# Multi Ray Model for Near-Ground Millimeter Wave Radar
## 2 Millimeter Wave Surface Reflectivity  
The reflection of an electromagnetic wave from a surface is characterized by the reflectivity:  
\[\rho_{0}=\left|\rho_{0}\right|\cdot e^{i\angle\rho_{0}} \tag{1}\]  
where \(\left|\rho_{0}\right|^{2}\) is the power reflectivity and \(\angle\rho_{0}\) is the phase shift due to the reflection. This is a frequency-dependent complex quantity determined by the dielectric constant \(\varepsilon^{\prime}\) of the surface, its electrical conductivity \(\sigma\), the wave incident angle \(\theta_{0}\), and its polarization [2]. Table 1 summarizes the dielectric constants of various types of ground at millimeter wave frequencies [12; 13; 14].  
Considering reflections from the ground, we define the horizontal polarization as the transverse electric mode (TE), and the vertical polarization as the tangential magnetic field (TM). Figure 1 illustrates four possible paths for radar wave propagation in which reflections from the ground are involved. The reflectivity of the ground is calculated using [PERSON]'s equations after introducing the Snell law [2], resulting in:  
\[\rho_{0}=\begin{cases}\dfrac{\eta\cos(\theta_{0})-\eta_{0}\sqrt{1-\dfrac{1}{ \varepsilon^{\prime}}\sin^{2}(\theta_{0})}}{\eta\cos(\theta_{0})+\eta_{0} \sqrt{1-\dfrac{1}{\varepsilon^{\prime}}\sin^{2}(\theta_{0})}}&\text{{TE}}\\ \\ \dfrac{\eta\sqrt{1-\dfrac{1}{\varepsilon^{\prime}}\sin^{2}(\theta_{0})}-\eta_{ 0}\cos(\theta_{0})}{\eta\sqrt{1-\dfrac{1}{\varepsilon^{\prime}}\sin^{2}( \theta_{0})}+\eta_{0}\cos(\theta_{0})}&\text{{TM}}\end{cases} \tag{2}\]  
Assuming that the magnetic permeability of the medium (air and ground) is identical to that of the vacuum \(\mu_{0}=4\pi\cdot 10^{-7}\) H/m, the impedance of the surface is \(\eta=\sqrt{\mu_{0}/(\varepsilon^{\prime}\varepsilon_{0})}\). Consequently, the expression for the field reflectivity for both polarizations can be simplified to:  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Radar Material** & **Dielectric Constant \(\varepsilon^{\prime}\)** & **Frequency** \\ \hline Soil (little moisture) & 2.9 & 94 GHz \\ Soil (moist) & 3.7 & 94 GHz \\ Acrylic 31 & 2.595 & 100 GHz \\ Ferroflow & 13 & 100 GHz \\ Glass, Pyrex & 4.33 & 100 GHz \\ Nylon & 2.993 & 100 GHz \\ PE (Polyethylene) & 2.306 & 100 GH \\ Asphalt & 3.18 & 94 GHz \\ \hline \hline \end{tabular}
\end{table}
Table 1: W-band dielectric coefficients of materials.  
\[\rho_{0}=\left\{\begin{array}{ll}\dfrac{\cos(\theta_{0})-\sqrt{e^{ \prime}-\sin^{2}(\theta_{0})}}{\cos(\theta_{0})+\sqrt{e^{\prime}-\sin^{2}(\theta _{0})}}&\quad TE\\ \dfrac{\sqrt{e^{\prime}-\sin^{2}(\theta_{0})}-e^{\prime}\cos(\theta_{0})}{ \sqrt{e^{\prime}-\sin^{2}(\theta_{0})}+e^{\prime}\cos(\theta_{0})}&\quad TM \end{array}\right. \tag{3}\]  
Figure 2 describes the millimeter wave (94 GHz) reflectivity as a function of range when both the transmitter and receiver are at identical heights \(h_{t}=h_{r}=10\) cm (Figure 2a) and \(h_{t}=h_{r}=1\) cm (Figure 2b) above ground. Graphs are drawn for both polarizations, while two types of soil are considered; soil with low moisture (\(e^{\prime}=2.9\)), and moist soil (\(e^{\prime}=3.7\)).  
Figure 1: Four main propagation paths in a radar scenario.  
Figure 2: _Cont._