## 4 The WADGPS Master Station Processes
### Satellite Ephemeris and Clock Corrections  
This section describes the MS procedures for satellite ephemeris and clock errors estimations. Figure 7 depicts the flow chart regarding the calculations of the satellites ephemeris and clock errors including the Common View Time Transfer (CVTT), ephemeris error estimation, satellite clock error estimation, and the User Differential Range Error (UDRE) estimation flow chart. The CVTT filter synchronizes the measurements with a common reference time and decouples the measurements sequentially for each satellite to eliminate the receiver clock bias. To find the difference of the clock biases between two RSs, CVTT filter obtains the synchronized pseudorange residuals from the first difference between the pseudorange residuals of two stations as shown in Equation (19), and the CVTT implementation is illustrated in Figure 8 [12]:  
\[\Delta_{i,I}^{j}=\Delta\rho_{i}^{j}-\Delta\rho_{I}^{j}=\Delta R^{j}\cdot\left( l_{i}^{j}-l_{i}^{j}\right)+\Delta b_{i,I}+\
u_{i,I}^{j} \tag{19}\]  
where:  
\(\Delta R^{i}\) is the ephemeris error,  
\(l_{i}^{j}\) is the unit line of sight vector from the \(i\) RS to the \(j\) satellite,  
\(l_{i}^{j}\) is the unit line of sight vector from the \(I\) RS to the \(j\) satellites,  
\(\left(l_{i}^{j}-l_{i}^{j}\right)\) is the line of sight difference,  
\(\Delta b_{i,I}\) is the clock difference, and  
Figure 7: Ephemeris and clock errors estimation flow chart. \(\
u_{i,I}^{j}\) is the measurement noise. Then, the clock bias difference, \(\Delta\hat{\rho}_{i,I}\), is described in Equation (20):  
\[\Delta\hat{\rho}_{i,I}=\frac{1}{k}\sum_{j=1}^{k}\Delta_{i,I}^{j} \tag{20}\]  
where \(k\) is the number of satellites in the common view of both RSs. Through the CVTT module, the pseudorange residuals from all RSs are synchronized based on a common clock, and the pseudorange residuals consist of satellite ephemeris and clock errors. The corrections to the ephemeris error and the clock error have to be sent frequently, and they occupy lots of bandwidth. To reduce the bandwidth, separating the satellite clock error term is necessary. Therefore, the single difference is used to remove the satellite clock error term as shown in the following equation:  
\[\Delta\tilde{\rho}_{i}^{j}-\Delta\tilde{\rho}_{m}^{j}=\Delta R^{j}\cdot\left( l_{i}^{j}-l_{m}^{j}\right)+\varepsilon_{i}^{j} \tag{21}\]  
where:  
\(\Delta R^{j}\) is ephemeris error which is this process solving for, and  
the subscript \"\(m\)\" denotes the key RS which has the smallest variance [12]. Then, the Equation (21) is re-written in matrix forms as follows:  
\[z=H_{e}\cdot\Delta R^{j}+\
u \tag{22}\]  
where:  
\[z= \begin{bmatrix}\Delta\tilde{\rho}_{i}^{j}-\Delta\tilde{\rho}_{m}^{j}\\ \vdots\\ \Delta\tilde{\rho}_{N-1}^{j}-\Delta\tilde{\rho}_{m}^{j}\end{bmatrix} ,\,H_{e}= \begin{bmatrix}l_{i}^{j}-l_{m}^{j}\\ \vdots\\ l_{N-1}^{j}-l_{m}^{j}\end{bmatrix} \]  
Figure 8: The common view time transfer flow chart. \(\Delta R^{j}\) is the ephemeris error which will be denoted as \(x\) in following discussions,  
\(N\) is number of synchronized RSs, and  
\(v\) is measurement noise with zero mean and variance of \(W\). As indicated in Equation (23), the satellite position errors are estimated by the minimum variance estimator [4]:  
\[\hat{x}_{M{V}^{\prime}}=(\Lambda^{-1}+H^{T}W^{-1}H)^{-1}H^{T}W^{-1}z \tag{23}\]  
where:  
\[\Lambda=E\Big{[}\,xx^{T}\,\Big{]},\ W=\mathrm{cov}\left(\
u\right)\]  
After estimating the ephemeris error by the minimum variance method, the clock error measurements for all satellites can be derived from the synchronized pseudorange residuals.