## 2 Methodology  
### Observation Models  
In our research, the ISL and GNSS measurements are both simulated as geometric distances between satellite positions or between a satellite and a ground station. The ranging errors are assumed to be zero-mean Gaussian white noise with selected standard deviations (marked as \(\sigma\)), which are added to geometric distances. This allows the simulation of various behaviors of the ISLs and GNSS ranges for each simulation scenario. Both ISL and GNSS measurements are also deteriorated by satellite-only or satellite and station clock errors, respectively. Clock errors were also simulated as white noise with defined standard deviation. In addition, ISL measurements are burdened with bias that is simulated as a constant value for each satellite per day. A more detailed description of the simulation parameters is presented in Section 2.2. In our simulation, we considered two types of possible connection between satellites (Figure 1) [12; 34]:  
1. One-way-satellite \(i\) establishes a link with satellite \(j\) (i.e., one range measurement at a time), and
2. Dual one-way-satellite \(i\) establishes a link with satellite \(j\) and satellite \(j\) establishes a link with satellite \(i\) (i.e., two range measurements at a time). The ISL pseudorange can be written as follows [12; 34]:  
\[\rho_{ij}(t)=\left|\overset{\rightarrow}{R}_{i}(t)-\overset{\rightarrow}{R} _{i}(t-\Delta t)\right|+c\times\delta t_{j}(t)-c\times\delta t_{i}(t-\Delta t)+ c\times\tau_{i}^{bias}+c\times\tau_{j}^{bias}+\epsilon_{ij}, \tag{1}\]  
where \(\overset{\rightarrow}{R}_{i}\) and \(\overset{\rightarrow}{R}_{j}\) represent the three-dimensional position of satellites \(i\) and \(j\), respectively, in Cartesian coordinates; \(\delta t_{i}\) and \(\delta t_{j}\) are the clock errors of the satellites; \(c\) is the speed of light; \(\Delta t\) is the signal travel time; \(\tau_{i}^{bias}\) and \(\tau_{j}^{bias}\) are the ISL range biases; and, \(\epsilon\) is the noise of the measurements. The emission time is given by the clock of the satellite that transmits in the current epoch and the reception time is given by the clock of the receiving satellite. Equation (1) represents the one-way measurement type and requires a satellite clock error estimation. In dual one-way ISLs, the two satellites send signals to each other in turn [5; 36]. The dual one-way observations need to be transformed to a common epoch for the satellite orbit determination, ensuring that the distance between two satellites is reduced to the same observation time, \(t_{0}\). The mean value computed from \(\rho_{ij}(t_{0})\) and \(\rho_{ji}(t_{0})\), given in Equation (2), contains only orbital parameters and biases, and the satellite clock error is eliminated:  
\[\frac{\rho_{ij}(t_{0})+\rho_{ji}(t_{0})}{2}=\frac{\left|\overset{\rightarrow} {R}_{j}(t_{0})-\overset{\rightarrow}{R}_{i}(t_{0})\right|+\left|\overset{ \rightarrow}{R}_{i}(t_{0})-\overset{\rightarrow}{R}_{j}(t_{0})\right|}{2}+ \epsilon_{ij}+\epsilon_{ji}+c\times\tau_{i}^{bias}+c\times\tau_{j}^{bias}. \tag{2}\]  
Exploitation of simultaneous dual one-way measurements is free of clock offsets and might be used to determine the orbit [37; 38]. For simulation of ground station-satellite measurements we used the following model:  
\[\rho_{\mathrm{r}}^{\mathrm{s}}(t_{1})=\left|\overset{\rightarrow}{R}_{ \mathrm{r}}(t_{1})-\overset{\rightarrow}{R}_{\mathrm{s}}(t_{1}-\Delta t_{1}) \right|+c\times\delta t_{\mathrm{r}}(t_{1})-c\times\delta t_{\mathrm{s}}(t_{1} )+c\times\tau_{\mathrm{tropo}}^{\mathrm{ZWD}}+\epsilon_{rs}, \tag{3}\]  
where \(\overset{\rightarrow}{R}_{\mathrm{r}}\) and \(\overset{\rightarrow}{R}_{\mathrm{s}}\) are three-dimensional position of the stations and satellites in Cartesian coordinates, \(\delta t_{\mathrm{r}}\) and \(\delta t_{\mathrm{s}}\) are the clock errors of the satellites, \(\tau_{\mathrm{tropo}}^{\mathrm{ZWD}}\) is the zenith wet delay  
Figure 1: Connection type: (**a**) one-way and (**b**) dual one-way [12; 34].