## 2 Materials and Methods  
### Gravity Frequency Shift  
General relativity theory proposes a correlation between a clock's frequency and the gravitational potential (GP) at the clock's location. Consider two clocks situated at disparate positions \(P\) and \(Q\), with their respective GP values being \(U_{P}\) and \(U_{Q}\), the frequencies \(f_{P}\) and \(f_{Q}\) of the two clocks conform to a particular equation as delineated in previous studies [22; 23]. \[U_{P}-U_{Q}=\frac{f_{P}-f_{Q}}{f}\cdot c^{2}+O(c^{-4}), \tag{1}\]  
where \(c\) represents the speed of light in a vacuum, and \(f\) signifies the average of frequencies \(f_{P}\) and \(f_{Q}\). Terms on the order of \(O(c^{-4})\) denote higher-order terms that may be neglected, specifically when the two stations are near Earth. By accurately measuring and comparing the clock frequencies \(f_{P}\) and \(f_{Q}\), one can derive the GP difference \(U_{P}-U_{Q}\) between positions \(P\) and \(Q\). Tackling geodetic issues, such as determining GP, using the clock comparison technique is called relativistic geodesy [24; 25]. Presently, three methods have been suggested for comparing clocks situated at distinct locations: (1) clock transportation [26; 27], (2) transmission of frequency signals via optical fiber links [28; 29; 30], and (3) transmission of frequency signals through free-space links and satellites [21; 31]. The first two approaches are applicable to terrestrial clock comparisons, while the third is specifically crafted for satellite-based comparisons. However, transferring frequency signals via satellites poses significantly more complexity than what is indicated in Equation (1). For instance, the high-speed motion of the satellite induces Doppler effects. Furthermore, space media, such as the ionosphere and troposphere, can cause frequency shifts during the propagation of microwave or optical signals. To mitigate these issues, [PERSON] et al. proposed a technique for transmitting microwave frequency signals between a satellite and a terrestrial site [32]. This method has been successfully applied to validate [PERSON]'s equivalence principle in the Gravity Probe A (GP-A) experiment [33; 34]. The crux of the frequency transfer method involves the concurrent connection of a ground site and a satellite through three microwave links, as depicted in Figure 1. A frequency signal \(f_{e}\) is transmitted by ground station \(E\) at time \(t_{1}\), an event denoted as the uplink (represented by a blue line). Upon receipt, satellite \(S\) retransmits the received signal \(f^{\prime}_{e}\) (downlink depicted by the blue line) and emits a new frequency signal \(f_{s}\) at time \(t_{2}\) (downlink represented by a dark-blue line). These signals are then received at the ground station at time \(t_{3}\) at position \(E^{\prime}\) as \(f^{\prime\prime}_{e}\) and \(f^{\prime}_{s}\). Here, \(\phi\) denotes the gravitational potential (GP). In this scenario, most medium influences and the first-order Doppler effect are nullified in the output beat frequency \(\Delta f\), defined as follows:  
\[\frac{\Delta f}{f_{e}}=\frac{f^{\prime}_{s}-f_{s}}{f_{e}}-\frac{(f^{\prime \prime}_{e}-f^{\prime}_{e})+(f^{\prime}_{e}-f_{e})}{2f_{e}}, \tag{2}\]  
where \(f_{e}\) and \(f_{s}\) represent frequencies of the signals emitted from the ground site and the satellite. These signals are then received as \(f^{\prime}_{e}\) and \(f^{\prime}_{s}\) at the satellite and the ground site, respectively. When the satellite receives the frequency signal \(f^{\prime}_{e}\), the signal is instantaneously transmitted and subsequently received at the ground site as \(f^{\prime\prime}_{e}\). For additional details, please refer to [33]. [PERSON]'s method was subsequently refined and applied to relativistic geodesy for GP determination [20; 21]. This refined method is commonly referred to as the satellite frequency signal transmission (SFST) method.