## 2 Methodology
### Satellite-Based Augmentation System (SBAS) Orbit and Clock Correction Evaluation Method
#### 2.1.2 SBAS-Derived Orbit and Clock Error Calculation  
The IGS precise orbit products are used as the references in the assessment of SBAS-derived orbits, therefore adequate care must be taken to ensure the compatibility of different types of orbit. Firstly, SBAS-derived orbits are referred to the antenna phase center (APC), while IGS precise orbit products are provided with respect to the center-of-mass (COM). Evidently, the comparison of these two types of orbits cannot be conducted directly. The IGS antenna information file provides the satellite antenna phase center offset (PCO) vectors in the spacecraft body-fixed frame. The coordinate transformation from COM to APC for IGS precise orbits can be computed with the known PCO vector \(d\mathbf{r}_{PCO}\) and the modeled spacecraft attitude, which can be given as [15; 16]  
\[\mathbf{r}_{APC}=\mathbf{r}_{COM}+\mathbf{R}_{spacecraft}^{ECEF}\cdot d\mathbf{r}_{PCO} \tag{6}\]  
where \(\mathbf{r}_{APC}\), \(\mathbf{r}_{COM}\) are satellite APC and COM Earth-centered Earth-fixed (ECEF) Cartesian coordinate vectors, \(\mathbf{R}_{spacecraft}^{ECEF}\) denotes the transformation matrix from the spacecraft body-fixed frame to the ECEF frame. The transformation matrix \(\mathbf{R}_{spacecraft}^{ECEF}\) is expressed with the unit vectors of \(x,\ y\) and \(z\) axis in the spacecraft body-fixed frame as follow [17]  
\[\mathbf{R}_{spacecraft}^{ECEF}=\Big{[}\ e_{spacecraft,x}\ \ \ \mathbf{e}_{spacecraft,y}\ \ \ \mathbf{e}_{spacecraft,z}\ \ \Big{]} \tag{7}\]  
where  
\[\left\{ \begin{array}{c}\mathbf{e}_{spacecraft,z}=-\frac{\mathbf{r}_{sat}}{|\mathbf{r}_ {sat}|}\\ \mathbf{e}_{spacecraft,y}=\mathbf{e}_{spacecraft,z}\times\mathbf{e}_{spacecraft,x}\end{array} \right. \tag{8}\]  
\(\mathbf{r}_{sat}\) is the ECEF Cartesian coordinate vector of satellite, and \(|*|\) denotes the norm of the vector. The unit vector of x axis \(\mathbf{e}_{spacecraft,x}\) can be obtained by rotating the unit vector of the spacecraft velocity around the \(z\)-axis by yaw angle. The yaw attitude model of GPS Block IIA, IIR and IIF satellites can be referred to [17; 18; 19]. In addition, SBAS-derived and IGS precise orbits are formally referred to different reference frames. The current IGS precise orbits are referred to IGS14, which is aligned with the latest International Terrestrial Reference Frame (ITRF) called ITRF2014 [20]. WAAS adopts WGS84 as the coordinate reference system to broadcast satellite orbit corrections [21], and the most recent WGS84 realization (G1762) agrees with ITRF2008 at the centimeter level [22]. As presented in the International Terrestrial Reference Service (ITRS) website ([[http://itrf.ensg.ign.fr/trans_para.php](http://itrf.ensg.ign.fr/trans_para.php)]([http://itrf.ensg.ign.fr/trans_para.php](http://itrf.ensg.ign.fr/trans_para.php))), the differences between ITRF2008 and ITRF2014 are at the millimeter-level. EGNOS reference frame, named EGNOS Terrestrial Reference Frame (ETRF), is periodically aligned on ITRF within a consistency of a few centimeters [23]. We cannot find any official documentation or information about the reference frames adopted for the MSAS, GAGAN, and SDCM. However, an alignment to WGS84 can be expected since the requirements of international air navigation must be fulfilled [5]. Therefore, the differences between reference frames mentioned above would not exceed a few centimeters, which can be neglected in the accuracy assessment of SBAS-derived orbits. Usually, the orbit errors are presented in the satellite along-track, cross-track, and radial directions. The transformation of orbit errors from ECEF to the satellite along-track, cross-track and radial frame is given by [24; 25]:  
\[\left[ \begin{array}{c}\Delta r_{along}\\ \Delta r_{cross}\\ \Delta r_{radial}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{e}_{along}& \mathbf{e}_{cross}&\mathbf{e}_{radial}\\ \end{array} \right]\cdot\left[ \begin{array}{c}\Delta r_{x}\\ \Delta r_{y}\\ \Delta r_{z}\end{array} \right] \tag{9}\]  
with  
\[\left\{