# Optimization of Protection Level of GBAS with Gaussian Mixture Model
## 5 Testing and Simulation
### Global GBAS Service Performance Simulation Verification  
The main error parameters referred to the RTCA DO245 for ground ranging error, airborne ranging error, tropospheric error, and ionospheric error [4]. (1) Ground ranging error  
Figure 4: Probability density of Beidou ground ranging error of the GBAS ground station. The root mean square of the ground ranging error can be expressed as follows:  
\[RMS_{Gud}(\theta_{i})=\sqrt{\frac{\left(a_{0}+a_{1}e^{-\theta_{i}/\theta_{0}} \right)^{2}}{M_{r}}+\left(a_{2}\right)^{2}} \tag{35}\]  
where \(\theta_{i}\) represents satellite elevation; \(M_{r}\) represents the number of ground reference receivers; and \(\theta_{0}\) is the model parameter. The ground ranging error in CAT-II/IIIa simulation was subject to the GAD_B model parameters. The GAD B model parameters \(a_{0}\), \(a_{1}\), \(\theta_{0}\) and \(a_{2}\) were 0.16 m, 1.07 m, 15.5\({}^{\circ}\) and 0.08 m, respectively. (2) Airborne ranging error  
The root mean square of the airborne ranging error can be expressed as follows:  
\[RMS_{Aad}(\theta_{i})=\sqrt{\left(a_{0}+a_{1}e^{-\theta_{i}/\theta_{0}}\right) ^{2}} \tag{36}\]  
where \(\theta_{i}\) represents satellite elevation and \(\theta_{0}\) is the model parameter. The airborne ranging error in CAT-II/IIIa simulation was subject to the AAD_B and AMD model parameters. \(a_{0}\), \(a_{1}\), \(\theta_{0}\) were 0.11 m, 0.13 m, and 4\({}^{\circ}\), respectively, in AAD_B, and 0.065 m, 0265 m, and 10\({}^{\circ}\), respectively, in AMD_B. (3)  
Topospheric error  
The standard variance of the tropospheric error can be expressed as follows:  
\[\sigma_{trop}(\theta)=\sigma_{N}h_{0}\frac{10^{-6}}{\sqrt{0.002+\sin^{2}( \theta)}}\left(1-e^{-\frac{\Delta t}{\theta_{0}}}\right) \tag{37}\]  
where \(\theta\) represents satellite elevation; \(\sigma_{N}\) represents the broadcast local tropospheric refraction uncertainty parameter; \(h_{0}\) represents the local tropospheric height of broadcasting and the typical value of \(h_{0}\) is 16,000 m; and \(\Delta t\) represents the altitude difference between the aircraft and the ground and is set to 30 m in CAT II/IIIa. (4)  
The standard variance of the ionospheric error can be expressed as follows:  
\[\sigma_{iono}(\theta)=F_{pp}\sigma_{vert\_iono\_grad}(x_{air}+2\tau v_{air}) \tag{38}\]  
where \(\theta\) represents satellite elevation; \(F_{pp}\) represents the vertical-to-slant obliquity factor; \(\sigma_{vert\_iono\_grad}\) represents the spatial gradient parameter of the GBAS and the typical value is 2 mm/km; \(x_{air}\) represents the distance between the aircraft and the GBAS reference station, taking a typical value of 5 km; \(\tau\) represents GBAS smoothing time, with the value of 100 s; and \(v_{air}\) represents the movement speed of the aircraft, and the typical value is 70 m/s. Airborne ranging error, tropospheric error, and ionospheric error basically obey Gaussian distribution, and the ground ranging error is affected by multipath and other factors, and follows the GMM. The above parametric model was used to simulate GBAS service performance in 2592 grids selected globally every 5 degrees of latitude and every 5 degrees of longitude, and each grid was simulated for a total of 24 h at intervals of 400 s. The simulated constellation was adopted from the Beidou-3 global system, including 3 GEO + 3 IGSO + 27 MEO, but only 3 IGSO + 27 MEO were used to realize the GBAS navigation and positioning function [24], and the satellite occlusion angle was set to 5\({}^{\circ}\)  
The comparison and analysis of the Gaussian over-bounding method and the GMM protection level optimization method were mainly carried out from two aspects: the ground ranging error protection level component and the total protection level. (1)  
Comparison of protection level components of ground ranging error  
The ground ranging error protection level component was the amount of ground ranging error contribution to the GBAS protection level.