# Estimating and Analyzing Long-Term Multi-GNSS Inter-System Bias Based on Uncombined PPP
## 2 Multi-GNSS Uncombined Model
### General GNSS Observation Model
\Big{)}^{T}\); Receiver clock: \(\delta\overline{t}_{r}^{S}=\delta t_{r}+D_{r,IF}^{S}+dD_{r,IF}^{S}\); ionosphere-free combination of receiver hardware delay: \(D_{r,IF}^{S}=\frac{u_{2}^{S}}{u_{2}^{5}-1}\cdot b_{r,1}^{S}-\frac{1}{u_{2}^{5 }-1}b_{r,2}^{S}\). Unlike the GPS system, which uses code division multiple access (CDMA), the GLONASS system uses frequency division multiple access (FDMA) technology, which introduces obvious differences between different satellite frequencies (IFB). It can be seen from Equation (2) that the receiver pseudo-range hardware delay \(b_{r,i}^{j,S}\) of GLONASS system is related to satellite number, while the \(b_{r,i}^{S}\) parameters of the other three GNSS systems are independent of satellite number. Generally, IFB can be divided into carrier IFB and pseudo-range IFB, in which the carrier IFB parameters are usually absorbed by the ambiguous parameters and can be ignored in the float solution of PPP. The pseudo-range IFB of GLONASS system is usually modeled as a function of frequency number, and then estimated together with other parameters. However, this may introduce a large number of IFB parameters related to satellite or satellite frequency numbers into the equation. Considering that there are already a large number of parameters to be estimated in the four-system uncombined PPP, the stability and efficiency of the normal equation will be significantly affected by this method. Studies show that the pseudo-range hardware delay of the GLONASS system is approximately linearly correlated with the frequency number \(\eta^{j}\)[26; 27], and the pseudo-range observations are generally given much smaller weights compared with carrier phase equations, so the pseudo-range IFB parameters can be approximately set as follows:  
\[IFB=b_{r,i}^{j,S}-b_{r,i}^{\beta,S}=\eta^{j}\cdot\Pi_{r,i}^{R} \tag{3}\]  
In Equation (3), the pseudo-range IFB parameter is obtained based on pseudo-range hardware delay of satellite \(\beta\) of which frequency number \(\eta^{j}\) is zero. Then the model of the GLONASS system can be expressed as follows:  
\[\left\{\begin{array}{l}\Delta P_{r,1}^{j,R}=-\gamma_{r,1}^{j,R}\cdot\Delta \mathbf{r}+c\cdot\delta\overline{t}_{r}^{R}+M_{w}^{R}\cdot T_{w}+u_{1}^{R} \cdot\overline{l}_{r,1}^{j,R}+\varepsilon_{1,P}^{j}\\ \Delta P_{r,2}^{j,R}=-\gamma_{r,2}^{j,R}\cdot\Delta\mathbf{r}+c\cdot\delta \overline{t}_{r}^{R}+M_{w}^{R}\cdot T \(\widetilde{M}_{r,1}^{j,R}\) represent the equivalent ambiguity of each frequency, the variables in Equation (4) are expressed as follows:  
\[\left\{ \begin{array}{l}\widetilde{l}_{r,1}^{j,R}=\frac{40.3}{\left(\ell_{1}^{R} \right)^{2}}\cdot sTEC_{r}^{j,R}-\frac{c}{u_{2}^{S}-1}\Big{(}DCB_{12}^{j,R}+DCB_ {r,12}^{R}\Big{)}+c\cdot\eta^{j}\cdot\Pi_{r,1}^{R}\\ \widetilde{M}_{r,i}^{j,R}=M_{r,i}^{j,R}+\frac{c}{\lambda_{i}^{K}\left(u_{2}^{R }-1\right)}\Big{(}DCB_{12}^{j,R}+DCB_{r,12}^{j,R}\Big{)}-\frac{c}{\lambda_{i}^ {K}}\Big{(}\frac{u_{2}^{R}}{u_{2}^{R}-1}\cdot b_{r,1}^{j,R}-\frac{1}{u_{2}^{R }-1}b_{r,2}^{j,R}\Big{)}\\ \quad\quad-\frac{c}{\lambda_{i}^{K}}\Big{(}\frac{u_{2}^{R}}{u_{2}^{R}-1}b_{1} ^{j,R}-\frac{1}{u_{2}^{R}-1}b_{2}^{j,R}\Big{)}+\frac{c\,u_{1}^{R}\cdot\eta^{ j}\cdot\Pi_{r,1}^{R}}{\lambda_{i}^{K}}\\ D\mathrm{CB}_{12}^{j,R}=b_{1}^{j,R}-b_{2}^{j,R};DCB_{r,12}^{S}=b_{r,1}^{j,R}-b_ {r,2}^{j,R}\\ \Delta\mathbf{r}=\Big{(}\ \Delta x\quad\Delta y\quad\Delta z\ \Big{)}^{T}\\ \delta\overline{t}_{r}^{R}=\delta t_{r}+D_{r,IF}^{R}+dD^{R}\\ D_{r,IF}^{R}=\frac{u_{2}^{R}}{u_{2}^{R}-1}\cdot b_{r,1}^{j,R}-\frac{1}{u_{2} ^{R}-1}b_{r,2}^{j,R}\\ \Pi_{r}^{R}=\Pi_{r,2}^{R}-u_{2}^{R}\cdot\eta^{j}\cdot\Pi_{r,1}^{R}\\ \end{array} \right. \tag{5}\]