# Testing Multi-Frequency Low-Cost GNSS Receivers for Geodetic Monitoring Purposes
## 2 Material and Methods
### Evaluation Approach of ZED-F9P
#### 2.4.1 Positional Quality  
To evaluate the positional quality of ZED-F9P, static observations were carried out at 1 Hz for 28 h from the 31 st to 34 th day of the year 2020 (from 31 st January until 3 rd February), where data from three different navigation systems, namely, GPS, GLONASS, and Galileo, were obtained. GNSS data were processed with open-source software RTKLIB (demo5_b33b) in hourly sessions. The processing  
Figure 3: GNSS receivers: (**a**) Reference station Leica GS 15 in pillar FGG2; (**b**) Rover ZED-F9P in pillar FGG1.  
Figure 2: Faculty of Civil and Geodetic Engineering building.  
parameters used in RTKLIB are shown in Table 1. The number of satellites was at least 15 in every session for both baselines. We initially used also Leica Infinity for processing of the GNSS data, but since it processed data from L1 frequency only and somehow ignored all L2 observations, we did not use it in further analysis.  
The results of GNSS data processing were estimated coordinates of the FGG1 point; for each hourly session the mean value of obtained coordinates was calculated, for both baselines, the G-baseline and the L-baseline. Baseline residuals were used to obtain some elementary statistics, such as minimum and maximum value, and Root-Mean-Square-Error (RMSE). No outliers were detected in GNSS data; all residuals were in the interval less than 3\(\sigma\), where the verification was done by \(\tau\)-test and Tukey's Method [28,29]. Residuals were, in most cases, less than 4 mm for all coordinate components (Y--east, X--north, and h--ellipsoidal height). To better express their differences, we followed a classification adopted from [PERSON] et al. [12], where five groups were defined. The first group represents all residuals in a range from 0 to 2 mm, the second from 2 to 4 mm, the third from 4 to 6 mm, the fourth from 6 to 8 mm, and the last one from 8 to 10 mm.  
Because GNSS data from two baselines were processed, two triplets of FGG1 coordinates were obtained for each session. One set of coordinates belongs to the G-baseline, denoted as \(Y_{\text{G}}\), \(X_{\text{G}}\), and \(h_{\text{G}}\), and the other one to the L-baseline, denoted as \(Y_{\text{L}}\), \(X_{\text{L}}\), and \(h_{\text{L}}\). The differences of estimated 1D, 2D and, 3D positions from both (L- and G-) baselines are determined as  
\[b_{1}=h_{L}-h_{G} \tag{1}\]  
\[b_{2}=\sqrt{\left(Y_{L}-Y_{G}\right)^{2}+\left(X_{L}-X_{G}\right)^{2}} \tag{2}\]  
\[b_{3}=\sqrt{\left(Y_{L}-Y_{G}\right)^{2}+\left(X_{L}-X_{G}\right)^{2}+\left(h_ {L}-h_{G}\right)^{2}} \tag{3}\]  
The differences in estimated coordinates from Equations (1)-(3) are mainly due to different GNSS instruments at both base stations, the geodetic GNSS receiver/antenna at FGG2 and low-cost GNSS receiver/antenna at FGG4 [12]. To compare the performance of different GNSS instruments at both base stations, the analysis of variance (ANOVA) is applied separately for each coordinate component (\(Y\), \(X\), and \(h\)). ANOVA is used to identify if there is a significant difference between the means of two groups, i.e., means of FGG1 coordinates obtained from both baselines [30]. The comparison of coordinate means is done by defining the following hypotheses:  
H\({}_{0}\): The use of different types of GNSS instruments doesn't affect coordinates;  
H\({}_{a}\): The use of different types of GNSS instruments affects coordinates.  
Results of the positional analysis are presented in the first part of Section 3, namely, in Section 3.1.