## 2 Materials and Methods
### Damage-Sensitive Features
#### 2.1.3 Energy-Based Indicators  
The wavelet decomposition exposes changes in the energy distribution of a signal in the frequency domain over time, especially in case of non-stationary earthquake signals [76]. The wavelet transform decomposes a time history into a sum of wavelets that are obtained by dilating, scaling and time shifting a mother wavelet. The DSFs investigated in this paper have first been introduced by [PERSON] et al. [50] as robust features. A first DSF, termed \(ME_{o,m}\), tracks the spread of the spectral energy as the ratio of the energy accumulated by wavelet coefficients in the region of the \(m^{th}\) natural frequency and the total energy accumulated by the structure at the DOF \(o\). To reduce the impact of sensor noise, the total energy, \(E_{o}(\psi_{ref})\), is calculated for discrete wavelet coefficients within a reduced frequency bandwidth, \([f_{m}/2,f_{m}]\), which is defined based on the frequency of a structural mode, \(f_{m}\), as shown in Equation (7):\[ME_{o,m}=\frac{E_{o}(\psi_{m})}{E_{o}(\psi_{ref})}\,, \tag{7}\]  
where \(\psi_{m}\) denotes the wavelet coefficients related to the (pseudo-)frequency of the \(m^{th}\) mode, \(\psi_{ref,m}\) refers to the reference frequency bandwidth (typically chosen as the range between half of the natural frequency of the \(m^{th}\) mode, \(0.5\cdot f_{m}\), and the natural frequency, \(f_{m}\)); and \(E\) refers the the energy of the signal, which is derived as the absolute amplitude of the complex wavelet coefficients. To account for the changing energy distribution in the frequency domain that characterizes the seismic ground motion and significantly violates white-noise (WN) assumptions, Equation (7) can be slightly rewritten in relative terms and expressed as a ratio with respect to the same quantity derived for the input signal \(\bar{x}_{i}\):  
\[MER_{io,m}=\frac{E_{o}(\psi_{m})\cdot E_{i}(\psi_{ref,m})}{E_{o}(\psi_{ref,m}) \cdot E_{i}(\psi_{m})}\,. \tag{8}\]  
Another feature that exploits the energy decay in the time domain in order to track the residual stiffness deterioration is \(CE_{io,m}\)[51]. According to Equation (9), \(CE_{io,m}\) is the time-domain centroid of the signal part beyond 95% of the cumulative input energy. At this point, the measured response comprises mainly free vibrations, as the strong shaking part of the earthquake is past, and thus reflects the post-earthquake dynamic characteristics of the studied structure and can be considered independent from the input shaking. Lower frequencies are associated with longer cycles that shift and result in later time values of the time-domain centroid:  
\[CE_{io,m}=\frac{\sum_{b=b_{5}}^{K}E_{f_{m}}(b)\cdot b\cdot t_{s}}{\sum_{b=b_{5 }}^{K}E_{f_{m}}(b)}-t_{95,i}\,, \tag{9}\]  
where \(t_{95},i\) designates the time to reach 95% of the cumulative input energy; \(E_{f_{m}}(b)\) is the output wavelet energy at (pseudo-)frequency \(f_{m}\) at time sample \(b\); \(b_{5}\) is the time sample corresponding to \(t_{95}\), \(i\), \(t_{s}\) is the sampling period and \(K\) reflects the total number of samples that are measured for the earthquake signal. An alternative formulation of a wavelet-based DSF in time domain is \(TS_{io,m}\), which is expressed in terms of the difference between the timespans needed to accumulate from 5 to 95% of the output and input signal, respectively [50]. This DSF is derived using Equation (10):  
\[TS_{io,m}=\frac{(t_{95,o}(\psi_{m})-t_{05,o}(\psi_{m}))-(t_{95,i}(\psi_{m})-t _{05,i}(\psi_{m}))}{t_{95,i}(\psi_{m})-t_{05,i}(\psi_{m})}\,, \tag{10}\]  
where \(t_{x,i}\) refers to the time required to accumulate \(\times\)% of the energy of the input signal and \(t_{x,o}\) is the timespan relating to the output signal. To reduce susceptibility to signal noise, the time for accumulation is calculated for the reconstructed signal of the wavelet coefficient \(\psi_{m}\), corresponding to the \(m^{th}\) mode.