## 3 Methods
### Optirrg Simulations
#### 3.2.2 Inversion Approach for Irrigation Detection  
The previous sections proposed two criteria (Section 3.1.2: \(\Psi\)SSM\({}_{\text{P}_{(1)}}>\)\(\Psi\)SSM\({}_{\text{G}_{(1)}}\) and Section 3.2.1: \(\Psi\)SSM\({}_{\text{P}_{(1)}}>\)\(\Psi\)SSM\({}_{\text{r}_{(1)}}\)) to seek the possible irrigation events \(\text{I}_{\text{(k)}}\) having taken place at a date j between t\({}_{\text{(I)}}\) and t\({}_{\text{(I)}}\), with an amount given by the k index. However, [PERSON] et al. [72] highlighted that the SSM is estimated with an accuracy of 5 vol.%, which applies in our case to the SSM values estimated at the plot scale (\(\mu\)SSM\({}_{\text{P}}\) = 5%). The uncertainty on \(\Psi\)SSM\({}_{\text{P}}\) will then depend on the uncertainties in the measurements of SSM\({}_{\text{P}}\) at t\({}_{\text{(I)}}\) and t\({}_{\text{(I)}}\) as follows [94]:  
\[\mu_{(1)}=\Psi\text{SSM}_{\text{P}(1)}\times\sqrt{\left(\frac{\mu\text{SSM}_{ \text{P}(1)}}{\text{SSM}_{\text{P}(1)}}\right)^{2}+\left(\frac{\mu\text{SSM}_{ \text{P}(i)}}{\text{SSM}_{\text{P}(i)}}\right)^{2}} \tag{5}\]For instance, at the SAR acquisition date t(i), a value of the SSM\({}_{\rm P}\) is recorded as SSM\({}_{\rm P(i)}\) = 25 vol.%. Furthermore, our approach focuses on detecting the change in SSM\({}_{\rm P}\) between t(i) and the previous image date t(i), where SSM\({}_{\rm P(i)}\) = 15 vol.%. As a result, the \(\Psi\)SSM\({}_{\rm P}\) value is determined using Equation (2) yielding \(\Psi\)SSM\({}_{\rm P(i)}\) \(\cong\) 0.67. Subsequently, the error value on \(\Psi\)SSM\({}_{\rm P(i)}\) (\(\mu_{\rm(i)}\)) is computed using Equation (5), with a calculated value of \(\mu_{\rm(i)}\) \(\cong\) 0.26. The presence of too large uncertainties may hamper the identification of irrigation events from the previous arguments involving the \(\Psi\)SSM\({}_{\rm P(i)}\) values (\(\Psi\)SSM\({}_{\rm P(i)}\) - \(\Psi\)SSM\({}_{\rm G(i)}\) \(>\) 0 and \(\Psi\)SSM\({}_{\rm P(i)}\) - \(\Psi\)SSM\({}_{\rm T(i)}\) \(>\) 0). The following two conditions thus have to be met:  
\[\Psi\)SSM\({}_{\rm P(i)}\) - \(\Psi\)SSM\({}_{\rm G(i)}\) \(>\)\(\mu_{\rm(i)}\) (6)  
\[\Psi\)SSM\({}_{\rm P(i)}\) - \(\Psi\)SSM\({}_{\rm T(i)}\) \(>\)\(\mu_{\rm(i)}\) (7)  
Moreover, [PERSON] et al. [48] and [PERSON] et al. [44] highlighted that the effect of a rain or irrigation event on the SAR backscattering coefficient can last up to 3 days before the total dry-out of the surface. It is essential to note that the duration of dry-out largely relies on the specific soil type and its infiltration characteristics, as well as the vegetation cover. Hence, an irrigation that occurred 3 days (\(n\) = 3) before t(i) can have an effect on the change observed in SSM between SSM\({}_{\rm P(i)}\) and SSM\({}_{\rm P(i)}\), and therefore, on the computed \(\Psi\)SSM\({}_{\rm P(i)}\). From this point of view, a light rain that occurred shortly before the acquisition could have the same signature as a significant rain occurring 3 days before would. This means searching for an irrigation event that occurred at the j date between t(i) - 3 and t(i). As for the possible irrigation amounts, we restrained the set to realistic values with three possibilities: k = 20, 30, or 40 mm. This produced the I\({}_{\rm[j,k)}\) combinations of dates and amounts to test, resulting in SSM\({}_{\rm R_{\rm[j,k]}}\) values simulated by Optirrig, and therefore, in \(\Psi\)SSM\({}_{\rm R_{\rm[j,k]}}\) values, for each I\({}_{\rm[j,k]}\) between t(i) and t(i):  
\[\Psi\)SSM\({}_{\rm R_{\rm[j,k]}}=\frac{\rm SSM_{R_{\rm[j]}}-SSM_{R_{\rm[j]}}}{ \rm SSM_{R_{\rm[j]}}} \tag{8}\]  
The most probable irrigation combination between t(i) and t(i) was the I\({}_{\rm[j,k]}\) that minimized the absolute value of the discrepancy between \(\Psi\)SSM\({}_{\rm R_{\rm[j,k]}}\) and \(\Psi\)SSM\({}_{\rm P(i)}\), which was denoted \(\mid\)\(\Delta\Psi\)\({}_{\rm[j,k]}\)\(\mid\).