## 3 Results and Discussion
### Performance of Four KDE Methods  
The results from the four kernels for the nine stations are presented in Table 3. The different KDE methods provide similar estimates, suggesting that the distribution is well defined and can be modeled using a variety of kernel functions. The estimated median rain heights for each station are approximately equal for each season, as shown in Table 3. The triangular kernel provides the sharpest estimate with the smallest ISE during the four seasons across all stations. However, the Gaussian kernel is most preferred due to its low optimum window width, which is an indication that the kernel has the smoothest estimate relative to the measured rain height. The measured PDF and the KDE methods fit well, indicating that the rain height estimates are reliable. Details of the seasonal variation of the kernels are discussed in the subsequent sub-sections.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Station**} & \multirow{3}{*}{**Season**} & \multicolumn{4}{c}{**Kernel**} \\ \cline{3-13}  & & \multicolumn{3}{c}{**Epanechnikov**} & \multicolumn{3}{c}{**Gaussian**} & \multicolumn{3}{c}{**Triangular**} & \multicolumn{3}{c}{**Rectangular**} \\  & & **h** & **Hr** & **ISE** & **h** & **Hr** & **ISE** & **h** & **Hr** & **ISE** & **h** & **Hr** & **ISE** \\  & & & & **(E-6)** & & **(E-6)** & & **(E-6)** & & **(E-6)** & & **(E-6)** & & **(E-6)** \\ \hline \multirow{4}{*}{Polokware} & Autumn & 243.39 & 4869.46 & 7 & 110.26 & 4869.46 & 6 & 184.11 & 4869.46 & 4 & 208.03 & 4869.46 & 9 \\  & Winter & 286.68 & 4530.86 & 1 & 129.86 & 4530.86 & 6 & 216.85 & 4530.86 & 3 & 245.03 & 4530.86 & 9 \\  & Spring & 185.67 & 4991.13 & 19 & 84.11 & 4991.13 & 17 & 140.44 & 4991.13 & 12 & 158.69 & 4991.13 & 22 \\  & Summer & 112.95 & 5271.97 & 13 & 51.17 & 5271.97 & 11 & 85.44 & 5271.97 & 7 & 96.54 & 5271.97 & 18 \\ \hline \multirow{4}{*}{Nelspruit} & Autumn & 209.76 & 5098 & 15 & 95.021 & 5098 & 14 & 158.66 & 5098 & 9 & 179.28 & 5098 & 18 \\  & Winter & 279.66 & 4426.43 & 12 & 126.68 & 4426.43 & 10 & 211.53 & 4426.43 & 7 & 239.20 & 4426.43 & 16 \\  & Spring & 276.01 & 4704.61 & 10 & 125.03 & 4704.61 & 9 & 208.77 & 4704.61 & 6 & 235.9 & 4704.61 & 12 \\  & Summer & 179.92 & 5133.91 & 10 & 81.5 & 5133.91 & 8 & 136.09 & 5133.91 & 4 & 153.78 & 5133.91 & 13 \\ \hline \multirow{4}{*}{Pretoria} & Autumn & 246.04 & 4776.72 & 9 & 111.45 & 4776.72 & 9 & 186.11 & 4776.72 & 7 & 210.29 & 4776.72 & 10 \\  & Winter & 279.94 & 4357.36 & 9 & 126.81 & 4357.36 & 8 & 211.75 & 4357.36 & 5 & 239.27 & 4357.36 & 11 \\  & Spring & 197.27 & 4899.91 & 9 & 89.36 & 4899.91 & 8 & 149.22 & 4899.91 & 4 & 168.61 & 4899.91 & 11 \\  & Summer & 115.55 & 5220.11 & 31 & 52.34 & 5220.11 & 25 & 87.44 & 5220.11 & 17 & 98.76 & 5220.11 & 41 \\ \hline \multirow{4}{*}{Mahikeng} & Autumn & 192.57 & 5095.07 & 11 & 87.23 & 5095.07 & 10 & 145.66 & 5095.07 & 5 & 164.59 & 5095.07 & 14 \\  & Winter & 265.84 & 4401.02 & 6 & 120.42 & 4401.02 & 6 & 201.08 & 4401.02 & 4 & 227.21 & 4401.02 & 6 \\  & Spring & 251.95 & 4718.87 & 10 & 114.13 & 4718.87 & 9 & 190.58 & 4718.87 & 5 & 215.34 & 4718.87 & 12 \\  & Summer & 144.47 & 5153.62 & 17 & 65.44 & 5153.62 & 13 & 109.28 & 5153.62 & 7 & 123.48 & 5153.62 & 22 \\ \hline \multirow{4}{*}{Upington}