# Multi Ray Model for Near-Ground Millimeter Wave Radar
## 3 Four-Ray Model for Near-Ground Radar  
Considering ground reflections, there are, in general, four possible propagation paths between the transmitter and the receiver, as shown in Figure 1. The number of paths is determined by the scattering characteristics of the target. All four paths exist if the target demonstrates a diffuse reflection, where the incident wave is reflected in many directions due to surface roughness scattering. Such a phenomenon emerges if the reflecting surface irregularities are comparable with radiation wavelength [15]. A Lambertian reflectance is one example, where the scattered wave presents equal brightness from all directions that lie in the half-space adjacent to the surface. The reflections are also at many angles if the target has a rounded shape (as in the bumper of a car). In the case of a flat smooth target, the reflection is specular; the angle of reflection equals that of the incident angle. The flat target perpendicular to the ground creates a 'corner reflector', resulting in two possible paths, as illustrated in Figure 3.  
The total signal received by the radar is a result of the summation of fields scattered from the target and arriving at the receiver via four paths. One path is the line of sight, the direct path. The other three involve reflections from the ground. Assuming that the reflectivity from the target is represented by \(\rho_{\rm target}\), and does not depend on the incident angle, the total field resulting from all four rays is given by the sum:  
Figure 3: Two main radar propagation paths from flat target close to ground. Figure 2: Soil reflectivity as a function of distance between the transmitter and receiver (**a**) when \(h_{t}=h_{r}=10\) cm; and (**b**) when \(h_{t}=h_{r}=1\) cm at frequency 94 GHz. Solid line—little moisture. Dashed line—moist soil. \[ \begin{split}&\widetilde{E}_{Total}=\underbrace{\frac{1}{2d}\cdot e^{-j \frac{2\pi f}{c}(2d)}\cdot\rho_{\text{target}}}_{Path1}+\underbrace{\frac{1}{d+d_{1} +d_{2}}\cdot e^{-j\frac{2\pi f}{c}(d+d_{1}+d_{2})}\cdot\rho_{\text{target}} \cdot\rho_{\text{ground}}}_{Path2}\\ &+\underbrace{\frac{1}{d+d_{1}+d_{2}}\cdot e^{-j\frac{2\pi f}{c} (d+d_{1}+d_{2})}\cdot\rho_{\text{ground}}\cdot\rho_{\text{target}}}_{Path3}+ \underbrace{\frac{1}{2d_{1}+2d_{2}}\cdot e^{-j\frac{2\pi f}{c}(2d_{1}+2d_{2})} \cdot\rho_{\text{target}}\cdot\rho_{\text{ground}}^{2}}_{Path4}\end{split}  \tag{4}\]  
The ratio between the total received power attained using the four-ray model and the power obtained from a single line of sight path is calculated by:  
\[ \begin{split}&\frac{P_{r}(\text{Four rays})}{P_{r}(\text{LOS})}=\frac{\left|\widetilde{E}_{Total} \right|^{2}}{\left|\widetilde{E}_{\text{LOS}}\right|^{2}}\\ &=\Big{|}\,1+\frac{4d}{d+d_{1}+d_{2}}\cdot e^{-j\frac{2\pi f}{c}(d _{1}+d_{2}-d)}\cdot\rho_{\text{ground}}+\frac{d}{d_{1}+d_{2}}\cdot e^{-j\frac{ 4\pi f}{c}(d_{1}+d_{2}-d)}\cdot\rho_{\text{ground}}^{2}\Big{|}^{2}\\ &=\Big{|}\,1+\Big{[}\frac{4d}{d+d_{1}+d_{2}}+\frac{d}{d_{1}+d_{2} }\cdot\rho_{\text{ground}}\cdot e^{-j\frac{2\pi f}{c}(d_{1}+d_{2}-d)}\Big{]} \cdot\rho_{\text{ground}}\cdot e^{-j\frac{2\pi f}{c}(d_{1}+d_{2}-d)}\Big{|}^{ 2}\end{split}  \tag{5}\]  
If the height \(h_{\text{radar}}\) of the radar antennas and the height \(h_{\text{target}}\) of the target above ground is substantially smaller than the distance, i.e., \(h_{\text{radar}}\), \(h_{\text{target}}<<d\), the total distance can be approximated by:  
\[d\cong d_{1}+d_{2}-\frac{2h_{\text{radar}}h_{\text{target}}}{d_{1}+d_{2}} \cong d_{1}+d_{2}-\frac{2h_{\text{radar}}h_{\text{target}}}{d} \tag{6}\]  
Substituting the last expression into the power ratio (5), results in:  
\[