## 3 Methods
### Herbaceous Vegetation Backscattering Model  
[PERSON] and [PERSON] developed a forest backscattering model for depicting the biometric characteristics and geometric distribution of forest canopy [55]. Three components are included in this model: crown, crown-ground and trunk-ground volume scattering. So, this model can be expressed as follows:  
\[\sigma^{0}_{\text{pq}}=\sigma^{0}_{\text{pq,c}}+\sigma^{0}_{\text{pq,cg}}+ \sigma^{0}_{\text{pq,tg}} \tag{1}\]  
where, \(\sigma^{0}_{\text{pq,c}}\), \(\sigma^{0}_{\text{pq,cg}}\) and \(\sigma^{0}_{\text{pq,tg}}\) are the crown, crown-ground and trunk-ground backscattering, respectively. p and q represent horizontal or vertical polarization.  
Figure 3: Flowchart for soil moisture retrieval in vegetation-covered regions based on remotely sensed SAR and optical data.  
The three scattering components can be expressed in the following equations [55]:  
\[\sigma^{0}_{\mathrm{Pq,c}}=k_{0}\frac{|\varepsilon_{\mathrm{w}}|^{1.3}}{\varepsilon _{\mathrm{w}}^{m}}\cos\theta\gamma_{\mathrm{Pq,c}}M_{\mathrm{veg}}\bigg{(}1- \mathrm{e}^{-k_{0}\varepsilon_{\mathrm{w}}^{m}\varepsilon_{\mathrm{w}}^{m} \sec\theta\beta_{\mathrm{Pq,c}}M_{\mathrm{veg}}}\bigg{)}, \tag{2}\]  
\[\sigma^{0}_{\mathrm{Pq,cg}}=k_{0}^{1.4}|\varepsilon_{\mathrm{w}}|^{1.3}e^{-k_{0 }^{2.2}\cos^{2}\theta}\Gamma_{\mathrm{Pq}}\gamma_{\mathrm{Pq,cg}}\times M_{ \mathrm{veg}}e^{-k_{0}\varepsilon_{\mathrm{w}}^{m}\sec\theta(\beta_{\mathrm{ Pq,c}}M_{\mathrm{veg}}+\beta_{\mathrm{Pq,c}}M_{\mathrm{veg}})}, \tag{3}\]  
\[\sigma^{0}_{\mathrm{Pq,tg}}=k_{0}\sin\theta|\varepsilon_{\mathrm{w}}|^{0.65}e^{ -k_{0}^{2.2}\cos^{2}\theta}\Gamma_{\mathrm{Pq}}\gamma_{\mathrm{Pq,tg}}\times M _{\mathrm{veg,t}}e^{-k_{0}^{m}\varepsilon_{\mathrm{w}}^{m}\sec\theta(\beta_{ \mathrm{Pq,c}}M_{\mathrm{veg}}+\beta_{\mathrm{Pq,t}}M_{\mathrm{veg,t}})}, \tag{4}\]  
where, \(k_{0}\) means the wave number. \(\varepsilon_{\mathrm{w}}=\varepsilon_{\mathrm{w}}^{\prime}-ie_{\mathrm{w}}^{m}\) is the dielectric constant of water, \(\varepsilon_{\mathrm{w}}^{\prime}\) and \(\varepsilon_{\mathrm{w}}^{\prime\prime}\) represent the real and imaginary part of dielectric constant. \(M_{\mathrm{veg}}\) is crown water content, which is same as VWC. \(M_{\mathrm{veg,t}}\) is trunk water content. \(\theta\) is incidence angle. \(s\) means the RMS height. \(\gamma_{\mathrm{Pq,c}}\), \(\beta_{\mathrm{Pq,c}}\), \(\gamma_{\mathrm{Pq,cg}}\), \(\beta_{\mathrm{Pq,t}}\) and \(\gamma_{\mathrm{Pq,tg}}\) are structural coefficients. \(\Gamma_{\mathrm{Pq}}\) is surface Fresnel reflectivity, which has a logarithmic relationship with backscatter coefficient of bare soil [56].  
However, this forest scattering model cannot be directly applied to characterize scattering mechanisms and structural parameters of herbaceous vegetation. Thus, [PERSON] et al. proposed a herbaceous vegetation backscattering model (HVBM) by eliminating the trunk-ground scattering component in the forest model [37]. HVBM was developed to provide solutions for the volume scattering from vegetation and bare soil. It considers the vegetation layer as a collection of homogeneous spherical particles and ignores the multiple scattering effects between vegetation and underlying surface. The crown-ground scattering part can be rewritten as follows:  
\[\sigma^{0}_{\mathrm{Pq,cg}}=k_{0}^{1.4}|\varepsilon_{\mathrm{w}}|^{1.3}e^{-k_ {0}^{2.2}\cos^{2}\theta}\Gamma_{\mathrm{Pq}}\gamma_{\mathrm{Pq,cg}}\times M_{ \mathrm{veg}}e^{-k_{0}\varepsilon_{\mathrm{w}}^{m}\sec\theta\beta_{\mathrm{ Pq,c}}M_{\mathrm{veg}}}, \tag{5}\]