## 2 RTK Mathematic Model
### DF Data-Aided AR  
Defining \(N_{1}\) and \(N_{2}\) as the carrier ambiguities on frequencies \(f_{1}\) and \(f_{2}\), respectively, the \(\rho\) on different frequencies can be formed by \(\
abla\Delta\varphi\) as follows:  
\[\left\{\begin{array}{l}\
abla\Delta\rho_{1}=(\
abla\Delta\varphi_{1}+\
abla \Delta N_{1})\lambda_{1}-\frac{A}{f_{2}^{2}}\\ \
abla\Delta\rho_{2}=(\
abla\Delta\varphi_{2}+\
abla\Delta N_{2})\lambda_{2}- \frac{A}{f_{2}^{2}}\end{array}\right. \tag{5}\]  
where \(N\)e is the number of electrons in unit area \(A=40.3\int_{S}\)_Neds_. The wide-lane ambiguity \(\
abla\Delta N_{w}\) with wavelength \(\lambda_{w}\) can be expressed as:  
\[\
abla\Delta N_{w}=\
abla\Delta\varphi_{1}-\
abla\Delta\varphi_{2}-\frac{1}{ \lambda_{w}}\bigg{(}\
abla\Delta\rho-\
abla\Delta T-\frac{f_{1}}{f_{2}}\
abla \Delta I-\
abla\Delta\varepsilon\bigg{)} \tag{6}\]For the atmosphere errors in the above equation, \(\
abla\Delta I\) cannot be ignored for long baselines due to the spatial difference. \(\
abla\Delta T\) varies from 2 to 20 m depending on the satellite elevation, but a higher cutoff angle reduces data utilization [40]. Thus, the timeliness of the fixed solution could not be guaranteed, which is crucial for dynamic RTK once the satellites are available [41; 42; 43; 44]. The proposed DF data-aided AR method is summarized as follows:  
\[\left\{\begin{array}{l}\
abla\Delta N_{w}=(\
abla\Delta\varphi_{1}-\
abla \Delta\varphi_{2})-\frac{(f_{1}\
abla\Delta\varphi_{1}+f_{2}\
abla\Delta \varphi_{2})}{\lambda_{w}(f_{1}+f_{2})}+\
abla\Delta\varepsilon\\ \
abla\Delta N_{1}=\frac{1}{m\lambda_{1}-\lambda_{2}}[\
abla\Delta\varphi_{1F }-m\
abla\Delta\varphi_{1}+n\
abla\Delta\varphi_{2}-n\lambda_{2}\
abla\Delta N _{w}]\\ \
abla\Delta N_{2}=\
abla\Delta N_{1}-\
abla\Delta N_{w}\end{array}\right. \tag{7}\]  
where \(\
abla\Delta\varphi_{1\text{F}}\) is the ionosphere-free combination for carrier observations, \(m=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}\) and \(n=\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}\), and \(\
abla\Delta\varepsilon\) represents the noise. The above method introduces the code wide-lane combination to invert \(\
abla\Delta N_{1}\) and the derivation can be found in Appendices A and B. The proposed method shows the following merits: (1) The positioning accuracy and AR success rate are improved as the influence of \(\
abla\Delta I\) error is eliminated. (2) The method is applicable for medium and long baselines as the limitation of geometry distance \(\rho\) is eliminated. (3) The \(\
abla\Delta N\) on each frequency can be inverted directly from the high precise ionosphere-free and wide-lane measurements. Moreover, the following moving average with \(n\) epochs is also adopted to reduce the effect result from noise in \(\
abla\Delta P\)[45]:  
\[\
abla\Delta\hat{N}_{w}(n)=\frac{\sum\limits_{i=k}^{k+n}Z^{i}\
abla\Delta N_{w }^{i}}{\sum\limits_{i=k}^{k+n}Z^{i}} \tag{8}\]  
where \(k\) is the start epoch of the observation arc without cycle slip. \(Z^{i}\) is the weight of \(i\)th epoch. The \(\
abla\Delta\hat{N}_{w}(n)\) could be fixed to \(round\big{(}\
abla\Delta\hat{N}_{w}(n)\big{)}\) by the integer rounding method [46] if the differential residual of \(\
abla\Delta\hat{N}_{w}\) between adjacent epochs meets the following constraint:  
\[\big{|}\
abla\Delta\hat{N}_{w}(n)-\
abla\Delta\hat{N}_{w}(n-1)\big{|}<0.1\; \;\text{cycles}\]  
Then, the corresponding \(\
abla\Delta N_{1}\), \(\
abla\Delta N_{2}\) and precise carrier measurements could be formed by Equation (7). In the following test, the moving average window width \(n\) is set to be five epochs.