## 2 Materials
### Optirrig Model Description  
Optirrig is a process-based crop growth and irrigation model that comprises an inner layer responsible for hydrological and agronomical calculations, and an outer layer dedicated to generating, analyzing, and optimizing irrigation scenarios. It is derived from the Pilote model [64; 65] developed by the same research team, with some adaptations regarding process modeling (and a thorough rewriting in the R language in a completely explicit, agile, and modular form that fits the modern international standards). Optirrig simulates crop growth and yield based on water and nutrient availability and consumption, emphasizing the identification of effective irrigation and fertilization practices and decision rules, particularly for cereals and horticulture [57; 84]. The model version used in this study focuses on field crops and is represented in Figure 2. It assumed no nitrogen stress due to sufficient fertilization on the study sites, and hence, the nitrogen cycle component was excluded from the model. Within its inner hydro-agronomic module, Optirrig conducts a daily water balance, considering factors such as infiltration, drainage, evaporation, transpiration, and changes in soil water reserves (R1, R2, R3) in different reservoirs (surface, root zone, and deep). Optirrig relies on four daily climatic inputs: precipitation (P), mean air temperature (T), potential evapotranspiration (ETo), and total global solar radiation (R\({}_{g}\)). Additionally, irrigation (I) can be incorporated as a management parameter. Crop growth is tracked through the leaf area index (LAI), which is determined by the thermal time (TT) and potential water stress (S\({}_{w}\)), as shown in Figure 2. The crop coefficient (Kc) and partition coefficient (Cp) are derived from the LAI, as they control the allocation of ETo into transpiration demand (Tp\({}_{0}\)) and evaporation demand (Es\({}_{0}\)). The actual transpiration (Tp) and evaporation (Es) are then calculated based on available water reserves, which are continually updated. Biomass accumulation (TDM) is determined by radiation interception and can also be influenced by water stresses. Finally, crop yield (\(\Upsilon\)) is evaluated using the harvest index (HI), which may deviate from its potential value if significant water stresses are observed throughout the cropping season. For more specific information, Table 3 provides additional details, particularly regarding the soil and crop parameters involved. \begin{table}
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**Region** & **Year** & **Plot** & **Number of Irrigations** & **Average Amount Per Irrigation** & **Sowing Date** & **Period of Irrigation** & **Harvest Date** & **Irrigation Method** \\ \hline Montpellier & 2017 & P1 & 10 & 30 mm & 15 April & 2 June-26 September & 25 September & Sprinkler \\ Tarbes & 2018 & P2 & 4 & 40 mm & 20 April & 5 August & 6 October & Sprinkler \\ Tarbes & 2019 & P3 & 4 & 40 mm & 01 May & 1 July-29 July & 1 October & Sprinkler \\ Tarbes & 2020 & P4 & 3 & 40 mm & 08 May & 9 July-6 August & 30 September & Sprinkler \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the irrigation and crop management practices for the experimental maize plots. The outer layer of Optirrig allows for the generation and analysis of multiple irrigation scenarios to enhance water efficiency and optimize crop yield. The model's structure offers the capability to incorporate independent information, treated as observations, pertaining to certain state variables. This feature can be utilized for various purposes, including classical tasks like model fitting or exploratory analyses. As stated above, Optirrig performs a daily water balance that encompasses infiltration and drainage, evaporation and transpiration, and changes in soil water reserves (R\({}_{1}\), R\({}_{2}\), R\({}_{3}\)) in the surface, root-zone, and deep reservoirs, respectively.