## 4 Ambiguity Resolution  
Ambiguity resolution is a crucial step in achieving GNSS data processing with phase observations. Successfully fixing phase ambiguities to integers in double-difference data processing can be challenging due to observation noise [23]. Double-difference ambiguities are generally categorized into wide-lane (WL) ambiguity and narrow-lane (NL) ambiguity. These two types of ambiguity need to be resolved sequentially. WL ambiguities are typically resolved using the Hatch-Melbourne-Wubbena (HMW) combination, followed by the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Multipath**} & \multicolumn{2}{c}{**B1I**} & \multicolumn{2}{c}{**B1C**} & \multicolumn{2}{c}{**B2a**} & \multicolumn{2}{c}{**B3I**} \\ \cline{2-9}  & **Mean** & **STD** & **Mean** & **STD** & **Mean** & **STD** & **Mean** & **STD** \\ \hline MET3 & 0 & 0.309 & 0 & 0.262 & 0 & 0.197 & 0 & 0.174 \\ MCHL & 0 & 0.279 & 0 & 0.266 & 0 & 0.165 & 0 & 0.164 \\ HOFN & 0 & 0.232 & 0 & 0.278 & 0 & 0.216 & 0 & 0.186 \\ VNDP & 0.007 & 0.230 & 0.003 & 0.231 & \(-\)0.002 & 0.186 & \(-\)0.001 & 0.166 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistics of multipath for BDS-3 different frequencies from four stations with different receiver types. Figure 3: The multipath from different BDS-3 frequencies for four stations with different receiver types. calculation of NL ambiguities using ionosphere-free and WL ambiguities. The formulas for calculating WL and NL ambiguities are presented below [23; 41]:  
\[ \begin{cases}\lambda_{WL}N_{WL}=\left(\frac{f_{i}}{f_{i}-f_{j}}\varphi_{i}-\frac{ f_{j}}{f_{i}-f_{j}}\varphi_{j}\right)-\left(\frac{f_{i}}{f_{i}+f_{j}}P_{i}-\frac{ f_{j}}{f_{i}+f_{j}}P_{j}\right)\\ \lambda_{NL}N_{NL}=N_{IF}-\frac{\lambda_{NL}}{f_{i}f_{j}-1}N_{WL}\end{cases}  \tag{2}\]  
\[ \begin{cases}\lambda_{WL}=\frac{\lambda_{i}\lambda_{j}}{\lambda_{j}-\lambda_{ i}}\\ \lambda_{NL}=\frac{\lambda_{i}\lambda_{j}}{\lambda_{i}+\lambda_{j}}\end{cases}  \tag{3}\]  
where \(f_{i}\) and \(f_{j}\) represent the two frequencies we selected for ionosphere-free combination. \(\varphi_{i}\) and \(\varphi_{j}\) are the phase observations while \(P_{i}\) and \(P_{j}\) are the code observations of the two frequencies. \(\lambda_{WL}\) and \(\lambda_{NL}\) represent the wavelength of WL and NL, which can be calculated by the wavelength \(\lambda_{i}\) and \(\lambda_{j}\). \(N_{WL}\) denote the WL ambiguity and \(N_{NL}\) denote the NL ambiguity. \(N_{IF}\) is the ionosphere-free combination ambiguity. When \(N_{WL}\) and \(N_{NL}\) are fixed to integers, the double-difference ambiguity is resolved successfully. Figure 4 presents the percentage of successfully resolved WL ambiguity and NL ambiguity using BDS-3 B1I/B3I and B1C/B2a combination. The figure indicates that the B1C/B2a achieves a higher percentage of ambiguity fixing compared with the B1I/B3I combination. In detail, the average percentage of fixed WL ambiguities is 89.1% for the B1I/B3I combination and increases to 93.8% for the B1C/B2a solutions. In the case of NL ambiguities, the mean percentage of fixed ambiguities is 68.8% for B1I/B3I and is notably higher at 79.1% for the B1C/B2a combination. Considering all the fixed ambiguities, including both WL and NL, the mean percentage is 78.9% for the B1I/B3I combination, while the B1C/B2a combination demonstrates a 7.6% improvement with a mean percentage of fixed ambiguities. Figure 4: Percentage of successfully fixed ambiguities for BDS-3 by using B1I/B3I and B1C/B2a combination for all days. In addition, we conducted a comparison of the residual distributions for WL ambiguity and NL ambiguity between B1I/B3I and B1C/B2a combinations, as illustrated in Figure 5. The black lines in Figure 5 denote the probability density function (PDF) of the residuals, which follows the residual distribution. The plots indicate that the PDFs of WL and NL for both B1I/B3I and B1C/B2a combinations closely adhere to normal distribution.