## 3 Application to a Simulated Fault  
To verify the reliability of our algorithm, we performed a blind test using two earthquake hypocenter distributions.  
The first one was characterized by a random normal density distribution of 5000 hypocenters in a volume (\(x\), \(y\), \(z\)) of 40 \(\times\) 40 \(\times\) 20 km, without any planar event distributions. In order to identify the possible presence of a fault in that volume, we produced and analyzed a density function diagram (Figure 3a) and a conditional angular probability map (Figure 3b). After randomly selecting a certain number of hypocenters (300), the algorithm looks for areas with a greater density of epicenters and produces, as a result, a diagram like the ones shown in Figure 3a, where no significant density emerges.  
In general, the absence of a fault is marked by a function which oscillates randomly without a clear trend and is characterized by various peaks with similar amplitudes (Figure 3a). Accordingly, the absence of a fault returns an angular conditional probability map (Figure 3b) without a clear pattern for any pairs of \(\vartheta\) and \(\varphi\) angles. Figure 3b represents the results of all possible rotations of the explorer parallelepiped around the pivot with the maximum number of counted hypocenters. The absence of a well-defined nucleus highlights the clear lack of a fault plane.  
In the second case (Figure 4), the presence of a fault immersed in a normal random distribution of 5,000 hypocenters is simulated (Figure 4a). We added to the previous normal distribution a simulated fault made of 500 hypocenters with a random distribution in the \(x\)-\(y\) dimension (10 \(\times\) 10 km) and a normal distribution, with variance of 200 m, in the \(z\)-dimension. The aligned earthquake distribution (i.e., the fault) initially on the \(z\) = 0 plane, is 35 degrees rotated with respect to the \(y\)-axis and 160 degrees rotated with respect to the  
Figure 3: **(a) Density diagram of random distribution of events: no fault detected. (b) Occurrence events as function of rotations around x and y axes (values shown in barcode).**  
\(x\)-axis. The algorithm converges towards the solution, which individuates the angles of the simulated fault.  
The results are shown in Figure 5a,b, where a clear density peak is shown which represents the presence of a well-defined fault. Figure 5b confirms the existence of a fault with a well-defined nucleus, which approximatively individuates the simulated fault angles.  
Figure 6a,b represent, respectively, the estimated dip and strike of the simulated fault as a function of the fault dimension and thickness. Both the dip and strike are individuated where the represented functions have stable behavior. In particular, the estimated dip value is approximately 40 degrees and the strike value about 124 degrees, which matches very well with the imposed simulated values.  
Figure 4: A random normal hypocenter distribution with a hidden fault. (**a**) The hypocenters belonging to the fault plane are hidden by the other hypocenters. (**b**) To make the fault visible, the hypocenters which individuate the fault plane are both enhanced in size by a factor of 4 and colored blue.  
Figure 5: (**a**) Density diagram (normal to the fault) of random distribution of events: fault detection. (**b**) Occurrence events as function of rotations around x and y axes (values shown in barcode).  
Figure 6: (**a**) Estimated dip and (**b**) strike as a function of the fault dimension and thickness.