Stochastic Non-Stationary Model for Ground Motion Simulation Based ...

Journal of Earthquake Engineering, 00:1–28, 2016 Copyright © Taylor & Francis Group, LLC ISSN: 1363-2469 print / 1559-808X online DOI: 10.1080/13632469.2016.1149894

Stochastic Non-Stationary Model for Ground Motion Simulation Based on Higher-Order Crossing of Linear Time Variant Systems ZAKARIYA WAEZI and FAYAZ R. ROFOOEI Civil Engineering Department, Sharif University of Technology, Tehran, Iran This article introduces a new time-varying model to generate synthetic non-stationary acceleration records using the modified Kanaii-Tajimi model with time-variant parameters. The proposed method can capture two different dominant frequencies per time which makes it suitable for synthesizing the near-field no-pulse earthquake records. A number of closed-form relationships are developed to describe the frequency dependent time-domain level crossings of the simulated records under white noise excitation. The model parameters are optimized using the crossing and Arias intensity properties of the synthetic and target records. The efficiency of the proposed model is demonstrated using a data base of 106 near-field records. Keywords Non-stationary Records; Ground Motion Simulation; Linear Time Variant; Zero Crossing; Positive Minimum; Negative Maximum

1. Introduction The simulation of earthquake records has always been a great challenge for engineers who want to assess the reliability of the designed structures subjected to real ground motion excitations [Naeim, 1989]. Even though there are great advances in the field of the physical modeling of the earthquakes, their cost in terms of both time and exclusivity makes them unappealing for engineering purposes [Douglas and Aochi, 2008]. Large amounts of research have been conducted to investigate the properties of the recorded ground motions as a guide to regenerate the synthetic yet realistic ground motion realizations. Recent advances in the field of nonlinear analysis have proved the insufficiency of the response spectrum method in simulating the ground motion excitations [Naeim, 1989]. This is due to the sensitivity of the nonlinear structural systems to the frequency content of the excitation. On the other hand, the stochastic methods could provide enough accuracy and flexibility for utilization in large-scale performance-based design applications. The record generation techniques using numerical and physical methods have proved to be insufficient for frequencies higher than 1–2 Hz due to lack of site soil information. Moreover, as a result of being dependent on the recorded accelerograms, the physical methods cannot be extended to other conditions [Alamilla et al., 2001]. Some researchers believe that the deterministic physical methods are more reliable for lower frequencies in heterogeneous models, because of the difficulty in accurately representing the incoherency of source radiation and wave propagation at higher frequencies [Douglas and Aochi, 2008], However, some Received 14 August 2015; accepted 14 January 2016. Address correspondence to Fayaz R. Rofooei, Civil Engineering Department, Sharif University of Technology, Azadi Ave., Tehran, 11155-4313, Iran. E-mail: [email protected] Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/ueqe.

1

2

Z. Waezi and F. R. Rofooei

other researchers have proved the efficiency of these model for frequencies up to 10 Hz. Nowadays, with the advantage of recent developments in the field of “hybrid modeling” and faster computers, powerful tools capable of broad-band simulation of the ground motions with efficient performance in the time-cost are within reach [Maechling et al., 2014]. Since stochastic site-based models yield realistic results for frequencies greater than 1 Hz [Sun et al., 2015] and the importance of introducing simple yet reliable models for engineering projects, the stochastic methods still look appealing for some problems. Furthermore, developing well-suited stochastic site-base models for the recorded ground motions can be useful for the “hybrid” methods that use physical models along with the stochastic-method. From the signal processing aspect, a signal can be categorized either as stationary or non-stationary according to time variation of its amplitude and frequency content. The amplitude non-stationarity is defined as the change in the amplitude of acceleration record versus time, while the frequency non-stationarity indicates the change of its power spectrum [Rezaeian and Der Kiureghian, 2008]. Usually stochastic methods to simulate the earthquake records manage the amplitude non-stationarity but not the frequency non-stationarity. Frequency non-stationarity of signals results from the dynamic nature of the ground motion that is mainly due to faster propagation speed of high frequency waves in the soil media. Many studies indicate the significance of the frequency content change on the seismic-induced response of linear and nonlinear structures [Conte and Peng, 1996; Papadimitriou, 1990; Saragoni and Hart, 1974; Yeh and Wen, 1990]. The frequency nonstationarity of ground motion is of higher importance for structures with nonlinear behavior when their fundamental frequencies reduce due to the stiffness degradation. The coincidence of this phenomena together with the arrival of low-frequency surface waves could lead to disastrous results and even collapse of these structures [Kiureghian and Crempien, 1989]. Many approaches have been introduced for considering the non-stationarity of the records, especially in frequency domain, with some of them having a large number of parameters involved [Beresnev and Atkinson, 1998; Motazedian and Atkinson, 2005; Papadimitriou, 1990; Pousse et al., 2006; Stafford et al., 2009; Yamamoto and Baker, 2013]. This article introduces a new method for generation of a non-stationary acceleration record based on the zero-crossing characteristics of a group of 106 selected accelerograms. This article improves the well-known Kanaii-Tajimi model by addition of another high-pass filter with time-dependent parameters to develop a frequency-wise, non-stationary process. Compared to the other existing methods, the proposed model does not strictly depend on sophisticated mathematical transformation and requires the least number of parameters for the synthesizing the earthquake records. In the following, first the concept of non-stationary records, linear time-variant (LTV) systems and the method of level crossing for identification of the model parameters will be briefly described. Then, the proposed LTV model as well as the procedure for using all this information for record generation will be explained.

