## 2 Materials and Methods
### Uncombined Dual- and Triple-Frequency Multi-GNSS Decoupled Clock Model
#### 2.1.2 Uncombined Dual- and Triple-Frequency DCM  
The model introduced in Equation (1) is singular and cannot be resolved. A reparameterization is therefore necessary to reduce the number of states to be estimated and to group some states together. The triple-frequency model used in this research was derived from first principles in [19]. The derivation is omitted in this manuscript, though a discussion of the equations and of the model is still included. The complete triple-frequency model used in this research is summarized in Equation (2). \[\left\{ \begin{array}{llllll}P_{1}^{s}&=\rho_{r}^{s}+c(\overline{dt}_{r}-dt^{s} )+\gamma_{1}\overline{I}_{1}+M_{r}^{s}.T_{r}&-b_{1}^{s}&+\varepsilon_{P_{1}}\\ P_{2}^{s}&=\rho_{r}^{s}+c(\overline{dt}_{r}-dt^{s})+\gamma_{2}\overline{I}_{1}+M _{r}^{s}.T_{r}&-b_{2}^{s}&+\varepsilon_{P_{2}}\\ P_{3}^{s}&=\rho_{r}^{s}+c(\overline{dt}_{r}-dt^{s})+\gamma_{3}\overline{I}_{1}+M _{r}^{s}.T_{r}&-b_{3}^{s}&+IFB_{r}&+\varepsilon_{P_{3}}\\ \Phi_{1}^{s}&=\rho_{r}^{s}+c(\overline{dt}_{r}-dt^{s})-\gamma_{1}\overline{I}_{1 }+M_{r}^{s}.T_{r}+&\lambda_{1}(\overline{N}_{1}^{s}&-B_{1}^{s})&+\varepsilon_ {\[PERSON]_{1}}\\ \[PERSON]_{2}^{s}&=\rho_{r}^{s}+c(\overline{dt}_{r}-dt^{s})-\gamma_{2}\overline{I}_ {1}+M_{r}^{s}.T_{r}+&\lambda_{2}(\overline{N}_{2}^{s}&-B_{2}^{s})&+\delta t_{1 2}&+\varepsilon_{\[PERSON]_{2}}\\ \[PERSON]_{3}^{s}&=\rho_{r}^{s}+c(\overline{\delta t}_{r}-dt^{s})-\gamma_{3}\overline {I}_{1}+M_{r}^{s}.T_{r}+&\lambda_{3}(\overline{N}_{3}^{s}&-B_{3}^{s})&+\delta t _{13}&+\varepsilon_{\Phi_{3}}\end{array} \right. \tag{2}\]  
Compared to Equation (1), Equation (2) contains reparameterized states, as well as additional states, some of which are specific to the DCM, while others are not:  
* Two receiver clock terms are present in the equation: a pseudorange clock, defined as: \(c\overline{dt}_{r}=cdt_{r}+\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}b_{r,1}- \frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}b_{r,2}\) and a carrier-phase clock defined as: \(c\overline{\delta t}_{r}=c\delta t_{r}+\lambda_{1}(\delta N_{1}^{ref}+B_{r,1} )-(b_{r,1}-b_{r,2})\). This decoupling of the clocks is a consequence of using the DCM, as both terms are biased versions of \(dt_{r}\) and \(\delta t_{r}\), the pseudorange and carrier-phase clocks, respectively. Both state terms are affected by receiver biases, with the carrier-phase clock being affected by the reference satellite's ambiguity as well, which is discussed in more detail later. * The ionospheric delay term absorbs the ionosphere-free combination of the receiver pseudorange biases in the form: \(\overline{I}_{1}^{s}=I_{1}^{s}-\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}(b_{r,1}- b_{r,2})\). When trying to use external ionospheric corrections, either global or regional corrections, the biases have to be separated from the ionospheric delay state term. The biases can be estimated as separate state terms, as long as additional observations of the ionospheric delays are present. * \(\overline{N}_{i}^{s}\) are the integer single-differenced ambiguities relative to the reference satellite. The single-differencing is performed implicitly without having to explicitly difference the measurements or ambiguities from different satellites. The implicit single-difference is a consequence of the decoupling of the clocks, as the phase measurements lose their datum--typically set by the pseudorange measurements through the receiver clock. The new datum is set by the reference satellite, as its ambiguities are fixed to arbitrary integers and they are not estimated. Doing so not only leads to all ambiguities being single-differenced, but also recovers the datum in the phase measurements, as each frequency's datum is being set by the reference satellite's ambiguity on that frequency. * An interfrequency bias term appears in the third frequency's pseudorange measurement equation, expressed as: \(IFB_{r}=\frac{f_{2}^{2}-f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}b_{r,1}-\frac{f_{1}^{2 }-f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}b_{r,2}+b_{r,3}\).