2 Linke atmospheric turbidity factor (-), m is the relative optical air mass (-), *δR*(*m*) the Rayleigh optical thickness at air mass m (-), and *δexp* is the solar incidence angle measured between the sun and a surface.

The diffuse component implemented in this model is empirically derived from European climate conditions. The model for estimating the clear-sky diffuse irradiance on a surface *D* [W/m2] is defined by the following equation [25]:

$$D = G\_0 T\_n (T\_{LK}) F\_d (h\_0) \left\{ F(\gamma\_N) (1 - K\_b) + K\_b \sin \delta\_{exp} / \sin h\_0 \right\} \tag{2}$$

where *Tn*(*TLK*) is a diffuse transmission function dependent on the Linke turbidity factor *TLK*, *Fd*(*h*0) is a diffuse solar altitude function dependent on the solar altitude *h*0, *F*(*γN*) is a function accounting for the diffuse sky irradiance dependent on surface inclination *γN*, and *Kb* is a measure of the amount of beam irradiance available. For surfaces in a shadow, we assume *δexp* = 0 and *Kb* = 0.

The ground reflected clear-sky irradiance received on a surface *R* (W/m2) is proportional to the total horizontal irradiance, which is a sum of beam and diffuse irradiance on a horizontal surface, mean ground albedo, and a fraction of the ground viewed by a surface [26]. The reflected radiation contributes to total radiation only by several percents in open areas and depends strongly on the reflectance of surrounding surfaces [8].
