# Integrity Monitoring of PPP-RTK Positioning: Part II: LEO Augmentation
## 4 LEO Augmentation
### Orbital Errors  
Due to the high altitudes of the GNSS satellites, the influences of orbital errors on baselines of a few hundreds of kilometers can be neglected when using high-precision real-time orbits with an accuracy of centimeters. For LEO satellites which have much lower altitudes, i.e., from a few hundreds of kilometers to more than 1000 km, the influences of the orbital errors need to be analyzed when the distance between the network stations and the user reaches a few hundreds of kilometers. The orbital errors in the radial (\(\delta_{\rm R}\)), along-track (\(\delta_{\rm S}\)) and cross-track directions (\(\delta_{\rm W}\)) are projected into the signal direction to a network (\(r\)) and a user station (\(v\)), denoted as \(\Delta\delta_{v}\) and \(\Delta\delta_{r}\), respectively. Their differences \(\Delta\delta_{rv}\) can be expressed as:  
\[ \begin{array}{l}\Delta\delta_{rv}=\Delta\delta_{v}-\Delta\delta_{r}\\ =\langle e_{\rm R},e_{\rm Lv}\rangle\delta_{\rm R}+\langle e_{\rm S},e_{\rm Lv }\rangle\delta_{\rm S}+\langle e_{\rm W},e_{\rm Lv}\rangle\delta_{\rm W}- \langle e_{\rm S},e_{\rm Lv}\rangle\delta_{\rm R}-\langle e_{\rm S},e_{\rm Lv }\rangle\delta_{\rm S}-\langle e_{\rm W},e_{\rm Lv}\rangle\delta_{\rm W}\\ =(\langle e_{\rm R},e_{\rm Lv}\rangle-\langle e_{\rm R},e_{\rm Lv}\rangle) \delta_{\rm R}+(\langle e_{\rm S},e_{\rm Lv}\rangle-\langle e_{\rm S},e_{\rm Lv }\rangle)\delta_{\rm S}+(\langle e_{\rm W},e_{\rm Lv}\rangle-\langle e_{\rm W },e_{\rm Lv}\rangle)\delta_{\rm W},\end{array}  \tag{9}\]  
where \(e_{\rm R}\), \(e_{\rm W}\) and \(e_{\rm S}\) are the unit vectors in the radial, along-track and cross-track directions of the LEO satellite, respectively. \(e_{\rm Lv}\) and \(e_{\rm Lv}\) denote the unit vectors from the LEO satellite to the two stations. \(\langle\cdot,\cdot\rangle\) forms scalar product of two vectors. When not considering the  
Figure 7: Networks of different sizes that are used for the network positioning. The stars represent the worst-location users in the three networks, which are used for the test results in user positioning and IM in Sections 4.3 and 4.4.  
correlation among the orbital errors in the three directions, the standard deviation of \(\Delta\delta_{rv}\), denoted as \(\sigma_{rv}\), is formulated using the variance law as:  
\[\sigma_{rv}=\sqrt{\left(\langle e_{\text{R}},e_{\text{L}v}\rangle-\langle e_{ \text{R}},e_{\text{L}v}\rangle\right)^{2}\sigma_{\text{R}}^{2}+\left(\langle e _{\text{S}},e_{\text{L}v}\rangle-\langle e_{\text{S}},e_{\text{L}v}\rangle \right)^{2}\sigma_{\text{S}}^{2}+\left(\langle e_{\text{W}},e_{\text{L}v} \rangle-\langle e_{\text{W}},e_{\text{L}r}\rangle\right)^{2}\sigma_{\text{W}}^{ 2}} \tag{10}\]  
for which \(\sigma_{\text{R}}\), \(\sigma_{\text{S}}\) and \(\sigma_{\text{W}}\) represent the standard deviation of the LEO satellite orbital errors in the radial, along-track and cross-track directions, respectively. Considering the sizes of the three simulated networks as introduced in [14], as shown in Figure 7, with the inter-station distance ranging from about 40 to 560 km, the influences of the LEO orbital errors are investigated for baselines with one station located at the red star in Beijing (see Figure 8a), and the other station illustrated with blue dots having an latitude or longitude differences ranging from 0.5 to 5 degrees, i.e., with an inter-station distance ranging from about 43 to 555 km. The tested LEO constellation introduced in Section 2.1 is used for the analysis. The standard deviations of the orbital errors, i.e., \(\sigma_{\text{R}}\), \(\sigma_{\text{S}}\) and \(\sigma_{\text{W}}\), are assumed to be 1 cm in each of the radial, along-track and cross-track directions for the reduced-dynamic precise orbit determination (POD), which normally exhibit an accuracy of centimeters using the post-processed GNSS products [45]. Note that using real-time GNSS products, the orbital accuracy of LEO satellites could be degraded depending on the products used [46].