## 3 Methods  
Our methodology focused on retrieving irrigation dates and amounts at the plot level by integrating surface soil moisture (SSM) data obtained from Sentinel-1 (SSM\({}_{\text{S1}}\), described in Section 3.1) into Optirrig using an inversion technique. The approach relied on comparing the change in the SSM\({}_{\text{S1}}\) between two S1 SAR acquisition dates, namely, t(l) and the previous date t(l), to the change in the SSM simulated by Optirrig when forcing combinations of irrigation dates and amounts (l(j,k); j accounting for the dates and k for the amounts) in the input files of the model. In return, Optirrig provides SSM values in its first, shallow reservoir of 10 cm depth, which exceeded by a few centimeters the penetration depth of the radar signal in soils. Due to this difference, comparing the SSM\({}_{\text{S1}}\) and Optirrig SSM simulations (SSM\({}_{\text{M}}\)) was not a straightforward task. We, therefore, proposed to instead compare the rates of change (YSSM\({}_{\text{S1}}\), YSSM\({}_{\text{M}}\)) between successive observation dates, seeking the irrigation date and amount that reduced the difference between those two rates, defining the rate of change between any two t(i) and t(i+1) dates as  
\[\Psi\text{SSM}=\frac{\text{SSM}_{\text{(i+1)}}-\text{SSM}_{\text{(i)}}}{\text{ SSM}_{\text{(i)}}} \tag{1}\]  
However, the SSM estimated from SAR data might be influenced by various factors, especially rainfall and irrigation: both lead to an increase in the backscattering coefficient, which affects the SSM values. Hence, it was mandatory to first distinguish between rainfall and irrigation events, which is expected from a combined analysis of changes in SSM values at the plot scale (SSM\({}_{\text{P}}\)) and the grid scale (SSM\({}_{\text{G}}\)).  
The method for detecting irrigations was based on identifying changes in the plot-scale surface soil moisture (SSM\({}_{\text{P}}\)) values between the current SAR acquisition at time t(i) and the previous SAR acquisition at time t(i). In essence, the approach assumed that an increase in SSM\({}_{\text{P}}\) between two consecutive SAR dates (t(i) and t(i)), resulting in a positive rate of change (\(\Psi\text{SSM}_{\text{P}}\)), was attributable to either rainfall or irrigation (Section 3.1.1). Conversely, an increase in SSM\({}_{\text{G}}\) values at the grid scale (10 km \(\times\) 10 km) likely denoted a rainfall event (Section 3.1.2). Consequently, a \(\Psi\text{SSM}_{\text{P}}\) value higher than the \(\Psi\text{SSM}_{\text{G}}\) value between t(i) and t(i) signaled the effect of local water application (which a priori excluded rainfall) and was considered indirect evidence of irrigation.  
Further verifications were performed using Optirrig. The evolution of SSM throughout the maize growing cycle was simulated using Optirrig without the presence of irrigation in so-called \"rainfeed\" conditions (SSM\({}_{\text{r}}\)), thus taking into account the sole effect of rainfall on the soil water content (Section 3.2.1). The \(\Psi\text{SSM}_{\text{r}}\) rate of change between t(i) and t(i) was computed and compared with \(\Psi\text{SSM}_{\text{P}}\). A \(\Psi\text{SSM}_{\text{P}}\) value lower than or equal to the \(\Psi\text{SSM}_{\text{r}}\) value meant that there was no effect of water application other than rainfall on the plot-scale surface soil moisture. Conversely, if the \(\Psi\text{SSM}_{\text{P}}\) value was higher than the \(\Psi\text{SSM}_{\text{r}}\) value, it was interpreted as evidence of irrigation between t(i) and t(i).  
The proposed methodology for irrigation detection consisted of (i) applying the above conditions to the SSM values derived from Sentinel-1 at plot and grid scales to determine whether irrigation took place, and (ii) if so, simulating the SSM values with Optirrig for different combinations of irrigation dates and amounts, seeking the combination that produced the best match between the rates of change of the observed and simulated SSM values.