## 2 Methodology
### Undifferenced Uncombined Single-Frequency PPP Model  
The ionospheric delay is a key issue influencing the performance of SF-PPP. Therefore, several models, such as the GIM data correction model [32], the single-frequency GRAPHIC (GROup And PHase Ionosphere Calibration) model [33], and the single-frequency ZIDE (Zenith Ionospheric Delay Estimation) PPP model [34], have been proposed. For example, [PERSON] [32] recommended using the GIM to correct the ionospheric delay on the pseudorange and carrier phase. The corresponding functions can be written as [35]. \[P_{r}^{s}=\rho_{r}^{s}+c\cdot dT_{r}+dTrop_{r}^{s}+dIon_{r}^{s}+\Delta_{P} \tag{6}\]  
\[L_{r}^{s}=\rho_{r}^{s}+c\cdot dT_{r}+dTrop_{r}^{s}-\frac{c}{f_{1}}N_{r}^{s}-dIon _{r}^{s}+\Delta_{L} \tag{7}\]  
where \(dT_{r}\) is the receiver clock bias; \(dTorp_{r}^{s}\) is the slant tropospheric delay; \(dIon_{r}^{s}\) is the slant ionospheric delay which is obtained from the GIM model (Centre for Orbit Determination in Europe, CODE). However, the drawbacks of the GIM-correction based SF-PPP model are that the residuals of each satellite's ionospheric delay and the un-disposed Differential Code Bias (DCB) of receiver will contaminate the performance of SF-PPP. To overcome these issues, [PERSON] [36] tried to use the GRAPHIC model [33] to eliminate the ionospheric delay on carrier phase instead of GIM data. In the GRAPHIC model, the characteristic that the ionospheric delays on pseudorange and carrier phase observations are equal in magnitude and opposite in sign for the same satellite is utilized. Hence, a linear ionospheric-free combination between pseudorange and carrier phase is adopted to form a new pseudorange plus carrier phase ionospheric-free combination [35]. \[P_{r}^{s}=\rho_{r}^{s}+cdT_{r}+dTrop_{r}^{s}+dIon_{r}^{s}+\Delta_{P} \tag{8}\]  
\[G_{r}^{s}=\frac{(P_{r}^{s}+L_{r}^{s})}{2} \tag{9}\]  
where \(G_{r}^{s}\) represents the linear ionospheric-free combination between pseudorange and carrier phase; the other symbols are the same as those described above. The noise of combined observation is much larger than the carrier phase observation because of the mixed pseudorange noise. Even so, the SF-PPP positioning accuracy of GRAPHIC still cannot satisfy the accuracy requirement because of the large noise of carrier phase in the GRAPHIC model [37]. Therefore, based on the GRAPHIC model, [PERSON] et al. [34; 38] proposed the ZIDE model to estimate the zenith ionospheric delay of each satellite as a parameter, which has been verified to promote the performance of SF-PPP. The corresponding functions could be described as  
\[ \begin{array}{l}P_{1}^{k}(t)=\rho^{k}(t)+c\cdot\delta t_{r,k}(t)-o^{k}-c\cdot \delta t^{k}(t-\tau_{r}^{k})+T_{r}^{k}(t)+l_{r,1}^{k}(t)+c(d_{r,1}+d_{1}^{k}) \\ \hskip 142.26378 pt+dm_{r}^{k}+ep_{r,1}\end{array}  \tag{10}\]  
\[ \begin{array}{l}L_{1}^{k}(t)=\rho^{k}(t)+c\cdot\delta t_{r,k}(t)-o^{k}-c\cdot \delta t^{k}(t-\tau_{r}^{k})+T_{r}^{k}(t)-l_{r,1}^{k}(t)+\lambda_{1}N_{r,1}^{k} \\ \hskip 142.26378 pt+\lambda_{1}(b_{r,1}-b_{1}^{k})+\delta m_{r}^{k}+\varepsilon _{L,1}\end{array}  \tag{11}\]  
where the symbols are the same as those defined in Equations (1) and (2). According to the works of [PERSON] et al. [35], [PERSON] et al. [39] and [PERSON] et al. [40], ZIDE SF-PPP could improve position accuracy and convergence time by adopting precise global and regional ionospheric models. However, the receiver DCB on pseudorange is ignored in the ZIDE model. In this paper, the SF-PPP model on the basis of Ionospheric delay and Receiver DCB Constraint (IRC) [41] is applied to acquire the epoch-by-epoch time series of positions and VTEC values. Here, to separate the receiver DCB from ionospheric delay [42], the GIM data are utilized to generate a virtual external observation for each ionospheric delay.