# Optimization of Protection Level of GBAS with Gaussian Mixture Model
## 5 Testing and Simulation  
### Verification of GMM using Data from an Airport  
The 54 th Research Institute of the China Electronics Technology Group deployed a GBAS ground station for category II/IIIa (CAT II/IIIa) landings at Xi'an Xianyang International Airport, and it carried out long-term data collection, analysis, and test verification. CAT II and CAT IIIa operations are both precision instrument approach and landing procedures with decision heights of less than 200 feet (60 m) and 100 feet (30 m), respectively, and runway visual ranges of 1200 feet (350 m) and at least 700 feet (200 m), respectively. The specific GBAS equipment used is shown in Figure 3.  
In order to verify the correctness and effectiveness of the typical parameters of the GMM, the GNSS data collected by the abovementioned GBAS ground station in a real airport environment were used to analyze the distribution characteristics of the GBAS ground ranging error. The GBAS B value of GBAS can reflect the ground ranging error level [23] as follows. The B-value is calculated as:  
\[B_{m}^{n}=PRc^{n}-\frac{1}{M_{r}-1}{\sum_{ \begin{subarray}{c}j=1\\ j\
eq m\end{subarray} }^{M_{r}}}\;PRc_{j}^{n} \tag{32}\]  
where  
\[ \begin{array}{l}PRc^{n}=\frac{1}{M_{r}}\sum_{j=1}^{M_{r}}\;PRc_{j}^{n}\\ PRc_{m}^{n}=SA^{n}+t^{n}+I^{n}+T^{n}+\Delta t_{m}+MP_{m}^{n}+\varepsilon_{m}^{ n}\end{array}  \tag{33}\]  
\(PRc_{m}^{n}\) is the pseudo range correction for satellite \(n\) and receiver \(m\); \(SA^{n}\) is the selective availability error for satellite \(n\); \(t^{n}\) is the clock bias of satellite \(n\); \(I^{n}\) is the ionosphere error for satellite \(n\); \(T^{n}\) is the troposphere error for satellite \(n\); \(MP^{n}\) is the multipath error for satellite \(n\) and receiver \(m\); \(\Delta t_{m}\) is the clock bias of receiver \(m\); and \(\varepsilon_{m}^{n}\) is the residual error for satellite \(n\) and receiver \(m\). Clearly, the common errors \(SA^{n}+t^{n}+I^{n}+T^{n}+\Delta t_{m}\) are eliminated in the B value calculations, and the B values are composed only of uncorrelated measurement noise \(MP_{m}^{n}+\varepsilon_{m}^{n}\), which are the main ranging errors of the GBAS. Moreover, the theoretical value of the statistical characteristics of the B-value is  
\[B_{m}^{n}\sim N\left(0,\frac{\left(RMS_{pr,gnd}\right)^{2}}{M_{r}-1}\frac{N_{ s}-1}{M_{r}N_{s}}\right) \tag{34}\]  
Figure 3: GBAS ground station central processing equipment and outdoor acquisition equipment. where \(M_{r}\) is the number of the receivers of the GBAS, \(N_{s}\) is the number of the satellites, \(RMS_{pr,gnd}\) is the root mean square of the GBAS ground ranging error. In consideration of the differences in satellite constellation configurations, the GBAS ground ranging error of the Beidou navigation satellite system (BDS) was analyzed by statistically analyzing the B values. The B values were calculated and normalized by the Ground Accuracy Designator B (GAD_B) model to verify that the GBAS ground ranging error obeys a GMM distribution and that the Gaussian over-bounding method can over-bound the heavy tail of the ranging error, as shown in Figure 4. The GAD_B model can be referred to in Equation (35). As shown in Figure 4, as the probability density of the actual error was higher than the Gaussian model of the core error, it proved that the ranging error obeyed a heavy tail distribution. To bound the heavy tail, the GMM model, which consisted of the Gaussian model of the core error, and the Gaussian over-bounding model, which was configured as GAD_B, were both proposed. It was clear that both models can bound the tail of the ranging error; therefore, the GAD_B model was used as the Gaussian over-bounding model of this GBAS and it could be expressed as \(N\left(0,RMS_{gad\_B}\right)\), where \(N\) represents the probability density function of the normal Gaussian distribution, and \(RMS_{gad\_B}\) can be referred to in Equation (35).