## 3 Research Method
### New Mathematical Scheme for Improving the GPS and GLONAS Solution in the PPP Method  
The new concept of PPP positioning from the GPS and GLONAS solution was based on the resultant linear combination model in which weighting factors \((a,b,c,d,e,f)\) were used. In the new approach, the aircraft's resultant position is determined in geometric XYZ coordinates based on a single PPP GPS and PPP GLONASS solution. The new algorithm's goal is to determine the best match between the PPP GPS and PPP GLONASS solution relative to the flight reference position calculated from the differential RTK technique. Moreover, the proposed algorithm determines the most optimal solution of the resultant position of the aircraft, i.e., it will be checked which measuring scales improve the performance of determining the resultant aircraft coordinates. The proposed calculation scheme of the resultant position of the aircraft is described below:  
\[\left\{ \begin{array}{l}X_{k}=\frac{a\cdot X_{g}+b\cdot X_{r}}{a+b}\\ Y_{k}=\frac{c\cdot Y_{g}+d\cdot Y}{c+d}\\ Z_{k}=\frac{e\cdot Z_{g}+f\cdot Z_{r}}{e+f}\\ \end{array} \right. \tag{6}\]  
where:  
\(X_{g}=X_{i}^{r}\) (see Equation (3)), position of aircraft from GPS solution along \(X\) axis,  
\(Y_{g}=Y_{i}^{r}\) (see Equation (3)), position of aircraft from GPS solution along \(Y\) axis,  
\(Z_{g}=Z_{i}^{r}\) (see Equation (3)), position of aircraft from GPS solution along \(Z\) axis,  
\(X_{r}=X_{i}^{r}\) (see Equation (3)), position of aircraft from GLONASS solution along \(X\) axis,\(Y_{r}=Y_{r}^{r}\) (see Equation (3)), position of aircraft from GLONASS solution along \(Y\) axis, \(Z_{r}=Z_{r}^{r}\) (see Equation (3)), position of aircraft from GLONASS solution along \(Z\) axis, \((a,b,c,d,e,f)\)--determined linear coefficients, dimensionless values, \((X_{k},Y_{k},Z_{k})\)--resultant position of aircraft. Equation (4) combines 2 independent GPS and GLONASS determinations of the aircraft position using the PPP method. In this way, the aircraft's resultant position is determined every given measurement epoch, typically in the time interval of 1 s. It should be noted that the critical parameters in Equation (4) are the values of linear coefficients. In the analyzed example, a weighting scheme based on the following weighting factors was proposed:  
\[\left\{ \begin{array}{l}a=c=e=\frac{1}{ns_{g}}\lor a=c=e=\frac{1}{CL_{r}^{2}} \\ b=d=f=\frac{1}{ns_{g}}\lor b=d=f=\frac{f}{CL_{r}^{2}}\end{array} \right. \tag{7}\]  
where:  
\(ns_{g}\)--number of GPS satellites,  
\(ns_{r}\)--number of GLONASS satellites,  
\(Cl_{g}\)--mean error of the determined coordinates from the PPP GPS solution,  
\(Cl_{r}\)--mean error of the determined coordinates from the PPP GLONASS solution. The calculation process of determining the resultant position of the aircraft according to Equation (4) uses weighting factors \((a,b,c,d,e,f)\). In the analyzed model, two approaches for determining the weighting factors \((a,b,c,d,e,f)\) were proposed. The first approach is based on a weighing scale as a function of the reciprocal of the number of GPS or GLONASS satellites. On the other hand, the second approach is based on the measurement weight as a function of the inverse of the square of the mean coordinate errors of the aircraft from the PPP GPS and PPP GLONASS solutions. On this basis, it is possible to optimize the calculation process of the aircraft's resultant position. It means that based on Equation(4), the resultant plane position for the PPP method will be calculated, and by the way, it will be shown which measurement weight is the best for this type of mathematical model, as in Equation (4). That is important, because we are looking for mathematical models in air navigation to improve the aircraft's position. All the more, determining the appropriate measurement weight here is crucial to improving the accuracy of PPP positioning in aeronautical navigation. The algorithm for determining the values in the \(Cl_{g}\) and \(Cl_{r}\) parameters is described below [41]:  
\[\left\{