# Integrity Monitoring of PPP-RTK Positioning: Part II: LEO Augmentation
## 3 PPP-RTK Processing Strategy  
The strategy of PPP-RTK processing is introduced in the first part of this series of papers [14]. It uses undifferenced and uncombined observations [37], and forms new estimable parameters based on the \(S\)-system theory [38] to avoid rank deficiencies. The tested LEO satellites are designed for navigation purposes and are expected to broadcast GNSS-like phase and code signals on L1 and L5. As with the GNSS signals, they experience errors such as noise, tropospheric delays, clock biases, hardware biases and ambiguities. The LEO satellites are located at the altitudes of about 300 to 1500 km [39], which are within or above the ionosphere layer that is present from about 50 to 1000 km above the Earth's surface. In our contribution, the altitudes of the tested LEO satellites are 975 and 1100 km (see Table 1). The LEO signals are thus assumed to go through the ionosphere and suffer from similar ionospheric delays as the GNSS satellite signals. The observation equations for the network processing can be formulated as follows:  
\[\mathrm{E}\Big{(}\Delta\varphi^{s}_{r,j}(t_{i})\Big{)}=(g^{s}_{r}(t_{i}))^{ \mathrm{T}}\Delta\tau_{r}(t_{i})+\tilde{dt_{r}}(t_{i})-\tilde{dt^{\mathrm{S}} }(t_{i})-\mu_{j}\widetilde{r}_{r}(t_{i})+\widetilde{\delta}_{rj}(t_{i})- \widetilde{\delta}^{s}_{j}(t_{i})+\lambda_{j}\widetilde{N}^{s}_{r,j}(t_{i}) \tag{4}\]  
\[\mathrm{E}\Big{(}\Delta p^{s}_{r,j}(t_{i})\Big{)}=(g^{s}_{r}(t_{i}))^{\mathrm{ T}}\Delta\tau_{r}(t_{i})+\tilde{dt_{r}}(t_{i})-\tilde{dt^{\mathrm{S}} }(t_{i})+\mu_{j}\widetilde{r}_{r}(t_{i})+\tilde{dt_{r}}(t_{i})-\widetilde{d}^ {s}_{j}(t_{i}), \tag{5}\]  
where \(\Delta\varphi^{s}_{r,j}\) and \(\Delta p^{s}_{r,j}\) indicate the phase and code observed-minus-computed (O-C) term, respectively. The superscript T denotes the matrix transposition. The \(g^{s}_{r}\) maps the zenith tropospheric delay (ZTD) \(\Delta\tau_{r}\) from the vertical to the slant direction. \(\lambda_{j}\) is the wavelength on frequency \(f_{i}\), and \(\mu_{j}=f_{1}^{2}/f_{j}^{2}\) is the coefficient for the ionospheric delay on frequency \(f_{i}\). The forms of the estimable receiver (\(d\widetilde{t_{r}}\)) and satellite clocks (\(d\widetilde{t^{\mathrm{S}}}\)), the estimable ionospheric delays on L1 (\(\widetilde{r_{i}}\)) and ambiguity (\(\widetilde{N}^{s}_{r,j}\)), the phase (\(\widetilde{\delta}_{r,j}\)) and code receiver hardware biases \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Constellation**} & \multicolumn{4}{c}{**Improvement Factor \(f_{\text{HDOP}}\)**} \\ \cline{2-5}  & **5\({}^{\circ}\)** & **10\({}^{\circ}\)** & **15\({}^{\circ}\)** & **20\({}^{\circ}\)** \\ \hline G & 62.6\% & 60.4\%/62.4\% & 54.5\%/60.8\% & 43.8\%/56.3\% \\ \hline E & 33.2\% & 33.0\%/34.2\% & 38.7\%/42.0\% & 47.9\%/53.3\% \\ \hline C & 20.1\% & 19.9\%/21.0\% & 19.6\%/22.8\% & 21.4\%/27.0\% \\ \hline GE & 23.7\% & 23.4\%/24.4\% & 23.0\%/25.9\% & 24.3\%/29.6\% \\ \hline GC & 20.1\% & 19.9\%/20.7\% & 19.3\%/21.6\% & 19.5\%/23.9\% \\ \hline EC & 17.8\% & 17.4\%/18.1\% & 16.9\%/18.9\% & 16.7\%/20.5\% \\ \hline GEC & 14.7\% & 14.5\%/15.0\% & 13.9\%/15.6\% & 13.6\%/17.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Improvement factors in HDOPs using L1/L5 satellites from different GNSS constellations and under different elevation mask angles. The slash sign separates the cases that the elevation mask for LEO satellites increases and does not further increase after 5 degrees. (\(\widetilde{d}_{r,j}\)), as well as their counterparts for the satellite (\(\widetilde{\delta}_{j,i}^{s}\), \(\widetilde{d}_{j,j}^{s}\)) are given in the first part of this series of papers (see Table 1 in [14]). \(\mathrm{E}(\cdot)\) is the expectation operator. Note that the \(d\widetilde{t}_{r}\), \(\widetilde{\delta}_{r,j}\) and \(\widetilde{d}_{r,j}\) are different for each constellation. The ZTDs, the hardware biases and the ambiguities are assumed linked in time by a random-walk noise.