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3rd Reading March 5, 2013 8:5 WSPC/1793-5369 244-AADA

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Advances in Adaptive Data Analysis Vol. 4, No. 3 (2012) 1250023 (25 pages) c World Scientific Publishing Company DOI: 10.1142/S1793536912500239

EMPIRICAL MODE DECOMPOSITION OF EARTHQUAKE ACCELEROGRAMS

S. T. G. RAGHUKANTH and S. SANGEETHA Department of Civil Engineering IIT Madras Received Revised Accepted Published This article analyzes the strong motion records of past earthquakes by empirical mode decomposition (EMD) technique. The recorded earthquake acceleration time histories are decomposed into a finite number of empirical modes of oscillation. The instantaneous frequency and amplitude of these modes and evolutionary power spectral density (PSD) is estimated from the Hilbert–Huang transform (HHT). Strong motion parameters such as spectral and temporal centroid, spectral and temporal standard deviation, Arias intensity, correlation coefficient of frequency and time are derived from the evolutionary PSD. The variation of these parameters with magnitude, distance and shear wave velocity of the recording station is reported. Empirical equations to estimate these six ground motion parameters are derived from the strong motion data by regression analysis. These equations can be used by engineers to estimate the design ground motion. Keywords: Empirical mode decomposition (EMD); Hilbert–Huang transform (HHT); intrinsic mode function (IMF); instantaneous frequency (IF); instantaneous amplitude; evolutionary power spectral density (PSD); strong motion parameters; ground motion prediction equations (GMPE).

1. Introduction The starting point in earthquake engineering and seismology is the modeling of the available strong motion data. A large number of records are available from the strong motion arrays operating in seismically active regions of the world. These records contain detailed information about the earthquake source process, the propagating medium and the local site condition. Sophisticated mechanistic models where in the complex earth structure and fault geometry can be included have been used to understand the strong motion data [Komatitsch and Tromp (2002a); Raghukanth and Iyengar (2008)]. These models are widely used to simulate ground motion during past earthquakes [Hartzell et al. (2011); Raghukanth (2008); Raghukanth and Bhanu Teja (2011)]. Given the rupture details and properties of the medium, these models can simulate ground motion reliably. These models would be an obvious choice to estimate ground motion in seismic hazard 1250023-1


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S. T. G. Raghukanth & S. Sangeetha

analysis, but uncertainties exist in choosing inputs regarding fault rupture process for future earthquakes. To cover all possible peculiarities in the rupture process, a large number of simulations are required to estimate ground motion realizations of scenario events which make these mechanistic models computationally expensive. The geometric details about faults and precise material properties are not available for many regions in the world. To circumvent these difficulties, development of empirical models is also pursued along with the mechanistic models to estimate ground motion. These models require very few input parameters such as magnitude, distance and local soil condition which can be fixed easily for a site. Several parameters required for engineering purposes has been identified from the earthquake acceleration time histories. These parameters also known as strong motion parameters characterize the ground motion. The most popular measures are peak ground acceleration (PGA), peak ground velocity (PGV) and Spectral acceleration (Sa) respectively. Ground motion prediction equations (GMPE) have been developed to obtain these measures based on the recorded strong motion data [Boore and Atkinson (2008)]. These empirical equations are routinely used in linear structural analysis. Nonlinear dynamic analysis demands acceleration time histories valid in all frequency ranges. For this purpose, stochastic process models for generating samples of acceleration time histories have also been developed [Iyengar and Iyengar (1969); Shinozuka and Sato (1967); Boore (1983)]. In these approaches, first a stationary Gaussian process of known power spectral density (PSD) is simulated. To account for temporal nonstationarity, the simulated time history is multiplied with a deterministic modulating function. The spectral nonstationarity is ignored in these approaches. It is well known that earthquake accelerograms have time varying frequency content due to different arrival times of various seismic waves. The primary, shear, surface and coda waves have different frequency content. The spectral nonstationarity has a substantial effect on the structural response [Papadimitriou (1990); Conte (1992)]. Due to these reasons, models which can include both temporal and frequency nonstationarity have also been developed in the past [Sabetta and Pugliese (1996); Conte and Peng (1997); Pousse et al. (2006); Rezaeian and Der Kiureghian (2010)]. In these models, the evolutionary PSD is computed from short-time Fourier transform (STFT) or wavelet transform. A functional form is assumed for the evolutionary PSD and the constants are estimated by regression analysis. The spectral analysis of earthquake accelerograms in these approaches traditionally relies on the Fourier decomposition of the processes. It is well known that for nonstationary processes whose duration is short and amplitude and frequency contents change with time, Fourier based approaches can yield distorted information. Huang et al. [1998] have shown that adaptive basis functions are required to understand nonlinear and nonstationary data. In this paper, the empirical mode decomposition (EMD) technique combined with Hilbert–Huang transform (HHT) is used to model strong motion data of past events. Important parameters of the evolutionary PSD are estimated from Hilbert spectral analysis. 1250023-2


