## 2 Approach of GPS/GALILEO RT PPP with Ionospheric Constraint  
### Function Models of GPS/GALILEO RT PPP  
In the RT PPP model, satellite clock and position are calculated by broadcast ephemerides with RT precise orbit/clock corrections products from CNES. The uncombined raw observable model for GPS/GALILEO PPP can be written as [3]:  
\[ \begin{array}{l}P_{r,i}^{sys,s}=D_{r,i}^{sys,s}+\gamma_{i}^{sys,s}I_{1}^{sys,s,s}+b_{r,p}^{sys}-b_{p}^{sys,s}+cdt_{r}^{sys}-cdt_{r}^{sys,s}\\ \lambda_{i}^{sys,s}L_{r,i}^{sys,s}=D_{r,i}^{sys,s}-\gamma_{i}^{sys,s}I_{1}^{ sys,s,s}+b_{r,L_{i}}^{sysl}-b_{L_{i}}^{sys,s}+cdt_{r}^{sys}-cdt_{r}^{sys,s}+ \lambda_{i}^{sys,s}W^{sys,s}-\lambda_{i}^{sys,s}N_{i}^{sys,s}\end{array}  \tag{1}\]  
where \(P_{r,i}^{sys,s}\) and \(L_{r,i}^{sys,s}\) are the raw pseudororange (or code) and phase measurements at frequency \(f_{i}\)\((i=1,2)\) to the system \"\(sys\)\" (for GPS and GALILEO) for the specific satellite \(s\) and receiver \(r\). The pseudorange measurements are expressed in meters, while phase measurements are expressed in cycles. \(\lambda_{i}^{sys,s}\) denotes the wavelength of phase observations at frequency \(f_{i}\)\((i=1,2)\). \(D_{r,i}^{sys,s}\) is the geometrical propagation distances between the satellite and receiver phase centers at each frequency including troposphere elongation, relativistic effects, etc. \(I_{1}^{sys,s}\) is the slant ionospheric delay at frequency \(f_{1}\) in meters. \(\gamma_{i}^{sys,s}=(f_{1}/f_{i})^{2}\) is a frequency-dependent scale factor. This elongation varies with the inverse of the square of the frequency and with opposite signs between phase and pseudorange [3]. \(b_{p_{i}^{sys,s}}^{sys,s}\) and \(b_{L_{i}}^{sys,s}\) are the frequency-dependent uncalibrated code delyas (UCD) and uncalibrated phase delays (UPD) of the satellite. \(b_{r,p_{i}}^{sys}\) and \(b_{r,L_{i}}^{sys}\) are the UCD and UPD of receiver. \(W_{r}^{sys,s}\) is the contribution of the wind-up effect in cycles. \(N_{i}^{sys,s}\) stands for the undifferenced ambiguities for each frequency \(f_{i}\) in cycle. \(dt^{sys,s}\) and \(dt_{r}^{sys}\) are satellite and receiver code clock offsets. In order to eliminate the UCD and UPD of satellite, the code biases \(\overline{b}_{P_{i}}^{s}\) and phase biases \(\overline{b}_{L_{i}}^{s}\) from CNES caster are applied to the raw observable model, the reparameterization of (1) can be written as [6]:  
\[ \begin{array}{l}P_{r,1}^{sys,s}+\overline{b}_{P_{i}}^{sys,s}&=\overline{D}_{ r}^{sys,s}+c\overline{d}_{r}^{sys}+\gamma_{1}^{sys,s}\overline{I}_{1}^{sys,s}\\ P_{r,2}^{sys,s}+\overline{b}_{P_{i}}^{sys,s}&=\overline{D}_{r}^{sys,s}+c \overline{d}_{r}^{sys}+\gamma_{2}^{sys,s}\overline{I}_{1}^{sys,s}\\ \lambda_{1}^{sys,s}\cdot\left(L_{r,1}^{sys,s}+\overline{b}_{L_{1}}^{s}\right)& =\overline{D}_{r}^{sys,s}+c\overline{d}_{r}^{sys}-\gamma_{1}^{sys,s} \overline{I}_{1}^{sys,s}+\lambda_{1}^{sys,s}W^{s}+\lambda_{1}^{sys,s}\overline {N}_{1}^{sys,s}\\ \lambda_{2}^{sys,s}\cdot\left(L_{r,2}^{sys,s}+\overline{b}_{L_{2}}^{s}\right)& =\overline{D}_{r}^{sys,s}+c\overline{d}_{r}^{sys}-\gamma_{2}^{sys,s} \overline{I}_{1}^{sys,s}+\lambda_{2}^{sys,s}W^{s}+\lambda_{2}^{sys,s}\overline {N}_{W}^{sys,s}\end{array}  \tag{2}\]  
With  
\[\left\{ \begin{array}{l}\overline{I}_{1}^{sys,s}&=I_{1}^{sys,s}+\beta_{12}^{ sys}DCB_{r,12}^{sys}\\ c\cdot\overline{d}_{r}^{sys}&=c\cdot dt_{r}^{sys}+b_{r,P_{i}}^{sys}\\ \lambda_{1}^{sys}\cdot\overline{N}_{1}^{sys,s}&=\lambda_{1}^{sys}\cdot\left( N_{1}^{sys,s}+b_{r,L_{1}}^{sys}\right)-b_{r,P_{i}}^{sys}+\beta_{12}^{sys}DCB_{r,12}^{ sys}\\ \lambda_{2}^{sys}\cdot\left(\overline{N}_{W}^{sys,s}+\overline{N}_{1}^{sys,s} \right)=\lambda_{2}^{sys}\cdot\left(N_{2}^{sys,s}+b_{r,L_{2}}^{sys}\right)-b_{r,P_{i}}^{sys}+\gamma_{2}^{sys}\beta_{12}^{sys}DCB_{r,12}^{sys}\\ DCB_{r,12}^{sys,s}=b_{r,P_{1}}^{sys}-b_{r,P_{2}}^{sys}&b_{r,P_{2}}^{sys}= \alpha_{12}^{sys}\cdot b_{r,P_{1}}^{sys}+\beta_{12}^{sys}\cdot b_{r,P_{1}}^{sys} \\ \alpha_{12}^{sys}=f_{1}^{2}/\left(f_{1}^{2}-f_{2}^{2}\right)&\beta_{12}^{sys}=-f_ {2}^{2}/\left(f_{1}^{2}-f_{2}^{2}\right)\end{array}