# GNSS RTK Positioning Augmented with Large LEO Constellation
## 2 Methods
### Combined GNSS/LEO RTK Observation Model  
The observation models for combined GNSS/LEO RTK in this study have been of the short, medium, and long single-baseline types. Hence, the processing methods of atmospheric delay must be taken into account when it comes to different-length baseline situations. This determines whether the ionospheric and tropospheric residuals are used as parameter estimations in the DD observation equations. After constructing between-station and between-satellite SD equations based on above-mentioned (1) and (2), the delays common to satellite and station part can be eliminated or ignored. Considering the existence of DD ionospheric and tropospheric delay, the DD pseudorange and carrier-phase observation equations can be described as:  
\[\left\{ \begin{array}{l}P_{r_{1}\tau_{2},f,i}^{G_{1}G_{2}}=\rho_{r_{1}\tau_{2}, i}^{G_{1}G_{2}}+I_{r_{1}\tau_{2},f,i}^{G_{1}G_{2}}+T_{r_{1}\tau_{2},i}^{G_{1}G_{2}}+e _{r_{1}\tau_{2},f,i}^{G_{1}G_{2}}\\ P_{r_{1}\tau_{2},f,i}^{L_{1}L_{2}}=\rho_{r_{1}\tau_{2},i}^{L_{1}L_{2}}+I_{r_{1} \tau_{2},f,i}^{L_{1}L_{2}}+T_{r_{1}\tau_{2},i}^{L_{1}L_{2}}+e_{r_{1}\tau_{2},f,i}^{L_{1}L_{2}}\end{array} \right. \tag{3}\]  
\[\left\{ \begin{array}{l}\Phi_{r_{1}\tau_{2},f,i}^{G_{1}G_{2}}=\rho_{r_{1}\tau_ {2},i}^{G_{1}G_{2}}-T_{r_{1}\tau_{2},f,i}^{G_{1}G_{2}}+T_{r_{1}\tau_{2},i}^{G_{ 1}G_{2}}+\lambda_{f,G}C_{r_{1}\tau_{2},f}^{G_{1}G_{2}}+\epsilon_{r_{1}\tau_{2},f,i}^{G_{1}G_{2}}\\ \Phi_{r_{1}\tau_{2},f,i}^{L_{1}L_{2}}=\rho_{r_{1}\tau_{2},i}^{L_{1}L_{2}}-I_{ r_{1}\tau_{2},f,i}^{L_{1}L_{2}}+T_{r_{1}\tau_{2},i}^{L_{1}L_{2}}+\lambda_{f,L}N_{r_{1} \tau_{2},f}^{L_{1}L_{2}}+\varepsilon_{r_{1}\tau_{2},f,i}^{L_{1}L_{2}}\end{array} \right. \tag{4}\]  
With  
\[\left\{ \begin{array}{l}\rho_{r_{1}\tau_{2},f,i}^{s_{1}g_{2}}=\left(\rho_{r_{ 2},i}^{s_{2}}-\rho_{r_{1},i}^{s_{2}}\right)-\left(\rho_{r_{2},i}^{s_{1}}-\rho _{r_{1},i}^{s_{1}}\right)\\ I_{r_{1}\tau_{2},f,i}^{s_{1}g_{2}}=\left(\rho_{r_{2},f,i}^{s_{2}}-\rho_{r_{1}, f,i}^{s_{2}}\right)-\left(I_{r_{2},f,i}^{s_{1}}-I_{r_{1},f,i}^{s_{1}}\right)\\ T_{r_{1}\tau_{2},i}^{s_{1}g_{2}}=\left(T_{r_{2},i}^{s_{2}}-T_{r_{1},i}^{s_{2}} \right)-\left(T_{r_{2},i}^{s_{2}}-T_{r_{1},i}^{s_{2}}\right)\\ N_{r_{1}\tau_{2},f}^{s_{1}g_{2}}=\left(N_{r_{2},f}^{s_{2}}-N_{r_{1},f}^{s_{2}} \right)-\left(N_{r_{2},f}^{s_{1}}-N_{r_{1},f}^{s_{1}}\right)\\ \epsilon_{r_{1}\tau_{2},f,i}^{s_{1}g_{2}}=\left(\epsilon_{r_{2},f,i}^{s_{2}}- \epsilon_{r_{1},f,i}^{s_{2}}\right)-\left(\epsilon_{r_{2},f,i}^{s_{1}}-\epsilon _{r_{1},f,i}^{s_{1}}\right)\\ \epsilon_{r_{1}\tau_{2},f,i}^{s_{1}g_{2}}=\left(\epsilon_{r_{2},f,i}^{s_{2}}- \epsilon_{r_{1},f,i}^{s_{2}}\right)-\left(\epsilon_{r_{2},f,i}^{s_{1}}-\epsilon _{r_{1},f,i}^{s_{1}}\right)\end{array} \right. \tag{5}\]  
where \(L\) and \(G\) refer to LEO and GNSS system, respectively. The reference station is denoted using subscript \(r_{1}\), the rover station is denoted with a subscript \(r_{2}\). The index \(G_{1}\) and \(L_{1}\) stand for reference satellites of each system, while the non-reference satellites are denoted using \(G_{2}\) and \(L_{2}\), respectively. The remaining symbols can refer to the meanings introduced in (1) and (2), but applicable for DD scheme here. Furthermore, it is worthy of indicating that all DD variables are expressed in meters, except the ambiguity which is expressed in cycles. And the label \(s\) is used to replace \(G\) and \(L\) for the sake of simply expression in (5). Usually in the medium-to-long baselines, the relative slant ionospheric delays can be estimated as a parameter at each measurement epoch. For the tropospheric delay, the dry component is precisely corrected by a priori model, the wet component is estimated through setting up a zenith tropospheric delay (ZTD). Then, the linear observation equations of (3) and (4) can be expressed as:  
\[\left\{