2. Basic Definitions 2.1. Frequency and Amplitude Non-Stationarities Basically two classic approaches toward estimation of time-varying spectrum of nonstationary signals were defined by Page [1952] and Priestley [1965]. Through definition of “instantaneous power spectrum,” Page extended the concept of energy density of

Stochastic Non-Stationary Model

3

stationary processes to the non-stationary ones and explained the change of spectrum content between the (0, t + δt) and (0,t) time intervals. For a non-stationary process {x (t)} “the instantaneous power spectrum” ρ (t, f ) at time t is defined as: ∞ x (t) x (t − τ ) cos (2π f τ ) dτ .

ρ (t, f ) = 2

(1)

0

However, Priestley’s “evolutionary (time-varying) power spectral density (EPSD)” aims to define the distribution of energy of a random process in the vicinity of time t more rigorously. If a non-stationary process {x (t)} can be written in the following form: ∞ m (t, ω) eiωt dz (ω) ,

a (t) =

(2)

−∞

where m (t, ω) is a complex modulating function and dz (ω) is an orthogonal process, the EPSD S (t, ω) can be defined as: S (t, ω) = |m (t, ω)|2 dμ (ω) ,

(3)

where E |dZ (ω)|2 = dμ (ω). Saragoni and Hart [1974] defined the EPSD for finite time regions while Liu [1970] developed a method for calculation of the evolutionary power spectrum for improved demonstration of the non-stationarity in earthquake accelerograms. Based on these concepts, one of the primary models to generate stochastic earthquake excitation is to use uniformly modulated models in the following form: a (t) = m (t) s (t) ,

(4)

where a(t), s(t), and m(t) represent ground acceleration process, stationary process, and a modulating function, respectively. The models introduced by Amin and Ang [1966] and Shinozuka and Sato [1967] belong to this category which are basically derived from evolutionary models of Priestly. Yeh and Wen’s model [Yeh and Wen, 1990] also considers the same form as in Eq. (4) but adjusts the dominant frequency of the ground motion using a time-scale modulator φ (t) such that a (t) = m (t) s (φ (t)). The s (φ (t)) is a frequency modulated random process whose instantaneous power spectrum is derived using Eq. (1). Kiureghian and Crempien [1989] used a combination of the processes in the form of Eq. (1) to obtain a more “flexible” EPSD for any degree of non-stationarity. Having based their model on Priestley’s EPSD, Conte and Peng [1996] modeled the earthquake records using a bundle of waves with different arrival times. Following Yeh and Wen’s model, Alamilla et al. [2001] used a frequency modulated process with time-varying Clough and Penzien’s [1993] model as the instantaneous power spectrum of the signal. Yamamoto and Baker [2013] used wavelet transform to approximately estimate the EPSD of the earthquakes. They calibrated the parameter of wavelet transform to relate them to the ground motion parameters. Papadimitriou used a second-order differential equation with time-variable coefficients which is subjected to white noise excitation [Papadimitriou, 1990]. He derived approximate expressions for second-order statistical characteristics of the response of the model under conditions of slow varying coefficients.

