# Automated Modal Analysis Using Stochastic Subspace Identification and Field Monitoring Data
## 4 Examples of Field Applications
### Verification Result Using Seven-Story RC Frame in Taichung  
The first dataset was collected from the RC building in Taichung, Taiwan, through its conventional SHM system. The SHM system includes 29 channels of accelerometers, as shown in Figure 4a; however, only the structural responses in the longitudinal and transverse directions were exploited to identify modal parameters. As a result, channels 12, 13, 14, 15, 18, 19, 21, and 22 were used in the following verification. Although it is not  
Figure 4: The seven-story and four-story RC frames used for verifying the proposed procedure. **(a)** the side view and the accelerometers of the first dataset; **(b)** the side view and the accelerometers of the second dataset. shown in the figure, channels 1, 2, and 3 are free-field sensors, which represent ground motions and facilitate the discrimination of nonstationary excitations. Consequently, the stationary parts can be treated as ambient excitations, and the structural responses between this period can be analyzed by SSI. First, in the data preprocessing step, waveforms that were clearly distorted or exhibited dropouts or glitches were excluded from the measurements. The normalized Arias intensity and AIC were sequentially implemented to identify the arrival of P-waves, as well as to determine stationary excitations, as shown in Figure 2a. Then, FFT was applied to the acceleration responses with a 200 Hz sampling rate to provide an overview of the structure's modal information, as shown in Figure 5. This example uses measurements collected on 21 September 1999 at 17:57:16 local time. It is worth noting that these measurements do not relate to the mainshock but rather the aftershock, by which point the seven-story RC frame had already undergone damage. After careful examination, the fundamental frequency of the seven-story RC building was about 3.2 Hz before the Chi-Chi earthquake and 2.3 Hz after the destructive earthquake. Moreover, the structure's highest mode of interest was less than 20 Hz, so the acceleration responses can be down-sampled to 50 Hz--which corresponds to the Nyquist frequency of 25 Hz--to increase analysis efficiency. The next step is matrix construction and decomposition. According to Equation (14), the number of rows, \(i\), can be selected as 8 (3.2 Hz) and 11 (2.3 Hz) and their multiples to secure the complete cycles of the lagged vectors for a clear decomposition. For example, Figure 6 shows the comparison of the normalized differences between the singular pairs with respect to different strategies. Clearly, the choice of the size of the Hankel matrix can affect the resultant decomposition performed by SVD, and the conventional method of selecting sequential numbers likely results in an incomplete decomposition, especially for the first mode, as shown in Figure 6a. Contrarily, the proposed procedure, according to the above-mentioned equation, can generate an elaborate isolation of every mode, providing well-separated singular pairs, as shown in Figure 6b. As stated in the previous section, the selection of system orders, \(n_{e}\), is important when realizing the system in the final step. Figure 7 illustrates the results of the four approaches for estimating system orders through singular values. In this example, the size of the Hankel matrix is assigned as 22 to ensure differentiation of energy. Correspondingly, the criteria from \(C_{1}\) to \(C_{4}\), defined in Equation (15) to Equation (18), are 0.05, 0.01, 0.7, and 5, respectively, and the final selections of all approaches come to 11. The odd number  
Figure 5: The Fourier spectrum of the structural responses measured from the seven-story RC frames. resulting from the selection actually indicates that at least one spurious pole is extracted in the following system realization. By assigning the system order as 11, Figure 8 depicts the stability diagram of the selected \(i\) together with the Fourier spectrum. Clearly, the identified frequencies are highly  
Figure 6: The comparison of normalized differences between the singular pairs.