## 3 Materials and Methods
### Proportional Compensation Method for Reproducing the Satellite Clock  
Considering the aforementioned analysis, the requisite compensation values for Equation (3) can be ascertained using Equation (1). Establishing a real-time active compensation system for the ground station carrier phase enables the phase difference between the transmitter and receiver to be detected after combining the correction values. The all-link phase difference can then be locked to 0 using uplink and downlink phase shifters. By denoting the compensation quantity as \(\Delta\phi\), after the system is locked to a steady state, we obtain the following equation:  
\[\begin{split}\Delta\phi=\phi_{AR}(t)-\phi_{AT}(t)-K_{\text{iono} }\left(\frac{TEC_{A}}{\omega_{uA}}+\frac{TEC_{A}}{\omega_{dA}}\right)\\ -(\omega_{uA}-\omega_{dA})\gamma_{A}\Sigma_{A},\end{split} \tag{6}\]  
where \(\Delta\phi\) is the total phase offset given in real-time.  
This analysis assumed that a dual-frequency GPS receiver is used to measure the TEC value at Station A in real-time. If the compensation is scaled directly without measurements, the ionosphere-related terms are deleted from the equation and a minor calculated time difference is introduced. All quantities in Equation (6) are known in real-time, and based on Equation (1), we can obtain the relationship between the compensation quantity and the upstream and downstream link delays as follows:  
\[\Delta\phi=\frac{\omega_{uA}\rho_{as}(t)}{c}+\phi_{S}+\frac{\omega_{dA}\rho_{ as}(t)}{c}. \tag{7}\]  
In this case, owing to the proposed phase-locked forwarding mechanism, the additional S phases (\(\phi_{S}\)) introduced by the original on-satellite crystal oscillator are eliminated. Then, by allocating the total compensation to the uplink and downlink phase shifters in proportion to the uplink and downlink carrier frequencies and by introducing corrections for the relativistic and ionospheric effects in the uplink, we can obtain the following:  
\[\Delta\phi_{up}=\frac{\omega_{uA}}{\omega_{uA+}\omega_{dA}}\Delta\phi+\frac{ K_{\text{iono}}\ TEC_{A}}{\omega_{uA}}+\omega_{uA}\gamma_{A}\Sigma_{A}; \tag{8}\]  
\[\Delta\phi_{down}=\frac{\omega_{dA}}{\omega_{uA+}\omega_{dA}}\Delta\phi+\frac {K_{\text{iono}}\ TEC_{A}}{\omega_{dA}}-\omega_{dA}\gamma_{A}\Sigma_{A}. \tag{9}\]  
Therefore, the principle of real-time phase compensation is as follows. First, relativistic corrections are computed based on satellite orbital data, while ionospheric corrections are derived from ionospheric models. Since these two corrections affect the uplink and downlink paths independently, they are applied separately to each link.  
After applying these preliminary corrections, the closed-loop feedback system is activated. It measures the phase difference between the received and transmitted signals and adjusts this difference to zero, thereby determining the total compensation value required by the system. According to the theoretical derivation, this total compensation is then proportionally distributed between the uplink and downlink paths based on the carrier phase. To ensure precise and synchronized compensation, two phase shifters are controlled simultaneously: the compensation values are pre-calculated by the host computer according to the carrier frequency and sent to both phase shifters simultaneously, with a common phase-shift enable signal to guarantee synchronous high-precision operation.  
Once the compensation amount (\(\Delta\phi_{up}\)) has been added to the uplink phase shifter, the resulting value can be obtained by combining the terms of Equations (3) and (8) as follows:  
\[\phi_{SR}(t)=\phi_{AT}(t). \tag{10}\]This result confirms that the received signal phase at the satellite matches the phase of the atomic frequency standard at the ground station. This methodology enables the GEO satellites to recover precise and stable atomic frequency standards from ground stations in real-time.