## 2 Ray Launching Technique and Simulation Scenario  
A 3D Ray Launching (RL) algorithm has been implemented in-house based on Geometrical Optics (GO) and Geometrical Theory of Diffraction (GTD). Different applications of this algorithm can be found in the literature, like the analysis of wireless propagation in closed environments [15, 16, 17, 18], interference analysis [19] or electromagnetic dosimetry evaluation in wireless systems [20]. RL techniques are based on identifying a single point on the wave front of the radiated wave with a ray that propagates along the space following a combination of optic and electromagnetic theories, as is illustrated in Figure 1. Each ray propagates in the space as a single optic ray. The electric field \(E\) created by an antenna with a radiated power \(P_{rad}\) with a directivity \(D_{t}(\theta_{t}\phi_{t})\) and polarization ratio (\(X^{\perp}\), \(X^{\parallel}\)) at a distance \(r\) in the free space is calculated by [21]:  
\[E_{l}^{\perp}=\sqrt{\frac{P_{rad}D_{t}(\theta_{t},\phi_{t})\eta_{0}}{2\Pi}}\; \frac{e^{-j\beta_{0}r}}{r}\;\mathrm{X}^{\perp}L^{\perp} \tag{1}\]  
\[E_{t}^{\parallel}=\sqrt{\frac{P_{rad}D_{t}(\theta_{t},\phi_{t})\eta_{0}}{2\Pi} }\;\frac{e^{-j\beta_{0}r}}{r}\;\mathrm{X}^{\parallel}L^{\parallel} \tag{2}\]  
where \(\beta_{0}=2\pi\!\!f_{c}\sqrt{\varepsilon_{0}\mu_{0}}\;,\;\varepsilon_{0}=8.854 \times 10^{-12}\), \(\mu_{0}=4\pi\times 10^{-7}\) and \(\eta_{0}=120\pi\). \(L^{\perp}\)\(\parallel\) are the path loss coefficients for each polarization. When this ray finds an object in its path, two new rays are created: a reflected ray and a transmitted ray. These rays have new angles provided by [PERSON]'s law [22]. The diffracted field is calculated by [23]:  
\[E_{UTD}=e_{0}\,\frac{e^{-jks_{1}}}{s_{1}}\;D^{\perp 1\!\!\!\!\perp}\sqrt{\frac{s_{1} }{s_{2}(s_{1}+s_{2})}}\;e^{-jks_{2}} \tag{3}\]  
where \(s_{1}\), \(s_{2}\) are the distances from the source to the edge and from the edge to the receiver point, respectively. \(D^{\perp 1\!\!\!\!\perp}\) are the diffraction coefficients given by [23, 24]. The rays considered in GO are only direct, reflected and refracted rays, leading to the existence of abrupt areas, corresponding to the boundaries of the regions where these rays exist. In order to enhance the GO approximation, the diffracted rays are introduced with the GTD and its uniform extension, the Uniform GTD (UTD). The purpose of these rays is to remove the field discontinuities and to introduce proper field corrections, especially in the zero-field regions predicted by GO. Ray launching is performed three-dimensionally, with angular resolution (horizontal and vertical planes) in a predefined solid angle that takes into account the radiation diagram of the transceivers sources. Spatial resolution when computing ray interaction is given by a uniform hexahedral mesh with cuboids of a given lateral dimension. A finite sample of the possible directions of the propagation from the transmitter is chosen and a ray is launched for each such direction. When the ray impacts with an obstacle, reflection, transmission and diffraction will occur, depending on the geometry and the electric properties of the object, as is depicted in Figure 2.  
A deterministic method based on an in-house developed 3-D Ray Launching code has been used to analyze the radio electric behavior of an inhomogeneous vegetation environment [25, 26]. The software has been implemented in-house based on Matlab programming environment. It is based on geometrical optics (GO) and the uniform geometrical theory of diffraction (UTD), taking into account electromagnetic phenomena like reflection, refraction and diffraction. The main principle of the  
Figure 1: Wavefront propagation with rays associated with single wave front points. algorithm is that power is modeled as a finite number of rays launched within a solid angle.