## 2 Methods  
In this section, the five-system integrated PPP model with BDS, GPS, GLONASS, Galileo and QZSS is described. In this contribution, we adopt the functional model based on ionospheric-free (IF) combined observables, which is usually employed by the analysis centers for precise orbit determination (POD) and precise clock estimation (PCE). There are uncalibrated code delays (UCDs) at both satellite and receiver ends in the code observations, and the satellite- and receiver-specific IF-based UCDs will be grouped into the satellite clock parameters and receiver clock parameters due to the strong correlation. The IF combined code and phase observations (i.e., \(P_{IF}\) and \(L_{IF}\)) from GPS, GLONASS, Galileo, BDS or QZSS satellites can be modeled as:  
\[\left\{ \begin{array}{l}P_{IF,r}^{S,O}=\rho_{r}^{S,O}+c\delta t_{r}^{O}&c \delta t^{S,O}+T_{r}^{S,O}+\varepsilon\left(P_{IF,r}^{S,O}\right)\\ L_{IF,r}^{S,O}=\rho_{r}^{S,O}+c\delta t_{r}^{O}&c\delta t^{S,O}+T_{r}^{S,O}+ \widehat{N}_{IF,r}^{S,O}+\varepsilon\left(L_{IF,r}^{S,O}\right)\end{array} \right. \tag{1}\]with  
\[\left\{ \begin{array}{l}cdt_{r}^{\Omega}=cdt_{r}+b_{IF,r}^{\Omega}\\ cdt^{\rm S,O}=cdt^{\rm S,O}+b_{IF}^{\rm S,O}\\ \overline{N}_{IF,r}^{\rm S,O}=N_{IF,r}^{\rm S,O}+d_{IF,r}^{\Omega}&d_{IF}^{\rm S,O}\\ d_{IF,r}^{\Omega}=B_{IF}^{\Omega}&b_{IF}^{\Omega}\\ d_{IF}^{\rm S,O}=B_{IF}^{\rm S,O}&b_{IF}^{\rm S,O}\end{array} \right. \tag{2}\]  
where the superscript \(s\) and the subscript \(r\) denote a GNSS/RNSS satellite and a receiver, respectively, the superscript \(O\) refers to \(G\), \(R\), \(E\), \(C\) or \(J\) which represents the satellite system GPS, GLONASS, Galileo, BDS or QZSS, the subscript \(IF\) refers to the dual-frequency IF combination which is formed with L1 and L2 signals for GPS and QZSS satellites, G1 and G2 signals for GLONASS satellites, E1 and E5a signals for Galileo satellites, and B1 and B3 signals for BDS (BDS-2/BDS-3) satellites in this study, respectively, \(\rho\) denotes the geometric range between the receiver and the satellite, \(cdt_{r}\) and \(cdt^{\rm S}\) denote the receiver and satellite clock offsets, respectively, \(T\) is the slant tropospheric delay, \(N_{IF}\) is the IF-based phase ambiguity, \(\varepsilon(P_{IF})\) and \(\varepsilon(L_{IF})\) denote the code and phase measurement noises including the multipath errors, respectively, \(b_{IF,r}\) and \(b_{IF}^{\rm S}\) denote the receiver and satellite UCDs, respectively, and \(B_{IF,r}\) and \(B_{IF}^{\rm S}\) denote the receiver and the satellite uncalibrated phase delays (UPDs), respectively. Following Equation (2), the estimable phase ambiguity \(\overline{N}_{IF,r}^{\rm S,O}\) absorbs the IF-based UCDs and UPDs of the receiver and the satellite (i.e., \(b_{IF,r}\), \(b_{IF}^{\rm S}\), \(B_{IF,r}\) and \(B_{IF}^{\rm S}\)). When the IF combination used for PPP processing is the same as that adopted by PCE, both the satellite clock offsets and satellite UCDs (i.e., \(cdt^{\rm S,O}\)) in Equation (1) can be cancelled out. As for the estimable receiver clock \(cdt_{r}^{\Omega}\), it is dependent on the satellite system due to the grouped receiver UCDs, and thus the setting of only a receiver clock parameter in the five-system integrated PPP model is obviously unreasonable. To solve this issue, we can estimate a respective receive clock parameter for each satellite system. Alternatively, by taking the receiver clock parameter of a selected satellite system as the reference, an ISB (i.e., inter-system bias) parameter can be introduced into the observation equations for other satellite systems. As GLONASS employs a frequency division multiple access (FDMA) technique, the receiver UCDs differ among different satellite signals. If we add many parameters to consider the inter-channel biases in the code observation equations, the strength of the functional model will be relatively weak. In this study, the GLONASS code observations are down weighted to mitigate the negative effects of inter-channel biases.