## 2 Principle of Radar Identification Probability Calculation  
### Criteria for Radar Identification in 3D Space  
As previously discussed, the challenges of radar identification caused by MGFs stem from two main aspects: radar detection blind zones and the RTA problem. To address the former, it is essential to ensure that the target is within the radar LOS. In terms of the RTA problem, it primarily depends on the radar resolution. Radar resolution can be categorized into range and angular resolutions in the spatial domain. Range resolution refers to the minimum distance between two adjacent targets that the radar can distinguish, primarily determined by the radar signal's bandwidth and pulse width. Angular resolution can be further divided into azimuth angle resolution and elevation angle resolution, which refer to the minimum angles for radar to distinguish two adjacent targets in horizontal and vertical directions, respectively. These are mainly dependent on the radar antenna's beamwidth. Existing methods [15; 16; 17; 18; 19; 20] have mostly simplified MGFs to point masses or relied solely on their 2D contour data, focusing mainly on radar range and azimuth angle resolution within a 2D plane. However, these methods did not fully consider the influence of elevation angle resolution. To extend the research scope from 2D to 3D for advanced high-altitude aircraft, this paper introduces radar elevation angle resolution and an MGF 3D model. When considering only the spatial domain and ignoring factors such as Doppler resolution, the RTA problem occurs when two or more objects are within the radar LOS and their differences in range, azimuth angle, and elevation angle relative to the radar are all smaller than the respective resolutions. To more precisely quantify and describe the criteria for the RTA problem in 3D space, this paper establishes a local Cartesian coordinate system with the due north direction as the \(X\)-axis, the due east direction as the \(Y\)-axis, the instantaneous sea surface as the \(XOY\) plane, and the \(Z\)-axis pointing vertically upward. Two targets are denoted as the points \(T_{1}(X_{T1},\,Y_{T1},\,Z_{T1})\) and \(T_{2}(X_{T2},\,Y_{T2},\,Z_{T2})\), and the radar position is denoted as the point \(S(X_{S},\,Y_{S},\,Z_{S})\). The schematic diagram of the relative positional relationship between these points is shown in Figure 1.  
In Figure 1, \(d_{ST1}\) and \(d_{ST2}\), \(A_{ST1}\) and \(A_{ST2}\), and \(E_{ST1}\) and \(E_{ST2}\) represent the distances, azimuth angles and elevation angles from point \(S\) to points \(T_{1}\) and \(T_{2}\), respectively. According to the definition of RTA, given the radar range resolution \(R_{res}\), azimuth angle resolution \(A_{res}\), and elevation angle resolution \(E_{res}\), when both \(T_{1}\) and \(T_{2}\) are within the radar LOS, the condition for the radar to misidentify \(T_{1}\) and \(T_{2}\) as a single target is that Equations (1)-(3) are simultaneously satisfied. \[\left|d_{ST_{1}}-d_{ST_{2}}\right|\leq R_{res} \tag{1}\]  
\[\left|A_{ST_{1}}-A_{ST_{2}}\right|\leq A_{res} \tag{2}\]  
\[\left|E_{ST_{1}}-E_{ST_{2}}\right|\leq E_{res} \tag{3}\]  
Figure 1: Schematic diagram of the positional relationship between radar and targets. The terms in Equations (1)-(3) are as follows:  
\[\left\{ \begin{array}{l}d_{ST_{1}}=\sqrt{\left(X_{S}-X_{T_{1}}\right)^{2}+\left(Y _{S}-Y_{T_{1}}\right)^{2}+\left(Z_{S}-Z_{T_{1}}\right)^{2}}\\ d_{ST_{2}}=\sqrt{\left(X_{S}-X_{T_{2}}\right)^{2}+\left(Y_{S}-Y_{T_{2}}\right)^{ 2}+\left(Z_{S}-Z_{T_{2}}\right)^{2}}\\ A_{ST_{1}}=\arctan\frac{Y_{T_{1}}-Y_{S}}{X_{T_{1}}-X_{S}}\\ A_{ST_{2}}=\arctan\frac{Y_{T_{2}}-Y_{S}}{X_{T_{2}}-X_{S}}\\ E_{ST_{1}}=\arctan\frac{\sqrt{\left(X_{S}-X_{T_{1}}\right)^{2}+\left(Y_{S}-Y_{ T_{1}}\right)^{2}}}{Z_{S}-Z_{T_{1}}}\\ E_{ST_{2}}=\arctan\frac{\sqrt{\left(X_{S}-X_{T_{2}}\right)^{2}+\left(Y_{S}-Y_{ T_{2}}\right)^{2}}}{Z_{S}-Z_{T_{2}}}\\ \end{array} \right.