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Empirical Mode Decomposition of Earthquake Accelerograms

2. Ground Motion Data A good quality strong motion data is a prerequisite to develop any model. Strong motion records can now be accessed easily over the Internet. There are several organizations which distribute strong motion records. These strong motion databases can be accessed through World Wide Web operated by Pacific Earthquake Engineering Research center (PEER, http://peer.berkeley.edu) or through the consortium of organizations for strong motion observation systems (http://db.cosmos-eq.org). The PEER strong motion database has been updated and expanded with records of all active regions in the world. This database also known as Next Generation of Ground-Motion Attenuation Models (NGA) project data is available from (http://peer.berkeley.edu/nga/). This database consists of 3,551 strong motion records generated by 173 shallow crustal events. There are total of 1,456 recording stations. The NGA strong motion database is shown in Fig. 1 as a function of magnitude and rupture distance. It can be observed that the moment magnitude (Mw ) of the events lie in between Mw 4.2 and 7.9. The closest distance to rupture (Rrup ) varies from 0.01 km to 200 km where as focal depth of the events lie in between

Fig. 1.

NGA data as a function of rupture distance (Rrup ). 1250023-3


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0–81 km. The local soil condition of the recording stations characterized in terms of average shear wave velocity in the top 30 m (Vs30 ) varies from 116 m/s to 2016 m/s, respectively. Since horizontal components are widely used in seismic design, both the fault normal and fault parallel components are used in this study to estimate ground motion parameters.

3. Empirical Mode Decomposition The Hilbert–Huang transform proposed by Huang et al. [1998] has emerged as a popular method in analyzing the nonstationary and nonlinear data. HHT consists of empirical mode decomposition (EMD) and Hilbert transform. The EMD decomposes the signal into a finite and often small number of empirical modes called intrinsic mode functions (IMFs). The main difference between HHT and the Fourier-based methods or the wavelet method is that HHT does not use predetermined function forms and thus is adaptive to data. An IMF is defined as a time series where number of extrema and number of zero crossings must either be equal or differ by one. Further the mean value of the envelopes defined by the local maxima and minima of the IMF should be zero. The obtained IMF is a narrow band time series with slowly varying frequency and amplitude. After its introduction by Huang et al. [1998], EMD-HHT has found lot of applications in earthquake engineering. Huang et al. [2001] and Zhang et al. [2003a] demonstrated the superiority of the HHT over the conventional Fourier approaches to analyze earthquake acceleration time histories. Wen and Gu [2009] and Pandey et al. [2011] used HHT to generate samples of acceleration time histories compatible with the design response spectrum. Zhang et al. [2003b] and Raghukanth [2010] used EMD to understand the earthquake rupture process. Since the original EMD technique suffers from mode mixing, Wu and Huang [2009] developed ensemble empirical mode decomposition (EEMD) to extract IMF’s from the data. In EEMD, a white noise of finite amplitude is added to the data and EMD is used to extract the data. The procedure for extracting IMF’s from strong motion accelerograms (SMA) by EEMD is as follows. First, add white noise of finite amplitude [w(t)] to the acceleration time history, a(t). Denote this new data as R(t). The extraction of IMF from the data depends on the sifting process which consists of an interpolation method for constructing envelopes and a stopping criterion. Identify the consecutive maxima’s and minima’s in R(t) and construct the lower and upper envelopes by cubic splines. The averages of the positive and negative envelopes are determined at every time step. This average which is the local mean of the data is subtracted from the data to get R1 (t) = R(t) − m0 (t). This new time series is treated as the data as in the previous step to get R2 (t) = R1 (t) − m1 (t). This process is repeated n times till the sieved data Rn (t) becomes an IMF. To extract the second IMF, the first IMF is subtracted from the data, R(t) and the whole sifting process is repeated. In a similar fashion, IMF2 , IMF3 , . . . are extracted until the sieved data shows no oscillations. After extracting all the IMF’s the entire EMD procedure is repeated 1250023-4


