# Estimating and Analyzing Long-Term Multi-GNSS Inter-System Bias Based on Uncombined PPP
## 2 Multi-GNSS Uncombined Model
### Multi-GNSS Uncombined Model Construction  
After selecting the receiver clock of the GPS system as the reference, the G + C + R + E four-system fusion observation model can be obtained as follows:  
\[\left\{\begin{array}{l}\Delta P_{r,i}^{j,G}=-\gamma_{r,i}^{j,G}\cdot\Delta \mathbf{r}+c\cdot\delta\overline{t}_{r}^{G}+M_{W}^{\mathrm{G}}\cdot T_{w}+u_{ i}^{G}\cdot\overline{l}_{r,1}^{j,G}+\varepsilon_{i,P}^{j,G}\\ \Delta P_{r,i}^{j,C}=-\gamma_{r,i}^{j,C}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{G}+c\cdot ISB_{r}^{C-G}+M_{W}^{\mathrm{C}}\cdot T_{w}+u_{i}^{ C}\cdot\overline{l}_{r,1}^{j,C}+\varepsilon_{i,P}^{j,C}\\ \Delta P_{r,i}^{j,E}=-\gamma_{r,i}^{j,E}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{G}+c\cdot ISB_{r}^{E-G}+M_{W}^{\mathrm{C}}\cdot T_{w}+u_{i}^ {E}\cdot\overline{l}_{r,1}^{j,E}+\varepsilon_{i,P}^{j,E}\\ \Delta P_{r,1}^{j,R}=-\gamma_{r,1}^{j,R}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{G}+c\cdot ISB_{r}^{R-G}+M_{W}^{\mathrm{R}}\cdot T_{w}+u_{1}^ {R}\cdot\overline{l}_{r,1}^{j,R}+\varepsilon_{1,P}^{j,R}\\ \Delta P_{r,2}^{j,R}=-\gamma_{r,2}^{j,R}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{G}+c\cdot ISB_{r}^{R-G}+M_{W}^{\mathrm{R}}\cdot T_{w}+u_{2}^ {s}\cdot\overline{l}_{r,1}^{j,R}+c\cdot\eta^{j}\cdot\Pi_{r}^{R}+\varepsilon _{2,P}^{j,R}\\ \end{array}\right. \tag{6}\]  
\[\left\{\begin{array}{l}\Delta L_{r,i}^{j,G}=-\gamma_{r,i}^{j,G}\cdot\Delta \mathbf{r}+c\cdot\delta\overline{t}_{r}^{G}+M_{W}^{\mathrm{G}}\cdot T_{w}+u_{ i}^{G}\cdot\overline{l}_{r,1}^{j,G}+\Lambda_{i}^{G}\cdot\overline{l}_{r,i}^{j,G}+ \varepsilon_{i,D}^{j,G}\\ \Delta L_{r,i}^{j,C}=-\gamma_{r,i}^{j,C}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{G}+c\cdot ISB_{r}^{C-G}+M_{W}^{\mathrm{C}}\cdot T_{w}-u_{i}^ {C}\cdot\overline{l}_{r,1}^{j,C}+\Lambda_{i}^{C}\cdot\overline{l}_{r,i}^{j,C}+ \varepsilon_{i,L}^{j,C}\\ \Delta L_{r,i}^{j,E}=-\gamma_{r,i}^{j,E}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{G}+c\cdot ISB_{r}^{E-G}+M_{W}^{\mathrm{E}}\cdot T_{w}-u_{i}^ {E}\cdot\overline{l}_{r,1}^{j,E}+\Lambda_{i}^{E}\cdot\overline{M}_{r,i}^{j,E}+ \varepsilon_{i,L}^{j,E}\\ \Delta L_{r,i}^{j,R}=-\gamma_{r,i}^{j,R}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{G}+c\cdot ISB_{r}^{R-G}+M_{W}^{\mathrm{R}}\cdot T_{w}-u_{i}^ {R}\cdot\overline{l}_{r,1}^{j,R}+\Lambda_{i}^{R}\cdot\overline{l}_{r,j}^{J,R}+ \varepsilon_{i,L}^{j,R}\\ \end{array}\right. \tag{7}\]  
From Equations (6) and (7), the specific forms of three groups ISB parameters can be determined as follows:  
\[\left\{\begin{array}{l}ISB_{r}^{C-G}=dD^{C}-dD^{G}+D_{r,IF}^{C}-D_{r,IF}^{G}\\ ISB_{r}^{E-G}=dD^{E}-dD^{G}+D_{r,IF}^{E}-D_{r,IF}^{G}\\ ISB_{r}^{R-G}=dD^{R}-dD^{G}+D_{r,IF}^{R}-D_{r,IF}^{G}\\ \end{array}\right. \tag{8}\]  
From Equation (8), it is evident that ISB is primarily composed of two parts, i.e., the difference of hardware delay between different systems and the error term, which is independent of the receiver and is caused by the selection of the reference clock of different systems. The estimated parameters of the Equations (6) and (7) can be expressed as follows:  
\[\mathbf{x}=\Big{[}\ \delta x\quad\delta y\quad\delta z\quad\delta\overline{t}_{r}^{G} \quad\Pi_{r}^{R}\quad T_{w}\quad ISB_{r}^{E-G}\quad ISB_{r}^{E-G}\quad ISB_{r}^{R- G}\quad\overline{l}_{r,1}^{S}\quad\cdots\quad\overline{l}_{r,1}^{S}\quad\overline{M}_{r,1}^{ 1,S}\quad\cdots\quad\overline{M}_{r,1}^{m,S}\quad\overline{M}_{r,2}^{m,S}\quad \cdots\quad\overline{M}_{r,2}^{m,S}\quad\Big{]}^{T} \tag{9}\]