## 4 The WADGPS Master Station Processes
### Ionospheric Corrections  
The major functions of the WADGPS master station (MS) are the ionospheric corrections model and the satellites ephemeris and clock errors estimation algorithms. After the dual-frequency carrier smoothing filter outputs the smoothed ionospheric delay, the MS then converts all ionospheric slant delays to the vertical delays at the Ionosphere Pierce Points (IPPs) by the Obliquity Factor (_OF_) [11]. By doing so, the ionospheric measurement is independent of the elevation angle, and it will be more convenient to use. The location of IPP is defined as the intersection of the line segment from the RS to the satellite and an ellipsoid with constant high above 350 km from earth's surface [16]. The next step is to create a vertical ionospheric delay model from the IPP measurements to estimate the ionospheric vertical delay at the Ionosphere Grid Points (IGP), \(\hat{I}_{G}\). The Grid Ionospheric Vertical Error (_GIVE_) is provided for each IGP which is a confidence bound of the corrected ionospheric delay residual at the IGP. The following Equations (15) and (16) estimate the ionospheric vertical delay at the IGP and _GIVE_ by the weighted least-squared algorithm, and the derivations of these equations and _OF_ are detailed in [11]:  
\[\hat{I}_{G}=I_{Klobuchar,G}\cdot\sum_{i=1}^{k}\Biggl{(}\frac{I_{measure,i}}{I_{ Klobuchar,i}}\cdot\omega_{i}\Biggr{)}/\sum_{i=1}^{k}\omega_{i} \tag{14}\]  
\[GIVE=3.29\,/\sum_{i=1}^{k}\omega_{i} \tag{15}\]  
where:  
\(\omega_{i}\) is the weight of the \(i^{th}\) IPP measurement,  
\(I_{Klobuchar,G}\) is the vertical ionospheric delay at the grid point using the Klobuchar model parameters [1],  
\(I_{Measure,i}\) is the vertical ionospheric delay measurement at the pierce point, and  
\(I_{Klobuchar,i}\) is the vertical ionospheric delay at the pierce point using the Klobuchar model parameters. The weight is calculated by the inverse of the vertical delay measurements variance according to the correlation distance between the grid point and the IPP as shows in Equation (17) [11]. Figure 4: The dual-frequency smoothing of ionospheric delay and pseudorange. \[\omega_{i}=\frac{\sigma_{i}}{\Delta} \tag{16}\]  
where:  
\(\sigma_{i}\) is \(i^{th}\) vertical ionospheric delay measurements variance [11], and  
\(\Delta\) is a function of the correlation distance of the ionosphere [11]. Specifically, this model scales the measurements using the Klobuchar model to transport the measurement from the IPP location to the location of the desired grid point through the relationship of latitude and longitude dependence provided by the Klobuchar model [16]. The generation process of this grid model is illustrated in the upper plot of Figure 5.  
Figure 5: The WADGPS ionospheric vertical delay grid model flow chart. The bottom plot of Figure 5 describes the ionospheric correction algorithm for WADGPS user receiver which uses the nearest IGPs around the IPP to estimate the vertical ionospheric delay at a specific IPP by the interpolation algorithm. The interpolation algorithm is expressed as:  
\[I_{IPP,Y,i}^{Eni}=\sum_{i=1}^{4}W_{i}\left(x_{IPP,i},y_{IPP,i}\right)\cdot I_{ IGP,Y,i} \tag{17}\]  
\[UIVE_{IPP,i}=\sum_{i=1}^{4}W_{i}\left(x_{IPP,i},y_{IPP,i}\right)\cdot GIVE_{i} \tag{18}\]  
where:  
\(I_{IPP,Y,i}^{Eni}\) is the vertical ionospheric delay at the \(i^{\text{th}}\) IPP, estimated with the broadcasted ionospheric corrections,  
\(W_{i}(x_{IPP,i}y_{IPP,i})\) is the weighting factor of the \(i^{\text{th}}\) IPP whose location is \((x_{IPP,i}y_{IPP,i})\)[8],  
\(I_{IGP,Y,i}\) is the broadcast vertical ionospheric delay at \(i^{\text{th}}\) IGP,  
\(UIVE_{IPP,i}\) is the user ionospheric vertical error (UIVE) which is a 99.9% confidence (error bound) on the post-correction ionospheric vertical delay residual [8], and  
\(GIVE_{i}\) is a confidence bound of the corrected ionospheric delay residual at the \(i^{\text{th}}\) IGP.