## 2 Methodology  
### Climatology of Research Locations and Data Source  
The selected locations used in this work are Port Elizabeth, Bloemfontein, Pretoria, Durban, Polokwane, Nelspruit, Kimberly, Mahikeng, and Cape Town. The study area map and the characteristics of the stations are presented in Figure 1 and Table 1, respectively. The stations were carefully selected to represent each of the nine provinces in South Africa. Figure 1: A map of South Africa showing the nine study stations. South Africa is situated at the south of the African continent where it meets the southern hemisphere of the world. It is bounded by longitudes 18-32\({}^{\circ}\) E and latitudes 23-34\({}^{\circ}\) S, as shown in Figure 1. It is a subtropical country classified as ITU Region 1 according to ITU regional classification. The four traditional annual weather seasons together with their features are highlighted in Table 2 below. The diversity in the country's climate is due to the relative complex topography, ocean proximity, and rainfall distribution of the constituent cities [23; 24]. For instance, the southern region including Port Elizabeth and Cape Town has a Mediterranean climate (hot dry summer and cool wet winter), while the north including Polokwane, Nelspruit, etc., has a subtropical highland climate (hot wet summer with thunderstorms and wet cool winter) [22]. Further details about the individual study stations and their local weather can be found on the South Africa Weather Service website [25]. The vertical profile of the atmospheric temperature and geopotential height at seven pressure levels (1000, 850, 700, 500, 300, 200, and 100 hPa) were retrieved from the ERA5 global reanalysis dataset of the CDS. The CDS is an open access database for retrieving a wide range of climate data. It is developed and being maintained by the European Centre for Medium-Range Weather Forecasts (ECMWF). It provides detailed global climate information to enhance research and decision making on climate adaptation and mitigation [27; 28]. The geometric metric height at which the atmospheric temperature becomes zero degrees Celsius, commonly referred to as the zero-degree isotherm height (ZDIH), was computed from the geopotential height and temperature using the interpolation technique. The data were processed to obtain the daily rain height \(h_{r}\) for each station using Equation (1) [11]. \[h_{r}=h_{o}+0.36\ (\mathrm{km}) \tag{1}\]  
where \(h_{o}\) is the zero-degree isotherm height. The rain height data were classified into the four dominant seasons in South Africa, as described in Table 2. KDE was applied to estimate the seasonal values of rain height across the study stations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Season** & **Period** & **Characteristics** \\ \hline
1 & Summer & Dec.–Feb. & Hot and wet with most of the annual rainfall \\
2 & Autumn & Mar.–May. & The transition period with gradually cooling temperatures \\
3 & Winter & Jun.–Aug. & Usually dry and cool with rainfall in the coastal areas \\
4 & Spring & Sep.–Nov. & A gradual rise in temperature which signifies the onset of the rainy season in the central and eastern parts of the country \\ \hline \hline \end{tabular}
\end{table} 
Table 2: South Africa weather seasons and their characteristics [26].