## 4 Integrity Monitoring
### FDE Procedure  
The FDE procedure used for the RTK is borrowed from the ARAIM algorithm and consists of the SST that is applied in the position domain and a confirmation check (sanity check) with the \(\chi^{2}\)-test applied in the observation domain [13; 14; 42; 46]. In this contribution, the positioning integrity monitoring is performed for the ambiguity-fixed solutions having the SST and the \(\chi^{2}\)-test performed before computing the protection levels. The \(\chi^{2}\)-test is also performed for the ambiguity-float solutions to exclude large outliers. The SST is applied to exclude outliers by comparing the differences between the all-in-view solution and the subset solution excluding suspected satellites with a test threshold. The threshold is calculated based on the formal standard deviation of the corresponding solution difference and the probability of false alert. The ambiguity-fixed all-in-view solution is calculated with \(\Delta{r_{0}}=S_{0}y_{x}\) as shown in Equation (28), and the corresponding subset solution \(\Delta\ddot{r}_{k}\) is computed excluding certain observations (\(k\)) using the least-squares adjustment:  
\[\Delta\ddot{r}_{k}=S_{k}\,y_{x_{k}} \tag{37}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Scenario** & **L1 Phase [m]** & **L2 Phase [m]** & **C1 Code [m]** & **C2 Code [m]** \\ \hline  & \(\mathbf{\hat{m}_{\phi,1}}\) & \(\mathbf{\hat{\sigma}_{\phi,1}}\) & \(\mathbf{\hat{m}_{\phi,2}}\) & \(\mathbf{\hat{\sigma}_{\phi,2}}\) & \(\mathbf{\hat{m}_{p,1}}\) & \(\mathbf{\hat{\sigma}_{p,1}}\) & \(\mathbf{\hat{m}_{p,2}}\) & \(\mathbf{\hat{\sigma}_{p,2}}\) \\ \hline Open-sky scenario & 0.003 & 0.004 & 0.003 & 0.003 & 0.08 & 0.51 & 0.11 & 0.49 \\ \hline Suburban scenario & 0.002 & 0.006 & 0.003 & 0.006 & 0.21 & 0.71 & 0.23 & 0.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Overbounding mean values (\(\mathbf{\hat{m}_{\phi,j}}\), \(\mathbf{\hat{m}_{p,j}}\)) and standard deviations (\(\mathbf{\hat{\sigma}_{\phi,j}}\), \(\mathbf{\hat{\sigma}_{p,j}}\)) for the weighted between-receiver phase (\(\delta\phi_{v,j}\)) and code (\(\delta p_{v,j}\)) residuals. Figure 6: Empirical and overbounding cumulative distribution functions (CDFs) for the weighted between-receiver code residuals on L1 \(\delta p_{v,1}\) in: (**a**) the open-sky scenario and (**b**) the suburban scenario. with  
\[S_{k}=\big{(}A_{x_{k}}^{T}Q_{y_{k}}^{-1}A_{x_{k}}\big{)}^{-1}A_{x_{k}}^{T}Q_{y_{k }}^{-1} \tag{38}\]  
where the rows in \(A_{x_{k}}\) and the columns and rows in \(Q_{y_{k}}\) related to the observations \(k\) are excluded during the calculation, and the corresponding column of zeros are then removed in \(A_{x_{k}}\). Note that these columns of zeros for the observations \(k\) are added in \(S_{k}\) to match its size with that of \(S_{0}\). Assuming that the observation errors are normally distributed, and accordingly the positioning errors, the test thresholds of SST in the north (\(q=1\)), east (\(q=2\)) and up (\(q=3\)) directions are expressed as:  
\[T_{k,q}=K_{FA,q}\times\sigma_{ss,k,q} \tag{39}\]  
with  
\[K_{FA,1} = K_{FA,2}=Q^{-1}\big{(}\frac{P_{FA,H}}{2\times 2\times N}\big{)} \tag{40}\] \[K_{FA,3} = Q^{-1}\big{(}\frac{P_{FA,V}}{2\times N}\big{)}\] (41) \[\sigma_{ss,k,q} = \sqrt{\sigma_{q}^{T}(S_{0}-S_{k})Q_{y}(S_{0}-S_{k})^{T}o_{q}} \tag{42}\]  
where vector \(o_{q}\) contains one for the \(q\)-th positioning element, and its other elements are set to zero. \(N\) is the number of fault modes, and \(Q^{-1}(\cdot)\) is the inverse right-folded CDF of a standard normal distribution. In contrast to aviation, the focus in ground-based applications is on the horizontal positioning. As a result, in this study, the probability of false alert allocated in the horizontal (\(P_{FA,H}\)) and the vertical (\(P_{FA,V}\)) domains are set to \(3\times 10^{-6}\) and \(10^{-6}\), respectively. Note that these values are not yet clearly defined for diverse road transport applications.