# Optimization of Protection Level of GBAS with Gaussian Mixture Model
###### Abstract  
The Gaussian mixture model (GMM) is commonly used to model the heavy tail of the ground-based augmentation system (GBAS) range error distribution. In practice, Gaussian over-bounding based on a GMM is used to over-bound the heavy tail of the ranging errors, but the GBAS protection levels (PLs) based on the Gaussian over-bounding tend to be overestimated. Based on the idea of solution separation and overcoming the shortcoming of its direct reference to GBAS, this paper analyses the constraint conditions and objective functions of the optimal protection level based on solution separation under a GMM distribution, and proposes that multi-hypothesis solution set classification can effectively reduce the computational complexity. At the same time, least squares optimization and dynamic allocation of integrity risk are used to further reduce the protection level. This paper verifies the validity of the parameters of the GMM based on actual airport GBAS data, performs simulation verification of the typical scenarios of CAT I and CAT II/IIIa global GBAS under the Beidou 3 constellation, and analyses the performance improvement effect under different solution set traversal depths. The results show that when the traversal depths of CAT I and CAT II/IIIa are 4 and 6, the vertical protection level component of the ground ranging error is reduced by 14% and the total vertical protection level is reduced by 10%. GBAS; separation of multiple hypothetical solutions; integrity; protection level; Beidou; Gaussian mixture +
Footnote †: journal: Electronics  
0000-0002-43  
integrity risk testing is difficult to obtain, and it is impossible to verify that the actual error distribution, particularly the tail distribution, obeys the Gaussian distribution [9]. To address the aforementioned issues, a number of measures have been implemented, the most important of which are:  
1. The first method, called the Gaussian over-bounding method, replaces the original GMM distribution with a simplified Gaussian distribution in satisfying the integrity risk based on the over-bounding principle of cumulative distribution function (CDF). [PERSON], [PERSON], and others have proposed an over-bounding method for estimating the error distribution in the measurement domain by dividing it into two parts, core and tail, which effectively handles the error's heavy tail characteristics [9; 10; 11]. Based on this premise, [PERSON] put forward the Gaussian over-bounding approach in the positioning domain. The findings indicated that when compared to the Gaussian over-bounding method in the measurement domain, this approach cuts down on the root mean square of the position error by 30% and efficiently decreases the protection level to fulfill the integrity requirements of GBAS [12; 13]. 2. The other method, called the mixed distribution method, calculates the protection level based on a GMM of GBAS ranging error. [PERSON] and others proposed the core over-bounding method [14], [PERSON] proposed the normal inverse Gaussian (NIG) distribution method [15], and [PERSON] proposed the use of the excess mass function method [16]. The above calculations are extremely complex and difficult to achieve in engineering. [PERSON] proposed a feasible calculation scheme. Assuming that the error distribution follows the GMM, and taking into account the tight core and heavy tail characteristics of the error, he used the Bayesian method to estimate the probability values of different distributions according to the error residual values to obtain the optimal protection level calculation method. The verification results using SBAS data showed that the method reduced the vertical protection level by 50% without losing the integrity risk, but the calculation is complex [17]. In summary, the Gaussian over-bounding method has the advantage of being easy to calculate and the disadvantage of giving a higher protection level, while the mixed distribution method has the advantage of giving a lower protection level and the disadvantage of being difficult to calculate.