# A Kriging Model Based Finite Element Model Updating Method for Damage Detection
## 2 Kriging Based Model Updating Method
### Construction of Kriging Model  
A Kriging model is a surrogate model based on a stochastic process. It was originally put forward in geostatistics and made its way into engineering design following the work of [PERSON], et al. [36]. For a given set of sample data (input), \(\mathbf{X}=\left\{\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{n}\right\}^ {\mathrm{T}}\), and the observed responses (output), \(\mathbf{Y}=\left\{y_{1},y_{2},\cdots,y_{n}\right\}^{\mathrm{T}}\), the expression of Kriging model that reflects the relationship between them is:  
\[y(\mathbf{x}_{i})=\mathbf{f}^{\mathrm{T}}(\mathbf{x}_{i})\mathbf{\beta}+z( \mathbf{x}_{i}),\ \ \ i=1,2,\cdots,n \tag{1}\]  
where \(\mathbf{f}(\mathbf{x})\) is a polynomial vector of the sample \(\mathbf{x}\), \(\mathbf{\beta}\) is the vector of the linear regression coefficients to be estimated and \(z(\mathbf{x})\) represents errors and is assumed to be a stochastic process that follows a normal distribution of \(N(0,\sigma^{2})\) with a zero mean and standard deviation \(\sigma\). It should be noted that the basic assumption of the [PERSON] model is that the same input will lead to an identical output. Therefore, the deviation between the output response and the polynomial regression part is only due to the modeling error itself, regardless of the measurement error and other random factors. This method does not depend on the simulated precision of the polynomial part to the response surface, but focuses on constructing the appropriate surrogate model by effective filling of the stochastic process part, which makes it more suitable for dealing with nonlinearity. Thus, the polynomial part is often taken as a constant in some other references. To estimate the stochastic process \(z(\mathbf{x})\), [PERSON] assumes that the true response surface is continuous, any two points will tend to have the same value as the distance in between approaches zero and it is the same for \(z(\mathbf{x})\) of two points. Thus, the correlation between \(z(\mathbf{x})\) of any two sample points can be expressed as a function of their spatial distance. The most widely used Gaussian correlation model is adapted here:  
\[R(z(\mathbf{x}_{i}),\ z(\mathbf{x}_{j}))=\exp\biggl{(}-\sum_{k=1}^{m}\theta_{ k}\Bigl{|}x_{i}^{k}-x_{j}^{k}\Bigr{|}^{2}\biggr{)} \tag{2}\]where \(x_{i}^{k}\) and \(x_{j}^{k}\) are the \(k\)th components of the two sample points \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\), \(m\) denotes the number of design variables, \(\theta_{k}\) controls the decay rate of correlation on different dimensions. And then the matrix of correlation functions between sample points is obtained as  
\[\mathbf{R}=\left[ \begin{array}{cccc}R(\mathbf{x}_{1},\mathbf{x}_{1})&\dots&R (\mathbf{x}_{1},\mathbf{x}_{n})\\ \vdots&\ddots&\vdots\\ R(\mathbf{x}_{n},\mathbf{x}_{1})&\dots&R(\mathbf{x}_{n},\mathbf{x}_{n})\\ \end{array} \right] \tag{3}\]  
The likelihood function of the sample point can then be written as:  
\[L=\frac{1}{\left(2\pi\sigma^{2}\right)^{m/2}\left|\mathbf{R}\right|^{1/2}} \exp\left[-\frac{\left(\mathbf{Y}-\mathbf{F}\mathbf{\beta}\right)^{\mathrm{T} }\mathbf{R}^{-1}(\mathbf{Y}-\mathbf{F}\mathbf{\beta})}{2\sigma^{2}}\right] \tag{4}\]  
where \(\mathbf{F}\) is a matrix of vector \(\mathbf{f}(\mathbf{x})\) for each sample point. \(\left|\mathbf{R}\right|\) is the determinant of \(\mathbf{R}\) which is a function of \(\theta_{k}\).