## 3 Results
### Consistency of Accuracy of Satellite and Terrestrial Measurements by the Traverse Method  
The secondary system (terrestrially measured traverse) is inserted into the primary system (determined by GNSS). An important prerequisite for the integration of measurements is their metrological correctness. For simplicity, assume a solution model (Figure 5) where the points of the secondary system are formed by a traverse between the points of the primary system. Assume further that the mutual distance of the points of the primary system is about \(S=1000\) m, the number of vertices of the traverse is \(n=7\) and the length of the sides in the traverse is about \(S_{i}=170\) m.  
If we use a simplified model for a straight equilateral traverse, we can calculate the longitudinal standard deviation from Equation (2), e.g., according to [23; 24]  
\[\sigma_{L}^{2}=(n-1)\sigma_{S}^{2} \tag{2}\]  
and the lateral standard deviation from Equation (3)  
\[\sigma_{q}^{2}=\frac{n(n-1)(2n-1)}{6}S_{i}^{2}\sigma_{\omega}^{2} \tag{3}\]  
where \(\sigma_{S}\) is the standard deviation of the length measurement and \(\sigma_{\omega}\) is the standard deviation of the angle measurement. For example, for a total length of a unilaterally connected and unilaterally oriented traverse \(S=1000\) m, number of vertices \(n=7\), standard deviation of length measurement \(\sigma_{S}=3\) mm and standard deviation of angle measurement \(\sigma_{\omega}=0.3\) mgon, the end point of the traverse will be:  
longitudinal standard deviation  
\[\sigma_{L}^{2}=(7-1)\sigma_{S}^{2} \tag{4}\]  
\[\sigma_{L}=7.3\text{ mm} \tag{5}\]  
and lateral standard deviation  
\[\sigma_{q}^{2}=\frac{7(7-1)(14-1)}{6}170^{2}\sigma_{\omega}^{2} \tag{6}\]  
\[\sigma_{q}=7.6\text{ mm}. \tag{7}\]  
The magnitude of the transverse standard deviation \(\sigma_{q}=7.6\) mm, or limit deviation 15.2 mm (for a confidence factor \(k=2\)) of the traverse endpoint, is significantly higher compared to the accuracy of the GNSS determination of the point of the primary system lateral standard deviation \(\sigma_{x,y}=5.0\) mm, or a limit deviation of 10 mm (for a confidence factor of \(k=2\)). Therefore, it is justifiable to use the higher accuracy of the GNSS determina  
Figure 5: Model example of primary (yellow) and secondary system configuration. tion of points of the primary system to fix (calculation invariance) the spatial position of the traverse. Fixing the coordinates of the two endpoints of a connecting traverse between the points of the primary system results in maximum deviations in the middle of the traverse. Longitudinal standard deviation for \(n=4\), \(\sigma_{S}=3\) mm (for 3 traverse legs, \(S=170\) m)  
\[\sigma_{L}^{2}=(4-1)\sigma_{S}^{2} \tag{8}\]  
\[\sigma_{L}=5.2\;\text{mm} \tag{9}\]  
and lateral standard deviation  
\[\sigma_{q}^{2}=\frac{4(4-1)(8-1)}{6}170^{2}\sigma_{\omega}^{2} \tag{10}\]  
\[\sigma_{q}^{2}=\frac{n(n-1)(2n-1)}{6}S_{i}^{2}\sigma_{\omega}^{2} \tag{3}\]0  
and positional standard deviation  
\[\sigma_{q}^{2}=\frac{n(n-1)(2n-1)}{6}S_{i}^{2}\sigma_{\omega}^{2} \tag{3}\]1  
Table 4 shows selected parameters of the Least Squares Method (LSM) calculation of the terrestrial measured traverse (secondary system) with respect to the GNSS determined points (primary system), where the primary system points are fixed during the calculation. The LSM calculation was performed for apriori standard deviation of the angle measurement of 0.3 mgon and apriori standard deviation of the length measurement of 3 mm. The results of the calculation of the accuracy characteristics correspond to the simplified theoretical model. The critical value of the unit standard deviation is only slightly exceeded for the second, third and fourth measured lines due to the effect of the uncertainty of the primary system points and the residual geometric inhomogeneity of the uniform transformation key. The root mean square standard deviation (RMS \(\sigma_{p}\)) of the secondary system points corresponds well to the homogeneity of the primary system.