# Optical Sensing Technologies for Cryo-Tank Composite Structural Element Analysis and Maintenance
## 4 Structural Element Analysis
### Methodology
#### 4.1.2 Wavelet Decomposition for Transient Dynamics by FBG  
In this technique, the same FBG is used at high frequency for monitoring the transient dynamic response associated with flaw emergence. In detail, the FBGs are interrogated via an FAE-400 instrument at a max data rate of 25 KHz. To this purpose, wavelet analysis is a suitable approach to extract tailored signal features. Typically, wavelets with a small number of vanishing moments have a better time resolution and can be used to detect abrupt changes (discontinuities) in the time domain. Conversely, wavelets with a large number of vanishing moments have a better frequency resolution, and can be used to detect periodicities in the time series. In the discrete-time domain, the implementation of the wavelet transform is based on a bank of discrete time filters. The discrete wavelet transform (DWT) is computed by successive low-pass and high-pass filtering, referred to as the Mallat algorithm [27]. The coefficients for the approximation signals (a\({}_{j,k}\)) and detail signals (d\({}_{j,k}\)), corresponding to the wavelet scaling index (j) of the mother wavelet \(\Psi\)(t) and shifting index (k), can be plotted. The acquisition is reported in Figure 11. The signal includes events characterized by different magnitude, along with the time. In particular, two specific events are synchronized with the initiation of the flaw (red curve).  
Figure 10: FBG strain slope tracking and first-order derivative analysis; the effects of damage onset are indicated in the dotted boxes.  
A wavelet multi-resolution analysis is performed by iteratively applying scaling and wavelet functions as low and high pass filters, respectively. The mother wavelet considered is biorthogonal (bior) 3.1. As shown in Figure 12, such a data process allowed the separation of the signal's main frequency components, preserving sufficient time resolution.  
The slow change in the low-frequency constituent captured by _cd_4 and _cd_3 provides evidence of damage occurrence, as a high local gradient event, and shows a certain potential in identifying damage time laps. In addition, a scalogram (Figure 13) can be potentially used to correlate the high local gradient with the corresponding energy value. The horizontal axis represents time. The vertical axis represents frequency. The color or intensity at each point (time, frequency) corresponds to the magnitude (or energy) of the wavelet coefficient at that scale and time. A threshold energy level (TEL) is arbitrarily estimated by plotting  
Figure 11: FBG dynamic high-strain history (red curve) and static-strain evolution (blue curve) showing synchronization.  
Figure 12: Wavelet decomposition of the biorthogonal 3.1 function at different scale resolutions: (**a**) scale coefficient cd4 at 400 samples; (**b**) scale coefficient cd3 at 800 samples; (**c**) scale coefficient cd2 at 2000 samples; (**d**) scale coefficient cd1 at 3500 samples.  
the scalogram for each decomposition and computing a simple mean value among them. The TEL can be set to filter eligible events.