## 2 The CNES PPP-WIZARD Demonstrator
### The User Side of the Demonstrator  
#### 2.3.1 Measurement Modeling in the User  
The user software is described in [31]. Its main goal is to compute the receiver trajectory. It is based on a Kalman filter that can be run in forward mode for real-time purposes or in forward and backward mode for post-processing. Ambiguity resolution is performed when phase biases are available and compatible. The four main constellations are supported. The software is compatible with the undifferenced multi-frequency bias representation, allowing instantaneous ambiguity resolution when possible and fast convergence. The code is portable and can be embedded in a hand-held device. The code and phase measurements are modeled with the undifferenced code and phase biases defined with respect to the IGS convention (code ionosphere-free clocks) as \begin{table}
\begin{tabular}{c l c c c} \hline
**Constellation** & **Nature** & **RTCM Message** & **IGS Message** & **Occurrence (s)** \\ \hline \multirow{3}{*}{GPS} & orbits/clocks & 1060 & 23 & 5 \\  & code biases & 1059 & 25 & 5 \\  & phase biases (L1, L2, L5), all ambiguities & 1265 & 26 & 5 \\ \hline \multirow{3}{*}{Glonass} & orbits/clocks & 1066 & 43 & 5 \\  & code biases & 1065 & 45 & 5 \\  & phase biases (yaw) & 1266 & 46 & 5 \\ \hline \multirow{3}{*}{Galileo} & orbits/clocks & 1243 & 63 & 5 \\  & code biases & 1242 & 65 & 5 \\  & phase biases (E1, E5a, E5b, E6), all ambiguities & 1267 & 66 & 5 \\ \hline \multirow{3}{*}{Beidou 2-3} & orbits/clocks & 1261 & 103 & 5 \\  & code biases & 1260 & 105 & 5 \\ \cline{1-1}  & phase biases (B1, B2, B3), widelane ambiguities & 1270 & 106 & 5 \\ \hline \multirow{3}{*}{Beidou 2-3} & Ionosphere VTEC & 1264 & 201 & 60 \\ \cline{1-1}  & & & & \\ \end{tabular}
\end{table} 
Table 1: List of real-time messages. described in Equations (14) and (17). In order to simplify the notations, the \(C_{ij}\) subscript of the IF clocks will be removed so that the measurements model reads:  
\[ \begin{split} C_{j}&=\rho_{r}^{s}+\gamma_{j}e+m(E^{s}) T_{Z}+h_{r,\text{IF}}^{s}+b_{r,\text{IF},C_{j}}^{s},\\ \lambda_{j}L_{j}&=\rho_{r}^{s}-\gamma_{j}e+m(E^{s}) T_{Z}+h_{r,\text{IF}}^{s}+b_{r,\text{IF},L_{j}}^{s}+\lambda_{j}W+\lambda_{j}N_{j} ^{s}.\end{split}  \tag{19}\]  
The aim of the network side of the demonstrator is to compute and disseminate the satellite part of the code and phase biases appearing in Equation (1) as well as the satellite part of the clock offset: \(b_{\text{IF},C_{j}}^{s}\), \(b_{\text{IF},L_{j}}^{s}\), and \(h_{\text{IF}}^{s}\). The user software can retrieve the satellite biases in the RTCM and the IGS standards. The receiver part of these biases has hence to be determined, along with its position. The model defined by Equations (14) and (17) suffers from a rank deficiency. To reduce this deficiency, the measurement equations are reparameterized to let only one receiver clock offset for all the frequencies appear. The code and phase bias definition on all other frequencies is then redefined. Choosing \(f_{1}\) as the reference frequency, it is possible to define a clock relative to the code measurement of the frequency \(f_{1}\) as \(\tilde{h}_{r,\text{IF},C_{1}}=h_{r,\text{IF}}+b_{r,\text{IF},C_{1}}\) This clock is relative to the frequency \(f_{1}\) in the sense that it embeds both the receiver clock and the receiver code bias of the frequency \(f_{1}\). With this definition, the code measurement model of the observables on the frequency \(f_{1}\) is rewritten as:  
\[ \begin{split}\tilde{C}_{1}=C_{1}+h_{\text{IF}}^{s}+b_{\text{IF},C _{1}}^{s}&=\rho_{r}^{s}+h_{r,\text{IF}}+b_{r,\text{IF},C_{1}}+e+m (E^{s})T_{Z},\\ &=\rho_{r}^{s}+\tilde{h}_{r,\text{IF},C_{1}}+e+m(E^{s})T_{Z},\\ \lambda_{1}\tilde{L}_{1}=\lambda_{1}L_{1}+h_{\text{IF}}^{s}+b_{ \text{IF},L_{1}}^{s}&=\rho_{r}^{s}+h_{r,\text{IF}}+b_{r,\text{IF},L_{1}}-e+m(E^{s})T_{Z}+\lambda_{1}N_{1}^{s}+\lambda_{1}W,\\ &=\rho_{r}^{s}+\tilde{h}_{r,\text{IF},C_{1}}+\tilde{b}_{r,\text{ IF},L_{1}}-e+m(E^{s})T_{Z}+\lambda_{1}N_{1}^{s}+\lambda_{1}W.\end{split}