## 2 Materials and Methods
### Characterization of Extreme Deep Convection Events  
Prior to the actual analysis of the PF dataset, a selection has been made by applying a \"noise\" filter and a snow cover filter for the 4 years considered within the domain of interest. Only the PFs with latitude between 30\({}^{\circ}\) N and 48\({}^{\circ}\) N, and longitude between 6\({}^{\circ}\) W and 36\({}^{\circ}\) E (latitude/longitude vectors in the PF dataset represent the geographical center of each PF), have been selected. Although the Black Sea and a small part of the Atlantic Ocean (north of Spain and west of France) are included in the selected area, hereafter it will be referred to as Mediterranean area. The \"noise\" filter is applied to eliminate from the database spurious data often attributed to radar \"mirror\" echoes at very high altitude that may occur in presence of very intense convective systems [7]. Following [PERSON] and [PERSON] [26], PFs having the MAXHT20 higher than 17 km, but minimum 89 GHz PCT warmer than 220 K, and less than 4 contiguous radar pixels (<80 km\({}^{2}\)) in size, are discarded from the analysis, since such radar reflectivity altitude is unrealistic. The second filter is applied to account for snow-covered surface contamination of the scattering signal due to the cloud. Snow-covered areas appear as BT depression areas in the lower frequency channels (e.g., the 37 GHz used in the analysis), and the use of the PCT is not very effective in removing this spurious effect. The scattering of snow at the ground, during the winter or in mountainous regions, may be erroneously interpreted as deep convection. Therefore, in order to exclude PFs likely associated with snow cover from the analysis, the monthly snow depth data obtained by the latest reanalysis produced by ECMWF ([[https://cds.climate.copernicus.eu/cdsapp#](https://cds.climate.copernicus.eu/cdsapp#)]([https://cds.climate.copernicus.eu/cdsapp#](https://cds.climate.copernicus.eu/cdsapp#))!/dataset/ reanalysis-era5-single-levels-monthly-means, accessed on 26 April 2021) has been considered. In particular, the ERA-5 monthly snow depth (expressed in meters of liquid water equivalent), available on a 0.25\({}^{\circ}\)\(\times\) 0.25\({}^{\circ}\) regular grid, is associated with each PF using a nearest-neighbor approximation. In order to identify the extreme events among all the selected elements, PFs are categorized from the top 10%, 1%, 0.1%, and 0.01% extreme (in terms of intensity) values of each parameters selected to study convection [16; 19; 26]. Then, the percentiles of the distribution of each parameter, corresponding to the cited percentages, have been calculated. In particular, for the parameters in which frequency in the distribution decreases as their value decreases (e.g., BTs or PCTs), these percentages correspond to the 0.01 th, 0.1 th, 1 th and 10 th percentiles of each distribution, respectively. For the parameters in which frequency decreases as their value increases (e.g., maximum radar echo top, or volumetric precipitation), these percentages correspond to the 99.99 th, 99.9 th, 99 th, and 90 th percentiles of each distribution, respectively. In this way, the most \"extreme\" values for a given parameter are those belonging to the top 0.01% (so, below the 0.01 th percentile for GMI-based parameters, and above the 99.99 th percentile for DPR-based parameters), with less and less extreme values being included as the top percentage increases. The distribution of each parameter reflects that of the PFs. Therefore, the number of PFs and the parameter values corresponding to each interval obtained by the top percentage categorization are determined, creating \"intensity classes\" for each parameter. These intensity classes depends on the considered variable, since each of them describes different convection-related properties.