## 3 Precise Positioning Using Uncombined PPP (UPPP) Algorithm  
With GNSS observations of pseudorange, carrier-phase and Doppler, the uncombined PPP model can be employed for precise positioning. The undifferenced GNSS pseudorange and carrier-phase observations for the satellite \(s\) and the receiver \(r\) on frequency \(j\) are as follows [41]:  
\[\begin{array}{c}E\Big{(}\Phi_{r,j}^{s}\Big{)}=\rho_{r}^{s}+T_{r}^{s}+cdt_{r} -cdt^{s}-\gamma_{j}!s_{r,1}^{s}+\lambda_{j}N_{r,j}^{s}+B_{r,j}-B_{j}^{s}\\ E\Big{(}\mathrm{P}_{r,j}^{s}\Big{)}=\rho_{r}^{s}+T_{r}^{s}+cdt_{r}-cdt^{s}+ \gamma_{j}!s_{r,1}^{s}+b_{r,j}-b_{j}^{s}\end{array}, \tag{5}\]  
where \(E\) is the mathematical expectation operator, \(\mathrm{P}_{j}\) and \(\Phi_{j}=\lambda_{j}\varphi_{j}\) denote the pseudorange and carrier-phase observations on the frequency \(j\) in meters, \(\rho\) is the geometric range between satellite and receiver as a function of the satellite and the receiver coordinates, \(T\) is the tropospheric delay (m) which can be spilt into dry and wet parts, \(\mathrm{c}\) is the vacuum speed of light (m/s), \(dt_{r}\) and \(dt^{s}\) are the receiver and satellite clock errors (s), respectively, \(I_{r,1}^{s}\) is the first-order slant ionospheric delay on frequency L1 (m), \(\gamma_{j}\)= \(f_{1}^{2}\Big{/}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.} \Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.  
pseudorange and carrier-phase hardware delays (biases), respectively, and \(b_{f}^{s}\) and \(B_{f}^{s}\) are the frequency-dependent satellite pseudorange and carrier-phase hardware delays (biases), respectively.  
The precise satellite clock errors provided by International GNSS Service (IGS) are based on the ionosphere-free (_IF_) linear combination of code observations on L1 and L2 frequencies, i.e., P1 and P2, as follows [42]:  
\[dt^{s,IF}=cdt^{s}+b_{IF(1,2)}^{s}, \tag{6}\]  
where \(b_{IF(1,2)}^{s}=\alpha_{IF}^{L1,L2}b_{1}^{s}+\beta_{IF}^{L1,L2}b_{2}^{s}\) is the satellite ionosphere-free code bias in which \(b_{1}^{s}\) and \(b_{2}^{s}\) are the satellite pseudororange hardware delays for P1 and P2, respectively. The coefficients \(\alpha_{IF}^{L1,L2}\) and \(\beta_{IF}^{L1,L2}\) are also of the following form:  
\[\alpha_{IF}^{L1,L2}=f_{1}^{2}\Big{/}\Big{(}f_{1}^{2}-f_{2}^{2}\Big{)}\text{ and }\beta_{IF}^{L1,L2}=-f_{2}^{2}\Big{/}\Big{(}f_{1}^{2}-f_{2}^{2}\Big{)}, \tag{7}\]  
The uncombined PPP model for L1 and L5 frequencies can then be rewritten as follows:  
\[\begin{split} E\Big{(}\mathrm{P}_{r,1}^{s}\Big{)}& =\mathrm{\rho}_{r}^{s}+T_{r}^{s}+\Big{(}cdt_{r}+b_{r,1}\Big{)}-cdt^ {s,IF}+I_{r,1}^{s}+(b_{IF(1,2)}^{s}-b_{1}^{s})\\ E\Big{(}\mathrm{\Phi}_{r,1}^{s}\Big{)}&=\mathrm{ \rho}_{r}^{s}+T_{r}^{s}+\big{(}cdt_{r}+b_{r,1}\big{)}-cdt^{s,IF}-I_{r,1}^{s}+[ \lambda_{1}N_{r,1}^{s}-b_{r,1}+\ B_{r,1}-B_{1}^{s}+b_{IF(1,2)}^{s}]\\ E\Big{(}\mathrm{P}_{r,3}^{s}\Big{)}&=\mathrm{ \rho}_{r}^{s}+T_{r}^{s}+\big{(}cdt_{r}+b_{r,1}\big{)}-cdt^{s,IF}+\gamma_{3}I_{r,1}^{s}+b_{r,3}-b_{r,1}+(b_{IF(1,2)}^{s}-b_{3}^{s})\\ E\Big{(}\mathrm{\Phi}_{r,3}^{s}\Big{)}&=\mathrm{ \rho}_{r}^{s}+T_{r}^{s}+\Big{(}cdt_{r}+b_{r,1}\Big{)}-cdt^{s,IF}-\gamma_{3}I_{r,1}^{s}+[\lambda_{3}N_{r,3}^{s}+B_{r,3}-B_{3}^{s}+b_{IF(1,2)}^{s}-b_{r,1}]\\ &\qquad\qquad