## 5 Discussion and Conclusions  
In order to obtain better GNSS positioning results, as in any geodetic process, observations should be weighted according to the measurement errors made. The simplest method is to weigh the observations by the elevation angle using a simple sine function [78]. In this study, the impact of different weighting functions on the performance of PPP for GPS, Galileo and GPS+Galileo was analyzed. Eight models have been selected for analysis: one of them assumes that all observations have the same precision, without dependence on the elevation angle; for the rest, selected elevation-dependent functions were used. In this paper, parameters that related to the quality of PPP solutions such as accuracy, convergence time, ZPD and ISB are analyzed. The results showed that for different solutions, different functions allowed for obtaining the best results. It is worth emphasizing that the weighting functions have the biggest impact for the Up component. This concerns the accuracy and convergence time. Additionally in convergence time, using different weighting functions also has an impact on the East component. For GPS-only, the highest accuracy was obtained for the Function_3, whereas for Galileo-only, the best result was obtained for Function_5. Both functions depend on the elevation angle, but for Function_3, the errors for lower satellites have less weight (Figure 12). This is different for Function_5, where observations at lower angles have more weight relative to most other functions. With higher signal strength and less multipath effect for low Galileo satellites, this improves Galileo-only solutions. It is worth noting that GPS observations from the newest satellites also have higher signal strength and less multipath effect, so it will be worthwhile to do the study again in the future for GPS, where the vast majority of satellites will be newer generation. For GPS+Galileo, we analyzed two solutions. The first one used the same precision of observations for GPS and Galileo (GE solution). The second one (GE1) used a slightly lower  
Figure 12: Standard deviations of carrier-phase of ionosphere-free linear combination with elevation angle for analyzed weighting functions. Bottom plot for Function_6 and Function_8. Top plot for the rest of the functions. precision for Galileo observations. Unlike the GPS-only and Galileo-only solutions, for the GE and GE1 solutions, the best accuracy were obtained for the same functions--Function_2. The shortest convergence time was obtained for Function_5. Considering that the difference in convergence time between Function_2 and Function_5 is less than a minute, it should be pointed out that the Function_2 proved to be the best. The accuracy for the GE solution was found to be better than that for the GE1. The worst results were obtained for Function_6 and Function_8. The solutions were even worse than the solution without weights (Function_1). Both functions give very low weights to low observations, an order of magnitude lower than for the other models analyzed. This directly translated into the worst result obtained. This shows that improper weighting of observations can degrade the results. The best agreement of the tropospheric solutions with the IGS solutions was obtained for the Galileo-only solution. Slightly worse results were obtained for the GPS+Galileo solutions and the worst were for the GPS-only. As shown, the weighting functions have an influence on the results obtained. For the Galileo-only solution, the best results were obtained for Function_5, whereas for the G-only solutions, the best results were obtained for Function_3, similar to accuracy and convergence time. For the GPS+Galileo solutions, the best results were obtained for Function_4, and again better results were for the GE solution than for the GE1 solution. When analyzing the ISB parameter, there was no significant dependence on the weighting functions used. Further research on a statistically larger sample of receivers is needed to document a reliable impact.