## 3 Model Construction  
### Integral Equation Model and Calibrated Integral Equation Model  
The IEM is the most commonly used physical model [54]. In the C-band, the validity domain of IEM [33] covers only a part of the range of roughness values commonly encountered for agricultural surfaces (\(k\)\({}^{*}\)\(Hrms\)\(\leq\) 3 corresponds to \(Hrms\)\(\leq\) 1.11 cm\({}^{-1}\) in C-band). The IEM describes the relationship between the backscattering coefficient with the characteristics of the sensor (incidence angle, polarization, and radar wavelength) and the soil properties (dielectric constant, \(Hrms\), correlation length, and correlation function) over bare soils in agriculturally bare areas. It can be expressed as follows:  
\[\begin{split}&\sigma_{pp}^{0}=\frac{k^{2}}{2}\big{|}f_{pp}\big{|} ^{2}e^{-4 Hrms^{2}k^{2}\cos^{2}\theta}\sum\limits_{n=1}^{+\infty}\frac{(AHrms^{ 2}k^{2}\cos^{2}\theta)^{n}}{n!}W^{(n)}(2k\sin\theta,0)\\ &+\frac{k^{2}}{2}\text{Re}(f_{pp}^{*}F_{PP})e^{-3 Hrms^{2}k^{2} \cos^{2}\theta}\sum\limits_{n=1}^{+\infty}\frac{(Hrms^{2}\cos^{2}\theta)^{n}} {n!}W^{(n)}(2k\sin\theta,0)\\ &+\frac{k^{2}}{8}\big{|}f_{pp}\big{|}^{2}e^{-2 Hrms^{2}k^{2}\cos^ {2}\theta}\sum\limits_{n=1}^{+\infty}\frac{(Hrms^{2}k^{2}\cos^{2}\theta)^{n}} {n!}W^{(n)}(2k\sin\theta,0)\end{split} \tag{1}\]  
The CIEM is a semiempirical calibration model, which was proposed based on IEM by [PERSON] et al. [35; 36]. The proposed CIEM adopts the empirical correlation length _lopt_ and reduces the number of input soil parameters of the IEM from three to two (\(Hrms\) and \(mv\) only, instead of \(Hrms\), \(l\) and \(mv\)). Correlation length \(l\) as the measured correlation length was replaced by a calibration parameter _lopt_ included in the CIEM. Calibration parameter _lopt_ (an empirical correlation length) is computed as a function of the \(Hrms\), radar frequency, incidence angle, and polarization to obtain a better fit between CIEM simulations and radar observations.  
The variable _lopt_ in CIEM is not the input parameter but can be calculated in CIEM. Furthermore, it integrates the true correlation length and the imperfections of the IEM so that the simulated data using the CIEM can be consistent with radar measurement data.  
For C-band SAR data, calibration parameter _lopt_ can be expressed as follows [35; 36]:  
\[l_{\text{{opt-HH}}}(Hrms,\theta,HH)=0.162+3.006(\sin 1.23\theta)^{-1.494}Hrms \tag{2}\]  
\[l_{\text{{opt-HV}}}(Hrms,\theta,HV)=0.9157+1.2289(\sin 0.1543\theta)^{-0.3139}Hrms \tag{3}\]  
\[l_{\text{{opt-VV}}}(Hrms,\theta,VV)=1.281+0.134(\sin 0.19\theta)^{-1.59}Hrms \tag{4}\]  
where \(\theta\) is in radians, and \(Hrms\) and _lopt_ are in centimeters.  
The CIEM has been successfully tested in many studies [11; 55; 56].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**Satellite Data Description**} \\ \hline
**Satellite Data** & **Polarization** & **Band** & **Acquisition Date** & **Incidence Angle (\({}^{\circ}\))** \\ \hline ENVISAT/ASAR & VV/VH & C & 11 July 2008 & 33.5\({}^{\circ}\) \\ ENVISAT/ASAR & VV/VH & C & 24 May 2008 & 26.3\({}^{\circ}\) \\ \hline \multicolumn{5}{c}{**Ground Measurements Data**} \\ \hline
**Min** & **Max** & **Average** & **Observation Time** \\ \hline
**Soil moisture** (\%) & 13.5 & 34.7 & 18.6 & 11 July 2008 \\
**Soil moisture** (\%) & 17.6 & 50.7 & 24.7 & 24 May 2008 \\
**\(Hrms\)** (cm) & 0.68 & 4.08 & 1.4 & 7 June 2008 \\
**Correlation length** (cm) & 53 & 69 & 63.2 & 7 June 2008 \\ \hline \hline \end{tabular}
\end{table}
Table 2: ASAR images and simultaneous ground-based measurement data.