# Optimization of Protection Level of GBAS with Gaussian Mixture Model
## 3 Appling Solution Separation Method to the GMM  
The GMM has N states, and for each state the satellite errors follow a Gaussian distribution. The observation equation of any state can be expressed as:  
\[Y=HX+V \tag{9}\]  
where \(Y\) is the \(M\) dimensional vector of measurements; \(H\) is the geometric matrix in the local east-north-up (ENU) Cartesian coordinates; \(X\) is the \(L\) dimensional vector of the states to be solved containing the user's location; \(V\) is the error information, which obeys the Gaussian distribution, and the parameters of the Gaussian distribution are determined by the state. Then, the positioning results of the state \(j\) can be expressed as:  
\[X^{j}=S^{j}Y \tag{10}\]  
\[S^{j}=(H^{T}W^{j}H)H^{T}W^{j} \tag{11}\]  
\(W^{j}\) is the weight matrix of the corresponding state, and \(S^{j}\) is the mapping matrix from the ranging domain to the localization domain. Since GBAS and SBAS focus on the vertical protection level performance, this paper mainly analyzed the vertical protection level, which can be expressed as [18; 19]:  
\[VPL^{j}=K_{ffmd}{}^{j}\sqrt{\sum_{i=1}^{M}{S^{j}_{3,i}}^{2}{\sigma^{j}_{i}}^{ 2}}+K_{fd}{}^{j}\sqrt{\sum_{i=1}^{M}{\Delta S^{j}_{3,i}}^{2}{\sigma^{j}_{i}}^ {2}} \tag{12}\]  
where \(VPL^{j}\) is the vertical protection level of the failure free hypothesis, \(K_{ffmd}{}^{j}\) is the multiplier determined by the allocated integrity risk, \(K_{fd}{}^{j}\) is the multiplier determined by the allocated continuity risk, \(S^{j}_{3,i}\) is the element of row 3, column \(i\) of the matrix of \(S^{j}\), \(\sigma^{j}_{i}\) is the standard variance of the corresponding satellite under the state \(i\) as shown in Equation (8), and \(\Delta S^{j}_{3,i}=S^{j}_{3,i}-S^{1}_{3,i}\). \(K_{ffmd}{}^{j}\) is determined by the allocated integrity risk \(P_{\text{int}}{}^{j}\) assigned to the state \(j\) and the probability of the state \(P^{j}\). \(K_{ffmd}{}^{j}\) can be calculated as:  
\[P_{md}{}^{j}=\frac{P_{\text{int}}{}^{j}}{p^{j}} \tag{13}\]  
\[K_{ffmd}{}^{j}=\Phi^{-1}(1-P_{\text{int}}{}^{j}) \tag{14}\]  
where \(\Phi^{-1}\) is the inverse function of the standard normal Gaussian distribution function. \(K_{fd}{}^{j}\) is determined by the allocated continuity risk assigned to the state \(j\) and the state probability \(P^{j}\). \(K_{fd}{}^{j}\) can be calculated as:  
\[P_{fa}{}^{j}=\frac{P_{cont}{}^{j}}{p^{j}} \tag{15}\]  
\[K_{fd}{}^{j}=\Phi^{-1}(1-\frac{P_{fa}{}^{j}}{2}) \tag{16}\]  
In summary, the elements of the solution for any of the states \(j\) includes:  
1. Gaussian distribution parameters, i.e., the standard variance parameters, \(\sigma_{i}^{j}=\left[\sigma_{1}^{j},\sigma_{2}^{j},\cdots,\sigma_{M}^{j}\right]\);
2. Probability of solution \(P^{j}\), allocated integrity risk \(P_{\text{int}}{}^{j}\), allocated continuity risk \(P_{cont}{}^{j}\), false alarms rate \(P_{fa}{}^{j}\), and missed alarms rate \(P_{md}{}^{j}\);
3. Protection level multiplier \(K_{ffmd}{}^{j}\) and \(K_{fd}{}^{j}\) with the mapping matrix \(S^{j}\);
4. The solution at \(X^{j}\) and the protection level \(VPL^{j}\). The continuity risk and integrity risk need to be satisfied as: \[\left\{ \begin{array}{l}\sum\limits_{j=1}^{M}P_{cont}{}^{j}\leq P_{con}\\ \sum\limits_{j=1}^{M}P_{\text{int}}{}^{j}\leq P_{\text{int}}\end{array} \right.\] (17) where \(P_{cont}\), \(P_{\text{int}}\) are the total continuity and integrity risks assigned to the protection level. The final protection level is the maximum protection level of all solutions as: \[VPL=\max(VPL^{j})\] (18) \[j\in[1,2,\cdots,N]\]  
The traditional solution separation algorithm applied to GMM over-bounding has two drawbacks:  
1. The design concept of the solution separation algorithm aims to detect the anomaly of satellite navigation, which is achieved by monitoring the difference between the different solutions below a threshold value, so continuity risk is introduced when the difference is higher than the threshold value.