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m times by adding a different white noise time series to the data. An ensemble of IMFs are obtained from EEMD. The final empirical modes are obtained by averaging the corresponding IMFs obtained from EMD. In Figs. 2(a) and 2(b), IMF’s of both horizontal components acceleration time histories recorded at TCU129 station during the Chi-Chi earthquake (Mw 7.6) are shown. The sum of the IMFs leads to the original data. After the EEMD, the earthquake acceleration time history can be expressed in terms of IMFs as follows, a(t) = ΣIMFi (t).

(a) Fig. 2. (a) Intrinsic mode functions of acceleration time histories recorded at TCU129 station during Chi-Chi earthquake (Component: Fault normal). (b) Intrinsic mode functions of acceleration time histories recorded at TCU129 station during Chi-Chi earthquake (Component: Fault parallel). 1250023-5


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(b) Fig. 2.

(Continued )

4. Hilbert–Huang Transform (HHT) Once the IMFs are extracted the evolutionary spectrum can be constructed by taking the Hilbert transform of the IMFs as Dj (t) = P

∞

−∞

IMFj (τ ) dτ , π(t − τ )

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(1)


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where P indicates the Cauchy principal value. Since IMFj and Dj form the complex conjugate pair, one can form the analytical function as Zj (t) = IMFj (t) + iDj (t) = Cj (t)eiθj (t) .

(2)

The instantaneous amplitude, Cj (t) of the jth IMF is defined as Cj (t) =

IMFj (t)2 + Dj (t)2

(3)

and the instantaneous phase can be expressed as θj (t) = tan−1

Dj (t) . IMFj (t)

(4)

The instantaneous frequency (IF) of the jth IMF can be estimated as ωj (t) = dθj (t)/dt.

(5)

The original acceleration time history can be expressed as the real part of the sum of the Hilbert transform of each IMF   n   Cj (t)eiθj (t) . a(t) = Re (6)   j=1

It can be observed that the final representation of HHT is a surface in threedimensional space of frequency, amplitude and time. The estimated instantaneous frequencies of the recorded acceleration time histories at TCU129 station for the Chi-Chi event are shown in Figs. 3(a) and 3(b). The instantaneous frequencies of the IMFs decrease in the higher modes. The mean frequency of each IMF is determined by averaging instantaneous frequency over time. In a similar fashion, the central frequency of the 10 IMFs for all the 3,551 acceleration time histories have been estimated by HHT. Earthquake ground motion in general can be decomposed into number of IMFs. As the variance of the higher modes is insignificant, only the first 10 IMFs are considered for further statistical analysis. Table 1 shows the mean and standard deviation of the mean frequency estimated from 3,551 time histories for both the horizontal components. In Figs. 4(a) and 4(b) the variation of the obtained mean IF of the first 10 IMFs with moment magnitude (Mw ), rupture distance (Rrup ) and average shear wave velocity in the top 30 m (Vs30 ) is shown. It can be observed that the mean of the instantaneous frequency remains constant and does not show any pattern with Mw , Rrup , and Vs30 . The percentage of variance explained by each IMF, or the contribution of each IMF to total variance of the data is determined for all the 3,551 time histories. The mean and standard deviation of the percentage of variance estimated from 3,551 samples is presented in Table 1 for both fault normal 1250023-7


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(a)

