## 2 Method
### Determination of the Height Difference between Two Ground Height Datum Stations  
Suppose we have two ground datum stations--the Chinese height datum station at Qingdao and the American height datum station at San Francisco (which were assumed)--denoted respectively as \(P\) and \(Q\). They are located on two continents that are connected to the same satellite via SFST links simultaneously, cf. Figure 3.  
Suppose the gravitational potential difference \(\Delta\phi_{PQ}\) between the datum points in China and the US has been determined using the SFST approach described in Section 2.2, then we may determine the geopotential difference \(\Delta W_{PQ}\) by the following equation:  
\[\Delta W_{PQ}=(\phi_{Q}-\phi_{P})+(Z_{Q}-Z_{P}), \tag{9}\]  
where \(Z_{P}\) and \(Z_{Q}\) are centrifugal force potentials at \(P\) and \(Q\), respectively, and \(Z\) is expressed as  
\[Z=\frac{1}{2}\omega^{2}(x^{2}+y^{2}), \tag{10}\]where \(\omega\) is the angular velocity of the Earth's rotation, and \(x\) and \(y\) are coordinates defined in the geocentric Earth-fixed Cartesian coordinate system \(o-xyz\) (e.g., ITRF2008, see [1]). Suppose the height of point \(P\) (denoted as \(H_{P}\)) is given, and the geopotential difference \(\Delta W_{PQ}\) has been measured by the SFST method; then, the height of point \(Q\) (denoted as \(H_{Q}\)) can be determined based on Equations (2) and (3), expressed as  
\[ \begin{split} H_{Q}&=\frac{W_{0}-W_{Q}}{\bar{g}_{Q }}=\frac{W_{0}-W_{P}-\Delta W_{PQ}}{\bar{g}_{Q}},\\ H_{P}&=\frac{W_{0}-W_{P}}{\bar{g}_{P}},\end{split}  \tag{11}\]  
where \(\bar{g}_{P}\) and \(\bar{g}_{Q}\) are the average gravity values along the plumb-lines \(PP^{\prime}\) and \(QQ^{\prime}\), respectively (see Figure 1). It should be noted that \(\bar{g}_{i}(i=P,Q)\) cannot be directly calculated by Equation (3), because we do not know the exact density distribution or the gravity distribution \(g(h)\) inside the Earth. We can see that, besides the influence of the given value of \(H_{P}\), the accuracy of the determined \(H_{Q}\) depends on that of \(\Delta W_{PQ}\); consequently, it is related to the stability of the optical atomic clocks. Since we cannot determine the \"mean value\" \(\bar{g}(i)\) precisely, in practical applications in plain regions, \(\bar{g}_{i}\) is usually replaced by the following formula [40]  
\[\bar{g}_{i}=g_{i}+4.24\times 10^{-5}H_{i}, \tag{12}\]  
where \(g_{i}\), in gas (cm/s\({}^{2}\)), is the gravity at ground point \(i\), which can be measured by an absolute gravimeter, and \(H_{i}\), in meters, is the height difference between \(i\) and \(i^{\prime}(i^{\prime}=P^{\prime},Q^{\prime})\) (see Figure 1). Therefore, according to Equations (11) and (12), we obtain a practical formula for determining \(H_{Q}\), expressed as  
\[H_{Q}=\frac{H_{P}\cdot(g_{P}+4.24\times 10^{-5}H_{P})-\Delta W_{PQ}}{g_{Q}+4.24 \times 10^{-5}H_{Q}}, \tag{13}\]  
where \(\Delta W_{PQ}\) applies the geopotential unit (m\({}^{2}\) s\({}^{-2}\), 1 m\({}^{2}\) s\({}^{-2}\) = 1000 gal m), and an iteration procedure can be applied if needed. Figure 3: Connection of China’s HS originating at the Qingdao datum and the US’ HS originating at the San Francisco datum via satellite frequency signal transmission. To connect VHSs, since the heights \(H_{P}\) and \(H_{Q}\) of the two height datum stations are relatively small (say less than 100 m), using Equation (13) is sufficient. For instance, suppose \(H_{P}=0\), \(\Delta W_{PQ}=-100,000\) mGal (which is equivalent to 100 m near Earth's surface), the maximum error caused by using Equation (13) will not exceed 0.4 mm. The reason is stated as follows. In the mentioned case, \(|H_{Q}|=\Delta W_{PQ}/(g_{Q}+0.00424)\). The error caused by the uncertainty \(\delta\tilde{g_{i}}\) of the chosen mean gravity \(\tilde{g_{i}}\) will not exceed 0.00424 gal. Then, we have \(|\delta H_{Q}|=(\Delta W_{PQ}/g_{Q}^{2})\delta\tilde{g_{i}}=100\) m \(\delta\tilde{g_{i}}/g_{Q}\leq 100\) m \(\times\) 0.00424 gal/1000 gal = 0.4 mm.