## 2 Methodology
### Satellite-Based Augmentation System (SBAS) Orbit and Clock Correction Evaluation Method
#### 2.1.1 Orbit and Clock Calculation with SBAS Corrections  
As previously mentioned, the SBASs provide long-term and fast corrections for satellite orbit and clock. The long-term correction message contains components to correct the satellite position, velocity, clock offset, and clock drift. The position and clock offset components are always broadcasted while the velocity and clock drift components are not always included in the message. There is an indicator in the message to show whether the velocity and clock drift components are available or not. When the velocity code in the message is not set, velocity and clock drift should be treated as zero. As for fast corrections, they are designed to correct the rapidly changing part of satellite clock errors and broadcast in the form of pseudo-range correction (PRC) [1].  
The corrected satellite position \(\mathbf{r}(t)\) at time \(t\) can be computed as  
\[\mathbf{r}(t)=\mathbf{r}_{broadcast}(t)+\left[\begin{array}{c}\delta x\\ \delta y\\ \delta z\end{array}\right]+\left[\begin{array}{c}\delta\dot{x}\\ \delta\dot{y}\\ \delta\dot{z}\end{array}\right]\cdot(t-t_{0}) \tag{1}\]  
where \(\mathbf{r}_{broadcast}(t)\) is the satellite position vector computed using the broadcast ephemeris, \(\left[\begin{array}{ccc}\delta x&\delta y&\delta z\end{array}\right]^{T}\) and \(\left[\begin{array}{ccc}\delta\dot{x}&\delta\dot{y}&\delta\dot{z}\end{array} \right]^{T}\) denote satellite position and velocity correction vectors, \(t_{0}\) is the time of day applicability of the long-term correction. It is noted that the issue of data (IOD) in the broadcast ephemeris and that in the long-term correction message must match with each other.  
The correction to the satellite clock using the long-term correction message can be given as:  
\[\delta t_{long}(t)=\delta a_{f_{0}}+\delta a_{f_{1}}\cdot(t-t_{0}) \tag{2}\]  
where \(\delta a_{f_{0}}\). and \(\delta a_{f_{1}}\) represent satellite offset and drift components in the message. As mentioned above, in addition to the long-term correction message, the fast correction message also contributes to the satellite clock correction, which is given as:\[\delta t_{\text{fast}}(t)=\Big{(}PRC\Big{(}t_{of}\Big{)}+RRC\Big{(}t_{of}\Big{)} \cdot\Big{(}t-t_{0f}\Big{)}\Big{)}/c \tag{3}\]  
where \(c\) and \(t_{of}\) denote velocity of light and time of day applicability of the fast correction, \(PRC\Big{(}t_{of}\Big{)}\) is the broadcast PRC value, and \(RRC\Big{(}t_{of}\Big{)}\) is the range-rate correction (RRC). However, RRC is not included in the fast correction message. It can be computed by differencing the current and previous PRC values as:  
\[RRC\Big{(}t_{of}\Big{)}=\Big{(}PRC\Big{(}t_{of}\Big{)}-PRC\Big{(}t_{of,previous} \Big{)}\Big{)}/\Big{(}t_{of}-t_{of,previous}\Big{)} \tag{4}\]  
where \(PRC\Big{(}t_{of,\text{ previous}}\Big{)}\) is the previous broadcast PRC value at the time of day applicability \(t_{of,\text{ previous}}\). After applying both long- and fast-term correction, the corrected satellite clock offset \(dT(t)\) for L1 C/A signal is given by:  
\[dT(t)=dT_{broadcast}(t)-T_{GD}+\delta t_{long}(t)+\delta t_{fast}(t) \tag{5}\]  
\(dT_{broadcast}(t)\) is the satellite clock offset at the time of \(t\) computed with the GPS broadcast navigation data, and \(T_{GD}\) is the satellite time group delay provided in the the GPS broadcast navigation data.