## 3 Research Method
### Positioning Model of the PPP Method for GPS and GLONASS Solutions  
The basic observation equation of the PPP measurement technique can be written as follows [33; 34; 36]:  
* for GPS or GLONASS: \[\left\{ \begin{array}{l}P_{3}=d+c\cdot(dtr-dts)+Trop+Rel+M_{P3}\\ L_{3}=d+c\cdot(dtr-dts)+Trop+B_{3}+Rel+\delta w+M_{L3}\end{array} \right.\] (1) where: \(P_{3}\) and \(L_{3}\)--observation equations of the GPS or GLONASS linear Ionosphere-Free combination; \(P_{3}=\alpha\cdot P_{1}+\beta\cdot P_{2}\); \(L_{3}=\alpha\cdot L_{1}+\beta\cdot L_{2}\); \((P_{1},P_{2})\)--GPS or GLONASS code observations; \((L_{1},L_{2})\)--GPS or GLONASS phase observations; \(\alpha=+\mathrm{f_{1}}^{2}/(\mathrm{f_{1}}^{2}-\mathrm{f_{2}}^{2})\), GPS or GLONASS linear coefficient; \(\beta=-\mathrm{f_{2}}^{2}/(\mathrm{f_{1}}^{2}-\mathrm{f_{2}}^{2})\), GPS or GLONASS linear coefficient; \((\mathrm{f_{1}},\mathrm{f_{2}})\)--frequencies of the GPS or GLONASS signals; \(d\)--geometric satellite-receiver distance in GPS or GLONASS system; takes into account the phase centre characteristics of the satellite antenna and the receiver [37], \[d=\sqrt{\left(X^{r}-X^{s}\right)^{2}+\left(Y^{r}-Y^{s}\right)^{2}+\left(Z^{r}- Z^{s}\right)^{2}}\] (2) \((X^{r},X^{r},Z^{r})\)--\(XYZ\) geocentric coordinates of the aircraft; \((X^{s},X^{s},Z^{s})\)--orbital coordinates of the GPS or GLONASS satellite; \(c\)--the speed of light; \(dtr\)--correction of the receiver's clock in the GPS or GLONASS system; \(dts\)--correction of the satellite's clock in the GPS or GLONASS system; \(Trop\)--Tropospheric delay in GPS or GLONASS measurements; \(Trop=SWD+SHD\); \(SHD\)--slant path tropospheric delay for the hydrostatic component; \(SWD\)--slant path tropospheric delay for the wet component; \(SHD=m\!f_{H}\cdot ZHD\); \(SWD=m\!f_{W}\cdot ZWD\); \((m\!f_{H},\,m\!f_{W})\)--mapping functions for the hydrostatic and wet paths of the tropospheric delay; \(ZHD\)--hydrostatic component of tropospheric delay; \(ZWD\)--wet component of tropospheric delay; \(Rel\)--relativistic effects in GPS or GLONASS measurements; \(B_{3}\)--float value of phase ambiguity in GPS or GLONASS measurements; \(\delta w\)--phase wind-up in GPS or GLONASS system; \(M_{P3}\)--multipath effect of GPS or GLONASS code observations; \(M_{L3}\)--multipath effect of GPS or GLONASS phase observations. The unknown parameters in the PPP measurement technique in the GPS and GLONASS solutions are, respectively [38], residuals to the approximate coordinates of the receiver \([\delta X,\delta Y,\delta Z]\), the receiver's clock correction \(dtr\), phase ambiguities \(B_{3}\) (determined for each visible GPS and GLONASS satellite) and the wet component of the tropospheric delay \(ZWD\). The above-mentioned unknown parameters are determined by the least-squares method in a sequential stochastic process in GPS and GLONASS systems, respectively, as follows [39]:  
\[Sx=\left(A^{T}\cdot P\cdot A+C_{x}^{-1}\right)^{-1}A^{T}\cdot P\cdot l \tag{3}\]  
where:  
\(Sx\)--vector of the unknown parameters,\(A\)--design matrix,  
\[A=\left[ \begin{array}{ccccccccc}\frac{X_{i-1}^{r}-X^{s}}{d_{i-1}}&\frac{Y_{i-1}^{ r}-Y^{s}}{d_{i-1}}&\frac{Z_{i-1}^{r}-Z^{s}}{d_{i-1}}&1&0&0&\dots&\dots&\dots&mf_{W}\\ \dots&\dots&\dots&\dots&1&1&0&\dots&\dots&\dots&\dots\\ \dots&\dots&\dots&\dots&1&0&0&\dots&\dots&\dots&\dots\\ \dots&\dots&\dots&\dots&1&0&1&\dots&\dots&\dots&\dots\\ \dots&\dots&\dots&1&0&0&\dots&0&0&\dots\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&1&0&\dots\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&0&0&\dots\\ \dots&\dots&\dots&\dots&1&0&0&\dots&0&1&mf_{W}\\ \end{array}