# Detection and Localization of Multiple Damages through Entropy in Information Theory
## 4 The Finite Element Models
### Input Definition and Simulation Settings  
The AVs are generally classified as microseismicity, induced by natural sources, predominantly between the 0-1 Hz range, and microtemors, generated by nearby human activities such as pedestrian, traffic, and machinery [45]. These latter ones cover the higher frequencies (1 to 10 \(\div\) 20 Hz) and are more subject to intra- and inter-day variability [46]. For the former group, for regions not too far inland, as in the Italian peninsula, the main natural phenomena originating the natural AVs are the waves impacting the coast (in the range 0.05-0.1 Hz according to [47], 0.5-1.2 Hz according to [48]), large scale meteorological perturbation on site (0.1-0.25 Hz [47], 0.16-0.5 Hz [48]), and large scale meteorological events on nearby seas/oceans (0.3-1 Hz [47], 0.5-3 Hz [48]). However, the current expert consensus indicates the range of engineering interest between 0.5 and 20 Hz [49], i.e., the frequency range is mainly affected by anthropic activities [50]. Thus, the following assumptions and considerations have been made:  
1. The AVs are applied as independent point-like sources of white Gaussian noise (WGN) time histories of forces;
2. The AVs are applied at each node of the boundaries of the SP;
3. The frequency and amplitude content of the AVs is stationary (i.e., not time-dependent);
4. In each direction, the WGN is assumed with zero mean and unit variance;
5. At each point-like source, the AVs are applied in the three main directions (\(x\)-, \(y\)-, and \(z\)-axes). Concerning the first assumption, the Gaussian distribution of independent random point-like sources is derived from the state of the art in AV numerical modeling [49]. The application of the forces on the edges of the SP is intended to mimic the random vibration of the surrounding soil. Concerning the third assumption, it is assumed that the measurements are performed in absence of abrupt variations during recording. This is assured for the small portion of microseismic noise in these interferences. Even for the larger anthropogenic portion, this is reasonable due to its slow variation rate, generally corresponding to the day/night and seasonal cycles (as experimentally validated in [23]). These timescales are much larger than the average acquisition duration. Concerning the amount of variance of the external force, it is worth stating that its definition is arbitrary and does not affect the results of the proposed approach. Indeed, it is well known that the standard deviation \(\sigma_{Y}\) of a linear transformation \(Y\) (with zero intercepts) of a random variable \(X\), i.e., \(Y=kX\) (with \(k\) constant of proportionality) is proportional to the standard deviation \(\sigma_{X}\) of the reference random variable \(X\), i.e., \(\sigma_{Y}=k\sigma_{X}\) (in the present work, \(X\) is a Gaussian distribution with unit variance or standard deviation, i.e., \(\sigma_{X}=1\), and \(\sigma_{Y}=k\)). Thus, being a linear model, the structural response will vary linearly for a uniform linear variation of the external force vector \(\mathbf{x}\). In this case, the external force vector is made up of the realization of the random variable \(X\).