# Investigation on Reference Frames and Time Systems in Multi-GNSS
## 3 Data Used and Adopted Model of the Pseudorange
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Satellite** & **Radial (m)** & **Along-Track (m)** & **Cross-Track (m)** & **Clock (ns)** & **Generation** & **Clock 3** \\ \hline E01 & \(0.01\pm 0.14\) & \(-0.13\pm 0.28\) & \(-0.04\pm 0.16\) & \(2.2\pm 1.3\) & FOC 2 & PHM \\ E02 & \(-0.01\pm 0.15\) & \(-0.06\pm 0.29\) & \(0.04\pm 0.13\) & \(1.3\pm 1.1\) & FOC 2 & PHM \\ E03 & \(0.01\pm 0.11\) & \(-0.10\pm 0.25\) & \(0.00\pm 0.17\) & \(2.1\pm 1.2\) & FOC 2 & PHM \\ E04 & \(-0.01\pm 0.11\) & \(-0.05\pm 0.36\) & \(0.01\pm 0.16\) & \(1.6\pm 1.2\) & FOC 2 & PHM \\ E05 & \(-0.03\pm 0.12\) & \(0.00\pm 0.25\) & \(0.01\pm 0.16\) & \(2.1\pm 1.1\) & FOC 2 & PHM \\ E07 & \(0.01\pm 0.12\) & \(-0.11\pm 0.28\) & \(0.05\pm 0.16\) & \(2.8\pm 1.3\) & FOC 2 & PHM \\ E08 & \(-0.01\pm 0.11\) & \(0.04\pm 0.24\) & \(0.04\pm 0.14\) & \(1.0\pm 1.2\) & FOC 2 & PHM \\ E09 & \(0.00\pm 0.13\) & \(0.05\pm 0.30\) & \(0.03\pm 0.17\) & \(1.7\pm 1.2\) & FOC 2 & PHM \\ E11 & \(0.01\pm 0.15\) & \(0.00\pm 0.35\) & \(-0.05\pm 0.18\) & \(0.7\pm 1.2\) & IOV 1 & PHM \\ E12 & \(0.00\pm 0.13\) & \(-0.04\pm 0.31\) & \(-0.04\pm 0.17\) & \(1.6\pm 1.4\) & IOV 1 & PHM \\ E14 & \(0.04\pm 0.48\) & \(0.04\pm 2.10\) & \(-0.14\pm 0.84\) & \(6.7\pm 1.3\) & FOC 2 & PHM \\ E18 & \(0.04\pm 0.37\) & \(-0.04\pm 2.05\) & \(-0.02\pm 0.50\) & \(4.1\pm 1.2\) & FOC 2 & PHM \\ E19 & \(-0.01\pm 0.18\) & \(-0.08\pm 0.44\) & \(-0.01\pm 0.23\) & \(1.1\pm 1.5\) & IOV 1 & PHM \\ E22 & \(-0.01\pm 0.16\) & \(0.03\pm 0.25\) & \(-0.05\pm 0.13\) & \(1.4\pm 1.5\) & FOC 2 & PHM \\ E24 & \(-0.02\pm 0.15\) & \(0.01\pm 0.25\) & \(-0.05\pm 0.17\) & \(7.8\pm 1.1\) & FOC 2 & PHM \\ E26 & \(0.01\pm 0.16\) & \(-0.09\pm 0.29\) & \(-0.06\pm 0.15\) & \(2.5\pm 1.3\) & FOC 2 & PHM \\ E30 & \(0.02\pm 0.15\) & \(-0.02\pm 0.22\) & \(0.02\pm 0.16\) & \(2.0\pm 1.2\) & FOC 2 & PHM \\ \hline \end{tabular}  
* In-Orbit Validation; 2 Full Operational Capability; 3 PHM: Passive Hydrogen Maser.  
\end{table} 
Table 5: Galileo mean values and standard deviations of coordinates and clock differences. solution is given in Equation (9). Term \(TSC_{X}\) is a GNSS-specific Time System Correction, which is dependent on the GNSS but independent of the receiver. Term \(dT_{R\alpha}\) is the receiver clock offset, which is independent of the GNSS. However we will see later that different receivers measure different \(TSC_{X}\) implying receiver biases which are dependent both on the receiver and the GNSS [16]. To monitor the GNSS specific time offset it is convenient to take \(TSC_{G}\) as the reference time scale and evaluate the difference \((TSC_{X}+dT_{R\alpha})-(TSC_{G}+dT_{R\alpha})\), with X referring to all the GNSS's different from GPS [17]. In such a way we obtain the sum of the system specific time offset relative to GPS and the receiver hardware inter-system bias (ISB) [17; 18; 19]. Table 8 lists these specific variables. \[p(t)=\varrho-c\cdot dt\big{(}t^{\prime}\big{)}+c\cdot(TSC_{X}+dT_{R\alpha})+ \frac{TZD}{\sin(El)} \tag{2}\]  
\[\varrho\varrho=\sqrt{\left[X(t^{\prime})+Y(t^{\prime})\cdot\omega_{e^{\prime}} t_{of}-x\right]^{2}+\left[Y(t^{\prime})-X(t^{\prime})\cdot\omega_{e^{\prime}} t_{of}-y\right]^{2}+\left[Z(t^{\prime})-z\right]^{2}} \tag{3}\]  
\[dt\big{(}t^{\prime}\big{)}=a_{0}+a_{1}\cdot\big{(}t^{\prime}-T_{\alpha c}-LS \big{)}+a_{2}\cdot\big{(}t^{\prime}-T_{\alpha c}-LS\big{)}^{2}+dT_{R\alpha} \tag{4}\]  
\[dT_{Rel}=-\frac{2\cdot\left(\overline{X}\cdot\overline{v}\right)}{c^{2}} \tag{5}\]  
\[t_{of}=\frac{\sqrt{\left(X-x\right)^{2}+\left(Y-y\right)^{2}+\left(Z-z\right) ^{2}}}{c} \tag{6}\]  
\[H=\left[