## 3 Methodology
### The Lunar Orbiter Scenarios
#### 3.1.2 Comparison with a Terrestrial Scenario  
To compare the parallelism assumption validity in a terrestrial scenario with respect to the space scenario of Figure 3, the ratio between the inter-users distance \(d_{ij,k}\), as shown  
Figure 3: Schematic of the GNSS scenario for VMMO and Pathfinder.  
in Figure 4, and a typical user-to-satellite range \(r_{*,k}^{s}\) is considered. The closer this ratio is to zero, the closer the assumption is to be valid. Table 1 presents typical values of this ratio in a terrestrial case with respect to the lunar scenario, considering the distances typical to the VMMO-Pathfinder cooperation and the GPS constellation. The numbers shown in Table 1 show that in a DGNSS case applied to a typical lunar scenario, the ratio becomes significantly higher when compared to terrestrial applications. In the long-baseline DGNSS terrestrial applications presented in the literature, the dominant error factor becomes the ionospheric and tropospheric delays between the cooperating users such that the effect of the bias introduced by the assumption in Equation (13) in the algorithm itself is not analyzed [68; 69; 70].  
On top of the introduction of a bias due to the large ISR values, Figure 4 also confirms that such a differential scenario in space is also characterized by large relative velocities between the involved users, as anticipated in Section 2.2.1. This causes the ISR variation rate to be large. Consequently, a residual error on the aiding user's observables extrapolation process might be more impactful on the final ISR estimation quality, when compared to terrestrial scenarios. The significant user dynamics also cause high pseudorange rates and hence high Doppler shift values. Figure 5 shows the Doppler shifts of the GNSS signals received by Pathfinder, with values up to 17 kHz when compared to around 2 kHz for typical terrestrial ground users. As a consequence, a residual error on the estimated time offset between a pair of measurements used for differential algorithms is likely to be amplified by the large Doppler shifts characteristic of lunar GNSS users.  
Figure 6 shows the GDOP and the visibility of the number of common GNSS signals acquired by both receivers, meaning the signals of which the \(C/N_{0}\) are higher than the receiver sensitivity. Figure 7 shows the \(C/N_{0}\) of the signals tracked by a simulated receiver in lunar orbit (VMMO) as simulated by STK. The formula used to calculate these values is to be found in [42]. And finally, Figure 8 represents the angle \(\theta_{ij^{\prime}}^{s}\) as depicted in Figure 1b between the two Line-of-Sight (LoS) vectors between each user and the GNSS satellite, for each visible GNSS satellite \(s\) over time, in parallel with the ISR truth. As expected, it shows that there is a high correlation between the ISR truth and the inval  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Typical Values** & **Terrestrial Scenario [16; 68]** & **Lunar Scenario** \\ \hline Inter-user range (km) & [0; 300] & [2500; 11,000] \\ \hline User-to-satellite distance (km) & 20,200 & 380,000 \\ \hline Validity ratio (\%\(\%\)) & [0; 15] & [6; 29] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the parallelism assumption validity between terrestrial and lunar scenarios.  
Figure 4: ISR and relative velocity between VMMO and Pathfinder over a 4 h test window.