## 4 Results
### Spring Recharge Area Identification by Using Isotope-Driven Model
This condition implies that the recharge area, relative to each spring discharge value and to each isotopic value obtained from the groundwater sample, is characterized by a fast decrease of areas with elevations lower than the average elevation of the recharge area detected by isotopic concentration. We have used intermediate discharge values, Q1 (08 November 2018) and Q2 (18 January 2019), in order to define also the model parameters relative to areas between the maximum and minimum recharge areas (Figure 5). The shape of Weibull coefficients \(\lambda\) and k, found for the four conditions mentioned before (Qmax, Q1, Q2 and Qmin) and reported in Table 3, show a good correlation with elevations and discharge values (Figure 10). With reference to the latter, Weibull coefficients show a good non-linear correlation. This condition permits us to use the two diagrams in Figure 10 as a tool to obtain the correlation between the discharge and average elevation by using the aforesaid coefficients as driver values and vice-versa. This correlation shows a good robustness that is reflected in the acceptable estimation of the discharge and of the recharge area average elevation, for all other values of the data-set (Figure 5). In this way, the use of both plots (Figure 10) allows us to obtain also the two shape coefficients used in Equation (5) to estimate the topographic area involved in the recharge area that supplies the spring discharge. Figure 9: Example of the Weibull-reached area distribution on topographic area distribution for maximum and minimum discharges and the other two intermediate values. Through knowledge of the recharge areas depending on elevation, it is possible to perform a map of the simulated distribution of these areas. Referring to the model setup, the map (Figure 11) can take into consideration the two different model conditions, one at a time. In particular, (i) different rainfall infiltration rate uniformly distributed with elevation; (ii) local storm events that involve limited zones within the elevations range (elevations classes). In this way, concerning the spring discharge minimum value, Figure 11 shows the map of the simulated distribution of the percentage of topographic elevation classes, in terms of area, involved in recharge, or, with a different interpretation, the map of the simulated distribution of the percentage of infiltration with respect to the maximum. Both the interpretations obtained coming from the application of the same Weibull cumulative distribution obtained by Equation (5). This map shows that for the spring discharge minimum value a large part of the area at the base of mountain aquifer (red area) is weakly or not engaged in recharging. By contrast, the area belonging to the mountain plateau upper 1100 m (green area) is totally engaged in the aquifer recharge. The simulated distribution of recharge areas, defined by the model, can be also related to the variability of rainfall intensity that involves the basin, too. Specifically, this variability takes into consideration the natural trend of the rainfall intensity increasing with elevation. In particular, for the rainfall-altitude relationships, we use the exponential relationships to obtain much more plausible values in the case of mountain basins [29]. This last relationship was used with the aim to quantify the infiltration volumes related to the maximum and minimum spring discharge (Figure 12). In particular, the rainfall-altitude relationships associated to maximum and minimum discharge were respectively assumed to refer to the average of the four wet months (October-January) and the four dry months (June-September), computed considering the historical data-set (40 years: 1959-1999). The thermo-pluviometric stations taken into consideration are Gaeta (45 m a.s.l. ), Itri (165 m a.s.l. ), Esperia (302 m a.s.l.) and SS Cosma e Damiano (206 m a.s.l.) (see you Figure 2). The regression laws relating to the aforementioned wet and dry period in the time frame of the case study (April 2018-June 2019) were recomputed.