## 2 Wide Range Two-Dimensional High-Altitude Refractivity Estimation
The equations of the IDW interpolation, in consideration of the Earth's curvature, can be defined through the following:  
\[M_{p}=\left\{ \begin{array}{l}\frac{\sum\limits_{i=1}^{N}w_{i}M_{i}}{\sum \limits_{i=1}^{N}w_{i}}\,\ ifd_{i}\
eq 0\\ \dot{M_{i}}\ \ \ \ \,\ if\d_{i}=0\end{array} \right., \tag{3}\]  
\[d_{i}=\sqrt{(x_{i}-x)^{2}+(y_{i}-y)^{2}}, \tag{4}\]  
\[d^{\prime}{}_{i}=d_{i}+0.157\times h, \tag{5}\]  
\[w_{i}=\frac{1}{(d^{\prime}{}_{i})^{k}}, \tag{6}\]  
where the index \(i\) is an integer number (1, 2,..., and 20), and \(d_{i}\) is the distance between the locations of \(M_{p}\) and \(M_{i}\), \(d^{\prime}{}_{i}\) is the distance in consideration of the Earth's curvature with the altitude, \(h_{i}\). \(w_{i}\) is the weight function for an inverse distance, \(d^{\prime}{}_{i}\), with a power parameter of \(k\), which was set to 2 in this research. For example, the interpolated value can be less affected by the long-distance interpolation points for the large value of \(k\) because the weight, \(w_{i}\), is close to zero. To verify the results of the IDW method, the refractivities at seven meteorological observatories in the Korean Peninsula were interpolated, one by one, using the data from the other nineteen nearby observatories, excluding their own exact data. Then, the RMSE of \(\
abla M\) between the exact and the interpolated data at the locations of the seven observatories was examined and compared with the results from other conventional interpolation methods: linear, cubic, and nearest-neighbor. In the interpolation, the refractivity data of the four seasons in 2019 at twenty observatories were used. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Site**} & **Meteorological** & \multirow{2}{*}{**Latitude**} & \multirow{2}{*}{**Longitude**} & \multirow{2}{*}{**Height**} & **Average** & **Number of** \\  & & & & & **Altitude Interval** & **Measurements per Day** \\ \hline S\({}_{1}\) & Baeknyeongdo & 37.97 & 124.63 & 158 m & 315.3 m & 2 \\ S\({}_{2}\) & Bukgangneung & 37.81 & 128.85 & 89 m & 303.1 m & 2 \\ S\({}_{3}\) & Osan & 37.10 & 127.03 & 52 m & 310.8 m & 4 \\ S\({}_{4}\) & Pohang & 36.03 & 129.38 & 6 m & 308.7 m & 2 \\ S\({}_{5}\) & Gwangju & 35.11 & 126.81 & 13 m & 308.5 m & 4 \\ S\({}_{6}\) & Heuksando & 34.68 & 125.45 & 69 m & 293.3 m & 2 \\ S\({}_{7}\) & Jeju National & 33.33 & 126.68 & 235 m & 296.8 m & 2 \\  & Typhoon Center & 33.76 & 120.25 & 7 m & 750.3 m & 2 \\ E\({}_{1}\) & Sheyang & 33.76 & 120.25 & 7 m & 746.7 m & 2 \\ E\({}_{2}\) & Qingdao & 36.06 & 120.33 & 77 m & 746.7 m & 2 \\ E\({}_{3}\) & Dalian & 38.90 & 121.63 & 97 m & 1284.8 m & 2 \\ E\({}_{4}\) & Jakangdo & 41.71 & 126.91 & 333 m & 1318.4 m & 2 \\ E\({}_{5}\) & Yanbian & 42.88 & 129.46 & 178 m & 1319.0 m & 2 \\ E\({}_{6}\) & Vladivostok & 43.26 & 132.05 & 82 m & 306.2 m & 2 \\ E\({}_{7}\) & Misawa & 40.70 & 141.38 & 39 m & 399.5 m & 2 \\ E\({}_{8}\) & Akita & 39.71 & 140.10 & 7 m & 345.1 m & 2 \\ E\({}_{9}\) & Tateno & 36.05 & 140.13 & 31 m & 326.6 m & 2 \\ E\({}_{10}\) & Wajima & 37.38 & 136.90 & 14 m & 291.7 m & 2 \\ E\({}_{11}\) & Matsue & 35.45 & 133.07 & 22 m & 290.4 m & 2 \\ E\({}_{12}\) & Fukuoka & 33.58 & 130.38 & 15 m & 322.2 m & 2 \\ E\({}_{13}\) & Kagoshima & 31.55 & 130.55 & 31 m & 319.5 m & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geographical coordinates of the weather stations with measurement setups. Figure 3a-d represents the comparisons of the RMSE results at seven meteorological observatories in terms of the interpolation methods for all seasons in 2019. To derive the RMSE of \(\
abla M\), we included a total number of 15,783 of the refractivity profile data to achieve the seasonal results. Black circle markers indicate the results of the IDW, while the green, blue, and red cross markers represent those with the nearest-neighbor, linear, and cubic interpolation, respectively. S\({}_{1}\) to S\({}_{7}\) on the \(x\)-axis represent the locations of seven meteorological observatories. Figure 3a shows the RMSEs for the spring season.