## 4 Results
### ISR Estimation
#### 4.1.2 Impact of Large Orbital Errors  
The results presented so far were simulated without satellite orbital errors in order to isolate the impact of the large-scale lunar DGNSS scenario on the performances of the ranging algorithm itself. However, it is important to analyze in such an environment if differential techniques keep their potential to cancel errors correlated among the involved users and, in particular, large satellite orbital errors in case the lunar GNSS receiver must use GNSS ephemerides from the past, as explained in Section 2.2.1. As mentioned in Section 2.2.2, the closer the parallelism assumption is to reality, the better will be the capacity of the ISR estimator to cancel orbital errors. Therefore, Figure 12 shows what would be the residual orbital error on each GNSS SD measurement in the presence of representative satellite orbital errors of 400 m norm as described in Section 2.2.1. This canceling potential is expressed analytically in Equation (23). The residuals individually depend on the original direction of the GNSS satellites orbital errors which was modeled as random in the present simulations. Hence, they vary significantly over the constellation. However, the overall behavior does present a non-negligible correlation with the respective angles depicted in Figure 8. In such a scenario, Figure 12 shows that the residuals of the orbital errors on the SD pseudorange measurements would be upper bounded by a value of 11 m. Despite the very large baseline characteristic of space GNSS ranging scenarios, single differencing the pseudorange measurements hence still has a significant potential to remove systematic errors common to both users.  
Figure 13 analyzes through RMSE the impact of large orbital errors on the inter-user ranging performances over time. In comparison with Figure 10c, no apparent effect is present for the terrestrial techniques, most probably due to the domination by the large bias already present. However, what stands out from this new result is the degradation of the APD technique after 02:00 a.m., or when the theta angles become larger overall. APD is overtaken by the differential algorithms, demonstrating the advantage of applying differential techniques at the measurement level to calculate the range information, instead of retrieving it at position-level like it is achieved in APD.  
Figure 11: ISV estimation over time.  
1.3 Impact of Extrapolating the Aiding User's Observables to the Aided User's Timestamp in the Presence of Receiver Clock Biases  
Finally, Figure 14 compares the results of the ISR estimation with and without differential clock bias between both users, using a comparison based on the simulated and the experimental results. It shows that such a differential clock bias dramatically impacts the differential techniques. As explained in Section 2.2.1, the large Doppler shifts characteristic of a lunar GNSS scenario makes the ranging algorithms much more sensitive to an error in time correction. Indeed, each receiver needs to estimate its own local clock bias in order to find its own timestamp in the system time frame, and successively, to be able to combine its local measurements with the aiding user. Since a time estimation error will be present for both users, the time offset between their respective set of GNSS measurements that is estimated to extrapolate the aiding measurements at the aided user's timestamps will be affected by errors as well. The large Doppler shifts might magnify these time estimation errors. Therefore, the ranging performances are degraded.  
Figure 12: Orbital error residuals on SD measurements.  
Figure 13: ISR RMSE over time with worst-case SISE.