# Comparison Analysis on the Accuracy of Galileo PPP Using Different Frequency Combinations in Europe
## 2 Methods  
The undifferenced, uncombined phase and range observation for PPP between receiver \(r\) and satellite \(s\) can be generally expressed as:  
\[\begin{array}{l}P_{r,f}^{s}=\rho_{r}^{s}+t_{r}-t^{s}+m_{r}^{s}\cdot\delta T_{r} +\beta_{f}\cdot I_{r,1}^{s}+\epsilon_{P}\\ L_{r,f}^{s}=\rho_{r}^{s}+t_{r}-t^{s}+m_{r}^{s}\cdot\delta T_{r}-\beta_{f} \cdot I_{r,1}^{s}-N_{r,f}^{s}+\epsilon_{L}\end{array} \tag{1}\]  
where \(P_{r,f}^{s}\) and \(L_{r,f}^{s}\) are the pseudo range and carrier phase observation on frequency \(f\); \(\rho_{r}^{s}\) is the geometric distance from the satellite to the receiver; \(t_{r}\) and \(t^{s}\) are the receiver and satellite clock offset; \(m_{r}^{s}\) and \(\delta T_{r}\) are mapping function and zenith tropospheric delay parameter; \(I_{r,1}^{s}\) is the slant ionospheric delay on the first frequency; \(\beta_{f}\) is the frequency-related constant; \(N_{r}^{s}\) denotes the float ambiguities in meters; \(\epsilon_{P}\) and \(\epsilon_{L}\) are the unmodeled errors for the range and carrier-phase observation. Note that the precise correction models such as the relativity effects, phase-up effects, antenna phase center corrections, and ocean loading are not included in the equations but should be corrected according to the existing models.  
The troposphere and the ionospheric parameters in Equation (1) are usually corrected in advance by an empirical model, then the residual errors are estimated together with other parameters with a proper epoch-wise constraint:  
\[\begin{array}{l}\delta T_{t}=\delta T_{t-1}+\epsilon_{T,t},\ \ \epsilon_{T,t}\sim N(0,\sigma_{T}^{2})\\ I_{t}^{s}=I_{t-1}^{s}+\epsilon_{I,t},\ \ \epsilon_{I,t}\sim N(0,\sigma_{T}^{2}) \end{array} \tag{2}\]  
where \(t\) and \(t-1\) indicate the current and previous epoch; \(\epsilon\) is assumed to be random errors; the corresponding variance is \(\sigma_{T}^{2}\) and \(\sigma_{T}^{2}\) for troposphere and ionosphere, respectively. If the external corrections are available to augment the PPP, the variance is important to the convergence and precision of PPP.  
Note that the satellite and receiver hardware delays are not presented in the equation. The phase hardware delays can be absorbed by the ambiguity parameter. The code hardware delays can be absorbed by the ionospheric parameter and receiver clock parameter for the dual-frequency observations. However, additional inter-frequency biases should be estimated when multi-frequency observations are used. The observation model for triple-frequency PPP can be expressed as:  
\[\begin{array}{l}P_{r,1}^{s}=\rho_{r}^{s}+t_{r}-t^{s}+m_{r}^{s}\cdot\delta T _{r}+\beta_{1}\cdot I_{r,1}^{s}+\epsilon_{P}\\ P_{r,2}^{s}=\rho_{r}^{s}+t_{r}-t^{s}+m_{r}^{s}\cdot\delta T_{r}+\beta_{2}\cdot I _{r,1}^{s}+\epsilon_{P}\\ P_{r,3}^{s}=\rho_{r}^{s}+t_{r}-t^{s}+m_{r}^{s}\cdotwere processed based on the in-house software that is capable of real-time PPP and clock estimation. The antenna phase center (PCO) and antenna phase variations (PCV) were corrected using the IGS antenna files. There is still no official receiver antenna PCO/PCV corrections for Galileo provided, therefore, the GPS phase center corrections were used as a replacement. The corrections of the GPS L1 signal were used for the Galileo E1 signal, and the corrections of the GPS L2 signal were used for the Galileo E5a, E5b, E6, and E5 signals. As the ambiguities are resolved as float in the PPP filter, the elevation-dependent receiver PCO variations can be absorbed by the ambiguities, thus not affecting the PPP coordinate solutions [20]. Note that the products for satellite orbit/clock corrections and differential code biases (DCBs) corrections are from different centers. However, this will not affect the solution because the code hardware delays can be absorbed by the ionospheric parameters in the uncombined PPP model. The simplified elevation-dependent stochastic modeling was used for the multi-frequency PPP, assuming that there were no correlations among the different observables.