## 5 Experiment Design and Test Results
### Positioning Solution
The AR validation test effectiveness was presented in a form of the epochs number for which the type I (false rejection, FR) and type II (false acceptance, FA) errors occurred. Positioning accuracy for fixed solutions has been presented in the form of Root Mean Square (RMS) errors for the horizontal (ne) and vertical (u) components, the percentage of epochs for which the 3D error in the relative reference position exceeds 5 mm and the maximum position error. In each of the tested scenarios, the empirical model results were better compared to the elevation model. In particular, the empirical model allowed for the correct AR for all epochs, which was impossible to achieve in the elevation model. Additionally, the positioning accuracy for the empirical model was apparently better. Incorporating the cross and temporal correlations into the model sequentially did not improve the positioning accuracy significantly, but it increased the effectiveness of the validation test for the elevation model. The best positioning performance was obtained for the individual satellite block models, but the difference is negligible compared to the uniform model. As an additional scenario, due to the small number of 5Q observations available, positioning was performed excluding these measurements. For such data, the best positioning results were obtained for the C/N\({}_{0}\)(b) model with errors \(\pm\)3.0 mm and \(\pm\)3.7 mm for horizontal and vertical components, errors below 5 mm at the level of 68% and a maximum error of 26.0 mm. The results obtained for GLONASS (Table 5) essentially confirm the previous conclusions, although compared to GPS, the quality of the solution is lower. In each scenario, the use of the empirical model gives greater efficiency of AR and lower position errors, but only the use of a fully populated VC matrix in C/N\({}_{0}\) model provides 100% effectiveness. A relatively large number of incorrect AR refers to the epochs in which observations from descending satellites were included in the model, which resulted in filter tuning taking several epochs. What is important, for the empirical model, the validation test incorrectly accepted all fixed ambiguities, which may indicate an overestimation of the measurement accuracy due to the presence of multipath, resulting in a decrease in the test significance level. Due to the incomplete satellite constellation for Galileo and BeiDou, a dual-system solution GPS + Galileo (see Table 6) and GPS + BeiDou (see Table 7) was performed. In both cases, the 5Q signal was excluded from the GPS observation; also, the 5Q observations were excluded from the Galileo system as they are directly dependent on already used 8Q measurements. For BeiDou system, two types of empirical models were applied: the model distinguishing between satellite blocks (C/N\({}_{0}\)(b)) and the model distinguishing only geostationary satellites ( C/N\({}_{0}\)(G)). For both dual-system solutions, similar results were obtained, which indicate a clear improvement in the positioning accuracy and reliability with the use of empirical stochastic models. In each scenario, considering the additional correlation parameters in the elevation model increased the effectiveness of AR and reduced position errors. The empirical model allowed to achieve a higher accuracy of horizontal components (\(\pm\)1.9 mm) and a lower maximum error (16 mm) for GPS-Galileo solution. For the GPS-BeiDou, smaller height errors (\(\pm\)3.8 mm) and a much larger number of epochs with errors less than 5 mm (71%) were obtained. For tested approaches, the use of satellite-block-dependent models only slightly increased the solution quality. The four system Multi-GNSS solution (Table 8) was performed for all signals, excluding GPS 5Q and Galileo 5Q measurements. For this test, the difference in the elevation model results compared to the empirical model was the greatest.