# 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 Parameters of Quality Positioning  
The analysis of the quality parameters of Multi-SBAS positioning in the presented research method is based on the determination of four parameters: accuracy, continuity, availability, and integrity of SBAS positioning. The values of the accuracy parameter of positioning the aerial vehicle were expressed in the ellipsoidal system BLh. For the purposes of this study, the following mathematical algorithms were used:  
* for the accuracy parameter: \[\left[ \begin{array}{c}dB\\ dL\\ dh\end{array} \right]=\left[ \begin{array}{c}B_{m}-B_{RTK}\\ L_{m}-L_{RTK}\\ h_{m}-h_{RTK}\end{array} \right]\] (4) where: \((dB,dL,dh)\)--accuracy of positioning the aerial vehicle [63], \((B_{m},L_{m},\ h_{m})\)--the resultant coordinates of the aerial vehicle determined with use of solution (1), \((B_{RTK},L_{RTK},h_{RTK})\)--the coordinates of the aerial vehicle calculated with use of the RTK (Real Time Kinematic) differential technique in the OTF (On The Fly) mode [61; 62]. Additionally, for the \((dB,\ dL,\ dh)\) parameters, the RMS (Root Mean Square) error was calculated [64]. It expresses the statistical measure of the accuracy of the determined resultant coordinates of the aerial vehicle. The value of the RMS error was calculated for the results of the \(dB,\ dL\), and \(dh\) parameters in the following way:  
\[\left\{ \begin{array}{l}RMS_{dB}=\sqrt{\frac{|dB|^{2}}{N}}\\ RMS_{dL}=\sqrt{\frac{|dL|^{2}}{N}}\\ RMS_{dh}=\sqrt{\frac{|dh|^{2}}{N}}\end{array} \right. \tag{5}\]where:  
\(RMS_{dB}\)--RMS error for the calculated accuracy of the determined resultant component \(B\) of the aerial vehicle,  
\(RMS_{dL}\)--RMS error for the calculated accuracy of the determined resultant component \(L\) of the aerial vehicle,  
\(RMS_{dh}\)--RMS error for the calculated accuracy of the determined resultant component \(h\) of the aerial vehicle,  
\(N\)--number of measurement epochs. Another essential factor for the determination of the positioning accuracy in 3D space is the determination of the resultant position error \(dr\). In the analysed example, the \(dr\) parameter will define the shift vector of the resultant coordinates of the aerial vehicle with respect to the reference trajectory of the flight. The \(dr\) parameter may be calculated from the mathematical Equation (6), as presented below:  
\[dr=\sqrt{dB^{2}\;+\;dL^{2}\;+\;dh^{2}} \tag{6}\]  
where:  
\(dr\)--resultant position error in 3D space. - for the availability parameter:  
\[A_{m}=\frac{A_{EGNOS}\cdot\alpha_{E}\;+\;A_{SDCM}\cdot\alpha_{S}}{\alpha_{E}\;+ \;\alpha_{S}} \tag{7}\]  
where:  
\(A_{m}\)--resultant availability of Multi-SBAS positioning,  
\(A_{EGNOS}\) --positioning availability from the SBAS/EGNOS solution,  
\(A_{EGNOS}=100\%\cdot\frac{t_{E}/residual}{t_{all}}\),  
\(A_{SDCM}\)--positioning availability from the SBAS/SDCM solution,  
\(A_{SDCM}=100\%\cdot\frac{t_{S}/residual}{t_{all}}\),  
\(t_{E/accident}\)--outage duration in the SBAS/EGNOS solution,  
\(t_{S/accident}\)--outage duration in the SBAS/SDCM solution,  
\(t_{all}\)--total time of functioning of the given SBAS system. The determined value of the availability parameter \(A_{m}\) is dimensionless; optionally it may be expressed as percentage.