## 3 Model Construction
### Analysis of Simulated Backscattering Coefficient as Functions of Roughness and Soil Moisture  
[PERSON] et al. [57] suggested that the backscattering coefficients of a bare soil surface could be presented as a product of two functions on a linear scale:  
\[\sigma_{pq}(\theta)=R_{pq}(\varepsilon_{s},\theta)\cdot Sr_{pq}(\theta,kHrms, kl) \tag{5}\]  
Figure 2: Backscattering coefficients calculated from the integral equation model (IEM) and calibrated integral equation model (CIEM) vs. the observations measured by synthetic aperture radar (SAR) at the C-band: (**a**) VV polarization, IEM; (**b**) VH polarization, IEM; (**c**) VV polarization, CIEM; (**d**) VH polarization, CIEM. or on a logarithmic scale [58]:  
\[\sigma_{pq}(\text{dB})=10\log[R_{pq}(\varepsilon_{s},\theta)\cdot Sr_{pq}(\theta, kHrms,kl)] \tag{6}\]  
One function is the dielectric function \(R_{pq}\) that reflects the soil moisture information; the other is the roughness function \(Sr_{pq}\) that describes the effect of the surface roughness at a different polarization. \(\sigma_{pq}\) is the backscattering coefficient, where subscripts \(q\) and \(p\) represent the polarization status. Meanwhile, \(R_{pq}\) and functions \(Sr_{pq}\) are independent of each other. Firstly, this paper discussed the effects of \(Hrms\) and the empirical correlation length _lopt_ on the backscattering coefficient using CIEM simulations in this section. Secondly, the effect of soil moisture \(mv\) on the backscattering coefficient using CIEM simulations was analyzed. Finally, the soil moisture inversion model was built using Equation (5) or Equation (6) based on the description of the model, which was built in Sections 3.3.1 and 3.3.2. 3.1 Effect of \(Hrms\), Empirical Correlation Length _lopt_ on the Backscattering Coefficient Using CIEM Simulations  
As far as we know [24; 40], two variables describe the surface roughness: \(Hrms\) and correlation length \(l\). Studies have shown that the measurement error of correlation length \(l\) is much larger than that of \(Hrms\) in the contact measurement techniques on roughness because the shorter length profile and large sampling interval in field roughness measurements can result in substantially undervaluing the correlation length [59]. The error between the measured backscattering coefficients and the simulations by CIEM is mainly determined by the inaccuracy of correlation length \(l\)[35]. If the effect of the correlation length on the backscattering coefficient is neglected, it may cause large errors in the estimation of the backscattering coefficient. Therefore, the CIEM in this study did not use the measured correlation length \(l\) but used the empirical correlation length _lopt_ instead. In building the model, variable incidence angle \(\theta\) and wavelength \(\lambda\) can be considered constant at the C-band. Therefore, the correlation between backscattering coefficient and surface roughness is as follows:  
\[\sigma_{pq}(\theta)=Sr_{pq}(\theta,kHrms,kl_{qpt})=Sr_{pq}(\theta,\frac{2\pi} {\lambda}Hrms,\frac{2\pi}{\lambda}l_{qpt})=Sr_{pq}(Hrms,l_{qpt}) \tag{7}\]  
The calibration correlation length _lopt_ in VV and HV polarizations can be expressed as Equations (3) and (4) in the CIEM. At the C-band, the incidence angle and frequency are known in VV and HV polarizations of the ENVISAT ASAR data, so variable incidence angle \(\theta\) and frequency \(f\) can be considered constant. Equations (3) and (4) can be expressed as follows:  
\[l_{optpq}=a+bHrms \tag{8}\]  
where \(a\) and \(b\) are constants. Therefore, combining Equations (5) and (7) yields the following expression:  
\[\sigma_{pq}(\theta)=Sr_{pq}(Hrms,l_{qpt})=Sr_{pq}(Hrms,a+bHrms)=Sr_{pq}(Hrms) \tag{9}\]  
When the backscattering coefficient was simulated in CIEM, the volumetric soil moisture content was assumed to be 18.6%, which is the mean of the measured soil moisture at the incidence angle of 33.5\({}^{\circ}\). Incidence angle \(\theta\) is set to 33.5\({}^{\circ}\), and frequency \(f\) is set to 5.405 GHz.