## 3 Methods
### Workflow for the Detection of Irrigation Events  
Figure 5 provides a comprehensive flowchart of the proposed irrigation detection methodology, highlighting its key steps and processes. The approach aimed at retrieving the irrigation dates and amounts between a SAR acquisition date (\(t_{(i)}\)) and the previous date (\(t_{(i)}\)). It is good to mention that SSM values derived from S1 SAR images acquired in the ascending and descending modes were used independently. For example, over the P1 plot and at \(t_{(i)}\) (SAR acquisition date) the detection approach was applied between the SSM derived from ascending SAR time series (MA1) at \(t_{(i)}\) and the date of the previous image \(t_{(i)}\) from the same time series, at a fixed interval of 6 days. For each plot and between \(t_{(i)}\) and \(t_{(i)}\), ten primary indicators could be derived for the plot, as presented in Table 4.  
The chain started with phase 1, which consisted of distinguishing the occurrence or not of an irrigation event. It was crucial to differentiate between rainfall and irrigation events, as both can cause changes in SSM values. To eliminate the ambiguity between irrigation and rainfall events, two filters were implemented. The first filter compared the change rate of SSM at the plot scale \(\forall\mathrm{SSM}_{P_{(i)}}\) between \(t_{(i)}\) and \(t_{(i)}\) with the rate of change of SSM at the grid scale (\(\forall\mathrm{SSM}_{G_{(i)}}\)). A higher value of \(\forall\mathrm{SSM}_{P_{(i)}}\) indicated the presence of water application apart from rainfall.  
The second filter involved a comparison between \(\forall\mathrm{SSM}_{P_{(i)}}\) and the simulated change rate of SSM without irrigation (\(\forall\mathrm{SSM}_{P_{(i)}}\)). Similarly, if \(\forall\mathrm{SSM}_{P_{(i)}}\) was higher than \(\forall\mathrm{SSM}_{P_{(i)}}\), it indicated the presence of a water application other than rainfall. Once it was confirmed that an irrigation event occurred, the approach moved to phase 2, which aimed to determine the irrigation date and amount between two SAR acquisitions.  
In phase 2, irrigation combinations of date j (ranging between \(t_{(i)}-3\) and \(t_{(i)}\)) and dose k (\(I_{(j,k)}\)) were entered into Optirrig between \(t_{(i)}\) and \(t_{(i)}\). This resulted in simulated SSMvalues generated by Optirrig, and corresponding change rates \(\mathbf{Y}\mathrm{SSM}_{\mathrm{R}_{\left(\left|\mathbf{k}\right\rangle_{\left( \right)}\right|_{\left(\right)}}}\) were computed. The objective was to find the most probable irrigation combination (\(\mathrm{I}_{\left(\left|\mathbf{j}\right\rangle_{\mathrm{opt}}\right)}\) between \(\mathrm{t}_{\left(\mathrm{i}\right)}\) and \(\mathrm{t}_{\left(\mathrm{i}\right)}\) by minimizing the absolute discrepancy between \(\mathbf{Y}\mathrm{SSM}_{\mathrm{R}_{\left(\left|\mathbf{k}\right\rangle_{ \left(\right)}\right|_{\left(\right)}}}\) and \(\mathbf{Y}\mathrm{SSM}_{\mathrm{P}_{\left(\right)}}\) (\(\left|\Delta\mathbf{Y}_{\left(\left|\mathbf{k}\right\rangle_{\left( \right)}\right|_{\left(\right)}}\right|\)). This process is elaborated in Section 3.2.2 and illustrated in Figure 5.  
Figure 5: Workflow of the methodology for the irrigation dates and amounts detection between two subsequent SAR acquisition dates (\(\mathrm{t}_{\left(\mathrm{i}\right)}=\mathrm{t}_{\left(\mathrm{i}\right)}+6\) days). The two dashed colored lines refer to the two phases of the irrigation retrieval approach described in Section 3.2: phase 1 in black and phase 2 in orange.