Energy-Based Analysis of Mechanisms of Earthquake-Induced Landslide Using Hilbert–Huang Transform and Marginal Spectrum

Abstract

Based on the Hilbert–Huang Transform and its marginal spectrum, an energy-based method is proposed to analyse the dynamics of earthquake-induced landslides and a case study is presented to illustrate the proposed method. The results show that the seismic Hilbert energy in the sliding mass of a landslide is larger than that in the sliding bed when subjected to seismic excitations, causing different dynamic responses between the sliding mass and the sliding bed. The seismic Hilbert energy transits from the high-frequency components to the low-frequency components when the seismic waves propagate through the weak zone, causing a nonuniform seismic Hilbert energy distribution in the frequency domain. Shear failure develops first at the crest and toe of the sliding mass due to resonance effects. Meanwhile, the seismic Hilbert energy in the frequency components of 3–5 Hz, which is close to the natural frequency of the slope, is largely dissipated in the initiation and failure processes of the landslide. With the development of dynamic failure, the peak energy transmission ratios in the weak zone decrease gradually. This study offers an energy-based interpretation for the initiation and progression of earthquake-induced landslides with the shattering-sliding failure type.

Appendix

The EMD will reduce the data into a collection of IMFs defined as functions which satisfy the following conditions:

• 1. The number of extreme values and the number of zero-crossings must either equal or differ at most by one in the whole data set, and

• 2. The mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero at any point.

After the EMD, the original signal a(t) is decomposed into n IMF components c _j and a residual r _n , which can either be the mean trend or a constant. The original data can be expressed as

$$a(t) = \sum\limits_{j = 1}^{n} {c_{j} + r_{n} } .$$

(1)

An EMD example of the original El Centro earthquake, which yields seven IMF components, is illustrated in Fig. 15.

Fig. 15

Original El Centro earthquake wave and EMD results

Having obtained the IMF components, one will have no difficulty in applying the Hilbert transform to each IMF component, and computing the instantaneous frequency of each IMF component according to the Hilbert transform. For an arbitrary time series X(t), its Hilbert transform Y(t) can be defined as:

$$Y(t) = \frac{1}{\pi }P\int_{ - \infty }^{\infty } {\frac{{X(t^{{\prime }} )}}{{t - t^{{\prime }} }}} {\text{d}}t^{{\prime }} ,$$

(2)

where P represents the Cauchy principal value, t represents time, with X(t) and Y(t) forming the complex conjugate pair. An analytical signal Z(t) can be defined as:

$$Z(t) = X(t) + iY(t) = a(t)e^{i\theta (t)} .$$

(3)

In which $a(t)$ and $\theta (t)$ are defined as:

$$a(t) = [X^{2} (t) + Y^{2} (t)]^{1/2} ,$$

(4)

$$\theta (t) = \arctan [Y(t)/X(t)],$$

(5)

where $\theta (t)$ is the instantaneous phase, and the instantaneous frequency $\omega$ is defined as:

$$\omega = \frac{{{\text{d}}\theta (t)}}{{{\text{d}}t}}.$$

(6)

The instantaneous frequencies of the IMFs of the original El Centro earthquake wave are calculated and illustrated in Fig. 16. After performing the Hilbert transform on each IMF component, the original data can be expressed as the real part, RP, in the following form:

$$X(t) = {\text{RP}}\sum\limits_{j = 1}^{n} {a_{j} (t)} \exp \left[ {i\int {\omega_{j} (t){\text{d}}t} } \right].$$

(7)

Fig. 16

Instantaneous frequencies of the IMFs of the original El Centro earthquake wave

This frequency–time–amplitude distribution in Eq. (7) is designated as the Hilbert amplitude spectrum, H(ω, t), or simply the Hilbert spectrum. The Hilbert spectrum of the original El Centro earthquake wave is computed and shown in Fig. 17. The Hilbert spectrum, i.e. Eq. (7), shows that the Hilbert energy and instantaneous frequency of the original signal are functions of time t in a three-dimensional plot, in which the Hilbert energy can be contoured on the frequency–time plane.

Fig. 17

Hilbert spectrum of the original El Centro earthquake wave

With the Hilbert spectrum defined, the marginal spectrum, $h(\omega )$, can be defined as

$$h(\omega ) = \int_{0}^{T} {H(\omega ,t)} {\text{d}}t.$$

(8)

The marginal spectrum of the original El Centro earthquake wave is shown in Fig. 18.

Fig. 18

Marginal spectrum of the original El Centro earthquake wave