# GPS Multipath Analysis Using Fresnel Zones
## 2 Multipath Theory
### Theoretical Prerequisites for Multipath Occurrence  
In order to formulate theoretical prerequisites for multipath occurrence, it is necessary to distinguish between _diffuse_ and _specular_ reflections. In Figure 11, both types of reflection are illustrated schematically.  
Basically, the type of reflection depends on the signal wavelength \(\lambda\), the satellite elevation angle \(el\) and the roughness \(\Delta H\) of the reflecting surface, where roughness is usually described as the standard deviation of the difference to a mean surface height. With these values, the _Rayleigh criterion_,  
\[\Delta H\leq\frac{\lambda}{8\sin el}, \tag{19}\]  
can be formulated and used to decide whether diffuse or specular reflection will occur. If the Rayleigh criterion is not fulfilled, the reflecting surface will lead to diffuse reflection, i.e., the signal will be scattered in different directions. Hence, this type of reflection has a rather random nature and can be denoted as less critical. Contrarily, if the Rayleigh criterion is fulfilled, i.e., the surface is rather smooth compared to the signal wavelength, mostly specular reflection will occur, leading to the phenomena described in Section 2.1. In Table 1, the \(\Delta H\)-values for different satellite elevation angles and the GPS-L1 carrier wavelength (\(\lambda=0.19\) m) are listed.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Elevation**[\({}^{\circ}\)] & 5 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \\ \hline \(\Delta H\) **[cm]** & 27.3 & 13.7 & 7.0 & 4.8 & 3.7 & 3.1 & 2.8 & 2.5 & 2.4 & 2.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Surface roughness for different satellite elevation angles computed from the Rayleigh criterion for GPS-L1 carrier wavelength (\(\lambda=0.19\) m).  
Figure 11: (**a**) Diffuse reflection at a rough surface. (**b**) Specular reflection at a smooth surface. In (**a**) and (**b**), the horizontal dashed line represents the mean surface height, and the vertical dashed lines are the differences from the mean surface height.  
Figure 10: Ray-tracing for determination of reflection point \(R\).  
Table 1 shows that the occurrence of specular reflection is more likely for lower elevation angles, since the surface roughness \(\Delta H\) can be bigger compared to higher elevation angles. Especially for the first Fresnel zone, the Rayleigh criterion should be taken into account. As described in Section 2.3, in this region, most of the signal energy will be reflected and thus, the surface roughness in this area is significant for distinguishing between diffuse and specular reflection. Therefore, as a first prerequisite for multipath occurrence, the reflecting surface should fulfill the Rayleigh criterion defined by Equation (19), or in other words, the reflecting surface has to be smooth enough compared to the signal wavelength.  
The second prerequisite is related to the size and location of the Fresnel zones with respect to the reflecting surface. Since the first Fresnel zone defines the region where most of the signal energy is reflected, at the same time, it defines the minimum size of the reflector. That means that for multipath to occur, the reflector needs to be larger than the first Fresnel zone, so that a sufficient amount of energy is reflected, and the Fresnel zone needs to be completely located on the reflecting surface