## 4 Construction of the Crack Kriging Surrogate Model  
A Kriging surrogate model is a statistics-based interpolation method and has been elaborated by [PERSON] et al. [59]. In this part, the crack Kriging surrogate model will be established for crack identification using the sensitive super-harmonic features. From the sensitive analysis above, the 2\(\times\) components at 1/3 critical speed obtained from measurement points A and B will be used for the crack Kriging model construction. The relationship between crack parameters and 2\(\times\) amplitudes at 1/3 critical speed can be written as: \(\mathbf{Y}=\mathbf{f}(\mathbf{l},\ \mathbf{d})=\mathbf{f}(\mathbf{x})\), where, \(\mathbf{l}\) and \(\mathbf{d}\) are vectors for crack locations and crack depths respectively and \(\mathbf{x}\) is the crack parameters set. Specifically, the relationships for measurement points A and B are \(\mathbf{Y}^{1}=\mathbf{f}^{1}(\mathbf{l},\ \mathbf{d})=\mathbf{f}^{1}( \mathbf{x})\), \(\mathbf{Y}^{2}=\mathbf{f}^{2}(\mathbf{l},\ \mathbf{d})=\mathbf{f}^{2}( \mathbf{x})\). Since two measurement points are required, two surrogate models corresponding to measurement points A and B should be constructed, respectively. Details for the crack Kriging surrogate model construction are as follows. For measurement point A, given \(n\) samples of crack parameters and the corresponding 2\(\times\) amplitudes:  
\[\mathbf{x}=\left[ \begin{array}{c}\mathbf{x}_{1}\\ \mathbf{x}_{2}\\ \mathbf{x}_{3}\\ \vdots\\ \mathbf{x}_{n}\end{array} \right]=\left[ \begin{array}{cc}l_{1}&d_{1}\\ l_{2}&d_{2}\\ l_{3}&d_{3}\\ \vdots&\vdots\\ l_{n}&d_{n}\end{array} \right],\qquad\mathbf{Y}^{1}=\left[ \begin{array}{c}y _{1}\\ y_{2}\\ y_{3}\\ \vdots\\ y_{n}\end{array} \right] \tag{4}\]  
their relationship can be written as a polynomial regression model \(F(\beta,\mathbf{x})\) and a normally distributed random process \(\epsilon(\mathbf{x}_{i})\) with mean value \(\mu\) and standard deviation \(\sigma\):  
\[y_{i}(\mathbf{x}_{i})=F(\beta,\mathbf{x})+\epsilon(\mathbf{x}_{i}) \tag{5}\]  
The correlation function between the \(i\)-th and \(j\)-th samples is \(R\left[\epsilon(\mathbf{x}_{i}),\epsilon\big{(}\mathbf{x}_{j}\big{)}\right]\), which is the Kriging function in this paper. For a \(k\)-variable design space, a Kriging function can be expressed as:  
\[R\left[\epsilon(\mathbf{x}_{i}),\epsilon\big{(}\mathbf{x}_{j}\big{)}\right]= \exp\left(-\sum_{h=1}^{k}\theta_{h}\Big{|}x_{i}^{h}-x_{j}^{h}\Big{|}^{p_{h}}\right) \tag{6}\]  
where \(\theta_{h}\) indicates the sensitivity of the variable \(h\) in a sample (as only the location and depth of a crack are involved, therefore \(h=1,2\). ), and \(p_{h}\) determines the smoothness of the correlation function. The likelihood to eliminate the modelling error can be expressed as a Probability Density Function:  
\[L(y_{1},\ y_{2},\cdots,\ y_{n}\big{|}\mu,\sigma)=\frac{1}{\left(2\pi\sigma^{2 }\right)^{n/2}|\mathbf{R}|^{1/2}}\exp\left(-\frac{\left(\mathbf{Y}^{1}- \mathbf{1}\mu\right)^{\text{T}}\mathbf{R}^{-1}\Big{(}\mathbf{Y}^{1}-\mathbf{1} \mu\Big{)}}{2\sigma^{2}}\right) \tag{7}\]  
and its logarithmic form is:  
\[\ln(L)=-\frac{n}{2}\ln(2\pi)-\frac{n}{2}\ln\left(\sigma^{2}\right)-\frac{1}{2 }\ln(|\mathbf{R}|)-\frac{\left(\mathbf{Y}^{1}-\mathbf{1}\mu\right)^{\text{T}} \mathbf{R}^{-1}\Big{(}\mathbf{Y}^{1}-\mathbf{1}\mu\Big{)}}{2\sigma^{2}} \tag{8}\]where **R** is the correlation matrix of observed samples and \(|\cdot|\) is the operator of determinant:  
\[\textbf{R}=\left( \begin{array}{cccc}R[\epsilon(\textbf{x}_{1}),\epsilon( \textbf{x}_{1})]&\cdots&R[\epsilon(\textbf{x}_{1}),\epsilon(\textbf{x}_{n})]\\ \vdots&\ddots&\vdots\\ R[\epsilon(\textbf{x}_{n}),\epsilon(\textbf{x}_{1})]&\cdots&R[\epsilon( \textbf{x}_{n}),\epsilon(\textbf{x}_{n})]\end{array} \right) \tag{9}\]  
Hence, an essential step to construct a surrogate model is to maximize the likelihood.