## 5 Conclusions  
We specifically designed a relative field antenna calibration method for low-cost GNSS antennas, whose PCC can be up to several centimeters. By employing the triple-differencing operation, we constructed a set of PCC measurements, which could be considered as a combination of relative PCC values at four different directions. The carrier-phase ambiguities were eliminated by the time-differencing operations. The time-differencing interval was set to a relatively long period such as one hour. The reference satellite in the between-satellite difference was selected according to the angular distance it passed over during the time-differencing interval. As for the phase pattern fitting process, we presented a weight setting method based on the area of a spherical quadrilateral, which was enclosed by the terminal points of four unit LOS vectors, the PCCs on whose directions were the components of the corresponding PCC measurements. The appropriate data collection duration was discussed with respect to GPS and BDS. The performance of the proposed method was evaluated with real GPS and BDS observations and eight simulated phase patterns, indicating that a calibration result with a mm-level RMSE could be obtained. The impact of cycle slip and elevation mask angle was also discussed, showing that a longer data collection duration and an appropriate elevation mask angle could be set under challenging conditions. Since the data collected from the low-cost GNSS antennas are more complicated than those collected from the high-grade antennas, such as lower signal strength and more cycle slips, an essential future work is to conduct field experiments with a variety of low-cost GNSS antennas in order to sufficiently validate the proposed antenna calibration method. Despite this vacuum, we hope that this work can still bring some insight to the antenna calibrations for low-cost antennas. Moreover, our proposed weight setting method only concerned the scatter of the unit LOS vectors; a conceivable direction for future work is to incorporate the elevation angle and the SNR into the weighting. Conceptualization and methodology, [PERSON], [PERSON] and T.H. ; software, data curation, and validation, [PERSON] and W.G. ; writing--original draft preparation, [PERSON] and T.H. ; writing--review and editing, [PERSON], T.H., [PERSON] and H.M.; supervision, [PERSON] and [PERSON]; funding acquisition, [PERSON] and X.S. All authors have read and agreed to the published version of the manuscript. This research was funded by the Young Elite Scientists Sponsorship Program by CAST, grant number 2022 QNRC001, and the National Natural Science Foundation for Young Scientists of China, grant number 62201028. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Elevation Mask Angle** & **a** & **b** & **C** & **d** & **e** & **f** & **g** & **h** \\ \hline \(0^{\circ}\) & 4.191 & 7.218 & 7.500 & 7.402 & 7.365 & 7.226 & 7.956 & 9.211 \\ \(10^{\circ}\) & 3.385 & 4.253 & 4.428 & 4.322 & 4.373 & 4.256 & 4.467 & 4.937 \\ \(20^{\circ}\) & 2.894 & 3.864 & 3.877 & 3.983 & 3.973 & 3.868 & 4.185 & 4.479 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RMSEs (mm) of GPS antenna calibration results with different elevation mask angle settings. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Elevation Mask Angle** & **a** & **b** & **c** & **d** & **e** & **f** & **g** & **h** \\ \hline \(0^{\circ}\) & 3.547 & 3.149 & 3.566 & 3.174 & 3.084 & 3.149 & 3.587 & 3.767 \\ \(10^{\circ}\) & 3.348 & 2.888 & 3.110 & 2.881 & 2.807 & 2.888 & 3.122 & 3.433 \\ \(20^{\circ}\) & 3.114 & 2.531 & 2.707 & 2.477 & 2.440 & 2.531 & 2.601 & 3.056 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: RMSEs (mm) of BDS antenna calibration results with different elevation mask angle settings.