(b) Fig. 3. (a) Instantaneous frequencies of acceleration time histories shown in Fig. 2(a) (Component: Fault normal). (b) Instantaneous frequencies of acceleration time histories shown in Fig. 2(b) (Component: Fault parallel).

and fault parallel components. It is observed that IMF3 with a central frequency of 4 Hz is the predominant mode contributing to 31% of the variability in the data. IMF4 with a central frequency of 1.8 Hz is the second most important mode in both the cases. The next important modes are IMF2 and IMF5 oscillating about 11 Hz and 0.9 Hz contributing to 14% variability in the data. These four dominant 1250023-8


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Empirical Mode Decomposition of Earthquake Accelerograms Table 1. Central frequency of the IMF’s in Hz and % variance contributed to total variability of the data. Fault normal

IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8 IMF9 IMF10

Fault parallel

Frequency mean ± STD

%Variance explained mean ± STD

Frequency mean ± STD

%Variance explained mean ± STD

24.38 ± 4.63 11.37 ± 2.61 3.99 ± 0.89 1.82 ± 0.45 0.91 ± 0.24 0.47 ± 0.13 0.25 ± 0.07 0.12 ± 0.04 0.06 ± 0.02 0.03 ± 0.01

8.51 ± 11.67 14.21 ± 14.95 32.02 ± 14.86 28.40 ± 14.87 14.83 ± 11.85 5.25 ± 5.95 1.66 ± 2.86 0.62 ± 1.61 0.32 ± 0.78 0.24 ± 1.37

24.37 ± 4.63 11.40 ± 2.61 4.01 ± 0.88 1.83 ± 0.45 0.91 ± 0.24 0.47 ± 0.13 0.24 ± 0.07 0.12 ± 0.04 0.06 ± 0.02 0.03 ± 0.01

8.64 ± 11.74 14.29 ± 13.17 31.85 ± 14.63 27.93 ± 14.46 15.03 ± 20.74 5.55 ± 10.26 1.81 ± 5.83 0.66 ± 2.15 0.35 ± 1.40 0.29 ± 3.43

modes explain about 87% variability in both the fault normal and fault parallel acceleration time histories. 5. Evolutionary Power Spectral Density The evolutionary PSD [G(t, ω)] of the acceleration time history can be constructed from HHT as [Liang et al. (2007)]. G(t, ω) =

n 1 j=1

2

δ[ω − ωj (t)]Cj2 (t).

(7)

The corresponding evolutionary PSD of both fault normal and fault parallel component is shown in Figs. 5(a) and 5(b). The instantaneous average power [Pa(t)], the central frequency [Fc (t)] and the frequency bandwidth [Fb (t)] can be estimated from the first three spectral moments as [Lai (1982)] ∞ G(t, ω)dω, (8) P a(t) = 0

∞ ωG(t, ω)dω Fc (t) = 0 ∞ , 0 G(t, ω)dω  ∞ 2 1/2 ∞ 2 ω G(t, ω)dω ωG(t, ω)dω  . Fb (t) =  0 ∞ − 0 ∞ G(t, ω)dω G(t, ω)dω 0 0

(9)

(10)

These three strong motion parameters are shown in Figs. 6(a) and 6(b) for both fault normal and fault parallel components recorded at TCU129 station for the 1999 Chi-Chi earthquake. It can be observed that the central frequency and frequency bandwidth oscillate around a constant mean and does not show any pattern with time where as instantaneous power decreases with time. 1250023-9


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6. Strong Motion Parameters Due to high complexity, developing a model which can exactly reproduce the observed evolutionary PSD of SMA is not possible. Large number of parameters are required to model the evolutionary PSD. Physical understanding of these parameters and linking them to earthquake source, path and site parameters is very difficult. In such situations, a few parameters which are of engineering interest can be identified from the evolutionary PSD. In this study, six parameters namely total energy of the acceleration time history (Eacc ) also known as Arias Intensity (AI), spectral centroid [E(ω)], temporal centroid [E(t)], spectral standard deviation [S(ω)], temporal standard deviation [S(t)] and the correlation of time and frequency

(a) Fig. 4. (a) Central frequency of all the IMF’s as a function of magnitude, rupture distance and Vs30 (Component: Fault normal). (b) Central frequency of all the IMF’s as a function of magnitude, rupture distance and Vs30 (Component: Fault parallel). 1250023-10


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(b) Fig. 4.

