## 2 Methodologies
### Seasonal Empirical Relationships between Potential Ewopotranspiration and Soil Moisture Content  
The amount of potential evapotranspiration (ET\({}_{0}\)) depends on net radiation, air temperature, relative humidity (RH), and wind speed. For example, with the same wind speed and RH, the rate of ET\({}_{0}\) will increase as net radiation and air temperature increase. This in turn leads to a change in the amount of water (moisture) contained in the soil. Indeed, potential evapotranspiration ET\({}_{0}\) (mm/day) and SMC (dimensionless) depend on the same climate variables (net insolation, air temperature, wind speed, relative humidity). These physical considerations then lead to an attempt to find empirical relationships that link ET\({}_{0}\) to soil moisture content, for the two geographic case studies and time frame (daily). The approach used is that of linear regression and is conducted on a seasonal basis, for each of the selected years of historical data. The purpose is to see if the coefficients of the fitting curves have consistency of magnitude and sign between the two different geographical locations chosen, over the four seasons considered, so that through fitting curve expressions, ET\({}_{0}\) rates (mm/day) can be directly estimated from soil moisture data (known from [29]). A first-degree polynomial is used as fitting curve. ET\({}_{0}\) values are not available in [29] and therefore must be derived. The rate of evapotranspiration depends not only on atmospheric factors, but also on the type of vegetation covering the surface. Thus, several works in literature evaluate evapotranspiration depending on the type of crop considered; some papers evaluate the evapotranspiration in case of a reference crop, with the following characteristics [36]: a fixed surface resistance of 69 s/m, a fixed height of 0.12 m, and an albedo of 0.23. Furthermore, it is an extensive surface cover that completely shadows the ground [36]. In this paper, the reference crop approach is used since it makes the results found comparable with other studies using reference crop, and since ET\({}_{0}\) for a specific crop can be assessed through crop coefficients available in the literature. In the literature, among the models used to assess reference crop ET\({}_{0}\), there is one recognized by the Food and Agriculture Organization (FAO): the Penman-Monteith formula (FAO56) [37], that allows ET\({}_{0}\) to be evaluated on a daily basis. However, this formula requires as inputs environmental data that are not present in common in situ weather stations or satellite measurements, thus making it difficult to implement the FAO56 in several geographical regions, especially in developing countries. Another model allows daily ET\({}_{0}\) estimates to be obtained close to those of FAO56 is that of Hargreaves-Samani (HS) [38], as described in [39; 40; 41]. The equation for this model requires maximum and minimum air temperature as the only climate data. The HS equation is reported below [40]:  
\[\text{ET}_{0}=\text{ C}_{1}\cdot\text{ R}_{\text{ext}}\cdot(\text{T}+17.8)\cdot\Delta\text{T}^{\text{C}_{2}}\left(\frac{\text{mm}}{ \text{day}}\right) \tag{3}\]  
where ET\({}_{0}\) (mm/day) is the reference potential evapotranspiration, C\({}_{1}\) is a constant equal to 0.0023, C\({}_{2}\) is another constant equal to 0.5, and \(\Delta\)T (\({}^{\circ}\)C) is the difference between the monthly average daily maximum and minimum temperatures; the sum of these two temperature values, divided by two, yields T (\({}^{\circ}\)C), which is thus an average temperature; R\({}_{\text{ext}}\) (mm/day) is the extraterrestrial radiation expressed in equivalent water evaporation; finally, 17.8 is an empirical factor introduced to account for the temperature units used in the original formulations. In (3) in addition to explicitly appearing atmospheric quantities on which ET\({}_{0}\) depends, ET\({}_{0}\) binding to RH is also expressed, implicitly, since \(\Delta\)T depends linearly on relative humidity [42].