# Inter-System Differencing between GPS and BDS for Medium-Baseline RTK Positioning
## 2 Combined GPS and BDS Observation Model
### Intra-System and Inter-System Observation Model
 \tag{3}\]  
where, the subscript '\(G\)' and '\(C\)' denote GPS and BDS, respectively. \(s=1,\cdots,m_{G}\) is the index of GPS satellites, where \(m_{G}\) denotes the number of GPS satellite; \(q=1,\cdots,m_{C}\) is the index of BDS satellites, where \(m_{C}\) denotes the number of BDS satellites; the GPS NL wavelength \(\lambda_{NL,G}\), the ionosphere-free ambiguity \(N_{IF,G}\), the ionosphere-free hardware phase delay \(\delta_{IF,C}\) and hardware code delay \(d_{IF,G}\) are expressed as  
\[\left\{ \begin{array}{l}\lambda_{NL,G}=\frac{c}{f_{1,G}+f_{2,G}}\\ N_{IF}^{s}=\frac{f_{1,G}}{f_{1,G}-f_{2,G}}N_{1}^{s}-\frac{f_{2,G}}{f_{1,G}-f_{ 2,G}}N_{2}^{s}\\ \delta_{IF,G}=\frac{f_{1,G}}{f_{1,G}-f_{2,G}}\delta_{1,G}-\frac{f_{2,G}}{f_{1, G}-f_{2,G}}\delta_{2,G}\\ d_{IF,G}=\frac{f_{1,G}^{s}}{f_{1,G}-f_{2,G}}\cdot d_{1,G}-\frac{f_{2,G}^{s}}{f_{1,G}^{s}-f_{2,G}^{s}}\cdot d_{2,G}\end{array} \right. \tag{4}\]  
where \(c\) is the light speed in vacuum; \(\lambda_{NL,C}\), \(N_{IF}^{q}\), \(\delta_{IF,C}\), and \(d_{IF,C}\) in the BDS observation equations can be obtained similarly. Based on the above SD ionosphere-free observation equations, the classical intra-system DD observations can be formed, where the receiver-dependent biases can be eliminated. If we fix \(s=1_{G}\) as the reference satellite for GPS, we can obtain  
\[ \begin{array}{l}\Delta\phi_{IF}^{1_{G}s}=\Delta\phi_{IF}^{s}-\Delta\phi_{IF }^{1_{C}}=\Delta\rho^{1_{G}s}+\lambda_{NL}\Delta N_{IF}^{1_{G}s}+\Delta T^{1_ {G}s}+\Delta e_{IF}^{1_{G}s}\\ \Delta p_{IF}^{1_{G}s}=\Delta p_{IF}^{s}-\Delta p_{IF}^{1_{G}}=\Delta\rho^{1_ {G}s}+\Delta T^{1_{G}s}+\Delta e_{IF}^{1_{G}s}\end{array}  \tag{5}\]  
For BDS, the intra-system DD observation equations with \(q=1_{C}\) as the reference satellite are  
\[ \begin{array}{l}\Delta\phi_{IF}^{1_{G}q}=\Delta\phi_{IF}^{q}-\Delta\phi_{IF }^{1_{C}}=\Delta\rho^{1_{G}q}+\lambda_{NL}\Delta N_{IF}^{1_{G}q}+\Delta T^{1_ {G}q}+\varepsilon_{IF}^{1_{G}q}\\ \Delta p_{IF}^{1_{G}q}=\Delta p_{IF}^{q}-\Delta p_{IF}^{1_{C}}=\Delta\rho^{1_{G} q}+\Delta T^{1_{G}q}+e_{IF}^{1_{G}q}\end{array}  \tag{6}\]  
The DD inter-system observation equations between GPS and BDS can also be built in a similar way, but the hardware delays cannot be eliminated. The corresponding models can be expressed as  
\[ \begin{array}{l}\Delta\phi_{IF}^{1_{G}q}=\Delta\phi_{IF}^{q}-\Delta\phi_{IF }^{1_{G}q}=\Delta\rho^{1_{G}q}+\lambda_{IF,C}\Delta\delta_{IF,GC}+(\lambda_{ IF,C}\Delta N_{IF}^{q}-\lambda_{IF,G}\Delta N_{IF}^{1_{G}})+\Delta T^{1_{G}q}+ \Delta e_{IF}^{1_{G}q}\\ \Delta p_{IF}^{1_{G}q}=\Delta p_{IF}^{q}-\Delta p_{IF}^{1_{G}}=\Delta\rho^{1_{G} q}+\Delta d_{IF,GC}+\Delta T^{1_{G}q}+\Delta e_{IF}^{1_{G}s}\end{array}  \tag{7}\]  
where, the phase DISB \(\Delta\delta_{IF,GC}\) and code DISB \(\Delta d_{IF,GC}\) between GPS and BDS are expressed as  
\[\left\{ \begin{array}{l}\Delta\delta_{IF,GC}=\Delta\delta_{IF,C}-\frac{ \lambda_{NL,G}}{\lambda_{NL,C}}\Delta\delta_{IF,G}\\ \Delta d_{IF,GC}=\Delta d_{IF,C}-\Delta d_{IF,G}\end{array} \right. \tag{8}\]  
From Equation (7) we can see, because \(\lambda_{j_{C}}\
eq\lambda_{j_{C}}\), the DD ambiguities between GPS and BDS cannot be estimated directly. We can reparameterize the phase equation in Equation (7) as  
\[\Delta\phi_{IF}^{1_{G}q}=\Delta\rho^{1_{G}q}+\lambda_{IF,C}\Delta\delta_{IF,GC} +\lambda_{IF,C}\Delta N_{IF}^{1_{G}q}+(\lambda_{IF,C}-\lambda_{IF,G})\Delta N_{ IF}^{1_{G}}+\Delta T^{1_{G}q}+\Delta e_{IF}^{1_{G}q} \tag{9}\]  
where \(\Delta N_{IF}^{1_{G}q}=\Delta N_{IF}^{q}-\Delta N_{IF}^{1_{G}}\). The mixed observation in Equation (9) is rank-defective, as it is impossible to simultaneously estimate any two of the differential phase ISB parameter, the inter-system DD ambiguities, and the SD ambiguity of the GPS reference satellite.