## 2 Damage Identification Approach  
### Damage Identification Index  
Practically, structural modal results can be available from measurements or tests. Thus, modal results are usually regarded as the structural damage indices for damage identification [47]. The equation for modal analysis can be described as follows:  
\[\left(\mathrm{K}-\omega_{i}^{2}\mathrm{M}\right)\varphi_{i}=0 \tag{1}\]  
where K and M are the structural overall stiffness matrix and mass matrix, respectively. \(\omega_{i}\) and \(\varphi_{i}\) represent the \(i\)th order structural frequency and the corresponding normal mode, respectively. When \(\varphi_{i}^{T}\) times both sides of Equation (1), we can get the following expression due to the orthogonality of mode shape:  
\[\omega_{i}^{2}=\varphi_{i}^{T}\mathrm{K}\varphi_{i}/\varphi_{i}^{T}\mathrm{M} \varphi_{i} \tag{2}\]  
Suppose the difference in the structural stiffness between the intact structure and the damaged structure is \(\Delta\mathrm{K}\), and the corresponding structural square frequency and mode will change by \(\Delta\omega_{i}^{2}\) and \(\Delta\varphi_{i}\), respectively. The equation can be reformulated as:  
\[\left(\mathrm{K}+\Delta\mathrm{K}-(\omega_{i}^{2}+\Delta\omega_{i}^{2})M)( \varphi_{i}+\Delta\varphi_{i})=0 \tag{3}\]  
According to Equation (3), the structural damage \(\Delta\mathrm{K}\) will result in a change in the structural square frequencies and normal modes. Thus, both the frequencies and the modes can be considered in damage indices. Ignore the higher order quantity, and Equation (3) can be simplified as:  
\[\Delta\omega_{i}^{2}=\varphi_{i}^{T}\Delta\mathrm{K}\varphi_{i}/\varphi_{i}^{ T}\mathrm{M}\varphi_{i} \tag{4}\]  
The square frequency of the damaged structure and the intact structure are defined as \(\omega_{d}^{2}\) and \(\omega_{u}^{2}\), respectively. The relative difference of the square frequency between the intact structure and the damaged structure can be described as follows:  
\[RSF_{i}=\frac{\omega_{ui}^{2}-\omega_{di}^{2}}{\omega_{ui}^{2}}=\frac{\Delta \omega_{i}^{2}}{\omega_{ui}^{2}}\approx\frac{\varphi_{i}^{T}\Delta\mathrm{K} \varphi_{i}}{\varphi_{i}^{T}\mathrm{K}\varphi_{i}} \tag{5}\]  
where \(RSF_{i}\) is the relative difference of the \(i\)th order square frequency between the intact and the damaged structures; it is almost proportional to the difference in the structural stiffness between the intact and damaged structures. Thus, it will be regarded as a damage index in the following. In addition, according to Equation (3), we know that structural damage will result in modal changes. So the modal differences between the intact and damaged structures can also be utilized in damage indices. The modal difference can be calculated as follows:  
\[\Delta\varphi_{i}=\varphi_{di}-\varphi_{ui} \tag{6}\]  
where \(\varphi_{di}\) and \(\varphi_{ui}\) are the \(i\)th order modes for the damaged structure and the intact structure, respectively. It is easier to identify structural damage in high-order modes than in low-order modes [48], but in practice, it is often difficult to obtain high-order modal results precisely, and measurement errors may lead to large output errors after calculation. Thus, this study assumes that the lowest six orders of the modal results in the \(x\)-, \(y\)-, and \(z\)-directions are available for more convincing calculation results. For marine structures, sensors can be placed at specified nodal locations below the waterline and provide measurements. These measurements are incomplete and can only provide part of the modal results which are called partial modal results in the following. The partial modal results are still difficult to handle directly because of the large quantity of data. If \(\alpha\) sensors are distributed at \(\alpha\) nodal positions of the submerged marine structure, and each sensor can collect \(\beta\) orders of the modal results in three directions, then there are \(\alpha\times\beta\times 3\) damage indices which are too heavy for damage identification purposes.