## 2 Long-Range Propagation Analysis of Air-to-Air Airborne Radars
### Target Detection Performance in Abnormal Atmosphere  
In general, many studies on target detection performance have been investigated considering RCS fluctuations with clutters [19], multipath effects [20], regional environments [21], and refractive  
Figure 6: The Advanced Refractive Effects Prediction System (AREPS) simulation results of path losses employing the linear models and the actual data regarding the modified refractivity: (**a**) trilinear model; (**b**) quad-linear model; (**c**) actual data. Figure 7: Path loss comparisons among the AREPS software, the PETOOL software, and the ITU-R P528 model in terms of the atmospheric conditions: (**a**) normal atmosphere; (**b**) abnormal atmosphere. profiles [22; 23; 24]. In this research, we calculate the probability of the target detection using thresholds derived from the free-space path loss levels to examine the target detection performance. To calculate the path loss threshold, the detectable range with regard to the detection probability is defined using the radar equation in free space, as written in Equation (3) [25]. \[R_{f_{\rm{S}}}(P_{d}) = \sqrt[4]{\frac{c_{0}^{2}P_{t}\sigma\tau G^{2}}{(4\pi)^{3}kT_{0}(D (P_{d}))f^{2}N_{f}L_{s}}}\ \ \ [m], \tag{3}\]  
where \(c_{0}\) is the speed of light in m/s, and \(P_{t}\) is the transmitting power in W. \(\sigma\) is the RCS in m\({}^{2}\), and \(\tau\) is the pulse length of the radar in s. \(G\) is the array antenna gain for the airborne radar, and \(k\) is the Boltzmann's constant. \(T_{0}\) is the absolute temperature in K, and \(D\) is the detectability factor. \(P_{d}\) is the detection probability, and \(f\) is the operating frequency in Hz. \(N_{f}\) is the noise figure of the radar, and \(L_{s}\) is the miscellaneous system loss. In this equation, all the airborne radar parameters are set as constant, except for the detectability factor \(D\), which is simply approximated by Blake [25]. This factor includes the probability of the target detection and of a false alarm. Thus, it can simply approximate the functional relationship between the target detection probability and the radar equation in free space. Therefore, the free-space propagation loss for the threshold can be defined as follows:  
\[L_{Th}(P_{d}) = 20\log\Bigl{(}R_{f_{\rm{S}}}(P_{d})\Bigr{)}+20\log(f)+20\log \Biggl{(}\frac{4\pi}{c_{0}}\Biggr{)}. \tag{4}\]  
To determine the detection probability in the air-to-air situation for the airborne radar, we obtain the path loss values using the AREPS simulation software, and then we calculate the probability by including the airborne radar system parameters. The detailed parameter values are listed in Table 1.  
Figure 8a,b illustrates the detection probability results for the air-to-air scenario in terms of the normal and abnormal duct atmospheres when the source height \(h_{s}\) is 4.8 km. The dashed lines indicate the wave propagation direction at \(\theta=0.5\theta_{HPBW}\), \(0^{\circ}\), and \(-0.5\theta_{HPBW}\), where \(\theta_{HPBW}\) is an HPBW angle of the airborne radar antenna radiation pattern. As can be seen in Figure 8b, the target detection performance is seriously degraded within the HPBW region as a result of the abnormally elevated duct. To more intuitively analyze the target detection performance, we propose a DPMA, which is defined as the ratio of the detectable area of the normal atmosphere to that of the abnormal atmosphere, as follows:  
\[DPMA = \frac{A_{d,atm}}{A_{d,nar}}, \tag{5}\]  
\[A_{d,atm} = \int_{-0.5\theta_{HPBW}}^{0.5\theta_{HPBW}}F_{atm}(\theta)d\theta, \tag{6}\]  
\[A_{d,nar} = \int_{-0.5\theta_{HPBW}}^{0.5\theta_{HPBW}}F_{nor}(\theta)d\theta, \tag{7}\]  
where \(A_{d,atm}\) and \(A_{d,nar}\) indicate the detectable areas for the abnormal and normal atmospheres, respectively. When the DPMA value is close to 1, the target detection performance is similar to that