## 2 Problem Statement
### Potential Hazards in Integrity and Availability  
Multi-constellation can increase the number of visible satellites and improve the geometry. Meanwhile, the increasing number of fault subsets also increases the potential hazards in integrity and makes it difficult to detect the fault.  
From the aspect of fault detection, the threshold is calculated in every fault subset. In the \(k\)th fault subset, the difference \(\Delta\hat{\mathbf{x}}^{(k)}\) between the all-in-view position solution \(\hat{\mathbf{x}}^{(0)}\) and the fault-tolerant position solution \(\hat{\mathbf{x}}^{(k)}\) will test in three components with the corresponding threshold \(T_{k,q}\):  
\[\left|\hat{\mathbf{x}}^{(k)}_{q}-\hat{\mathbf{x}}^{(0)}_{q}\right|\leq T_{k,q} \tag{7}\]  
where the subscript \(q\) is equal to 1, 2 and 3, which represents the east, north, and up components, respectively. The threshold is defined by:  
\[T_{k,q}=K_{fa,q}\sigma^{(k)}_{ss,q} \tag{8}\]  
where \(\sigma^{(k)}_{ss,q}\) is the standard deviation of the difference \(\Delta\hat{\mathbf{x}}^{(k)}\) under nominal conditions. \(K_{fi}\) is the threshold of the standard normal distribution. Different components correspond to different \(K_{fa}\). Then,  
\[\left\{\begin{aligned} K_{fa,1}=K_{fa,2}=Q^{-1}\Big{(}\frac{P_{FA, IOR}}{4N_{faultmodes}}\Big{)}\\ K_{fa,3}=Q^{-1}\Big{(}\frac{P_{FA,VERT}}{2N_{faultmodes}}\Big{)} \end{aligned}\right. \tag{9}\]  
where \(P_{FA,HOR}\) and \(P_{FA,VERT}\) represent the continuity budget allocated to the horizontal and vertical mode respectively. \(Q\) is defined as the right-hand side cumulative distribution function of a zero-mean unit Gaussian and \(Q^{-1}\) is the inverse of the \(Q\) function, also known as the inverse of the tail probability of a zero mean unit normal distribution. The step-by-step specification about the above solution separation calculation can be found in [25].  
The relationship between \(K_{fa}\) and the number of fault modes are shown in Figure 3. With the increase in fault modes, \(K_{fa}\) in three components will increase synchronously. It then results in the expansion of threshold. It affects the fault detection and is easy to lead to missed detection. If the number of fault modes can be limited into the interval [1; 2; 3; 4; 5; 6; 7; 8; 9; 10], it can effectively control the expansion of threshold.  
Figure 3: Relationship between \(K_{fi}\) and number of fault modes.  
At the same time, the multi-constellation will affect the ARAIM availability further. EMT is one of the criteria for vertical positioning performance described by the SARPs (Standards and Recommended Practices) and accepted by the ARAIM as one of LPV-200 requirements on the vertical position. It is based on operational trials that showed that an additional requirement was needed beyond the normal requirements of vertical accuracy and integrity [14]. The EMT can be defined as the maximum of the detection thresholds of faults that have a prior equal or above \(P_{EMT}\) (Probability used for the calculation of EMT). It is computed as follows:  
\[\text{EMT}=\max_{k|p_{fault,k\geq P_{EMT}}}T_{k,3}=\max_{k|p_{fault,k\geq P_{ EMT}}}\left(K_{fa,3}\sigma^{(k)}_{ss,3}\right) \tag{10}\]  
where \(T_{k,3}\) is the solution separation threshold in the up component, and it is the product of \(K_{fa,3}\) and \(\sigma^{(k)}_{ss,3}\). It has shown above that \(K_{fa,3}\) becomes larger with the increase of fault modes. The EMT at the current epoch will get larger and directly threaten the availability of ARAIM.