4


TABLE 1 Some of the modulating functions used by researchers Authors

Model Name

Expression q (t, α, β, γ ) = α {exp (−βt) − exp (−γ t)} β < γ

Bolotin [1960]; Shinozuka Exponential Model and Sato [1967] Amin and Ang [1966]

Piece-wise modulating

q (t, α) =

Saragomi and Hart [1974] Gamma Function

Arias et al. [1976]

Beta Model

Kiureghian and Crempien [1989] Yeh and Wen [1990]

Piece-Wise linear

⎧0 T0 ≥ t ⎪

2 ⎪ ⎨ o = α1 Tt−T T 0 ≤ t ≤ T1 1 −To q(t, a) = ⎪ = α T ⎪ 1 1 ≤ t ≤ T2 ⎩ = α1 exp[−α2 (t − T2 )α3 ] T2 ≤ t 0 = α1 (t − T0 )α2 −1 exp [−α3 (t − T0 )] βk

γk mk (t) = αk tt 1 − tt f

T0 ≥ t T0 ≤ t

f

mk (t) = αk (t − ti ) + βki , ti < t < ti+1 t1 < t2 < . . . < tn −1 −C t A tB D + tE e

—

In their model, Rezaeian and Der Kiureghian [2008, 2010] used a filtered white noise whose non-stationarity is separated into uncorrelated amplitude and frequency nonstationarities. By means of a time-varying filter with a specific EPSD, they generated a time-varying filter and obtained a non-stationary output. They have used the same idea that Yeh and Wen [1990] adopted to identify the parameters of the model. On the other hand, a modulating function is usually used for considering the amplitude non-stationarity. A number of well-known modulating functions are presented in Table 1. Although the procedure to determine the parameters of each model is different, but they are all based on minimization of the Arias intensity difference between the target and simulated records. Even though Amin and Ang’s piecewise model needs more parameters than the others, it could effectively capture wider range of amplitude non-stationarity for recorded accelerograms and is opted by many researchers. Some other methods have been proposed by researchers consisting moving average techniques to better consider the amplitude change. However, in this article the former versions whose finite parameters and deterministic approach are well suited for the purpose of scenario-based simulation are considered. 2.2. LTV Systems The main versions of the LTV system models were based on filtering a white noise with a linear time-invariant SDOF system whose response can be calculated using the following convolution integral: t h (t − τ ) w (τ ) dτ ,

f (t) =

(5)

−∞

where h (t − τ ) is the unit impulse response function (IRF) of the time-invariant filter at time t to the impulse at time τ , and w (τ ) is a Gaussian white noise. Despite being simple, this method cannot consider non-stationary characteristics of earthquake records. The LTV systems are identified using a time varying impulse function h (t − τ , τ ) that is depended on both impulse incidence time (τ ) as well as elapsed time (t = t − τ ). Thus, for a linear time invariant (LTI) system one could have h (t, τ1 ) = h (t, τ2 ) = h (t), while a LTV


5

system results in h (t, τ1 ) = h (t, τ2 ). Beside generating non-stationary signals, these models have the ability of treating the nonlinear systems as a LTV model given the system characteristics are well-defined [Verhaegen and Yu, 1995]. An LTV system whose output to white noise includes both types of non-stationarities can be defined by adjusting Eq. (5): t h (t − τ , τ ; λ (τ )) w (τ ) dτ .

f (t) =

(6)

−∞

In this model with an IRF in the form of h (t − τ , τ ), λ (τ )) is a vector consisting of model’s time-varying parameters and indicates that the dependence on the impulse incidence time (τ ) is conveyed to the model through making its own parameters variable. This feature can guarantee a time-varying frequency content as the IRF changes. A time-varying transfer function or evolutionary transfer function (ETF), H (ω, τ ) can be defined using Fourier transform of h (t − τ , τ ): 1 H (ω, τ ) = 2π

∞ h (θ , τ ) exp (−iωθ ) dθ.