(Continued )

[ρ(t, ω)] are used to characterize the evolutionary PSD of all the 3,551 acceleration time histories. These parameters are estimated from the evolutionary PSD as Eacc =

∞

∞

G(t, ω)dωdt,

(11)

∞ ∞ ωG(t, ω)dωdt E(ω) = 0 ∞ 0 ∞ , G(t, ω)dωdt 0 0

(12)

0

0

∞ ∞ (ω − E(ω))2 G(t, ω)dωdt S (ω) = 0 0 ∞ ∞ , 0 0 G(t, ω)dωdt 2

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(13)


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(a)

(b) Fig. 5. (a) Evolutionary PSD of acceleration time histories shown in Fig. 2(a) for the fault normal component. (b) Evolutionary PSD of acceleration time histories shown in Fig. 2(b) for the fault parallel component.

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∞ ∞ tG(t, ω)dωdt , E(t) = 0∞ 0∞ 0 0 G(t, ω)dωdt ∞ ∞ (t − E(t))2 G(t, ω)dωdt 2 S (t) = 0 0 ∞ ∞ , G(t, ω)dωdt 0 0 ∞ ∞ (t − E(t))(ω − E(ω))G(t, ω)dωdt ∞ ∞ ρ(t, ω) = 0 0 . S(t)S(ω) 0 0 G(t, ω)dωdt

(14)

(15)

(16)

These parameters are estimated for both fault normal and fault parallel components. Figures 7(a) and 7(b) shows the spectral centroid and its standard deviation

(a) Fig. 6. (a) Instantaneous power, central frequency and frequency bandwidth estimated from evolutionary PSD (Component: Fault normal). (b) Instantaneous power, central frequency and frequency bandwidth estimated from evolutionary PSD (Component: Fault parallel). 1250023-13


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(b) Fig. 6.

(Continued )

as a function of magnitude, closest distance to rupture and Vs30 for both the horizontal components. It can be observed that spectral centroid decreases with increase in magnitude and closest distance to rupture. It is well known that SMA’s of large magnitude events will be rich in low frequency content. The low frequency waves can propagate up to large distances where as energy at high frequencies gets damped by the medium. As the shear wave velocity increases spectral centroid moves towards high frequencies which means SMA’s recorded at rock type sites will be rich in high frequencies compared to soil sites. The temporal parameters are shown in Figs. 8(a) and 8(b) for both the horizontal components. The temporal centroid and standard deviation increases with increase in magnitude and distance to rupture where as it decreases with increase in shear wave velocity in top 30 m. The duration of SMA’s 1250023-14


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(a) Fig. 7. (a) Spectral centroid and spectral standard deviation of the evolutionary PSD as a function of Mw , Rrup and Vs30 (Component: Fault normal). (b) Spectral centroid and spectral standard deviation of the evolutionary PSD as a function of Mw , Rrup and Vs30 (Component: Fault parallel). 1250023-15


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(b) Fig. 7.

(Continued )

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(a) Fig. 8. (a) Temporal centroid and temporal standard deviation of the evolutionary PSD as a function of Mw , Rrup and Vs30 (Component: Fault normal). (b) Temporal centroid and temporal standard deviation of the evolutionary PSD as a function of Mw , Rrup and Vs30 (Component: Fault parallel). 1250023-17


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(b) Fig. 8.

(Continued )

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Fig. 9. Energy and correlation coefficient of the evolutionary PSD as a function of Mw , Rrup and Vs30 (Component: Fault normal).

will increase with magnitude and distance. The duration of SMA’s on soil sites will be more when compared to rock sites. Figures 9 and 10 show the variation of Arias intensity and correlation coefficient of time and frequency with the magnitude, distance and Vs30 . 1250023-19


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Fig. 10. Energy and correlation coefficient of the evolutionary PSD as a function of Mw , Rrup and Vs30 (Component: Fault parallel).

