# Recent Progress on Vegetation Remote Sensing Using Spaceborne GNSS-Reflectometry
## 4 Model Simulations and Discussion
### Theoretical Models  
The Bi-MIMICS (Bitstaic-Michigan Microwave Canopy Scattering Model) model is based on the first-order solution of the radiative transfer equation. In the model, the forest stand is divided into canopy layer, trunk layer, and rough surface layer (Figure 1). The first-order scattering includes the single scattering of each layer and double scattering between layers. The canopy is simulated as a dielectric cylinder (representing needles or branches) and a disc (representing leaves). The trunk layer is composed of a uniform radius of perpendicularly facing dielectric cylinders. Regarding the circular polarization characteristics of GNSS-R, the Bi-MIMICS model needs to be modified by the method of polarization synthesis, so that it can calculate the scattering characteristics of any polarization. The improved model can calculate the scattering characteristics of any polarization under any observation geometry [48]. There are eight scattering mechanisms in the model: G-C-G, C-G, DC, G-C, G-T, DG, T-G, and the S-G. Traditional microwave electromagnetic scattering models or radiation models are mainly aimed at the linear polarization characteristics of radars or radiometers. Therefore, the traditional models pay less attention to the circular polarization characteristics. In order to overcome the impact of the ionosphere, GNSS satellites emit signals in RHCP polarization. After this kind of signal is reflected on the ground, the polarization characteristics will change. Therefore, if we conduct research on GNSS-R, it is necessary to improve the traditional model so that it can calculate the scattering characteristics of various polarizations. Regarding the first-order radiation transfer equation model used in this paper, the wave synthesis technique is used to modify the model. It should be noted that before the wave synthesis technique is used, a 4 \(\times\) 4 transformation matrix is needed to be constructed in the model. The corresponding formula of the wave synthesis technique is like the following:  
\[\sigma_{rt}(\psi_{r},\chi_{r},\psi_{t},\chi_{t})=4\pi\tilde{Y}_{m}^{r}M_{m}Y_{m }^{t} \tag{1}\]  
where \(M_{m}\) is the first-order fully polarized transformation matrix, i.e., the normalized Mueller matrix. \(\tilde{Y}_{m}^{r}\)\(\Upsilon_{m}^{r}\) and \(Y_{m}^{t}\) are the normalized Stokes vectors characterizing the transmitter and receiver polarization. By using this technique, we can obtain the bistatic scattering cross section \(\sigma_{rt}\) for any combination of transmit and receive polarizations. The subscript r and t represent the polarization of receiver and transmitter, respectively. As  
Figure 1: Scattering mechanisms in the first-order Bi-MIMICS model, including G-C-G (ground reflection and crown scattering and ground reflection), C-G (crown scattering and ground reflection), DC (direct crown bistatic scattering), G-C (ground reflection and crown scattering), G-T (ground reflection and trunk scattering), DG (direct ground), and T-G (trunk scattering and ground reflection). The specular ground reflection is not shown in the figure. Crown layer depth = d, trunk layer depth = H.  
can be seen from Equation (1), the final bistatic scattering cross section is a function of the orientation and ellipticity angles (\(\psi\) and \(\chi\)), as shown in Figure 2.  
The two angles \(\psi\) and \(\chi\) are sufficient to specify the polarization completely. The normalized Stokes vectors can be represented like the following:  
\[Y_{m}=\left[ \begin{array}{c}\frac{1}{2}(1+\cos 2\psi\cos 2\chi)\\ \frac{1}{2}(1-\cos 2\psi\cos 2\chi)\\ \sin 2\psi\cos 2\chi\\ \sin 2\chi\end{array} \right] \tag{2}\]  
Table 1 presents the modified Stokes vectors corresponding to several commonly used polarizations. During our simulations in Section 4.4, we employed Aspen as a representative. The detail input parameters are shown in Tables 2 and 3.