## 4 Integrity Monitoring
### Protection Level  
Protecting against the onset of satellite faults, the probability of two or more faults is calculated as [14]:  
\[P_{n}=1-\prod_{j=1}^{N}(1-p_{j})-\prod_{j=1}^{N}(1-p_{j})\times\sum_{i=1}^{N} \frac{p_{j}}{1-p_{j}} \tag{45}\]  
where \(p_{j}\) represents the probability of the onset of satellite fault for satellite \(j\). \(N\) is the number of the fault modes, here the number of the satellites minus \(1\). Note that as only the GPS observations are used, the constellation fault and fault of the reference satellite are not feasible for a consistency check, and are thus not protected in this study. The \(P_{n}\) amounts to, e.g., about \(10^{-9}\) to \(4.5\times 10^{-9}\) in the case that \(N\) equals \(5\) to \(10\). Assuming a horizontal probability of hazards misleading information (PHMI\({}_{H}\)) of \(10^{-5}\) per approach, the \(P_{n}\), which is considered as the probability of non-monitored faults, is at an insignificant level. The probabilities of the monitored faults in the north (\(q=1\)) and east (\(q=2\)) directions, could then be computed as:  
\[\mathrm{PHMI}_{1}=\mathrm{PHMI}_{2}\approx\frac{\mathrm{PHMI}_{H}}{2} \tag{46}\]  
As in the ARAIM algorithm [13; 14; 42], the north (\(q=1\)) and east (\(q=2\)) protection levels can be computed (as given in these references):  
\[2\times(1+\epsilon)^{n_{0}}\times Q\Bigg{(}\frac{\mathrm{HPL}_{q}-\bar{b}_{0,q }}{\bar{\sigma}_{0,q}}\Bigg{)}+\sum_{k=1}^{N}p_{k}\times(1+\epsilon)^{n_{k}} \times Q\Bigg{(}\frac{\mathrm{HPL}_{q}-T_{k,q}-\bar{b}_{k,q}}{\bar{\sigma}_{k,q }}\Bigg{)}=\mathrm{PHMI}_{q} \tag{47}\]  
where \(n_{0}\) and \(n_{k}\) are the numbers of the between-receiver observations in the all-in-view and subset cases (\(k\)), respectively. \(\bar{b}_{k,q}\) and \(\bar{\sigma}_{k,q}\), including the all-in-view case, i.e., \(k=0\), are calculated based on the overbounding standard deviations and mean values as follows (Section 4.1):  
\[\bar{\sigma}_{k,q} = o_{q}^{T}\sqrt{\mathrm{diag}\big{(}(A_{x_{k}}^{T}\bar{Q}_{y_{k} }^{-1}A_{x_{k}})^{-1}\big{)}} \tag{48}\] \[\bar{b}_{k,q} = o_{q}^{T}\underbrace{[(A_{x_{k}}^{T}\bar{Q}_{y_{k}}^{-1}A_{x_{k }})^{-1}A_{x_{k}}^{T}\bar{Q}_{y_{k}}^{-1}]}_{S_{k}}\bar{m}_{DD,k} \tag{49}\]  
where \(\mathrm{diag}(\cdot)\) retrieves the diagonal elements of the matrix contained in (\(\cdot\)) as a vector. Recall that the \(\epsilon\) is the excess mass of the EMC (Equations (35) and (36)), and a total mass \(1+\epsilon\) needs to be considered for all the between-receiver observations [44]. \(\bar{Q}_{y_{k}}\) is calculated with Equations (2)-(4) excluding the rows and columns for observations \(k\) and using \(\bar{\sigma}_{\phi,j}\) and \(\bar{\sigma}_{p,j}\) instead of \(\sigma_{\phi,j}\) and \(\sigma_{p,j}\). The vector \(\bar{m}_{DD,k}\) contains the maximum nominal biases for the DD observations, and can be expressed as:  
\[\bar{m}_{DD,k}=\Bigg{(}\frac{\bar{m}_{\phi,1}}{\sqrt{w_{\phi,1}^{12}}},\cdots,\frac{\bar{m}_{\phi,1}}{\sqrt{w_{\phi,1}^{10}}},\frac{\bar{m}_{\phi,2}}{\sqrt {w_{\phi,2}^{12}}},\cdots,\frac{\bar{m}_{\phi,2}}{\sqrt{w_{\phi,2}^{10}}}, \frac{\bar{m}_{p,1}}{\sqrt{w_{p,1}^{12}}},\cdots,\frac{\bar{m}_{p,1}}{\sqrt{w _{p,1}^{10}}},\frac{\bar{m}_{p,2}}{\sqrt{w_{p,2}^{12}}},\cdots,\frac{\bar{m}_ {p,2}}{\sqrt{w_{p,2}^{10}}}\Bigg{)}^{T} \tag{50}\]  
where the items corresponding to observations \(k\) are excluded.