## 4 Comparison of Light-Tube Optical Properties under Arbitrary and Homogenous Skies
### Clear and Overcast Skies
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Time (UTC)** & \(A_{S}\) **(deg)** & \(z_{S}\) **(deg)** & \(G_{h}\) **(W.m\({}^{-2}\))** & \(D_{h}\) **(W.m\({}^{-2}\))** & \(t\) (\({}^{\circ}\)C)** & **RH (\%)** & \(C_{F}\) **(-)** \\ \hline
7:00 & 120.04 & 81.56 & 113.77 & 38.03 & 2.5 & 61 & 0 \\
8:00 & 132.55 & 73.50 & 265.25 & 52.69 & 4 & 56 & 0 \\
9:00 & 146.49 & 67.00 & 388.61 & 63.88 & 6.9 & 49 & 0 \\
10:00 & 161.93 & 62.64 & 476.91 & 68.31 & 6.9 & 48 & 0 \\
11:00 & 178.43 & 60.92 & 505.98 & 66.43 & 7.7 & 47 & 0 \\
12:00 & 195.03 & 62.10 & 484.00 & 68.31 & 8.4 & 45 & 0 \\
13:00 & 210.72 & 66.00 & 404.05 & 61.76 & 9.4 & 44 & 0 \\
14:00 & 224.95 & 72.14 & 282.90 & 58.68 & 8.8 & 43 & 0 \\
15:00 & 237.71 & 79.95 & 134.30 & 46.88 & 7.9 & 54 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Data collected at the Slovak Hydro-meteorological Institute (Bratislava, Slovakia) on 16 February 2015 under clear sky conditions: \(A_{S}\)—azimuth of the sun, \(z_{S}\)—solar zenith angle, \(G_{h}\)—global horizontal irradiance, \(D_{h}\)—diffuse horizontal irradiance, \(t\) —air temperature, \(RH\)—relative humidity, \(C_{F}\)—cloud fraction. Time is given in the coordinated universal time (UTC). Figures 6 and 7 demonstrate that for the homogeneous skies, there is not much difference between the optical properties of the light guides simulated using the UMRP and [PERSON]'s model. In general, [PERSON]'s model works well and the small differences found are due to its physical limits or variability of input parameters (e.g., altitude of the cloud base in the case of the overcast sky; see Figure 7). It is symptomatic for the clear sky that the optical efficiency of a vertical tubular light guide peaks when the sun is high. This feature has been reproduced by the UMRP and [PERSON]'s model in a consistent way. On the other hand, the illumination scene below a straight light pipe is normally conserved under overcast sky conditions. However, [PERSON]'s model is unable to mimic changes in cloud altitudes. We have computed the average cosine, a parameter that characterizes the directionality of light at the exit from the bottom interface of a light guide. For more details on the fundamentals of the average cosine, \(<\)cos\(>\), see [31]. \(<\)cos\(>\) approaches unity if all the photons propagate axially and thus illuminate the small circular area just below the pipe. \(<\)cos\(>\) monotonically decreases when the angle from the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Time (UTC)** & \(h_{C}\) **(km)** & \(r_{C}\) **(km)** & \(\tau_{a}\) **(-)** & \(g^{\prime}\) **(-)** & \(\omega\) **(-)** & \(\Delta\) **(-)** & \(\epsilon\) **(-)** & **Dev. (\%)** \\ \hline
7:00 & 1.1 & 2.4 & 0.01 & 0.4 & 0.4 & 0.099 & 1.000 & 1.80 \\
8:00 & 0.6 & 1.0 & 0.01 & 0.4 & 0.4 & 0.107 & 1.000 & 0.34 \\
9:00 & 0.7 & 0.9 & 0.01 & 0.4 & 0.4 & 0.136 & 1.000 & 1.05 \\
10:00 & 0.7 & 0.9 & 0.01 & 0.4 & 0.4 & 0.118 & 1.000 & 1.90 \\
11:00 & 1.4 & 0.6 & 0.01 & 0.4 & 0.4 & 0.199 & 1.000 & 1.49 \\
12:00 & 1.3 & 0.7 & 0.01 & 0.4 & 0.4 & 0.192 & 1.000 & 1.93 \\
13:00 & 0.7 & 0.9 & 0.01 & 0.4 & 0.4 & 0.134 & 1.000 & 1.70 \\
14:00 & 0.8 & 0.6 & 0.01 & 0.4 & 0.4 & 0.148 & 1.000 & 0.44 \\
15:00 & 0.5 & 1.0 & 0.01 & 0.4 & 0.4 & 0.106 & 1.000 & 0.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The same as in Table 3, but for 24 February 2015 (overcast sky).