## 2 Wide Range Two-Dimensional High-Altitude Refractivity Estimation
At the three coastal area sites (S\({}_{1}\), S\({}_{6}\), and S\({}_{7}\)), the average RMSEs of the IDW, linear, cubic, and nearest-neighbor were 36.7, 38.1, 39.9, and 42.9, respectively. Meanwhile, the average RMSEs at the four inland area sites (S\({}_{2}\), S\({}_{3}\), S\({}_{4}\), and S\({}_{5}\)) of the IDW, linear, cubic, and nearest-neighbor were 17.1, 20.8, 22.2, and 23.5, respectively, which are lower than those from the three coastal area sites. For the summer season, as shown in Figure 3b, the average RMSEs of the IDW, linear, cubic, and nearest-neighbor at the four inland area sites were 29.1, 33.0, 35.0, and 41.0, respectively. The resulting values of the summer season were higher than those of the spring season. This is because physical phenomena such as inversion layers, which often occur during the summer, cause very irregular atmospheric conditions, resulting in increased estimation errors. In autumn, the average RMSEs of the IDW, linear, cubic, and nearest-neighbor at four inland area sites were 25.5, 29.8, 31.9, and 33.9, respectively. The winter season had the lowest average RMSEs of the IDW, linear, cubic, and nearest-neighbor interpolations, as shown in Figure 3d, which were 13.3, 16.1, 17.2, and 18.6, respectively. During the winter, the cold air temperature with little up-down convection maintains the most stable atmospheric condition from all seasons, which results in steady refractivity along the altitude. Note that the proposed IDW interpolation method has the lowest RMSEs of \(\
abla M\) when estimating the refractivity over all seasons. Since the IDW method had the lowest RMSEs, we used the IDW for estimating the refractivity distribution across entire regions in Korea. We also evaluated the RMSE of \(M\) at fixed altitudes (0 m and 5000 m), using the refractivity data set from 2019, according to the four interpolation methods. The lowest RMSE value was 11.4, which was obtained for the average of the IDW interpolation method. The highest RMSE value was 69.7 for the average of the nearest-neighbor method. The RMSE results were also averaged in terms of the heights from 0 m to 5000 m to more obviously compare the interpolation performances. The average results demonstrate that the IDW method had more accurate performances than the nearest-neighbor method in all observatories. The detailed RMSE of \(M\) results for the seven observatories in Korea are listed in Table 2.  
Figure 2: Inverse distance weighting (IDW) method in consideration of the Earth’s curvature. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Site**} & \multirow{2}{*}{**Meteorological Observatory**} & \multicolumn{2}{c}{**IDW**} & \multicolumn{2}{c}{**Linear**} & \multicolumn{2}{c}{**Cubic**} & \multicolumn{2}{c}{**Nearest**} \\  & & **(0 m/5000 m/Avg)** & **(0 m/5000 m/Avg)** & **(0 m/5000 m/Avg)** & **(0 m/5000 m/Avg)** \\ \hline S\({}_{1}\) & Baeknyeongdo & 62.6/30.0/31.1 & 64.3/29.0/30.0 & 64.1/29.3/30.3 & 65.0/29.5/33.2 \\ S\({}_{2}\) & Bukgangneung & 20.1/27.7/27.8 & 18.4/38.5/34.0 & 18.7/42.4/36.5 & 24.5/26.0/29.6 \\ S\({}_{3}\) & Osan & 18.3/12.0/17.6 & 24.9/16.1/19.8 & 26.6/18.6/21.7 & 24.4/26.0/29.6 \\ S\({}_{4}\) & Pohang & 11.4/41.9/34.5 & 13.2/43.6/36.3 & 14.6/44.6/37.2 & 18.7/48.3/41.1 \\ S\({}_{5}\) & Gwangju & 14.4/34.9/31.1 & 17.3/44.7/38.5 & 19.2/49.0/42.3 & 25.1/58.1/51.3 \\ S\({}_{6}\) & Heuksando & 22.4/52.1/46.6 & 22.2/54.3/48.0 & 23.2/55.9/49.2 & 24.5/58.2/51.4 \\ S\({}_{7}\) & Jeju National Typhoon Center & 30.0/56.7/55.0 & 31.0/57.7/55.4 & 32.3/61.0/57.6 & 34.8/69.7/65.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: RMSE of \(M\) results for seven observatories. Figure 3: _Cont._Figure 4a-d presents the two-dimensional regional interpolation results of the atmospheric refractivity, \(M\), for all methods at a fixed altitude of 5000 m inside the Korean Peninsula on 15 February 2019. The atmospheric refractivities, \(M\), were estimated across the entire Korean Peninsula using the atmospheric data collected from twenty meteorological observatories.