# Optimization of Protection Level of GBAS with Gaussian Mixture Model
## 4 Optimizing the GMM Protection Level  
For problems (1) and (2), this section mainly adopts the optimized least squares parameters to achieve the same position for different solutions to eliminate continuity risk by avoiding the monitoring of the difference between the different solutions, so the protection level can be reduced by removing \(K_{fd}{}^{j}\sqrt{\sum_{i=1}^{M}\Delta{S_{3,j}^{j}}^{2}\sigma_{i}^{j}{}^{2}}\), which is determined by continuity risk, in Equation (12). On this basis, the classification of solutions is achieved to significantly reduce the operation complexity, and the protection level is further reduced based on the optimal allocation of integrity risk. The vertical protection level minimization with controlled computational complexity is finally achieved as:  
\[ \begin{array}{c}\min\bigl{(}\max\bigl{(}VPL^{j}\bigr{)}\bigr{)}\\ j\in[1,2,\cdots,N]\end{array}  \tag{19}\]  
where \(VPL^{j}=K_{ffmd}{}^{j}\sqrt{\sum_{i=1}^{M}{S_{3,i}^{j}}^{2}\alpha_{i}^{j}{}^{2}}\) and \(VPL^{j}\) is the vertical protection level. Obviously, the protection level of solution \(j\) is affected by \(K_{ffmd}{}^{j}\), \(S_{3,i}^{j}\), and \(\alpha_{i}^{j}\), which differ from solution to solution, and the final protection level is the maximum of all protection levels, as shown in Equation (18). Therefore, it is necessary to rationally configure \(K_{ffmd}{}^{j}\) and \(S_{3,i}^{j}\) among all the solutions so that the protection levels of different solutions are the same, as much as possible, in order to reduce the maximum value of all protection levels, i.e., the final protection level. The algorithm should take into account three factors:  
1. Least squares parameter unification  
The different corresponding weighted least squares \(S^{j}\) matrices for different distribution states lead to different localization solutions \(X^{j}\) and the resulting computational complexity of repeated operations. The continuity risk is mainly used to bound the difference monitoring between different \(X^{j}\). The least squares parameter unification aims to provide a unified \(S\) matrix for all the solutions to make all the position solutions \(X^{j}\) be the same, to simplify the operation process and to eliminate the continuity risk. 2. Solution set classification  
As each satellite error distribution has two different states and the satellite number is \(M\), the objective function of protection level minimization \(\min(\max(VPL^{j}))\) needs to consider \(N=2^{M}\) cases under the constraints, which is difficult to calculate. The set of solutions is classified as a set of \(K\) complementary and mutually exclusive solutions. For any solution set \(Set_{l}\), \(l\) is the number of the solution set, whose values range from 1 to \(K\), and its key parameters include:  
1. The set of solutions \(Set_{l}=[l_{1},l_{2},\cdots,l_{N_{l}}]\), where \(l_{j}\) represents the number of solutions included in \(Set_{l}\) and \(N_{l}\) represents the number of solutions included in \(Set_{l}\);
2. Gaussian distribution parameters, i.e., the standard variance parameters of the solution set \(\sigma_{set}^{l}=\left[\sigma_{set,1}^{l},\sigma_{set,2}^{l},\cdots,\sigma_{ set,M}^{l}\right]\). As there are \(N_{l}\) solutions in the solution set \(Set_{l}\), the standard variance of satellite \(i\), \(\sigma_{set,i}^{l}\), in \(Set_{l}\) is configured as the maximum value among all the standard variance in \(Set_{l}\) as \(\sigma_{set,i}^{l}=\max\Bigl{(}\sigma_{i}^{l_{j}}\Bigr{)}\) with \(l_{j}\in[l_{1},l_{2},\cdots,l_{N_{l}}]\);
3. The probability of solution set \(P_{set}^{l}\), the integrity risk of solution set \(P_{int\_set}^{l}\), the missing alarm rate of solution set \(P_{mid\_set}^{l}\);  
The state probability of the solution set is the sum of the state probabilities of all the solutions included as:  
\[\left\{ \begin{array}{c}P_{set}^{l}=\sum\limits_{l_{j}\in Set_{l}}P_{l}^{l_{j} }\\ P_{mid\_set}^{l}=\frac{P_{mid\_set}^{l}}{P_{mid\_set}^{l}}\end{array} \right.