# Detection and Localization of Multiple Damages through Entropy in Information Theory
## 7 Conclusions  
This paper described the use of Wiener entropy for pipeline integrity management and structural health monitoring of buried steel pipes.  
In general terms, the undergoing establishment of an entropic framework for SHM strongly supports the main concepts of entropy measurements as an indicator for health state development. A quantifiable decrease in the WE signal entropy in operational conditions can be directly linked to a localized decrease of material stiffness when strain measurements are used as a structural sensing response.  
For unchanged boundary conditions, this can be further linked to developing one or more damages in the structure under investigation.  
Regarding the specific case study discussed here, the motivations for using the proposed procedure to underground metallic structures have been highlighted. As long as the input spectral flatness is guaranteed by the use of ambient vibrations, the use of a spectral flatness measurement on the system output is entirely logical. This allows for precise assessment for very low amplitude output signals, where traditional techniques struggle or completely fail, thanks to the non-deterministic behavior of the target system. Two FE models were employed and several damage patterns were applied to them.  
The WE-based procedure was applied for several SHM tasks, defined accordingly to the classic Rytter's hierarchy. The main findings can be summarized as follows:  
* The small extension and severity of some cases highlighted the capabilities of the proposed indicator to detect stiffness reduction at an early stage, thus avoiding the insurgence of latent damages;
* The proposed approach can detect and assess multiple damages of different sizes and severities;
* The simple health state indicator proved the robustness of the monitoring of WE of strain also in the presence of artificially added measurement noise;
* The proposed approach performed well both under ideal (yet unrealistic) conditions and with initial flaws and soil inhomogeneities.  
As expected, it was noted that the WE of the simulated output signals (axial strains) locally decreases for local stiffness reductions. It was also noticed that it increases immediately outside the damaged area; this becomes more pronounced as the damage increases. This aspect will be further addressed in future works. Further studies will be needed to better address specific tasks, such as damage classification, which strictly requires pattern recognition techniques in a supervised learning fashion. Nevertheless, this study proved the feasibility of Wiener entropy as a damage-sensitive feature for vibration-based inspection, particularly well-suited for output-only measurements of underground structures.  
Conceptualization, R.C., M.C., G.M. and E.L.; methodology, R.C., M.C., G.M. and E.L.; software, G.M., M.C. and E.L.; validation, G.M., M.C. and E.L.; resources, R.C.; data curation, G.M., M.C. and E.L.; writing--original draft preparation, M.C., G.M. and E.L.; writing--review and editing, R.C., G.M. and E.L.; visualization, M.C., G.M. and E.L.; supervision, R.C. and C.S.; project administration, R.C. and C.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data that support the findings of this study are available from the authors upon reasonable request.