# Detection and Localization of Multiple Damages through Entropy in Information Theory
###### Abstract  
According to recent works, entropy measures, and more specifically, spectral entropies, are emerging as an efficient method for the damage assessment of both mechanical systems and civil structures. Specifically, the occurrence of structural system alterations (intended in this work as stiffness reduction) can be detected as a localized change in the signal entropy. Here, the Wiener Entropy (also known as the Spectral Flatness) of strain measurements is proved as a viable tool for single and multiple damage assessment, including damage detection, localization, and severity assessment. A case study from oil & gas engineering, i.e., a finite element model of a buried steel pipeline, is utilized for this aim.  
Wiener entropy; structural health monitoring; pipeline integrity management; damage detection; damage localization; damage severity assessment; multiple damages; information theory; steel pipes; oil & gas engineering +
Footnote †: journal:Some entropy-based SHM approaches reported in the last years in the scientific literature include the use of power spectral entropy [6] and multiscale cross-sample entropy [7]. In [8], the authors proposed an interesting interpretation of the insurgence and development of fatigue, corrosion, wear, radiation, and creep damage as irreversible thermodynamics phenomena, thus from an entropic point of view.  
Recently, in [9] benchmarked Shannon, Renyi, permutation, sample, approximate, and spectral entropy for SHM purposes. The authors of [10] performed a similar comparison between Wiener and Shannon spectral entropy for the application to mixed brick-stone masonry structures, considering experimental evidence from three case studies damaged from previous seismic events.  
Indeed, the basic principles of the procedure reported here were set in [10; 11]. In the present paper, these principles are expanded, further detailed, and validated on a numerically simulated case study of a buried steel pipe. This application, relevant in the context of oil & gas engineering, presents several differences in comparison with the masonry buildings analyzed in [10; 11]. Underground structures are inherently challenging due to:  
1. The impossibility to directly excite the structure with a controlled input; and
2. The very stiff boundary conditions.  
These aspects are shared by many geotechnical applications, such as deep foundation piles, which are very difficult to investigate for structural integrity [12; 13]. This is mainly due to the very small amplitude of the output displacements, naturally induced by ambient vibrations (AVs) and recorded with a grid of sensing devices. In fact, it is notoriously difficult to extract reliable DSFs from output-only recordings from weakly-excited structures (see, e.g., a case study for cultural heritage preservation in [14]). As will be shown here, entropy measures are instead particularly well-suited for dealing with low-amplitude, random signals.  
In this work, the potentialities of this framework were methodically investigated for the first 2/3 main tasks of SHM, defined according to the classic [PERSON]'s hierarchy [15; 16].  
The remainder of this paper is organized as follows: Section 2 describes the theoretical framework of the proposed approach; in Section 3, the context of this application is briefly outlined; the numerical case study is detailed in Section 4; the results are discussed in the following Section 5; and Section 6 (Conclusions) ends this paper.