## 2 Methodology
### Satellite-Based Augmentation System (SBAS) Orbit and Clock Correction Evaluation Method
#### 2.1.2 SBAS-Derived Orbit and Clock Error Calculation
\begin{array}{c}\mathbf{e}_{along}=\frac{\mathbf{v}_{IGS}}{|\mathbf{v}_{IGS}|} \\ \mathbf{e}_{cross}=\frac{\mathbf{r}_{ICS}\times\mathbf{v}_{IGS}}{|\mathbf{r}_{IGS}\times\mathbf{v} _{IGS}|}\\ \mathbf{e}_{radial}=\mathbf{e}_{along}\times\mathbf{e}_{cross}\end{array} \right. \tag{10}\]  
where \(\mathbf{r}_{IGS}\), \(\mathbf{v}_{IGS}\) are satellite position and velocity vector in the IGS orbital frame. \(\Delta r_{x}\), \(\Delta r_{y}\) and \(\Delta r_{z}\) are the orbit errors in the x-, y- and \(z\)-axis of IGS orbital frame, which are computed by SBAS-derived orbits minus IGS precise orbits. It is noted that the differences between COM and APC should be removed according to Equation (6). Similar to orbit comparison, SBAS-derived clocks and IGS precise clock products cannot be compared directly. SBAS-derived clock offsets computed with Equation (5) are referred to C1 (L1 C/A signal) [1]. However, the ionosphere-free combination of P1 (L1 P signal) and P2 (L2 P signal) observations are employed in the generation of IGS precise clock products [26; 27]. A corresponding correction should be applied to eliminate the inconsistency between signal combinations adopted by SBAS-derived clocks and IGS precise clock products. The hardware delay difference between C1 and P1/P2 ionosphere-free combination can be given as [28; 29]:  
\[ \begin{array}{ll}B_{\text{C1}}-B_{P1/P2}&=B_{\text{C1}}-\left(\frac{f_{1} ^{2}}{f_{1}^{2}-f_{2}^{2}}B_{P1}-\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}B_{P2} \right)\\ &=\left(B_{\text{C1}}-B_{P1}\right)-\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}(B_{ P1}-B_{P2})\\ &=DCB_{P1-\text{C1}}+\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}DCB_{P1-P2}\end{array}  \tag{11}\]  
where \(B_{\text{C1}}\), \(B_{P1}\) and \(B_{P2}\) are the hardware delays of C1, P1 and P2 respectively, \(B_{P1/P2}\) represents the hardware delay of P1, P2 ionosphere-free combination, \(f_{1}\), \(f_{2}\) are the frequency values of L1 and L2, \(DCB_{P1-\text{C1}}\) denotes differential code bias between P1 and C1, and \(DCB_{P1-P2}\) represents differential code bias between P1 and P2. The monthly mean values of \(DCB_{P1-\text{C1}}\) and \(DCB_{P1-P2}\) can be obtained from the Center of Orbit Determination in Europe (CODE) analysis center ([[ftp://ftp.aiub.unibe.ch/CODE/](ftp://ftp.aiub.unibe.ch/CODE/)]([ftp://ftp.aiub.unibe.ch/CODE/](ftp://ftp.aiub.unibe.ch/CODE/))). Correspondingly, the relationship between two clock offsets referred to C1 and P1/P2 can be expressed as:  
\[dT_{\text{C1}}=dT_{P1/P2}+DCB_{P1-\text{C1}}+\frac{{f_{2}}^{2}}{{f_{1}}^{2}- f_{2}^{2}}DCB_{P1-P2} \tag{12}\]  
Theoretically, the last term on the right side should be equal to \(-T_{GD}\) in Equation (5) [29]. Therefore, the last term in Equation (12) and the term of \(-T_{GD}\) in Equation (5) can be omitted in the clock comparisons. In addition, there exist differences in the underlying realization of the SBAS and IGS precise clock time scales. These differences, known as time reference offsets, affect all satellites of a constellation with the same value and vary from one epoch to another. To remove the time reference offsets, the average SBAS-minus-IGS clock differences of satellites in a constellation is computed at each epoch, and then the computed average is subtracted from all individual SBAS-minus-IGS clock differences [30].