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7. Ground Motion Prediction Equations After characterizing the evolutionary PSD of SMA data, the next step is to develop empirical equations for predicting the six ground motion parameters. Ground motion equation is a key component in seismic hazard analysis. A review of some of the general ground motion relations and the methods appropriate for regression analysis are already available in the literature. Several functional forms of ground motion attenuation have been proposed in the literature reflecting salient aspects of the spread of ground motion [Pousse et al. (2006); Raghukanth (2008); Boore and Atkinson (2008); Yamamoto (2011)]. Since all the proposals are empirical the particular form selected is justified heuristically and sometimes by limited comparison with instrumental records. After reviewing the various available forms of equations, it has been decided to develop an empirical relation for both the horizontal components in the form 2 + h2 Rrup ln(y) = β1 + β2 Mw + β3 ln(Mw ) + β4 exp(Mw ) + β5 2 + β6 ln Rrup + h2 + β7 ln(Vs30 ) + ln(ε) (17) where y, Mw and Rrup refer to ground motion parameter, moment magnitude and closest distance to rupture respectively, h, is the pseudo focal depth to control the ground motion parameters close to the fault rupture. ln(ε) is the error associated with the regression. Since ‘y’ can be taken to be a lognormal random variable, ln(y) would be normally distributed with the average of ln(ε) being zero. Hence with ε = 1, Eq. (17) represents the mean hazard estimation formula for ground motion. This is also nearly the median value. Due to nonuniform sampling of the recorded data (Fig. 1) there can be spatial aliasing in determining the regression coefficients in the attenuation equation (Eq. (17)). To address this issue, stratified regression is carried out on the SMA data to obtain the coefficients. The advantage of this method is that it decouples the determination of magnitude dependence from the determination of distance dependence. In this procedure Eq. (17) is first written as ln(y) =

n

Ei li + β5

2 + h2 + β ln 2 + h2 + β ln(V Rrup Rrup 6 7 s30 ) + εd . (18)

i=1

Here n is the number of earthquakes = 900, Ei is a constant for the ith earthquake and li is a dummy variable which takes value 1 for the ith earthquake and 0 otherwise. εd is the error in the regression associated with distance dependence. The unknowns (β5 , β6 , β7 , Ei ) in the above equation are determined from linear regression analysis. The obtained (β5 , β6 , β7 ) for both the horizontal components are reported in Tables 2 and 3. Once the distance and soil coefficients are known, the coefficients (β1 , β2 , β3, β4 ) are determined from Ei as, Ei = β1 + β2 Mw + β3 ln(Mw ) + β4 exp(Mw ) + εm , 1250023-21

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S. T. G. Raghukanth & S. Sangeetha Table 2. Fault parallel Eacc (g 2 -s) E(ω) (Hz) S(ω) (Hz) E(t) (s) S(t) (s) ρ(t, ω)

β1

Eacc (g 2 -s ) E(ω) (Hz) S(ω) (Hz) E(t) (s) S(t) (s) ρ(t, ω)

β2

β3

β4

−30.02 −2.28 24.49 0.0 2.5600 −0.1754 0 0 2.2610 −0.1037 0 0 1.0210 0 0 0.0006 0.3972 0 0 0.0004 0.0256 0.0078 0 0

Table 3. Fault normal

Coefficients in Eq. (17) for Fault parallel component.

β1

β5

β6

β7

h

σ(εd )

σ(εm )

0.00 −1.92 −0.89 — 8 0.87 −0.0067 0.1593 0.3749 4 0.3640 0.3575 −0.0051 0.0973 0.2485 5 0.2954 0.2739 −0.0011 0.2511 −0.2060 5 0.2809 0.3564 −0.0003 0.1836 −0.3015 3 0.3206 0.3073 0.0005 −0.0444 0.0574 4 0.0865 0.0685

Coefficients in Eq. (17) for Fault normal component. β2

β3

β4

−28.84 −1.89 22.04 0.00 2.5381 −0.1714 0 0 2.2271 −0.0914 0 0 1.0359 0 0 0.0006 0.4232 0 0 0.0005 0.0074 0.0120 0 0

