# Galileo Single Point Positioning Assessment Including FOC Satellites in Eccentric Orbits
## Materials and Methods  
### Methodology  
The methodology used to achieve single-point positioning with GNSS pseudoranges is well known and widely described [16; 17]. However, a brief recall is provided here to describe the algorithms used to implement the software developed according to flowchart shown in Figure 2.  
GNSS receivers provide four types of measurements for each frequency: pseudorange, Doppler, carrier phase, and carrier-to-noise. The first three measurements are used to achieve receiver position [7], while the carrier-to-noise density provides an indication of the accuracy of the tracked GNSS satellite observations and the noise density as seen by the receiver's front-end [9]. It also indicates the level of noise present in the measurements. The pseudorange measurement \(P\) is expressed by the following relation:  
\[P=\rho+c\delta t_{RX}-c\delta t_{S}+T+I+M_{P}+c\delta t_{Sag}+\epsilon_{p} \tag{2}\]  
where \(\rho\) is the geometrical distance, \(c\delta t_{RX}\) is the receiver clock offset, \(c\delta t_{S}\) is the satellite clock offset including the time group delay \(Tgd\), \(T\) is the tropospheric delay, \(I\) is the ionospheric delay, \(M_{P}\) is the code multipath, \(\delta t_{Sag}\) is the delay due to the [PERSON] effect, and \(\epsilon_{p}\) comprises the receiver hardware delays and receiver noise of the code measurements. Satellite clock offset correction parameters are  
Figure 2: Flowchart representation of the algorithm developed for the determination of receiver position and Dilution Of Precision (DOP) parameters; TGD stands for Time Group Delay, while brdc for broadcast. transmitted in navigation data, while ionosphere and troposphere delays were estimated by using the Klobuchar and Hopfield models, respectively. To achieve the receiver coordinates and offset, at least four equations that express the distance between unknown receiver position and the known \(i\)th satellite position are necessary; they can be linearized around an a priori value obtaining:  
\[\underline{\Delta\rho}=H\cdot\underline{\Delta X}+\varepsilon \tag{3}\]  
where: \(\underline{\Delta\rho}\) (\(n\)\(x\) 1) is the difference vector between the \(n\) raw pseudoranges and a priori information (with \(n\) number of Galileo observations), \(H(nx4)\) is the design matrix, \(\underline{\Delta X}(4x1)\) is the state vector, and \(\underline{\varepsilon}\) is the residuals vector, which includes all unmodeled errors (measurement noise, multipath, and so on). The explicit forms of \(H\) and \(\underline{\Delta X}\) are the following:  
\[H= \begin{bmatrix}\frac{X_{a}-X_{1}}{\rho_{a}^{2}}&\frac{Y_{a}-Y_{1}}{\rho_{a}^ {2}}&\frac{Z_{a}-Z_{1}}{\rho_{a}^{2}}&1\\ \frac{X_{a}-X_{2}}{\rho_{a}^{2}}&\frac{Y_{a}-Y_{2}}{\rho_{a}^{2}}&\frac{Z_{a}- Z_{2}}{\rho_{a}^{2}}&1\\ \vdots&\vdots&\vdots&\vdots\\ \frac{X_{a}-X_{i}}{\rho_{a}^{2}}&\frac{Y_{a}-Y_{i}}{\rho_{a}^{2}}&\frac{Z_{a}- Z_{i}}{\rho_{a}^{2}}&1\\ \vdots&\vdots&\vdots&\vdots\\ \frac{X_{a}-X_{a}}{\rho_{a}^{2}}&\frac{Y_{a}-Y_{i}}{\rho_{a}^{2}}&\frac{Z_{a}- Z_{a}}{\rho_{a}^{2}}&1\end{bmatrix} \]  
\[\underline{\Delta X}= \begin{bmatrix}\Delta X&\Delta Y&\Delta Z&\Delta(c\delta T _{RX})\end{bmatrix} \]  
where \(n\) is the number of Galileo simultaneous measurements, the subscript indicates the \(i\)th Galileo satellite used, \((X_{a},Y_{a},Z_{a})\) denotes the approximate position of the receiver, \((X_{i},Y_{i},Z_{i})\) denotes the position of the the \(i\)th Galileo satellite, \(\rho_{a}^{i}\) is the approximate a priori range between the \(i\)th Galileo satellite and receiver, \(\Delta X\),\(\Delta Y\),\(\Delta Z\) are the differences between \"true\" and approximate receiver X, Y, and \(Z\) coordinates, and \((c\delta T_{RX})\) is the receiver clock offset relative to the time scale of the Galileo system.