## 3 Materials and Methods
### Ionospheric Phase Delay Analysis and Correction  
The ionosphere introduces nonlinear effects in proportional compensation. For uplink and downlink carriers, the propagation medium is divided into intra- and extra-atmospheric parts. In the extra-atmospheric part, approximated as a vacuum, the refractive index is 1, making time delays carrier frequency-independent. Within the atmosphere, carriers pass through various layers where the refractive index varies with wavelength due to dispersion effects. As noted in some GNSS textbooks, the troposphere can be approximated as non-dispersive for frequencies up to 15 GHz. Based on this approximation, we have neglected tropospheric dispersion in our modeling, considering only the phase delays proportional to the carrier frequency. However, the ionosphere differs, as it contains free electrons generated by solar ionization. These alter the refractive index and, due to electron-wave interactions, introduce time delays that are dependent on the carrier frequency. Proportionally allocating (in real-time) the total ionospheric delay to the upstream and downstream links inevitably introduces errors. For a carrier beam of frequency \(f\), the propagation delay within the ionosphere satisfies the following relationship [28]:  
\[ \begin{split} I_{p}=\int S\ \left(n_{p}-1\right)ds&=\int S \ \frac{a_{1}}{f^{2}}ds\\ =-\frac{40.28}{f^{2}}\int S\ \ N_{e}(s)ds&=-\frac{40.28 TEC}{f^{2}}\end{split}  \tag{5}\]  
where \(I_{p}\) denotes the ionospheric delay in meters, \(n_{p}\) denotes the ionosphere's phase refractive index for electromagnetic waves, and TEC denotes the total electron content, which characterizes the number of free electrons per unit volume of the circular projection. The TEC is usually measured in TECU, with 1 TECU being equivalent to 1 \(\times\) 10\({}^{16}\) el/m\({}^{2}\). Consequently, we can estimate the level of residual asymmetry after proportional compensation. Global ionospheric map products based on the International GNSS Service are available from several organizations. The historical ionospheric dataset, sourced from the updated International Reference Ionosphere (IRI) 2016 model (from [[https://ccmc.gsfc.nasa.gov/models/IRI-2016/](https://ccmc.gsfc.nasa.gov/models/IRI-2016/)]([https://ccmc.gsfc.nasa.gov/models/IRI-2016/](https://ccmc.gsfc.nasa.gov/models/IRI-2016/)), accessed on 31 March 2025), was used to calculate the discrepancy between the estimated ionospheric delay using the proportional compensation model and the actual ionospheric delay. This analysis, with Beijing in September 2020 as the case study, determined the discrepancy between the modeled and actual ionospheric delays across multiple bands. After proportional compensation, the residual ionospheric delay in the uplink was \(<\)1 ns for the C-S band and \(<\)20 ps for the Ku band, as calculated from the typical satellite-terrestrial radio transmission bands. In the Ka-band, after proportional compensation, in the Beijing area, the residual asymmetric delay was in the order of picoseconds based on historical ionospheric data, as shown in Figure 5.  
Meanwhile, using the global ionospheric data corresponding to Xi'an--where our laboratory also operates a ground station--we performed additional calculations, which yielded results of the same order of magnitude, as shown in Figure 6.  
Moreover, in future picosecond-level experiments, the ionospheric asymmetry could be measured in real-time using the differential pseudo-code deviation method based on the GNSS constellation and corrected in real-time [29]. Figure 5: \(Cont.\)  
Figure 5: Residual time delays of the signals reproduced on the satellites in the three bands. (**a**) The electron content (\(TEC\)) of the Beijing ionosphere fluctuated over three days. (**b**) Uplink C-band (4.75 GHz), downlink S-band (3 GHz). (**c**) Uplink Ku-band (14.3135 GHz), downlink Ku-band (12.5635 GHz). (**d**) Uplink Ka-band (31.75 GHz), downlink Ka-band (30 GHz). Figure 6: Residual time delays of the signals reproduced on the satellites in the three bands.