(7)

−∞

Moreover, if the system is slow-varying, the frequency response and bandwidth concepts can be generalized too [William et al., 2004]. On this basis, one could extend this idea to generate much more complicated non-stationary signals. Thus, the IRF can determine the instantaneous properties of the output signal. The Fourier Transform of IRF for Rezaeian and Der Kiureghian’s [2008] model can be obtained as: H (ω, τ ) = 2π

1

2 2

−1 + β(τ )

+ 4β(τ )2 ξf (τ )2

,

(8)

where τ is the time of the applied impulse to the system and β (τ ) = ω/ωf (τ ). It is obvious that ωf and ξf determine the dominant frequency and the bandwidth of the IRF [Rezaeian and Der Kiureghian, 2008]. Because of having an IRF dependent on filter frequency ωf and damping ξf , the characteristics of this model’s output signals are strictly determined by the filter frequency and damping. The identification of the LTI and LTV dynamic systems has been studied using various methods in order to determine their important characteristics. The output-only identification methods can be categorized as frequency domain and time-domain approaches. Frequency domain methods are essentially originated from Fourier transform such as DFT, FFT, etc. They can provide valuable information about the frequency content of the signal as well as system’s parameters, but they are unable to track the time variation of nonlinear and stochastic signals. Also, most of frequency domain methods fail in identification of local properties of the signals since these are based on the transforms that give the average value of the functions in time and for that matter time-domain methods have been developed for specific cases. The autoregressive moving average methods (ARMA) [Chang et al., 1982; Jachan et al., 2007; Mobarakeh et al., 2002; Polhemus and Cakmak, 1981], random decrement techniques (RDT) [Ibrahim, 1977], and Eigensystem realization algorithm (ERA) [Juang and Pappa, 1985] are examples of such methods [Bao et al., 2009]. Various time-frequency identification approaches have been introduced for non-stationary signals from which STFT, spectrogram, wavelet transform, Hilbert-Huang transform, and

6


Wigner-Ville transform have gained more popularity [Boashash, 2003]. The quadratic timefrequency distribution, e.g., Wigner-Ville Distribution, include the transformations that lead to time-frequency domain data, when using quadratic integral transformations. These methods have the advantage of eliminating some cross terms in the output that makes the identification of frequency variation of the signals much easier [Boashash, 1992]. However, they are not very suitable for multi component signals obtained from LTV systems since the cross terms cannot be completely eliminated and the identification procedure may fail. A number of Hilbert-based transformations are introduced to extract the timefrequency domain of the signals. These are all based on Hilbert-Huang transform which decomposes the signal into several empirical modal functions using empirical mode decomposition (EMD) algorithm for applying Hilbert transform [Feldman, 2011a, 2011b, 2012]. If the input and output of systems are known, adaptive algorithms can be applied to determine the model parameters. Among them, the least square method (LMS) and recursive least square method (RLS) use the steepest descent algorithm to determine the unknown model parameters [Feldbauer et al., 2007]. However, for the identification of a LTV output-only systems, the large computational cost of these algorithms make them inefficient for general practices. To properly consider the time-variance characteristics of LTV systems, some adjustments have been introduced by researchers so far. The main idea of these adjustment lie on the concept of basis expansion model(BEM) in which the unknown time-varying IRF of the system is described as the superposition of the response function of multiple linear time-invariant systems (single input multiple output system (SIMO) [Abed-Meraim et al., 1997; Moulines et al., 1995; Tsatsanis and Giannakis, 1996, 1997]. Rofooei et al. used the averaged zero-crossing rate of the acceleration signal to identify the non-stationary Kanaii-Tajimi model’s ground frequency ωg (t) while keeping the ground damping value ξg (t) constant [Rofooei et al., 2001]. Despite implementing the same non-stationary model as those of Rofooei et al. [2001], Amiri et al. [2014] used wavelet decomposition to track the dominant instantaneous frequency and identify the model parameters. Rezaeian and Kiureghian [2008, 2010] used statistical characteristics of the signal to identify the model parameters. They used the zero-crossing rate and number of positive minima/negative maxima of the output signal to identify the system parameters ωf and ξf that are addressed next. 2.3. Level Crossings of the Signals The zero crossing of a signal contains valuable information about the signal properties. It can be used to extract some information about the dominant frequency and other dynamic characteristics of signals without using spectral analysis whose efficiency for nonstationary signals is yet to be explored [Baykut and Akgül, 2010]. Assuming the number of zero-crossings and local maxima and minima be defined as ZC1 and ZC2 , respectively, it is obvious that ZC1 is bigger for signals with bigger dominant frequencies. Consider two harmonic signals with frequencies ω1 and ω2 . Assuming ω2 > ω1 , it can be readily seen that the signal with the higher frequency (ω2 ) has larger ZC1 in a specific time since the value of ZC1 is related to the harmonic’s frequency. In other words, ZC1 represents the equivalent Dirac delta function’s location on the frequency axis with the same total energy and 2nd spectral moment about vertical axis as that of the original signals. However, ZC2 can be used to obtain more information about the high frequency content of the signals [Kedem, 1986]. Also, a combination of ZC1 and ZC2 could provide valuable information about the intermediate- and higher-frequency content of a signal. According to Wiener-Khintchine theorem, for a stationary Gaussian stochastic discrete process Zt of length N with mean


