## 4 Results and Discussion
### Overall Performance of the Statistical Approaches
#### 4.1.2 Multi-Incidence and Multi-Polarization Estimates of C-Band Backscattering Coefficients  
The results obtained in the C-band are presented in Figure 8, distinguishing the performance associated with each state of polarization. As in the previous section, the results obtained with MLR are based on variables with significant \(p\)-values. For signals acquired in co-polarization states (i.e., HH and VV), the estimates are based on the incidence angle and topsoil moisture, while for cross-polarization states, the following descriptors have \(p\)-values less than 0.05: the topsoil moisture, the fraction of clay, the standard deviation of roughness heights and the autocorrelation length collected in the direction parallel to the tillage orientation regarding the surface roughness. As previously mentioned, all the surface and satellite descriptors are considered for RF. Compared with the X-band results, the levels of accuracy are lower, with correlation values of 0.65/0.73, 0.65/0.71 and 0.58/0.70 for the two tested algorithms (i.e., MLR/RF) and for the HH, VV and VH polarizations, respectively (focusing on validation performance). The estimates are characterized by a near-zero bias, regardless of the considered configuration. Statistical models associated with co-polarizations (i.e., HH and VV) have similar levels of error, with RMSE values between 1.53 and 1.75 dB (Figure 8a-d). The error is slightly higher with cross-polarization (VH), with values of 2.22 dB for MLR (Figure 8e) and 1.95 dB for RF (Figure 8f) (comparable results are obtained with HV polarization, not shown here). Moreover, the RF algorithm provides results with higher stability, as evidenced by the quasi-similar levels of the statistical indices between the training and validation phases, while the performance is slightly degraded with MLR. Such behaviors have already been pointed out in previous studies [58; 59], with RF algorithms being associated with high predictive ability (showing a high level of performance in the multifactorial context in modeling non-linear relationships) and results with high stability (a characteristic mainly explained by the bootstrap aggregating procedure). As previously stated for the X-band, the results obtained in the C-band in the HH polarization state can be compared with those obtained with semi-empirical models on the same dataset [30]. The performance associated with the initial models proposed by [PERSON] et al. [11] and [PERSON] [12] or [PERSON] et al. [10] is lower than that obtained with the two statistical algorithms, as evidenced by the values of correlation (between 0.31 and 0.44) or corresponding errors (between 2.27 and 3.66 dB). After the improvement procedure (taking into account biases with regard to surface descriptors), the accuracy of the semi-empirical models is close (r = 0.65, RMSE = 1.74 dB) to that obtained with MLR but remains lower than that obtained with RF (r = 0.73 and RMSE = 1.60 dB for RF). Figure 7: Comparison between simulated and observed backscattering coefficients in the X-band with HH polarization using MLR (**a**) or RF (**b**) statistical algorithm. The gray dots represent the estimates obtained on the training subset, while the black dots come from the validation subset. Figure 8: Comparison between simulated and observed backscattering coefficients in the C-band in three polarizations states (HH (**a**,**b**), VV (**c**,**d**) and VH (**e**,**f**)) using MLR (**left**) or RF (**right**) statistical algorithm. The gray dots represent the estimates obtained on the training subset, while the black dots come from the validation subset. 1.3 Evaluation of the Statistical Algorithms Compared to Models Developed in the Literature--Which Approach to Retain? This section compares the performance obtained in this paper to that obtained in previous studies [24; 29; 30; 60; 61; 62; 63]. It focuses on statistical criteria (i.e., correlation coefficient and RMSE) obtained for satellite signals acquired in the X- and C-bands in HH, VV and VH polarization states (Figure 9).