# Vegetation Loss Measurements for Single Alley Trees in Millimeter-Wave Bands
## 2 Additional Attenuation of Vegetation  
### Radiative Transfer Equation Perspective  
Foliage can be modeled as a medium of random scatterers where the electromagnetic field has two components: coherent and incoherent [12; 18; 19]. The coherent component is the average field with a known direction of propagation and a defined polarization. It is susceptible to both absorption and scatter, and as a result, the attenuation of this component is high. The incoherent component is the field with zero mean, which is the result of the scattering of the coherent component. Therefore, it has no specific direction and is partially depolarized. In transport theory [20], the fundamental quantity is the specific intensity \(I(\overline{r},\overline{s})\)[21], defined as  
\[I(\overline{r},\overline{s})=\frac{\mathrm{d}^{2}P(\overline{r},\overline{s})}{ \mathrm{d}a\mathrm{d}\Omega}. \tag{1}\]  
It is defined as the power slice \(\mathrm{d}^{2}P\) per unit area \(a\) and per unit solid angle \(\Omega\) (compare with Figure 1) propagating at the point \(\overline{r}\) in the direction \(\overline{s}\). Absorption and scattering by the woodland medium reduce the power propagating in the direction \(\overline{s}\):  
\[\mathrm{d}I_{1}=-(\sigma_{a}+\sigma_{s})I(\overline{r},\overline{s})\mathrm{d }l, \tag{2}\]  
where \(\mathrm{d}I_{1}\) is the instantaneous power reduction rate, \(\sigma_{a}\) is the absorption cross section per volume, and \(\sigma_{s}\) is the scatter cross section per volume. \(l\) is the length such that volume \(dV=\mathrm{d}a\cdot\mathrm{d}l\). A fraction of the power propagating in other directions \(\overline{s}^{\prime}\) will contribute to the main propagating component in the direction \(\overline{s}\). Then, the increase of intensity \(I(\overline{r},\overline{s})\) by fraction of the power specific for the medium \(p(\overline{s},\overline{s}^{\prime})\) is equal to  
\[\mathrm{d}I_{2}=\frac{1}{4\pi}\sigma_{s}\iint_{4\pi}p(\overline{s},\overline {s}^{\prime})I(\overline{r},\overline{s}^{\prime})\,\mathrm{d}\Omega^{\prime} \,\mathrm{d}l, \tag{3}\]  
where \(\mathrm{d}I_{2}\) describes the change in the contribution from the so-called medium's back-lighting, \(p(\overline{s},\overline{s}^{\prime})\) is the _phase function_ of the scattering medium that describes the angular distribution of power scattered by the medium, and \(\mathrm{d}\Omega^{\prime}\) is a dummy integration variable. It is the differential element of the solid angle used for the purpose of the integration over all spatial directions \(\overline{s}^{\prime}\). Since the reciprocity of the channel ensures that the same results will be obtained when the transmitter and receiver are interchanged, therefore \(p(\overline{s},\overline{s}^{\prime})=p(\overline{s}^{\prime},\overline{s})\). Then, the total variation of the intensity \(I\) over length \(\mathrm{d}l\) is the following:  
\[\mathrm{d}I=[\overline{s}\cdot\mathrm{grad}[I(\overline{r},\overline{s})]] \mathrm{d}l=\mathrm{d}I_{1}+\mathrm{d}I_{2}, \tag{4}\]  
so it is the sum of intensities from Equations (2) and (3) where \(\mathrm{grad}[I(\overline{r},\overline{s})]\) is the gradient of the intensity function at point \(\overline{r}\) and direction \(\overline{s}\). Hence, when (2) and (3) are put in (4), then:  
\[\overline{s}\cdot\mathrm{grad}[I(\overline{r},\overline{s})]+(\sigma_{a}+ \sigma_{s})I(\overline{r},\overline{s})=\frac{\sigma_{s}}{4\pi}\iint_{4\pi}p (\overline{s},\overline{s}^{\prime})I(\overline{r},\overline{s})\,\mathrm{d }\Omega^{\prime} \tag{5}\]  
Equation (5) is known as the scalar form of the transport equation. When modeling the attenuation in the foliage, the simplified model can be taken into account. The woodland is a homogeneous medium of random scatterers, where the scatter elements are large compared to the wavelengths in the millimeter-wave bandwidth. Then, the forward-scattering component \(p(\gamma)\) can be modeled as a Gaussian lobe with a width of \(\Delta\gamma_{s}\) (Figure 2):  
Figure 1: Geometry for normally incident rays.