# Integrity Monitoring of GNSS Constellations with Only LEO-PNT Satellites +
## 3 User-Based Integrity with LEO-PNT Formulation (Option 2)
### User Equations  
#### 3.4.1 Basic LEO Barrier Formulation  
MEO to LEO Observations  
The observation residual at a given epoch for MEO i-LEO l is expressed as follows:  
\[y_{i,l}=\mathbf{v}_{i,l}^{T}\cdot\mathbf{x}_{i}+\mathbf{e}_{i,l} \tag{1}\]  
where  
\(\mathbf{v}_{i,l}^{T}\) is the unit vector from the LEO satellite \(l\) to the MEO satellite i. \(\mathbf{x}_{i}\) is the error in the MEO's position and clock according to the SiS. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Concept** & \begin{tabular}{c} **Budget** \\ **(1 Sigma, m)** \\ \end{tabular} & **Justification** \\ \hline A & \begin{tabular}{c} Accuracy of real-time LEO \\ position solution \\ \end{tabular} & 0.1 & This is considered feasible \\ B & \begin{tabular}{c} Accuracy of real-time LEO \\ synchronization solution \\ \end{tabular} & 0.15 & using the HAS service [3] \\ C & \begin{tabular}{c} Phase noise \\ \end{tabular} & 0.005 & This value is typical for \\ C & \begin{tabular}{c} Phase noise \\ \end{tabular} & 0.005 & \
\begin{tabular}{c} on-board receivers such as \\ PODRIX in the iono-free combination [4] \\ \end{tabular} \\ \hline \hline \multicolumn{4}{c}{**Derived LERE Budget (From A, B, C, and D)**} \\ \hline E & Phase LERE & 0.18 (0.23) &
\begin{tabular}{c} LERE for phase users \\ \(\left(\sqrt{A^{2}+B^{2}+C^{2}}\right)\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error budget. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Inputs for Barrier Design** & \\ \hline \multirow{2}{*}{**MDB**} &
\begin{tabular}{c} Minimum Detectable Bias to be detected \\ with the probability of missed detection \\ \end{tabular} & 2 m \\
**Pmd** &
\begin{tabular}{c} Probability of missed detection \\ \end{tabular} & \(1\times 10^{-5}\) \\
**Pfa** & \begin{tabular}{c} Probability of false alarm \\ \end{tabular} & \(1\times 10^{-5}\) \\ \hline \hline \multicolumn{4}{c}{**Output Barrier Performance Indicators**} \\ \hline \(\mathbf{T_{mon}}\) & \
\begin{tabular}{c} Monitoring Threshold: the threshold at \\ which the barrier is triggered \\ \end{tabular} & 1.13 m \\ \(\mathbf{\sigma_{mon,i}}\) &
\begin{tabular}{c} Monitoring Accuracy or sigma monitor, \\ derived from MDB, Pfa, and Pmd \\ \end{tabular} & 0.19 m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance indicators for LEO-PNT integrity monitor. \(\epsilon_{i,l}\) is the LEO to MEO observation noise considering the LEO PNT's Rx and environment, e.g., LERE. It is modelled as a normal distribution \(\epsilon_{i,l}\sim N\Big{(}0,w_{i,l}^{-1}\Big{)}\), with \(w_{i,l}^{-1}\) being the covariance of the MEO i-LEO i observation noise. This situation is depicted in Figure 1.  
Basic LEO Barrier  
A user will collect from the LEO satellites all the MEO-LEO residuals and run one barrier per MEO satellite. Stacking up the equations for each MEO-LEO observation yields  
\[ \begin{split}&\mathbf{y}_{i}=\begin{bmatrix}y_{i,1}\\ \dots\\ y_{i,L}\end{bmatrix}=\begin{bmatrix}\mathbf{v}_{i,1}^{T}\\ \dots\\ \mathbf{v}_{i,L}^{T}\end{bmatrix}\mathbf{x}_{i}+\begin{bmatrix}\epsilon_{i,1}\\ \dots\\ \epsilon_{i,L}\end{bmatrix}=\mathbf{C}_{t}\cdot\mathbf{x}_{i}+\mathbf{\epsilon}_{i}\\ &\\ & l=\{1,L\}\;\text{LEOs in view of the User and of }\;\text{MEO i}\end{split}  \tag{2}\]  
Now this user will run a barrier for the MEO satellite i. Here, a weighted LSQ barrier is taken as a reference:  
\[\hat{\mathbf{x}}_{i}=\Big{(}\mathbf{C}_{i}^{T}\mathbf{W}_{L_{i}}\;\mathbf{C}_{i}\Big{)}^{-1} \mathbf{C}^{T}\mathbf{W}_{L}\mathbf{y}_{l} \tag{3}\]  
with \(\mathbf{\mathrm{E}}\big{[}\mathbf{e}_{i}\mathbf{e}_{i}^{T}\big{]}=\mathbf{W}_{L_{i}}\). The user will perform a check on \(\hat{\mathbf{x}}_{i}\) (e.g., \(\left|\mathbf{u}_{i}^{T}\hat{\mathbf{x}}_{i}\right|<T_{mon}\)) and decide whether to include or exclude the satellite.