## 2 Methodology
### Traditional Observation Model for Medium-Long Baseline RTK  
The uncombined DD observation model is adopted for medium-long baseline RTK in this study. The basic DD pseudo-range and carrier phase equations are as follows:  
\[P_{rb,k}^{ij}=\rho_{rb}^{ij}+\mu_{k}I_{rb}^{ij}+(\alpha_{r}^{j}-\alpha_{r}^{i})T _{r}-(\alpha_{b}^{j}-\alpha_{b}^{i})T_{b}+\varepsilon_{P,k}, \tag{1}\]  
\[L_{rb,k}^{ij}=\rho_{rb}^{ij}-\mu_{k}I_{rb}^{ij}+(\alpha_{r}^{j}-\alpha_{r}^{i}) T_{r}-(\alpha_{b}^{j}-\alpha_{b}^{i})T_{b}+\lambda_{k}N_{rb,k}^{ij}+ \varepsilon_{L,k}, \tag{2}\]  
where \(L\) and \(P\) represent the DD pseudo-range and carrier phase observations in units of length, respectively; \(\rho\) is the geometric range between the satellite and the receiver antenna phase center; \(T\) denotes the zenith troposphere delay that can be converted into the slant delay with the mapping coefficient \(\alpha\); \(\lambda\) is the wavelength; \(k\) is the signal frequency number; \(N\) is the integer ambiguity; \(\mu\) is the coefficient that can convert the ionosphere delay on frequency 1 into frequency \(k\) with \(\mu_{k}=f_{1}^{2}/f_{k}^{2}\); \(\varepsilon_{P,k}\) and \(\varepsilon_{L,k}\) represent the observation noise of the pseudo-range and the carrier phase, respectively; \(r\) and \(b\) denote the rover and base station, respectively; \(j\) is the non-reference satellite; and \(i\) is the reference satellite.  
For medium-long baseline, the DD residual wet tropospheric delay can be approximately represented by a function of relative ZTD (RZTD) and a mapping function [2,34]. This means that the troposphere parameters \(T_{r}\) and \(T_{b}\) in (1) and (2) can be combined into a single parameter as follows:  
\[(\alpha_{r}^{j}-\alpha_{r}^{i})T_{r}-(\alpha_{b}^{j}-\alpha_{b}^{i})T_{b} \cong\alpha^{ij}\cdot T_{rztd}, \tag{3}\]  
where \(\alpha^{ij}\) is the coefficient of RZTD \(T_{rztd}\) and \(\alpha^{ij}=\alpha^{j}-\alpha^{i}\), the mapping coefficient \(\alpha\) is the function of satellite elevation angle, and the elevation angle parameters for \(\alpha^{i}\) and \(\alpha^{j}\) are the average elevation angles of two receivers for satellites \(i\) and \(j\), respectively [34]. The global mapping function (GMF) [35] is used in this study.  
The RTK observation equations--in which the baseline components, the ambiguities, and the slant ionosphere and zenith troposphere delays are unknown parameters--are given as follows:  
\[E\left[\begin{array}{c}P\\ L\end{array}\right]=\left[\begin{array}{ccc}A&\Lambda&B&0\\ A&-\Lambda&B&C\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right], \tag{4}\]  
where \(E[\cdot]\) denotes the expectation operator; \(P\) and \(L\) denote the DD pseudo-range and the carrier phase vector, respectively; \(a\) is the baseline vector and the corresponding coefficient matrix \(A\) contains the line-of-sight unit vectors between the two receivers; \(b\) denotes the DD ionosphere delay vector and the corresponding matrix \(\Lambda\) contains the coefficient \(\mu_{k}\) for all frequencies; \(c\) is the RZTD parameter and the corresponding vector \(B\) contains mapping coefficients between two receivers; and \(d\) is the DD ambiguity vector and the corresponding coefficient matrix \(C\) contains the wavelength vectors for carrier phase observations. For multi-constellation signals, we apply system-specific DD and thus have one pivot satellite per system [14]; the baseline vector and RZTD parameters are common, and the ionosphere and ambiguity parameters are specific to the respective system.