## 3 Methods
### Ortho Vertical Dataset Processing for Best Fitting Axial-Symmetric Model
Considering the Mogi model results (Figure 5), we show the best fitting models obtained by varying the parameter d; also in this case, we see that even the best fitting model, for d = 2700 m, systematically underestimates the uplift at distances lower than about 3500 m and overestimates uplift at higher distances. The statistic we used to evaluate the goodness of fit is \(\mathrm{R}^{2}\), the 'coefficient of determination', defined as \(\mathrm{R}^{2}=1-\mathrm{SS}_{\mathrm{res}}/\mathrm{SS}_{\mathrm{tot}}\), where \(\mathrm{SS}_{\mathrm{res}}\) is the sum of squared residuals between the theoretical model and the data, and \(\mathrm{SS}_{\mathrm{tot}}\) is the sum of the squared differences from the average data value. It is worth noting that the closer to 1 the \(\mathrm{R}^{2}\) value is, the better the model fits to the data. The \(\mathrm{R}^{2}\) parameters for the two models, which indicate the goodness of fit, are \(\mathrm{R}^{2}=0.9724\) for the Gaussian function and \(\mathrm{R}^{2}=0.9543\) for the Mogi model, respectively. Figure 4: Best fitting Gaussian models for different sigma values. Figure 5: Best fitting Mogi models for different \(\mathrm{Z}\) values (\(\mathrm{Z}=\mathrm{d}/1000\)). It is clear that simple functions with two parameters, or even the most used model for volcanic pressure sources, are not able to give a very good fit to the axial-symmetrical shape of the data. So, we proceed to test a completely general axial-symmetric function, which is represented by a polynomial of a suitable degree. We have to stress that we searched for an axial-symmetric function giving the best possible fit to the data. The analysis and computation procedures involved the use of R software. The distances to the deformation center (denoted as r), which were calculated using ArcGIS Pro, were imported into R along with the vertical uplift values (z) that were measured by satellites at each specific target point. Then, a series of polynomial curves describing theoretical radial ground deformation with a progressively higher accuracy were generated by using the imported dataset of r and z values (Table 2). These polynomials were derived up to the 20 th degree; in Figure 6, we show only curves up to 6 th degree due to almost fully overlapping for curves of degrees higher than 6. Coefficients for polynomial curves are shown in Table 3.  
The curve of the 2 nd degree shows a significant geometric discrepancy from the data (Figure 6a), while the 3 rd degree curve shows some mismatches for distances greater than 5 km (Figure 6b); the 4 th, 5 th and 6 th degree curves show only minor differences for distances greater than 7-8 km (Figure 6c-e). We also computed best fitting parameters, i.e., adjusted R2 (coefficient of determination) and RMSE values, for polynomial functions from the 2 nd to 20 th order (Table 2). We observed that the adjusted R2 and RMSE value become quite stable at the fourth degree and that very small improvements are obtained by considering higher degrees. The selection of the best polynomial degree was made by considering the RMSE and by choosing the minimum order over which the improvement in the RMSE can be neglected, i.e., one order of magnitude lower than the intrinsic data error. Thus, we chose the fourth degree to best fit the vertical uplift. \begin{table}
\begin{tabular}{c c c} \hline
**Polynomial Degree** & **Adjusted R2** & **RMSE** \\ \hline
20 th degree & 0.9894 & 8.2356 \\
10 th degree & 0.9893 & 8.2927 \\
9 th degree & 0.9892 & 8.3092 \\
8 th degree & 0.9892 & 8.3109 \\
7 th degree & 0.9892 & 8.3315 \\
6 th degree & 0.9891 & 8.3536 \\
5 th degree & 0.9879 & 8.7886 \\
4 th degree & 0.9871 & 9.0848 \\
3 rd degree & 0.9830 & 10.4242 \\
2 nd degree & 0.8999 & 25.3370 \\ \hline \end{tabular}
\end{table} 
Table 2: R2 and RMSE values for calculated polynomial curves.