## 2 HYDE Algorithm Description  
The positions of small and large earthquake hypocenters, in areas where sequences of earthquakes occur, are used in a novel algorithmic method to clusterize earthquakes and accurately detect the seismogenic faults, identifying the strike and the dip parameters. As already mentioned, the fault plane can be simplified as an ideal flat surface (\(\pi\)). However, it can be considered to be a volume V characterized by two main dimensions, plus a third smaller dimension of the same order of magnitude of the localization mean error (\(\pm\sigma\)). The algorithm, named HYDE (HYocenter DEnsity), a flowchart for which is reported in Appendix A, works as a fast and efficient explorer in a five-dimensional space (x, y, z, \(\tau\), \(\phi\)), thanks to a series of roto-translation operations. Let N be the number of all available earthquake hypocenters contained in the investigated space, whose coordinates are expressed in the UTM system; we randomly select a finite number (k << N) of hypocenters, hereafter pivot points. The pivot points are distributed in 3D, with a probability linearly dependent on the hypocenter's density. These pivots' volumetric distribution guarantees that they are mainly located where the largest number of earthquake hypocenters occur. Each exploration is carried out by translating the reference system so that the origin coincides with a pivot point position. At this stage, the algorithm explores a set of volumes of rotations around the new origin and saves, for each V (nearly bidimensional and assumed to be a possible fault plane), its geometric characteristics (dimensions, rotation angles) and number of contained earthquake hypocenters. The operations just described are repeated for each pivot point. From the comparison between all of the investigated Vs, the best candidates emerge to be considered as fault planes based on the number of earthquakes. In the following, we describe the input, the fundamental steps and the outputs produced by the proposed method. The algorithm requires an input data matrix X made of N rows (seismic events) and M columns (latitude, longitude and depth). The data input to the algorithm are related to seismic events from an earthquake catalog. The input data format requires the latitude and longitude in UTM or GAUSS coordinates and the depth in meters to guarantee the orthonormality of the reference system. Moreover, HYDE requires an initial minimum thickness and fault extension values  
Our method consists of four different steps:  
Step 1. The algorithm randomly extracts K (K \(\ll\) N) pivot hypocenters (P\({}_{i}\), i = 1, 2,..., K) from the N earthquake hypocenters (see Figure 1). It then translates the coordinate system so that the origin coincides with P\({}_{i}\), which is assumed to be the rotation center for a hypothetical fault plane defined as a volume (V) (see Figure 2). Pivot points are selected using the Monte Carlo method [12; 13] from the set of hypocenters, based on the knowledge that the fault plane must pass through them. Moreover, the K pivots follow the density distribution of the N hypocenters and are primarily located in regions with a higher probability of constituting a fault plane. This approach accelerates the search process and ensures that many pivots lie on the fault plane. Step 2. We define a parallelepiped, the barycenter of which is a pivot hypocenter (cartesian coordinate 0,0,0), with thickness \(t\) (Figure 1) and volume \(V_{j}\) (Figure 2), such that the two dimensions, orthogonal to \(t\), are much greater than \(t\). The \(t\) value is assumed to be the mean error of the hypocenter localization (\(\sigma\)), but it can be increased to take into account slight deviations from the perfect planarity. Then, the algorithm counts the number of hypocenters which fall into the defined volume \(V_{j}\). Step 3.