## 3 Research Method
### New Mathematical Scheme for Improving the GPS and GLONAS Solution in the PPP Method
\begin{array}{l}\frac{1}{CL_{g}^{2}}=\frac{1}{mx_{g}^{2}+my_{g}^{2}+ mz_{g}^{2}}\\ \frac{1}{CL_{r}^{2}}=\frac{1}{mx_{g}^{2}+my_{g}^{2}+mz_{r}^{2}}\end{array} \right. \tag{8}\]  
where:  
\((mx_{g},my_{g},mz_{g})\)--mean coordinate errors determined from the GPS variance-covariance matrix \(C_{x}\) (see Equation (2)),  
\(\left\{\begin{array}{l}mx_{g}=C_{x}(1,1)\\ my_{g}=C_{x}(2,2)\quad\quad from\quad PPP\quad GPS\quad solution\text{ (see Equation (\ref{eq:  
The scheme of the procedure for this stage was written in Equation (7):  
\[\left\{ \begin{array}{l}V_{g}=\left\{\begin{array}{l}V_{g,x}=X_{g}-X_{k}\\ V_{g,y}=Y_{g}-Y_{k}\\ V_{g,z}=Z_{g}-Z_{k}\\ V_{r,x}=X_{r}-X_{k}\\ V_{r,y}=Y_{r}-Y_{k}\\ V_{r,z}=Z_{r}-Z_{k}\end{array}\right.\\ \\ \sigma_{x}=\sqrt{\frac{\left[V_{g,x}^{T}\cdot P_{g}\cdot V_{g,x}+V_{r,x}^{T} \cdot P_{r}\cdot V_{r,y}\right]}{n-1}}\\ \sigma_{y}=\sqrt{\frac{\left[V_{g,y}^{T}\cdot P_{g}\cdot V_{g,y}+V_{r,y}^{T} \cdot P_{r}\cdot V_{r,y}\right]}{n-1}}\\ \sigma_{z}=\sqrt{\frac{\left[V_{g,x}^{T}\cdot P_{g}\cdot V_{g,x}+V_{r,y}^{T} \cdot P_{r}\cdot V_{r,z}\right]}{n-1}}\\ M_{x}=\frac{\sigma_{x}}{\sqrt{n}}\\ M_{y}=\frac{\sigma_{y}}{\sqrt{n}}\\ M_{z}=\frac{\sigma_{z}}{\sqrt{n}}\end{array} \right. \tag{9}\]  
where:  
\(P_{g}\)--matrix of weights for PPP GPS solution, \(P_{g}=\frac{1}{n_{S}}\ \vee\ P_{g}=\frac{1}{CI_{g}^{2}}\) (see Equation (5)),  
\(P_{r}\)--matrix of weights for PPP GLONASS solution, \(P_{r}=\frac{1}{n_{S}}\lor P_{r}=\frac{1}{CI_{r}^{2}}\) (see Equation (5)),  
\(V_{g}\)--all residuals between PPP GPS solution and resultant aircraft position,  
\(V_{g,x}\)--residuals between PPP GPS solution and resultant aircraft position along \(X\) axis,  
\(V_{g,y}\)--residuals between PPP GPS solution and resultant aircraft position along \(Y\) axis,  
\(V_{g,z}\)--residuals between PPP GPS solution and resultant aircraft position along \(Z\) axis,  
\(V_{r}\)--all residuals between PPP GLONASS solution and resultant aircraft position along \(Y\) axis,  
\(V_{r,y}\)--residuals between PPP GLONASS solution and resultant aircraft position along \(Y\) axis,  
\(V_{r,z}\)--residuals between PPP GLONASS solution and resultant aircraft position along \(Z\) axis,  
\(n\)--number of solutions, \(n=2\),  
\((\sigma_{x},\sigma_{y},\sigma_{z})\)--mean errors of measurement along \(XYZ\) axis,  
\((M_{x},M_{y},M_{z})\)--mean error of the arithmetic mean along \(XYZ\) axis. Then, the calculations are verified in the form of the global chi-square test, in accordance with the algorithm [41]:  
\[\left\{ \begin{array}{l}\sigma_{x}^{2}\cdot(n-1)=V_{gx}^{T}\cdot P_{g}\cdot V _{g,x}+V_{r,x}^{T}\cdot P_{r}\cdot V_{r,x}\leq\chi_{f,1-\alpha}^{2}\\ \sigma_{y}^{2}\cdot(n-1)=V_{gy}^{T}\cdot P_{g}\cdot V_{g,y}+V_{r,y}^{T}\cdot P _{r}\cdot V_{r,y}\leq\chi_{f,1-\alpha}^{2}\\ \sigma_{z}^{2}\cdot(n-1)=V_{g,z}^{T}\cdot P_{g}\cdot V_{g,z}+V_{r,z}^{T}\cdot P _{r}\cdot V_{r,z}\leq\chi_{f,1-\alpha}^{2}\end{array} \right. \tag{10}\]  
where:  
\(\chi_{f,1-\alpha}^{2}\)--the table value of the chi-square test,  
\(f=n-1\)--number of degrees of freedom,  
\(1-\alpha=0.95\)--confidence level. The chi-square test is performed independently for the designated coordinates along each \(XYZ\) axis. The calculations' verification is determined with a confidence level of 0.95 for degrees of freedom equal to a number \(f=n-1\). The chi-square test determines the internal reliability of the presented research method. The values \(\chi^{2}_{f,1-\alpha}\) are taken from the mathematical table of the chi-square test. For the chi-square test to be fulfilled, numerical values of parameters: \(\sigma^{2}_{x}\cdot(n-1)\), \(\sigma^{2}_{y}\cdot(n-1)\) and \(\sigma^{2}_{z}\cdot(n-1)\) must be equal or less than the tabular value of \(\chi^{2}_{f,1-\alpha}\). If it did not happen, then the test of internal reliability of calculations is not fulfilled, and the algorithm with the confidence level of 0.95 does not meet the requirements.