# Modelling and Assessment of a New Triple-Frequency IF1213 PPP with BDS/GPS
## 2 Materials and Methods
### TF-C: IF1213 PPP Model with PIFCB Estimation  
The TF-C model is an IF1213 PPP model that fully considers the time-varying part of the phase hardware bias of the receiver and satellite. The GNSS precise satellite clock products are based on the dual frequency IF combined code and phase observations solved for the first and second frequencies (e.g., GPS L1/L2 and BDS B1I/B3I) [33]. Thus, the precise satellite clock is a linear combination of the time-varying components of the dual-frequency code and phase hardware bias and is expressed as follows:  
\[dt_{IF12}^{s,Q}=dt^{s,Q}+d_{IF12}^{s,Q}+\delta\varphi_{IF12}^{s,Q} \tag{7}\]  
Combining Equation (6) with Equation (7) and considering the consistency of the receiver clock difference, the TF-C model can be deduced after correcting for the satellite clock difference  
\[\left\{\begin{array}{lcl}p_{r,IF12}^{s,Q}&=&\mu_{r}^{s,Q}\cdot x+d\overline{ t}_{r}+m^{s,Q}\cdot Z_{r}+B_{12}+\varepsilon_{r,IF12}^{s,Q}\\ \par l_{r,IF12}^{s,Q}&=&\mu_{r}^{s,Q}\cdot x+d\overline{t}_{r}+m^{s,Q}\cdot Z_{ r}+\overline{N}_{r,IF12}^{s,Q}+\tilde{\varepsilon}_{r,IF12}^{s,Q}\\ \par p_{r,IF13}^{s,Q}&=&\mu_{r}^{s,Q}\cdot x+d\overline{t}_{r}+m^{s,Q}\cdot Z_{ r}+IFCB_{r}^{s,Q}+B_{13}+\varepsilon_{r,IF13}^{s,Q}\\ \par l_{r,IF13}^{s,Q}&=&\mu_{r}^{s,Q}\cdot x+d\overline{t}_{r}+m^{s,Q}\cdot Z_{ r}-IFCB_{r}^{s,Q}+\overline{N}_{r,IF13}^{s,Q}+\tilde{\varepsilon}_{r,IF13}^{s,Q} \end{array}\right. \tag{8}\]  
where  
\[\left\{\begin{array}{lcl}d\overline{t}_{r}&=&dt_{r}+d_{r,IF12}+\delta\varphi_ {r,IF12}\\ \overline{N}_{r,IFij}^{s,Q}&=&\alpha_{ij}\cdot\overline{N}_{r,i}^{s,Q}+\beta_{ ij}\cdot\overline{N}_{r,j}^{s,Q}+M_{1j}\\ \par B_{1j}&=&\left\{\begin{array}{l}\beta_{12}\cdot DPB_{r,12}^{s,Q}- \delta\varphi_{r,12}^{s,Q},\quad j=2\\ -\beta_{13}\cdot DPB_{r,13}^{s,Q}+2\cdot\beta_{12}\cdot DPB_{r,12}^{s,Q}- \delta\varphi_{r,1}^{s,Q},\quad j=3\\ \par M_{1j}&=&\left\{\begin{array}{l}\beta_{12}\cdot DCB_{r,12}^{s,Q}- \delta\varphi_{r,12}^{s,Q},\quad j=2\\ -\beta_{13}\cdot DCB_{r,13}^{s,Q}+2\cdot\beta_{12}\cdot DCB_{r,12}^{s,Q}- \delta r_{r,1}^{s,Q},\quad j=3\\ \par CIFCB&=&\beta_{12}\cdot DCB_{r,12}^{s,Q}-\beta_{13}\cdot DCB_{r,13}^{s,Q} \\ \par PIFCB&=&\beta_{12}\cdot DPB_{r,12}^{s,Q}-\beta_{13}\cdot DPB_{r,13}^{s,Q} \end{array}\right.\end{array}\right. \tag{9}\]  
and \(d\overline{t}_{r}\) and \(\overline{N}_{r,IFij}^{s,Q}\) are the reparametrized receiver clock difference and ambiguity parameters, respectively. \(B_{1j}\) is the combined time-varying part of the receiver and satellite phase hardware bias, which can be absorbed by the code observation residuals, \(M_{1j}\) is the combined receiver and satellite code hardware bias, which is constant throughout the day and will be completely absorbed by the ambiguity. \(IFCB_{r}^{s,Q}\) denotes IFCB, which consists mainly of CIFCB and PIFCB, corresponding to DCB synthesis and DPB synthesis for receivers and satellites, respectively. Additionally, the time-varying part of the hardware bias gives rise to PIFCB.  
All the estimated parameters in TF-C PPP models include  
\[X=[x\quad d\overline{t}_{r}\quad Z_{r}\quad IFCB_{r}^{s,Q}\quad\overline{N}_{r, IF12}^{s,Q}\quad\overline{N}_{r,IF13}^{s,Q}] \tag{10}\]  
It is worth noting that the TF-C model does not need to be corrected by DCB, as the CIFCB in IFCB contains the DCB of the satellite and the receiver.