## 2 Background
### Characteristics of a Space DGNSS Scenario
#### 2.2.2 Parallelism Assumption Used for Terrestrial Applications  
In a lunar DGNSS scenario, the ISR can become significantly large with respect to the ranges separating the users from the GNSS satellites in contrast to a terrestrial DGNSS scenario. This difference can impact the performance of DGNSS ranging in space. The present section presents these differences, exposes the inherent issues and proposes potential solutions. **Definition of the bias introduced by the parallelism assumption**  
On top of the measurements modeling differences explained in the previous section, the performances of DGNSS in space are also subject to variations with respect to the performances presented for terrestrial applications. This is due to the assumption taken to establish the relationship presented in Equation (13). Indeed, as stated in [23; 39; 40; 44; 60], the system of equations presented in Equation (10) comes from the assumption that the difference in true ranges \(\Delta r_{ij,k}^{s}\) can be approximated by the inner product between the  
Figure 2: Schematic of the GNSS transmitted signals in deep space. The main lobe beamwidth value was taken from the interoperable GNSS SSV book [8]. baseline vector \(\mathbf{d}_{ij,k}\) and the unit steering vector \(\mathbf{h}^{s}_{i,k}\) or \(\mathbf{h}^{s}_{j,k}\) between either user \(i\) or \(j\) and the satellite \(s\). \[\Delta r^{s}_{ijk}\approx-\Big{[}\mathbf{h}^{s}_{*,k}\Big{]}^{T}\mathbf{d}_{ij,k} \tag{13}\]  
This operation is depicted in Figure 1a, in which the projection of the baseline vector \(\mathbf{d}_{ij,k}\) on the direction \(\mathbf{h}^{s}_{*,k}\) is exactly equal to the difference in true ranges \(\Delta r^{s}_{ij,k}\). This assumption holds while the true ranges \(r^{s}_{i,k}\) and \(r^{s}_{j,k}\) from satellite \(s\) to users \(i\) and \(j\) are much larger than the distance \(d_{ij,k}\) between \(i\) and \(j\), where in such a case their respective unit steering vector \(\mathbf{h}^{s}_{i,k}\) and \(\mathbf{h}^{s}_{j,k}\) are nearly parallel to each other, as depicted in Figure 1a. However, the typical distances of a DGNSS scenario for spaceborne users are easily two orders of magnitudes larger than for terrestrial applications. Hence, this parallelism assumption is likely to introduce a bias in the ISR estimation process, as initially presented in [61] for long baseline Doppler positioning based on Signal of Opportunity (SoP) from LEO. As depicted in Figure 1b, when the distance between the users and the navigation satellites is not large enough with respect to the baseline length for the unit steering vectors originating from both users to be considered parallel, a bias term \(\varepsilon(\Delta r^{s}_{ij,k})\) is introduced:  
\[\Delta r^{s}_{ij,k}=-\Big{[}\mathbf{h}^{s}_{*,k}\Big{]}^{T}\mathbf{d}_{ij,k}+ \varepsilon_{*}(\Delta r^{s}_{ij,k}), \tag{14}\]  
where, if the unit steering vectors from the _aided user_\(\mathbf{h}^{s}_{i,k}\) are used in the SD LS \(\mathbf{H}\) design matrix, the bias term can be written as  
\[\varepsilon_{i}(\Delta r^{s}_{ij,k}) =r^{s}_{i,k}-r^{s}_{j,k}\Big{[}\mathbf{h}^{s}_{j,k}\Big{]}^{T} \Big{[}\mathbf{h}^{s}_{i,k}\Big{]}+\Delta r^{s}_{ij,k} \tag{15}\] \[=r^{s}_{j,k}-r^{s}_{j,k}\Big{[}\mathbf{h}^{s}_{j,k}\Big{]}^{T} \Big{[}\mathbf{h}^{s}_{i,k}\Big{]}\] \[=r^{s}_{j,k}(1-\Big{[}\mathbf{h}^{s}_{j,k}\Big{]}^{T}\Big{[} \mathbf{h}^{s}_{i,k}\Big{]}).\]  
If this correction term Equation (15) is inserted directly in Equation (14) after the replacement of:  
* unknown ranges \(r^{s}_{j,k}\) by the corrected pseudoranges \(\rho^{s}_{corr,j,k}\) from the SPP estimate and,
* the range differences \(\Delta r^{s}_{ij,k}\) by the SD measurements \(S^{s}_{ij,k}\),  
a new equation can be obtained:  
\[S^{s}_{ij,k}-\hat{\rho}^{s}_{corr,j,k}(1-\Big{[}\mathbf{h}^{s}_{j,k}\Big{]}^{ T}\Big{[}\mathbf{h}^{s}_{i,k}\Big{]})=-\Big{[}\mathbf{h}^{s}_{*,k}\Big{]}^{T} \mathbf{d}_{ij,k}+\Delta b_{ij,k}, \tag{16}\]  
and replaced in Equation (10).