## 2 Multipath Modelling and Mathematic Tools for Analysis  
### Multipath Models  
The equation for GNSS carrier phase observations is given by  
\[\phi^{s}_{r,i}=\rho^{s}_{r}+c\cdot(\delta t_{r}-\delta t^{s})-I^{s}_{r,i}+T^{s} _{r}+\lambda_{i}N^{s}_{r,i}+b_{r,\phi_{i}}-b^{s}_{\phi_{i}}+M^{s}_{r,\phi_{i}}+ \varepsilon^{s}_{r,\phi_{i}} \tag{1}\]  
where \(\phi^{s}_{r,i}\) is the carrier phase observation value on frequency point \(i\); the superscript \(s\) represents the GNSS satellite PRN (Pseudo-Random Noise) number; the subscript \(r\) represents the receiver identifier; \(\rho^{s}_{r}\) is the geometric distance between the satellite and the receiver; \(c\) is the speed of light in a vacuum; \(\delta t\), and \(\delta t^{s}\), respectively, represent the receiver clock errors and satellite clock errors; \(I^{s}_{r,i}\) is the ionospheric delay; \(T^{s}_{r}\) denotes the tropospheric delay; \(\lambda_{i}\) is the carrier wavelength and \(N^{s}_{r,i}\) is the ambiguity; \(b_{r,\phi_{i}}\) and \(b^{s}_{\phi_{i}}\) are the receiver phase bias and satellite phase bias, respectively; \(M^{s}_{r,\phi_{i}}\) is the multipath errors; and \(\varepsilon^{s}_{r,\phi_{i}}\) is the noise term and other unmodeled errors.  
Double-difference is a commonly adopted strategy in short baseline positioning, which effectively mitigates correlated errors at both the receiver and satellite ends. Consequently, the carrier phase observation equation is transformed as follows.  
\[\
abla\Delta\phi^{s_{1},s_{2}}_{r_{1},r_{2},i}=\
abla\Delta\rho^{s_{1},s_{2}}_{ r_{1},r_{2},i}+\lambda_{i}\
abla\Delta N^{s_{1},s_{2}}_{r_{1},r_{2},i}+\
abla \Delta\sigma^{s_{1},s_{2}}_{r_{1},r_{2},\phi_{i}}+\
abla\Delta\varepsilon^{s_{ 1},s_{2}}_{r_{1},r_{2},\phi_{i}} \tag{2}\]  
where \(\
abla\Delta\) is the double-difference operator, \(s_{1}\) and \(s_{2}\), respectively, represent any two satellites from the same GNSS system, and \(r_{1}\) and \(r_{2}\) are the two receivers located at the endpoints of the baseline. If the station coordinates are known with high precision, the baseline vector and integer ambiguity can be solved, leaving only the residual term primarily composed of the multipath effect and measurement noise.  
\[\
abla\Delta I^{s_{1},s_{2}}_{r_{1},r_{2},i}=\
abla\Delta M^{s_{1},s_{2}}_{r_ {1},r_{2},\phi_{i}}+\
abla\Delta\varepsilon^{s_{1},s_{2}}_{r_{1},r_{2},\phi_{i}} \tag{3}\]  
To avoid the impact of reference satellite changes on the residuals, we applied a \"zero-mean\" constraint to obtain the single-difference carrier-phase residuals from double-difference residuals [26]. We therefore have  
\[\begin{bmatrix}w_{1}&w_{2}&w_{3}&\cdots&w_{n}\\ 1&-1&0&\cdots&0\\ 1&0&-1&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&0&0&\cdots&-1\end{bmatrix}\begin{bmatrix}\Delta I^{s_{1}}_{r_{1},r_{2}}\\ \Delta I^{s_{1},r_{2}}_{r_{1},r_{2}}\\ \Delta I^{s_{1},r_{2}}_{r_{1},r_{2}}\\ \Delta I^{s_{1},r_{2}}_{r_{1},r_{2}}\end{bmatrix}=\begin{bmatrix}0\\ \
abla\Delta I^{s_{1},s_{2}}_{r_{1},r_{2}}\\ \vdots\\ \
abla\Delta I^{s_{1},s_{2}}_{r_{1},r_{2}}\end{bmatrix} \tag{4}\]  
where \(w_{1}\), \(w_{2}\), \(\ldots\), \(w_{n}\) refer to the weight of the satellite; and \(\Delta I^{s_{1}}_{r_{1},r_{2}}\), \(\Delta I^{s_{2}}_{r_{1},r_{2}}\), \(\ldots\), \(\Delta I^{s_{n}}_{r_{1},r_{2}}\) are the single-difference residuals which will be used to construct the multipath models.