## 5 Discussion
Regarding the noise introduced by the atmospheric link, while we consider that all media, except for the ionosphere, are non-dispersive, there still exists some frequency dependence, even for lower frequency bands. According to the Liebe model, the German Federal Institute for Physical Technology (PTB) simulated the link delay asymmetry of RF signals below 100 GHz under different atmospheric temperature and humidity conditions. The study, combined with measured data from the ground station, calculated the time delay in the troposphere over 300 days for the Ku-band. The experimental results show that, in the Ku-band used, the propagation delay asymmetry caused by the troposphere between the PTB ground station and the IS-4 geostationary satellite's uplink and downlink varies with humidity, but is always less than 10 ps [30]. This work gives an order of magnitude of the time delay asymmetry of the uplink and downlink. The time delay asymmetry would be even lower in the bands used in the ground experiment. Next, we further consider the extrapolation effects of the ionospheric and relativistic corrections. For the ionosphere, based on the calculations presented earlier, we estimate that the residual ionospheric delay in the Ku-band under our compensation method without any model-based correction. According to the IGS official website, the accuracy of ionospheric maps is typically within 2-8 TECU. Using this level of accuracy to compute corrections via an ionospheric delay model, the residual ionospheric asymmetry after proportional compensation can be reduced to the tens-of-picoseconds level. While discrepancies exist between the IRI model and actual ionospheric conditions, the prior literature has demonstrated the effectiveness of ionospheric corrections. Specifically, [PERSON] et al. investigated the discrepancy between time and frequency transfer results corrected using ionospheric maps and models, and those obtained from GPS CP measurements, within the context of TWSTFT applications [31]. Their findings confirm the efficacy of these model-based corrections, yielding long-term stability performance in the order of \(10^{-16}\). This demonstrates the feasibility of applying this ionospheric correction model in similar time and frequency transfer scenarios. If real-time ionospheric measurements are performed using a GNSS dual-frequency receiver, the correction accuracy can be further improved. As for the relativistic correction of the onboard virtual clock based on Einstein synchronization, the uncertainty of the correction depends on the orbital determination accuracy of the GEO satellite. In the following section, we evaluate the effect of orbit determination errors on the relativistic correction term. Taking the predicted orbit data of the Zhongxing-10 GEO communication satellite provided by China Satcom as an example, the orbit determination accuracy is approximately on the order of 10 m. Assuming the satellite position in the Earth-Centered Inertial (ECI) frame is \((a,b,c)\), the ground station position is \((x,y,z)\), and the Earth's angular velocity is \(\omega_{0}\), the onboard Einstein-synchronized virtual clock definition (as previously given in Equation (4)) leads to the following expression for the special relativistic asymmetry in uplink and downlink delay:  
\[\Sigma=-\frac{\sqrt{a^{2}+b^{2}}\sqrt{x^{2}+y^{2}}\omega_{0}\text{sin}\Big{(} \text{arctan}\frac{y-b}{x-a}\Big{)}}{c^{2}-\sqrt{a^{2}+b^{2}}\sqrt{x^{2}+y^{2} }\omega_{0}^{2}\text{cos}\Big{(}\text{arctan}\frac{y-b}{x-a}\Big{)}}. \tag{13}\]  
By taking the partial derivative of the geostationary satellite coordinates, an uncertainty propagation formula can be derived:  
\[\sigma^{2}=\gamma\Bigg{[}\bigg{(}\frac{\partial\Sigma}{\partial x}\bigg{)}^{ 2}\sigma_{x}^{2}+\bigg{(}\frac{\partial\Sigma}{\partial y}\bigg{)}^{2}\sigma_ {y}^{2}\Bigg{]}. \tag{14}\]  
where the definition of \(\gamma\) has been given in Equation (4).