## 3 Computation and Analysis
### Unified Covariance Function Parameters for Short and Long-Term Multipath Correction Analysis  
Using the MATE_MAT1 baseline's GPS L1 and Galileo E1 frequencies as examples, we applied both the grid method and the autocorrelation-based LSC using unified covariance function parameters in the multipath mitigation. The calibration, carried out from DOY 039 to 365 in 2023, employs a 30-day interval. The moving neighborhood parameter is set at 0.02 rad, and the covariance function parameters \(C_{0}\) and \(d_{0}\) are averaged across all frequencies from both baselines. The specific strategies for multipath correction are detailed in Table 2.  
The standard deviations of the SD residuals before and after applying the multipath correction strategies are detailed in Tables 3 and 4. On DOY 039 of 2023, the performance of the grid methods, labeled Grid_G and Grid_GE, are initially comparable. However, Grid_G's performance declines more rapidly over time. In contrast, the LSC method demonstrates robust initial performance, maintaining superiority over Grid_G and showing slightly lesser performance only compared to the combined GPS/Galileo grid method after 120 days. The LSC_GE strategy, specifically modeled on DOY 029-038, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frequencies to be Corrected** & **Method** & **Modeling Date** & **Frequencies Used for Correction** & **Abbreviations** \\ \hline GPS L1 & Grid & 2023 DOY (029–038 & GPS L1 & Grid, Gcf for L1 \\  & & 2023 DOY 239–248 & GPS L1 & Grid,G for L1 (emodel) \\  & Autocorrelation-based LSC & 2023 DOY (029–038 & GPS L1 & LSC_G for L1 \\  & & & & & \\  & & & & & \\  & & & & & \\  & & & & & \\ \end{tabular}
\end{table} 
Table 2: Multipath correction strategies in this study. Figure 19: Repetition rate of GPS and Galileo satellite orbits between DOY 039–365 in 2023 before and after the GPS/Galileo combination. consistently offers superior performance compared to all other strategies modeled on later dates (DOY 239-248). Regarding the Galileo E1 frequency, the combined GPS/Galileo correction strategy proves to be as effective or slightly more so than the Galileo-only approach. This effectiveness persists even after 210 days of modeling, where the LSC_GE strategy consistently surpasses the grid method's performance. This trend is maintained irrespective of whether the correction strategy involves only Galileo or a GPS/Galileo combination, with the combined LSC method achieving the best results. Thus, the study confirms the advantage of employing unified covariance function parameters for diverse baselines and frequency signals, which simplifies the computational process and enhances the effectiveness of multipath error mitigation. Then, the residual variance reduction rate calculated by \(\left(1-\sigma_{1}^{2}/\sigma_{0}^{2}\right)\times 100\%\), where \(\sigma_{1}^{2}\) is the remaining SD residual variance after multipath correction and \(\sigma_{0}^{2}\) denotes the uncorrected SD residual variance, is shown in Figure 20. For GPS L1, the variance reduction rate on DOY 039 of 2023 using the LSC_GE strategy shows improvements of 0.13%, 11.05%, and 11.16% over LSC_G, Grid_GE, and Grid_G, respectively. By DOY 365 of 2023, these improvements markedly increase to 14.24%, 11.25%, and 29.59%, respectively. This suggests that the LSC_GE strategy not only enhances multipath correction immediately after model application but maintains superior correction capabilities over extended periods. Additionally, the LSC_GE's variance reduction consistently exceeds that achieved by the other strategies, even following repeated modeling. For the Galileo E1 frequency, the LSC_GE strategy improves the variance reduction rate by 0.75%, 12.27%, and 11.58% over LSC_E, Grid_GE, and Grid_E respectively on the first day. By DOY 365, these improvements rise to 1.5%, 15.69%, and 14.96%, respectively. Notably, the grid method's performance post-re-modeling remains inferior to the pre-re-modeling performance of the LSC method.