## 1 Introduction  
Recent dramatic advances in radar systems have resulted in the extensive use of a variety of long-range airborne radars, including active electronically scanned array (AESA) radars, synthetic aperture radars (SARs), and airborne early warning (AEW) radars [1; 2; 3; 4; 5; 6]. Such radars are generally required to exhibit extreme high-performance in order to accurately detect long-range targets in the case of even a low radar cross-section (RCS). However, regardless of the radar performance, the target detection probability can be decreased by various factors, for example, external noise, clutter, atmospheric gas attenuation, polarization mismatches, multipath interference, and the atmospheric refractive index [7]. In particular, the gradient of the atmospheric refractive indices along the altitude usually changed by the temperature, atmospheric pressure, and dew-point temperature, causes the refraction of the wave propagation, that is, sub-, super-, normal-, and duct refraction. These propagation characteristics make it difficult to estimate the long-range target position [8], and it is essential to model the atmospheric refractivity along the altitude to precisely predict the propagation direction, path loss, and propagation factor. Thus, many studies involving the modeling of atmospheric refractive indices at low altitudes of less than 1 km have been conducted by estimating the duct phenomena through the measured path loss values [9], the clutter power values [10], and statistical analyses based on stored atmospheric data [11]. In addition, various studies have also been carried out to estimate the wave propagation characteristics when considering the oversea atmospheric characteristics in ground-to-air and ground-to-ground scenarios. For example, substantial low-altitude atmospheric data concerning certain coastal areas in the United States, Europe, and China have been used to analyze the path loss for wave propagation characteristics. Although previous studies have shown a sufficient analysis of the radar propagation characteristics in a low-altitude situation, it still needs in-depth research regarding the propagation characteristics at high altitudes of over 5 km in air-to-air situations considering the atmospheric effects. In this paper, we analyze the target detection performance of air-to-air airborne radars using long-range propagation simulations with a novel quad-linear refractivity model under abnormal atmospheric conditions. Both the radar propagation characteristics and the target detection performance are simulated using the Advanced Refractive Effects Prediction System (AREPS) software, as based on a propagation hybrid model [12; 13]. In the AREPS simulation conditions, the refractivity along the altitude, the antenna radiation pattern, and the digital terrain elevation data (DTED) are considered to be inputs to obtain the path loss of the wave propagation. The quad-linear lines are employed to model the modified refractive indices for the abnormal atmospheres associated with various elevated ducts. In addition, the radiation pattern is calculated based on a \(32\times 32\) array antenna with a triangular array configuration used for modeling the airborne radar antenna. Then, the path loss results obtained using the quad-linear model are compared to the results obtained using the conventional trilinear refractivity model and to the actual refractivity data. Subsequently, the resulting path loss and the other radar parameters are used to calculate the radar target detection probability in order to estimate the target detection performance of the airborne radar. Using the calculated target detection probability, we propose a detection performance metric in terms of the atmosphere (DPMA) for intuitively examining the long-range propagation characteristics in air-to-air situations. To investigate the effects of abnormal atmospheric conditions on the radar performance, the DPMA map is examined by varying the height and thickness of the elevated duct.