# Detection and Localization of Multiple Damages through Entropy in Information Theory
## 2 Wiener Entropy Measurements for SHM Purposes  
Out of the several definitions of 'entropy' currently reported in the scientific literature, the Wiener entropy (WE, also known as spectral flatness) was found to be particularly well-suited for metallic structures in previous studies [17]. The main motivation resides in its greater sensitivity to small alterations in comparison to the more common Shannon spectral entropy. This is relatively inconvenient for masonry or concrete structures, where the material inhomogeneity and variability may cause damage-unrelated fluctuations. In turn, these cause a higher risk of false-positive identifications [10], which are a common threat to vibration-based SHM [18]. On the other hand, for more homogeneous building materials, this greater accuracy can instead be exploited to detect small cracks in their earlier extension phases.  
The WE can be mathematically defined in a discretized fashion as:  
\[S_{w}=B\frac{\sqrt[b]{\prod_{i=1}^{B}\left|H\right|_{i}^{2}}}{\sum_{i=1}^{B} \left|H\right|_{i}^{2}} \tag{1}\]  
where \(B\) represents the total number of frequency bins, that is to say, the number of discrete measurements in the frequency domain, while \(\left|H\right|^{2}\) is the (discretized) power spectrum of the recorded vibrational time history, evaluated at frequency \(f=f_{1},f_{i},\dots,\ f_{B}\).  
The convenience of entropy measurements over other, more classic, approaches directly derives from the specific conditions of underground pipeline monitoring.  
The first aspect regards the amplitude of the system input and output. As hinted above, the AV input is very low, and thus the system is only weakly excited. Due to this minimal excitation level and the increased stiffness from the confining soil, the expected output amplitude is even lower in amplitude. Generally, this is an inconvenient condition.  
One can consider the following: the system behaves more deterministically for increasing input energy. In fact, the energy of the system's response is naturally amplified more at the natural frequencies than at other frequencies. Thus, increasing the system energy increases this disproportion and, therefore, inherently reduces the flatness of the frequency spectrum. In contrast, at low energy levels, the actual response tend to become indistinguishable from the measurement noise, the system behaves mostly non-deterministically, and the responses will have inherently higher entropy.  
The second aspect regards the frequency content of the input. For system identification (SI) purposes, it is preferable to have a controlled input in order to excite only specific portions of the system's global dynamics. This is, e.g., the case of chirp or stepped sines, which allow to isolate and follow a specific resonance frequency even in nonlinear conditions. However, the Gaussianity of ambient dynamic loads (due to the central limit theorem) guarantees an (almost) flat distribution of energy in the frequency domain, which is ideal for entropy measurements [19].  
Thus, these two aspects--the natural spectral flatness of the input signal and the very low amplitude of the output response--make the use of the Wiener entropy/spectral flatness particularly well-suited for buried metallic structures.  
The rest of this discussion aims to verify the use of entropy measurements as a health state indicator for the several tasks included in a complete SHM diagnosis. These include, according to the classic hierarchical structure proposed by [PERSON] [15]:  
1. (single- or multi-) Damage Detection: to identify if one or more damage(s) are present or not in the structure;
2. (single- or multi-) Damage Localization: to define the most probable position of damage;
3. (single- or multi-) Damage Assessment: to evaluate the severity of the damage(s);
4. (single- or multi-) Damage Prediction: to establish the expected future growth of damage.  
The last step is of less interest here since it will require further statistical information concerning the expected remaining useful life (e.g., prognosis).