## 2 Methods
### GPS PPP Observation Models
#### 2.1.1 Standard Single-frequency GPS PPP Observation Model  
The distance from the satellite to the receiver can be measured while using pseudorange and carrier phase observations, with the following expression [34]:  
\[\left\{ \begin{array}{ll}P_{r}^{s}&=\rho_{r}^{s}+\text{c}\cdot(dt_{r}-dt^{s})+ I_{r}^{s}+M_{w}\cdot Z_{w}+M_{h}\cdot Z_{h}+d_{r}-d^{s}+\varepsilon_{r}^{s}\\ L_{r}^{s}&=\lambda_{1}\cdot\Phi_{r}^{s}\\ &=\rho_{r}^{s}+\text{c}\cdot(dt_{r}-dt^{s})-I_{r}^{s}+M_{w}\cdot Z_{w}+M_{h} \cdot Z_{h}+\lambda_{1}\cdot(N_{r}^{s}+b_{r}-b^{s})+\xi_{r}^{s}.\end{array} \right. \tag{1}\]  
where, \(P_{r}^{s}\) and \(L_{r}^{s}\) are the original pseudorange and the carrier phase \(\Phi_{r}^{s}\) is multiplied by wavelength \(\lambda_{1}\) for the specific receiver \(r\) and satellite \(s\); \(\rho_{r}^{s}\) is the geometric distance from the satellite to the receiver; c is the speed of light in vacuum; \(dt_{r}\) and \(dt^{s}\) are the clock error of the receiver and satellite; \(I_{r}^{s}\) is the slant ionospheric delay on the first frequency f\({}_{1}\); \(M_{w}\) is the wet mapping function; \(Z_{w}\) is the zenith wet delay; \(M_{h}\) is the hydrostatic mapping function; \(Z_{h}\) is the zenith hydrostatic delay; \(d_{r}\) is the frequency-dependent receiver uncalibrated code delay (UCD) with respect to satellite \(s\); \(d^{s}\) is the frequency-dependent satellite UCD; \(N_{r}^{s}\) is the integer phase ambiguity; \(b_{r}\) and \(b^{s}\) are the uncalibrated phase delays (UPDs) for the receiver and satellite, which is frequency-dependent; and, \(\varepsilon_{r}^{s}\) and \(\xi_{r}^{s}\) are the sum of measurement noise caused by the pseudorange and carrier phase observations and the error caused by the multipath effect. Other errors have been modeled in advance. The original pseudorange and carrier phase observations equations are linearized, and the receiver clock offset only absorbs common part of frequency dependent receiver UCDs. Supposing that satellites are tracked simultaneously at a certain epoch by the receiver \(r\), the standard SF PPP model can be written, as follows [35]:  
\[\left[ \begin{array}{c}p_{r}^{1}\\ l_{r}^{1}\\ \vdots\\ p_{r}^{m}\\ l_{r}^{m}\end{array} \right]=\left[ \begin{array}{ccccc}-\mu_{r}&\mathbf{I}_{t}&M_{w }&\mathbf{K}&\mathbf{R}\\ \end{array} \right]\left[ \begin{array}{c}\mathbf{x}\\ \tilde{d}_{T}\\ Z_{w}\\ \mathbf{I}_{r}^{s}\\ \overline{\mathbf{N}_{r}}\end{array} \right]+\left[ \begin{array}{c}\varepsilon_{ r}\\ \varepsilon_{r}\\ \overline{\mathbf{N}_{r}}\end{array} \right],\quad\mathbf{Q}_{L}\, \tag{2}\]  
with  
\[\left\{ \begin{array}{c}\tilde{d}_{r}=dt_{r}+d_{r}\\ \overline{N}_{r}^{s}=\lambda_{1}\cdot(N_{r}^{s}+b_{r}-b^{s})+d_{IF}^{s}-d_{r}, \end{array} \right. \tag{3}\]  
where \(\mu_{r}\) is the unit vector of the coordinate component between the receiver and the satellite; \(\mathbf{x}\) is the vector of the receiver position increments relative to the priori position; and, \(\mathbf{I}_{t}\) is a unit vector of \(2\cdot m\) rows and one column, corresponding to the receiver clock parameter \(\tilde{d}_{r}\). In matrix \(\mathbf{K}\), the elements for the corresponding \(p_{r}^{s}\) and \(l_{r}^{s}\) are \(1\) and \(-1\), respectively, corresponding to the ionospheric parameter \(\mathbf{I}_{r}^{s}\). \(\mathbf{R}\) is the matrix corresponding to the ambiguity parameters \(\overline{\mathbf{N}_{r}}\), and the elements for the corresponding \(p_{r}^{s}\) and \(l_{r}^{s}\) are \(0\) and \(1\), respectively. \(\mathbf{Q}_{L}\) is the stochastic model of the observed minus computed (OMC) observables.