## 2 Methodology
### MEMD-Based Identification Methods
#### 2.2.2 MEMD-Based FBFFT (MFBFFT) Method  
Based on [PERSON] theorem, several modal identification methods are proposed to determine the most probable value (MPV) of modal parameters as well as the posterior uncertainties [33]. Notably, this theorem is originally based on the assumption of the white noise excitation and stationary process. Whereas, the wind load or earthquake load, as the wide-band excitation, does not generally satisfy the condition of the stationary white noise process. Fortunately, under this condition, the FBFFT also proves to be credible and robust [34]. The FBFFT method is thus adapted to identify the modal parameters under the nonstationary process in this study.  
In the FBFFT method, the measured acceleration response \(\hat{\mathbf{X}}\) is modeled as the practical part \(\tilde{\mathbf{X}}\) and the error \(\varepsilon\) resulted from the measurement or noise:  
\[\hat{\tilde{\mathbf{X}}}=\tilde{\mathbf{X}}(\mathbf{\theta})+\varepsilon \tag{7}\]  
where \(\mathbf{\theta}\) denotes the modal parameters including the natural frequency \(f\), damping ratio \(\varsigma\), spectrum values of modal force and error.  
Define an augmented vector \(\{\mathbf{Z}_{k}\}=[\mathbf{F}_{k}\,\mathbf{G}_{k}]^{T}\), in which \(\mathbf{F}_{k}\) and \(\mathbf{G}_{k}\) are the real and imaginative part of the FFT of the \(\hat{\tilde{\mathbf{X}}}\) and \(k\) is the number of associated frequencies. By virtue of the [PERSON] theorem, the posterior probability density function of \(\mathbf{\theta}\) given by \(\mathbf{Z}_{k}\) is proportional to the likelihood function.  
\[P(\mathbf{\theta}|\{\mathbf{Z}_{k}\}\,)\propto P(\{\mathbf{Z}_{k}\}|\mathbf{\theta}) \tag{8}\]  
For the recorded response data in a high sampling rate and long duration, \(\{\mathbf{Z}_{k}\}\) is proved to be asymptotically independent and zero mean Gaussian vector. Hence the likelihood function can be given as  
\[P(\,\{\mathbf{Z}_{k}\}|\mathbf{\theta})=(2\beta)^{-(N_{q}-1)/2}\left[\prod_{k=2}^{N_ {q}}det\mathbf{C}_{k}(\mathbf{\theta})\right]^{-1/2}\exp\left[-\frac{1}{2}\sum_{k=2}^ {N_{q}}\mathbf{Z}_{k}^{T}\mathbf{C}_{k}(\mathbf{\theta})^{-1}\mathbf{Z}_{k}\right] \tag{9}\]  
where \(\mathbf{C}_{k}\) is the covariance function of \(\mathbf{Z}_{k}\), referenced to [PERSON] et al. [35]; \(N_{q}\) is the number of frequency point.  
For the purpose of simplifying the procedure of optimization, the negative log-likelihood function is defined as \(L(\theta)\propto-lnP(\theta|\{\mathbf{Z}_{k}\})\) and \(L(\theta)\) can be expressed as  
\[L(\theta)=-ln(P(\{\mathbf{Z}_{k}\}|\theta))=\frac{1}{2}\left[\sum_{k}ln(det \mathbf{C}_{k}(\theta))+\sum_{k}\mathbf{Z}_{k}^{T}\mathbf{C}_{k}(\theta)^{-1} \mathbf{Z}_{k}\right] \tag{10}\]  
The MPV of the modal parameter can be obtained by the minimization of \(L(\theta)\). However the direct usage of Equation (10) is not convenient and efficient until [PERSON] and [PERSON] [34] proposed a fast algorithm for well-separated modes. Meanwhile, the posterior uncertainty can be defined as the coefficient of variation (COV) (coefficient of variation = standard deviation/MPV). Since seismic excitation is a typical non-stationary process, the MEMD is also adopted as a pre-processing tool to the FBFFT method for identifying the modal parameters from the seismic-induced response.