## 2 Materials and Methods
### Determination of Gravitational Potential along the Target Satellite Orbit  
Supposing the GP value of a terrestrial station is known, the GP values for a satellite can be ascertained by establishing SFST links. By determining the GP distribution across a TS sphere (a sphere defined by the orbits of satellites similar to the GOCE or GRACE types), the Earth's external gravitational field can be deduced correspondingly [35]. However, due to the proximity of the satellite's orbit for gravitation measurement (for instance, the altitude of the GRACE satellite is around 500 km), only a short arc length of the orbit is visible to a specific terrestrial station. Ascertaining the GP distribution across the TS sphere would necessitate hundreds of terrestrial datum stations with known GP values to ensure the satellite can continuously connect to at least one terrestrial station at any given time [36]. Furthermore, the majority of Earth's surface is composed of oceans, which further complicates the task of establishing a sufficient number of terrestrial datum stations to meet the observation objectives. Consequently, our capacity to deploy the necessary infrastructure for continuous GP measurement of the target satellite is considerably restricted. Figure 1: The satellite frequency signal transmission (SFST) technique for determining the gravitational potential difference (GP) between a satellite and a ground site. The SFST method described in Section 2.1 was initially conceived for determining the GP difference between a terrestrial station and a satellite. Nevertheless, with some modifications, it can also be adapted to determine the GP difference between two satellites [36]. Imagine a scenario in which a target satellite (TS) in a low Earth orbit is linked to a Geosynchronous Equatorial Orbit (GEO) satellite, herein referred to as GS for simplicity, situated in a higher Earth orbit. The configuration of their inter-satellite SFST links is described as follows. Referring to Figure 2, a frequency signal \(f_{\mathrm{G}}\) is generated by an emitter on the GS at a specific time \(t_{1}\). This signal is subsequently received by the TS at time \(t_{2}\). Upon receiving \(f_{\mathrm{G}}\), the TS transmits it back as \(f^{\prime}_{\mathrm{G}}\) and simultaneously emits a new frequency signal \(f_{T}\). The receiver on the GS, now designated as \(GS^{\prime}\), intercepts both signals (\(f^{\prime}_{\mathrm{G}}\) and \(f^{\prime}_{T}\)) from the TS at a later time \(t_{3}\). During the period of signal emission and reception, the GEO satellite's position in space transitions from \(GS\) to \(GS^{\prime}\). The target satellite transmits and emits its signals at the same instant it receives the signal. As such, its position in the signal links is designated as the point \(TS\) at time \(t_{2}\). By setting \(f_{\mathrm{G}}=f_{T}\), the gravitational potential difference between the GS and the TS can be expressed as:  
\[\frac{\Delta\phi_{\mathrm{G}T}}{c^{2}}\equiv\frac{\phi_{T}-\phi_{\mathrm{G}}} {c^{2}}=\frac{\Delta f}{f_{\mathrm{G}}}-\frac{v_{T}^{2}-v_{G}^{2}}{2c^{2}}- \sum_{i=1}^{4}q^{(i)}+\Lambda f+O(c^{-5}), \tag{4}\]  
where the subscripts \(G\) and \(T\) represent the \(GS\) and \(TS\) respectively, and the beat frequency \(\Delta f\) is given as:  
\[\frac{\Delta f}{f_{\mathrm{G}}}=\frac{f^{\prime}_{T}-f_{T}}{f_{\mathrm{G}}}- \frac{(f^{\prime\prime}_{\mathrm{G}}-f^{\prime}_{\mathrm{G}})+(f^{\prime}_{ \mathrm{G}}-f_{\mathrm{G}})}{2f_{\mathrm{G}}}. \tag{5}\]  
The transmission process might introduce a slight latency, leading to slight discrepancies in the satellite's position between the signal emission and reception times. Considering a signal transponder delay of approximately 800 ns [37] and a typical orbital height of the TS of around 500 km (with a velocity of about 7.6 km/s), the TS moves a mere 0.61 cm between the emission and reception of signals. This minute displacement is negligible for observations requiring a centimeter scale precision level.