## 4 Results
### Detection of Irrigation Dates and Amounts
For instance, an occurrence of irrigation was identified between t\({}_{\rm(j)}\) = 31 July 2017 and t\({}_{\rm(0)}\) = 6 August 2017, with ascending SAR image dates (Figure 6a). A detailed analysis was carried out in Figure 6b to determine the specific irrigation date and amount that probably occurred between t\({}_{\rm(j)}\)\(-\)3 and t\({}_{\rm(j)}\). This investigation involved examining the changes in \(\Delta\Psi_{\rm(j,k)}\) for different irrigation dates j ranging from 28 July 2017 to 5 August 2017, and k amounts of 20, 30, and 40 mm to compare \(\Delta\Psi_{\rm(j,k)}\) values to the error on \(\Psi\)SSM\({}_{\rm P}\) (\(\mu_{\rm(l)}\)). It is important to note that the error on \(\Psi\)SSM\({}_{\rm P}\) between t\({}_{\rm(j)}\) and t\({}_{\rm(l)}\) was \(\mu_{\rm(l)}\) = 0.068. Concerning the plot P1 (Figure 6a,b), on the date j = 31 July 2017, the observed values of \(\Delta\Psi_{\rm(j,k)}\) ranged from \(-\)0.71 for k = 20 mm to \(-\)0.76 for k = 30 and 40 mm, all of which were lower than \(-\mu_{\rm(l)}\). This discrepancy could be attributed to significantly lower \(\Psi\)SSM\({}_{\rm R_{\rm(j,k)}}\) values compared with \(\Psi\)SSM\({}_{\rm P}\) (\(\Delta\Psi_{\rm(j,k)}<-\mu_{\rm(l)}\)) observed at j in Figure 6a. Furthermore, on j + 1 = 1 August 2017, the \(\Delta\Psi_{\rm(j+1,k)}\) values exhibited positive values greater than \(-\mu_{\rm(l)}\) (\(\Delta\Psi_{\rm(j+1,k)}=0.22>-\mu_{\rm(l)}\)) for k = 20, 30, and 40 mm. This satisfied one of the criteria for identifying the most likely irrigation event (I\({}_{\rm(j,k)}\)), as stated in Equation (12) (\(\Delta\Psi_{\rm(j,k)}<-\mu_{\rm(l)}\) and \(\Delta\Psi_{\rm(j+1,k)}\geq-\mu_{\rm(l)}\)). Therefore, it can be concluded that an irrigation event occurred between j = 31 July 2018 and j + 1 = 1 August 2017. To determine the most probable irrigation date (I\({}_{\rm(j,k)_{opt}}\)) the irrigation combination (I\({}_{\rm(j,k)}\)) leading to the smallest absolute value between \(\Delta\Psi_{\rm(j,k)}\) and \(\Delta\Psi_{\rm(j+1,k)}\) was identified, as described in Equation (14) (I\({}_{\rm(j,k)_{opt}}=\min(\left|\Delta\Psi_{\rm(j,k)}\right|,\left|\Delta\Psi _{\rm(j+1,k)}\right|)\)). Accordingly, since \(\left|\,\Delta\Psi_{\rm(j+1,k)}\,\right|\) = 0.22 (regardless of the amount k) is smaller than \(\left|\,\Delta\Psi_{\rm(j,k)}\,\right|\) = 0.71 for k = 20 mm and \(\left|\,\Delta\Psi_{\rm(j,k)}\,\right|\) = 0.76 for k = 30 and 40 mm, the irrigation date j + 1 = 1 August 2017 was identified as the most probable date of I\({}_{\rm(j,k)_{opt}}\). However, the analysis of \(\Delta\Psi_{\rm(j+1,k)}\) did not reveal any discernible variations across different irrigation amounts k = 20, 30, and 40 mm, making it impossible to identify irrigation amounts beyond k = 20 mm. In Figure 6b, the actual irrigation date, marked in blue, was 3 August 2017. Therefore, the detected irrigation date (1 August 2017) had a time difference of 2 days compared with the actual date. Based on the reasoning above, for the remaining three cases over P2 (Figure 6c,d), P3 (Figure 6e,f), and P4 (Figure 6g,h), the actual irrigation dates were successfully detected with a time lapse of 1 day from the actual irrigation dates:  
* For P2: The detected irrigation date was 26 June 2018 and the actual date was 27 June 2018;
* For P3: The detected irrigation date was 25 July 2019 and the actual date was 24 July 2019;
* For P4: The detected irrigation date was 8 July 2020 and the actual date was 9 July 2020. Furthermore, it is noteworthy that the detected irrigation amounts for the P2, P3, and P4 plots were consistently k = 20 mm. This demonstrates that the analysis did not identify any discernible differences in the \(\Delta\Psi_{\rm(j,k)}\) values when testing irrigations with different amounts (k = 20, 30, and 40 mm) at the detected irrigation dates.