## 2 Methods
### PPP-RTK Model  
The raw observations of the GNSS pseudo-range and carrier phase can be expressed as follows [39]:  
\[ \begin{array}{c}P_{r,f}^{s}=\rho_{r}^{s}+t_{r,sys}+\alpha_{r}^{s}T_{Z}+\frac{4 0.3}{f^{2}}\gamma_{r}^{s}I_{r}^{s}-b^{s,f}+b_{r,f}+\varepsilon_{p}\\ \Phi_{r,f}^{s}=\rho_{r}^{s}+t_{r,sys}+\alpha_{r}^{s}T_{Z}-\frac{40.3}{f^{2}} \gamma_{r}^{s}I_{r}^{s}+\lambda N_{r,f}^{s}+\varepsilon_{\Phi}\end{array}  \tag{1}\]  
in which \(P_{r,f}^{s}\) and \(\Phi_{r,f}^{s}\) are the pseudo-range and carrier phase at frequency \(f\) corresponding to receiver \(r\) and satellite \(s\) in length units, respectively; \(\rho_{r}^{s}\) is the geometric distance between receiver \(r\) and satellite \(s\); \(t_{r,sys}\) is the receiver clock error corresponding to the system \(sys\in(G\ C)\) in the length units, respectively; \(\alpha_{r}^{s}\) and \(\gamma_{r}^{s}\) are the mapping functions of the tropospheric and ionospheric delays, respectively; \(T_{Z}\) and \(I_{r}^{s}\) stand for the zenith tropospheric delay and the zenith total electron content (TEC); \(b^{s,f}\) and \(b_{r,f}\) denote the hardware delay for satellite \(s\) and receiver \(r\), respectively; \(\lambda\) and \(N_{r,f}^{s}\) are the carrier phase wavelength and float ambiguity; \(\varepsilon_{p}\) and \(\varepsilon_{\Phi}\) represent the measurement noise of pseudo-range and carrier phase including the unmodeled multipath error, respectively. Additionally, it is assumed that other errors, such as satellite orbit and clock errors and relativistic effects, are corrected in advance. After correcting the hardware delay for the satellite and linearization, Equation (1) can be written as  
\[ \begin{array}{c}\Delta P_{r,f}^{s}=\mathbf{h}_{s}^{s}\delta\mathbf{x}_{\text{GNSS}}^ {c}+t_{r,sys}+\alpha_{r}^{s}\delta T_{w}+\frac{40.3}{f^{2}}\gamma_{r}^{s}I_{r} ^{s}+b_{r,f}+\varepsilon_{p}\\ \Delta\Phi_{r,f}^{s}=\mathbf{h}_{s}^{s}\delta\mathbf{x}_{\text{GNSS}}^{c}+t_{r,sys}+ \alpha_{r}^{s}\delta T_{w}-\frac{40.3}{f^{2}}\gamma_{r}^{s}I_{r}^{s}+\lambda N _{r,f}^{s}+\varepsilon_{\Phi}\end{array}  \tag{2}\]  
where \(\Delta P_{r,f}^{s}\) and \(\Delta\Phi_{r,f}^{s}\) are the OMC (observed-minus-computed) of the pseudo-range and carrier phase, respectively; superscript \(\cdot^{e}\) represents the \(e\)-frame (earth-centered earth-fixed frame); \(\delta\mathbf{x}_{\text{GNSS}}^{c}\) and \(\mathbf{h}_{r}^{s}\) are the correction vectors of the receiver position and the corresponding direction cosine vector; \(\delta T_{w}\) denotes the residual of the zenith tropospheric wet delay. Additionally, the DESIGN (deterministic plus stochastic ionosphere model for GNSS) model is adopted in this study as [39; 40]  
\[ \begin{array}{c}I_{r}^{s}=a_{0}+a_{1}dL+a_{2}dB+a_{3}dL^{2}+a_{4}dB^{2}+r_{r }^{s}\\ \widetilde{I}_{r}^{s}=a_{0}+a_{1}dL+a_{2}dB+a_{3}dL^{2}+a_{4}dB^{2}+r_{r}^{s}+ \varepsilon_{\widetilde{I}_{r}}\end{array}  \tag{3}\]  
in which \(\widetilde{I}_{r}^{s}\) is the virtual observation of ionospheric delay and can be obtained from the ionospheric delay prior models of high-precision ionospheric products; \(a_{i}\) (\(i\) = 0, 1, 2, 3, 4) describes the spatial distribution of the ionospheric delay; and \(dL\) and \(dB\) represent the difference in longitude and latitude between the approximate location of the station and the ionospheric pierce point (IPP), respectively. \(r_{r}^{s}\) describes the stochastic behavior of the ionospheric delay in the time domain and \(\varepsilon_{\widetilde{I}_{r}^{s}}\) is the corresponding noise of the virtual observation.