# Inter-System Differencing between GPS and BDS for Medium-Baseline RTK Positioning
## 3 Stability Analysis of GPS-BDS DISBs
### Stability Analysis of DISBs  
The uncombined DISB estimation model with zero and ultra-short baselines can be easily derived by simplifying Equation (11). In order to test the variation of DISBs with time, the estimation is carried out epoch-by-epoch. For the zero and ultra-short baselines with known baseline components, the intra-system DD ambiguities can be solved reliably ahead of the determination of DISB parameters. Similar to Equation (11), the estimable phase DISB parameter will absorb not only the DD ambiguity between the two reference satellites, but also the SD ambiguity of the GPS reference satellite. Thus, the phase DISB estimators will change as the reference satellite of GPS or BDS changes. For convenience, we only need to analyze the fractional part of the phase DISBs. The DD ambiguity between the two reference satellites only affects the integer part. When the BDS reference satellite changes, the fractional \begin{table}
\begin{tabular}{l l} \hline \hline
**Options** & **Processing Strategy** \\ \hline Ephemeris & GPS and BDS broadcast ephemeris \\ \hline Signals & GPS L1/L2, BDS B1/B2 \\ \hline Elevation cutoff angle & 10\({}^{\circ}\) \\ \hline \multirow{4}{*}{Troposphere modeling} & Corrected using GPT2w model \\  & For the baseline longer than 30 km: RZTD estimated as a random walk \\  & process (Mapping function: GMF [31]; Initial value : 0; Initial variance: 0.1 m \(\times\) 0.1 m; Spectral density: 10\({}^{-9}\) m\({}^{2}\)/s) \\ \hline \multirow{2}{*}{Observation weighting} & Elevation-dependent weighting [32] \\  & Relative precision of code and phase observations: 100:1 \\ \hline \multirow{2}{*}{Coordinates of rover station} & Kinematic: estimated and modeled as a random walk process \\  & Spectral density of coordinates: 10\({}^{2}\) m\({}^{2}\)/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Additional data processing models and strategies used in this paper. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Baseline** & **Receiver Type 1** & **Receiver Type 2** & **Baseline Length/m** & **Remark** \\ \hline
1 & CUT0-CUTB & Trimble NETR9 & Trimble NETR9 & 4.3 & with the same \\
2 & CUT1-CUAI & Septentrio POLARX4 & Septentrio POLARXS & 8.4 & receiver types \\
3 & CUT3-CUBB & Javad TRE\_G3 TH & Javad TRE\_G3 TH & 4.3 & \\ \hline
4 & CUT0-CUT1 & Trimble NETR9 & Septentrio POLARX4 & 0 & \\
5 & CUT0-CUBB & Trimble NETR9 & Javad TRE\_G3 TH & 4.3 & with different \\
6 & CUT1-CUBB & Septentrio POLARX4 & Javad TRE\_G3 TH & 4.3 & receiver types \\
7 & SEU1-SEU2 & ComNav-K508 & Unicore-UB380 & 2.9 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Baselines and the corresponding receiver types in the experiments. part of the phase DISB will not be affected. However, the SD ambiguity of the GPS reference satellite affects both the integer part and fractional part. That is, when the GPS reference satellite changes, the fractional part of the phase DISB will also change. We set a virtual SD ambiguity \(\Delta N^{ref}\) of the GPS reference satellite to eliminate this influence. Table 3 illustrates the procedures that the GPS pivot satellite changes from \(1_{G}\) to \(s\), then changes from \(s\) to \(r\), where the initial virtual SD ambiguity is set at 0 at the beginning. Then, when the GPS reference satellite changes, it can be transformed accordingly with the integer DD ambiguity between the two reference satellites from the two consecutive epochs. With the updated virtual SD ambiguity, the phase DISB estimators \(\Delta\widetilde{\delta}_{j}^{GC}\) with single epoch can be transformed to the new phase DISB series \(\Delta\widetilde{\delta}_{j}^{GC}\) by absorbing the same SD ambiguity, thus the phase DISBs with the continuous fractional part can be obtained. Figures 1 and 3 depict the dual-frequency code and phase DISB time series, respectively, for the three baselines at Curtin University with the same receiver types. We can see, regardless of the random terms caused by observation noises, the DISBs estimators are very stable.