## 2 BDS Multipath
### Single-Difference Multipath Modeling  
The satellite with the maximum elevation angle is selected as the pivot satellite, which can reduce the influence of the error term of the pivot satellite. The double-difference observation equation will frequently change the pivot satellite, making the double-difference residuals more complicated. Moreover, the satellite ORT is not a constant, which results in the inconsistent time nodes of the pivot satellite in the two revisit periods. Furthermore, using the double-difference residuals as the multipath models will have many limiting factors. Therefore, the double-difference residuals are converted into the single-difference residuals using the transformation matrix, which follows the \"zero mean\" assumption (\(\sum w_{i}s_{AB}^{i}=0\)). The reconstructed single-difference residuals can be expressed as:  
\[\left[\begin{array}{c}s_{AB}^{1}\\ s_{AB}^{2}\\ s_{AB}^{3}\\ \cdots\\ s_{AB}^{n}\end{array}\right]=\left[\begin{array}{cccc}\sum w_{i}s_{AB}^{i} \\ dd_{AB}^{13}\\ dd_{AB}^{13}\\ \cdots\\ dd_{AB}^{1n}\end{array}\right]\left[\begin{array}{cccc}w_{1}&w_{2}&w_{3}& \cdots&w_{n}\\ 1&-1&0&\cdots&0\\ 1&0&-1&\cdots&0\\ &&\cdots&&\\ 1&0&0&\cdots&-1\end{array}\right]^{-1} \tag{4}\]  
where \(s_{AB}^{n}\) are the reconstructed single-difference residuals; \(dd_{AB}^{1n}\) are the extracted double-difference residuals; and \(w_{i}\) is the weight factor of the \(i\)th satellite observation that obeys the elevation angle stochastic model \(w_{i}=\sin^{2}(\theta)\).  
Once the double-difference residuals are converted to the single-difference residuals, multipath modeling is performed on the single-difference residuals using ASF and SD-MHM to mitigate the BDS multipath. Moreover, the multipath mitigation capabilities of the two methods in BDS-2/BDS-3 are analyzed. Figure 1 shows the framework of the two methods for multipath mitigation.  
The multipath modeling process comprises three steps: (1) The average baseline vector is calculated and substituted into the double-difference linear observation equation to obtain the double-difference residual. (2) The double-difference residuals are converted into single-difference residuals based on the \"zero mean\" assumption. (3) The single-difference residuals are denoised based on a low-pass filter to complete the single-difference modeling. The ASF modeling process first needs to calculate all satellite ORTs of the current satellite constellation, and the single-difference model after denoising is time-aligned by the time shift of ORT. Then, the aligned single-difference residuals are transformed into the double-difference residuals to mitigate the multipath. The SD-MHM method requires only the single-difference residuals of the current constellation into the grid corresponding to the elevation angle and the azimuth angle. In the SD-MHM processing stage, the elevation and the azimuth angles of the satellites are calculated, the nearest grid position is searched based on the k-nearest neighbor (KNN) search [35], and the multipath model of the corresponding position is obtained. Finally, the double-difference residuals are transformed to mitigate the multipath in real-time.