## 2 Method
### SFST Method  
According to the general theory of relativity, we have the following relationship between the geopotentials \(W_{P}\) and \(W_{Q}\) and the clock frequencies \(f_{P}\) and \(f_{Q}\) at two points \(P\) and \(Q\)[16; 42]  
\[\frac{f_{P}^{2}}{f_{Q}^{2}}=\frac{1-2W_{Q}/c^{2}}{1-2W_{P}/c^{2}}, \tag{4}\]  
where \(c\) is the speed of light in a vacuum. Since the Earth's gravity field is weak, we have the following approximation  
\[W_{P}-W_{Q}=\frac{f_{P}-f_{Q}}{f}c^{2}+O(c^{-4}), \tag{5}\]  
where \(f=(f_{P}+f_{Q})/2\), and \(O(c^{-4})\) denote higher order terms that can be neglected if the stations \(P\) and \(Q\) are stationary near the Earth's surface. Equation (5) is sufficient for fiber link measurement between two clocks located on the ground. However, the case is much more complex when we use microwave links to compare two clocks located on-board a satellite and at a ground station. For example, a satellite might move at a high velocity, bringing significant Doppler effects. In addition, the ionosphere and troposphere will influence the frequency of microwaves propagating  
Figure 1: The red dashed curve denotes the global geoid, and the two solid blue curves denote the \(W=W_{P}\) and \(W=W_{Q}\) surfaces, respectively. The bold blue curve denotes the plumb-line, along which the height integration is executed. in space within a medium, and the rotation and tidal effects of the Earth will change the status and environment of a ground station. To address these problems, we formulated the SFST approach for determining the geopotential difference between a satellite and a ground station or between two ground stations [33]. We briefly introduce the main idea and formulations as follows, the details of which refer to [33; 34]. Referring to Figure 2, the SFST contains three microwave links. An emitter at a ground station \(E\) emits a frequency signal \(f_{e}\) at time \(t_{1}\). When the signal is received by satellite \(S\) at time \(t_{2}\), it immediately transmits the received signal \(f^{\prime}_{e}\) and simultaneously emits a frequency signal \(f_{s}\). The two signals simultaneously transmitted and emitted from the satellite are received by a receiver at ground station \(E\) at time \(t_{3}\) and are denoted as \(f^{\prime\prime}_{e}\) and \(f^{\prime}_{s}\), respectively. During the emitting and receiving period, the position of the ground station in space changes from \(E\) to \(E^{\prime}\). The satellite transmits signals simultaneously as it receives signals, so its position in the signal links is supposed to be point S at time \(t_{2}\). There might be a small amount of latency during transmission, since the satellite's position is slightly different at the time it receives and emits the signals. However, the unsynchronization influence is minimal and can be neglected for the SFST, as explained in [34]). The output frequency shift \(\Delta f\) is expressed by a combination of three frequencies as [43; 44]  
\[\frac{\Delta f}{f_{e}}=\frac{f^{\prime}_{s}-f_{s}}{f_{e}}-\frac{(f^{\prime \prime}_{e}-f^{\prime}_{e})+(f^{\prime}_{e}-f_{e})}{2f_{e}}, \tag{6}\]  
Figure 2: Ground station \(E\) emits a frequency signal \(f_{e}\) at time \(t_{1}\), denoted by an uplink (blue line). Satellite \(S\) transmits the received signal \(f^{\prime}_{e}\) (the downlink denoted by the blue line) and emits a new frequency signal \(f_{s}\) at time \(t_{2}\) (the downlink denoted by the dark blue line). The ground station receives signals \(f^{\prime\prime}_{e}\) and \(f^{\prime}_{s}\) at time \(t_{3}\) at position \(E^{\prime}\). \(\phi\) is the gravitational potential (GP), \(\vec{r}\) is the position vector. The beat frequency \(\Delta f\), as expressed by Equation (6), cancels out the first-order Doppler effect due to the relative motion between the satellite and the ground station.