# Integrity Monitoring of PPP-RTK Positioning: Part II: LEO Augmentation
## 4 LEO Augmentation
### User Positioning
#### 4.3.1 Ambiguity Resolution  
The PAR is enabled at the user side with ambiguities from all constellations resolved together. Figure 10 illustrates the number of fixed decorrelated ambiguities in PAR for different constellation combinations at the user side. The dotted lines represent the cases with LEO augmentation. Note that the full ambiguity resolution (FAR) is achieved at the 60 th epoch for all the lines in Figure 10a. The number of fixed ambiguities is shown in order to illustrate the fact that under LEO-augmentation, an increasing number of total ambiguities (see the ambiguity numbers reaching FAR at the 60 th epoch in Figure 10a) does not hamper the speed of ambiguity resolution. Figure 10a corresponds to the case of using corrections from Network 1 with the smallest size among the three networks and under a relatively quiet ionospheric condition, i.e., with the standard deviation of the between-station ionospheric signals (\(\sigma_{i_{ij,\mathrm{lim}}}\)) set to 1.5 mm at 1 km. In such a case, the variance of the between-station combined network corrections tends to be small due to the precise ionospheric interpolation (see Figure 5 in [14]). The fix rate rises quickly to 1 in almost all cases except for the GPS-only scenario (see the solid red line in Figure 10a) due to the limited number of GPS satellites transmitting on L1/L5 at the moment and those observed by the user. From Figure 10a it can be observed that the total number of ambiguities has been significantly increased under the LEO augmentation (dotted lines), which does not slow down the ambiguity resolution. For single-constellation scenarios (see the red, magenta and dark purple lines) with limited satellite numbers, LEO augmentation has accelerated the ambiguity resolution due to the increased model strength.  
In Figure 10b, network corrections are processed in Network 3 which has a larger size and under a more active ionospheric condition with \(\sigma_{i_{ij,\mathrm{lim}}}\) set to 5 mm. For the same constellation combinations, compared with Figure 10a, it is clear that a lower precision of the combined network corrections (at the between-satellite level) slows down the ambiguity resolution. However, under such a case, the benefits of the LEO augmentation become more evident for the single-constellation scenarios. Taking the GPS- and Galileo-only cases (see the red and magenta lines in Figure 10b) as examples, without LEO augmentation, the FAR is not reached until the 60 th epoch, while adding the LEO constellation enables the FAR at about the 30 th to the 40 th epochs. From Figure 10 it can also be observed that the LEO-only scenario (see the gray lines) does not behave worse than the GPS- and  
Figure 10: Total number of fixed (decorrelated) ambiguities for different constellation combinations using: (**a**) Network 1 with a \(\sigma_{i_{ij,\mathrm{lim}}}\) of 1 mm; (**b**) Network 3 with a \(\sigma_{i_{ij,\mathrm{lim}}}\) of 5 mm. “#” means the number.  
Galileo-only scenarios for L1/L5 satellites. For user processing without the need to estimate satellite-specific parameters such as satellite clocks and phase biases, the rapid geometry change strengthens the model and accelerates the ambiguity resolution under the LEO augmentation. Recall that the elevation cut-off angle of LEO satellites was set to 5 degrees due to their stronger signal strength, while for GNSS satellites it is set to 10 degrees.  
It is also noted that ambiguity resolution here is performed based on simulated data, which delivers formal results without consideration of the mis-modeled effects. The results aim to show the improvement in the efficiency of the ambiguity resolution when having the LEO augmentation. With mis-modeled effects, the number of resolved ambiguities could be reduced at the same processing time. As such, the positioning solutions and the HPLs are assessed in both the ambiguity-float and -fixed cases, for which the former case represents an extreme situation that no ambiguities are fixed.