# Detection and Quantification of Damage in Metallic Structures by Laser-Generated Ultrasonics
## 4 Artificial Damage Detection and Quantification for Aluminum Alloy Plates
### Calculation of Embedding Parameters  
The proposed technique attempts to detect the changes in the geometric figure in the state space resulting from damage. First, the baseline and current-line attractors are reconstructed by the response data recorded from both the undamaged and damaged structures. Notably, the attractors are reconstructed using the same embedding parameters. An accurate reconstruction requires a proper choice of embedding parameters, as mentioned above. Generally, the choice of the embedding dimension \(m\) should guarantee that the reconstructed attractor is unfolding in the state space. In addition, the choice of the time delay \(T\) depends on the appearance location of the first minimum value of the auto-covariance function. Here, the AMI function for time delay \(T\) and Cao's function for dimension \(m\) are used. For the recorded response data \(x(n)\) (\(n=1\),\(2\),...,\(N\)), the AMI quantifies the average amount of information obtained about \(x(n+T)\) through \(x(n)\). If \(x(n+T)\) and \(x(n)\) are completely independent, the AMI value becomes zero. However, in practice, the value of \(T\) corresponding to the first minimum of the AMI function is considered sufficient, which means that the first minimum value of \(T\) can be chosen as the proper time delay. Figure 8 shows the AMI values for the eight plates' responses. When \(T\) reached 12, all the AMI values were reaching their minimum values. Hence, the time delay was chosen to be 12 for this experiment. A proper choice of embedding dimension \(m\) should guarantee that no false neighbors exist in the reconstructed state space. [PERSON]'s functions for the eight plates' responses are provided in Figure 9a. Figure 9b is [PERSON]'s function for the plate whose minimum embedding dimension is largest compared with the others'. \(E_{1}\) and \(E_{2}\) are two quantities defined by [PERSON] [29] for determining the minimum embedding dimension of a scalar time series. Suppose a time series, \(x_{1}\), \(x_{2}\),... \(x_{N}\). The attractor \(X_{i}(m)\)can be reconstructed as follows [29]:  
\[X_{i}(m)=(x_{i},x_{i+T},\ldots,x_{i+(m-1)T}),\quad i=1,2,\ldots,N-(m-1)T. \tag{6}\]  
Define:  
\[a(i,m)=\frac{\|X_{i}(m+1)-X_{n(i,m)}(m+1)\|}{\|X_{i}(m)-X_{n(i,m)}(m)\|}\quad i =1,2,\ldots,N-mT, \tag{7}\]  
where \(|\ |\cdot|\ |\) is some measurement of Euclidian distance and is given by the maximum norm, i.e.,  
\[\|X_{k}(n)-X_{l}(n)\|=\max\Bigl{|}x_{k+jT}-x_{l+jT}\Bigr{|}\ (0\leq j\leq n-1). \tag{8}\]  
Figure 7: The recorded ultrasonic responses in the time domain: **(a)** Reference undamaged plate; **(b)** currently undamaged plate;**(c)**damage width: 0.05 mm; **(d)** damage width: 0.1 mm; **(e)** damage width: 0.2 mm; **(f)** damage width: 0.4 mm; **(g)** damage width: 0.6 mm; **(h)** damage width: 0.8 mm.  
Define:  
\[E(m)=\frac{1}{N-mT}\sum_{i=1}^{N-mT}a(i,m). \tag{9}\]  
Then  
\[E_{1}(m)=E(m+1)/E(m). \tag{10}\]  
Define:  
\[E^{*}(m)=\frac{1}{N-mT}\sum_{i=1}^{N-mT}\Bigl{|}x_{i+mT}-x_{n(i,m)+mT}\Bigr{|}. \tag{11}\]  
Then  
\[E_{2}(m)=E^{*}(m+1)/E^{*}(m). \tag{12}\]  
Equations (10) and (12) are the definitions of \(E_{1}\) and \(E_{2}\), respectively. Notably, the values of \(E_{1}\) and \(E_{2}\) stopped changing after \(m\) exceeded 15 and an embedding dimension of 16 was selected for this experiment. The detailed selection procedure can be found in [29].  
Figure 8: The average mutual information (AMI) function for time delay \(T\).  
Figure 9: Cao’s function for embedding dimension \(m\): (**a**) Cao’s function for the eight plates’ responses; (**b**) Cao’s function for the plate with the largest embedding dimension.