## 3 Simulation Experiments
### The Simulated Observations of GP along the TS Orbit
The velocity uncertainty of a spacecraft can reach 0.1 mm/s [50], corresponding to an approximate \(e_{vol}\approx 10^{-17}\) in relative frequency. The position errors of satellites influence the determined GP over the TS's orbit since the satellite's GPs are observed at positions divergent from the assumed positions. The magnitudes of these errors can be estimated using the EGM2008 model by determining the GPs of two slightly different positions. The position uncertainty of a GEO or LEO satellite is around 1 dm currently [51; 52]. However, it is highly probable that if these satellites are equipped with high-precision atomic clocks, the position uncertainty can be improved to 1 cm. Therefore, in our experiments, we suppose the satellites' position uncertainty is around 1 cm, corresponding to a maximum of \(8.5\times 10^{-2}\) m\({}^{2}\)/s\({}^{2}\) in GP (\(e_{pos}<10^{-18}\) in relative frequency). Similarly, considering the precision of GP values determined by EGM2008 is also about 1 cm in height, the magnitude of potential error \(e_{pot}\) is less than \(10^{-18}\) in relative frequency. The ionosphere's effect on the frequency shift of a one-way signal is presented as [43]:  
\[\Delta f_{ion}=-\frac{f}{c}\frac{\mathrm{d}}{\mathrm{d}t}\int_{L}(n_{i}-1)\, \mathrm{d}s, \tag{9}\]  
where \(\Delta f_{ion}\) is the frequency shift caused by ionosphere, \(L\) shows the path of the signal's propagation, and \(n_{i}\) denotes the phase index of refraction of ionosphere expressed as [53]:  
\[n_{i}=1-40.3\frac{N_{e}}{f^{2}}+O(f^{-3}), \tag{10}\]  
where \(N_{e}\) represents the electron density per cubic meter and \(O(f^{-3})\) designates high-order terms, which are omitted as they are at least two orders of magnitude smaller than the \(f^{-2}\) term [54]. The distribution of electron density, \(N_{e}\), can be acquired from the IRI model, and the path of the frequency signals can be deduced from the satellite orbit data. Therefore, the \(\Delta f_{ion}\) value can be accurately modeled, and only the residual values need to be considered. Furthermore, the electron density is relatively low (the maximum density is at about 350 km height [55]), as the signals between the TS and the GSs are higher than 500 km from Earth's surface. Consequently, in this satellite-to-satellite case, the magnitude of the ionosphere error, \(e_{ion}\), is estimated as \(10^{-19}\) in relative frequency. This magnitude is significantly smaller than the satellite-to-ground case estimated in our previous studies [21]. Figure 7: The total Allan deviation for three different clocks. The instabilities of the clocks are \(10^{-13}\tau^{-1/2}\), \(10^{-15}\tau^{-1/2}\) and \(10^{-17}\tau^{-1/2}\) for case 1, case 2 and case 3, respectively. Tidal effects can be corrected and modeled by the Tidal Potential library. The tidal effects on a satellite are smaller and can be estimated more precisely than on a ground station. We assume that the magnitude of the tidal error, \(e_{tide}\), is approximately \(10^{-18}\) in relative frequency because the tidal effects can be modeled to a precision of several centimeters at a ground station. Table 2 illustrates all errors (or residual errors) magnitudes. Except for clock errors, there are no established mathematical models that simulate the characteristics of other error sources, such as residuals of ionospheric and tidal corrections. Consequently, we implemented a general error model that includes systematic (initial) offset, drift, and white Gaussian noise for each of these error sources, as follows:  
\[e_{j}(t)=a_{j}+b_{j}\cdot t+c_{j}\cdot w_{i}(t).\quad(j=ion,pos,vel,pot,tide) \tag{11}\]  
In this equation, \(a_{j}\), \(b_{j}\), and \(c_{j}\) represent constant coefficients that are randomly determined based on the magnitudes of each error source. Although Equation (11) is a simplified model and may not accurately reflect the characteristics of various error sources, it is deemed acceptable because these errors' magnitudes are much smaller than clock errors.