# 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
The standard deviation of the estimated ambiguities should be  
\[\
abla\Delta P_{(m,n,l)}=(m+n+l)(\
abla\Delta\rho+\
abla\Delta T)+\beta_{(m,n,l )}\frac{\
abla\Delta K}{f_{1}^{2}}+\
abla\Delta\varepsilon_{P_{(m,n,l)}} \tag{12}\]2  
\[\
abla\Delta P_{(m,n,l)}=(m+n+l)(\
abla\Delta\rho+\
abla\Delta T)+\beta_{(m,n,l )}\frac{\
abla\Delta K}{f_{1}^{2}}+\
abla\Delta\varepsilon_{P_{(m,n,l)}} \tag{12}\]3  
Assuming that the triple-frequency code and carrier phase observations have the same standard deviations (STD),  
\[\
abla\Delta P_{(m,n,l)}=(m+n+l)(\
abla\Delta\rho+\
abla\Delta T)+\beta_{(m,n,l )}\frac{\
abla\Delta K}{f_{1}^{2}}+\
abla\Delta\varepsilon_{P_{(m,n,l)}} \tag{12}\]4  
\[\
abla\Delta P_{(m,n,l)}=(m+n+l)(\
abla\Delta\rho+\
abla\Delta T)+\beta_{(m,n,l )}\frac{\
abla\Delta K}{f_{1}^{2}}+\
abla\Delta\varepsilon_{P_{(m,n,l)}} \tag{12}\]5  
Under the circumstances of a short-range baseline, we can set the DD ionospheric error (\(\Delta\
abla I\) in Equation (23) to be 10 cm [30]. According to the error propagation, the standard deviation of the estimated ambiguity is 0.148 cycle and 0.172 cycle, respectively. Consequently, relatively reliable ambiguities of the extra-wide lanes can be fixed using the rounding method in a single epoch. After the AR of the extra-wide lane, we can use \(\
abla\Delta\phi_{(0,1,-1)}\) and \(\
abla\Delta\phi_{(1,0,0)}\) to fix the ambiguity \(\
abla\Delta N_{(1,0,0)}\). However, this combination amplifies the observation noise and the ionospheric delay residual, which will significantly affect the ambiguity resolution. To overcome these issues, we chose \(\
abla\Delta\phi_{(1,-1,0)}\); its ambiguity can be calculated as \(\
abla\Delta\mathrm{N}_{(1,-1,0)}=-5\
abla\Delta\mathrm{N}_{(0,1,-1)}+\
abla \Delta\mathrm{N}_{(1,4,-5)}\). Thus, we have  
\[\left\{ \begin{array}{c}\
abla\Delta\phi_{(1,-1,0)}=\Delta\
abla\rho+\Delta \
abla\delta_{trop}-\beta_{(1,-1,0)}\frac{\Delta\
abla K}{f_{1}^{2}}-\lambda_{ (1,-1,0)}\Delta\
abla N_{(1,-1,0)}+\Delta\
abla\varepsilon\phi_{(1,-1,0)}\\ \
abla\Delta\phi_{(1,0,0)}=\Delta\
abla\rho+\Delta\
abla\delta_{trop}-\beta_{( 1,0,0)}\frac{\Delta\
abla K}{f_{1}^{2}}-\lambda_{(1,0,0)}\Delta\
abla N_{(1,0, 0)}+\Delta\
abla\varepsilon\phi_{(1,0,0)}\end{array} \right. \tag{26}\]Then, the ambiguity can be expressed as  
\[\
abla\Delta N_{(1,0,0)}=\frac{\
abla\Delta\phi_{(1,-1,0)}-\
abla\Delta\phi_{(1,0,0)}+\lambda_{(1,-1,0)}\
abla\Delta N_{(1,-1,0)}}{\lambda_{(1,0,0)}} \tag{27}\]  
where \(\
abla\Delta\phi_{(1,-1,0)}\) and \(\
abla\Delta\phi_{(1,0,0)}\) are the DD carrier phase observation of the two wide-lane combinations in metres, \(\Delta\
abla\delta_{trop}\) is the DD tropospheric delay, \(\frac{\Delta\
abla K}{f_{2}^{2}}\) is the ionospheric delay, and \(\beta\), \(\lambda\), \(\Delta\
abla N\), and \(\Delta\
abla\varepsilon\) are the mapping function, wavelength, DD ambiguity and the observation noise of the two wide-lane combinations, respectively. In this combination, the standard deviation of the estimated ambiguity is only 0.209 cycle. Thus, the success rate of the AR of \(\
abla\Delta\phi_{(1,0,0)}\) is also very high in theory.