# A New GNSS Single-Epoch Ambiguity Resolution Method Based on Triple-Frequency Signals
## 4 Experiments and Discussion
### Ambiguity Resolution of BDS Triple-Frequency Signals  
Due to the large wavelength, we can fix the ambiguity of the extra-wide-lane combination easily. In this study, the ambiguity of the BDS DD carrier phase observation is fixed based on Equation (25). Generally, the estimated ambiguity is more reliable when the difference between the float ambiguity and the integer is smaller. Figure 3 shows the distribution of the difference (DF) between float ambiguity and the integer of Geostationary Earth Orbit (GEO), Inclined Geosynchronous Satellite Orbit (IGSO) and Medium Earth Orbit (MEO) satellite observation. From this figure, we can see that more than 98% of the DF of the GEO satellites floats in the range of \(-0.2\) to \(-0.1\) cycles. Consequently,  
Figure 2: Number of visible satellites of BDS/GPS/GLONASS.  
using the rounding method to fix the ambiguity of GEO observation is reliable enough. For IGSO and MEO satellites, the DF mainly varies around \(\pm 0.3\). However, some DF reach approximately 0.5 cycles. On this occasion, using the rounding method will result in mis-fixing. The reliability of the ambiguity resolution will be relatively low.  
Generally, the difference between the float ambiguity and the integer is decided by the ionospheric delay and the noise of the DD carrier phase observation. This difference of GEOs is smaller than that of IGSOs, and MEOs mean that the noise of the GEO carrier phase observation is smaller than that of the other kinds of satellites. Similar results are also shown in the paper [32], revealing that the amplitudes of the ionosphere-free and geometry-free time series of GEOs is smaller than that of IGSOs' and MEOs' carrier phase measurement.  
To verify the ambiguity at each epoch, we put the ambiguity into the DD ionosphere-free combination and simplify it as  
\[V=A\hat{X}-L \tag{34}\]  
The unknown parameter \(\hat{X}\) here includes the correction of the unknown coordinates \(\delta x\), \(\delta y\), \(\delta z\), the zenith tropospheric delay, and the ambiguity. Based on the least-square estimation, we can measure the accuracy of the unknown parameter with the mean square error \(\sigma\) (Equation (34)).  
\[\sigma=\pm\sqrt{\frac{[vv]}{n-t}} \tag{35}\]  
In this equation, \(n\) represents the number of observations at each epoch, and \(t\) is the necessary number of observations. Figure 4 shows the variation of the absolute value of the mean square error at each epoch. The measurement noise of the ionosphere-free combination contributes most of the error  
Figure 3: Variation of distribution of the difference (DF) of three kinds of BDS satellites. GEO = Geostationary Earth Orbit; IGSO = Inclined Geosynchronous Satellite Orbit; MEO = Medium Earth Orbit. (**a**) GEO; (**b**) IGSO; (**c**) MEO.  
in Equation (34). In general, the carrier-phase measurement noise is \(\pm\)2 mm. According to the law of error propagation, the measurement noise of the ionosphere-free combination should be \(\pm\)1.2 mm, and the limit error should be 3.8 mm (3 times sigma). From Figure 4, we can see that the mean square error of some epochs is beyond 10 mm and the maximum reaches 53 mm. It is obvious that the integer AR at these epochs is wrong. The main reasons are that (1) the mis-fixing in the AR of the extra-wide lane and (2) the measurement noise and the ionospheric delay residual affect the AR of the BDS carrier phase observation.