## 2 Wide Range Two-Dimensional High-Altitude Refractivity Estimation  
Figure 1 shows the locations of twenty meteorological observatories within (indicated by red triangles) or nearby (indicated by yellow circles) the Korean Peninsula, where the observatories are capable of measuring weather data up to the altitude of the troposphere. As shown in Figure 1, there are only seven meteorological observatories in Korea, and their locations can be regionally categorized into two groups: coastal area sites (S\({}_{1}\), S\({}_{6}\), and S\({}_{7}\)) and inland sites (S\({}_{2}\), S\({}_{3}\), S\({}_{4}\), and S\({}_{5}\)). Since the seven meteorological observatories are not sufficient to estimate the refractivity of the entire Korean Peninsula, we collected additional high-altitude atmospheric data from other overseas meteorological observatories near the Korean Peninsula. The collected atmospheric condition data, as a function of the altitude, included air pressure, temperature, and dew-point temperature. These data were measured 2 or 4 times a day, and the average altitude interval was 507.84 m. The detailed  
geographical coordinates and measurement conditions are listed in Table 1. From these data, the dimensionless refractivity, \(N\), can be calculated using the following equation [18]:  
\[N=\left(77.6\times 10^{-6}\times\frac{P}{T}+0.373\times\frac{e}{T^{2}}\right) \times 10^{6}, \tag{1}\]  
where \(T\) is the absolute temperature in K, \(e\) is the water vapor pressure in millibars, and \(P\) is the air pressure in millibars. The refractivity, \(N\), can then be replaced by the modified refractivity, \(M\), according to the altitude on a hypothetical flat Earth and in consideration of the Earth's curvature, as follows [18]:  
\[M=\ N+0.157\times h, \tag{2}\]  
where \(h\) indicates the altitude at which the modified refractivity, \(M\), is calculated. Due to physical atmospheric conditions, the gradient of modified refractivity, \(\
abla M\), according to the altitude, changes naturally, which causes the propagation direction to bend depending on \(\
abla M\). The wave refractions are generally classified into four types according to \(\
abla M\)[19]. The sub-refraction (\(\
abla M\geq 157\)) occurs when the wave direction is bent further away from the surface of the Earth than in normal atmospheric conditions. Normal refraction occurs when the gradient of the refractivity is \(78\leq\
abla M\leq 157\). Super-refraction (\(0\leq\
abla M\leq 78\)) occurs when the direction of the wave propagation is bent further toward the Earth's surface. When the gradient \(\
abla M\) becomes negative, the atmospheric state is called the duct, and the wave propagation is often trapped at a certain altitude, as if propagating in a waveguide. Note that it is well-known that the refraction of the wave propagation is generally affected by the \(\
abla M\) along the altitude [30]. The \(\
abla M\), which changes with altitude, can also significantly vary depending on the location (latitude and longitude). Therefore, in order to more accurately predict wave propagation characteristics, it is necessary to have dense refractivity data for the location; however, as mentioned earlier, there is a limited number of meteorological observatories in Korea. Thus, we employed IDW interpolation to achieve the dense refractivity data, depending on the location (latitude and longitude), for the entire Korean Peninsula. Figure 1: Locations of the twenty meteorological observatories inside and around the Korean Peninsula. Figure 2 shows a conceptual figure of the IDW interpolation method, in consideration of the Earth's curvature. The red triangle specifies interpolated data of \(M_{p}\) at a given altitude, \(h\), with the location of (\(x\), \(y\)). The yellow circles represent the refractivity data of \(M_{i}\) at the locations (\(x_{i}\), \(y_{i}\)) as a function of an altitude, \(h\), collected from the twenty meteorological observatories.