## 4 Comparison of Light-Tube Optical Properties under Arbitrary and Homogenous Skies
### Partly Cloudy Days  
It is not uncommon that the model of the homogeneous skies works well under stable cloudless and overcast conditions. However, the sky state is usually heterogeneous in many regions with arbitrary cloud configurations. This makes the availability of direct subbeams very irregular. In addition, the diffuse component of the ground-reaching radiation may alter due to the spatial arrangement of isolated clouds or cloud arrays that reflect the sunlight depending on their position relative to the sun. It is difficult, or rather impossible, to model all combinations of the cloud arrangements that can be realized in nature. The simplest way to express the cloud statistics is to implement the concept of the cloud fraction (CF). CF is a scalar value ranging from zero (cloudless sky) to one (overcast sky). Instead of using CF, [PERSON] uses sky clearness that is close to one for the overcast skies and increases monotonically as the sky state transitions to a very clear one (\(\varepsilon>6.2\)). However, the concept based  
Figure 6: Light tube efficiency (**a**) and average cosine (**b**) computed for \(H\) = 1.5 m, \(r\) = 0.26 m, \(\rho\) = 0.934 using data recorded during clear sky conditions on 16 February 2015 (see Tables 1 and 3). The dashed lines are for the UMRP model, while the solid lines are for [PERSON]’s all-weather model. Figure 7: The same as in Figure 6, but for the overcast sky, 24 February 2015.  
on \(\varepsilon\) is not capable of distinguishing between the effects of low and high clouds, optically dense and optically thin clouds, or situations when clouds accumulate near the horizon or are moving from/to a specific quadrant rather than being scattered randomly throughout the sky vault. It is therefore uncertain how inaccurate the optical properties of the light guides could be when predicted from the model of the homogeneous skies. This is why we have performed a set of numerical tests on the light-tube efficiency (LTE) and \(<\)cos\(>\) using meteorological and optical data both recorded in Bratislava. The all-seasons modeling is important e.g., because sun altitudes in the winter time typically differ from those in summer months. Furthermore, the cloud types are closely linked to the type of air mass, so it is not surprising that the clouds prevailing in winter do not necessarily mask the sky in summer. Although the positions of the clouds constantly change, the LTE computed from [PERSON]'s model of a homogeneous sky was found to coincide with more accurate UMRP computations (see Figures 8a, 9a, 10a, 11a and 12a). The largest differences between the LTE data (up to 10%) are found in the winter months and are most likely due to the prevalence of low sun elevations. In general, [PERSON]'s all-weather model underestimates LTE, but the LTEs predicted from both models tend to have the same patterns. The light tube efficiency is a non-directional quantity and only expresses the fraction of exterior light used for the illumination of interior spaces. The light guide harvests light from all directions, so it is not a surprise that the integral of the sky luminance over the definition domain of the zenith angles and azimuth angles averages all imperfections and efficiently suppresses the contribution from local luminance peaks. However, the average cosine encapsulates the information on preferred directions, thus implying the combination of the LTE and \(<\)cos\(>\) is a preferred concept to characterize the light guides. This approach is advantageous as \(<\)cos\(>\) is also a single scalar value, so simple two-valued records are sufficient to distinguish between two skies that do not differ in CF (or \(\varepsilon\)), but produce different illuminance patterns due to different cloud configurations. In principle, [PERSON]'s model is unable to mimic a large variability of \(<\)cos\(>\), therefore the differences between the homogeneous, all-weather model and exact UMRP computations can be large (as shown in Figures 8b, 9b, 10b, 11b and 12b).