## 3 Computation and Analysis
### Orbit Shift over a Long Time Span  
Due to various perturbations, satellite orbits exhibit drift, with variations across different satellites and systems. Figures 14 and 15 depict the satellite tracks of all GPS and Galileo satellites observable at the MATE station over a four-year period, from 1 January 2020 to 31 December 2023. In these figures, the satellite tracks for 2020 are shown in red, 2021 in blue, 2022 in orange, and 2023 in purple, visually representing the orbit drift across the years. Within the Galileo system, satellites E14 and E18 show more pronounced orbit drift compared to others, indicating less stable orbits. Conversely, many GPS satellites exhibit significant orbit drift, notably satellite G02 in 2022 and 2023, G20 in 2021, and G21 in 2020, with their drift far exceeding that observed in the Galileo system. Specifically, substantial orbital drift was observed for G02 during July-November 2022 and April-June 2023, G20 during March-July 2021, and G21 during May-November 2020. Figure 16 illustrates the variation trends of selected orbital parameters--namely eccentricity, Right Ascension of the Ascending Node (RAAN), orbital inclination, argument of perigee, and semi-major axis--for these three satellites during the noted periods. Significant adjustments in these orbital parameters occurred during times of pronounced deviations in satellite trajectories. Figure 13: Variance reduction rates of SD residuals after multipath correction using different parameter values. The results of the parameters \(C_{0}\) and \(d_{0}\) by averaging all frequency parameters across two baselines (mean of the two baselines), averaging parameters for each baseline (mean of the respective baselines), and for each frequency individually (self-value) are compared. The significant changes in the semi-major axis correspond to a substantial variation in the satellite's orbit repeat period. Consequently, if substantial changes occur in the orbital parameters such as the semi-major axis, the residual data from previous periods may not accurately represent the current multipath errors due to significant spatial discrepancies between the expected and actual satellite positions, leading to inaccuracies in multipath error correction. Utilizing orbital data from DOY 029-038 of 2023 as reference orbits for both the GPS and Galileo systems, the average unit spherical distance from the positions of satellites G01-G32 and E01-E36 during DOY 039-365 of 2023 to their nearest reference orbit was calculated. Figure 17a,b demonstrates these deviations. The average deviation distance for Galileo satellites remains relatively stable, peaking at only 0.004 rad, while that for GPS satellites shows a marked increase, with satellite G02 reaching a maximum deviation of 0.029 rad. Interestingly, around DOY 249, the average deviation for GPS satellites stabilizes. Figure 14: Satellite tracks of all GPS satellites observable at the MATE location over the four-year period from 2020 to 2023. The tracks are color-coded by year: 2020 in red, 2021 in blue, 2022 in orange, and 2023 in purple. By recalculating the average unit spherical distance using orbital data from DOY 239-248 as a new reference for the period DOY 249-365, the results, shown in Figure 18, reveal a reduction in the maximum average deviation for GPS satellites to 0.017 rad, which is significantly lower than earlier measurements. This adjustment indicates that using more recent data as reference orbits leads to more accurate multipath error correction, suggesting that outdated historical data may not effectively correct for current GPS multipath errors. Based on the analysis in Section 3.2, there is a notable correlation in multipath errors between the GPS and Galileo systems, with overlapping or narrowly-spaced frequencies showing very strong correlation. Consequently, integrating these two systems could enhance the density of orbit trajectories, potentially mitigating the negative impacts of GPS orbit shifts on long-term multipath correction.