## 5 Conclusions  
In this research, we developed a methodology for calculating the gravitational potentials (GPs) along the orbits of a Low Earth Orbit (LEO) satellite, leveraging precise clocks and inter-satellite frequency signal transfer. We further showcased a potential application of determining the Earth's gravitational field based on the calculated GPs using the satellite frequency signal transfer (SFST) technique. When the precision of the optical atomic clocks (OACs) is less than \(10^{-17}\tau^{-1/2}\), the accuracy of the observed GPs is primarily influenced by the OACs. For example, according to our simulated experiments, the GPs over a target satellite's (TS) orbit can be calculated with precision levels of approximately \(12.815.256\) m\({}^{2}\)/s\({}^{2}\), \(128.086\) m\({}^{2}\)/s\({}^{2}\), and \(1.662\) m\({}^{2}\)/s\({}^{2}\), using clocks with stabilities of \(10^{-13}\tau^{-1/2}\), \(10^{-15}\tau^{-1/2}\) and \(10^{-17}\tau^{-1/2}\), respectively. We can use these GP observations to derive recovered Earth Gravitational Models (REGMs). The precisions of GPs calculated by the REGMs improved approximately one order of magnitude at the TS's position compared to the direct observations, reaching \(0.143\) m\({}^{2}\)/s\({}^{2}\) (equivalent to \(1\)\(\sim\)\(2\) cm) if the clock instability is \(10^{-17}\tau^{-1/2}\). Presently, the stability of a satellite's onboard clock is approximately \(10^{-13}\tau^{-1/2}\). However, precise optical atomic clocks have achieved a stability level of \(9.7\times 10^{-18}\tau^{-1/2}\) under laboratory conditions [15]. It is foreseeable that the stability of onboard atomic/optical clocks could attain a similar level in the near future. At this clock stability level, the precision of the GPs ascertained by the inter-satellite SFST method can reach \(1.662\) m\({}^{2}\)/s\({}^{2}\) (see Table 3 case 3, equivalent to \(1\)\(\sim\)\(2\) decimeters), which presents a novel method for obtaining GP data around the Earth, with broad potential applications in geoscience. Suppose the clock stability surpasses \(10^{-17}\tau^{-1/2}\), reaching levels such as \(10^{-18}\tau^{-1/2}\) or even \(10^{-19}\tau^{-1/2}\); then various other sources of error, such as satellite velocity errors and ionospheric correction  
Figure 9: The divergence between the recovered harmonic expansion coefficients and EGM2008. The clock instabilities of various experiments are (**a**) \(10^{-13}\tau^{-1/2}\); (**b**) \(10^{-15}\tau^{-1/2}\); (**c**) \(10^{-17}\tau^{-1/2}\). Figures (**d**,**e**) depict two additional experiments conducted to establish the total error magnitude for an SFST link of \(10^{-18}\) and \(10^{-19}\), respectively. Figure (**f**) demonstrates the recovered coefficients derived from the true values of GP distribution over the TS-sphere, as defined by EGM2008. residual errors, will become the principal factors affecting the determination of GPs. This instance necessitates the development of more precise error correction models. Directly applying the GP data around the Earth involves deriving an Earth Gravitational Model (EGM). In this study, we merely demonstrated a rudimentary method for obtaining the REGM without any optimization or fine-tuning. It is advisable to derive an EGM from various data sources and methods in a practical setting. The GP data of a LEO satellite, obtained via the SFST method, is a valuable addition to our current resources, considering we lack a robust method for measuring a satellite's GP. Conceptualization, Z.S. and W.S. ; methodology, Z.S. ; software, Z.S. and X.X. ; validation, X.X. and S.Z. ; formal analysis, T.Z. ; investigation, L.W. ; resources, L.H. ; data curation, L.H. ; writing--original draft preparation, Z.S. ; writing--review and editing, Z.C. ; visualization, S.X. ; supervision, W.S. and S.Z. ; project administration, W.S. ; funding acquisition, Z.S. and W.S. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (NSFC) (Grant Nos.