# Designation of the Quality of EGNOS+SDCM Satellite Positioning in the Approach to Landing Procedure
## 2 Research Method
### Mathematical Model of Multi-SBAS Solution for Aircraft Position using EGNOS and SDCM Data  
The presented approach employs the multi-system Multi-SBAS solution. The analysed example refers to the EGNOS and SDCM supporting systems. This will constitute the basis for the determination of the model of the resultant position of an aerial vehicle:  
\[\left\{\begin{array}{l}B_{m}=\frac{\alpha_{E}-B_{EGNOS}+\alpha_{S}-B_{SDCM}} {\alpha_{E}+\alpha_{S}}\\ L_{m}=\frac{\alpha_{E}-B_{EGNOS}+\alpha_{S}-L_{SDCM}}{\alpha_{E}+\alpha_{S}}\\ h_{m}=\frac{\alpha_{E}-B_{EGNOS}+\alpha_{S}-B_{SDCM}}{\alpha_{E}+\alpha_{S}} \end{array}\right. \tag{1}\]  
where:  
\(B_{EGNOS}\)--geodetic latitude from an individual SBAS/EGNOS solution [5; 10; 62],  
\(B_{SDCM}\)--geodetic latitude from an individual SBAS/SDCM solution [5; 10; 62],  
\(L_{EGNOS}\)--geodetic longitude from an individual SBAS/EGNOS solution [5; 10; 62],  
\(L_{SDCM}\)--geodetic longitude from an individual SBAS/SDCM solution [5; 10; 62],  
\(h_{EGNOS}\)--ellipsoidal height from an individual SBAS/EGNOS solution [5; 10; 62],  
\(h_{SDCM}\)--ellipsoidal height from an individual SBAS/SDCM solution [5; 10; 62],  
\(\alpha_{E}\)--measurement weight from the SBAS/EGNOS solution,  
\(\alpha_{S}\)--measurement weight from the SBAS/SDCM solution.  
The measurement weight was defined as a function of the number of tracked GPS satellites, for which EGNOS and SDCM corrections were recorded, as presented below:  
\[\alpha_{E}=\frac{1}{ns_{EGNOS}}\quad\mathrm{i}\quad\alpha_{s}=\frac{1}{ns_{ SDCM}} \tag{2}\]  
where:  
\(ns_{EGNOS}\) is the number of GPS satellites for which EGNOS corrections were defined in the SBAS/EGNOS solution,  
\(ns_{SDCM}\) is the number of GPS satellites for which EGNOS corrections were defined in the SBAS/SDCM solution.  
Equation (1) describes the weighted mean model, taking into account the measurement weights \((\alpha_{E},\alpha_{s})\) that enable the integration of individual EGNOS and SDCM navigation solutions. Additionally, it should be noted that the resultant model of the position of the aircraft is expressed in ellipsoidal coordinates \(BLh\) (\(B\)-Latitude, \(L\)-Longitude, and \(h\)-ellipsoidal height). The standard deviations for coordinates (\(B_{m}\), \(L_{m}\), and \(h_{m}\)) will also be calculated for the determined resultant position of the aerial vehicle, in the form:  
\[\left\{\begin{array}{l}\delta B=\sqrt{\frac{\mathbb{P}_{B,ECNOS}\cdot\mathbb{ a}_{E}\cdot\mathbb{v}_{B,ECNOS}+\mathbb{v}_{B,SDCM}\cdot\mathbb{a}_{S}\cdot \mathbb{v}_{B,SDCM}}{n-1}}\\ \delta L=\sqrt{\frac{\mathbb{P}_{L,ECNOS}\cdot\mathbb{a}_{E}\cdot\mathbb{v}_{ L,ECNOS}+\mathbb{v}_{L,SDCM}\cdot\mathbb{a}_{S}\cdot\mathbb{v}_{L,SDCM}}{n-1}}\\ \delta h=\sqrt{\frac{\mathbb{P}_{L,ECNOS}\cdot\mathbb{a}_{E}\cdot\mathbb{v}_{ B,ECNOS}+\mathbb{v}_{B,SDCM}\cdot\mathbb{a}_{S}\cdot\mathbb{v}_{B,SDCM}}{n-1}} \end{array}\right. \tag{3}\]  
where:  
\(v_{B}\)--corrections along the \(B\) axis,  
\(v_{B,EGNOS}=B_{ECNOS}-B_{m}\),  
\(v_{B,SDCM}=B_{SDCM}-B_{m}\),  
\(v_{L}\)--corrections along the \(L\) axis,  
\(v_{L,EGNOS}=L_{EGNOS}-L_{m}\),  
\(v_{L,SDCM}=L_{SDCM}-L_{m}\),  
\(v_{h}\)--corrections along the \(h\) axis,  
\(v_{h,EGNOS}=h_{ECNOS}-h_{m}\),  
\(v_{h,SDCM}=h_{SDCM}-h_{m}\),  
\(n\)--number of measurements, \(n=2\),  
\(n-1\)--number of degrees of freedom.  
The standard deviations calculated for the resultant position of the aerial vehicle will determine the internal accuracy of the presented mathematical Equation (1).