## 2 Materials and Methods
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\#} & \multicolumn{2}{c}{**Site**} & \multirow{2}{*}{**Location**} & \multirow{2}{*}{\(\varphi\) **(deg)**} & \multirow{2}{*}{\(\lambda\) **(deg)**} & \multirow{2}{*}{**z (m asl)**} & \multicolumn{2}{c}{**Surface**} & \multirow{2}{*}{**Topography**} & \multirow{2}{*}{**I/II**} & \multirow{2}{*}{**Period**} \\  & & & & & & & & & & \\ \hline
1 & Athens & & & & & & & & \\  & (ATH) & Greece & 37.97 N & 23.72 E & 107 & & & & \\
2 & Boulder (BOU) & USA & 40.05 N & 105.01 W & 1577 & grass & flat & I & 1992–2016 \\
3 &
\begin{tabular}{c} Carpentras \\ (CAR) \\ \end{tabular} & France & 44.08 N & 5.06 E & 100 & cultivated & hilly & I & 1996–2018 \\
4 & De Aar (DAA) & South & & & & & & \\  & Africa & 30.67 S & 23.99 E & 1287 & sand & flat & I & 2000–present \\
5 & Gandhinagar & India & 23.11 N & 72.63 E & 65 & shrubs & flat & II & 2014–present \\  &
\begin{tabular}{c} Huancayo \\ \end{tabular} & & & & & & & & & \\
6 & Observatory & Peru & 12.05 S & 75.32 W & 3314 & grass & & \\  & (OHY) & & & & & & & \\
7 & Ilorin (ILO) & Nigeria & 8.53 N & 4.57 E & 350 & shrubs & flat & I & 1992–2005 \\
8 & Kishney (KIS) & Moldova & 47.00 N & 28.82 E & 205 & grass & flat & II & \\
9 & Lerwick (LER) & UK & 60.14 N & 1.18 W & 80 & grass & hilly & I & 2001–present \\
10 &
\begin{tabular}{c} Lindenberg \\ (LIN) \\ \end{tabular} & Germany & 52.21 N & 14.12 E & 125 & cultivated & hilly & I & 1994–present \\
11 & Payerne (PAY) & Switzerland & 46.82 N & 6.94 E & 491 & cultivated & hilly & I & 1992–present \\
12 & Regina (REG) & Canada & 50.21 N & 104.71 W & 578 & cultivated & flat & I & 1995–2011 \\
13 &
\begin{tabular}{c} Sonholick \\ (SON) \\ \end{tabular} & Austria & 47.05 N & 12.96 E & 3109 & rocks & & & & & \\  & & & & & & & & \\  & & & & & & & \\  & & & & & & & \\
14 & \
\begin{tabular}{c} Solar Village \\ (SOV) \\ \end{tabular} &
\begin{tabular}{c} Saudi \\ Arabia \\ \end{tabular} & 24.91 N & 46.41 E & 650 & & & & & & & \\  & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected sites for the application of the mathematical determination of the \(\mathrm{k_{du}}\) and \(\mathrm{k_{dl}}\) limits. \(\varphi\) and \(\lambda\) are the geographical latitudes and longitudes of the sites, respectively; \(\varphi\) is given in the northern (N) or the southern (S) hemisphere, and \(\lambda\) east (E) or west (W) of the Greenwich meridian. The values of both \(\varphi\) and \(\lambda\) have been rounded to the second decimal digit. In column 9, I denotes rural and II denotes urban environment. The period of measurements is given in the last column. The selection of the sites and their solar radiation data were based on the BSRN (Baseline Surface Radiation Network), except for Athens (ATH); in this case, data from the Actinometric Station of the National Observatory of Athens not belonging to BSRN were used. The abbreviations of the sites (except for Athens) are those provided by the BSRN topology. Description of the BSRN operation is found in [16]. routine, named XRONOS.bas in Basic programming language (xronos means time in Greek with x being spelled as ch), was used to derive the solar altitude (\(\gamma\), in deg) at 30 min past the hour LST. For example, the value of any parameter in the data base at 10.00 LST denotes its average value between 09.01 LST and 10.00 LST. The XRONOS.bas algorithm calculated \(\gamma\) at 09.30 LST and this value was assigned to all parameters at 10.00 LST. Finally, the hourly values of k\({}_{\rm d}\) were computed from concurrent values of H\({}_{\rm g}\) and H\({}_{\rm d}\), i.e., k\({}_{\rm d}\) = H\({}_{\rm d}\)/H\({}_{\rm g}\). In cases that H\({}_{\rm d}\) and/or H\({}_{\rm g}\) contained zero values, no value was assigned to k\({}_{\rm d}\). Correspondingly, no value was assigned to H\({}_{\rm bm}\), too. In plotting the (x, y) data pairs, i.e., (k\({}_{\rm d}\), H\({}_{\rm bm}\)), the hourly values of these two parameters were considered for all \(\gamma>5\) deg.