# 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  
To analyze the possibility of displacement detection, a mechanical device (Figure 4) that can impose controlled movements with sub-millimeter accuracy in the horizontal plane was used. The rover antenna was placed on a specially designed iron ground plane with a 15-cm diameter mounted on the \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **RTKLIB** \\ \hline Elevation Angle & 10\({}^{\circ}\) \\ Observations & L1, L2 \\ Session Length & 1 h, 30 min \\ Constellations & GPS, GLONASS, Galileo \\ Troposphere & Saastamoinen \\ Ephemeris & Broadcast \\ Ambiguity & Fix and hold (LAMBDA) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Used RTKLIB parameters for data processing. device (Figure 4), which has already been proved to improve the results of the survey with low-cost antennae [20]. Measurements were performed from 52 nd until 55 th day of the year 2020 (from 21 st February until 24 th February), with sampling rate set to 1 Hz. The survey consisted of a series of 30-min sessions, where the rover was moved by 5 mm in-between two consecutive sessions. All movements were imposed in horizontal plane only. According to [PERSON] et al. [31], 30-min sessions can be considered as suitable for detecting displacements. The same survey was then repeated for all the following days and, as a result, 270 displacements, classified into 10 groups, were obtained. The first group represents a displacement of 5 mm, the second a displacement of 10 mm, the third of 15 mm, and so on. The last, 10 th, group is represented with a displacement of 50 mm. Consecutive groups are therefore mutually different for 5 mm, from 5 up to 50 mm. This classification is done to analyze the minimum value of displacements which can be detected by using only low-cost instruments or low-cost instruments in combination with geodetic receivers on base station. In order to determine if the displacements of the rover are significant, statistical tests are applied [32; 33; 28; 34]. To test point stability between two different sessions, the following hypotheses are used:  
H\({}_{0}\): \(d=0\); Point remained stable between two sessions;  
H\({}_{a}\): \(d\
eq 0\); Point did not remain stable between two sessions--the displacement is detected. The test statistic that is used to verify the rejection of the null hypothesis is defined as follows [34]  
\[T=\frac{d}{\sigma_{d}} \tag{4}\]  
The verification of the null hypothesis is done by comparing calculated test values with the critical values. The test is dependent on the displacement dimension, i.e., for 1D, 2D, or 3D (see Equations (1)-(3)), however, the significance level of 5% (\(\alpha\) = 0.05) was set for all cases [34]. The displacement between two sessions, session \(i\) and session \(j\) (\(j\
eq i\)), for all three dimensions (1D, 2D, and 3D) is as follows  
\[d_{1}=\Delta h=h_{j}-h_{i} \tag{5}\]  
Figure 4: Specially constructed device for control movements and ground plane. \[d_{2}=\sqrt{\Delta Y^{2}+\Delta X^{2}}=\sqrt{\left(Y_{j}-Y_{i}\right)^{2}+\left(X _{j}-X_{i}\right)^{2}} \tag{6}\]  
\[d_{3}=\sqrt{\Delta Y^{2}+\Delta X^{2}+\Delta t^{2}}=\sqrt{\left(Y_{j}-Y_{i} \right)^{2}+\left(X_{j}-X_{i}\right)^{2}+\left(h_{j}-h_{i}\right)^{2}} \tag{7}\]  
where:  
\(Y_{i},X_{i},h_{i}\)--FGG1 coordinates from the \(i\)th session;  
\(Y_{j},X_{j},h_{j}\)--FGG1 coordinates from the \(j\)th session. Based on the error propagation law, the variance of the considered displacements was estimated as follows [35]  
\[\alpha_{d}^{2}=\mathbf{J}_{d}\mathbf{\Sigma}_{ij}\mathbf{J}_{d}^{T} \tag{8}\]  
Jacobi matrices (\(\mathbf{J}_{d}\)), design matrices in case of error propagation law, are defined for 1D, 2D, and 3D displacements, respectively [33]  
\[\mathbf{J}_{d_{1}}=\left[ \begin{array}{cc}\frac{\partial d_{1}}{\partial\mathbf{h}_ {i}}&\frac{\partial d_{1}}{\partial\mathbf{h}_{j}}\end{array}