## 3 Interval Analysis Method for Consideration of Uncertainties  
The updated ESP is prone to uncertainty since uncertainties (noises) in the obtained vibration data are unavoidable. The uncertainties in the measured modal data are considered to be independent, normally distributed random variables with zero means and a particular covariance, as indicated in [19]. As a result, the eigenvalues and mode shapes may be written in the following [19]:  
\[\lambda_{i}^{E}=\lambda_{i,0}^{E}(1+X_{\lambda i}),\;i=1,2,\ldots,n_{m}, \tag{6}\]  
\[\phi_{i}^{E}=\phi_{i,0}^{E}\big{(}1+X_{\phi i}\big{)},\;i=1,2,\ldots,n_{m}, \tag{7}\]  
where the subscript 0 represents the true values, \(X_{\lambda i}\) and \(X_{\phi i}\) indicate relative random noises in the measured frequencies and mode shapes, respectively. The mean value of vector \(X\) is zero and the standard deviation indicates the noise level. The interval bounds can provide supports for structural health monitoring under uncertain conditions [41]. To account for epistemic and aleatory uncertainties, the main concept of interval mathematics is implemented by providing the upper and lower boundaries of input parameters to generate the upper and lower boundaries of the output parameters. The stiffness reduction factor (SRF) represents the changes in the stiffness parameter for each element of the structure, as given in:  
\[SRF=1-\frac{\alpha_{d}}{\alpha_{u}}, \tag{8}\]  
where \(\alpha_{d}\) is ESP value of the damaged state and \(\alpha_{u}\) is ESP value of the undamaged state. The intervals of the ESPs, including the natural frequencies and mode shapes, for the undamaged and damaged state, can be formulated as follows [42]:  
\[[\underline{a}]\approx\Big{[}\underline{\lambda};\underline{\phi}\Big{]}= \text{ESP value lower bound}, \tag{9}\]\[[\overline{\alpha}]\approx[\overline{\lambda};\overline{\phi}]=\text{ESP value upper bound}, \tag{10}\]  
Therefore, the interval bounds for each parameter can be derived as:  
\[\lambda_{c}^{I}=\left[\lambda_{c^{I}}^{I}\overline{\lambda_{c}^{I}}\right]= \left\{\lambda_{c1}^{I},\lambda_{c2}^{I},\cdots,\lambda_{ci}^{I}\right\}^{T}, \lambda_{ci}^{I}=\left[\lambda_{c\bar{c}^{I}}^{I}\overline{\lambda_{ci}^{I}} \right], \tag{11}\]  
\[\phi_{c}^{I}=\left[\underline{\phi_{c}^{I}},\overline{\phi_{c}^{I}}\right]= \left\{\left(\phi_{c1}^{I}\right)^{T},\left(\phi_{c2}^{I}\right)^{T},\cdots, \left(\phi_{cj}^{I}\right)^{T}\right\}^{T},\phi_{cij}^{I}=\left[\underline{ \phi_{cji}^{I}},\overline{\phi_{cji}^{I}}\right], \tag{12}\]  
\[\alpha_{c}^{I}=\left[\underline{\alpha_{c}^{I}},\overline{\alpha_{c}^{I}} \right]=\left\{\alpha_{c1}^{I},\alpha_{c2}^{I},\cdots,\alpha_{ck}^{I}\right\}^ {T},\alpha_{ck}^{I}=\left[\underline{\alpha_{ck}^{I}},\overline{\alpha_{ck}^{ I}}\right] \tag{13}\]  
where the subscript \(c\) is the index of damage cases, the subscripts \(i,\ j\) and \(k\) are the number of modes, the number of structural nodes, and the number of segments of structures, respectively. The middle values of the parameters are denoted as:  
\[x^{c}=m(x)=\frac{(\overline{x}+\overline{x})}{2}, \tag{14}\]  
where \(x\) denotes the exact values of the modal parameters (frequencies and mode shapes) and the output parameters (ESPs). The upper and lower bars, respectively, represent the upper and lower boundaries of \(x\). The uncertainties are coupled with the natural frequencies (\(\lambda\)) and mode shapes (\(\phi\)) in terms of the interval bounds. The ESPs are used as variables of the optimization algorithm, and natural frequencies and mode shapes are used as in MDLAC as damage indexes. Thus, two ESMA models, which include the lower bound and upper bound analyses, are provided. The expressions are summarized in Table 1.  
The variable \(\omega\) represents the modal data's level of uncertainty, with varying values for natural frequencies and mode shapes. The \(+\) and \(-\) values of the uncertainty in two separate ESMA models--ESMA 1 and ESMA 2--are used to apply the boundaries (lower and upper bounds) of the input parameters of MDLAC.