# 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  
For GPS and BDS, the time difference between GPST and BDST will be eliminated by between-receiver difference. The small coordinate difference between GPS and BDS can also be neglected for medium-baseline relative positioning of centimeter-level accuracy [19]. For medium baselines, the differential atmospheric delays between receivers need to be considered. The between-receiver SD observation equations for GPS or BDS can be expressed as  
\[ \begin{array}{l}\Delta\phi_{j}^{s}=\Delta\rho^{s}+\Delta dt+\lambda_{j}( \Delta\delta_{j}+\Delta N_{j}^{s})+\Delta T^{s}-\mu_{j}\Delta I_{1}^{s}+ \Delta e_{j}^{s}\\ \Delta p_{j}^{s}=\Delta\rho^{s}+\Delta dt+\Delta d_{j}+\Delta T^{s}+\mu_{j} \Delta I_{1}^{s}+\Delta e_{j}^{s}\end{array}  \tag{1}\]  
where, \(\Delta\) is the between-receiver SD operator; \(\phi\) is the carrier observation and \(p\) is pseudorange observation; \(j\) is the frequency of GPS or BDS; \(s\) is the index of GPS or BDS satellites; \(\rho\) is the distance between satellite and receiver; \(dt\) denotes the receiver clock error; \(\lambda\) denotes the wavelength at the corresponding frequency; \(\delta\) denotes the hardware phase delay, which also contains the initial phase in the receiver; \(N\) denotes the integer phase ambiguity; \(T\) denotes the tropospheric delay, and \(I\) denotes the ionospheric delay; \(\mu\) is the ionospheric scale factor; \(d\) denotes the hardware code delay in the receiver for GPS or BDS; \(\varepsilon\) and \(e\) are the noise for carrier phase and pseudorange measurements, respectively. The atmospheric errors, including the tropospheric delay and ionospheric delay, could not be ignored for medium baselines. The DD tropospheric delay can be corrected with an empirical troposphere model, e.g., the GPT2w model [20]. If the baseline is longer than 30 km or has a large height difference between two stations, a relative zenith troposphere delay (RZTD) parameter can be estimated as a random-walk process to absorb the residual tropospheric biases [21]. For the ionospheric delay in dual-frequency data, there are usually two processing models. The first one is to form the ionosphere-free combination to eliminate the first-order ionospheric delay. In this model, the wide-lane (WL) ambiguities are usually fixed firstly using the geometry-free and ionosphere-free Melbourne-Wubbena (MW) combination [22; 23]. One can easily choose the subset of WL integer ambiguities to recover the integer property of narrow-lane (NL) ambiguity in the subsequent step since the MW combination is implemented for each satellite pair. This is particularly important for BDS due to the existence of satellite-induced code bias and the systematic multipath effects in GEO satellites [24; 25]. The other model is to parameterize the slant ionospheric delay for each satellite (pair), which is also called the uncombined model. Theoretically, the uncombined model is more rigorous than the ionosphere-free model since it has more redundancy. However, it also introduces much more unknowns and needs more complex operations for DISB parameters in the inter-system differencing model. In this paper, we adopt the ionosphere-free model due to its simplicity. Based on Equation (1), the SD ionosphere-free observation equations for GPS can be expressed as  
\[ \begin{array}{l}\Delta\phi_{IF,G}^{s}=\Delta\rho^{s}+\Delta dt+\lambda_{NL, G}\Delta\delta_{IF,G}+\lambda_{NL,G}\Delta N_{IF}^{s}+\Delta T^{s}+\Delta e_{IF}^{s}\\ \Delta p_{IF,G}^{s}=\Delta\rho^{s}+\Delta dt+\Delta d_{IF,G}+\Delta T^{s}+ \Delta e_{IF}^{s}\end{array}  \tag{2}\]Similarly, the SD ionosphere-free observation equations for BDS can be expressed as  
\[ \begin{array}{l}\Delta\phi_{IF,C}^{q}=\Delta\rho^{q}+\Delta dt+\lambda_{NL,C} \Delta\delta_{IF,C}+\lambda_{NL,C}\Delta N_{IF}^{q}+\Delta T^{q}+\Delta e_{IF}^{ q}\\ \Delta p_{IF,C}^{q}=\Delta\rho^{q}+\Delta dt+\Delta d_{IF,C}+\Delta T^{q}+ \Delta e_{IF}^{q}\end{array}