## 2 Materials and Methods
### Construction of PCC Measurements with Triple-Difference  
For low-cost antennas, the PCCs can be up to centimeters, severely reducing the success rate of integer ambiguity resolution, which is incorporated in the classical double-differencing approach to eliminate ambiguities. To address this problem, we introduced a triple-differencing procedure. Figure 1 shows eight carrier-phase observations used to construct a triple-differenced PCC measurement. They were measured from two antennas 1 and 2, at two epochs \(t\) and \(t-T_{diff}\), where \(T_{diff}\) denotes the time-differencing interval. Each carrier-phase measurement could be expressed similar to (2), in which only \(\phi\) was the GNSS observable that could be directly measured, and all other components except \(\xi\) needed to be eliminated as much as possible. The differencing operations between two antennas can eliminate most space-dependent errors like tropospheric and ionospheric delays; the satellite clock delay can also be eliminated. According to Equation (3), four relative PCC measurements can be constructed as  
\[ \begin{array}{l}\Xi^{s}(t)=\lambda\cdot\phi_{12}^{s}(t)-r_{12}^{s}(t)-c \delta t_{12}(t)-\lambda N_{12}^{s}(t)-\varepsilon_{12}^{s}(t)\\ \Xi^{r}(t)=\lambda\cdot\phi_{12}^{r}(t)-r_{12}^{r}(t)-c\delta t_{12}(t)-\lambda N _{12}^{r}(t)-\varepsilon_{12}^{r}(t)\\ \[PERSON]^{s}\Big{(}t-T_{diff}\Big{)}=\lambda\cdot\phi_{12}^{s}\Big{(}t-T_{diff} \Big{)}-r_{12}^{s}\Big{(}t-T_{diff}\Big{)}-c\delta t_{12}\Big{(}t-T_{diff} \Big{)}-\lambda N_{12}^{s}\Big{(}t-T_{diff}\Big{)}-\varepsilon_{12}^{s}\Big{(} t-T_{diff}\Big{)}\\ \[PERSON]^{r}\Big{(}t-T_{diff}\Big{)}=\lambda\cdot\phi_{12}^{r}\Big{(}t-T_{diff} \Big{)}-r_{12}^{r}\Big{(}t-T_{diff}\Big{)}-c\delta t_{12}\Big{(}t-T_{diff} \Big{)}-\lambda N_{12}^{r}\Big{(}t-T_{diff}\Big{)}-\varepsilon_{12}^{r}\Big{(} t-T_{diff}\Big{)}\end{array}  \tag{4}\]  
The receiver clock delay and other receiver-dependent errors can be further eliminated by between-satellite differencing operations. The four relative PCC measurements in Equation (4) can then be transformed to two between-antenna and between-satellite double-differenced PCC measurements, expressed as \[\begin{split}&\Xi^{sr}(t)=\lambda\cdot\phi_{12}^{sr}(t)-t_{12}^{sr}(t)- \lambda N_{12}^{sr}(t)-\epsilon_{12}^{sr}(t)\\ &\Xi^{sr}\Big{(}t-T_{diff}\Big{)}=\lambda\cdot\phi_{12}^{sr} \Big{(}t-T_{diff}\Big{)}-r_{12}^{sr}\Big{(}t-T_{diff}\Big{)}-\lambda N_{12}^{ sr}\Big{(}t-T_{diff}\Big{)}-\varepsilon_{12  
rate of ambiguity resolution can be affected in the centimeter-level phase bias and single-frequency case. From the last row of Table 1, we can also see that a multi-epoch method can achieve a relatively high ambiguity resolution success rate. However, since the size of the PCCs of low-cost antennas can be comparable with the carrier wavelength, PCCs might be absorbed in the ambiguities, resulting in incorrect solutions. In this work, we preferred to simply avoid the ambiguity resolution and directly eliminate the ambiguities with the time difference. Consequently, we must deal with the cycle slip before the time-differencing operations. Cycle slip is an unpredictable but frequently encountered carrier-phase error that occurs when the receiver loses track of the number of the carrier-phase cycles [22; 23; 24; 25]. There are two ways to cope with cycle slips: detection and correction. The cycle-slip correction is more complicated, time-consuming, and computationally intensive [26]. For triple-differenced measurements, even a small cycle slip can introduce a deviation comparable to wavelength, so we only detected and then rejected the observations with cycle slips. Since many low-cost antennas are single-frequency antennas, without loss of generality, we adopted three cycle-slip detection methods that only utilized single-frequency observations. Specifically, the first method we used was to detect cycle slips through the loss of lock indicator (LLI), which was provided by the receiver and available in the RINEX observation file [27] (p. 67).