7

value E [Zt ] = μ and constant covariance, there always exits a strictly ascending function F (ω) for which one could have: +π γk =

cos (kω) dF (ω) .

(9)

−π

Defining the correlation function as ρk = γγ0k , k = 0, ±1, . . . it can be proved that the mathematical expectation of zero-crossing is [Kedem, 1986]:

π E [ZC1 ] ρ1 = cos N−1

π

=

−π

cos (ω) dF (ω) π . −π dF (ω)

(10)

Moreover, it can be proved that for Gaussian output process obtained from application of a linear filter L with transfer function H, the total number of zero-crossings ZCH1 can be obtained [Kedem, 1986]:

π E [ZCH1 ] cos N−1

π

=

−π

cos (ω) |H (ω)|2 dF (ω) π . |H (ω)|2 dF (ω) −π

(11)

This definition can be extended to determine the of ∇Zt ≡ Zt − Zt−1 , ∇ 2 Zt =

zero-crossing k k ∇ (∇Zt ) ≡ Zt − 2Zt−1 + Zt−2 and ∇ k Zt ≡ (−1)j Zt−j which are called ZC2 , ZC3 , j j=0 and ZCk+1 , respectively, and defined as:

ω 2k π dF (ω) π E ZCk+1 −π cos (ω) sin 2 cos . = π ω 2k N−1 sin dF (ω) −π

(12)

2

These relationships can be used to determine the spectral content of any discrete signal in a procedure explained by Kedem [1986]. On the other hand, earthquake accelerograms are based on continuous time functions and it is worthy to express the zero crossings of such signals in a closed form equation. Consider a zero mean Gaussian, twice differentiable, stationary signal {y (t)} for −∞ < t < ∞, with autocovariance and autocorrelation functions, R (τ ) and ρ (τ ), respectively. The zero crossing rate, i.e., the total number of zero-crossing in unit time, can be determined according to Rice’s Formula [Rice, 1944]: 1 E [ZC1 ] = π

ρ (0) 1 σY˙ 1 − = = ρ0 π σY π

∞

ω2 S (ω) dω

∞

0

0

S (ω) dω

12 ,

(13)

where S(ω), σY˙ and σY are the power spectral density and the standard deviation of {˙y (t)} and {y (t)} processes, respectively. It should be noted that in this study, instead of total zero crossing, the zero crossing with positive rate is used as an identification tool which is denoted as ZC+ . It can be easily shown that for a Gaussian signal, ZC+ is equal to the half of the total ZC obtained by Eq. (13). Even though the prescribed formula is obtained via the critical assumption of stationary signal, Lutes and Sarkani [2004] used a similar procedure to obtain the formula for a non-stationary signal y (t) with the stationary frequency content as:

8


1 − ρY Y˙ (t, t)2 ZC (t) = 2π +

12

1 1 − ρY2 Y˙ (t, t) 2 σY˙ (t) ωc . = σY (t) 2π

(14)

