## 3 Extraction of the Ground Vibration Components
### Comparison of EMD and MEEMD Based Long Period Component Extraction  
The HHT is a time-frequency domain analysis method for non-stationary signals [18; 19; 20]. Compared with the commonly used analog signal processing methods, it is completely free from the constraints of linearity and smoothness, completely adaptive, and not subject to [PERSON]'s uncertainty principle. The HHT transform is divided into two steps: first, the signal is subjected to Empirical Mode Decomposition (EMD); then, the decomposed intrinsic mode function is subjected to Hilbert Spectral Analysis (HSA). The EMD decomposition process adopts the \"sieve\" algorithm, which is mainly divided into three steps:  
Step 1. Determine all the local extreme points of the ground vibration signal \(x(t)\) and connect all the locally very large and very small values with a cubic spline to form the upper envelope \(x_{max}(t)\) and the lower envelope \(x_{min}(t)\), respectively. Step 2. Find the mean value of the upper and lower envelopes, denoted as \(m_{11}(t)\), and subtract this mean value from the original ground vibration time-range curve \(x(t)\) to obtain the new data series \(h_{11}(t)\). \[m_{11}(t)=[x_{max}(t)+x_{min}(t)]/2 \tag{2}\]  
\[h_{11}(t)=x(t)-m_{11}(t) \tag{3}\]  
Determine whether \(h_{11}(t)\) contains the two conditions of IMF; if not, repeat the above process with \(h_{11}(t)\) as the original data until the new data sequence is obtained. \[h_{1k}(t)=h_{1(k-1)}(t)-m_{1k}(t) \tag{4}\]  
The satisfies the two conditions of the IMF, which gives the first IMF component \(c_{1}(t)\):  
\[c_{1}(t)=h_{1k}(t) \tag{5}\]  
Step 3. Remove \(c_{1}(t)\) from the in-situ vibration \(x(t)\) to obtain the remaining sequence \(r_{1}(t)\). \[r_{1}(t)=x(t)-c_{1}(t) \tag{6}\]  
Take this as the new signal sequence, and follow the above steps to obtain the second, third,..., up to the nth intrinsic mode function; the EMD decomposition is finished when the  
Figure 7: Ground motion acceleration response spectra: (**a**) ordinary ground shaking acceleration response spectra; (**b**) far-field long-period ground shaking acceleration response spectra; (**c**) near-field long-period ground shaking acceleration response spectra. residual is a monotonous function of time or is less than a certain predetermined value. The original ground vibration time profile can be expressed as the sum of the IMF components and the final residuals, i.e.,  
\[x(t)=\sum_{j=1}^{n}c_{j}(t)+r_{n}(t) \tag{7}\]  
HSA is performed for each IMF component to obtain the corresponding Hilbert spectrum, which is summed to obtain the Hilbert spectrum of the original signal \(H(\omega,t)\). \[H(\omega,t)=Re\!\!\sum_{j=1}^{n}a_{j}(t)e^{i\int\omega_{t}(t)dt} \tag{8}\]  
Although EMD methods have been widely used in various fields, there are some drawbacks, one of which is mode aliasing. Modal aliasing refers to time scales with similar characteristics between IMFs. Mode aliasing tends to cause confusion in the time-frequency distribution. In order to eliminate the mode aliasing in the EMD, [PERSON] et al. [21] proposed an integrated empirical mode decomposition (EEMD) method by introducing white noise into the original signal. Different white noises are added to the original signal for decomposition, and the final IMF components are obtained by averaging the results of multiple decompositions. [PERSON] et al. [22] proposed a complementary coefficient-integrated empirical modal decomposition (CEEMD) to solve the problem of the poor completeness of the EEMD. This method adds two opposite white noise signals to the original signal and then averages the decomposition results overall to eliminate the effect of white noise. The CEEMD can reduce the reconstruction error caused by white noise, while ensuring that the decomposition effect is equivalent to the EEMD.