## 4 Results
### Using the Synthetic Database
#### 4.1.1 Neural Networks Using the VV Alone (NN1, NN1_dry, and NN1_wet)
Using a priori information on the \(mv\) with dry to slightly wet soil conditions, NN1_dry slightly improves the bias for an \(Hrms\) lower than 1.4 cm (less than 1 vol %), whereas the improvement varies from 1.1 vol % for an \(Hrms\) equal to 1.4 cm to 2.3 vol %, for an \(Hrms\) equal to 3.8 cm (Figure 4a). The use of a priori information with very wet soil conditions (NN1_wet) improves the bias significantly for an \(Hrms\) lower than 2 cm (improvement decreases from 4.9 to 1.2 as the \(Hrms\) increases from 0.5 to 2 cm) and slightly improves the bias for an \(Hrms\) higher than 2 cm (improvement is less than 1 vol %) (Figure 4a). In using NN1_dry, the bias by approximately 1.5 vol % for all NDVI and \(\theta\) values (Figure 4b,c). In using NN1_wet, the improvement in the bias varies between 1.5 vol % for an NDVI = 0 to 2.6 vol % for an NDVI = 0.75 (Figure 3(b)), and it is approximately 1.8 vol % for all \(\theta\) values (Figure 3(c)). The analysis of the RMSE on the \(mv\) estimates in the case of no a priori information on the \(mv\) (NN1) with a reference \(mv\) between 2 and 25 vol % shows that the RMSE decreases slightly from 4.4 vol % to 3.6 vol % when the \(Hrns\) increases from 0.5 to 1.2 cm. Beyond 1.2 cm, the RMSE increases greatly from 3.6 to 7.5 when the \(Hrns\) increases from 1.2 to 3.8 cm (Figure 3(d)). Moreover, the results show that the RMSE increases slightly from 5.0 to 6.4 vol % when the NDVI increases from 0 until 0.75 (Figure 3(e)). The RMSE seems not depends on \(\theta\) with an RMSE on the \(mv\) of approximately 5.5 vol % for a \(\theta\) between 20 and 45\({}^{\circ}\) (Figure 3(f)). In the case of no a priori information on the \(mv\) with a reference \(mv\) between 25 and 40 vol %, the RMSE on the \(mv\) decreases from 11.7 vol % to 4.3 vol % when the \(Hrns\) increases from 0.5 to 3.8 cm (Figure 3(d)). Moreover, the results show that the RMSE has a slight increase of approximately 1.3 vol % (from 6.5 to 7.8 vol %) when the NDVI increases from 0 to 0.75 (Figure 3(e)). Finally, the RMSE increases slightly from 6.2 to 6.9 vol % for \(\theta\) between 20 and 45\({}^{\circ}\) (Figure 3(f)). In the case of a priori information on the \(mv\) with dry to slightly wet soil conditions, NN1_dry improves the RMSE (calculated using the validation database with the \(mv\) between 2 and 25 vol %) by less than 1 vol % for an \(Hrns\) less than 1.6 cm, by 1.6 vol % for an \(Hrns\) between 1.6 and 2.8 cm, and by approximately 2.3 vol % for an \(Hrns\) higher than 2.8 cm (Figure 3(d)). In addition, the RMSE on the \(mv\) estimates is improved by approximately 1.5 vol % for an NDVI between 0 and 0.75 and for \(\theta\) between 20\({}^{\circ}\) and 45\({}^{\circ}\) when using NN1_dry, compared to the results obtained with NN1 (Figure 3(e),f). The use of a priori information on the \(mv\) with very wet soil conditions (NN1_wet) improves the RMSE on the \(mv\) estimates (calculated using the validation database with an \(mv\) between 25 and 40 vol %). This improvement is significant for a low \(Hrns\) (4.6 vol% for \(Hrns\) = 0.5 cm) and decreases when the \(Hrns\) increases (0.2 vol % for \(Hrns\) = 3.8 cm) (Figure 3(d)). In comparison to NN1, the network NN1_wet improves the RMSE from 1.5 vol % for an NDVI = 0 to 2.6 vol % for an NDVI = 0.75 (Figure 3(e)). In addition, Figure 3(f) shows that the decrease of the RMSE on the \(mv\) estimates is in the same order (approximately 1.8 vol %) for all \(\theta\) between 20\({}^{\circ}\) and 45\({}^{\circ}\) (RMSE = 6.9 vol % with NN1 and RMSE = 5.1 vol % with NN1_wet). Figure 3: \(Cont.\)  
Figure 4: _Cont_. Figure 3: Estimated \(mv\) as a function of the reference \(mv\) when the VV is used alone. (**a**) NN1 (no a priori on the \(mv\)), (**b**) NN1_dry (a priori information on the \(mv\), dry to slightly wet); (**c**) NN1_wet (a priori information on the \(mv\), very wet).