## 3 Methods
### Optirrg Simulations
#### 3.2.2 Inversion Approach for Irrigation Detection
Consequently, our goal was to find the irrigation event I\({}_{\rm[j,k]}\) that minimized the difference between the \(\Psi\)SSM\({}_{\rm R_{\rm[j,k]}}\) and \(\Psi\)SSM\({}_{\rm P(i)}\) values. By contrast, the highest \(\mid\)\(\Delta\Psi\)\({}_{\rm[j,k]}\)\(\mid\) values for a hypothesized I\({}_{\rm[j,k]}\) event indicated the lowest probability for I\({}_{\rm[j,k]}\) to have occurred:  
\[\left|\Delta\Psi\)\({}_{\rm[j,k]}\)\(\mid=\)\(\left|\Psi\)SSM\({}_{\rm R_{\rm[j,k]}}-\Psi\)SSM\({}_{\rm P(i)}\right|\) (9)  
When trying to minimize the \(\mid\)\(\Delta\Psi\)\({}_{\rm[j,k]}\)\(\mid\) values, seeking \(\mid\)\(\Delta\Psi\)\({}_{\rm[j,k]}\)\(\mid\) values as close to zero as possible, it is very likely that both negative and positive values of \(\Delta\Psi\)\({}_{\rm[j,k]}\) will be encountered. A negative \(\Delta\Psi\)\({}_{\rm[j,k]}\) (\(\Delta\Psi\)\({}_{\rm[j,k]}\)\(<\) 0) occurs when the tested irrigation I\({}_{\rm[j,k]}\) yields a \(\Psi\)SSM\({}_{\rm R_{\rm[j,k]}}\) lower than \(\Psi\)SSM\({}_{\rm P(i)}\). The negative value indicates a shortfall in the impact of I\({}_{\rm[j,k]}\) on \(\Psi\)SSM\({}_{\rm R_{\rm[j,k]}}\) when compared with \(\Psi\)SSM\({}_{\rm P(i)}\). Conversely, a positive \(\Delta\Psi\)\({}_{\rm[j,k]}\) suggests that the tested irrigation I\({}_{\rm[j,k]}\) outperforms the \(\Psi\)SSM\({}_{\rm P(i)}\) by leading to \(\Psi\)SSM\({}_{\rm R_{\rm[j,k]}}\) values higher than \(\Psi\)SSM\({}_{\rm P(i)}\). This indicates a desire to make the \(\Delta\Psi\)\({}_{\rm[j,k]}\) of an I\({}_{\rm[j,k]}\) atj and \(\Delta\Psi\)\({}_{\rm[j+1,k]}\) of an I\({}_{\rm[j,k]}\) atj + 1 comparable and a change in sign occurs between \(\Delta\Psi\)\({}_{\rm[j,k]}\) and \(\Delta\Psi\)\({}_{\rm[j+1,k]}\) values, from negative to positive (10) or from positive to negative (11):  
\[\Delta\Psi\)\({}_{\rm[j,k]}<\) 0 And \(\Delta\Psi\)\({}_{\rm[j+1,k]}\) \(\geq\) 0 (10)  
\[\Delta\Psi\)\({}_{\rm[j,k]}>\) 0 And \(\Delta\Psi\)\({}_{\rm[j+1,k]}\) \(\leq\) 0 (11)This change in sign is a \"necessary but not sufficient condition\" for the detection of an irrigation event having taken place between \(t_{(i)}\) and \(t_{(i)}\), as the magnitude of the change in \(\Delta\Psi_{(j,k)}\) values also has to be compared with the intrinsic uncertainties, which requires  
\[\Delta\Psi_{(j,k)}<-\mu_{(i)}\;\mathrm{And}\;\Delta\Psi_{(j+1,k)}\geq-\mu_{(i)} \tag{12}\]  
\[\Delta\Psi_{(j,k)}>\mu_{(i)}\;\mathrm{And}\;\Delta\Psi_{(j+1,k)}\leq\mu_{(i)} \tag{13}\]  
In the last step, Equations (12) and (13) establish the conditions for detecting an irrigation event between two specific time points, namely, \(t_{(i)}\) and \(t_{(i)}\). These conditions involve comparing the magnitudes of \(\Delta\Psi_{(j,k)}\) values with the intrinsic uncertainties represented by \(\mu_{(i)}\). Equation (12) states that if \(\Delta\Psi_{(j,k)}\) is less than the negative intrinsic uncertainty (\(-\mu_{(1)}\)), and \(\Delta\Psi_{(j+1,k)}\) is greater than or equal to \(-\mu_{(i)}\), it suggests the possibility of an irrigation event that occurred between \(t_{(i)}\) and \(t_{(i)}\). In addition, Equation (13) implies that when the difference \(\Delta\Psi_{(j,k)}\) is greater than the positive intrinsic uncertainty (\(\mu_{(i)}\)), and the subsequent difference \(\Delta\Psi_{(j+1,k)}\) is less than or equal to \(\mu_{(i)}\), it indicates the potential occurrence of an irrigation event between time points \(t_{(i)}\) and \(t_{(i)}\). Therefore, if either condition (12) or (13) is met, the magnitudes of \(\mid\Delta\Psi_{(j,k)}\mid\) and \(\mid\Delta\Psi_{(j+1,k)}\mid\) are compared. The one with the smallest value represents the most probable combination of irrigation date and amount (\(I_{(j,k)}\)).