## 3 Results
### Simulated Case Study
#### 3.1.2 Damage Detection and Quantification  
The first level of damage identification consists in detecting the onset of damage (shift in stiffness) or nonlinearity (i.e., regions where the response lies outside the linear elastic range). Damage detection is first executed in a global sense. This implies that the DSFs presented in Section 2.1 are derived for the entire structure, with input acceleration measured at the base and output acceleration acquired at top storey (root) level. This setup is most suitable for low-rise buildings, which are generally dominated by the fundamental vibration mode. If higher modes bear a more prominent role, for instance in high-rise buildings, the output (measured) acceleration can be assigned at an intermediate level, where higher modes contribute further to the deformation. The potential of different DSFs for damage identification is evaluated using four performance indicators (PIs), as illustrated in Figure 5:  
1. The correlation between a DSF and an EDP, such as maximum transient roof displacement or the DI by Park&Ang [87], is defined using Equation (19): \[DI_{PA}=\frac{d_{max}-d_{y}}{d_{u}-d_{y}}+\beta\frac{E_{h}}{d_{u}V_{y}}\,,\] (19) where \(d_{max}\) is the absolute maximum transient displacement the structure undergoes during an earthquake, \(d_{u}\) is the ultimate displacement capacity of the structure, \(d_{y}\) is the idealized yield displacement of the structure, \(E_{h}\) refers to the hysteretic energy that is dissipated by the structure, \(V_{y}\) is the idealized yield force, and \(\beta\) is a weighting factor that characterizes the influence of energy dissipation on the overall damage. Herein, it is assumed that \(\beta=0.15\). 2. The capacity of a DSF to distinguish between building responses of linear buildings (\(DI_{PA}<0.025\)) and buildings that sustained significant nonlinearity (\(DI_{PA}>0.50\)). This is quantified as the probability of exceeding the \(DI_{PA}\) thresholds given a DSF value, \(p(di\geqslant DI\mid DSF)\), which is computed using the so-called _IM-based_ method, as described by [PERSON] [88] and illustrated in Figure 5a,b. The efficacy in distinguishing between separate damage states is evaluated as the probability of exceeding a \(DI_{PA}\) of \(0.50\), \(p(di\geqslant 0.50\mid DSF)\) for the DSF value that produces a probability of \(90\%\) to exceed \(DI_{PA}=0.025\). 3. The third PI evaluates the probability of erroneous damage detection. To this end, the number of GMs, for which the detection threshold (derived using reference AV data) is exceeded, is calculated. This procedure is described in Figure 5c. 4. Finally, the fourth PI is related to the minimum value of EDP/\(DI_{PA}\), for which the DSF detects damage (see Figure 5d). This PI is evaluated as the mean EDP/\(DI_{PA}\) value, at which the DSF crosses the detection threshold. Similarly to the previous PI, this is defined using reference (\"healthy\") AV data. In both cases, the detection threshold is set to the \(DSF^{h}\) value corresponding to a cumulative probability of \(0.95\) for the healthy reference distribution. TAC values are derived for a specific frequency bandwidth, which influences the results, as described in Appendix Figure A1. The results of Figure 5 are obtained for the full bandwidth (\(fb=4\)-\(25\) Hz) that is tightly defined around the first and second mode. In addition, the TAC is obtained by averaging the values of four-second windows and a time-bandwidth parameter of the Morse wavelet set to \(100\) (see Appendix B Figure A1). For the robust detection of damage or tracking of the manifestation of nonlinearities during earthquake loading, the DSFs should scale with the amount of nonlinearity sustained by the structure and be invariant to the characteristics of the earthquake excitation. This forms a challenge as increasing shaking amplitudes naturally lead to higher amounts of nonlinearity, and an often-biased correlation to earthquake amplitudes may occur.