# GPS/BDS Medium/Long-Range RTK Constrained with Topospheric Delay Parameters from NWP Model
## 2 Materials and Methods
### GPS/BDS Medium/Long-Range RTK Algorithm Constrained with NWP Model  
In long-range baselines RTK, measurement errors (tropospheric delay, ionospheric delay, relativistic effects, solid earth tides, polar tides and ocean loading tides) should be considered. The tides and other geophysical related errors can be estimated using methods and models. As a consequence, the main errors that affect the performance of GPS/BDS RTK are the tropospheric delay and ionospheric delay. In this study, we focus on the tropospheric delay, and use the ionospheric delay estimation method for medium/long-range baselines we developed to reduce the ionospheric delay [11]. After the ionospheric delay and other measurement errors are removed, the GPS/BDS RTK processing model can be expressed as follows:  
\[\left[ \begin{array}{c}v_{\Delta P_{Gi}}\\ v_{\Delta\left|\overline{\varphi}\right|_{Gi}}\\ v_{\Delta P_{Bj}}\\ v_{\Delta\left|\overline{\varphi}\right|_{Bj}}\end{array} \right]=\left[ \begin{array}{ccc}A&C&0&0\\ A&C&\lambda_{Gi}\cdot E&0\\ A&C&0&0\\ A&C&0&\lambda_{Bj}\cdot E\end{array} \right]\left[ \begin{array}{c}B\\ T\\ \Delta N_{Gi}\\ \Delta N_{Bj}\end{array} \right]-\left[ \begin{array}{c}\Delta P_{Gi}\\ \Delta\overline{\varphi}\right|_{Gi}\\ \Delta P_{Bj}\\ \Delta\overline{\varphi}\right|_{Bj}\end{array} \right] \tag{3}\]  
where \(v_{\Delta P_{Gi}}\) and \(v_{\Delta\left|\overline{\varphi}\right|_{Gi}}\) are the residual vectors of the double difference GPS pseudo-range and phase observation; \(v_{\Delta P_{Bj}}\) and \(v_{\Delta\left|\overline{\varphi}\right|_{Bj}}\) are the residual vectors of the double difference BDS pseudo-range and phase observation; \(i,j\) denote the frequency; \(A\) is the design matrix; \(C\) is the coefficient corresponding to \(T\); \(\lambda_{Gi}\) and \(\lambda_{Bj}\) are the wavelength of the corresponding GPS and BDS observation, respectively; \(E\) is the unit matrix; \(B\), \(T\) are the vectors of unknown parameters of baseline components and double difference slant total delay,  
Figure 2: Tropospheric delay residual of the integral method and model estimation method. respectively; \(\Delta N_{Gi}\) and \(\Delta N_{Bj}\) are the GPS and BDS double difference ambiguity, respectively; \(\Delta P_{Gi}\), \(\Delta\overline{\varphi}_{Gi}\), \(\Delta P_{Bj}\) and \(\Delta\overline{\varphi}_{Bj}\) are the double difference GPS and BDS pseudo-range and phase observation, respectively. The double difference slant total delay \(T\) can be described as the sum of the double difference hydrostatic and non-hydrostatic orwet components [12]:  
\[\Delta T=\Delta_{\text{ZHD}}\cdot\text{mfh}+\Delta_{\text{ZWD}}\cdot\text{ mfnh} \tag{4}\]  
where \(\Delta_{\text{ZHD}}\) and \(\Delta_{\text{ZWD}}\) denote the double difference zenith hydrostatic and double difference non-hydrostatic orwet delays, respectively; mfh and mfnh are the double difference hydrostatic and non-hydrostatic mapping functions, respectively; and we used the GMF Global Mapping Function (GMF) in this research [37]. In this study, four RTK scenarios are applied concerning the approach for tropospheric delay modeling and GPS/BDS integration positioning: the standard RTK (GPS alone), the standard GPS/BDS RTK, the RTK constrained with NWP (GPS alone) and the GPS/BDS RTK constrained with NWP ZTD. For the standard RTK processing (both GPS alone and GPS/BDS RTK), the \(\Delta_{\text{ZHD}}\) is calculated by using the Hopfield model in this study [14]. Owing to the high variability of the water vapor distribution, the \(\Delta_{\text{ZWD}}\) is estimated as an unknown parameter in the adjustment together with the other parameters, such as the station coordinates and the ambiguities. All of the unknown parameters are adjusted in a sequential least squares filter.