## 2 Materials and Methods
### Construction of PCC Measurements with Triple-Difference
The LLI flag would be set if the receiver suffered from a loss of lock. The second method we adopted was the high-order phase differencing method; see [25]. The third method was to detect cycle slips by evaluating the triple-differencing residuals, which were obtained by differencing the double-differenced residuals at two adjacent epochs; if the time-differencing interval was shorter than one minute, the triple-differencing residual was normally at only the millimeter level [28; 29]. Other cycle-slip detection methods could also be incorporated to achieve a more comprehensive detection. In Equation (6), \(r_{12}^{sr}\Big{(}t,t-T_{diff}\Big{)}\) is a combination of geodetic distances between receiver and satellite antennas:  
\[ \begin{split} r_{12}^{sr}\Big{(}t,t-T_{diff}\Big{)}& =\big{(}r_{1}^{s}(t)-r_{2}^{s}(t)\big{)}-\Big{(}r_{1}^{s}\Big{(}t- T_{diff}\Big{)}-r_{2}^{s}\Big{(}t-T_{diff}\Big{)}\Big{)}\\ &-\big{(}r_{1}^{r}(t)-r_{2}^{r}(t)\big{)}+\Big{(}r_{1}^{r}\Big{(} t-T_{diff}\Big{)}-r_{2}^{r}\Big{(}t-T_{diff}\Big{)}\Big{)}\end{split}  \tag{7}\]  
Each component can be calculated with the receiver positions, which were precisely predetermined, and satellite positions, which were calculated with GNSS orbit products provided by the Multi-GNSS Experiment (MGEX) of the International GNSS Service (IGS) [30]. The orbit products, also known as precise ephemeris, are available at the IGS website in the Standard Product 3 (SP3) format with a sampling of five minutes [31]. The precise ephemeris can be interpolated to derive the position of satellites at signal emitting time; see [32; 33]. The accuracy of the orbit products is typically within a few centimeters, which can help improve the precision of the desired triple-differenced PCC measurements in Equation (6). In the process of constructing the triple-differenced PCC measurements, we needed to ensure that each observation was used only once. And to better recover PCCs from the triple-differenced PCC measurements, the reference satellite needed to be properly selected. The further the distance between two points on the antenna hemisphere, the more likely it is that the difference between the corresponding PCCs will be greater. Therefore, among \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Size of Phase Bias (cm)** & **0** & **0-1** & **0-2** & **0-3** & **0-4** & **0-5** & **0-6** \\ \hline Single-frequency, single-epoch & 41.0 & 11.7 & 1.7 & 0.9 & 0.8 & 0.8 & 0.8 \\ Single-frequency, multi-epoch & 79.2 & 79.0 & 78.0 & 75.4 & 73.4 & 71.9 & 67.8 \\ \hline \end{tabular}
\end{table} 
Table 1: Example of ambiguity resolution success rate (%) under different sizes of phase biases at a baseline of around 4 m. The sizes of the phase biases were random values within the specified intervals. The data of the multi-epoch case were derived using a Kalman filter-based method. all satellites that have been continuously locked for \(T_{diff}\) epochs, it makes sense to set the reference satellite to the one with largest angular distance between its line-of-sight (LOS) vectors at epoch \(t\) and epoch \(t-T_{diff}\). The angular distance \(\Delta\theta^{s}\) can be calculated by an inner product operation:  
\[\Delta\theta^{s}=e^{s}(t)\bullet e^{s}\Big{(}t-T_{diff}\Big{)} \tag{8}\]  
where \(e^{s}\) denotes the unit LOS vector of satellite \(s\), expressed as  
\[e^{s}=\Big{[} \begin{smallmatrix}x^{s}-x_{1}\\ r_{1}^{s}\end{smallmatrix} \quad\frac{y^{s}-y_{1}}{r_{1}^{s}}\quad\frac{z^{s}- z_{1}}{r_{1}^{s}}\Big{]} \tag{9}\]  
where \( \begin{bmatrix}x^{s}&y^{s}&z^{s}\end{bmatrix} \) denotes the satellite position derived from the precise ephemeris, and \( \begin{bmatrix}x_{1}&y_{1}&z_{1}\end{bmatrix} \) denotes the receiver antenna position that is measured in advance and keeps fixed during the data collection process.