## 2 Principle of Radar Identification Probability Calculation
### Error Analysis  
In addition to the limitations of radar LOS and resolution, potential errors can also affect radar identification. This paper focuses on errors from two sources: the MGF and the maritime radar (aircraft). For MGF data, the errors are mainly due to the accuracy of geospatial surveys. There are planar position errors (\(\Delta X_{B}\), \(\Delta Y_{B}\)) and elevation data error \(\Delta Z_{B}\). (\(\Delta X_{B}\), \(\Delta Y_{B}\)) are assumed to follow a 2D normal distribution with independent directions, an expected value of 0, and a standard deviation of \(\sigma_{BXY}\) for both directions. \(\Delta Z_{B}\) is assumed to follow a normal distribution with an expected value of 0 and a standard deviation of \(\sigma_{BZ}\):  
\[(\Delta X_{B},\Delta Y_{B})\sim\text{N}(0,\sigma_{B_{XY}};0,\sigma_{B_{XY}};0) \tag{5}\]  
\[\Delta Z_{B}\sim\text{N}(0,\sigma_{B_{Z}}) \tag{6}\]  
The errors from the radar (aircraft) mainly stem from the radar system's hardware precision and the navigation errors of the aircraft and its launch platform. They primarily consist of three aspects: the planar position error of the aircraft launch platform (\(\Delta X_{L}\), \(\Delta Y_{L}\)), the planar position error (\(\Delta X_{S}\), \(\Delta Y_{S}\)), and the elevation error \(\Delta Z_{S}\) of the radar activation point. Similarly, these errors are assumed to follow the normal distributions:  
\[(\Delta X_{L},\Delta Y_{L})\sim\text{N}(0,\sigma_{L};0,\sigma_{L};0) \tag{7}\]  
\[(\Delta X_{S},\Delta Y_{S})\sim\text{N}(0,\sigma_{S_{XY}};0,\sigma_{S_{XY}};0) \tag{8}\]  
\[\Delta Z_{S}\sim\text{N}(0,\sigma_{S_{Z}}) \tag{9}\]  
To simplify the calculations, this paper combines (\(\Delta X_{S}\), \(\Delta Y_{S}\)), (\(\Delta X_{L}\), \(\Delta Y_{L}\)), and (\(\Delta X_{B}\), \(\Delta Y_{B}\)) as shown in Equation (10), so that (\(\Delta X_{S}\), \(\Delta Y_{S}\)) can be equivalently considered to follow the 2D normal distribution as shown in Equation (11):  
\[\sigma_{m_{XY}}^{\ 2}=\sigma_{S_{XY}}^{\ 2}+\sigma_{L}^{2}+\sigma_{B_{XY}}^{\ 2} \tag{10}\]  
\[(\Delta X_{S},\Delta Y_{S})\sim\text{N}(0,\sigma_{m_{XY}};0,\sigma_{m_{XY}};0) \tag{11}\]Similarly, \(\Delta Z_{B}\) and \(\Delta Z_{S}\) are combined as follows:  
\[{\sigma_{m_{Z}}}^{2}={\sigma_{S_{Z}}}^{2}+{\sigma_{B_{Z}}}^{2} \tag{12}\]  
\[\Delta Z_{S}\sim\mathrm{N}(0,{\sigma_{m_{Z}}}) \tag{13}\]  
In summary, the errors from both the MGF and the radar (aircraft) are combined and equivalently regarded as the errors of \(S\), whose planar error (\(\Delta X_{S}\), \(\Delta Y_{S}\)) and elevation error \(\Delta Z_{S}\) follow the normal distributions shown in Equations (11) and (13), respectively. Based on that, the actual radar activation point with errors \(S^{\prime}(X_{S^{\prime}}\), \(Y_{S^{\prime}}\), \(Z_{S^{\prime}})\) and its distribution are shown in Equations (14)-(16), and the schematic diagram of that is shown in Figure 2.  
\[\left\{\begin{array}{l}X_{S}^{\prime}=X_{S}+\Delta X_{S}\\ Y_{S}^{\prime}=Y_{S}+\Delta Y_{S}\\ Z_{S}^{\prime}=Z_{S}+\Delta Z_{S}\end{array}\right. \tag{14}\]  
\[(X_{S}^{\prime},Y_{S}^{\prime})\sim\mathrm{N}(X_{S},{\sigma_{m_{XY}}};Y_{S},{ \sigma_{m_{XY}}};0) \tag{15}\]  
\[Z_{S}^{\prime}\sim\mathrm{N}(Z_{S},{\sigma_{m_{Z}}}) \tag{16}\]  
In Equations (14)-(16) and Figure 2, \(S(X_{S}\), \(Y_{S}\), \(Z_{S})\) is the theoretical radar activation point.  
With the error analysis completed, the detailed steps for calculating the radar identification probability using the Monte Carlo method would be presented.