## 3 Numerical Assessment of the New Methodology Performance  
The validation of the efficacy and performance of the proposed methodology for identifying and quantifying the modal parameters (NF and DR), the phase I of the benchmark problem developed by the IASC-ASCE structural health monitoring task group is used [51]. It consists of a FE-based code of a four-story 2 \(\times\) 2 bay 3D steel frame constructed at the University of British Columbia (UBC), located at Vancouver, the British Columbia, Canada. Figure 4 shows the overall dimensions of the structure and the location of the sixteen sensors used to measure vibration data during the excitation. The structure was excited using broadband input signals applied in both horizontal directions (\(x\) and \(y\)). Using a sampling frequency of 1000 Hz, 20 s of the structure response were acquired resulting in 20,000 samples. Every mode has a damping ratio of 1%. The measured vibration response of sensors 7 (\(x\)-direction) and 6 (\(y\)-direction) are selected to carry out the analysis. Figure 5 depicts the acquired signals in the \(x\)- and \(y\)-directions by the abovementioned sensors, respectively. Following the proposed methodology, which is implemented in MATLAB, the measured vibration signals of sensors 7 and 6 are converted to their free-decay response employing NExT method. Recalling from Section 2, the method uses reference signals. [PERSON] [52] suggests that these signals should be far away of the in-test ones. In this regard, sensors 15 (\(x\)-direction) and 14 (\(y\)-direction) fulfill with the aforementioned suggestion and, hence, they are used as the references. In this example, only 2000 samples of obtained signal are selected because they allow obtaining the free-decay response  
Figure 4: Benchmark structure used and the location of the used sensors. Figure 5: Measured signals in the: (**a**) \(x\)- and (**b**) \(y\)-directions. Table 2 summarizes modal parameters calculated using the proposal and the ones estimated by the FEA model. Additionally, the estimated natural frequencies and damping ratios using the methodologies proposed by [PERSON] et al. [34], [PERSON] [35], and [PERSON] et al. [11] are presented. In general, the results presented in this table show that the proposal can identify and estimate the first eight modes detect by the FEM model in an accurate way. In particular, the NF value is very close to the theoretical one, as the maximum deviation is found to be about 0.5%. These results show that the objective function, described by Equation (12), can be considered as an adequate maximization function. On the other hand, the results obtained by [PERSON] et al. [11] and [PERSON] et al. [34] show that the same eight NF are also found with their proposals. It is worth noting that the former uses the complex mode indicator function as a signal processing tool; however, the maximum deviation was found to be about of 23%, indicating that this function may not be able to deal with noisy and time-variant signal properly. On the contrary, the results estimated by the former show that this methodology can be considered as accurate, as the deviations are quite similar to the ones obtained by [PERSON] et al. [11]. Figure 8: Convergence graphs of the: (**a**) natural frequency; (**b**) damping ratio; and (**c**) correlation value for the \(y\)-direction modes. Figure 9: Estimated modes by the GA-based approach in the: (**a**) \(x\)- and (**b**) \(y\)-directions. to the ones obtained by the proposal. Even when the maximum deviation (0.33%) is lower than the one obtained with this proposal, more NF modes have a significant lower deviation compared with the aforementioned methodology. The work reported by [PERSON] and [PERSON] [35] use the sparse component analysis, a blind source separation technique, to analyze the response. It is evident that they cannot estimate the third NF of the \(y\)-direction. By the presented results, it is seen that this methodology cannot be considered as accurate, as the maximum deviation is higher than 10%.