# GNSS Performance Modelling and Augmentation for Urban Air Mobility +
## 3 GNSS Performance Characterization
### Error Modelling
#### 3.1.5 Ground Reflection  
The first-order specular ground reflection model is illustrated in Figure 6. Assuming a locally flat ground plane, the so-called image theory from fundamental electromagnetics can be employed to determine the relative delay [31].  
The delay in this case is the distance \(\delta\) from the point of reflection to the image of the receiver antenna about the ground plane given by:  
\[\delta=2h\text{sin}\theta \tag{16}\]  
where \(h\) is the height of the antenna and \(\theta\) is the elevation of the satellite with respect to the ground plane. Significant specular reflection will occur only for reflectors with sizes comparable to the first Fresnel zone radius. The first Fresnel zone radius \(r\) is approximated by:  
\[r\approx\sqrt{\lambda dr} \tag{17}\]  
Figure 6: Ground reflection.  
for cases where the distance between the transmitter and reflector is much larger than the distance from the reflector to the receiver (\(d_{r}\)). \(\lambda\) is the wavelength of the incident ray (\(\approx 19\) cm for L1 in this case). A critical time delay \(\tau_{crit}\) can be derived for different types of receivers (different in terms of correlator spacing). Multipath delays exceeding \(\tau_{crit}\) will be rejected by the receiver. Standard receivers with a spacing of one Pseudo-Random Noise (PRN) bit between early and late correlators have a value of \(\tau_{crit}\approx 1500\) nanoseconds. This corresponds to a distance of \(\tau_{crit}\times c=450\) m. This value reduces to 1000 nanoseconds (300 m) for more advanced narrow correlator receivers with spacings of 0.1 PRN bit. The relative amplitude is a function of the reflection coefficients of the reflector material, which are in turn, dependent on the permittivity of the material and the angle of incidence of the incoming ray. The Fresnel reflection coefficient is resolved into perpendicular and parallel components as given by [PERSON] and [PERSON] [31] for dielectrics:  
\[r_{\perp}=\frac{\cos\theta_{i}-\sqrt{\epsilon_{r}^{*}-sin^{2}\theta_{i}}}{\cos \theta_{i}+\sqrt{\epsilon_{r}^{*}-sin^{2}\theta_{i}}} \tag{18}\]  
\[r_{\parallel}=\frac{\varepsilon_{r}^{*}\cos\theta_{i}-\sqrt{\epsilon_{r}^{*}- sin^{2}\theta_{i}}}{\varepsilon_{r}^{*}\cos\theta_{i}+\sqrt{\epsilon_{r}^{*}- sin^{2}\theta_{i}}} \tag{19}\]  
where \(\varepsilon_{r}^{*}=\varepsilon_{r}-j\frac{\sigma}{\omega\varepsilon_{0}}\); \(\theta_{i}\) is the angle of incidence (relative to the surface normal); \(\varepsilon_{r}\) is the relative permittivity of the lossy media; \(\varepsilon_{0}\) is the absolute permittivity of free space; \(\sigma\) is the conductivity of the media. The reflectance is then:  
\[R=\frac{\left(\left|r_{\perp}\right|^{2}+\left|r_{\parallel}\right|^{2}\right)}{2} \tag{20}\]  
The following parameters (\(\varepsilon_{r}=2.0,\ \ \sigma=0.12\)) corresponding to asphalt were chosen to model the ground reflectance and are plotted in Figure 7 as a function of incident angle. A satellite elevation mask of \(10^{\circ}\) is assumed.  
It is also assumed that signals attenuated greater than 20 dB cannot be acquired by the receiver. This corresponds to a multipath region of \(10^{\circ}\)-\(38^{\circ}\). Assuming an operational airspace sector from 30 m AGL to 150 m AGL, this corresponds to echo delays in the range 34-615 nsec.  
Figure 7: Ground reflectance region.