## 2 Methodology
### TDCP Model  
GPS and BDS carrier phase measurements on frequency L1/B1 can be expressed as follows:  
\[\lambda^{j}\varphi^{j}=\rho^{j}+c(t_{G}-t^{j})-I^{j}+T^{j}+\lambda^{j}N^{j}+e^{j} \tag{1}\]  
\[\lambda^{k}\varphi^{k}=\rho^{k}+c(t_{C}-t^{k})-I^{k}+T^{k}+\lambda^{k}N^{k}+ \varepsilon^{k} \tag{2}\]  
where the superscript \(j\) stands for a GPS satellite and \(k\) for a BDS satellite, \(\varphi\) is the carrier phase measurement, is the carrier phase wavelength, \(\lambda\) is the geometric range, \(c\) is the speed of light in vacuum, \(t_{G}\) and \(t_{C}\) are the receiver clock biases for GPS and BDS, respectively, while \(t^{j}\) and \(t^{k}\) are the satellite clock biases, \(I\) and \(T\) are the slant ionospheric and tropospheric delay, respectively, \(N\) is the integer ambiguity, and \(\varepsilon\) includes multipath and receiver noise. The BDS receiver clock bias can be expressed as the sum of the GPS receiver clock bias and the inner-system bias (ISB) between GPS and BDS; thus:  
\[t_{C}=t_{G}+ISB_{C-G} \tag{3}\]  
where \(ISB_{C-G}\) is the ISB between GPS and BDS defined by their clock products. Substituting Equation (3) into Equation (2) gives:  
\[\lambda^{k}\varphi^{k}=\rho^{k}+c(t_{G}+ISB_{C-G}-t^{k})-I^{k}+T^{k}+\lambda^{ k}N^{k}+\varepsilon^{k} \tag{4}\]  
The TDCP method adopts the single differences in time of the one-way carrier phase measurements. The TDCP measurements are the time differences of successive carrier phases to the same satellite. According to Equations (1) and (4), the difference between carrier phase measurements at two successive epochs \(t_{m}\) and \(t_{m-1}\) is:  
\[\lambda^{j}\Delta\varphi^{j}_{t_{m-1},t_{m}}=\lambda^{j}(\Delta\varphi^{j}_{t _{m}}-\Delta\varphi^{j}_{t_{m-1}})=\Delta\rho^{j}+c(\Delta t_{G}-\Delta t^{j} )-\Delta I^{j}+\Delta T^{j}+\Delta\varepsilon^{j} \tag{5}\]  
\[\lambda^{k}\Delta\varphi^{k}_{t_{m-1},t_{m}}=\lambda^{k}(\Delta\varphi^{k}_{t _{m}}-\Delta\varphi^{k}_{t_{m-1}})=\Delta\rho^{k}+c(\Delta t_{G}+\Delta ISB_{C -G}-\Delta t^{k})-\Delta I^{k}+\Delta T^{k}+\Delta\varepsilon^{k} \tag{6}\]  
where \(\Delta\) represents the differencing operation. For example, \(\Delta\rho^{j}=\rho^{j}_{t_{m}}-\rho^{j}_{t_{m-1}}\) is the change in the geometric range between two epochs, and the other terms in Equations (5) and (6) are defined accordingly. The integer ambiguity is not included in Equations (5) and (6) as long as a cycle slip does not occur. To linearize Equations (5) and (6), the individual carrier phase measurements have to be corrected. Corrections for satellite clock errors are extracted from ephemeris data, while the ionospheric and tropospheric models are used to mitigate the influence of ionospheric and tropospheric effects, respectively. Therefore, satellite clock bias, ephemeris errors and slant ionospheric and tropospheric delay terms become time differences of residual errors and are negligible [21]. Therefore, the compensated TDCP measurement is:  
\[\lambda^{j}\Delta\tilde{\varphi}^{j}_{t_{m-1},t_{m}}=\Delta\rho^{j}+c\Delta t _{G}+\Delta\varepsilon^{j} \tag{7}\]  
\[\lambda^{k}\Delta\tilde{\varphi}^{k}_{t_{m-1},t_{m}}=\Delta\rho^{k}+c(\Delta t _{G}+\Delta ISB_{C-G})+\Delta\varepsilon^{k} \tag{8}\]  
ISBs consist of the difference in time scales between two constellations and the difference in hardware delays, and are quite stable [23]. ISBs have been applied to ensure the compatibility andinteroperability of multi-GNSS [24]. The 1-s interval GPS and BDS receiver clock biases of 5 sites on day of the year (DOY) 353 in 2015 were calculated with back-smoothed precise point positioning (PPP). The ISBs for each site were then calculated by differencing the GPS and BDS receiver clock biases. Furthermore, the 1-s interval ISB values are differenced between successive epochs and converted to the related distance. These statistics are shown in Figure 1.  
Figure 1 shows that the 1-s interval TDISB has an average absolute value of 2 mm with an STD of 3 mm for each station.