# A New Propagation Channel Synthesizer for UAVs in the Presence of Tree Canopies
## 2 Theoretical Background
### Multiple Scattering Theory (MST)  
MST in [2] follows [4] and [5], which are mainly based on [10] and [11]. The corresponding theoretical background applicable to radiowave propagation was provided first in [12; 13; 14] and used also later in [15] or [16] for the case of a slab filled with branches and leaves. As MST inputs also include physical characteristics in the form of branches and leaves dimensions, complex permittivity, number densities, and spatial orientation statistics [4; 5; 6], it represents a very suitable candidate for site-specific modeling. However, its complexity and non-trivial software implementation together with unwieldy computation demands, especially for the case of the coherent scattered field calculations [4; 5; 6], prevent its widespread usage. On the other hand, the incoherent scattered field dominating in other than the forward scatter direction can be obtained in a more convenient way. The reason is that, in this case, the corresponding tree canopy can be spatially sampled by voxels of sides exceeding half the wavelength [6], thus, significantly decreasing the computation time. We should note, however, that shadowing behind a tree canopy results only from the coherent field [4; 5; 6]. Considering the incident plane wave \(\mathbf{E}_{\text{inc}}\), the coherent field for the point \(\mathbf{r}^{\prime}\) inside the canopy is, e.g., [4; 5; 6; 7; 10]:  
\[\mathbf{E}_{\text{coh}}^{\text{in}}\left(\mathbf{r}^{\prime}\right)=\mathbf{E} _{\text{inc}}\exp\left\{-j(K-k_{0})s_{1}\left(\mathbf{r}^{\prime}\right)\right\} \tag{5}\]  
where \(K\) denotes the complex effective propagation constant inside the canopy. For a slab of scatterers, we have [15]:  
\[K=K^{\prime}-jK^{\prime\prime}=k_{0}\sin(\theta)+\frac{2\pi}{k_{0}\sin(\theta) }F^{eq}\Big{(}\mathbf{\hat{k}}_{\text{inc}},\mathbf{\hat{k}}_{\text{inc}} \Big{)} \tag{6}\]  
where \(\theta\) denotes the incident slant angle. In Equation (5), \(s_{1}(\mathbf{r}^{\prime})\) is the distance through the canopy to \(\mathbf{r}^{\prime}\) along the direction of \(\mathbf{\hat{k}}_{\text{inc}}\) (see Figure 2), and in Equation (6), \(F^{eq}(\mathbf{\hat{k}}_{\text{inc}},\mathbf{\hat{k}}_{\text{inc}})\) is the canopy's equivalent scattering amplitude per unit volume in the forward scatter direction [5; 11]. Similar to [7], as a simplification, we suppose the incident direction is always perpendicular to the surface of the  
Figure 1: The common PO approach. The incident plane wave fields \(\mathbf{E}_{\text{inc}}\) and \(\mathbf{H}_{\text{inc}}\) result into surface equivalent currents \(\mathbf{J}_{\text{eq}}\) and \(\mathbf{M}_{\text{eq}}\) on the illuminated outer surface of the object. In turn, these currents provide scattered fields resulting into the blockage fields behind the object and cause the final coherent fields behind the object to be \(\mathbf{E}_{\text{B}}\) and \(\mathbf{H}_{\text{B}}\) from Equations (3) and (4). canopy, i.e., \(\theta=90^{\circ}\) in Equation (6). For a point \(\mathbf{r}\) outside the canopy, the incoherent scattered field (only power without the phase information) in the \(\mathbf{\hat{k}}_{\text{out}}\) direction can be obtained as:  
\[\langle|\mathbf{E}_{\text{scat}}(\mathbf{r})|^{2}\rangle=\int_{V}|V\big{(} \mathbf{r},\mathbf{r}^{\prime}\big{)}|^{2}\Big{|}\mathbf{E}_{\text{coh}}^{ \text{in}}\big{(}\mathbf{r}^{\prime}\big{)}\Big{|}^{2}dV^{\prime} \tag{7}\]  
where \(\langle\cdot\rangle\) denotes the statistical average, \(dV^{\prime}\) represents the volume of each individual canopy's voxel and \(V(\mathbf{r},\mathbf{r}^{\prime})\) is an operator given as:  
\[V(\mathbf{r},\mathbf{r}^{\prime})=F^{eq}(\mathbf{\hat{k}}_{\text{out}}, \mathbf{\hat{k}}_{\text{inc}})\frac{\exp(-jk_{0}|\mathbf{r}-\mathbf{r}^{ \prime}|-j(K-k_{0})s_{2}(r^{\prime}))}{|\mathbf{r}-\mathbf{r}^{\prime}|}. \tag{8}\]  
Here, \(s_{2}(r^{\prime})\) is the distance through the canopy from \(\mathbf{r}^{\prime}\) along \(\mathbf{\hat{k}}_{\text{out}}\) (see Figure 2). For more details refer to [4; 5; 6; 10].