## 3 Methods
### Ortho Vertical Dataset Processing for Best Fitting Axial-Symmetric Model  
The Ortho Vertical dataset is formed by 23,970 points, characterized by values of total vertical deformation during 2016-2021 ranging from \(-\)82.2 to +547.2 mm with a mean of +21.77 mm with a standard deviation of 69.77 due to the wide range of data values that are not clustered around the mean value. The measurement uncertainties in annual velocity \begin{table}
\begin{tabular}{c c c c c} \hline
**EGMS** & **Number of** & **Mean Velocity** & **Velocity** & **Mean RMSE** \\
**Dataset** & **Points** & **(mm/yr)** & **Range** & **Mean RMSE** \\ \hline Ortho Vertical & 23,970 & + 3.62 & \(-\)13.7 to + 91.2 & 1.18 \\ Ortho East/West & 23,970 & \(-\)0.16 & \(-\)36.9 to + 40.5 & 1.29 \\ \hline \end{tabular}
\end{table} 
Table 1: Characteristics of ortho EGMS datasets (EGMS). can be estimated by the RMSE (root mean square error) values, having a mean value of 1.18 mm (ranging from 0.4 to 8) with a standard deviation of 0.88. The Ortho Vertical dataset was implemented by calculating--using ArcGisPRO 2.3.1. software--the distance r to the caldera's center of maximum deformation for each point, as defined by several authors [20; 84; 101; 102]. Values of r lower than 500 m have not been considered for the calculation of the polynomial curves, considering that many sectors within this distance to the deformation center fall within the sea, and few uplift values are available. Our goal was to subtract the axial-symmetrical part from the ground deformation, so we searched for the axial-symmetric function that gave the best fit to the vertical deformation data. Since there is not a scientific consensus about a given model for source of the actual unrest, as explained before, we tested several functions, as well as the most used volcanological models. Among several models tested, we report here the fit with the simple function which has the most similar shape to the data, as well as the most commonly used model to fit ground deformation within volcanoes. They are the Gaussian function and the Mogi model, respectively [103]. The analysis and computation procedures involved the use of both Microsoft Excel (ver. 2502 for Microsoft 365 MSO) and R (ver. 4.2.0) software. The Gaussian function, in our case, takes the following form:  
\[h(r)=h(0)e^{-\frac{r^{2}}{2r^{2}}} \tag{1}\]  
where \(h(r)\) is the vertical displacement at a distance r from the point of maximum displacement \(h(0)\), and \(\sigma\) is a parameter (in the original formulation, the standard deviation) which determines the decay of the curve. The Mogi [103] formula, which represents the effect of a point source of pressure increase and has been, for many decades, the most used model to fit and interpret ground deformation data at volcanoes, has the following form:  
\[h(r)=K\frac{d}{\left(d^{2}+r^{2}\right)^{\frac{3}{2}}} \tag{2}\]  
where \(K\) is a constant, depending on the source volume, overpressure, and medium rigidity; \(d\) is the source depth; and r is the distance from the point of maximum displacement. Each one of the two functions has two unknown parameters: they are \(h(0)\) and \(\sigma\) for the Gaussian function and \(K\) and \(d\) for the Mogi model. A least squares solution can easily be found for each one of these functions using a numerical search procedure in the two-variable space. The resulting best fitting Gaussian curve has the following parameters: \(h(0)=0.550\) m, \(\sigma=2.0\). The resulting best fitting Mogi model has the following parameters: \(K=4009.5\) m\({}^{3}\), \(d=2700\) m.  
Figure 4 shows the best fitting Gaussian curves obtained by varying the parameters. It is evident that the Gaussian shape is not very similar to the data decay as a function of distance. In particular, even the best fitting curve, obtained for \(\sigma=2.0\), almost systematically overestimates the uplift at a short distance, while it underestimates uplift above 4500 m of distance. Nor varying the shape parameters can give a better fit to the observed data.