# GPS Multipath Analysis Using Fresnel Zones
## 4 Multipath Analysis from a Spatially-Limited Reflector Using Fresnel Zones
### Multipath Analysis
Finally, the trend of the simulated time series, which was modeled by a polynomial fit, was subtracted to determine  
Figure 18: Color coded \(\delta\)-SNR time series of G14 observed at antenna A (**top** panel) and antenna B (**bottom** panel). The black line shows the elevation angle of G14 during the observation period. the simulated \(\delta\)-SNR time series. Since the spatial limitation of the reflector plane was not taken into account during the simulation, from a certain point onwards, the simulated and observed \(\delta\)-SNR values were expected differ from each other. In Figures 20 and 21, the observed and simulated \(\delta\)-SNR time series of G14 and G19 are shown, respectively. For reasons of clarity, in both figures, the values are not color coded, but the percentage of overlap \(P_{FZ}\) between the Fresnel zones and the roof is again grouped into the three phases: (1) \(P_{FZ}=100\%\), (2) \(50\%\leq P_{FZ}\geq 100\%\), and (3) \(P_{FZ}\leq 50\%\). Figure 19: Color coded \(\delta\)-SNR time series of G19 observed at antenna A (**top** panel) and antenna B (**bottom** panel). The black line shows the elevation angle of G19 during the observation period. Figure 20: Observed (blue) and simulated (ocher) \(\delta\)-SNR time series of G14 for antenna A (**top** panel) and antenna B (**bottom** panel). The black line shows the elevation angle of G14 during the observation period. For both satellites, the simulated and observed \(\delta\)-SNR time series show very good agreement for \(P_{FZ}=100\%\). This corresponds to the theoretical assumption that multipath will occur when the first Fresnel zones are completely located on the reflecting surface. The best agreement can be found for elevation angles lower than \(30^{\circ}\). In relation to the Rayleigh criterion and the smoothness of the reflecting surface, the assumption that the influence of diffuse reflections increases for elevation angles above \(26^{\circ}\) is confirmed. On the other hand, the simulated and observed SNR values differ greatly when the reflection point is not located on the roof and \(P_{FZ}\) is smaller than \(50\%\). Thus, the previous assumption that the variation in the \(\delta\)-SNR time series during this phase shows random observation noise holds true. Furthermore, this corresponds to the propagation theory described in Section 2.1. Although Fresnel zones are not considered here, it is assumed that the reflection point is always on the reflector plane if multipath occurs. However, since the overlap between the Fresnel zones and the roof is always less than \(50\%\), the Fresnel theory from Section 2.3 is also confirmed. In contrast to the results of the comparison for \(P_{FZ}=100\%\) and \(P_{FZ}\leq 50\%\), the comparison of the simulated and observed \(\delta\)-SNR values for the transition phase show partly ambiguous results. In the rising phase of G19 (cf. Figure 21), at both antennas, a good agreement between simulation and observation can be found. The same holds for G14 during the rising phase at antenna B and the setting phase at antenna A (cf. Figure 20). In the remaining transition phases, the agreement is less distinct. Nevertheless, the comparison reveals that especially during the transition phases with lower elevation angles, multipath occurs. Hence, it can be concluded that a Fresnel zone that is completely located on the reflecting surface does not necessarily need to be fulfilled as a theoretical prerequisite for multipath. In fact, the percentage of overlap between the Fresnel zone and reflector seems to be more important. In order to numerically analyze the accordance of the simulated and observed \(\delta\)-SNR time series, the time-dependent correlation \(C_{t}\) was determined for a sliding window of 15 min length. The results are shown in Figure 22. Furthermore, the observation periods where the Rayleigh criterion was fulfilled are highlighted in grey, and the borders of the three phases of \(P_{FZ}\) are denoted by the dashed and solid red lines.