# 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
Referring to [15], we can further change the form of the DD ambiguity as follows  
\[N_{IF}^{1_{G}q}=(N_{IF}^{q}-N_{IF}^{1_{C}})+(N_{IF}^{1_{C}}-N_{IF}^{1_{G}})=N_{ IF}^{1_{G}q}-N_{IF}^{1_{G}1_{C}} \tag{10}\]By Equation (10), we can reparameterize the inter-system DD ambiguities into ambiguities relative to the BDS reference satellite, plus the ambiguity of the BDS reference satellite with respect to the GPS reference satellite. This last ambiguity and the SD ambiguity of the GPS reference satellite are then estimable jointly with the differential phase DISB. Therefore, we can get the full-rank inter-system observation model between GPS and BDS as  
\[ \begin{split}\Delta\phi^{\text{L}cq}_{IF}&=\Delta \rho^{\text{L}cq}+\lambda_{IF,C}\Delta\overline{\delta}_{IF,GC}+\lambda_{IF,C} \Delta N^{\text{L}cq}_{IF}+\Delta T^{\text{L}cq}+\Delta\epsilon^{\text{L}cq}_{ IF}\\ \Delta p^{\text{L}cq}_{IF}&=\Delta\rho^{\text{L}cq}+ \Delta d_{IF,GC}+\Delta T^{\text{L}cq}+\Delta e^{\text{L}cs}_{IF}\end{split}  \tag{11}\]  
where, \(\overline{\delta}_{IF,GC}=\delta_{IF,GC}+N^{\text{L}c1_{C}}_{IF}+(1-\frac{ \lambda_{NL,G}}{\lambda_{NL,C}})N^{\text{L}c}_{IF}\). In Equation (11), the estimable DD ambiguities \(\Delta N^{\text{L}cq}_{IF}\) are still the intra-system ambiguities among BDS satellites. The difference between the inter-system model and the intra-system is the DISB terms, i.e., \(\Delta\overline{\delta}_{IF,GC}\) and \(\Delta d_{IF,GC}\), which are eliminated in the intra-system model. If the DISB parameters can be assigned with prior information, we can expect better positioning performance with the inter-system model. From Equation (11), we can see that the new integrated phase DISB parameter is tightly related to the GPS reference satellite and BDS reference satellite. Thus, in a real-time estimation process from epoch to epoch, when the reference satellite of GPS or BDS changes, besides the estimable DD ambiguities, the integrated phase DISB parameter in the filter also needs to be transformed correspondingly, so that the phase DISB can be updated continuously. Assuming that no cycle slips occur between two consecutive epochs \(t\) and \(t+1\), when the reference satellite of GPS changes from \(1_{G}\) to \(s\), the phase DISB parameter can be transformed as follows  
\[ \begin{split}\Delta\overline{\delta}_{IF,GC}(t)&= \Delta\delta_{IF,GC}+\Delta N^{\text{L}c1_{C}}_{IF}+(1-\frac{\lambda_{NL,G}}{ \lambda_{NL,C}})\Delta N^{\text{L}c}_{IF}\\ \Delta\overline{\delta}_{IF,GC}(t+1)&=\Delta \overline{\delta}_{IF,GC}(t)-\frac{\lambda_{IF,GC}}{\lambda_{IF,GC}}\Delta N^{ \text{L}c}_{IF}=\Delta\delta_{IF,GC}+\Delta N^{\text{s1}c}_{IF}+(1-\frac{ \lambda_{IF,G}}{\lambda_{IF,C}})\Delta N^{\text{s}}_{IF}\end{split}  \tag{12}\]  
Similarly, when the reference satellite of BDS changes from \(1_{C}\) to \(q\), the phase DISB parameter can be transformed  
\[ \begin{split}\Delta\overline{\delta}_{IF,GC}(t)&= \Delta\delta_{IF,GC}+\Delta N^{\text{L}c1_{C}}_{IF}+(1-\frac{\lambda_{NL,G}}{ \lambda_{NL,C}})\Delta N^{\text{L}c}_{IF}\\ \Delta\overline{\delta}_{IF,GC}(t+1)&=\Delta \overline{\delta}_{IF,GC}(t)+\Delta N^{\text{L}cq}_{IF}=\Delta\delta_{IF,GC}+ \Delta N^{\text{L}cq}_{IF}+(1-\frac{\lambda_{IF,G}}{\lambda_{IF,C}})\Delta N^{ \text{L}c}_{IF}\end{split}  \tag{13}\]  
By Equations (12) and (13), when the pivot satellite of GPS or BDS changes, the phase DISB estimator can be transformed to be always consistent with the current epoch, so that the continuous parameter estimation can be carried out.