## 3 Simulation Experiments
### The Simulated Observations of GP along the TS Orbit  
The next step involves determining the GP values at the TS's orbit once the input data have been established. We consider three GSs, each denoted as \(GS_{i}\) (\(i=1,2,3\)), respectively. By taking an SFST measurement every 1 s, we can derive an observed GP difference value, \(\Delta\hat{\phi}_{GTi}(t)\), according to Equation (4). Assuming the GP of \(GS_{i}(t)\) is provided (denoted as \(\phi_{GS_{i}}(t)\)), the observed GP of the TS (denoted as \(\hat{\phi}_{\rm T}(t)\)) can be obtained as \(\hat{\phi}_{\rm T}(t)=\phi_{GS_{i}}(t)-\Delta\hat{\phi}_{GTi}(t)\). Practically, the absolute GP values of the GSs can be calculated using an EGM, such as EGM2008. Given their considerable distance from Earth, the precision of the calculated GPs is relatively high, exceeding 0.1 m\({}^{2}\)/s\({}^{2}\). The error in the absolute GP values along the orbits of the GSs is denoted as \(e_{pot}\) in Equation (6). The observed values \(\hat{\phi}_{T}(t)\) differ from the true GP value \(\phi_{\rm T}(t)\) due to the influence of various sources of error. In this simulation experiment, we accounted for clock error (\(e_{clk}\)), errors in the satellite's position and velocity (\(e_{pos}\) and \(e_{vel}\)), ionospheric residual error (\(e_{ion}\)), potential errors (\(e_{pot}\)), and tidal correction residual error (\(e_{tide}\)). The errors above are treated as noises, necessitating their simulation and subsequent addition to the true values based on respective error models. The cumulative errors, \(e_{all}\), are expressed as follows:  
\[e_{all}=e_{clk}+e_{ion}+e_{pos}+e_{vel}+e_{pot}+e_{tide}, \tag{6}\]  
and the observed values \(\hat{\phi}_{T}(t)\) can be expressed as  
\[\frac{\hat{\phi}_{T}(t)}{c^{2}}=\frac{\phi_{C}(t)}{c^{2}}+\frac{\Delta f(t)}{f _{\rm G}}-\frac{v_{T}(t)^{2}-v_{G}(t)^{2}}{2c^{2}}-\sum_{i=1}^{4}q^{(i)}+ \Lambda f(t)+e_{all}(t). \tag{7}\]  
The nature and magnitude of such errors play pivotal roles in this study. Therefore, it is necessary to investigate various error models corresponding to different sources of errors in order to bring the simulation case closer to the real-world scenario. Initially, we establish the clock error magnitude of \(e_{clk}\) as \(1.0\times 10^{-13}\tau^{-1/2}\), which is currently attainable. Given the best contemporary laboratory clocks, which demonstrate stability of \(9.7\times 10^{-18}\tau^{-1/2}\)[15], we reduced the magnitude of \(e_{clk}\) to \(1.0\times 10^{-15}\tau^{-1/2}\) and \(1.0\times 10^{-17}\tau^{-1/2}\) to simulate specific cases under varying clock precision levels. Despite numerous types of random noises affecting atomic clock signals [48], white frequency modulation and random walk frequency modulation are the most significant components [49]. Accordingly, the behaviors of clock errors are modeled as follows:  
\[e_{clk}(t)=a_{clk}+b_{clk}\cdot t+c_{clk}\cdot w(t)+d_{clk}\cdot\int_{0}^{t }\xi(t)\:\mathrm{d}t, \tag{8}\]  
where \(a_{clk}\), \(b_{clk}\), \(c_{clk}\), and \(d_{clk}\) denote constant coefficients, while \(w(t)\) and \(\xi(t)\) both correspond to standard white Gaussian noises. Each term on the right side of Equation (8) carries a distinct physical implication: \(a_{clk}\) signifies the initial frequency difference, \(b_{clk}\cdot t\) denotes the drift term, \(c_{clk}\cdot w(t)\) represents the white noise component, and \(d_{clk}\cdot\int_{0}^{t}\xi(t)\:\mathrm{d}t\) indicates the random walk effect. A series of frequency comparison data with embedded errors can be generated by assigning suitable values to the constant coefficients based on the performance of OACs, as outlined in [13]. The statistical properties of the three clock error series are illustrated in Figure 7.  
The magnitudes of other error sources are substantially smaller when compared to clock errors. These are briefly discussed below, while more detailed explanations can be found in [21]. The impact of satellites' velocity errors is primarily determined by the \((v_{T}^{2}-v_{G}^{2})/(2c^{2})\) term in Equation (4).