# Testing Multi-Frequency Low-Cost GNSS Receivers for Geodetic Monitoring Purposes
## 2 Material and Methods
### Evaluation Approach of ZED-F9P
#### 2.4.2 Displacement Detection
\right] \tag{9}\]  
\[\mathbf{J}_{d_{2}}=\left[\frac{\partial d_{2}}{\partial Y_{i}}\frac{\partial d_{ 2}}{\partial X_{i}}\frac{\partial d_{2}}{\partial Y_{j}}\frac{\partial d_{2} }{\partial X_{j}}\right] \tag{10}\]  
\[\mathbf{J}_{d_{3}}=\left[\frac{\partial d_{3}}{\partial Y_{i}}\frac{\partial d_{ 3}}{\partial X_{i}}\frac{\partial d_{3}}{\partial h_{i}}\frac{\partial d_{3} }{\partial Y_{j}}\frac{\partial d_{3}}{\partial X_{j}}\frac{\partial d_{3}}{ \partial h_{j}}\right] \tag{11}\]  
Variance-covariance matrices (\(\mathbf{\Sigma}_{ij}\)) for all three dimensions have the following form [33]  
\[1\text{D}:\ \mathbf{\Sigma}_{ij}=\left[ \begin{array}{cc}\sigma_{h_{i}}^{2}&0\\ 0&\sigma_{h_{j}}^{2}\end{array} \right] \tag{12}\]  
\[2\text{D}:\ \mathbf{\Sigma}_{ij}=\left[\begin{array}{cccc}\sigma_{Y_{i}}^{2}& \sigma_{Y_{i}X_{i}}&0&0\\ \sigma_{  
the following cases (2D and 3D tests) because observations from positional quality assessment surveys (data from Section 2.4.1) were also used. The horizontal coordinates of point FGG1 were also tested in order to detect horizontal displacement. During the survey, 270 possible displacements were imposed, which ranged from 5 to 50 mm. Differences between sessions of two identical positions were not tested. In this case, we are testing only differences that correspond to truly imposed displacements and may, therefore, determine the minimum level of the displacement for low-cost GNSS instruments. The following hypothesis are set:  
H\({}_{0}\): \(d_{2}=0\); Point didn't move horizontally between two sessions;  
H\({}_{a}\): \(d_{2}\
eq 0\); Point moved horizontally between session--the horizontal displacement is detected. Verification of rejection of the null hypothesis is done by comparing test statistics with the critical value (\(t_{\text{critical}}=2.45\)), determined from \(\chi^{2}\) distribution with two degrees of freedom. The final critical value is determined as a square root of theoretical \(\chi^{2}\) critical value. In reality, the displacements of points are expected to be spatial (in 3D) in any direction. In this case, for 3D displacements, the following hypotheses are set:  
H\({}_{0}\): \(d_{3}=0\); Point did not move spatially between two sessions;  
H\({}_{a}\): \(d_{3}\
eq 0\); Point moved spatially between two sessions--the spatial displacement is detected. As in the case of 2D, 270 movements were imposed in total, where the defined critical value (\(t_{\text{critical}}=2.80\)) is used to verify the rejection of tested hypotheses. This was again obtained from \(\chi^{2}\) distribution with three degrees of freedom (and again a square root value). The procedure of displacements' detection is shown graphically in Figure 5.  
A mechanical device that imposes displacements provides them with sub-millimeter accuracy, and therefore true values of all displacements are known. In the horizontal plane, the displacements are in steps of 5 mm, from 0 to 50 mm, whereas in vertical directions there are truly no displacements, all their values are equal to 0 mm. For all estimated displacements (1D, 2D, and 3D), we also estimated Mean Absolute Error (MAE). In this way, important information will be acquired, namely, the level of accuracy of estimated displacements with respect to their true values that may be achieved with low-cost GNSS receivers. The MAE values will be determined for both baselines, for the G-baseline as well as for the L-baseline. The results are presented in the second part of the following Section 3. Figure 5: Displacement detection.