# Inter-System Differencing between GPS and BDS for Medium-Baseline RTK Positioning
## 6 Conclusions  
In this contribution, we studied the inter-system differencing between dual-frequency GPS and BDS for medium-baseline RTK positioning. The tightly combined ionosphere-free observation model with real-time estimation of DISBs was proposed. In the proposed model, the estimable ionosphere-free phase DISB is reparameterized to absorb not only the DD ambiguity between GPS and BDS reference satellites, but also the SD ambiguity of the GPS reference satellite. Thus, a full-rank inter-system differencing model can be obtained. The stability of dual-frequency (i.e., L1-B1 and L2-B2) phase and code DISBs is analyzed using several baselines with the same or different receiver types. Our studies find that for the baselines with the same receiver types, both the phase and code DISBs are very stable. Although the phase DISBs may have small variations for different receiver types, fortunately, these variations are very slow and smooth. The amplitudes are within 0.1 cycles for tens of hours to several days. In practical use of the proposed inter-system differencing model, a random walk process with a secure spectrum density (e.g., 0.05 \(\times\) 0.05 cycle\({}^{2}\)/h) is recommended to model this slow variation. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Number of Satellites**} & \multicolumn{6}{c}{**RMSs of Positioning Errors (cm)**} & \multirow{3}{*}{**Improvement (\%)**} \\ \cline{2-2} \cline{4-9}  & \multicolumn{2}{c}{**Classical Differencing**} & & \multicolumn{2}{c}{**Inter-System Differencing**} & & & \\ \cline{2-9}  & **N** & **E** & **U** & **N** & **E** & **U** & **N** & **E** & **U** \\ \hline
7 & 1.93 & 1.33 & 11.25 & 1.25 & 0.96 & 9.90 & 35.2 & 27.6 & 11.9 \\
8 & 1.49 & 1.03 & 8.08 & 1.05 & 0.82 & 6.97 & 29.7 & 20.7 & 13.7 \\
9 & 1.04 & 0.78 & 5.02 & 0.95 & 0.74 & 4.88 & 8.5 & 4.7 & 2.7 \\
10 & 0.91 & 0.69 & 2.77 & 0.85 & 0.68 & 2.71 & 6.1 & 2.4 & 2.2 \\
11 & 0.77 & 0.65 & 2.63 & 0.72 & 0.64 & 2.60 & 6.2 & 0.6 & 1.1 \\
12 & 0.73 & 0.56 & 2.44 & 0.68 & 0.56 & 2.44 & 6.7 & 1.5 & 0.3 \\ All & 0.68 & 0.51 & 2.27 & 0.65 & 0.51 & 2.27 & 4.4 & 0.2 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Positioning comparison for the classical differencing model and the inter-system differencing model with different numbers of visible satellites (75.9-km baseline). Compared with the classical intra-system differencing model, although only the same number of DD ambiguities can be fixed to integers in the proposed model, the stability of the DISBs can still be used. Once the ambiguities are correctly fixed for the first time, we can obtain the high-precision DISB estimators. These high-precision DISB estimators will become useful information to strengthen the positioning model for the subsequent epochs, especially for obstructed environments with few visible satellites. Our experiments show that for the 22.4-km baseline and the 75.9-km baseline, when only seven satellites are available, the positioning accuracies in the three directions can be improved by 23.9%/19.6%/34.1% and 35.2%/27.6%/11.9%, respectively, with the proposed model. Generally, as the number of visible satellites increases, the positioning performances with the two models become closer. However, the proposed inter-system differencing model always outperformed (or was at least equivalent with) the classical differencing model. This work is partially supported by the National Natural Science Foundation of China (Grant No. 41574026), the Primary Research & Development Plan of Jiangsu Province (BE2016176), and the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBJ)1635). The authors gratefully acknowledge GNSS Research Centre of Curtin University and the Multi-GNSS Experiment and Pilot Project (MGEX) for providing the multi-GNSS data. Thanks also go to the China Scholarship Council (CSC) for the funding of the first author's living expenses during his study at the University of Nottingham in the UK. [PERSON] conceived the idea and designed the experiments with [PERSON].