## 2 Methodology
### Satellite's Coordinates  
The satellite's coordinates for the considered low-orbit satellite are given in the Earth-centered, Earth-fixed coordinate (ECEF) system. The Earth-centered, Earth-fixed coordinate system (known also as the geocentric coordinate system) is a Cartesian spatial reference framework for representing, e.g., satellite coordinates referring to Earth's center of mass [14].  
The considered satellite has an orbital period of approximately 180 min with an average speed of 4.2 km/s, and maintains an average distance from Earth of 631.7 km.  
The considered errors in the satellite's coordinates due to acquisition and orbital propagation errors are taken into account [15], and the absolute values considered for every direction are listed in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Direction** & **Value (km)** \\ \hline X & 0.354 \\ Y & 0.729 \\ Z & 0.661 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Absolute error for satellite’s coordinates for each direction.  
Figure 2: Model.  
In this context, the given satellite's ECEF coordinates are transformed into the satellite's coordinates in the East-North-Up (ENU) reference system; the reference frames are depicted in Figure 3. The origin of the ENU reference frame results in ships' positions (commonly determined by a Global Navigation Satellite System (GNSS) [16]) and the two axes, \(Y\) and \(X\), are tangential to Earth's surface at that point [17]. This coordinate transformation is applied to convey the satellite's position with respect to the ATP system's position on Earth.  
The transformation between satellite ECEF and ENU coordinates is presented in [18]; below, the equations will be presented considering the following application scenario.  
The GNSS provides the positions of ground points in geodetic coordinates:  
\[\mathrm{P}_{\mathrm{GEO}}=(\varphi,\lambda,\mathrm{h}) \tag{1}\]  
where, in this case, \(\varphi\), \(\lambda\) and \(h\) are, respectively, the latitude, longitude and altitude of the considered ground point.  
Assuming that P represents the geodetic coordinates of the ship's position:  
\[\mathrm{P}_{\mathrm{GEO}}=(\varphi,\lambda,0) \tag{2}\]  
it is possible to determine the ship's coordinates in the ECEF reference frame:  
\[\mathrm{P}_{\mathrm{ECEF}}=(\mathrm{P}_{X},\mathrm{P}_{Y},\mathrm{P}_{Z}) \tag{3}\]  
by following the geodetic-to-ECEF coordinates' transformation presented in [19].  
The satellite's coordinates in the ECEF reference frame can be expressed as in Equation (4).  
\[\mathrm{S}_{\mathrm{ECEF}}=(\mathrm{S}_{X},\mathrm{S}_{Y},\mathrm{S}_{Z}) \tag{4}\]  
The satellite coordinates transformation from the ECEF reference frame, and the ENU reference frame can be expressed as follows:  
\[\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}=\begin{bmatrix}-\sin(\lambda)&\cos(\lambda)&0\\ -\sin(\varphi)\cos(\lambda)&-\sin(\varphi)\cos(\lambda)&\cos(\varphi)\\ \cos(\varphi)\cos(\lambda)&\cos(\varphi)\sin(\lambda)&\sin(\varphi)\end{bmatrix} \begin{bmatrix}S_{X}-P_{X}\\ S_{Y}-P_{Y}\\ S_{Z}-P_{Z}\end{bmatrix} \tag{5}\]  
in which X, Y and Z are the satellite's coordinates in the ENU reference frame as defined in Equation (6).  
\[\mathrm{S}_{\mathrm{ENU}}=(\mathrm{X},\mathrm{Y},\mathrm{Z}) \tag{6}\]  
Figure 3: ECEF and ENU reference frames.  
Considering the ENU reference frame as the Earth reference frame, the transformation matrix from Earth to satellite \(M_{E}^{\mathrm{S}}\) can be expressed through Equation (6).  
\[\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}=M_{E}^{\mathrm{S}}\begin{bmatrix}S_{X}-P_{X}\\ S_{Y}-P_{Y}\\ S_{Z}-P_{Z}\end{bmatrix} \tag{7}\]