## 6 Damage Detection of IASC-ASCE Structural Health Monitoring Benchmarks
### First PhaseComparing the Performance of PoDE and PDE  
In order to validate the proposed method, a comparison between PoDE and PDE indices is performed, as shown in Table 3. PDE values have been calculated based on the probabilistic surrogate method presented by [PERSON] et al. [49]. The acceleration response of full benchmark models with 12 and 120 DOF is used for damage cases 4 and 5. To consider the measurement noise, the acceleration responses of the structure are contaminated with Gaussian white noise (WGN). For the noise levels considered, the signal-to-noise ratios (SNRs) is: (i) 5%, SNR 26 dB. The SNR is expressed as:  
\[SNR=20\log_{10}\biggl{(}\frac{1}{n}\biggr{)}. \tag{20}\]  
where \(n\) is the noise level.  
Figure 6: Floor Plan for Benchmark Structure.  
As shown in Table 3, the PoDE is a more accurate damage index, which results in generating smaller errors in both damage cases 4-5. For example, in damage case 4-pattern 4, the braces of Y direction in story 1 are undamaged, however, the PDE value is 8% compared to a 0% PoDE value. The same pattern occurred in damage case 5-pattern 5, where the undamaged braces of Y direction in story 1 show 0% damage, and the probabilistic scheme indicates a 13% probability of damage. It is also indicated that in both cases, the proposed approach provide higher PoDE values compared to the PDE value for the damaged story. The key reason for the higher accuracy of the proposed method in identifying damage elements is the smaller number of surrogate models used in this approach, compared with the probabilistic method presented in previous studies, which in turn results in reducing the prediction error.  
Furthermore, as shown in Table 3, all of the actually damaged elements can be successfully identified by the proposed method. Note that \(\theta_{i,y}=\theta_{i,+y}=\theta_{i,-y}\) and \(\theta_{i,x}=\theta_{i,+x}=\theta_{i,-x}\) in this Phase.  
Table 4 shows the results of comparing between PoDE, PDE, and direct FE model in terms of computational cost and accuracy. To compute process time when using a surrogate model, data generation time, training and testing time, and WPD implementation time are considered together (core(tm) i7 2.67 GHz CPU).  
It can be concluded from Table 4 that the idea of using a WWLS-SVM model as a surrogate of FE model substantially reduces the computation time of uncertainty based damage detection and maintaining the acceptable detection accuracy. Furthermore, as stated before only two WWLS-SVM models are needed to predict the lower and the lower bounds of ESP. So employing fewer surrogate models in non-probabilistic method reduces the overall prediction error as shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Damage Case**} & \multirow{2}{*}{**Damage**} & \multirow{2}{*}{**Patten**} & \multirow{2}{*}{**Story 1**} & \multirow{2}{*}{**Story 2**} & \multirow{2}{*}{**Story 3**} & \multirow{2}{*}{**Story 4**} \\ \cline{3-3} \cline{5-10}  & & & & & & & & & \\ \cline{5-10}  & & & & & & & & & \\ \hline \multirow{4}{*}{Target} & 1 & 100 & 100 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\  & 2 & 100 & 100 & 0.00 & 0.00 & 100 & 100 & 0.00 & 0.00 \\ Damage & 3 & 0.00 & 100 & 0.00 & 0.00 & 0.00