# Optimization of Protection Level of GBAS with Gaussian Mixture Model
## 2 The GMM of the GBAS Ranging Errors  
The ranging error of satellite navigation obeys a heavy tail distribution. Its core distribution is approximately Gaussian distribution, and its tail distribution is unknown, but the probability density of the tails is much higher than that of the core Gaussian distribution model. The actual measurement data of GBAS and SBAS [12; 13; 17] show that the GMM is a distribution model that can effectively describe the actual error of satellite navigation. The GMM is a probabilistic model for representing normally distributed subpopulations within an overall population. As shown in Figure 1, the GMM of the ranging error consists of two Gaussian distribution models weighted to include a Gaussian distribution with a small mean square deviation obeyed by the core error distribution and a Gaussian distribution with a large mean square deviation obeyed by the tail errors [12; 13; 17]. Hence, the probability density function of the GMM can be expressed as [12; 13; 17]:  
\[f_{GM}=(1-\varepsilon)*N(\mu_{c},\sigma_{c})+\varepsilon*N(\mu_{t},\sigma_{t}) \tag{1}\]  
where \(N(\mu,\sigma)\) represents the probability density function of the normal Gaussian distribution with mean \(\mu\) and standard variance \(\sigma\); \(\mu_{c}\) and \(\sigma_{c}\) represent the mean and standard variance parameters of the Gaussian distribution dominated by the core distribution; \(\mu_{t}\) and \(\sigma_{t}\) represent the standard variance of the Gaussian distribution dominated by the tail distribution; and \((1-\varepsilon)\) and \(\varepsilon\) represent the probability that the actual errors obey the Gaussian distribution dominated by the core distribution and the Gaussian distribution dominated by the tail distribution. The actual measurement error has different parameters depending on the application scenario, user equipment, and error type, and the typical parameters. The correction error of the GBAS ground reference receiver [12] (referred to as typical parameters in the following) can be expressed as:  
\[\left\{ \begin{array}{l}\mu_{c}=\mu_{t}=0\\ \varepsilon=0.15\\ \sigma_{c}=0.75\sigma,\ \sigma_{t}=1.82\sigma\end{array} \right. \tag{2}\]  
It should be noted that the above parameters \(\sigma_{c}\) and \(\sigma_{t}\) usually represent the standard variance of the core and tail distributions, and \(\sigma\) represents the base level of the standard variance of \(\sigma_{c}\) and \(\sigma_{t}\). The GMM includes two different types of Gaussian distributions, which can be assumed to have the actual error in a core Gaussian distribution state 0 and a tail Gaussian distribution state 1, with probabilities \((1-\varepsilon)\) and \(\varepsilon\), respectively. The set of satellites available to the user can be expressed as:  
\[Sat=[s_{1},s_{2},\cdots,s_{M}] \tag{3}\]  
where \(M\) is the number of satellites in the available satellite set at \(Sat\) and \(s_{i}\) represents the satellite number. Each satellite error distribution has two different states, and the Gaussian distribution of the set of satellites \(Sat\) has \(N=2^{M}\) states. The set of states of the satellite error distribution \(\phi\) can be expressed as:  
\[\phi=[\phi_{1},\phi_{2},\cdots,\phi_{N}] \tag{4}\]  
Figure 1: The probability density of the GMM of the GBAS. The arbitrary error distribution state can be expressed as:  
\[\phi_{i}=\left[\alpha_{1}^{\phi_{i}},\;\alpha_{2}^{\phi_{i}},\cdots,\;\alpha_{M} ^{\phi_{i}}\right] \tag{5}\]  
\(\alpha_{j}^{\phi_{i}}\) can take the value 0 or 1, the probability of  
\[\left\{ \begin{matrix}p_{\alpha_{j}^{\phi_{i}}}\Big{(}\alpha_{j}^{\phi_{i}}=0 \Big{)}=1-\varepsilon\\ p_{\alpha_{j}^{\phi_{i}}}\Big{(}\alpha_{j}^{\phi_{i}}=1\Big{)}=\varepsilon \end{matrix} \right. \tag{6}\]  
The corresponding state probabilities are:  
\[P_{\phi_{i}}=\prod_{i=1}^{M}p_{\alpha_{j}^{\phi_{i}}} \tag{7}\]  
The Gaussian distribution of the corresponding satellite error parameters obeys:  
\[\left\{