# Multipath Error Fusion Modeling Methods for Multi-GNSS
## 2 Materials and Methods
### Multi-Point Hemispherical Grid Model for Multi-GNSS Multipath Correction  
If the receiver antenna was above an infinite plane and was only affected by a single reflected signal, the error caused by the multipath effect on the carrier phase observation can be described as follows [25]:  
\[\Delta\ =\ arctan\frac{\alpha\sin\left(4\pi\frac{H}{\lambda}sine\right)}{1+ \alpha\cos\left(4\pi\frac{H}{\lambda}sine\right)}, \tag{3}\]  
where \(H\) is the height of the reflector from the antenna phase center, \(\varepsilon\) is the incident angle of the reflected signal, \(\alpha\) is the reflection coefficient of the reflective objects, and \(\lambda\) is the wavelength of the satellite signal. Figure 1: Scatter plots of the daily time shifts for GPS, BDS, and Galileo in 2018 and 2020 based on the broadcast ephemeris data. (**a**) Time shifts for the MEO satellite of GPS. (**b**) Time shifts for the GEO and IGSO satellites of BDS2 and BDS3. (**c**) Time shifts for the MEO satellite of BDS2 and BDS3. (**d**) Time shifts for the MEO satellite of Galileo. Assuming that the receiver antenna and the surrounding environment remain unchanged, the signals of the same wavelength were reflected and entered the receiver antenna from the same incident angle when the different satellites passed through the same position in the sky. According to (3), the multipath error of the carrier phase observations should be the same [8]. From the frequency setting of each GNSS, it can be found that the frequency of the GPS carrier signal L1, Galileo carrier signal E1, and BDS carrier signal B1C was 1575.420 MHz, and B1 of BDS had a slight difference with the above frequency, which was 1561.098 MHz. Therefore, it is theoretically feasible to use the observations of multi-GNSS for integrated modeling of the multipath error. Based on the above analysis on the frequencies of different GNSS signals, we established a multipath modeling method for multi-GNSS, and the precise description of multi-path error sources in the environment around the station was realized in the space domain. The residuals of the double-differenced carrier phase observations in the ambiguity-fixed period were used as the input information. Then, a multi-point hemispherical grid model at each station was established and finally the model of each station could be obtained after the grid points of each model were parameterized and solved by the estimator [26]. First, the hemisphere was established with the antenna phase center of each station as the origin and was divided into grids according to the elevation angle and azimuth angle. The minimum value of the elevation angle was set as \(E_{0}\), the maximum value was set to \(E_{1}\), and the grid point parameter was set in the zenith direction. The interval of the hemisphere partition could be set as \(c\) and \(d\), and the value of \(c\) and \(d\) determined the density of the grid partition. The smaller the partition interval, the more parameters to be estimated and the more computing resource consumption, but the higher the precision of the model. Figure 2 shows an example of the grid partition for a hemisphere with \(E_{0}=0^{\circ}\), \(E_{1}=60^{\circ}\), \(c=d=30^{\circ}\). Given the double-differenced residuals \(v_{pq}^{st}\) with ambiguity-fixed solutions between two stations \(p\) and \(p\) and two satellites \(s\) and \(t\). The residual corresponding to satellite \(s\) on station \(p\) is \(v_{p}^{s}\). Supposing the parameters of the four grid points involved in the grid point of the satellite are \(Q_{p1}^{s}\), \(Q_{p2}^{s}\), \(Q_{p3}^{s}\) and \(Q_{p4}^{s}\), there is a case involving three grid point  
Figure 2: An example of a grid partition for a hemisphere. parameters when the elevation angle is greater than \(E_{1}\).