# Multipath Error Fusion Modeling Methods for Multi-GNSS
## 2 Materials and Methods
### Sidereal Filtering for Multi-GNSS Multipath Correction  
In order to reduce the complexity of the algorithm, the orbit repeat period of the same type of satellites in the same system is generally averaged to obtain the overall optimal SF models. Among the existing GNSSs, MEO satellites are used in GPS, BDS, and Galileo. However, the orbit repeat period is different because of the difference of the orbit parameters. The orbit repeat period of the GPS satellite is about 1 day and the Galileo satellite is about 10 days. BDS has adopted a mixed constellation of GEO, IGSO, and MEO satellites. The orbit repeat period of the GEO and IGSO satellites is about 1 day, and that of the MEO satellite is about 7 days. Therefore, the independent orbit repeat periods need to be set for these three systems. The orbit repeat period \(T_{s}\) of the above systems can be calculated from the relevant parameters in the broadcast ephemeris and [PERSON]'s third law, as given [21],  
\[T_{s}=kT=\frac{2k\pi}{n_{0}+\Delta n}=\frac{2k\pi}{\sqrt{GM}a^{-3/2}+\Delta n} \tag{1}\]  
where \(\sqrt{GM}\) is the square root of the corresponding standard gravity parameters of each GNSS for Earth, \(a\) is the semi major axis of the satellite orbit, and \(\Delta n\) is the correction of the satellite angular velocity parameter. \(k\) is the number of orbital revolutions of the satellite in its orbit repeat period. The calculation results of T\({}_{s}\) are relatively large. Generally, for the convenience of description and use, T\({}_{s}\) is further expressed as the time shift \(t\) of the days corresponding to the orbit repetition period [10; 22],  
\[t=n\cdot 86400-T_{s} \tag{2}\]  
where \(n\) is the integer day corresponding to the satellite orbit repeat period, and the values of \(n\) and \(k\) for the different constellations in each system are listed in Table 1.  
The time shifts of the orbit repeat period on days of year (DOY) 179\(\sim\)250 in 2018 and DOYs 1\(\sim\)100 in 2020 for each satellites were calculated. According to the statistical results of the time shifts of the orbit repeat period for different satellites in each system in Figure 1, the orbit repeat periods of the different satellites with the same constellation type in the same system were relatively consistent. However, there were differences among satellites from same type of constellations in different systems. Taking the MEO satellites as an example, the time shifts of the orbit repeat period for GPS, BDS, and Galileo were still significantly different, so it was necessary to set the parameters for each system separately. The time shifts of the orbit repeat period for the satellites with the same constellation types were relatively stable in the long-term statistical results in Figure 1. The time shifts of the GPS varied between 240 and 250 s. There were slight differences between all of the satellites, but the long-term changes of each satellite were relatively stable. The orbit repeat period of GPS was about 1 day, and the time shift was set as 245 s because of the statistical results of this kind of satellite; this parameter was as close as the result of 246 s obtained by [PERSON] et al. using the autocorrelation of the station coordinate time-series [5]. As shown in Figure 1b, the corresponding shifts of the GEO and IGSO satellites of the BDS varied between 215 s and 265 s, and there was a gradual decreasing trend, but the time shifts increased with a jump at regular intervals. It was found that this was due to the orbital maneuver, and the parameter SatH1 could be found in the broadcast ephemeris [23]. When the orbital maneuver phenomenon occurred, it changed from 0 to 1. In addition, the orbit repeat periods of the satellites with the same type of constellation from BDS2 and the global BeiDou navigation satellite system (BDS3) were basically the same, so the unified time shifts could be adopted in the multipath error correction of SF.