## 2 Medium Baseline RTK Algorithm
### Triple-Frequency Ambiguity Resolution  
As shown in Equation (1), the raw ambiguity parameter is related to the slant ionosphere delay parameter and requires a long time to be resolved. To address this, the ambiguity of EWL and WL is first fixed through combined observations of EWL and WL. Using geometry-free and ionosphere-free (GFIF) combination, the float EWL ambiguity can be  
\[\
abla\Delta\hat{N}_{(0,1,-1)}=\frac{\
abla\Delta L_{(0,1,-1)}-\
abla\Delta P_{(0,1,1)}}{\lambda_{(0,1,-1)}} \tag{6}\]  
where \(\
abla\Delta\hat{N}_{(0,1,-1)}\) is the float EWL ambiguity. Due to the long wavelength, the float EWL ambiguity is fixed by rounding to the nearest integer. After the EWL ambiguity fixed, the EWL phase observation equation can be given as  
\[\
abla\Delta L_{(0,1,-1)}-\lambda_{(0,1,-1)}\
abla\Delta N_{(0,1,-1)}=\
abla \Delta p_{rb}^{st}+\varepsilon_{rb,(0,1,-1)}^{st} \tag{7}\]  
As shown in Equation (7), the ambiguity-fixed EWL phase observation can be regarded as a high-precision code observation, thus accelerating the WL AR. This is also the main reason why the triple-frequency RTK converges faster than the traditional dual-frequency RTK. Here, the DD ionosphere delay and troposphere delay residuals are ignored due to their small magnitude [19; 20]. Due to the shorter wavelength compared to the EWL ambiguity, it is prone to wrongly fix when directly rounding WL ambiguity. Thus, the geometry-based approach is adopted:  
\[ \begin{cases}\
abla\Delta L_{(1,-1,0)}=\
abla\Delta p_{rb}^{st}+\lambda_{(1,-1,0)}\
abla\Delta\hat{N}_{(1,-1,0)}+\varepsilon_{rb,(1,-1,0)}^{st}\\ \
abla\Delta L_{(0,1,-1)}-\lambda_{(0,1,-1)}\
abla\Delta N_{(0,1,-1)}=\
abla \Delta p_{rb}^{st}+\varepsilon_{rb,(0,1,-1)}^{st}\end{cases}  \tag{8}\]  
As shown in Equation (8), the ambiguity-fixed EWL phase observation is used for fast WL AR. It is worth noting that, for traditional dual-frequency RTK, in Equation (8), there are no high-precision ambiguity-fixed EWL phase observations available, which are replaced by \(\
abla\Delta P_{1}\) and \(\
abla\Delta P_{2}\), as shown in [17], resulting in slow convergence. Also, the DD ionosphere delay and troposphere delay residuals are ignored; the rationale for this is explained in the next section. When obtaining the float WL ambiguity \(\
abla\Delta\hat{N}_{(1,-1,0)}\), the least-square ambiguity decorrelation adjustment (LAMBDA) is used for AR [21]. After resolving WL ambiguity, the ambiguity-fixed WL phase observation can be regarded as a code observation again, but with significantly reduced noise compared to the raw code observation. Then, the float N1/N2/N3 ambiguities can be solved as follows:  
\[ \begin{bmatrix}V_{\
abla\Delta P_{1}}\\ V_{\
abla\Delta P_{2}}\\ V_{\
abla\Delta P_{3}}\\ V_{\
abla\Delta L_{1}}\\ V_{\
abla\Delta L_{2}}\\ V_{\
abla\Delta L_{3}}\\ V_{\
abla\Delta L_{(1,-1,0)}}\end{bmatrix} = \begin{bmatrix}\mathbf{e}&m^{st}&\alpha_ {1}&0&0&0\\ \mathbf{e}&m^{st}&\alpha_{2}&0&0&0\\ \mathbf{e}&m^{st}&\alpha_{3}&0&0&0\\ \mathbf{e}&m^{st}&-\alpha_{1}&1&0&0\\ \mathbf{e}&m^{st}&-\alpha_{2}&0&1&0\\ \mathbf{e}&m^{st}&-\alpha_{3}&0&0&1\\ \mathbf{e}&m^{st}&\frac{f_{1}}{f_{2}}&0&0&0\end{bmatrix}\begin{bmatrix}\mathbf{X} \\ \
abla\Delta P_{1}-\
abla\Delta P_{1}^{0}\\ \
abla\Delta P_{2}-\
abla\Delta P_{2}^{2}\\ \
abla\Delta P_{3}-\
abla\Delta P_{3}^{0}\\ \
abla\Delta L_{1}-\
abla\Delta L_{1}^{0}\\ \Delta L_{2}-\
abla\Delta L_{2}^{2}\\ \
abla\Delta L_{3}-\
abla\Delta L_{3}^{0}\\ \
abla\Delta L_{(1,-1,0)}-\lambda_{(1,-1,0)}\
abla\Delta N_{(1,-1,0)}-\
abla \Delta L_{(1,-1,0)}^{0}\end{bmatrix}  \tag{9}\]  
where \(V\) refers to the code or phase noise; \(\mathbf{e}\) is the geometry vector; \(\mathbf{X}\) is the baseline coordinate vector; and \(\
abla\Delta P_{i}^{0}\) and \(\
abla\Delta L_{i}^{0}\) denote the computed code and phase observation, respectively. Therefore, the last column of Equation (9) represents the OMC (observed minus computed) vector.