## 2 Materials and Methods
### Uncombined Dual- and Triple-Frequency Multi-GNSS Decoupled Clock Model
#### 2.1.3 Reference Satellite and Multi-GNSS Considerations  
As mentioned in the previous section, one satellite has to be chosen as reference to set the datum in the carrier-phase measurements that lost their datum when decoupling the receiver clocks. Doing so results in additional parameters needing to be estimated in the form of \(\delta t_{12}\) and \(\delta t_{13}\). When dealing with multiple constellations, the PPP user has to be mindful of intersystem biases. Indeed, all receiver clocks and biases are constellation-dependent due to constellations being processed by different processing units in the receivers, as well as due to the presence of intersystem time biases [20; 21]. It is therefore typical to estimate separate clock/bias terms for each constellation, or to take one constellation as a reference for which the clock/bias is estimated, and then estimate the difference between other constellation clocks/biases and the reference clock/bias [16; 22]. In this research, the first strategy is adopted: one set of receiver clocks and biases is estimated for each constellation, and all those state terms are treated as clock-like terms by being estimated as white noise processes. The complete set of states to be estimated, as well as the number of states to be estimated depending on the number of constellations and frequencies to be processed is summarized in Table 1.  
Given the dependency of the receiver clock and bias states on the constellation, and knowing that the reference satellite's ambiguities are directly correlated with the receiver clocks and biases (as was shown in the previous section), it is necessary to select one reference satellite for each constellation. Fortunately, having as many reference satellites as there are constellations being processed does not negatively impact the degrees of freedom of the system. Indeed, let us consider one constellation with \(N\) frequencies being processed. Compared to the classical float PPP model, the DCM contains the additional receiver phase clock (one state term, common to all frequencies), and the receiver phase bias terms \(\delta t_{12}\) and \(\delta t_{13}\). In dual-frequency mode, the receiver phase clock \(\overline{\delta t}_{t}\) and \(\delta t_{12}\) need to be estimated, which is compensated for by the fact that the two ambiguities for the reference satellite are not estimated and they are fixed to arbitrary integers. In triple-frequency mode, on top of \(\overline{\delta t}_{r}\) and \(\delta t_{12}\), \(\delta t_{13}\) is estimated too; all three are compensated for by the reference satellite's three ambiguities not needing to be estimated. These observations apply to the multi-GNSS context, as one reference satellite is selected per constellation, and a set of receiver clocks and biases has to be estimated for each constellation as well. \begin{table}
\begin{tabular}{l c c c c c c c} \hline  & \multicolumn{2}{c}{**State Term Dependency (One Per \_)**} & \multicolumn{2}{c}{**Measurement Type**} & \multicolumn{2}{c}{**Frequency Band**} \\ \cline{2-7}  & **Receiver** & **Constellation** & **Satellite** & **Code** & **Phase** & **1** & **2** & **3** \\ \hline Receiver coordinates & X & & & X & X & X & X \\ Wet tropospheric delay & X & & X & X & X & X & X \\ Receiver pseudorange clock & & X & & X & & X & X & X \\ Receiver carrier-phase clock & & X & & X & X & X & X \\ Receiver L2 phase bias & & X & & X & & X & \\ Receiver L3 phase bias & & X & & X & & & X \\ Receiver IFB & & X & & X & & & X \\ \hline Slant ionospheric delay & & & X & X & X & X & X \\ L1 ambiguity & & & X & & X & & \\ L2 ambiguity & & & X & & X & & X \\ L3 ambiguity & & & X & & X & & X \\ \hline \end{tabular}
\end{table} 
Table 1: States and number of states to be estimated for each constellation and frequency. A cross in state term dependency column means that a state term is either receiver-, constellation-, or satellite-dependent. A cross in measurement type column specifies whether pseudorange or carrier-phase measurements are used to estimate state.