# Estimating and Analyzing Long-Term Multi-GNSS Inter-System Bias Based on Uncombined PPP
## 2 Multi-GNSS Uncombined Model
### General GNSS Observation Model  
The raw GNSS observation equation can be expressed as follows:  
\[\left\{ \begin{array}{l}p_{r,i}^{j,S}=\rho_{r}^{j,S}+c\cdot\left(\delta t_{r} -\delta t^{jS}\right)+T_{r}^{j,S}+u_{i}^{S}\cdot t_{r,1}^{j,S}+c\cdot B-c\cdot b _{i}^{j,S}+d_{rel}^{j,S}+\varepsilon_{i,P}^{j,S}\\ L_{r,i}^{j,S}=\rho_{r}^{j,S}+c\cdot\left(\delta t_{r}-\delta t^{jS}\right)+T_{r }^{j,S}-u_{i}^{S}\cdot t_{r,1}^{j,S}+\lambda_{i}^{S}\cdot M_{r,i}^{j,S}+d_{rel}^ {j,S}+\varepsilon_{i,P}^{j,S}\end{array} \right. \tag{1}\]  
where, \(j,r,i,S\) denote the satellite pseudo random noise code (PRN)number, receiver, frequency (\(i=1,2\)) and system (G/E/C/R), respectively. If (S = R), then \(B=b_{r,i}^{j,S}\), if S = (G, C, E), then \(B=b_{r,i}^{S}\); \(p_{r,i}^{j,S}\) and \(L_{r,i}^{j,S}\) represent pseudorange and carrier phase observations in meters, respectively, \(p_{r}^{j,S}\) indicates the geometric distance from the satellite to the receiver and \(\delta t_{r}\) and \(\delta t^{j,S}\) are the receiver and satellite clock differences, respectively. \(c\) represents the speed of light in a vacuum, \(T_{r}^{j,S}\) indicates tropospheric delay, \(u_{i}^{S}=\left(\frac{f_{i}^{S}}{f_{i}^{S}}\right)^{2}\) represents the ionospheric coefficient factor and \(\dot{I}_{r,1}^{j,S}=\frac{40.3}{\left(f_{i}^{S}\right)^{2}}sTEC_{r}^{j,S}\) represents the ionospheric parameters of the first frequency. \(M_{r,i}^{j,S}\) represents the ambiguity including hardware delay and initial phase deviation of the satellite and receiver on the frequency \(i\) of satellite \(j\), \(\lambda_{i}^{S}\) denotes the wavelength of the \(i\)-th frequency, \(b_{r,i}^{j,S}\) and \(B\) are pseudorange hardware delays at the receiver and satellite, respectively, \(d_{rel}^{j,S}\) represents the relativistic effect error and \(\varepsilon_{i,P}^{j,S}\), \(\varepsilon_{i,L}^{j,S}\) represent pseudorange and carrier phase observation noise including multipath error, respectively. When using the precise ephemeris products provided by IGS, the relationship with the true satellite clock difference is presented as [25]: \(\delta\overline{t}^{j,S}=\delta t^{j,S}-\frac{u_{i}^{S}}{u_{i}^{S}-1}\cdot b _{1}^{j,S}+\frac{1}{u_{i}^{S}-1}b_{2}^{j,S}+dD^{S}\), where, \(dD^{S}\) represents the bias of the reference clock, which is independent of the receiver and introduced by the precise clock product. Models were used to correct the tropospheric dry component, relativistic effect and tides effect. Then, the calculated corrections were subtracted from the observations to obtain the following expression (\(S=G,C,E\)) based on Equation (1):  
\[\left\{ \begin{array}{l}\Delta P_{r,i}^{j,S}=-\gamma_{r,i}^{j,S}\cdot\Delta \mathbf{r}+c\cdot\delta\overline{t}_{r}^{S}+M_{w}^{S}\cdot T_{w}+u_{i}^{S} \cdot\overline{l}_{r,1}^{j,S}+\varepsilon_{i,P}^{j}\\ \Delta L_{r,i}^{j,S}=-\gamma_{r,i}^{j,S}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{S}+M_{w}^{S}\cdot T_{w}-u_{i}^{S}\cdot\overline{l}_{r,1}^{j,S}+\lambda_{i}^{S}\cdot\widetilde{M}_{r,i}^{j,S}+\varepsilon_{i,L}^{j}\end{array} \right. \tag{2}\]  
The variables in Equation (2) are expressed as follows: The equivalent ionosphere at the first frequency: \(\overline{l}_{r,1}^{j,S}=\frac{40.3}{\left(f_{1}^{5}\right)^{2}}\cdot sTEC_{r} ^{j,S}-\frac{c}{u_{2}^{3}-1}\Big{(}DCB_{12}^{j,S}+DCB_{r,12}^{j,S}\Big{)}\); The equivalent ambiguity parameter: \(\widetilde{M}_{r,i}^{j,S}=M_{r,i}^{j,S}+\frac{c}{\lambda_{i}^{S}\left(u_{2}^{ 5}-1\right)}\Big{(}DCB_{12}^{j,S}+DCB_{r,12}^{j,S}\Big{)}-\frac{c}{\lambda_{i} ^{S}}\Big{(}\frac{u_{2}^{S}}{u_{2}^{5}-1}\cdot b_{r,1}^{j,S}-\frac{1}{u_{2}^{5 }-1}b_{r,2}^{j,S}\Big{)}-\frac{c}{\lambda_{i}^{S}}\Big{(}\frac{u_{2}^{S}}{u_{2 }^{5}-1}b_{r,1}^{j,S}-\frac{1}{u_{2}^{5}-1}b_{r,2}^{j,S}\Big{)}\); difference phase bias (DCB) parameters of satellite and receiver: \(DCB_{12}^{j,S}=b_{1}^{j,S}-b_{2}^{j,S};DCB_{r,12}^{j,S}=b_{r,1}^{j,S}-b_{r,2}^{ j,S}\); vector of position deviation parameter: \(\Delta\mathbf{r}=\Big{(} \begin{array}{cc}\Delta\mathbf{x}&\Delta\mathbf{y}& \Delta\mathbf{z}\end{array}