# Performance Assessment of Multi-GNSS PPP Ambiguity Resolution with LEO-Augmentation
## 3 Methods  
### LEO-Enhanced GNSSs PPP  
The observation of GNSSs PPP mainly included code pseudorange \(P_{r,j}^{s,K}\) and carrier phase \(L_{r,j}^{s,K}\), and its observation equation was expressed as follows:  
\[P_{r,j}^{s,K}=\rho_{r}^{s,K}+c\Big{(}dt_{r}-dt^{s,K}\Big{)}+T_{r}^{s,K}+I_{r,j}^{ s,K}+b_{r,j}^{K}-b_{j}^{s,K}+\varepsilon_{P_{r,j}^{s,K}}, \tag{1}\]  
\[L_{r,j}^{s,K}=\rho_{r}^{s,K}+c\Big{(}dt_{r}-dt^{s,K}\Big{)}+T_{r}^{s,K}-I_{r,j}^ {s,K}+\lambda_{j}^{s,K}\Big{(}N_{r,j}^{s,K}+B_{r,j}^{K}-B_{j}^{s,K}\Big{)}+ \varepsilon_{L_{r,j}^{s,K}}, \tag{2}\]  
where the subscript \(r\) is the station number, \(j\) represents the frequency point, the superscript \(s\) is the satellite number, and \(K\) represents GPS, GLONASS, BDS-3, or Galileo. \(\rho_{r}^{s,K}\) indicates the geometric distance between the station and satellite. \(c\) is the speed of light in a vacuum, \(dt_{r}\) is the receiver clock offset, and \(dt^{s,K}\) is the satellite clock offset. \(T_{r}^{s,K}\) is the tropospheric delay and \(I_{r,j}^{s,K}\) is the ionospheric delay at the frequency point \(j\). \(b_{r,j}^{s,K}\) and \(b_{j}^{s,K}\) are the  
Figure 1: The daily average number of visible satellites in the LEO constellation on 1 January 2022. Different colors in the color bar represent different numbers of visible satellites: (a) L; (b) M; (c) L+M. code hardware delays at the receiver and satellite ends, respectively. \(B_{r,j}^{K}\) and \(B_{j}^{s,K}\) are the phase hardware delays at the receiver and satellite ends, respectively. \(N_{r,j}^{s,K}\) is the integer ambiguity at the frequency point \(j\). The corresponding wavelengths \(\lambda_{j}^{s,K}\). \(\epsilon_{r,j}^{p_{r,k}}\) and \(\varepsilon_{L_{r,j}^{s,K}}\) are the measurement noise errors of the code pseudorange and carrier phase, respectively. There were also some potential error items, such as Earth rotation, satellite and receiver antenna phase center variations and offsets, antenna wind up, solid tide, oceanic tide, atmospheric tide, and relativistic effect, which were corrected according to existing models [31]; therefore, they have not been listed in the above equation. To eliminate the first-order ionospheric delay of the code pseudorange and carrier phase observation, the ionospheric-free (IF) combination of \(P_{r,\rm IF}^{s,K}\) and \(L_{r,\rm IF}^{s,K}\) is usually used to estimate PPP parameters, and the combination mode of IF is  
\[P_{r,\rm IF}^{s,K}=\frac{(f_{1}^{s,K})^{2}}{(f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2} }P_{r,1}^{s,K}-\frac{(f_{2}^{s,K})^{2}}{(f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2}}P_{ r,2}^{s,K}, \tag{3}\]  
\[L_{r,\rm IF}^{s,K}=\frac{(f_{1}^{s,K})^{2}}{(f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2} }L_{r,1}^{s,K}-\frac{(f_{2}^{s,K})^{2}}{(f_{1}^{s,K})^{2}-(f_{2}^{s,K})^{2}}L_{ r,2}^{s,K}, \tag{4}\]  
where \(f_{1}^{s,K}\) and \(f_{2}^{s,K}\) denote the different frequency points. In this study, GPS used L1 and L2, GLONASS used G1 and G2, BDS-3 used B1 and B3, and Galileo used E1 and E5a, which was consistent with the International GNSS Service (IGS) analysis center. The observation equation after the combination is  
\[\left\{ \begin{array}{l}P_{r,\rm IF}^{s,K}=\rho_{r}^{s,K}+c\left(dt_{r}-dt^{s, K}\right)+T_{r}^{s,K}+b_{r,\rm IF}^{k}-b_{\rm IF}^{s,K}+\varepsilon_{P_{r,\rm IF }^{s,K}}\\ L_{r,\rm IF}^{s,K}=\rho_{r}^{s,K}+c\left(dt_{r}-dt^{s,K}\right)+T_{r}^{s,K}+ \lambda_{\rm IF}^{s,K}(N_{r,\rm IF}^{s,K}+B_{r,\rm IF}^{K}-B_{\rm IF}^{s,K})+ \varepsilon_{L_{r,\rm IF}^{s,K}}\end{array} \right., \tag{5}\]  
where \(b_{r,\rm IF}^{K}\) and \(b_{\rm IF}^{s,K}\) are the code hardware delays of the IF combination at the receiver and satellite ends, respectively. For frequency division multiple access systems, such as GLONASS, \(p_{r,\rm IF}^{K}\) represents the average hardware delay at the receiver end. \(B_{r,\rm IF}^{K}\) and \(B_{\rm IF}^{s,K}\) are the phase hardware delays of the ionospheric-free combination at the receiver and satellite ends, respectively.