# Integrity Monitoring of PPP-RTK Positioning: Part II: LEO Augmentation
## 2 Measurement Geometry under LEO Augmentation
### Hdop  
The position dilution of precision (PDOP) is a unitless indicator to assess the measurement geometry [33], where lower PDOP values correspond to better positioning precision. For road transport users, the horizontal dilution of precision (HDOP) is of higher interest and will be discussed in this section. In this contribution, the HDOP is calculated considering an elevation-dependent weighting function \(P=\left(1+10\times\exp\left(-\frac{\theta^{*}_{r}}{10^{*}}\right)\right)^{-2}\)[34], where \(\theta^{*}_{r}\) denotes the elevation angle from receiver \(r\) to satellite \(s\) in degrees. This elevation-dependent weight function is often used for the GNSS processing [35; 36]. Figure 6 shows  
Figure 5: The daily average number of visible L1/L5 satellites of different constellations in Beijing under different elevation masks. The average number of satellites at an elevation mask angle of 5 degrees amounts to about 5.1, 7.2, 9.9, and 9.4 for GPS, Galileo, BDS and LEO satellites sending signals on L1/L5, respectively. the daily mean HDOPs without (solid lines) and with the augmentation of LEO satellites (dashed lines) at Beijing. The indicator \"G\", \"E\", \"C\" and \"L\" represents the GPS, Galileo, BDS and the tested LEO constellation, respectively. High HDOPs larger than 20 are not considered by calculating the daily mean values. As the signals of LEO satellites have whitened multipath effects and could penetrate infrastructures (see the last paragraph of Section 2.2), HDOPs of one more option are illustrated with the dashed-dotted lines in Figure 6, i.e., the case assuming that the elevation cut-off angle for LEO satellites does not increase with the x-axis after 5 degrees, while the elevation mask angles for other constellations still increase after 5 degrees. The GPS-only HDOPs (see the solid blue line in Figure 6a) do not increase with the elevation mask after 10 degrees due to the increasing number of excluded epochs with low satellite numbers and high HDOPs. From Figure 6, it can be observed that the improvements of the LEO augmentation to the GNSS-based HDOPs are significant. The improvement factors \(f_{\rm HDOP}\) are given for different elevation masks in Table 2, which are formulated as:  
\[f_{\rm HDOP}=\frac{\rm HDOP_{0}-\rm HDOP_{L}}{\rm HDOP_{0}}, \tag{3}\]  
where \(\rm HDOP_{0}\) and \(\rm HDOP_{L}\) represent the HDOP without and with the LEO augmentation, respectively. From both Figure 6 and Table 2 it can be seen that the LEO augmentation improves the horizontal positional geometry for all the listed GNSS-based single-, dual- and triple-constellations by more than 10%, where the increase has reached around or more than 20% for the three single-constellation scenarios. In the cases where the GNSS satellites can only be effectively used with a high elevation cut-off angle (20 degrees for example) while the mask for LEO satellites remains a low value (5 degrees for example), the HDOP improvements can be further increased by more than 3% (see the values in Table 2 after the slash signs). Comparing the values for \"GL\", \"EL\" and \"CL\" with the blue, green and red dashed-dotted lines in Figure 6a and those for dual-frequency HDOPs with solid lines in Figure 6b, it can be observed that the LEO-augmented single-constellation HDOPs can reach a similar level to the dual-constellation GNSS HDOPs, i.e., about 1 to 1.5. The same applies to the LEO-augmented dual-constellation HDOPs (see the blue, green and red dashed-dotted lines in Figure 6b), which are at a similar level to the triple-frequency GNSS HDOPs (see the solid magenta line in Figure 6b). This indicates that if one or two GNSS constellations are not available for the L1/L5 positioning, the suggested LEO constellation  
Figure 6: Daily mean HDOPs using L1/L5 satellites from (**a**) single GNSS constellation; (**b**) dual and triple GNSS constellations, without and with the augmentation of LEO satellites.