# A Novel Method to Mitigate the Multipath Error for BDS-2 Dam Deformation Monitoring
## 2 Materials and Methods
### Multi-Point Hemispherical Grid Model (MHGM)  
For a certain wavelength of signal, the multipath error depends upon the reflection coefficient and the angle of incidence of the reflected signal, as well as the spatial relationship between the antenna and the reflector. If the antenna is fixed during the observation process, the spatial relationship between the antenna and the reflector always remains unchanged. However, the propagation path of the reflected signal varies with the direction of the satellite signal, and the incidence angle and reflection coefficient change accordingly to affect the multipath error. Since the multipath effect is related to the observation environment around the station, the observation environment changes as the time span of multipath modeling increases (weather, surrounding vegetation, etc.). Thus, using the observation data of nearby periods as much as possible for multipath modeling helps to reduce the impact caused by environmental changes. According to the above characteristics of the multipath effect in the spatial domain, a hemisphere coordinate can be defined with the antenna's position as the center, and the position of a point can be described by a pair of angular coordinates (i.e., elevation and azimuth). Assuming that the environment around the station does not change with time, the multipath effect of the point in the hemispheric coordinate is only related to the elevation and azimuth of the satellite, but not to the observation time of the signal or the pseudorandom noise (PRN) code of the satellite. Based on the above assumptions, a novel multi-point hemispherical grid model was studied. With the signal puncture point of all satellites, the observation data are used for overall modeling. It is not required that only the same satellite is modeled after the corresponding orbit repetition period [19; 23]. This method can isolate multipath error using DD residuals between stations, and is consistent across stations. For each station, taking the elevation angle as the latitude and the azimuth angle as the longitude, a polar coordinate is constructed to describe the multipath error. The minimum latitude value of grid point is set to B0, and the maximum value is set to B1. According to the range of azimuth angle, the longitude of grid points is set from 0 to 360 degrees. The interval of the latitude and longitude components is \(d_{b}\) and \(d_{l}\), respectively, which determine the number of grid point parameters to be estimated in the model. The more grid points that are estimated, the more detailed the description of multipath error. In addition, for the same grid point parameter, when the satellite passes near the corresponding direction of the grid point, it can contribute to the modeling of the multipath error of the grid point. Therefore, the effectiveness of the modeling data can be improved. An example of how the grid can be divided for a set of stations is given below (\(B_{0}=0^{\circ}\), \(B_{1}=45^{\circ}\), \(d_{b}=45^{\circ}\), \(d_{l}=45^{\circ}\)). For each set of fixed DD ambiguities \(\Delta\
abla N\), the four zero-differenced (ZD) observed minus calculated residuals in the corresponding period can be obtained, and then the DD residuals of carrier-phase observation can be calculated as:  
\[DD\ residuals=\left(ZD_{m}^{j}-ZD_{m}^{k}\right)-\left(ZD_{n}^{j}-ZD_{n}^{k} \right)-\lambda\cdot\Delta\
abla N \tag{2}\]  
where indexes \(m\) and \(n\) refer to the stations, and \(j\) and \(k\) refer to the satellites. Figure 1 shows that the multipath error between the station \(m\) and the satellite \(j\) is assumed to involve the grid points. The number of grid points is determined by the satellite elevation angle. If the elevation angle of satellite is higher than \(B_{1}\), there are three \((Q_{m1}^{j},\)\(Q_{m2}^{j},\)\(Q_{m3}^{j})\) points parameters, or four \((Q_{m1}^{i},\)\(Q_{m2}^{i},\)\(Q_{m3}^{i},\)\(Q_{m4}^{i})\) points parameters.