# Optimization of Protection Level of GBAS with Gaussian Mixture Model
## 4 Optimizing the GMM Protection Level
\tag{20}\]The multiplier of protection level \(K_{ffmd\_set}l\);
* The protection level of the solution set \(Set_{l}\) is the maximum protection level of all the included solutions as: \[VPL_{Set^{l}}=\max(VPL^{l_{j}})l_{j}\in Set_{l}\] (21)  
As the continuity risk is eliminated in the least squares parameter unification, the vertical protection level can be re-expressed as:  
\[VPL^{l_{j}}=K_{ffmd\_set}l\sqrt{\sum\limits_{i=1}^{M}S_{3,i}2\sigma_{i}^{l_{j} ^{2}}} \tag{22}\]  
\[VPL=\max(VPL_{set}^{l})\;l\in[1,2,\cdots,K] \tag{23}\]  
(3) Dynamic allocation of integrity risk to solution set  
Dynamic allocation of integrity risk is mainly to reasonably allocate the total integrity risk, so that the integrity risk of all solution sets is less than the total integrity risk \(P_{\text{int}}\) as:  
\[\sum\limits_{i=1}^{K}P_{set}^{l}\leq P_{\text{int}} \tag{24}\]  
Based on the above three factors, the integrity protection level is the function of the unified matrix \(S\), the number of the solution set \(K\) and the solution set \([Set_{1},Set_{2}\cdots,Set_{K}]\)  
\[VPL=f(S,[Set_{1},Set_{2}\cdots,Set_{K}],K) \tag{25}\]  
The protection level optimization aims to determine \(S\) and \([Set_{1},Set_{2}\cdots,Set_{K}]\) according to the case where the operational complexity constrains the value of \(K\) such that the protection level is minimized as:  
\[ \begin{array}{l}\min(\max(f(S,[Set_{1},Set_{2}\cdots,Set_{K}],K)))\\ st\;\sum\limits_{i=1}^{K}P_{set}^{l}=P_{\text{int}}\end{array}  \tag{26}\]  
In this paper, hierarchical optimization was used to simplify \(\min(\max(f(S,[Set_{1},Set_{2}\cdots,Set_{K}],K)))\), as shown in Figure 2. The specific steps and methods were as follows:  
(1) Solution set determination  
The principle of solution set determination is that the calculation amount is controllable, the protection level of all the solutions in the solution set is similar, and the solution probability is the same. Considering that the solution with a large standard variance of  
Figure 2: Hierarchical optimization method steps. ranging error has a great impact on the protection level optimization objective function, this paper determined the solution set according to the number of satellites with a standard variance of ranging error that needed to be considered separately. The specific methods were:  
1. The standard variance of GBAS ground ranging error of all \(M\) measurements calculated by GBAS positioning and protection level was sorted from large to small;
2. The solution sets were determined according to the traversal depth \(dth\), i.e., the standard variance of the \(M-dth\) observations with smaller standard variance ranking in (1) was determined by the Gaussian over-bounding method, and the remaining observations were determined according to the GMM; then, a total of \(2^{dth}\) sets of solutions could be determined. It can be seen that the Gaussian over-bounding method and the mixed Gaussian method can be regarded as two special cases of this method, and their traversal depths were 0 and \(M\), respectively. 1. Least squares parameter \(S\) optimization  
The principle of least squares parameter optimization is to improve the consistency of the protection level of different solutions in the solution set as much as possible, and reduce the protection level of the maximum solution in the solution set to reduce the protection level of the solution set. Considering that the maximum protection level in the solution set is mainly determined by the satellite error distribution following the tail distribution corresponding term, a simplified objective function can be formed as follows:  
\[\min(\max(S_{i}^{2}*\sigma_{t,i}{}^{2})) \tag{27}\]  
The solution of this equation is similar to that of NOIRAIM. The solvable methods include heuristic method based on weight estimation [21] and analytic method [22]. Among them, the analytical method based on one-dimensional search has the least amount of calculation and has little difference with the optimal solution.