β5

β6

β7

h

σ(εd )

σ(εm )

0.00 −1.62 −0.81 — 7 0.87 −0.0069 0.1641 0.3748 2 0.3513 0.3437 −0.0050 0.0829 0.2351 1 0.2893 0.2617 −0.0010 0.2418 −0.2165 5 0.2760 0.3638 −0.0001 0.1659 −0.3247 3 0.3160 0.3030 0.0006 −0.0454 0.0652 3 0.0829 0.0665

where εm is the error in the regression associated with magnitude dependence. The constants of regression are reported in Tables 2 and 3. The standard deviation of the ground motion which is required in seismic hazard analysis can be expressed as (20) σ(ln ε) = σ(εm )2 + σ(εd )2 . These values are also reported in the above tables. These results can be used to construct the mean and (mean+sigma) ground motion parameters of the evolutionary PSD on any type site condition in shallow active regions in the world. In Fig. 11 the attenuation of Arias intensity for both the horizontal components is shown for different magnitude values. It can be observed that as distance increases the predicted arias intensity is almost same for both the horizontal components. 8. Summary and Conclusion This article explores the application of EMD–HHT technique for characterizing earthquake acceleration time histories. A total of 3,551 horizontal components generated by 173 worldwide events from NGA database have been used in the analysis. The EMD technique shows some interesting features of SMA data. It is observed that earthquake accelerations can be represented as a sum of ten independent modes. The mean instantaneous frequency of these modes estimated from time averaging does not show any pattern with magnitude, distance to rupture and the site shear wave velocity. The contribution of the individual modes to the total variability of the data is reported in Table 1. The third IMF with mean instantaneous frequency of 4 Hz explains the maximum variability of the data in both the fault normal and fault parallel components. The evolutionary PSD of the strong motion data is constructed from HHT. Since modeling the evolutionary PSD is complex, spectral 1250023-22


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Fig. 11.

Ground motion relation for Arias intensity as a function of Rrup given Vs30 = 1 km/s.

and temporal centroid, spectral and temporal standard deviation, Arias intensity and correlation coefficient have been extracted from the evolutionary PSD. The variation of these parameters with magnitude, rupture distance and site condition has been reported in Figs. 6–10. The spectral centroid decreases with increase in magnitude and rupture distance and increases with increase in shear wave velocity. The temporal centroid which characterizes duration of the ground motion increases with magnitude and rupture distance and decreases with increase in shear wave velocity. These patterns are consistent with the seismological concepts. Empirical equations to predict these six parameters have been derived by two-step regression analysis for both the fault parallel and fault normal components. These equations can be used to estimate the mean values of the six ground motion parameters for a given magnitude, rupture distance and the site condition. Sabetta and Puglise [1996], Pousse et al. [2006] and Yamamoto [2011] have modeled the evolutionary PSD as a lognormal density function. The six strong motion parameters estimated in this study can be used as an input in these studies and an ensemble of acceleration time histories can be constructed by the spectral representation method of Liang et al. [2007]. 1250023-23


