# Evaluation of BDS-2 and BDS-3 Satellite Atomic Clock Products and Their Effects on Positioning
## 2 Methods
### Ionosphere-Free PPP and Receiver Clock ISB  
For GNSS dual-frequency data, ionosphere-free (IF) linear combination observations are widely utilized in PPP to eliminate the first-order ionospheric delay, as shown in the following expression:  
\[ \begin{array}{ll}P_{r,IF}^{s}=\rho_{r}^{s}+c\cdot\left(dt_{r}+b_{r,IF}\right) -c\cdot\left(dt^{s}+b_{IF}^{s}\right)+a_{r}^{s}\cdot T_{z}&+\varepsilon_{P,IF} \\ \phi_{r,IF}^{s}=\rho_{r}^{s}+c\cdot\left(dt_{r}+b_{r,IF}\right)-c\cdot\left(dt ^{s}+b_{IF}^{s}\right)+a_{r}^{s}\cdot T_{z}+N_{r,IF}^{s}&+\varepsilon_{\phi,IF} \end{array}  \tag{7}\]  
where the subscript \(r\) refers to receivers, \(P_{r,IF}^{s}\) and \(\phi_{r,IF}^{s}\) are ionosphere-free combination observations in length units of pseudorange and carrier phase, respectively, \(\rho_{r}^{s}\) denotes the range between the receiver and the satellite with antenna phase center corrections for both \(P_{r,IF}^{s}\) and \(\phi_{r,IF}^{s}\), and phase wind-up corrections for \(\phi_{r,IF}^{s}\). \(c\) is the speed of light, \(dt_{r}\) and \(dt^{s}\) are the receiver and satellite clock offsets, respectively. \(b_{r,IF}\) and \(b_{IF}^{s}\) are the hardware delay of the ionosphere-free function in the receiver and satellite. \(T_{z}\) is the zenith tropospheric delay and \(\alpha_{r}^{s}\) is corresponding mapping function. \(N_{r,IF}^{s}\) is the float ionosphere-free ambiguity in units of length, and \(\varepsilon_{P,IF}\) and \(\varepsilon_{\phi,IF}\) denote observation noise. The effects of relativity, solid earth tide, ocean tide and earth rotation are corrected as well, based on existing models. Additionally, the satellite-induced, elevation-dependent code bias variation for BDS-2 is removed with the model provided by [PERSON] et al. (2016), while the effect is negligible and hence ignored for BDS-3. B1I/B3I IF combination is adopted in the strategy of BDS-2 and BDS-3 precise clock products estimation for IGS ACs. Hence, we employed the same B1I/B3I IF combination in PPP for both BDS-2 and BDS-3. In general, the satellite IF hardware delay \(b_{IF}^{s}\) is assimilated to estimated precise clock offsets. Meanwhile, considering the receiver hardware delay can be absorbed by receiver clock offsets, Equation (7) can be rewritten as follows:  
\[ \begin{array}{ll}P_{r,IF}^{s}=\rho_{r}^{s}+c\cdot d\overline{t}_{r,IF}-c \cdot\overline{d}_{IF}^{s}+\alpha_{r}^{s}\cdot T_{z}&+\varepsilon_{P,IF}\\ \phi_{r,IF}^{s}=\rho_{r}^{s}+c\cdot d\overline{t}_{r,IF}-c\cdot d\overline{t}_ {IF}^{s}+\alpha_{r}^{s}\cdot T_{z}+N_{r,IF}^{s}&+\varepsilon_{\phi,IF}\end{array}  \tag{8}\]  
with  
\[ \begin{array}{ll}d\overline{t}_{r,IF}=dt_{r}+b_{r,IF}\\ d\overline{t}_{IF}^{s}=dt^{s}+b_{IF}^{s}\end{array}  \tag{9}\]  
where \(d\overline{t}_{r,IF}\) and \(d\overline{t}_{IF}^{s}\) denote the modified receiver clock and satellite clock offset, respectively. \(d\overline{t}_{IF}^{s}\) can be eliminated directly by precise clock products. Nevertheless, it should be noted that the BDS-2 and BDS-3 broadcast clock refers to the frequency at B3I, which means it contains single B3I hardware delay \(b_{B3I}^{s}\) that is different to the B1I/B3I IF combination counterpart [28], as follows:  
\[b_{IF}^{s}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{3}^{2}}b_{B1I}^{s}+\frac{f_{3}^{2}}{ f_{1}^{2}-f_{3}^{2}}b_{B3I}^{s} \tag{10}\]  
in which \(b_{B1I}^{s}\) is the hardware delay at B1I, \(f_{1}\) and \(f_{3}\) denote the frequencies on B1I and B3I of the BDS, respectively. BDS provides the TGD\({}_{B1I/B3I}^{s}=b_{B1I}^{s}-b_{B3I}^{s}\) parameter to correct the satellite clock from B3I to B1I.