## 4 Data Processing
As we set proper values of constant coefficients following the performance of OACs in [19], we were able to generate a series of frequency comparison data with errors embedded. For other error sources, their magnitudes are discussed in detail in [34] and listed in Table 2. Most of them are reduced to small magnitudes, since errors in uplink and downlink signals are canceled out in the SFST method. In addition, though the residual influences of the ionosphere and troposphere for the SFST method are at the centimeter level (corresponding to a frequency shift at the \(10^{-18}\) level), we established correction models [34] to reduce their influences to the millimeter level. The tidal effects could reach up to 60 cm at their maximum [55]. Still, we can limit the residual error in the vertical direction to 2 mm after corrections for the solid Earth tide [56] and 8 mm for ocean tide loading [57]. The experiment's other time-varying gravity field changes, such as atmospheric loading, ice loading, etc., were not considered, since their residuals are at least one magnitude smaller than the residuals of tidal effects. Since there are no mature mathematical models for the errors mentioned above, and their influences are much smaller than the clock errors (see Table 2), we adopted a general error model which contains systematic (initial) offset, drift, and white Gaussian noise values for each of the error sources, expressed as the following equation:  
\[e_{j}(t)=a_{j}+b_{j}\cdot t+c_{j}\cdot\phi_{i}(t),\ \ \ \ (j=ion,tro,pos,vel, tide,asy) \tag{17}\]  
where \(a_{j}\), \(b_{j}\), and \(c_{j}\) are constant coefficients, which are randomly set following the error magnitudes listed in Table 2.  
According to Equations (16) and (17), we were able to generate noise signals based on the magnitudes and nature of the error sources at any time. Then, these noises were added to relevant true values, and we obtained a set of relevant \"Observed\" values, based on which the geopotential difference \(\Delta\hat{W}_{QD-SF}(t)\) was determined using Equations (8) and (14). The next step was to convert the geopotential difference to the corresponding height difference. Without a loss of generality, assuming the zero-height surface of China's VHS is just coinciding with the \(W_{0}\) surface, based on China's VHS, the height of SFDS can be calculated by Equation (13), expressed as  
\[\hat{H}_{SF}(t)=\frac{H_{QD}\cdot(g_{QD}+0.0424H_{QD})-\Delta\hat{W}_{QD-SF}(t )}{g_{SF}+0.0424H_{SF}(t)}, \tag{18}\]  
where \(H_{Q}=72.260\) m is the height of the QDDS in China's VHS. In this case, the observed VHS difference between China and US can be obtained as  
\[\Delta\hat{H}_{VHS}(t)=\hat{H}_{SF}(t)-H_{SF}, \tag{19}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Influential Factor** & **(Residual) Error Magnitude in \(\Delta\hat{ff}_{e}\)** \\ \hline ionospheric correction residual & \(\delta f_{int}\sim 5.5\times 10^{-19}\)\({}^{a}\) \\ tropospheric correction residual & \(\delta f_{tro}\sim 1.9\times 10^{-19}\)\({}^{b}\) \\ tidal correction residual & \(\delta f_{tide}\leq 10^{-18}\) (\(0.1\) m\({}^{2}\)/s\({}^{2}\)) \\ position and velocity & \(\delta f_{geup}\sim 3.4\times 10^{-19}\) (\(10\) mm and \(0.1\) mm/s\({}^{c}\)) \\ asynchronism & \(\delta f_{delay}\sim 10^{-19}\) (below \(1\) ms) \\ clock error & \(\delta f_{osc}\sim 4.8\times 10^{-17}\) \\ \hline \hline \end{tabular} \({}^{a}\) 10% of the ionospheric frequency shift after tri-frequency combination; \({}^{b}\) 5% of the tropospheric frequency shift after tri-frequency combination; \({}^{c}\) satellite position errors are assumed to be 10 mm [58], velocity errors are assumed to be 0.1 mm/s [59].  
\end{table} 
Table 2: Error magnitudes of different error sources when determining the gravitational potential difference between a satellite and a ground station. They are transformed to relative frequencies (modified from [34]). where \(H_{SF}=109.126\) m is the height of the SFDS in the US's VHS, and the unification of the two VHSs can be realized.