# Performance Assessment of Multi-GNSS PPP Ambiguity Resolution with LEO-Augmentation
## 3 Methods
### LEO-Enhanced GNSSs PPP
\(\lambda_{\rm IF}^{s,K}\) and \(N_{r,\rm IF}^{s,K}\) are the carrier wavelength and ambiguity of the IF combination, respectively. \(\varepsilon_{P_{r,\rm IF}^{s,K}}\) and \(\varepsilon_{L_{r,\rm IF}^{s,K}}\) are the measurement noise error of the code pseudorange and the carrier phase of the IF combination, respectively. The ambiguity, code, and phase hardware delays of the IF combination are as follows. \[b_{r,\rm IF}^{K}=\left((f_{1}^{s,K})^{2}b_{r,1}^{K}-(f_{2}^{s,K})^{2}b_{r,2}^ {K}\right)/\left((f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2}\right), \tag{6}\]  
\[b_{\rm IF}^{s,K}=\left((f_{1}^{s,K})^{2}b_{1}^{s,K}-(f_{2}^{s,K})^{2}b_{2}^{s, K}\right)/\left((f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2}\right), \tag{7}\]  
\[B_{r,\rm IF}^{K}=c\left(f_{1}^{s,K}B_{r,1}^{K}-f_{2}^{s,K}B_{r,2}^{K}\right)/ \lambda_{\rm IF}^{s,K}\left((f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2}\right), \tag{8}\]  
\[B_{\rm IF}^{s,K}=c\left(f_{1}^{s,K}B_{1}^{s,K}-f_{2}^{s,K}B_{2}^{s,K}\right)/ \lambda_{\rm IF}^{s,K}\left((f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2}\right), \tag{9}\]  
\[N_{r,\rm IF}^{s,K}=c\left(f_{1}^{s,K}N_{r,1}^{s,K}-f_{2}^{s,K}N_{r,2}^{s,K} \right)/\lambda_{\rm IF}^{s,K}\left((f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2}\right), \tag{10}\]  
To reduce satellite orbit and clock errors, precision ephemeris and precision satellite clock products are generally used in PPP. Because the precision satellite clock error is generated from the IF combined pseudorange observation data of the GNSS ground tracking network, the \(dt^{s,K}\) provided by the IGS absorbed the code hardware delay \(b_{\rm IF}^{s,K}/c\) of the IF combination at the satellite end. The code hardware delay at the receiver end varied with different satellite systems (such a difference is called intersystem bias), and it will be absorbed by the receiver clock error; therefore, the receiver-clock offsets of different satellite systems are different. The ambiguity parameter is closely related to the phase hardware delay. They all have constant characteristics, which vary between different satellites, making it difficult to separate them from each other. Simultaneously, owing to the introduction of the satellite clock error based on pseudorange observation, the code hardware delay at the satellite end was introduced into the phase observation equation, being then absorbed by the ambiguity parameter. Because the pseudorange observation equation and phase observation equation work together to estimate the receiver clock error, the code hardware delay at the receiver end was also introduced into the phase observation equation and absorbed by the ambiguity parameter. Therefore, because the satellite clock error was corrected by the precise product, Equation (5) could be rewritten as  
\[\left\{ \begin{array}{l}P_{r,\mathrm{IF}}^{s,K}=\rho_{r}^{s,K}+c\overline{d}t _{r}^{K}+T_{r}^{s,K}+\varepsilon_{r,\mathrm{IF}}^{s,K}\\ L_{r,\mathrm{IF}}^{s,K}=\rho_{r}^{s,K}+c\overline{d}t_{r}^{K}+T_{r}^{s,K}+ \lambda_{\mathrm{IF}}^{s,K}\overline{N}_{r,\mathrm{IF}}^{s,K}+\varepsilon_{L _{r,\mathrm{IF}}^{s,K}}\end{array} \right., \tag{11}\]  
where the reparameterized receiver clock error and ambiguity are shown in Equations (12)-(15). In addition, the three-dimensional position of the receiver and the tropospheric wet delay in the zenith direction have been estimated:  
\[c\overline{d}t_{r}^{K}=cdt_{r}+b_{r,\mathrm{IF}}^{K}, \tag{12}\]  
\[\overline{N}_{r,\mathrm{IF}}^{s,K}=N_{r,\mathrm{IF}}^{s,K}+d_{r,\mathrm{IF}}^ {s,K}-d_{\mathrm{IF}}^{s,K}, \tag{13}\]  
\[d_{r,\mathrm{IF}}^{K}=B_{r,\mathrm{IF}}^{K}-b_{r,\mathrm{IF}}^{K}/\lambda_{ \mathrm{IF}}^{s,K}, \tag{14}\]  
\[d_{\mathrm{IF}}^{s,K}=B_{\mathrm{IF}}^{s,K}-b_{\mathrm{IF}}^{s,K}/\lambda_{ \mathrm{IF}}^{s,K}, \tag{15}\]  
where \(d_{r,\mathrm{IF}}^{K}\) is the receiver uncalibrated phase delay (UPD), and \(d_{\mathrm{IF}}^{s,K}\) is the satellite UPD. As seen in the above formula, the IF combined ambiguity parameter \(\overline{N}_{r,\mathrm{IF}}^{s,K}\) contains the hardware delay of the receiver and satellite and loses the integer characteristic.