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References Boore, D. M. (1983). Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra. Bull. Seismol. Soc. Am., 73(6): 1865– 1894. Boore, D. M. and Atkinson, G. M. (2008). Ground-motion prediction equations for the average horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.01 s and 10.0 s. Earthquake Spectra, 24, 99–138. Conte, J. (1992). Nonstationary ARMA modeling of seismic motions. Soil Dyn. Earthquake Eng., 11: 411–426. Conte, J. P. and Peng, B. F. (1997). Fully nonstationary analytical earthquake groundmotion model. J. Eng. Mech., 123(1): 15–24. Hartzell, S., Frankel, A., Liu, P., Zeng, Y. and Rahman, S. (2011). Model and parametric uncertainty in source-based kinematic models of earthquake ground motion. Bull. Seismol. Soc. Am., 101(5): 2431–2452. Huang, N. E., Shen, Z., Long, S. R., Wu, C. M., Shih, H. H., Zheng, Q., Yen, N. C., Tung, C. C. and Liu, H. H. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis. Proc. R. Soc. Lond. A, 454: 903–995. Huang, N. E., Chern, C. C., Huang, K., Salvino, L. W., Long, S. R. and Fan, K. L. (2001). A new spectral representation of earthquake data: Hilbert spectral analysis of station TCU129, Chi-Chi, Taiwan, 21 September 1999. Bull. Seismol. Soc. Am., 91(5): 1310–1338. Iyengar, R. N. and Iyengar, K. T. S. R. (1969). A nonstationary random process model for earthquake acceleration. Bull. Seismol. Soc. Am., 59(3): 1163–1188. Komatitsch, D. and Tromp, J. (2002a). Spectral-element simulations of global seismic wave propagation — I. Validation. Geophys. J. Int., 149: 390–412. Lai, S. P. (1982). Statistical characterization of strong ground motions using power spectral density function. Bull. Seismol. Soc. Am., 72(1): 259–274. Liang, J., Chaudhuri, S. R. and Shinozuka, M. (2007). Simulation of nonstationary stochastic processes by spectral representation. J. Eng. Mech., 133(6): 616–627. Ni, S. H., Xie, W. C. and Pandey, M. (2011). Application of Hilbert–Huang transform in generating spectrum — Compatible earthquake time histories. Int. Scholarly Research Network (ISRN ) Signal Processing, doi:10.5402/2011/563678. Papadimitriou, K. (1990). Stochastic characterization of strong ground motion and applications to structural response. Report No. EERL 90-03. Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, California. Pousse, G., Bonilla, L. F., Cotton, F. and Margerin, L. (2006). Nonstationary stochastic simulation of strong ground motion time histories including natural variability: Application to the K-net Japanese database. Bull. Seismol. Soc. Am., 96(6): 2103–2117. Raghukanth, S. T. G. (2008). Modeling and synthesis of strong ground motion. J. Earth Syst. Sci., 117: 683–705. Raghukanth, S. T. G. (2008). Simulation of strong ground motion during the 1950 great Assam earthquake by hybrid green functions. Pure Appl. Geophys., 165: 1761–1787. Raghukanth, S. T. G. and Iyengar, R. N. (2008). Strong motion compatible source geometry. J. Geophys. Res.-Solid Earth, 113 B04309, doi:10.1029/2006JB004278. Raghukanth, S. T. G. (2010). Intrinsic mode functions of earthquake slip distribution. Adv. Adapt. Data Anal., 2(2): 193–215. Raghukanth, S. T. G. and Bhanu Teja, B. (2011). Ground motion simulation for January 26, 2001 Gujarat earthquake by spectral finite element method. J. Earthquake Engin., 16(2): 252–273. 1250023-24


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Rezaeian, S. and Der Kiureghian, A. (2010). Stochastic modeling and simulation of ground motions for performance-based earthquake engineering. PEER Report No. 2010/02. Sabetta, F. and Pugliese, A. (1996). Estimation of response spectra and simulation of nonstationary earthquake ground motions. Bull. Seismol. Soc. Am., 86(2): 337–352. Shinozuka, M. and Sato, Y. (1967). Simulation of nonstationary random process. J. Eng. Mech. ASCE, 93: 11–40. Wu, Z. and Huang, N. E. (2009). Ensemble emprical mode decomposition: A noise-assisted data analysis method. Adv. Adapt. Data Anal., 1(1): 1–41. Wen, Y. K. and Gu, P. (2009). HHT-based simulation of uniform hazard ground motions. Adv. Adapt. Data Anal., 1(1): 71–87. Yamamoto, Y. (2011). Stochastic Model for Earthquake Ground Motion Using Wavelet Packets. Ph.D. Thesis, Department of Civil and Environmental Engineering, Stanford University, Stanford, California. Zhang, R. R., Ma, S. and Hartzell, S. (2003a). Signatures of seismic source in EMDBased characterization of the 1994 Northridge, California, earthquake recordings. Bull. Seismol. Soc. Am., 93(1): 501–518. Zhang, R. R., Ma, S., Safak, E. and Hartzell, S. (2003b). Hilbert–Huang transform analysis of dynamic and earthquake motion recordings. J. Engin. Mech., 129(8): 861–875.

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