## 2 Methodology
### Ionospheric Correction Evaluation Method
#### 2.2.1 SBAS Ionospheric Correction Calculation  
According to RTCA DO-229D, the SBAS ionospheric correction message provides ionospheric vertical delays for specified IGPs at an altitude of 350 km. In order to facilitate flexibility in the location of IGPs, a fixed definition of IGP locations is used. The predefined IGPs are divided into in 11 bands (numbered 0 to 10), Bands 0-8 are vertical bands, and Bands 9-10 are horizontal bands. The distribution of all predefined IGPs are shown in Figure 1. It should be noted that part of IGPs in Bands 9 and 10 are at the same location as IGPs in Bands 0 to 8. Each SBAS system only broadcast part of the bands to cover its service area and, even for the broadcasted bands, a mask is broadcast to define the IGP locations for providing the most efficient model of the ionosphere [1]. The vertical delays at IGPs are used to interpolate the vertical delay at the IPP of the observed satellite. The position of IPP is defined as the intersection of the ionospheric layer at an altitude of 350 km and a straight line from the user's location to the satellite. Usually, four IGPs defining a cell or three IGPs defining a triangle that surrounds the IPP are selected to carry out the interpolation according to the IGP mask information. The detailed selection process can further refer to [1]. If any of the selected IGPs is identified as \"Do Not Use\", there is no ionospheric correction available. Four IGPs around the IPP are preferentially selected to compute the ionospheric delay at IPP. However, if one of the four is identified as \"Not Monitored\", then the remaining three corrections are used if the IPP is within the triangular region defined by three IGPs. If there are less than three IGPs identified as \"Monitored\", the ionospheric correction is also not available. It is noted that four selected IGPs must be identified as \"Monitored\" for the IPP south of 75\({}^{\circ}\)S or north of 75\({}^{\circ}\)N.  
For the four-point interpolation, denoting the vertical delay at four IGPs as \(VIC_{i}\)\((i=1,\,2,\,3,\,4)\), the interpolated vertical IPP delay \(VIC\) can be expressed as a function of IPP latitude \(\varphi_{IPP}\) and longitude \(\lambda_{IPP}\):  
\[VIC_{IPP}(\varphi_{IPP},\,\lambda_{IPP})=\sum_{i=1}^{4}W_{i}(x_{IPP},\,y_{IPP} )VIC_{i} \tag{14}\]  
with  
\[\left\{ \begin{array}{ll}W_{1}(x_{IPP},\,y_{IPP})&=x_{IPP}y_{IPP}\\ W_{2}(x_{IPP},\,y_{IPP})&=(1-x_{IPP})y_{IPP}\\ W_{3}(x_{IPP},\,y_{IPP})&=(1-x_{IPP})(1-y_{IPP})\\ W_{4}(x_{IPP},\,y_{IPP})&=x_{IPP}(1-y_{IPP})\\ \end{array} \right. \tag{15}\]  
For IPPs between 85\({}^{\circ}\)N and 85\({}^{\circ}\)S,  
\[\left\{ \begin{array}{ll}x_{IPP}=\frac{\lambda_{IPP}-\lambda_{1}}{\lambda_{ 2}-\lambda_{1}}\\ y_{IPP}=\frac{\varphi_{IPP}-\varphi_{1}}{\varphi_{2}-\varphi_{1}}\\ \end{array} \right. \tag{16}\]  
where \(\lambda_{1}\) denotes the longitude of IGPs west of IPP, \(\lambda_{2}\) is the longitude of IGPs east of IPP, \(\varphi_{1}\) represents the latitude of IGPs south of IPP, \(\varphi_{2}\) is the latitude of IGPs north of IPP. Note that if \(\lambda_{1}\) and \(\lambda_{2}\) cross 180\({}^{\circ}\) of longitude, the calculation of \(x_{IPP}\) should account for the discontinuity in longitude. For IPPs north to 85\({}^{\circ}\)N or south to 85\({}^{\circ}\)S,  
\[\left\{ \begin{array}{ll}y_{IPP}=\frac{|\varphi_{IPP}|-85^{\circ}}{10^{\circ} }\\ x_{IPP}=\frac{\lambda_{IPP}-\lambda_{2}}{90^{\circ}}\cdot(1-2y_{IPP})+y_{IPP} \\ \end{array} \right. \tag{17}\]  
\(\lambda_{3}\) is longitude of the closest IGP to the west of the IPP. For the three-point interpolation between 75\({}^{\circ}\)S and 75\({}^{\circ}\)N, a similar mathematical formulation can be used:  
\[VIC_{IPP}(\varphi_{IPP},\,\lambda_{IPP})=\sum_{i=1}^{3}W_{i}(x_{IPP},\,y_{IPP })VIC_{i} \tag{18}\]  
Figure 1: The predefined global ionospheric grid point (IGP) grids. with  
\[\left\{ \begin{array}{c}W_{1}(x_{IPP},\,y_{IPP})=y_{IPP}\\ W_{2}(x_{IPP},\,y_{IPP})=1-x_{IPP}-y_{IPP}\\ W_{3}(x_{IPP},\,y_{IPP})=y_{IPP}\end{array} \right.