## 4 Results
### Using the Synthetic Database
#### 4.1.1 Neural Networks Using the VV Alone (NN1, NN1_dry, and NN1_wet)  
Figure 3 shows the performance of the neural networks that take \(\sigma^{\circ}\)vv, \(\theta\), and NDVI as input parameters. For NN1 (no a priori information on the \(mv\)), the training and validation were performed using all training and validation database elements, respectively (\(mv\) between 2 and 40 vol %). NN1_dry was trained using the noisy training database elements with \(mv\) between 2 and 30 vol % and next was validated using the noisy validation database elements with \(mv\) between 2 and 25 vol %. For NN1_wet, the training was performed using the noisy training database elements with \(mv\) between 20 and 40 vol %, and the validation was realized using the noisy validation database elements the \(mv\) between 25 and 40 vol %. The neural network built without the use of a priori information on the \(mv\) (NN1) shows a non-biased estimation of \(mv\) and an RMSE of 6.1 vol % (Figure 3a). Moreover, errors on the \(mv\) estimated from NN1 was analyzed for a reference \(mv\) lower and higher than 25 vol %. For a reference \(mv\) lower than 25 vol %, the results show that NN1 yielded an overestimation of the reference \(mv\) of 2.7 vol % and an RMSE equal to 5.5 vol %. For a reference \(mv\) higher than 25 vol %, an underestimation of \(mv\) (bias = \(-4.1\) vol %) and an RMSE equal to 6.9 vol % were obtained with NN1. For a reference \(mv\) lower and higher than 25 vol %, the accuracy of \(mv\) estimates (bias and RMSE) is improved with the consideration of a priori information on the \(mv\) (dry to slightly wet soil conditions \"NN1_dry\" and very wet \"NN1_wet\") (Figure 3b,c). Indeed, for a reference \(mv\) lower than 25 vol %, the use of NN1_dry yields a smaller bias \"estimated \(mv\)--reference \(mv\)\" (1.2 vol %) and an RMSE (4.0 vol %) in comparison to NN1 (no a priori information on the \(mv\)) (Figure 3b). Similarly, for a reference \(mv\) between 25 and 40 vol %, the use of NN1_wet improves the bias by 1.8 vol % (bias = \(-2.3\) vol %) and the RMSE by 1.8 vol % (RMSE = 5.1 vol %) (Figure 3c). Furthermore, the evolution of errors (bias and RMSE) on the \(mv\) estimated from the NN1, NN1_dry, and NN1_wet are analyzed according to the \(Hrms\), NDVI, and \(\theta\) (Figure 4). In general, the results show that the bias varies with the \(Hrms\) and remains constant with the NDVI and \(\theta\). In the case of no a priori information on the \(mv\) (NN1) and for a reference \(mv\) between 2 and 25 vol % (Figure 4a), the bias varies from \(-2.2\) vol % to 6.2 vol % as the \(Hrms\) increases from 0.5 to 3.8 cm (underestimation for an \(Hrms\) lower than 1.2 cm and overestimation for an \(Hrms\) higher than 1.2 cm). However, the bias remains constant (approximately 2.7 vol %) with the NDVI for NDVI between 0 and 0.75 and with \(\theta\) for \(\theta\) between 20\({}^{\circ}\) and 45\({}^{\circ}\) (Figure 4b,c). In the case of no a priori information (NN1), and for a reference \(mv\) between 25 and 40 vol %, the \(mv\) estimates are highly underestimated for an \(Hrms\) equal to 0.5 cm (bias = \(-10.5\) vol %). This underestimation decreases when the \(Hrms\) increases to reach approximately 0.3 vol % for an \(Hrms\) = 3.8 cm (Figure 4a). Figure 4b,c show that the difference between the estimated and reference \(mv\) is approximately \(-4.1\) vol % regardless of the NDVI and \(\theta\) values in the case of NN1 and for very wet soil conditions. The use of a priori information on the \(mv\) (dry to slightly wet \"NN1_dry\" and very wet \"NN1_wet\") yields a similar bias behavior according to the \(Hrms\), \(NDVI\) and \(\theta\), as in the case of no a priori information (Figure 4). However, the bias level is lower with NN1_dry and NN1_wet than with NN1.