# Weighting of Multi-GNSS Observations in Real-Time Precise Point Positioning
## 4 Discussion  
We propose five different GPS + GLONASS + Galileo + BeiDou weighting schemes, including two schemes with no intra-system differences i.e. the same weights were applied for all the systems, and three schemes that are based on signal noise and/or quality of satellite orbits expressed using SISRE. For 12 diverse IGS stations distributed worldwide, we obtained six different PPP solutions over two test periods, one week long each. We performed the GPS-only solution as well as five multi-GNSS solutions representing each weighting scheme. Then, we analyzed formal errors of coordinates, coordinate repeatability, and solution convergence time, in order to assess the quality of each solution, and, consequently, the correctness of weighting schemes.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & **G\_L10** & **GREC\_L10** & **GREC\_L50** & **GREC\_GxN** & **GREC\_GxS** & **GREC\_N\&S** \\ \hline Horizontal [h] & 3:34 & 2:31 & 10:03 & 2:44 & 2:11 & 7:16 \\ Vertical [h] & 5:43 & 3:34 & 18:17 & 4:12 & 3:01 & 16:52 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Mean convergence time in hours to reach 0.01 m level for all of the stations during both test periods.  
Figure 8: Convergence time to the level of 0.01 m for horizontal components.  
Figure 9: Convergence time to the level of 0.01 m for the vertical component.  
The performed analyses show that using observations of a multi-GNSS constellation in real-time PPP allows for us to significantly reduce formal errors of estimated coordinates and to greatly shorten the convergence time. The improvement of the coordinate repeatability is limited, which is consistent with the post-processing case, in which the addition of other GNSS to GPS static PPP does not significantly improve the daily coordinate repeatability. The reason is that the quality of 24 h GPS-only PPP is already at the centimeter level, and the addition of new constellations does not improve the accuracy significantly [45]. On the other hand, we noticed that improper weighting of multi-GNSS observations leads to a significant degradation of the coordinate repeatability, even though there is still an improvement in convergence time and formal errors of coordinates. It proves that the improved satellite geometry is the main factor reducing formal errors of estimated parameters, and, consequently, solution convergence time, as it simply gives much more observations into the system of normal equations. However, only with a proper weighting scheme the precision of the daily multi-GNSS solutions can be at least maintained at the level when compared to the GPS-only solution.  
We found that the simplest approach, in which equal weights are applied for all the systems, is incorrect and in the best case causes a degradation of coordinate repeatability by 50% when compared to the GPS-only solution. Additionally, if pseudorange to carrier-phase weighting rate is too small, e.g., pseudoranges are overweighted, the solution will totally fail, leading to the degradation of all measures: formal errors, convergence time and coordinate repeatability. It proves that the optimum stochastic model for real-time multi-GNSS solutions should take into account both intra-system and intra-observation diversity.