# Comprehensive Analysis of PPP-B2b Service and Its Impact on BDS-3/GPS Real-Time PPP Time Transfer
## 2 Methods
### Algorithms of B2b-RTPPP  
The observation equations of the raw pseudorange \(P_{i}\) and raw carrier phase \(L_{i}\) at \(i\)th frequency are expressed as follows [27; 28]:  
\[P_{i}=\rho+c(dt_{r}-dt^{s})+T+\gamma_{i}I_{1}+c(d_{r,i}-d_{i}^{s})+\varepsilon_ {i} \tag{5}\]  
\[L_{i}=\rho+c(dt_{r}-dt^{s})+T-\gamma_{i}\cdot I_{1}+\lambda_{i}(b_{r,i}-b_{i}^{ s}+N_{i})+\zeta_{i} \tag{6}\]  
where the symbols \(r\) and \(s\) denote the receiver and satellite, respectively; \(\rho\) denotes the distance from the satellite to the receiver (in meters); \(dt_{r}\) and \(dt^{s}\) represent the receiver and satellite clock error (in seconds), respectively; \(T\) denotes the tropospheric wet delay (meters); \(\gamma_{i}=f_{1}^{2}/f_{2}^{2}\) is the frequency amplification factor; \(I_{1}\) is the first-order ionospheric delay (in meters); \(d_{r}\) and \(b_{r}\) are the frequency-dependent pseudorange and carrier phase hardware delay at the receiver end, respectively; \(d^{s}\) and \(b^{s}\) denote pseudorange and carrier phase hardware delay at the satellite end, respectively; \(\lambda_{i}\) is the wavelength on \(f_{i}\) (meters); \(N_{i}\) is the float ambiguity (cycle); and \(\varepsilon_{i}\) and \(\zeta_{i}\) indicate the unmodeled noises (meters). Note that the phase window effect, tidal effect, relativistic effect, and satellite antenna APC were corrected. In this paper, 1, 3, 4, and 5 denote the B1I, B3I, L1, and L2 signals, respectively. The following notations are also used:  
\[\left\{ \begin{array}{l}\alpha_{mn}=\frac{f_{2}^{2}}{f_{m}^{2}-f_{n}^{2}}\\ \beta_{mn}=-\frac{f_{2}^{2}}{f_{m}^{2}-f_{n}^{2}}\\ DCB_{mn}^{s}=d_{m}^{s}-d_{n}^{s}\\ DCB_{r,mn}^{s}=d_{r,m}^{s}-d_{r,n}\\ dt_{r,I}F_{mn}=dt_{r}+\alpha_{mn}d_{r,m}+\beta_{mn}d_{r,n}\\ dt_{s,I_{Fmn}^{s}}=dt^{s}+\alpha_{mn}d_{m}^{s}+\beta_{mn}d_{n}^{s}\end{array} \right. \tag{7}\]where \(\alpha\) and \(\beta\) are frequency factors. In this work, the B1I/B3I and L1/L2 combination observations were employed to realize the BDS-3 and GPS B2b-RTPPP time transfer, respectively. As mentioned above, the systematic bias of GPS satellites in PPP-B2b clock offset is the hardware delay of the L1/L2 IF combination, while that of BDS-3 satellites is the B3I. That is  
\[\left\{ \begin{array}{l}d^{s}_{IF_{45}}=dt^{s}+d^{s}_{IF_{45}}\\ dt^{s}_{brd}=dt^{s}+d^{s}_{3}\end{array} \right. \tag{8}\]  
By applying the PPP-B2b products, Equations (5) and (6) can be rewritten as [29]  
\[\left\{ \begin{array}{l}P_{i}=\rho+c(dt_{r}+d_{r,i}-dt^{s}_{brd})-cDCB^{s}_{i 3}+T+I_{i}+\varepsilon_{1}\\ L_{i}=\rho+c(dt_{r}+d_{r,i}-dt^{s}_{brd})+T-I_{i}+\lambda_{i}(b_{r,i}-b^{s}_{ i}+N_{i})+\zeta_{i}\end{array} \right.,i=1,3 \tag{9}\]  
\[\left\{ \begin{array}{l}P_{j}=\rho+c(dt_{r,IF_{45}}-dt^{s}_{IF_{45}})+T+I_{j} +\varepsilon_{j}\\ L_{j}=\rho+c(dt_{r,IF_{45}}-dt^{s}_{IF_{45}})+T-I_{j}+\lambda_{j}\overline{N}_ {j}+\zeta_{j}\end{array} \right.,j=4,5 \tag{10}\]  
with  
\[\overline{N}_{j}=(N_{j}+b_{r,j}-b^{s}_{j})-\frac{c}{\lambda_{j}}(d_{r,IF_{45} }-d^{s}_{IF_{45}}),\;j=4,5 \tag{11}\]  
Based on Equations (9)-(11), the B1I/B3I and L1/L2 IF pseudorange function \(P_{\text{IF}_{13}}\) and \(P_{\text{IF}_{45}}\) and carrier phase function \(L_{\text{IF}_{13}}\) and \(L_{\text{IF}_{45}}\) can be expressed as  
\[\left\{ \begin{array}{l}P_{\text{IF}_{13}}=\alpha_{13}P_{1}+\beta_{13}P_{3}= \rho+c(dt_{r,IF_{13}}-dt^{s}_{brd})-\alpha_{13}cDCB^{s}_{13}+T+\varepsilon_{13 }\\ L_{\text{IF}_{13}}=\alpha_{13}L_{1}+\beta_{13}L_{3}=\rho+c(dt_{r,IF_{13}}-dt^{ s}_{brd})+T+\lambda_{13}\overline{N}_{13}+\zeta_{13}\end{array} \right. \tag{12}\]  
\[\left\{ \begin{array}{l}P_{\text{IF}_{45}}=\alpha_{45}P_{4}+\beta_{45}P_{5}= \rho+c(dt_{r,IF_{45}}-dt^{s}_{IF_{45}})+T+\varepsilon_{45}\\ L_{\text{IF}_{45}}=\alpha_{45}L_{4}+\beta_{45}L_{5}=\rho+c(dt_{r,IF_{45}}-dt^{ s}_{IF_{45}})+T+\lambda_{45}\overline{N}_{45}+\zeta_{45}\end{array} \right.