## 3 Computation and Analysis
### Parameters of Covariance Function  
The covariance function used in this study is [PERSON]'s function by Equation (11), where \(C_{0}\) and \(d_{0}\) determine the function. These parameters for different frequencies are initially calculated by fitting the theoretical function to the empirical covariance function and then compared across the frequencies.  
The computed and fitted ECVF curves for the GPS L1 and Galileo E1 frequencies, from the MATE_MAT1 and KERG_KRGG baselines on DOY 038 in 2023, are shown in Figure 10 as an example. The curves of other frequencies are similar to these two frequency signals. Figure 11 presents the computed values of \(C_{0}\) and \(d_{0}\) over a 10-day modeling period for both the GPS and Galileo frequencies for the two baselines.  
The \(C_{0}\) values for the MATE_MAT1 baseline are slightly larger than those for the KERG_KRGG baseline, though the ranges for both baselines overlap significantly. In contrast, the \(d_{0}\) parameter for the MATE_MAT1 baseline is notably larger than for KERG_KRGG, which is expected due to the higher spatial frequency of multipath observed in the KERG_KRGG baseline.  
Additionally, the deviations of parameters relative to the averages of each single baseline and both baselines combined are computed and shown in Figure 12. Due to the absence of low-elevation data with large residuals at the E5a frequency for the KERG_KRGG baseline, the calculated covariance parameters are not representative of the multipath errors, leading to the exclusion of this frequency from the analysis. As anticipated, the differences between the original parameter values and the averages for each baseline are relatively small (Figure 12), while the deviations from the combined averages of both baselines are slightly larger. Nonetheless, the differences in \(C_{0}\) for this case remain small, around \(1\times 10^{-5}\), and the \(d_{0}\) differences stay within 0.02.  
Figure 10: Theoretical covariance functions and ECVFs of GPS L1 and Galileo E1 for MATE_MAT1 (**a**,**b**) and KERG_KRGG (**c**,**d**) on DOY 038 of 2023. The blue dot curve is the ECVF derived with the GPS L1 (**a**,**c**) and Galileo E1 (**b**,**d**) SD residuals. The red curve is the fitted curve.  
The parameters \(C_{0}\) and \(d_{0}\) of all frequency residuals are unified by averaging all frequency parameters across two baselines. These unified parameters are then applied in a spatial autocorrelation-based method for multipath mitigation. To evaluate performance, GPS L1 frequency observations from both baselines are analyzed. The residual variance reduction rates after applying the unified parameters are displayed in Figure 13, alongside rates using average parameters for each baseline and for each frequency individually.  
Figure 11: Daily \(C_{0}\) and \(d_{0}\) parameters for MATE_MAT1 (**a,b**) and KERG_KRGG (**c,d**) on DOY 028-039 of 2023.  
Figure 12: Deviations of the parameters respective to the average of each single baseline (**a–d**) and both baselines (**e–f**) are computed, separately, for MATE_MAT1 (**a,b,e,f**) and KERG_KRGG (**c,d,g,h**).  
The results demonstrate that the variance reduction rates achieved with the unified parameters are comparable to those obtained with baseline-specific averages. This suggests that using unified covariance function parameters for different baselines and frequency signals is effective, simplifying the computational process by eliminating the need to estimate the parameters \(C_{0}\) and \(d_{0}\) individually. Moving forward, these unified parameters will be employed in subsequent research efforts.