# A New GNSS Single-Epoch Ambiguity Resolution Method Based on Triple-Frequency Signals
## 3 GNSS Single-Epoch Ambiguity Resolution
### Ambiguity Resolution of BDS Triple-Frequency Signals  
Supposing that there are three carrier phase frequencies f1, f2, f3, the general form of linear observation equations for the GNSS pseudo-range and carrier phase observation combination can be expressed as Equations (11) and (12) (unit: metre). Although most of the atmospheric effects in the code and carrier phase observation can be cancelled by DD, the linear combination may increase these effects. Thus, the tropospheric delay and the ionospheric delay still exist in the observation combination equation below. \[\
abla\Delta\phi_{(i,j,k)}=\
abla\Delta\rho+\
abla\Delta T-\beta_{(i,j,k)} \frac{\
abla\Delta K}{f_{1}^{2}}-\lambda_{(i,j,k)}\
abla\Delta N_{(i,j,k)}+ \
abla\Delta\varepsilon_{\phi_{(i,j,k)}} \tag{11}\]  
\[\
abla\Delta P_{(m,n,l)}=(m+n+l)(\
abla\Delta\rho+\
abla\Delta T)+\beta_{(m,n,l )}\frac{\
abla\Delta K}{f_{1}^{2}}+\
abla\Delta\varepsilon_{P_{(m,n,l)}} \tag{12}\]  
where  
\[\
abla\Delta\phi_{(i,j,k)}=\frac{i\cdot f_{1}\cdot\
abla\Delta\phi_{1}+j\cdot f _{2}\cdot\
abla\Delta\phi_{2}+k\cdot f_{3}\cdot\
abla\Delta\phi_{3}}{i\cdot f _{1}+j\cdot f_{2}+k\cdot f_{3}} \tag{13}\]  
\[\
abla\Delta P_{(m,n,l)}=m\
abla\Delta P_{1}+n\
abla\Delta P_{2}+l\
abla \Delta P_{3} \tag{14}\]  
where \(\
abla\Delta\phi_{(i,j,k)}\) and \(\
abla\Delta P_{(m,n,l)}\) are the DD carrier phase and pseudo-range observation combination in metres, respectively. \(i,j,k\) are the combination coefficients, which are integers. \(m,n,l\) are the corresponding combination coefficients, which are real numbers. The corresponding ambiguity, virtual frequency and wavelength of the combination is  
\[\
abla\Delta\text{N}_{(i,j,k)}=i\cdot\
abla\Delta N_{1}+j\cdot\
abla\Delta N_{ 2}+k\cdot\
abla\Delta N_{3} \tag{15}\]  
\[f_{(i,j,k)}=i\cdot f_{1}+j\cdot f_{2}+k\cdot f_{3} \tag{16}\]  
Figure 1: Structure of the new single-epoch Global Navigation Satellite Systems ambiguity resolution (GNSS AR) method for a short-range baseline based on triple-frequency signals. \[\lambda_{(i,j,k)}=\frac{c}{f_{(i,j,k)}}=\frac{\lambda_{1}\lambda_{2}\lambda_{3}}{i \cdot\lambda_{2}\lambda_{3}+j\cdot\lambda_{1}\lambda_{3}+k\cdot\lambda_{1}\lambda _{2}} \tag{17}\]  
\(c\) denotes the speed of light. \(\lambda_{i}\) and N\({}_{i}\) indicate the wavelength and ambiguities of each carrier phase observation. The combined ionospheric delay factor is  
\[\beta_{(i,j,k)}=\frac{f_{1}^{2}\left(\frac{i}{f_{1}}+\frac{j}{f_{2}}+\frac{k}{f_ {3}}\right)}{f_{(i,j,k)}} \tag{18}\]  
The DD combination observation noise is  
\[\
abla\Delta\epsilon_{\phi_{(i,j,k)}}=\frac{i\cdot f_{1}\
abla\Delta\epsilon_{ \phi_{1}}+j\cdot f_{2}\
abla\Delta\epsilon_{\phi_{2}}+k\cdot f_{3}\
abla \Delta\epsilon_{\phi_{3}}}{f_{(i,j,k)}} \tag{19}\]  
Due to the large wavelength, the ambiguity of the extra-wide lane of \(\
abla\Delta N_{(0,1,-1)}\), \(\
abla\Delta N_{(1,4,-5)}\) can be fixed using (take the extra-wide lane of BDS as an example)  
\[\
abla\Delta P_{(m,n,l)}=(m+n+l)(\
abla\Delta\rho+\
abla\Delta T)+\beta_{(m,n,l )}\frac{\
abla\Delta K}{f_{1}^{2}}+\
abla\Delta\varepsilon_{P_{(m,n,l)}} \tag{12}\]0  
\[\
abla\Delta P_{(m,n,l)}=(m+n+l)(\
abla\Delta\rho+\
abla\Delta T)+\beta_{(m,n,l )}\frac{\
abla\Delta K}{f_{1}^{2}}+\
abla\Delta\varepsilon_{P_{(m,n,l)}} \tag{12}\]1  
where \([]\) represents the rounding-off operator.