## 3 Multi-GNSS PPP Mathematical Model  
### Uncombined Model  
The mathematical models of PPP mainly include ionosphere-free combined [13; 14] and uncombined. However, for most of the time, there are still not enough dual frequency satellites received by the Huawei Mate30 for ionosphere-free combined model even adding Galileo satellites. This is because only a few Galileo satellites such as PRN 33 observed by the Huawei Mate30 can be logged with dual frequency observations, and the most Galileo satellites are still logged with single frequency observations. In addition, for the uncombined PPP model based on SD between satellites, the GNSS satellite with the highest elevation angle is generally selected as the reference satellite index. Although the satellite with the highest C/N\({}_{0}\) can be selected as the reference satellite index for smartphone positioning to ensure the quality of measurements of the reference satellite, the reference satellite may not have the L5 observations. Either the GPS L5 band observations are not used or the satellite with L5 frequency band is directly selected as the reference satellite index. Considering the scarcity of GPS L5 satellites and the advantage of dual frequency observations, the undifferenced and uncombined PPP model was used in this experiment. The multi-GNSS PPP model can be written as:  
\[P_{i}^{Q}=\rho^{Q}+cdt_{r}+cdt_{D}^{Q}-cdt^{Q}+T^{Q}+\frac{f_{1}^{2}}{f_{i}^{ 2}}I^{Q}+B_{P_{i}}^{r,Q}-B_{P_{i}}^{Q}+\varepsilon_{P_{i}}^{Q} \tag{6}\]  
\[L_{i}^{Q}=\rho^{Q}+cdt_{r}+cdt_{D}^{Q}-cdt^{Q}+T^{Q}-\frac{f_{1}^{2}}{f_{i}^{ 2}}I^{Q}-\lambda_{i}^{Q}N_{i}^{Q}+B_{L_{i}}^{r,Q}-B_{L_{i}}^{Q}+\varepsilon_{ L_{i}}^{Q} \tag{7}\] \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{4}{c}{**Pseudorange Residuals RMS (m)**} \\ \cline{2-5} \multicolumn{1}{c}{\(C\)**/N\({}_{0}\) (dB-Hz)**} & \multicolumn{1}{c}{**GPS**} & \multicolumn{1}{c}{**BDS**} & \multicolumn{1}{c}{**GLONASS**} \\ \cline{2-5} \multicolumn{1}{c}{} & **P1** & **P5** & **P1** & **P1** \\ \hline \(C\)/N\({}_{0}\leq\) 30 & 5.388 & 5.730 & 6.964 & 10.604 \\ \(30<C\)/N\({}_{0}\leq\) 35 & 4.291 & 4.786 & 3.160 & 8.483 \\ \(35<C\)/N\({}_{0}\leq\) 40 & 3.307 & N/A & 2.762 & 7.497 \\ \(40<C\)/N\({}_{0}\) & 2.774 & N/A & 4.320 & 5.032 \\ \hline \end{tabular}
\end{table} 
Table 1: Root mean square (RMS) of GPS, BDS, and GLONASS pseudorange residuals of the Huawei Mate30 in different C/N\({}_{0}\) ranges. where the superscript \(Q\) denotes the satellite system, \(cdt_{D}^{Q}\) is time bias between \(Q\) satellite system and GPS, and the other symbols are the same as before. By Equations (6) and (7), the error equation of multi-GNSS observation can be written as:  
\[V=H\cdot X+l \tag{8}\]  
where the \(V\) is GNSS observation residual, \(H\) is the coefficient matrix of \(X\), and the basic parameters can be expressed as:  
\[X=\left[ \begin{array}{cccccccccccccccc}x,&y,&z,&cdt_{r},&cdt_{D^{\prime}}^{ B},&cdt_{D^{\prime}}^{R},&T,&\hat{I}_{1},&\hat{N}_{1}^{G},&\hat{N}_{5}^{G},&\hat{N}_{1}^{B },&\hat{N}_{1}^{R}\end{array} \right] \tag{9}\]  
where the \(x\), \(y\), \(z\) are three-dimensional coordinates of the receiver, \(cdt_{D}^{B}\) is BDS system time bias, \(cdt_{D}^{R}\) is GLONASS system time bias, \(\hat{I}_{1}\) is the ionospheric delay of L1, \(\hat{N}_{1}^{G}\), \(\hat{N}_{5}^{G}\), \(\hat{N}_{1}^{B}\), and \(\hat{N}_{1}^{R}\) are the integer ambiguity parameters of GPS, BDS, and GLONASS, respectively. The coefficient matrix \(H\) can be expressed as:  
\[H=\left[