# Unmanned Aerial Vehicle (UAV) Robot Microwave Imaging Based on Multi-Path Scattering Model
## 2 Model Description  
### Multi-Path Scattering Model  
In this study, the multi-path scattering model for virtual vegetation is established based on Distorted Born Approximation (DBA) [14]. The virtual vegetation is composed of dielectric cylinders and disks which stand for the trunk, branches and leaves, respectively. The ground surface, which is described by the standard deviation of the height, correlation length and dielectric constant, is divided into many small units equally. The total scattering fields are added coherently from all the scatterers from vegetation and ground surface. It is assumed that there are \(N\) scatterers in the vegetation area and \(M\) ground blocks. The total scattering fields include five components, in which four components are about vegetation scatterers with transmission paths inherited from Born approximation and one component is related to ground surface, as shown in Figure 1. Direct scattering from the scatterers in the vegetation is denoted as \(\overline{E}_{n}^{ds}(k_{i},x)\). The single bounce of the scattering with a path of transmitter-scatterer-ground-receiver is denoted as \(\overline{E}_{n}^{s}(k_{i},x)\), and the scattering with a path of transmitter-ground-scatterer-receiver is denoted as \(\overline{E}_{n}^{s}(k_{i},x)\). Furthermore, the double bounce of the scattering with a path of transmitter-ground-scatterer-ground-receiver is denoted as \(\overline{E}_{n}^{gg}(k_{i},x)\). Lastly, direct scattering from the ground is denoted as \(\overline{E}_{m}^{fg}(k_{i},x)\). Therefore, the total scattering field is the coherent summation of each component from all the azimuth observation positions \(x\) and wave numbers \(k_{i}\), which can be expressed as  
\[\overline{E}(k_{i},x)=\sum_{n=1}^{N}\left(\overline{E}_{n}^{ds}(k_{i},x)+ \overline{E}_{n}^{sg}(k_{i},x)+\overline{E}_{n}^{sg}(k_{i},x)+\overline{E}_ {n}^{sg}(k_{i},x)\right)+\sum_{m=1}^{M}\overline{E}_{m}^{fg}(k_{i},x) \tag{1}\]where \(n\) denotes the scatterer in the vegetation area, and m denotes the ground block. Each component can then be expressed as  
\[\overline{E}_{n}^{ds}(k_{i},x) =\frac{e^{ikr}}{4\pi r}e^{i\varphi_{n}}A_{t}(\theta_{i},\phi_{i})A_ {r}(\theta_{r},\phi_{r})S_{n}^{ds}\overline{E}_{0}^{i} \tag{2}\] \[\overline{E}_{n}^{sg}(k_{i},x) =\frac{e^{ikr}}{4\pi r}e^{i\varphi_{n}}A_{t}(\theta_{i},\phi_{i})A_ {r}(\theta_{r},\phi_{r})S_{n}^{sg}\overline{E}_{0}^{i}\] (3) \[\overline{E}_{n}^{gs}(k_{i},x) =\frac{e^{ikr}}{4\pi r}e^{i\varphi_{n}}A_{t}(\theta_{i},\phi_{i})A_ {r}(\theta_{r},\phi_{r})S_{n}^{sg}\overline{E}_{0}^{i}\] (4) \[\overline{E}_{n}^{sgg}(k_{i},x) =\frac{e^{ikr}}{4\pi r}e^{i\varphi_{n}}A_{t}(\theta_{i},\phi_{i})A _{r}(\theta_{r},\phi_{r})S_{n}^{sg}\overline{E}_{0}^{i}\] (5) \[\overline{E}_{n}^{dg}(k_{i},x) =\frac{e^{ikr}}{4\pi r}e^{i\varphi_{n}}A_{t}(\theta_{i},\phi_{i})A _{r}(\theta_{r},\phi_{r})S_{n}^{dg}\overline{E}_{0}^{i} \tag{6}\]  
where \(A_{t}(\theta_{i},\phi_{i})\) and \(A_{r}(\theta_{r},\phi_{r})\) denote the antenna patterns of the transmitter and receiver, respectively. \((\theta_{i},\phi_{i})\) and \((\theta_{r},\phi_{r})\) are the transmitting and receiving angles. \(\varphi_{n}\) is phase compensation term of the \(n_{th}\) scatterer, which is the phase shift from the local to the global coordinate system. Here \(\varphi_{n}=(\overline{k}_{i}-\overline{k}_{s})\cdot\overline{r}_{m}\), where \(\overline{r}_{m}\) denotes position vector of the \(n_{th}\) scatterer in global coordinate system. \(\overline{k}_{i}\) and \(\overline{k}_{s}\) are the wave vectors of incidence and scattering wave, respectively. In addition, \(\varphi_{m}=(\overline{k}_{i}-\overline{k}_{s})\cdot\overline{r}_{m}\), where \(\overline{r}_{m}\) denotes the position vector of the \(n_{th}\) ground block.