## 2 BDS Multipath  
### Multipath Extraction  
For short baseline positioning, the double-difference observation equations can be established to eliminate or significantly reduce various errors such as orbit error, receiver and satellite clock errors, tropospheric delay, and ionospheric delay using the double-differenced technique. However, multipath is only related to the geometric relationship of the receiver, reflection, and satellite positions. Since there is no correlation among the stations, the double-differenced technique cannot mitigate the multipath. When two stations receive \(n+1\) common-view satellites, there are \(n\) double-difference observation equations of B1 carrier phase observations that can be expressed as:  
\[\lambda\
abla\Delta\Phi_{ip}=\
abla\Delta\rho_{ip}+\lambda\
abla\Delta N_{ip} +\
abla\Delta\delta_{\text{mth}}+\
abla\Delta\varepsilon \tag{1}\]  
where \(\lambda\) is the wavelength of the B1 carrier phase; \(\
abla\Delta\) is the double-difference operator; \(\
abla\Delta\Phi_{ip}\), \(\
abla\Delta\rho_{ip}\), \(\
abla\Delta N_{ip}\), \(\
abla\Delta\delta_{\text{mth}}\), and \(\
abla\Delta\varepsilon\) are the carrier phase, the distance between satellites and stations, the integer ambiguities, the multipath values after double-difference, and the random noise, respectively; \(i\) denotes the \(i\)th satellite; and \(p\) denotes the pivot satellite.  
To calculate the baseline vector conveniently, it is necessary to linearize the double-difference observation equation. The linearized observation equation of the carrier phase can be expressed as:  
\[\mathbf{V}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{y}-\mathbf{L} \tag{2}\]  
where \(\mathbf{x}\) is a \(3\times 1\) order unknown baseline vector; \(\mathbf{y}\) is the \(n\times 1\) order integer ambiguity after double-difference; \(\mathbf{A}\) and \(\mathbf{B}\) are the design matrices of order \(n\times 3\) and \(n\times n\), respectively; \(\mathbf{L}\) is the \(n\)\({}^{*}\) 1 order the constant term of carrier phase; and \(\mathbf{V}\) is the double-difference residuals. In short baseline, the double-difference residuals are mainly the multipath and the noise.  
As shown in Equation (2), the multipath with noise can be obtained by determining the baseline vector and the ambiguities of the linear observation equation. Therefore, the fixed integer ambiguities are essential. According to the principle of the least-square ambiguity de-correlation adjustment (LAMBDA) algorithm, fixing all the ambiguities is not necessary for positioning, and it is almost impossible to fix all the ambiguities [33]. Moreover, the increase in the number of ambiguities increases will reduce both the searching space and computational efficiency. Therefore, it is recommended to use the partial ambiguity resolution (PAR) [34] to fix the integer ambiguities. The PAR has the advantages of simplicity and a high success rate for fixing the ambiguities. Meanwhile, the search space is reduced when there are many common-view satellites improving the operational efficiency of ambiguities searching.  
The double-difference residuals can be obtained by first calculating the baseline vector values and the integer ambiguities, and then substituting them into Equation (2). Under the least square criterion, the solution of Equation (2) is expressed as:  
\[\min\Bigl{\{}\parallel\mathbf{Ax}+\mathbf{By}-\mathbf{L}\parallel_{\mathbf{Q}^{-1}}^{2}\mathbf{x} \in\mathbf{R},\mathbf{y}\in\mathbf{Z}\Bigr{\}} \tag{3}\]  
where \(\mathbf{Q}\) is the variance-covariance matrix of the baseline vector and integer ambiguity; \(\mathbf{R}\) is the real value of the baseline vector; and \(\mathbf{Z}\) is the real value of the double-difference integer ambiguities.