# Construction Progress and Aviation Flight Test of BDSBAS
## 2 Overview of BDSBAS
### BDSBAS SF Solution
#### 2.3.1 Long-Term Correction and Fast Correction Calculation  
Limited by the data broadcast rate of SBAS satellites, the ephemeris corrections can be divided into long-term corrections and fast corrections. Among them, the long-term corrections are used to describe the slow change part of the satellite orbit and clock, and the broadcast period is 120 s; the fast corrections are used to describe the fast change part of the satellite clock, and the broadcast period is 6 s ([PERSON] et al., 2012).  
Long-Term Correction  
The BDSBAS SF long-term corrections are broadcast by the augmentation message type 25 (MT25), including the long-term corrections of satellite clock and the long-term corrections of satellite orbit, which are used to separately correct the GPS satellite clock and orbit.  
Figure 4: BDSBAS SF solution flow.  
Figure 3: BDSBAS workflow.  
The satellite clock long-term correction \(\delta\Delta t_{SV}(t)\) at the current observation time \(t\) is calculated by:  
\[\delta\Delta t_{SV}(t)=\delta a_{f0}+\delta a_{f1}(t-t_{0}) \tag{1}\]  
where \(\delta a_{f0}\) is the satellite clock deviation. \(\delta a_{f1}\) is the satellite clock deviation rate. \(t_{0}\) is the reference time of the corrections. After that, the corrected satellite clock error \(B_{i}\) can be obtained as follows:  
\[B_{i}=c\Big{(}\Delta t_{i}^{(s)}+\delta\Delta t_{SV,i}\Big{)} \tag{2}\]  
where index \(i\) refers to the \(i\)-th satellite. \(\Delta t_{i}^{(s)}\) is the satellite clock error before correction. \(c\) is the speed of light.  
The satellite orbit correction \(\big{[}\delta x_{k}(t)\quad\delta y_{k}(t)\quad\delta z_{k}(t)\big{]}^{\rm T}\) is calculated by:  
\[\left[\begin{array}{c}\delta x_{k}(t)\\ \delta y_{k}(t)\\ \delta z_{k}(t)\end{array}\right]=\left[\begin{array}{c}\delta x\\ \delta y\\ \delta z\end{array}\right]+\left[\begin{array}{c}\delta\dot{x}\\ \delta\dot{y}\\ \delta\dot{z}\end{array}\right](t-t_{0}) \tag{3}\]  
where \(\big{[}\delta x\quad\delta y\quad\delta z\big{]}^{\rm T}\) is the satellite orbit deviation. \(\big{[}\delta\dot{x}\quad\delta\dot{y}\quad\delta\dot{z}\big{]}^{\rm T}\) is the satellite orbit deviation rate. After that, the corrected satellite orbit coordinate \(\big{[}\hat{x}_{k,i}\quad\hat{y}_{k,i}\quad\hat{z}_{k,i}\big{]}^{\rm T}\) of satellite \(i\) can be obtained as follows:  
\[\left\{\begin{array}{c}\hat{x}_{k,i}=x_{k,i}+\delta x_{k,i}\\ \hat{y}_{k,i}=y_{k,i}+\delta y_{k,i}\\ \hat{z}_{k,i}=z_{k,i}+\delta z_{k,i}\end{array}\right. \tag{4}\]  
where \(\big{[}x_{k,i}\quad y_{k,i}\quad z_{k,i}\big{]}^{\rm T}\) is the satellite orbit coordinate before correction. \(\big{[}\delta x_{k,i}\quad\delta y_{k,i}\quad\delta z_{k,i}\big{]}^{\rm T}\) is the satellite orbit correction.  
The BDSBAS SF fast corrections are currently broadcast by the augmentation message type 2-4 (MT2-4). Let \(FC_{i,\rm current}\) be the most recent fast correction that user received, and \(FC_{i,\rm previous}\) be a previous fast correction that user received. Then, the range rate correction (RRC) for satellite \(i\) can be calculated by the fast corrections as follows:  
\[RRC_{i}=\frac{FC_{i,\rm current}-FC_{i,\rm previous}}{t_{i,0t}-t_{i,0t\_\rm previous}} \tag{5}\]  
where \(t_{i,0t}\) is the time of applicability of \(FC_{i,\rm current}\), which is the start of the epoch of the SBAS network time (SNT) second that is coincident with the transmission at the SBAS satellite of the first symbol of the message block: \(t_{i,0t\_\rm previous}\) is the time of applicability of \(FC_{i,\rm previous}\).  
As a result, user's pseudo-range correction (PRC) for satellite \(i\) can be further obtained by  
\[PRC_{i}=FC_{i,\rm previous}+RRC_{i}\times(t-t_{i,0t}) \tag{6}\]  
where \(t\) denotes the current calculation time of user.