## 4 Results
### Spring Recharge Area Identification by Using Isotope-Driven Model
This area is computed in term of weighted average, on the elevation classes (n. 15 in this case), as follows:  
\[\rm{A_{eq}}=\frac{\sum_{1}^{n}(A_{i}\cdot h_{i})}{\sum_{1}^{n}(h_{i})} \tag{3}\]  
Figure 7: (**a**) Elevations classes distribution of Capodacqua di Spigno Spring basin; (**b**) number of finite squared elements (FSE) of \(100\times 100\) m for each elevation class. where \(A_{i}\) is the area related to ith-elevation hi. In this model, we consider the maximum discharge of the time series, associated to the complete topographic elevation distribution of the area, as wholly involved in the recharge area. Therefore, considering Equation (3), it is possible to associate for this condition an equivalent average area which is distributed on all the altitudes, \(A_{eq\_max}=372.83\) n. FSE having the same elevation, detected by isotopic rate referred to the maximum discharge. Considering, in addition, the linear relation showed in Figure 5, it is possible to assume that minimal spring discharge is associated with a smaller recharge equivalent area (\(A_{eq\_min}\)), having a higher elevation detected by isotopic ratio values, by using the general following equation:  
\[A_{eq\_j}=\frac{A_{eq\_max}\cdot h_{q\_max}}{h_{q\_j}} \tag{4}\]  
where \(A_{eq\_j}\) is the generic recharge area associated to the generic jth-discharge, which corresponds to an average elevation \(h_{q\_j}\), obtained by isotopic concentration values. In this way, it has been obtained that the equivalent minimal recharge area, \(A_{eq\_min}\), is of 268.79 n. FSE associated with the maximum average recharge area elevation of \(h_{q\_min}\) of 878 m (Figure 8). Having obtained the generic couple values of \(A_{eq\_j}\) and \(h_{q\_j}\), for this average equivalent aquifer condition, the model provides a way to perform a simulated distribution of the recharge area on the geometrical elevations classes by considering conservative recharge areas at higher elevations. This condition matches the natural rainfall trend, which is typical of mountain aquifers. The simulated distribution, which is shown in Figure 8 for minimal spring discharge, is demanded by a Weibull cumulative probability function; this function is widely used to explain the distribution of natural events [26; 27; 28] because it can be associated with cumulative normal or non-normal distribution of many natural parameters. Specifically, we use the two-parameters Weibull function as follows:  
\[F(h_{i},\ A_{i},k,\lambda)_{i}=A_{i}\left(1-e^{-\left(\frac{h_{i}}{\lambda} \right)^{k}}\right) \tag{5}\]  
where \(h_{i}\) is the central value of the class elevation associated to the topographic ith-class area \(A_{i}\), while \(k\) and \(\lambda\) are the shape modelling parameters of the function. Figure 8: Maximum and minimum recharge area distribution defined by model. These shape parameters are found by iterative trial; specifically, the values change, in a limited range, up to the convergence of the following condition:  
\[\text{Aeq\_j}=\text{ Aeq\_weibull}\rightarrow\frac{\text{Aeq\_max}\cdot\text{h}_{ \text{q\_max}}}{\text{h}_{\text{q\_j}}}\;=\;\frac{\sum_{1}^{n}\text{A}_{\text{ j}}\bigg{(}1-\text{e}^{-(\frac{\text{h}_{\text{i}}}{\lambda})\text{ k}}\bigg{)}\text{h}_{\text{i}}}{\sum_{1}^{n}(\text{h}_{\text{i}})} \tag{6}\]  
where the \(A_{eq\_weibull}\) is the weighted average area computed using the Weibull function. By resolving Equation (5) for other intermediate values of discharge and their isotopic concentration, Figure 9 shows that the model progressively involves lower elevations areas, at the same time as discharges increases. In particular, the recharge areas having elevations over 1155 m are always quasi-totally involved in any spring discharge values. In addition, out of the maximum discharge Qmax, it is possible to notice that the average elevation of the recharge area, detected by using isotopic concentrations, is in accordance with the upper \"rollover\" of the Weibull curves.