## 4 Results  
### Model Fitting and Comparison  
A total of 727 soil moisture and backscattering coefficient data pairs from 15 observation sites on 49 imaging days are available for model fitting. Based on the Equation (2), three LME models were developed using \(\sigma^{0}\) of different polarization combinations: VV polarization, VH polarization, and VV + VH polarizations. Table 3 shows the calibrated parameters and performance statistics of these LME models. For the VV polarized model, the fixed effects of the intercept and slope (\(\sigma^{0}\)) are statistically significant with \(\alpha\) = 33.36 (\(p\) < 0.001) and \(\beta\) = 0.33 (\(p\) < 0.001), respectively. The standard deviations of the day-specific intercepts and slopes are 2.04 and 0.13, respectively, with that of the site-specific intercepts being 6.83. For the VH polarized model, while the fixed term of the intercept is still significant with \(\alpha\) = 32.54 (\(p\) < 0.001), the fixed effect of the slope (\(\sigma^{0}\)) is nonsignificant with \(\beta\) = 0.13 (\(p\) = 0.175). The VV polarized model gains a slightly higher R\({}^{2}\) and smaller RMSE and MPE when compared with the VH polarized model. Combining both VV and VH polarizations in the LME model, the performance of the VV + VH polarized model is slightly worse than the single polarized models, obtaining the lowest R\({}^{2}\) and the largest RMSE and MPE. In the VV + VH polarized model, while the fixed effects of the intercept and the VV polarized slope (\(\sigma^{0}_{vv}\)) are statistically significant, with \(\alpha\) = 32.98 (\(p\) < 0.001) and \(\beta_{vv}\) = 0.34 (\(p\) < 0.001), respectively, the fixed effect of the VH polarized slope (\(\sigma^{0}_{vh}\)) is nonsignificant with \(\beta_{vh}\) = \(-\)0.03 (\(p\) = 0.749). This is probably because VH polarization is more sensitive to volume scattering from vegetation canopies and multiple surface scattering from rough surfaces [55]. Although all of the three LME models meet the ESA accuracy requirement of 5% [32] for GMES soil moisture product, VV polarization achieves the best performance, and is hence used in the following LME models. Based on the Equation (3), a linear fixed effects model was developed using the same set of soil moisture and backscattering coefficient data pairs. The performance statistics of this model are shown in the Table 4. Without considering the spatial differences in the SM-\(\sigma^{0}\) relationship, although the linear fixed effects model obtains a satisfying temporal and spatial R\({}^{2}\), the MPE achieved is very large. This results in the relatively low overall R\({}^{2}\) and large RMSE, and the latter fails to meet the 5% accuracy requirement. Nevertheless, with a temporal R\({}^{2}\) of 0.56, the fixed effects model is supposed to capture the temporal variations of soil moisture to some extent. In contrast, the full LME model achieves both higher temporal and spatial R\({}^{2}\) and lower MPE, significantly improving the overall R\({}^{2}\) and RMSE to 0.89 and 2.53%, respectively. To further study the sensitivity of the SM-\(\sigma^{0}\) relationship to site locations, the site-specific term \(s_{i}\) was removed from the full LME model. As shown in Table 4, the spatial R\({}^{2}\) of the LME model without the site-specific term drops dramatically to 0.052, leading to the lowest overall R\({}^{2}\) and largest RMSE and MPE of the three models. The large differences between \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Fixed Slop** & **Fixed Intercept** & **Overall R\({}^{2}\)** & **RMSE (\%)** & **MPE (\%)** \\ \hline VV & 0.33 \({}^{*}\) & 33.36 \({}^{*}\) & 0.894 & 2.53 & 1.87 \\ VH & 0.13 & 32.54 \({}^{*}\) & 0.892 & 2.56 & 1.90 \\ VV + VH & VV: 0.34 \({}^{*}\); & 32.98 \({}^{*}\) & 0.860 & 3.12 & 2.45 \\  & VH: \(-\)0.03 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The calibrated parameters and performance statistics for three LME models with different polarizations.