## 2 Methodology
### Model of RTK Measurement  
There are a number of differential GNSS (DGNSS) positioning methods available, but RTK positioning is one of the most popular. Operating in a situation with an open sky can achieve an accuracy of centimeters. In order to mitigate errors along the signal path, a base station's aiding information can be utilized. Thus, it is possible to make double-difference measurements of both the pseudorange and carrier phase with this method. Our study has led to a standardization framework for the multi-constellation RTK positioning based on the undifferenced pseudorange and carrier-phase models, which can be summarized as follows:  
\[\left\{ \begin{array}{l}P=\rho+c(dt_{r}-dt^{s})+Iono+Trop+\varepsilon_{p}\\ \lambda\Phi=\rho+c(dt_{r}-dt^{s})+\lambda N-Iono+Trop+\varepsilon_{\Phi}\end{array} \right. \tag{1}\]  
where \(P\) and \(\Phi\) represent the pseudorange and carrier-phase measurements, respectively. The variable \(\rho\) is the geometric distance between the satellite and ground receiver, and the variables \(dt^{s}\) and \(dt_{r}\) are the offsets of the satellite and receiver clocks, respectively;\(\lambda\) and \(N\) relate to the wavelengths and ambiguities in carrier-phase measurements, respectively. The _Iono_ and _Trop_ terms also indicate the delays caused by the ionosphere and troposphere, respectively. The terms \(\varepsilon_{P}\) and \(\varepsilon_{\varTheta}\) encompass the measurement noise and other unmodeled errors. The double-differenced measurement model can be obtained using the following method between the base station (\(b\)) and the rover station (\(r\)):  
\[\left\{ \begin{array}{l}\
abla\Delta P_{r,b}^{S_{i}S_{j}}=\
abla\Delta\rho_{r,b}^{S_{i}S_{j}}+\
abla\Delta_{\varPhi,b}^{S_{i}S_{j}}\\ \lambda\
abla\Delta\Phi_{r,b}^{S_{i}S_{j}}=\
abla\Delta\rho_{r,b}^{S_{i}S_{j}}+ \lambda\
abla\Delta\Phi_{r,b}^{S_{i}S_{j}}+\
abla\Delta_{\varepsilon_{\varPhi r,b }^{S_{i}S_{j}}}\end{array} \right. \tag{2}\]  
where \(\left\{\
abla\Delta(.)_{r,b}^{S_{i}S_{j}}=\left((.)_{r}^{S_{i}}-(. )_{b}^{S_{i} }\right)\ -\ \left((.)_{r}^{S_{j}}-(. )_{b}^{S_{i}}\right)\right\}\) denotes the double-differenced operator. It is possible to effectively eliminate clock offsets, \(dt^{s}\), and \(dt_{r}\), by using the double-differenced operations described in Equation (2). It is also possible to ignore the remaining double-differenced ionospheric and tropospheric delays, _Iono_ and _Trop_, if the short baseline is less than 5 km [34]. This simplification assumes that, over short distances, the atmospheric conditions experienced by both the base station and the rover are similar enough that their differential effect can be neglected. This is a common assumption in RTK positioning to reduce the complexity and focus on the primary sources of error that can be more significant over short distances. GNSS receivers come equipped with phase-lock loops (PLLs), but these PLLs have the following limitation: they cannot directly establish values for the carrier-phase measurements, a parameter known as ambiguity \(N\). Consequently, this leads to the uncertain measurements of the carrier phase, introducing ambiguity. For precise carrier-phase measurements, it is most important to accurately address and resolve this ambiguity [35]. Achieving a high level of accuracy in RTK positioning depends on effectively dealing with this ambiguity. A technique called least-squares ambiguity decorrelation adjustment (LAMBDA) [36] has been used to solve this problem. This method adopts an integer least-squares approach to estimate the ambiguities. This technique optimizes the subsequent objective function using the least-squares methodology, as shown below:  
\[f(\
abla\Delta N)=min\Big{\{}\big{(}\
abla\Delta\hat{N}-\
abla\Delta N\big{)}^ {T}\!P_{\
abla\Delta\hat{N}}\big{(}\
abla\Delta\hat{N}-\
abla\Delta N\big{)} \Big{\}} \tag{3}\]  
Here, \(\
abla\Delta\hat{N}\) and \(P_{\
abla\Delta\hat{N}}\) represent the real-valued estimates for the ambiguity and its corresponding covariance matrix, respectively.