## 2 Long-Range Propagation Analysis of Air-to-Air Airborne Radars  
### Abnormal Atmospheric Conditions and Propagation Characteristics  
In general, the atmosphere is usually expressed by the refractive index \(n\) according to the height, and it can be changed into the modified refractive index \(M\) on a hypothetical flat Earth when considering the Earth's curvature, as expressed in Equation (1) [14]. \[M\ =\ (n-1)10^{6}+0.157h \tag{1}\]  
In fact, with regard to radio wave propagation, the gradient of the modified refractive index \(M\) in terms of the height considerably affects the propagation direction. As shown in Figure 1, there are four typical refractions associated with the wave propagations of an air-to-air airborne radar: sub (\(157<\
abla M\)), normal (\(79<\
abla M\leq 157\)), super refraction (\(0<\
abla M\leq 79\)), and duct (\(\
abla M<0\)). In particular, the duct condition represents the most abnormal phenomenon, which causes the wave to become trapped in the atmosphere as if it propagates in a waveguide. Figure 2 illustrates the modified refractive indices of the elevated duct atmospheres according to the altitude at the Heuksando meteorological observatory on 4 September 2010, 5 June 2015, and 10 December 2015, which are all calculated using the air pressure, temperature, and dew point temperature. Such atmospheric data can be approximated by combinations of linear lines, and there is a conventional method (i.e., a trilinear model) for modeling the refractivity using three linear lines. However, at a high altitude of over 5 km, we find that many elevated duct cases require an additional linear line to accurately model the modified refractivity. Thus, we propose a quad-linear model of the modified refractive indices alongside the higher altitude with four linear lines in order to approximate the elevated duct condition, as shown in Figure 3. The first to third linear lines have heights of \(h_{1}\), \(h_{2}\), and \(h_{3}\), as well as slopes of \(d_{1}\), \(d_{2}\), and \(d_{3}\). The fourth line is modeled by a height of \(h_{max}\) and a slope of \(d_{4}\). Herein, the height and thickness of the elevated duct are indicated by \(h_{1}\) and \(\Delta l\), respectively. Figure 1: Four refractive conditions of the wave propagation of an air-to-air airborne radar with the gradient of the modified refractivity \(M\) according to the height. Figure 2: Modified refractive indices of the elevated duct atmospheres according to the altitude at the Heuksando meteorological observatory in Korea: (**a**) 4 September 2015; (**b**) 5 June 2015; (**c**) 10 December 2015. Figure 4 presents comparisons of the quad-linear and trilinear modeling of the actual refractivity data obtained at the Osan meteorological observatory on 1 February 2015. The detailed values of the quad-linear model parameters are \(h_{1}=4.98\) km, \(h_{2}=5.16\) km, \(h_{3}=5.5\) km, \(h_{max}=10\) km, \(d_{1}=132.5\), \(d_{2}=-817.8\), \(d_{3}=587\), and \(d_{4}=142\). The quad-linear modeling result exhibits a good level of agreement with the actual refractivity data when compared with the conventional trilinear model result, where the deviations of each model can be calculated by the equation as follows:  
\[RMSD\ =\ \sqrt{\sum_{i=1}^{I}\frac{\left|M_{act}[i]-M_{model}[i]\right|}{I}}, \tag{2}\]  
where \(RMSD\) is the root-means-square deviation of the refractivity. \(M_{act}\) indicates the refractivity along the altitude, and \(M_{model}\) specifies the refractivity in terms of the tri- or quad-linear model along the altitude. The \(RMSD\) values are 7.2 M-unit for the quad-linear model and 60.2 M-unit for the trilinear model, respectively. To observe the propagation characteristics in an air-to-air situation, we include both these refractivity models and the actual data in the analysis involving the AREPS simulation software, while the radiation pattern of a conventional airborne array antenna is also applied as a simulation parameter.