## 4 Discussion and Conclusions
Vesuvius for MD \(\geq\) 2.0 [14], as well as a Brune radius in the range 30-136 m. This last value, the only one \(>\) 100 m, corresponds to the deepest (>4 km) earthquake that occurred on 2019/05/18 and which has the maximum M0 (6.1 \(\times\) 10\({}^{12}\) Nm) of the whole dataset. In the hypothesis of a rectangular fault, the relative stress drop is about 1.5 MPa, as expected for the deepest VTs [14]. We used the estimated focal mechanisms and the source spectra parameters to calculate the static displacement field according to the approach of Okada [32], in correspondencewith the elevation of each tiltmeter. The model of Okada for seismic source assumes the approximation of the propagation medium as homogeneous isotropic half-space, whereas Mt. Vesuvius has significant topographic relief and structural variations from the volcanic edifice to the deeper underlying crust. However, this model is often used with satisfactory results in the presence of significant topography and medium inhomogeneity e.g., [34; 38]. Once we obtained the deformation field in correspondence of the tiltmeters positions, we faced with the question of how to make the comparison with the ground tilt. The possibility of comparing the two observables came from the observation of clear transient signals recorded in correspondence with occurrence of the earthquakes. MD = 2.0 is the earthquake size threshold which allowed us to detect such transients. Once we individiudated the transients, we checked if any other possible spurious sources were present in proximity of the sensors. Since tiltmeters are very sensitive instruments (less than \(5\times 10^{-9}\) radians), we ensured that other exogenous forces, such as landslides, storms or other exceptional meteorological phenomena, did not occur, since these phenomena could generate tilt transients with an amplitude comparable with the earthquakes. The only external forces to still be considered were tides [39]. Due to their harmonic nature, tides can be removed by estimating the tilt trends over long time windows; time windows whose length is equal to an integer multiple of the tidal constituents' proper period (in our case diurnal, T = 24 h, and semidiurnal, T = 12 h). We performed several tests to calibrate the window length over which to estimate this long-term tilt. A 10-day window containing each seismic event gave the best choice. Once we removed the tides and excluded the presence of any other spurious exogenous disturbance, the residual tilt trend is likely to be mainly determined by the seismic force. For each tiltmeter, the estimated tilt trend vector was then reported on the plots of co-seismic static displacement generated by Coulomb software. The results, summarized in Table 3, show a very satisfactory agreement. The stations closer to the Mt. Vesuvius crater, IMB and CMT, seem more sensitive to the VIs' effect, while the two farthest stations are often insensitive. We also performed two formal tests estimating the effective derivative of the static displacement in a neighborhood of the point of the displacement grid closer to the tiltmeter position, using the selected earthquakes. We found that the two vectors, namely the long-term tilt trend and the derivative of the seismic static displacement, have a comparable size and direction (Figure 6). The discrepancies between the two vectors are likely ascribable to the propagation of uncertainties thorough the various passages (i.e., inaccuracies from simplifying assumptions) and to the presence of spurious ground motion sources that are not trivial to deal with. The aim of this study was to find a procedure to compare the co-seismic displacement and the ground tilt, and, for this, we showed a possible approach. By applying a series of well-established seismological methodologies and formulating a new approach to perform the comparison among these two quantities, we found that the tiltmeters installed at Mt. Vesuvius are likely able to render the co-seismic static displacement well.