# Vegetation Loss Measurements for Single Alley Trees in Millimeter-Wave Bands
## 2 Additional Attenuation of Vegetation
### Radiative Transfer Equation Perspective
\[p(\gamma)=\alpha\bigg{(}\frac{2}{\Delta\gamma_{s}}\bigg{)}^{2}e^{-\big{(}\frac{ \gamma}{\Delta\gamma_{s}}\big{)}^{2}}+(1-\alpha), \tag{6}\]  
where \(\gamma\) represents angle deviation to the main axis \(\gamma=0\), \(\Delta\gamma_{s}\) is the beamwidth, and \(\alpha\) is the forward-to-total scattered power ratio. As the transmitter is located sufficiently far from the woodland edge, the intensity entering the medium with zero boundary conditions for the \(z\) axis can be represented as  
\[I(0;\theta;\phi)=S_{p}\frac{\delta(\theta-\theta_{p})}{\sin\theta_{p}}\delta( \phi-\phi_{p}), \tag{7}\]  
where \(\delta(\cdot)\) is the delta function and \(S_{p}\) is the Poynting vector, while \(\theta_{p}\) and \(\phi_{p}\) are the angles of the incident intensity and \(\theta\), \(\phi\) are the angle counted from the positive \(Z\)-direction and the projected angle in planes \(Z=constant\), respectively. Please note the formula is valid for \(\theta\in(0,\pi/2)\). Specific intensity consists of two parts, as shown in (8). The first one, \(I_{\text{ri}}\), represents the reduced intensity connected with the coherent component. The second part, \(I_{\text{d}}\), is related to the diffuse intensity, which can be interpreted as the incoherent field component propagation, i.e., it represents the specific intensity of the zero mean field. It is created by the scattering of the coherent component. \[I(0;\theta;\phi)=I_{\text{ri}}(z;\theta;\phi)+I_{\text{d}}(0;\theta;\phi). \tag{8}\]  
We split the diffuse intensity into two parts:  
\[I_{d}(z;\theta;\phi)=I_{1}(z;\theta;\phi)+I_{2}(z;\theta), \tag{9}\]  
where \(I_{1}\) is determined primarily by the forward lobe of the scatter function, and \(I_{2}\) is the reminder of \(I_{d}\) connected with the isotropic background. The solution of the radiative transfer equation is the following:  
\[ \begin{split}\frac{P_{\text{R}}(\tau;\mu_{\text{R}};\phi_{\text{ R}})}{P_{\text{max}}}=\underbrace{\exp\left(-\bigg{(}\frac{\gamma_{\text{RP}}}{ \Delta\gamma_{\text{R}}}\bigg{)}^{2}-\frac{\tau}{\mu_{\text{P}}}\right)}_{ \text{Coherent component attenuation and scattering}}\\ \underbrace{+\frac{\Delta\gamma_{\text{R}}^{2}}{4}\bigg{\{} \bigg{[}\exp\left(-\frac{\hat{\tau}}{\mu_{\text{P}}}\right)-\exp\left(-\frac{ \hat{\tau}}{\mu_{\text{P}}}\right)\bigg{]}\overline{q}_{M}(\gamma_{\text{RP }})}}_{\text{I}_{1}\text{ incoherent forward scattering}}\\ \underbrace{+\exp\left(-\frac{\tau}{\mu_{p}}\right)\sum_{m=1}^{M }\frac{1}{m!}\frac{\alpha\Psi_{\text{T}}}{\mu_{\text{P}}}^{m}\overline{[q} _{m}(\gamma_{PR})-\overline{q}_{M}(\gamma_{PR})]\bigg{\}}_{\text{I}_{1}\text{ incoherent forward scattering}}\\ +\underbrace{\frac{\Delta\gamma_{\text{R}}^{2}}{2}\bigg{\{}-\exp \left(-\frac{\hat{\tau}}{\mu_{p}}\right)\frac{1}{P_{N}}}}_{\text{I}_{2}\text{ incoherent isotropic background scattering}}+\underbrace{\sum_{k=\frac{N+1}{2}}^{N}\bigg{[}A_{K}\exp\left(-\frac{\hat{\tau}}{S_{k}} \right)\sum_{n=0}^{N}\frac{1}{1-\frac{u_{n}}{S_{k}}}\bigg{]}}_{\text{I}_{2} \text{ incoherent isotropic background scattering}},\end{split}  \tag{10}\]  
where  
\[\overline{q}_{m}(\gamma_{PR})=\frac{4}{\Delta_{\gamma_{R}}^{2}+m\Delta_{\gamma_{S}} ^{2}}\exp\left(-\frac{\gamma_{PR}^{2}}{\Delta_{\gamma_{R}}^{2}+m\Delta_{\gamma S }^{2}}\right), \tag{11}\]  
\[P_{\text{Max}}=P_{\text{R}}(0;\mu_{p},\phi_{p})=\frac{\lambda_{o}^{2}}{4\pi}G_{ R}(0)S_{p}. \tag{12}\]  
The attenuation coefficients \(S_{k}\) are determined by the following characteristic equation:  
\[\frac{\hat{W}}{2}\cdot\sum_{n=0}^{N}\frac{P_{n}}{1-\frac{\mu_{n}}{5}}, \tag{13}\]  
where \(N\) is the odd integer chosen as a compromise for computing time, \(P_{n}=\sin(\frac{\pi}{N})\sin(\frac{n\pi}{N})\), for (n = 1,..., N - 1) and \(\hat{W}=\frac{(1-\alpha)W}{1-\alpha W}\). Note that \(S_{0,\sim N/2}=-S_{N,\sim(N+1)/2}\).