# Integrity Monitoring of PPP-RTK Positioning: Part II: LEO Augmentation
## 4 LEO Augmentation
### User Positioning
#### 4.3.2 Positioning Results  
The addition of LEO satellites also impacts the user positioning precision. As an example, Figure 11 shows the user positioning results based on the simulated data from Network 1 under quiet ionospheric conditions (with \(\sigma_{i_{j,1\text{Em}}}\) set to 1.5 mm). The positioning errors in the Galileo-only scenario (see the gray lines) and the Galileo-LEO-scenario (see the light green lines) are generally bounded by their corresponding 3\(\sigma\)-lines, in agreement with the 99.7% theoretical bounding (see the black and dark green lines). Reductions in the formal positional precision can be seen for both cases without and with the PAR enabled. This is especially obvious at the beginning phase of the ambiguity-float processing (see Figure 11a), where sharp convergence happens in the formal precision of the positioning. When being augmented with the LEO constellation, the improvement in the formal horizontal precision, denoted as \(f_{\text{H}}\), is formulated as:  
\[f_{\text{H}}=\frac{\sqrt{\sigma_{\text{N0}}^{2}+\sigma_{\text{E0}}^{2}}- \sqrt{\sigma_{\text{N}}^{2}+\sigma_{\text{E}}^{2}}}{\sqrt{\sigma_{\text{N0}}^ {2}+\sigma_{\text{E0}}^{2}}} \tag{11}\]  
Figure 11: User positioning results (based on simulations) and three times the formal standard deviations (3\(\sigma\)) of the positional components using Galileo (and LEO) satellites from Network 1 with \(\sigma_{i_{j,1\text{Em}}}\) set to 1.5 mm in (**a**) the ambiguity-float case and (**b**) with the PAR enabled. where \(\sigma_{\rm N0}\) and \(\sigma_{\rm E0}\) denotes the formal standard deviation of the north and the east positional component without the LEO augmentation, respectively. Their counterparts with the LEO augmentation are denoted by \(\sigma_{\rm N}\) and \(\sigma_{\rm E}\). Figure 12 shows the improvement (\(f_{\rm H}\)) in the formal horizontal precision by adding the LEO augmentation for Network 1 with a \(\sigma_{i_{ij,\rm km}}\) of 1 mm (panel a) and for Network 3 with a \(\sigma_{i_{ij,\rm km}}\) of 5 mm (panel b). The solid and dotted lines denote the cases with and without the PAR enabled, respectively. In general, the improvement is more significant in the ambiguity-float case, especially within the first 1000 epochs. For both the ambiguity-fixed and -float cases, the improvements for single-constellation scenarios, e.g., using low-cost receivers, are greater than those for dual- and triple-constellation scenarios. In general, the improvement in the formal horizontal precision amounts to about 10% to 40% after resolving the ambiguities, while it reaches about 40% to 70% for the ambiguity-float solutions within the first 1000 epochs. For ambiguity-float solutions, the convergence time to a good precision is of great interest to users. Figure 13a shows the convergence time needed to let the formal horizontal precision reach 0.02 m in the ambiguity-float case, given here for demonstration purposes only for formal results, where typically 10 cm is sufficient for ambiguity-float real data when mis-modelled effects are considered. The two scenarios distinguish between high and low precision of the combined network corrections, i.e., from Network 1 with a \(\sigma_{i_{ij,\rm km}}\) of 1.5 mm (Scenario 1) and from Network 3 with a \(\sigma_{i_{ij,\rm km}}\) of 5 mm (Scenario 2). It can be observed that with the LEO augmentation, the convergence time is significantly shortened for all constellation combinations, especially for the single-constellation cases and for Scenario 2 (see the dark and light green bars). The convergence time is shortened to within 3 min for all cases, even under Scenario 2 (see the light green bars in Figure 13a). As shown in Table 3, the improvement has reached over 85% for the single-constellation scenarios under Scenario 2. Note that the results discussed here are the formal precision based purely on the model, aiming to show the impact of the strengthened model with LEO augmentation.