# Modelling and Assessment of a New Triple-Frequency IF1213 PPP with BDS/GPS
## 2 Materials and Methods
### General Observation Model  
The code and carrier phase observations on a single frequency are as follows [31]:  
\[\left\{ \begin{array}{ll}p_{r,j}^{s,Q}=\mu_{r}^{s,Q}\cdot x+dt_{r}-dt^{s,Q}+m^ {s,Q}\cdot Z_{r}+\gamma_{j}\cdot l_{r,1}^{s,Q}+d_{r,j}-d_{j}^{s,Q}+\varepsilon_ {r,j}^{s,Q}\\ l_{r,j}^{s,Q}=\mu_{r}^{s,Q}\cdot x+dt_{r}-dt^{s,Q}+m^{s,Q}\cdot Z_{r}-\gamma_{j }\cdot l_{r,1}^{s,Q}+N_{r,j}^{s,Q}+\varphi_{r,j}-\varphi_{j}^{s,Q}+\xi_{r,j}^ {s,Q}\end{array} \right. \tag{1}\]  
where the superscript \(s\), \(Q\) and subscript \(r\) represent the satellite, constellation and receiver, respectively; in this paper \(Q\) can be G, C2, C3 and C for GPS, BDS-2, BDS-3 and BDS, respectively, where BDS means that BDS-2 and BDS-3 are considered as one constellation; \(p_{r,j}^{s,Q}\) and \(l_{r,j}^{s,Q}\) are the code and phase observed-minus-computed (OMC) values, respectively; \(j\) denotes the frequency (\(j=1,2,3\)); \(\mu_{r}^{s,Q}\) is the unit vector of direction; \(x\) represents the vector of position correction to the a priori position; and \(dt_{r}\) and \(dt^{s,Q}\) indicate the receiver and satellite clock offsets, respectively. Furthermore, \(Z_{r}\) is the wet troposphere delay at zenith; \(\gamma_{j}=f_{1}^{2}/f_{j}^{2}\) is the ionospheric factor; \(f\) indicates the carrier phase frequency; \(l_{r,1}^{s,Q}\) denotes the slant ionospheric delay at the first frequency; and \(N_{r,j}^{s,Q}\) represents the integer phase ambiguity. The parameters \(d_{r,j}\) and \(d_{j}^{s,Q}\) are the code hardware delays from the receiver and satellite, respectively. \(\varphi_{r,j}\) and \(\varphi_{j}^{s,Q}\) are the receiver-dependent and satellite-dependent carrier phase hardware delays, respectively. \(\varepsilon_{r,j}^{s,Q}\) and \(\xi_{r,j}^{s,Q}\) are the measurement noise of the code and carrier phase, respectively. Other error items include the phase center offset (PCO) and variation (PCV), dry slant troposphere delay, phase wind-up, and relativistic effect. For simplicity, they are precisely corrected with their corresponding models and are not listed in the equations. It should be noted that in the case of lacking the precise PCO/PCV information of the third frequency, we use the PCO/PCV corrections of the first frequency instead due to the adjacent frequency. For code and phase hardware bias, the code hardware bias is generally considered to be relatively stable and can be considered constant over the course of a day [32]. The phase hardware bias has a clear time-varying character and can be decomposed into a constant part and a time-varying part [13,14]. \[\left\{ \begin{array}{ll}\varphi_{r,j}=\overline{\varphi}_{r,j}+\delta \varphi_{r,j}\\ \varphi_{j}^{s,Q}=\overline{\varphi}_{j}^{s,Q}+\delta\varphi_{j}^{s,Q}\end{array} \right. \tag{2}\]  
where \(\overline{\varphi}_{r,j}\) and \(\overline{\varphi}_{j}^{s,Q}\) are the phase hardware bias constant parts of the receiver and satellite, respectively; \(\delta\varphi_{r,j}\) and \(\delta\varphi_{j}^{s,Q}\) are the corresponding time-varying parts. The constant part \begin{table}
\begin{tabular}{c c c c} \hline
**Number** & **GPS** & **BDS-2** & **BDS-3** \\ \hline
1 & L1 (1575.42) & B1I (1561.01) & B1I (1561.01) \\
2 & L2 (1227.60) & B3I (1268.52) & B3I (1268.52) \\
3 & L5 (1176.45) & B2I (1207.14) & B1C (1575.42) \\ \hline \end{tabular}
\end{table} 
Table 1: BDS/GPS frequency number. can be completely absorbed by the ambiguity \(\overline{N_{r,j}^{s,Q}}=N_{r,j}^{s,Q}+\overline{\varphi}_{r,j}+\overline{\varphi }_{j}^{s,Q}\). Moreover, the following variables are defined herein for ease of expression:  
\[\left\{ \begin{array}{l}\alpha_{ij}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{j}^{2}}\\ \beta_{ij}=-\frac{f_{1}^{2}}{f_{1}^{2}-f_{j}^{2}}\\ DCB_{r,ij}=d_{r,i}-d_{r,j}\\ DCB_{ij}^{s,Q}=d_{i}^{s,Q}-d_{j}^{s,Q}\\ DPB_{r,ij}=\delta\varphi_{r,i}-\delta\varphi_{r,j}\\ DPB_{ij}^{s,Q}=\delta\varphi_{i}^{s,Q}-\delta\varphi_{i}^{s,Q}\\ DCB_{r,ij}^{s,Q}=DCB_{r,ij}-DCB_{ij}^{s,Q}\\ DPB_{r,ij}^{s,Q}=DPB_{r,ij}-DPB_{ij}^{s,Q}\end{array} \right.