## 2 Materials and Methods
### Characterization of Extreme Deep Convection Events
So, it is not obvious for the same PFs to belong to the same intensity class for different parameters: an intense deep convective storm, that has the MAXHT40 belonging to the top 0.01% extreme values, might have the MIN37 PCT between the top 0.1% and the top 1% extreme values. It is worth pointing out this aspect because such an approach allows analyzation of the deep convection intensity (and the associated phenomena) using several perspectives and exploiting all the available proxies related to this type of phenomenology. The distributions of the PFs obtained enable to focus, in particular, on the geographical distribution of extreme classes. Once PFs belonging to a certain top percentage of each proxy are selected (intense PFs), the seasonal distribution within the domain, and the diurnal and seasonal cycle, are mapped and analyzed. Since different PF subsets are selected with the use of different proxies, it is fundamental to study the dispersion of the intensity classes based on one parameter as a function of the others. This is an aspect which will be clarified in the next two sections. For the second part of the work, focused on the detection of PFs associated with overshooting tops (OPFs) and potential hailstorms, criteria similar to those applied in previous studies are followed [27; 28]. In particular, the OPFs are identified as those PFs with the cloud top height (defined by maximum radar echo top heights found for different reflectivity thresholds) higher than the tropopause level (see Section 3.3 for details). For this latter quantity, two definitions have been adopted:  
* the lapse-rate tropopause, based on the WMO definition as the lowest height at which the lapse-rate decreases to 2 \({}^{\circ}\)C/km or less, with the average lapse-rate between this level and all higher levels within 2 km does not exceed 2 \({}^{\circ}\)C/km (\(Z_{LRT}\));
* the coldest point tropopause, obtained by the height of the coldest point of the temperature profile (\(Z_{CP}\)). The temperature profile is provided for each PF in the dataset. It is obtained from spatial and temporal interpolation of the ERA-Interim reanalysis available every 6 h at 0.75\({}^{\circ}\) horizontal resolution, at 37 pressure levels between 1000 hPa and 1 hPa [41]. In the PF dataset, the temperature profile is reported at 27 levels, between 1000 hPa (0.10 km) and 100 hPa (16.50 km). In accordance with Reference [27], besides the snow-cover filter and the mirror echo filter already described, for OPFs identification only PFs with at least 4 radar contiguous pixels with precipitation rate exceeding 0.1 mm/h at the surface are considered in order to have more significant precipitating events. Then, all those PFs with a MAXHT20 above 17 km and a MIN85 PCT above 200 K and with the difference between maximum 15 dBZ and 20 dBZ radar echo top heights greater than 5 km, are excluded (following the idea of Reference [27]). This ensures that the OPFs selection is carried out only for radar reflectivity vertical profile free of spurious effects that could increase false OPFs identification. On the other hand, PFs associated with potential hailstorms are detected by applying two different criteria separately, one radar-based and one radiometer-based, following the analysis carried out by [PERSON] et al. [28]. These criteria are associated with:  
* air temperature at the 44 dBZ radar echo top height (indicated as T_MAXHT44) less than or equal to \(-\)22 \({}^{\circ}\)C (DPR-based criterion);
* a minimum PCT at 37 GHz (MIN37 PCT) less than 230 K (GMI-based criterion). In order to apply the DPR-based criterion, the ERA-Interim temperature vertical profiles are linearly interpolated at 0.5 km intervals, between 0.5 km and 16.50 km. The 44 dBZ radar echo maximum heights in the Ku-band radar reflectivity profile are searched within each PF profile and associated with the corresponding temperature value.