# GPS Multipath Analysis Using Fresnel Zones
## 4 Multipath Analysis from a Spatially-Limited Reflector Using Fresnel Zones
### Multipath Analysis  
In preparation for the multipath analysis, the smoothness of the reflecting surface was determined. Therefore, a plane was fitted to the TLS point cloud of the roof and the standard deviation \(\sigma_{\Delta H}\) of the residuals was computed as \(\sigma_{\Delta H}=0.055\) m. By inserting \(\sigma_{\Delta H}\) into Equation (8), the corresponding elevation angle was determined, leading to \(26^{\circ}\) for the GPS-L1 carrier phase wavelength. That means that for satellite signals with elevation angles lower than \(26^{\circ}\), specular reflection could be expected. However, the Rayleigh criterion should not be interpreted as a sharp border between specular and diffuse reflection [18], but in the subsequent multipath analysis, it should be taken into account that the propability for diffuse scattering increases for signals with higher elevation angles. In the first step of the multipath analysis, for every satellite and every observation epoch, the reflection points were determined according to Equations (16)-(18). Herein, the required normal vector of the reflector plane \(\mathbf{n}\) was estimated during a plane fit in the Gauss-Helmert model [42]. Afterwards, the semi-axes and the orientation of the Fresnel zones were computed from Equation (14). In Figure 17, the Fresnel zones and the respective reflection points of every hundredth epoch are  
Figure 16: Colored point cloud of the building roof and the positions of GPS antennas A and B.  
shown for both antennas in a topocentric coordinate system for satellites G14 and G19, exemplarily. Furthermore, the satellite tracks of G14 and G19 are visualized in a skylplot. On the one hand, Figure 17 visualizes the theoretical considerations from Section 2.3. The Fresnel zones (red and cyan lines) form ellipses in the reflector plane with the reflection points (green and magenta dots) located in the middle of the ellipses. Depending on the elevation of the satellite signal, the shape of the ellipse becomes elongated, and the reflection points move away from the antenna. Furthermore, it becomes obvious that for horizontal reflection planes, the orientation of the Fresnel zones is defined by the azimuth angle of the respective satellite. On the other hand, the Fresnel zones of satellites G14 and G19 illustrate the basic idea of the experiment. Satellite G14 rises at an azimuth angle of approximately \(320^{\circ}\), passes both antennas in the Western direction and sets at an azimuth of around \(235^{\circ}\). Due to the spatially limited and rectangular shape of the roof, Fresnel zones are only completely located on the roof for certain directions and elevations. For antenna B, this is especially the case during the rising phase of G14. Since antenna A is placed nearer to the edge of the roof, the Fresnel zones determined for the same satellite positions are only partially located on the roof or they are completely disjoint. Satellite G19 rises at an azimuth of \(145^{\circ}\) and passes the antennas in the Eastern direction before it sets at an azimuth of \(45^{\circ}\). In this case, the location of the Fresnel zones with respect to the reflecting surface are nearly identical, and the multipath analysis should lead to similar results. Thus, for each antenna, it becomes possible to analyze the relationship between the size and location of the Fresnel zones and multipath occurrence. For this purpose, SNR observations are used, since they reflect the characteristics of multipathed signals, especially after the trend from the direct signal has been removed (cf. Section 2.2). In Figure 18, the \(\delta\)-SNR time series of G14 is shown for antennas A and B. The values are color coded according to the percentage \(P_{FZ}\) of overlap between the Fresnel zone and the roof. Between the solid red lines, both, the Fresnel zones and the reflection points are completely located on the roof (\(P_{FZ}=100\%\)).