## 3 Simulation Experiments
### Input Data  
Section 2.2 demonstrates that a pair of GSs can maintain consistent SFST links to a TS. However, employing three evenly distributed GSs positioned above the equator is preferable for enhanced reliability in real-world applications. For our experiments, we selected the following satellites to serve as GSs: the Chinese communication satellite CHINASAT-1A (at 130.0\({}^{\circ}\)E), the European Union meteorological satellite METEOSAT-9 (at 9.2\({}^{\circ}\)E), and the US communication satellite ECHOSTAR-10 (at 110.2\({}^{\circ}\)W). The GRACE-FO 1 satellite, with an orbit height of approximately 500 km, was chosen as the TS. The configuration for this experiment is illustrated in Figure 5. As the TS orbits the Earth, it maintains SFST links with at least one or two GSs. Leveraging the gravitational potentials known at the GSs, we can effectively determine the gravitational potential at the TS's orbit.  
Figure 4: The scheme of the simulation experiment.  
Figure 5: The target satellite (TS) retains consistent connections with three geosynchronous equatorial orbit satellites (GSs).  
The orbital period of the TS, specifically GRACE-FO 1, is approximately 1.6 h, with an inclination of about 89\({}^{\circ}\). In order to acquire GP data across its orbital sphere with a resolution of 30\({}^{\prime}\times 30^{\prime}\), it is necessary to conduct continuous observations for at least 576 h (or 24.0 days), with the observation interval not exceeding 8.0 s. Consequently, we have established an observational period of 30.0 days with observation interval of 1 s, thereby fully meeting the resolution requirements (see Figure 6). It is pertinent to note that minor unobserved areas exist at the poles due to the TS's inclination not being exactly 90\({}^{\circ}\). However, our computations (detailed in Section 4) indicate that this factor has negligible influence on establishing an EGM. The TS is linked to the closest observable GS for every observation via SFST links. The base frequency, \(f_{G}\), transmitted by the three GSs is alternately set as 2.8 GHz, 3.0 GHz, and 3.2 GHz. Orbit data for the four satellites (one TS and three GSs) are derived from two-line element (TLE) set data, utilizing the Simplified General Perturbations 4 (SGP4) model [41]. Subsequently, the GPs at the orbits of these satellites can be determined using the EGM2008 model. All these data are treated as true values. Thus, errors associated with orbit data and GP are only considered when we conduct simulated observations.  
The ionospheric and tropospheric conditions significantly impact the frequency of microwave signals. To ascertain the values for electron density, we employed the International Reference Ionosphere (IRI) model [42] as a means to approximate the ionospheric effects [43]. Given the TS's altitude, roughly 500 km, which surpasses the tropospheric layer's extent (typically up to 60 km from the ground), the influence of the troposphere can be considered negligible. The GP at the satellites' orbits is also subject to periodic tidal effects. However, these effects are well modeled [44] and can be effectively mitigated with robust software tools such as Tsoft [45] or ETERNA [46]. Our experiment used a Python library Tidal Potential [47] to construct and analyze tidal signals. The input data are summarized in Table 1.  
\begin{table}
\begin{tabular}{l l} \hline
**Entities** & **Values of Parameters** \\ \hline GS Satellite & METEOSAT-9 (\(f_{G}=2.8\) GHz) \\  & CHINASAT-1A (\(f_{G}=3.0\) GHz) \\  & ECHOSTAR-10 (\(f_{G}=3.2\) GHz) \\ TS Satellite & GRACE-FO 1 \\ Gravity field model & EGM2008 \\ Ionospheric model & International Reference Ionosphere \\ Tide correction & Tidal Potential \\ Observation duration & 1\(\sim\)30 January 2023 \\ Measurement interval & 1 s \\ \hline \end{tabular}
\end{table}
Table 1: The input data utilized in simulation experiments.  
Figure 6: The trace of target satellite (TS) in Earth-centered, Earth-fixed (ECEF) coordinate for (a) 1 day and (b) 5 days.