## 3 Methods
### Metrics Associated with Detection Issues
\begin{table}
\begin{tabular}{c c} \hline
**Indicator** & **Description** \\ \hline SSM\({}_{\{\mathrm{P}_{(\mathrm{i})}\}}\) & SSM value at the plot scale \\ SSM\({}_{\{\mathrm{G}_{(\mathrm{i})}\}}\) & SSM value at grid scale over plot’s area (10 km) \\ SSM\({}_{\{\mathrm{G}_{(\mathrm{i})}\}}\) & SSM simulated by Optiring with the absence of irrigation \\ SSM\({}_{\{\mathrm{R}_{(\mathrm{j,k})\}}_{(\mathrm{i})\}}\) & SSM simulated by Optiring through the injection of I\({}_{\{\mathrm{j,k}\}}\) \\ SSM\({}_{\{\mathrm{R}_{(\mathrm{j,k})\}}_{(\mathrm{i})\}}\) & (irrigation of date j between t\({}_{\{\mathrm{i}\}}\) and t\({}_{\{\mathrm{i}\}}\)) and dose k) \\ \(\Psi\)SSM\({}_{\{\mathrm{P}_{(\mathrm{i})}\}}\) & Rate of change between SSM\({}_{\{\mathrm{P}_{(\mathrm{i})}\}}\) and SSM\({}_{\{\mathrm{P}_{(\mathrm{i})}\}}\) \\ \(\Psi\)SSM\({}_{\{\mathrm{G}_{(\mathrm{i})}\}}\) & Rate of change between SSM\({}_{\{\mathrm{G}_{(\mathrm{i})}\}}\) and SSM\({}_{\{\mathrm{G}_{(\mathrm{i})}\}}\) \\ \(\Psi\)SSM\({}_{\{\mathrm{F}_{(\mathrm{i})}\}}\) & Rate of change between SSM\({}_{\{\mathrm{F}_{(\mathrm{i})}\}}\) and SSM\({}_{\{\mathrm{F}_{(\mathrm{i})}\}}\) \\ \(\Psi\)SSM\({}_{\{\mathrm{R}_{(\mathrm{j,k})\}}_{(\mathrm{i})\}}\) & Rate of change between SSM\({}_{\{\mathrm{R}_{(\mathrm{j,k})\}}_{(\mathrm{i})\}}\) and SSM\({}_{\{\mathrm{R}_{(\mathrm{j,k})\}}_{(\mathrm{i})\}}\) \\ \(\Delta\Psi_{(\mathrm{j,k})_{(\mathrm{i})}}\) & Difference between \(\Psi\)SSM\({}_{\{\mathrm{R}_{(\mathrm{j,k})\}}_{(\mathrm{i})\}}\) and \(\Psi\)SSM\({}_{\{\mathrm{F}_{(\mathrm{i})}\}}\) \\ \(\mu_{(\mathrm{i})}\) & Uncertainty in the \(\Delta\Psi_{(\mathrm{j,k})_{(\mathrm{i})}}\) values \\ \hline \end{tabular}
\end{table} 
Table 4: Summary of the primary indicators estimated and calculated for the irrigation detection approach between a SAR acquisition date (t\({}_{\{\mathrm{i}\}}\)) and the previous date (t\({}_{\{\mathrm{i}\}}\)). date detection accuracy. The \"Precision\" (Pn) was the second metric, defined as the ratio of TP to TP + FP (all detected events) (Equation (16)). False positive detections decrease Pn, affecting the precision. The third metric calculated the harmonic mean of recall and precision, resulting in the F-score (Equation (17)). The F-score enables comparison of the harmonic mean of Re and Pn in detecting irrigation dates across multiple cases, providing a balanced evaluation of the performance:  
\[\text{Recall}=\frac{\text{True Positive}}{\text{True Positive}+\text{ False negative}} \tag{15}\]  
\[\text{Precision}=\frac{\text{True Positive}}{\text{True Positive}+\text{ False positive}} \tag{16}\]  
\[\text{F-score}=\frac{2\times\text{Recall}\times\text{Precision}}{\text{ Recall}+\text{Precision}} \tag{17}\]  
Once the probable dates of irrigation were retrieved, additional investigations were conducted to evaluate the detection of irrigation amounts applied at the detected dates of irrigation (t(m)). To assess the accuracy of the estimation, the mean absolute error percentage (MAE%) was calculated by comparing the detected irrigation amounts (A\({}_{D}\)) with the corresponding actual (real) irrigation amounts (A\({}_{R}\)) over the t(m) for the four reference plots. The MAE% quantifies the average deviation between the estimated and actual irrigation amounts, allowing for a seasonal assessment of the performance of the estimation method:  
\[\text{MAE}\%=\frac{1}{\text{n}}\times\frac{\sum_{t(m)}^{\text{t}}\lvert A_{ \text{D}_{m}}-A_{\text{R}_{m}}\rvert}{\overline{A_{R}}}\times 100 \tag{18}\]  
where t(m) is the detected irrigation date; n is the total number of detected irrigation dates (t(m)); \(\overline{A_{R}}\) is the average of the actual irrigation amounts; and \(A_{\text{D}_{m}}\) and \(A_{\text{R}_{m}}\) are the detected and actual irrigation amounts, respectively, at t(m). A lower MAE% indicates a higher level of accuracy in estimating the irrigation amount, as the probable amounts closely align with the actual amounts applied. Conversely, a higher MAE% suggests a larger discrepancy between the probable and actual amounts.