## 2 Overview of Multi-Constellation GNSS PPP Technique
Carrier-phase and code pseudo-range observations on \(i\)th frequency for satellite \(j\) can be expressed as in [5]:  
\[P_{i}^{j}=\rho^{j}+cdt-cdT^{j}+d_{orb}^{j}+d_{trop}^{j}+d_{ion/L_{i}}^{j}+b_{P_{ i}}^{j}+\varepsilon_{P_{i}}^{j} \tag{1}\]  
\[\Phi_{i}^{j}=\rho^{j}+cdt-cdT^{j}+d_{orb}^{j}+d_{trop}^{j}-d_{ion/L_{i}}^{j}+B_{ i}^{j}+\varepsilon_{\Phi_{i}}^{j} \tag{2}\]  
where \(j\): any GNSS satellite (G, R, E, or C), \(P_{i}\) and \(\Phi_{i}\): GNSS pseudorange and phase measurements on \(i\)th frequency (m), \(\rho\): true geometric range (m), \(c\): speed of light in vacuum (m/s), \(dt\): receiver clock error (s), \(dT\): satellite clock offset (s), \(d_{orb}\): satellite orbit error (m), \(d_{trop}\): tropospheric delay (m), \(d_{ion/L_{i}}\): ionospheric delay on \(i\)th frequency (m), \(B_{i}\): phase ambiguity term on \(i\)th frequency (m), which includes the receiver and satellite phase biases and phase hardware delay biases, \(b_{P_{i}}\): hardware delay bias in the code observations on \(i\)th frequency (m), and \(\varepsilon_{P_{i}}\) and \(\varepsilon_{\Phi_{i}}\): code and phase observations noises including multipath (m), respectively. On the other hand, the PPP technique uses ionosphere-free combined observables to eliminate the first-order ionospheric delay errors. In this case, after applying the MGEX precise satellite orbit and clock corrections, the ionosphere-free combined code and carrier-phase observables for a satellite \(j\) are expressed as [5]:  
\[P_{IF}^{j}=\Big{(}f_{1}^{2}.P_{1}^{j}-f_{2}^{2}.P_{2}^{j}\Big{)}/\Big{(}f_{1}^ {2}-f_{2}^{2}\Big{)}=\rho^{j}+cdt+d_{trop}^{j}+\varepsilon_{P_{IF}}^{j} \tag{3}\]  
\[\Phi_{IF}^{j}=\Big{(}f_{1}^{2}.\[PERSON]_{1}^{j}-f_{2}^{2}.\[PERSON]_{2}^{j}\Big{)}/\Big{(} f_{1}^{2}-f_{2}^{2}\Big{)}=\rho^{j}+cdt+d_{trop}^{j}+B_{IF}^{j}+\varepsilon_{\Phi_{IF}}^{j} \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Institution/ID** & **Constellations** & **Products and Intervals** \\ \hline CNES/CLS & G * + R * + E * & SP3 *(15 min);CLK * (30 s) \\ GRGOMGXFIN & ERP *; SNX * \\ CODE & & SP3 * (5 min);CLK * (30 s/5 min) \\ CODMGXFIN & G * + R * + E * + C * + J * & ERP * (12 h); BIA * (1 d) \\ GZ & G * + R * + E * + C * + J * & SP3 * (15 min);CLK * (30 s/5 min) \\ GFZOMGXRAP & G * + R * + E * + C * + J * & ERP * (1 d); BIA * (1d) \\ JAXA & G * + R * + J * & SP3 * (5 min);CLK * (30 s) \\ JAXOMGXFIN & ERP *; SNX * \\ SHAO & G * + R * + E * + C * & SP3 * (15 min) \\ SHAOOMGXRAP & CLK * (5 min) \\ TUM & G * + C * + J * * & SP3 * (5 min) \\ TUMOMGXRAP & & SP3 * (15 min);CLK * (5 min) \\ Wuhan Univ. & & \\ WUMOMGXFIN & G * + R * + E * + C * + J * & ERP * (1 d) \\ \hline \hline \end{tabular}  
* *: BRD: broadcast ephemerides; SP3: precise ephemerides; CLK: clock products; ERP: Earth orientation parameters; SNX: IGS stations coordinates; BIA: biases; G: GPS; R: GLONAS; E: Galileo; C: BDS; J: QZSS.  
\end{table} 
Table 1: The analysis centers of MGEX’s and available products [7]. where \(P_{IF}\) and \(\Phi_{IF}\): ionosphere-free combination of pseudo-range and phase measurements (m); \(f_{1}\) and \(f_{2}\): two carrier-phase frequencies (Hz). The code hardware delay biases (\(b_{P_{i}}\)) seen in the original observation equations do not appear in the ionosphere-free observables (as \(b_{P_{IF}}\)) because they are absorbed into the receiver clock error for the GPS and into the system time difference for the other satellite constellations [5]. For an ambiguity-float PPP solution using ionosphere-free combined observations given in Equations (3) and (4), the ionosphere-free ambiguity parameter is estimated as a real-value constant. In order to get the cm-level accurate positioning in multi-constellation GNSS PPP, the satellite and receiver antenna offsets and directional variation corrections; solid earth, ocean tide loading displacement effects; phase wind-up effects; relativistic effects; polar motion; and the others, should also be considered.