# Experimental Study of Accuracy of High-Rate GNSS in Context of Structural Health Monitoring
## 3 Results and Discussion
### Evaluation of Accuracy of the 100-Hz Displacement Time Series
#### 3.2.3 Sinusoidal Motion Test  
As shown in Figure 2, a sinusoidal motion was simulated over three time periods. The motion had multiple vibration frequencies. Figure 8 shows the displacement times series from the different satellite systems and positioning techniques. The GPS-derived displacements from all the techniques are significantly biased due to the limited number of visible satellites, resulting in the RMSE being 4.59 cm, 3.60 cm, and 2.20 cm for PPP, PPP-AR, and RTK techniques, respectively. Similar to the quasi-static motion results, the BeiDou results were better than the GPS results due to much more satellites being tracked. Solutions from multi-GNSS observations (the last two rows in Figure 8) were improved noticeably compared to solutions from single satellite systems. A millimeter-level accuracy could be achieved with the RMSE of the solutions from the PPP, PPP-AR, and RTK techniques being 0.92 cm, 0.62 cm, and 0.35 cm, respectively. The histograms and spectra of the displacement errors from the different techniques are given in Figures 9 and 10, respectively. The displacement errors from PPP and GPS-only  
Figure 8: Sinusoidal displacement time series from the LDT (purple) and different satellite systems derived from (left column) PPP, (middle column) PPP-AR, and (right column) RTK techniques (starting from 08:32:00). From top to bottom, results, respectively, were computed from G (red), C (dark blue), GC (orange), and GCER (yellow) observations, respectively. Figure 7: Amplitude-frequency spectra of displacement errors from quasi-static motion test computed with observations from the different satellite systems G, C, GC, and GCER, respectively. From left to right, results from PPP, PPP-AR, and RTK. observations distribute fairly evenly over the range of \(-\)8.71-1.20 cm, while a uni-modal structure centered around \(\pm\)0.5 cm can be observed for displacement errors from Beidou-only and multi-GNSS observations. The distribution patterns of the errors from PPP-AR and PPP are similar, although the error ranges of the PPP-AR results are narrower. The errors of the RTK results concentrate much more around 0 except for the GPS-only solutions that have a bias. The results in Figure 10 show the FFT spectra of the displacement errors. The RTK results have the best performance, especially for those in the low-frequency band (less than 1-Hz), followed by the PPP-AR technique. For different GNSS observation combinations, in terms of the low-frequency band, better results can be observed based on Beidou-only and multi-GNSS observations, while GPS-only solutions have the worst performance due to the biases (also see Figure 9). In addition, the results from different observations systems have similar accuracy in the high-frequency band. The peak frequency and vibration amplitude of the results from each positioning technique computed with Fast Fourier Transform (FFT) are shown in Figure 11 and Table 2. The frequency and amplitude derived from the LDT solutions are regarded as benchmarks. The length and resolution of the FFT were 30,000 and 0.0033 Hz, respectively. The results indicate that the two peak frequencies could be well detected by all the positioning techniques, i.e., PPP, PPP-AR, and RTK; although, there are slight differences in the derived amplitudes. Figure 10: Amplitude–frequency spectra of displacement errors from sinusoidal motion test computed with observations from the different satellite systems G, C, GC, and GCER, respectively. From left to right, results from PPP, PPP-AR, and RTK. Figure 9: Histograms of displacement errors. From top to bottom, results correspond to PPP, PPP-AR, and RTK techniques while from left to right, results correspond to G, C, GC, and GCER observations, respectively. The maximum, minimum, and median values of each time series are given in each plot (in cm). The BeiDou-only results provided more accurate amplitude estimations than the GPS-only results, which is likely due to the BeiDou satellites being more available.