## 2 Materials and Methods
### Mathematic Resume  
In signal processing, cross-correlation is a mathematical operator measuring the rate of similarity of two series of data, as a function of the displacement of one set relative to the other. It is commonly used for searching a long signal for a shorter, known feature. For vectors defined on the set of reals R, the cross correlation of two finite sequences \(v1\) and \(v2\) is given by Equation (1):  
\[(v_{1}*v_{2})[i-1]=\sum\
olimits_{i=2}^{ns}v_{1}[i]v_{2}[i-1], \tag{1}\]  
In this application, the vectors \(v1\) and \(v2\) are spatially shifted (one sensor gauge lag) strain readouts of the same fiber optic sensor. The operation is implemented in Matlab code.  
The vectors \(v1\) and \(v2\) are obtained by the mean value (in time domain) of the original matrix raw data (En). The interrogation system provides a matrix set where each row is a timestamp of the current strain readouts at a predefined sample rate, and the columns is the time evolution history of each sensor readout.  
A damage index (DI) can be then estimated. Generally speaking, given two signals of different structural configurations (baseline and current), a DI could be simply defined as follows:  
\[\text{DI}=\frac{\text{CS}-\text{BS}}{\text{BS}}, \tag{2}\]  
where CS is the current signal and it represents the strain vector of the signal response after an impact or more generally an operational life. BS is the baseline signal and it represents the strain vector of the structure when it is supposed in healthy state, generally after component assembly or after manufacturing. According to this definition, with a DI close to zero, the structure is in a healthy state; when the DI overcomes a certain threshold, this means that damage appears; the greater the index, the more serious the damage would be.  
In this application, having fixed a baseline configuration, the readout is the relative measured output with respect to their initial state. So, the fiber optic readout (\(\Delta\varepsilon\)) is actually a differential strain vector. The Equation (2) can be specialized as follows:  
\[DI_{FO}=\frac{\widetilde{\Delta\varepsilon}}{\text{max}\!\!\left(\widetilde{ \Delta\varepsilon}\right)}, \tag{3}\]  
where, \(\widetilde{\Delta\varepsilon}\) is the cross-correlated differential strain readout. The normalized damage index definition provides information on which sensing paths is closely located to a damaged area. In addition, it does also return the absolute position of the damage and its extension all along the fiber optic length. For the \(DI_{FO}\) a threshold level (\(TL\)) can be set. At this stage, the authors arbitrarily decided to define a \(TL\) according to Equation (4):  
\[TL=\mu+k\sigma \tag{4}\]  
where \(\mu\) is the mean value of the \(DI_{FO}\), \(\sigma\) its standard deviation, and \(k\) is a dimensionless constant value herein set to 3, indicating the probability the measure is within the normal distribution with the 98% [30]. In the end, the \(TL\) is drawn and used to plot \"eligible\" sensors among those exceeding the \(TL\) value. The cluster of eligible sensors position along the fiber optic is used to indicate the damage position and extension onto the structural component under test.