# Hilbert-Huang-Transform-Based Seismic Intensity Measures for Rocking Response Assessment
## 2 Methods  
### Intensity Measures  
#### 2.1.1 HHT-Based IMs  
HHT is a powerful signal processing technique used to analyze complex, nonstationary, and nonlinear data sets, such as earthquake ground motion records [48]. The whole process results in the amplitude-frequency-time distribution of the signal.  
The HHT algorithm is a two-step procedure consisting of (a) the empirical mode decomposition (EMD) and (b) the Hilbert spectral analysis (HSA). Performing the EMD algorithm, the original signal X(t) is decomposed into a finite set of intrinsic mode functions (IMFs) as follows:  
\[\mathrm{X(t)}=\sum_{j=1}^{n}\varsigma_{j}(t)+r_{n} \tag{1}\]  
where n is the number of IMF, \(\varsigma_{j}(t)\) the IMF of mode j, and \(r_{n}\) the residual, which is a monotonic function. Subsequently, the Hilbert transform \(y_{j}(t)\) is applied to each IMF:  
\[y_{j}(t)=\frac{p}{\pi}\int\limits_{-\infty}^{\infty}\frac{c_{j}(\tau)}{t-\tau} dt \tag{2}\]  
where p refers to the Cauchy principal value. Thus, the analytical signal \(z_{j}(t)\) of an IMF is defined as  
\[z_{j}(t)=c_{j}(t)+y_{j}(t)i=a_{j}(t)e^{i\theta j(t)} \tag{3}\]  
where \(a_{j}(t)\) and \(\theta_{j}(t)\) are the instantaneous amplitude and phase, respectively, defined as follows:  
\[a_{j}(t)=\sqrt{c_{j}^{2}(t)+y_{j}^{2}(t)}\text{ and }\theta_{j}(t)=\tan^{-1} \Big{(}y_{j}(t)/c_{j}(t)\Big{)} \tag{4}\]  
The instantaneous frequency f(t) of each IMF is derived as follows:  
\[f_{j}(t)=\frac{1}{2\pi}\frac{d\theta_{j}(t)}{dt} \tag{5}\]  
Finally, the initial signal can be reformed as follows:  
\[X(t)=\sum_{j=1}^{n}a_{j}(t)\cos\left(\int 2\pi f_{j}(t)dt\right) \tag{6}\]In the present study, similar to [PERSON] et al. [50; 51; 52], 12 new HHT-based ground motion IMs are proposed. Primarily, the first parameter is defined by the volume V\({}_{\rm{HHT}}\) of each spectrum given by the following equation:  
\[\rm{V}_{\rm{HHT}}=\int_{0}^{\rm{max}}\int_{0}^{\rm{max}}E(f,t)dtdf \tag{7}\]  
where E(f, t) = a\({}^{2}\)(f, t) signifies the instantaneous energy, which corresponds to the instantaneous frequency f at a time equal to t, and f\({}_{\rm{max}}\) and t\({}_{\rm{max}}\) are the maximum instantaneous frequency and the total duration of the analytical signal, respectively.  
The upper surface S\({}_{\rm{HHT}}\) of each Hilbert spectrum is considered as the second IM, defined as follows:  
\[\rm{S}_{\rm{HHT}}=\int\limits_{0}^{\rm{max}}\int\limits_{0}^{\rm{max}}\sqrt{1+ \left(\frac{\partial E(f,t)}{\partial f}\right)^{2}+\left(\frac{\partial E(f,t )}{\partial t}\right)^{2}}dtdf \tag{8}\]  
Moreover, as additional parameters, the maximum and mean values of instantaneous energy and their ratio are considered:  
\[\rm{A}_{\rm{max,HHT}}=\rm{max}(E(f,t)) \tag{9}\]  
\[\rm{A}_{\rm{mean,HHT}}=\rm{mean}(E(f,t)) \tag{10}\]  
\[\rm{A}_{\rm{ratio,HHT}}=\rm{\frac{A_{\rm{mean,HHT}}}{A_{\rm{max,HHT}}}} \tag{11}\]  
Subsequently, the volume V\({}_{\rm{Pos,HHT}}\), as well as the upper surface S\({}_{\rm{Pos,HHT}}\) of the segment of Hilbert spectrum above the horizontal plane that intersects with the point of mean amplitude are also evaluated:  
\[\rm{V}_{\rm{Pos,HHT}}=\int_{0}^{\rm{max}}\int_{0}^{\rm{max}}E(f,t)dtdf\;where \;E\geq E_{\rm{mean}} \tag{12}\]  
\[\rm{S}_{\rm{Pos,HHT}}=\int\limits_{0}^{\rm{max}}\int\limits_{0}^{\rm{max}} \sqrt{1+\left(\frac{\partial E(f,t)}{\partial f}\right)^{2}+\left(\frac{ \partial E(f,t)}{\partial t}\right)^{2}}dtdf\;where\;E\geq E_{\rm{mean}} \tag{13}\]  
Finally, the following additional IMs are defined as a combination of the aforementioned parameters:  
\[\rm{A}_{\rm{HHT}}=\rm{\frac{V_{\rm{HHT}}}{S_{\rm{HHT}}}} \tag{14}\]  
\[\rm{SA}_{\rm{mean}}=\rm{S}_{\rm{HHT}}\rm{A}_{\rm{mean,HHT}} \tag{15}\]  
\[\rm{VA}_{\rm{mean}}=\rm{V}_{\rm{HHT}}\rm{A}_{\rm{mean,HHT}} \tag{16}\]  
\[\rm{VA}_{\rm{max}}=\rm{V}_{\rm{HHT}}\rm{A}_{\rm{max,HHT}} \tag{17}\]  
\[\rm{A}_{\rm{pos,HHT}}=\rm{\frac{V_{\rm{pos,HHT}}}{S_{\rm{pos,HHT}}}} \tag{18}\]