## 3 Results
### The Colosseum Case Study  
The velocity-entropy-energy approach is also adopted to analyze another case study, i.e., the Colosseum, chosen for its importance in the area, considering its artistic, historical and cultural significance.  
The period for which the data are available (2011-2019) does not include the excavation operations on the tunnel in the proximity of this structure, which are more recent. Nevertheless, it is possible to evaluate the application of the entropy-energy approach to a monumental structure with distinctive features to identify the outliers. In addition, since the area has not yet been subjected to excavation, it is possible to compare the dispersions in the two periods identified above to verify that no significant variations have occurred. It allows validating the correlation between the parameters' variation and the effects of excavation on the building studied in previous sections.  
The first analysis concerns the entropy-energy distribution for the available period, shown in Figure 17. Defining the average model and the thresholds related to the standard deviation makes it possible to identify the outliers. It is observed that these points fall mainly on the high values of entropy and low energy. The localization of these points on the Colosseum, reported in Figure 18, highlights that they are distributed mainly on the base and in a cluster on the west end of the structure. Therefore, although further investigation is required, the presence of these outliers could be due to a differential behavior of the west end with respect to the whole system.  
Analogously to the previous case study, the velocity, entropy and energy dispersions in the first and second half of the reference period are studied to highlight possible changes (Figures 19-21). In addition, the parameters for the mean trends and standard deviation curves, respectively, are reported (Figures 22-24).  
The entropy-velocity (\(S\)-\(v\)) scatter in Figures 18(a) and 18(a) shows a slight decrease in entropy for velocities with high modulus, represented by the change in the first parameter of the mean curve (Figure 21(a)). The energy-velocity (\(E\)-\(v\)) dispersion in Figures 18(b) and 18(b) shows minimal changes in the average behavior, as shown by the parameters in Figure 23, though, in the second period, there are points showing higher velocity modules in the negative, as well as the positive section.  
Figure 17: Entropy-energy dispersion and mean model for the Colosseum, from 2011 to 2019.  
Figure 18: Map showing the outliers evaluated from the entropy-energy PDF for the Colosseum.  
Figure 19: Correlation between LOS velocity and entropy-energy dispersion for the Colosseum in the period from 2011 to 2015: (**a**) Entropy-velocity dispersion; (**b**) Energy-velocity dispersion.  
Figure 20: Correlation between LOS velocity and entropy-energy dispersion for the Colosseum in the period from 2015 to 2019: (**a**) Entropy-velocity dispersion; (**b**) Energy-velocity dispersion.  
Figure 23: Fitting parameters of the energy-velocity dispersions in the two periods for the Colosseum: (**a**) parameters of the mean model; (**b**) parameters of the standard deviation model.  
Figure 22: Fitting parameters of the entropy-velocity dispersions in the two periods for the Colosseum: (**a**) parameters of the mean model; (**b**) parameters of the standard deviation model.  
Finally, the entropy-energy (\(S\)-\(E\)) dispersions in Figure 21 exhibit similar characteristics, highlighted by the parameters in Figure 24. It can be observed that there is no significant change in the standard deviation of the points from the mean models. Moreover, as expected, the overall variation in the mean models is lower than in the previous case study. It should be noted that the Colosseum is only analyzed on an anthropogenic basis. Therefore, it would be appropriate to extend the analysis to also consider other aspects because of its complex hydrogeological configuration, which could lead to further data perturbations.