## 4 Results  
We first estimate the precision of the observed GP distributions over the TS sphere. The observed disturbing potentials from three different experimental cases are displayed in Figure 8, with Figure 8a exhibiting the disturbing potential of EGM2008, taken as the reference true values. For case 1, featuring a clock instability of \(10^{-13}\tau^{-1/2}\), the outcomes appear to be non-beneficial (refer to Figure 8b). However, a substantial improvement is discerned in the results of case 2 (see Figure 8c) and the outcomes for case 3, which is characterized by a clock instability of \(10^{-17}\tau^{-1/2}\), align closely with the true values. The standard deviation (SD) and the mean offset between the true and observed values can be estimated by \(\phi_{T}(t)\) and \(\phi_{T}(t)\), which are outlined in Table 3. Specifically, the SDs of GP distribution, equivalent to height over the TS sphere, are 12,815.256 m\({}^{2}\)/s\({}^{2}\), 128.086 m\({}^{2}\)/s\({}^{2}\), and 1.662 m\({}^{2}\)/s\({}^{2}\) for cases 1, 2, and 3, respectively. Figure 8: The disturbing potentials (m\({}^{2}\)s\({}^{-2}\)) of (**a**) EGM2008 as true values; (**b**) experiment case 1, utilizing clocks with instability of about \(10^{-13}\tau^{-1/2}\); (**c**) experiment case 2, using clocks with instability of about \(10^{-15}\tau^{-1/2}\); (**d**) experiment case 3, utilizing clocks with instability of about \(10^{-17}\tau^{-1/2}\). Then, we evaluate the precisions of the REMGs drawing from the GP distributions over the TS sphere. The procedure outlined in Section 3.3 was executed, which led to the procurement of the coefficients for the spherical harmonics expansion formula, as defined by Equation (12). Consequently, the REMs were formulated. The difference between the coefficients of the REGMs and those of EGM2008 is visually represented in Figure 9.  
The outcomes suggest that clock instability greater than \(10^{-17}\tau^{-1/2}\), as in cases 1 and 2, leads to diminished precision in the REGMs. However, when the clock instability attains \(10^{-17}\tau^{-1/2}\) (case 3), a relatively accurate set of coefficients for orders and degrees lower than 50 can be achieved. The REGMs' efficacy can also be evaluated by computing the GPs at the TS sphere. The mean offset and SD between the GP distribution over the TS sphere calculated by EGM2008 and REGMs are shown in Table 4. The SDs of the GP distribution over the TS sphere are, respectively, 1320.475 m\({}^{2}\)/s\({}^{2}\), 13.231 m\({}^{2}\)/s\({}^{2}\), and 0.143 m\({}^{2}\)/s\({}^{2}\) for cases 1, 2, and 3. A comparison of cases 1, 2, and 3 in Table 3 reveals that the SD in Table 4 is approximately one order of magnitude smaller for each corresponding case. This improvement suggests that the GPs calculated by the REGMs demonstrate superior precision compared to directly observed GPs. This can be attributed to the least-squares approach, which operates by smoothing the observations. However, each case has no noticeable improvement in the mean offset. In scenarios in which the clock instabilities are improved to the level of \(10^{-18}\tau^{-1/2}\) or even \(10^{-19}\tau^{-1/2}\), other sources of error, such as velocity error, become the dominant factor in limiting the precision of the recovered coefficients. In these instances, further detailed analysis and corrections are required for various error sources to formulate more effective correction models. As there could be a substantial period for adjusting onboard clocks to the \(10^{-18}\tau^{-1/2}\) instability level, we have reserved these investigations for future research. Nonetheless, to underscore the potential of this method, we performed two simplified experiments in which the total error (the summation of clock errors and various other error sources) of SFST links was established at \(10^{-18}\) and \(10^{-19}\), respectively (as shown in Figure 9d,e). We observed that by decreasing the total error magnitudes to \(10^{-19}\), the recovered harmonic expansion coefficients demonstrated commendable quality, closely matching