## 4 Results  
### Detection of Irrigation Dates and Amounts  
The proposed methodology for detecting irrigation involved two main steps. First, the SSM values derived from Sentinel-1 were analyzed at both the plot and grid scales using specific conditions, as described in Section 3.2.2, to determine whether irrigation took place between two subsequent SAR images (phase 1). If irrigation was detected, the second step involved simulating the SSM values with Optirrig, considering various combinations of irrigation dates and amounts. The goal was to find the combination that best matched the observed and simulated SSM value changes (phase 2). Phase 1 consisted of applying two filters to distinguish between irrigation and rainfall events. The first filter compared the plot-scale SSM change rate (\(\text{\text{\text{\text{\text{\text{\  
precipitation data (shown as blue bars) and evapotranspiration (ETo, depicted as yellow bars) obtained from nearby weather stations are included in the figures. These additional data help to understand the relationship between the \(\Psi\)SSM\({}_{\rm G}\) and \(\Psi\)SSM\({}_{\rm r}\) values and rainfall events. Figure 6a,c,e,g illustrate that when there was low or no precipitation, both \(\Psi\)SSM\({}_{\rm G}\) and \(\Psi\)SSM\({}_{\rm r}\) exhibited a slight decrease or remained stable. Concurrently, in all four cases documented, \(\Psi\)SSM\({}_{\rm P}\) demonstrated higher values than both \(\Psi\)SSM\({}_{\rm G}\) and \(\Psi\)SSM\({}_{\rm r}\). This observed difference, which exceeded the error margin of \(\Psi\)SSM\({}_{\rm P}\) (\(\mu_{(\rm I)}\)), confirming the conditions in Equations (6) and (7) were met, which served as evidence of irrigation occurrence in the four examples. Figure 6: _Cont._  
In phase 2, when an irrigation occurrence was identified between t\({}_{(i)}\) and t\({}_{(j)}\), further investigations were conducted to determine the probable date and amount of irrigation. This involved simulating the \(\mathbb{VSSM}_{R_{(j,k)}}\) (black lines) for each irrigation combination (I\({}_{(j,k)}\)), where the date j ranged from t\({}_{(i)}-3\) to t\({}_{(j)}\), and the tested amount k was set to 20, 30, and 40 mm. The purpose of these simulations was to compare the \(\mathbb{VSSM}_{R_{(j,k)}}\) values with the \(\mathbb{VSSM}_{\mathrm{P}}\) and minimize the difference between them (\(\left|\Delta\Psi_{(jk)}\right|=\left|\mathbb{VSSM}_{R_{(j,k)}}-\mathbb{VSSM}_{ \mathrm{P}}\right|\), Equation (9)) in order to derive insights about the most likely I\({}_{(j,k)}\). The evolution of \(\Delta\Psi_{(jk)}\)  
Figure 6: Irrigation detection using the evening SAR acquisition over P1 for t\({}_{(j)}\) = 6 August 2017 **(a,b)**, P2 for t\({}_{(j)}\) = 30 August 2018 (**c,d**), P3 for t\({}_{(j)}\) = 30 July 2019 **(e,f)** and the morning SAR acquisition over P4 for t\({}_{(j)}\) = 13 July 2020 **(g,h)**. In (**a,c,e,g**), the calculated \(\mathbb{VSSM}_{R_{(j,k)}}\) (rate of change of \(\mathrm{SSM}_{R_{(j,k)}}\) between t\({}_{(j)}\) and t\({}_{(j)}\)) as a function of the date j (between t\({}_{(j)}\) and t\({}_{(j)}-3\)) of the injected irrigation I\({}_{(j,k)}\) is represented by the black lines for k = 20, 30, and 40 mm. The vertical dash lines in dark blue represent the real irrigation dates (d\({}_{\mathrm{Re}}\)). The light blue, green, and red horizontal dash lines represent the \(\mathbb{VSSM}_{\mathrm{P}}\), \(\mathbb{VSSM}_{\mathrm{G}}\), and \(\mathbb{VSSM}_{\mathrm{G}}\), respectively. In (**b,d,f,h**), the difference \(\Delta\Psi_{(j,k)}=\mathbb{VSSM}_{R_{(j,k)}}-\mathbb{VSSM}_{\mathrm{P}}\) is represented by the white values in the matrix for each t\({}_{(j,k)}\). The date j in blue indicates the d\({}_{\mathrm{Re}}\) and \(\mu_{(j)}\) indicates the value of the error on \(\mathbb{VSSM}_{\mathrm{P}}\). for different I\({}_{\rm(j,k)}\) values of date j and dose k is presented in Figure 6b,d,f,h for each of the examples where irrigation occurrence was suspected in phase 1 (Figure 6a,c,e,g).