Later on in this study a relation for ZC will be developed for signals with non-stationary frequency content where the effect of ρY Y˙ will be ignored due to its insignificant role in the final results as a result of slowly varying parameters. Contrary to the zero-crossing of signals which its main outcome is the central frequency of the signal, the frequency bandwidth of the signal is of higher importance and can be related to physical properties of the system too. For an LTI, SDOF system, the bandwidth of response of the system to white noise input is depended on its damping ratio. It can be proved that the number of positive minima/negative maxima of a signal corresponds to the frequency bandwidth of the signal. For a stochastic Gaussian function of time a (t), a probability density function of the stochastic parameters can be defined as p (ξ , η, ζ ) in which ξ = a (t), η = a˙ (t), and ζ = a¨ (t). According to central limit theorem, it can be stated that the distributions of ξ , η, and ζ approach the normal distribution. The second moment of this distribution can be estimated via the prescribed definitions for ξ , η, and ζ as well as the following correlation functions: E ξ 2 = ψ0 , E η2 = −ψ0 , E [ξ η] = 0, E [ηζ ] = 0, E [ξ ζ ] = ψ0 , E ζ 2 = ψ0(4) , (15) where E[.] is the mathematical expectation operator. For every even number k it can be deduced that: ψ0(k)

= (−1)

k 2

∞ ωk S (ω) dω.

(16)

0

Thus, the covariance matrix can be defined as in [Rice, 1944], given the covariance matrix ⎧ ⎫ ⎨ X1 ⎬ .. of random vector X = as Mij = cov Xi , Xj = E (Xi − μi ) Xj − μj where μi = . ⎩ ⎭ XN E (Xi ), the multivariate normal distribution can be described as: fx (x1 , . . . , xn ) = √

1 1 T −1 M (x − μ) . − μ) exp − (x 2 (2π)n |M|

In which |M| is the determinant of matrix M defined as: ⎡

ψ0 ⎣ 0 M= ψ0

0 −ψ0 0

⎤ ψ0 0 ⎦. ψ0(4)

(18)

The probability distribution function (PDF) of the maxima can now be computed according to the multivariate normal distribution: p (x, 0, ζ ) = (2π )− 2 |M|− 2 exp[ 3

1

1 M11 x2 + M33 ζ 2 + 2M13 xζ . 2 |M|

(19)


9

The probability of x having a maximum in the time domain of [t, t + dt], between x and x+dx is equal to: 0 p (x, 0, ζ ) ζ dζ .

dP = −dt dx

(20)

−∞

Integrating Eq. (20) over a ≤ t ≤ b would lead to the number of maxima in that time period. Now, for the total number of positive minima in the time domain [t, t + dt] we can write: ∞ PosMin = dt

∞ p (x, 0, ζ ) ζ dζ .

dx 0

(21)

0

Derivation of both sides of the above equations with respect to time would yields the rate of negative maxima occurrence which can be easily calculated using assumed PDF described by Eq. (19): ⎛ ⎞ ψ0(4) ψ0 d (PosMin) 1 ⎝ d (NegMax) − − − ⎠ . = = dt dt 4π ψ0 ψ0

(22)

Since the signal is Gaussian, it can be assumed that the number of NegMax and PosMin are equal and therefore total rate of NegMax and PosMin is: ⎛ ⎞ ψ0(4) ψ0 1 ⎝ d (NegMax&PosMin) − − − ⎠ = dt 2π ψ0 ψ0 =

1 2π

∞ 0∞ 0

ω4 S (ω) dω ω2 S (ω) dω

−

∞ 2 0 ω S (ω) dω ∞ 0 S (ω) dω

(23)

.

This relation, obtained through Gaussian signal assumption, will be used in the rest of this study.

3. New Model to Generate Non-Stationary Signals In this section a new model is introduced to simulate the earthquake records capable of capturing two dominant frequencies. As already mentioned, the main parameters of model by Rezaeian and Der Kiureghian [2008], i.e., ωf and ξf , cannot be used to consider existing earthquake records with multiple dominant frequencies per time. Red boxes in Fig. 1 show the frequency regions that are not captured in the synthetic record using their method. The underestimated regions in the vicinity of 0.4 Hz and 3–6 Hz observed in the synthetic records may drastically affect the large-period part of the displacement response spectrum. This proves the necessity of developing a new model for records having multiple dominant frequencies. In order to generate a model with the prescribed multiple dominant frequencies using LTV model of Eq. (6), the following evolutionary transfer function is considered for the IRF:

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FIGURE 1 The comparison of the Fourier Amplitude of Kobe 1995, Nishi Akashi record versus a simulated one produced using Rezaeian and Der Kiureghian’s method [Rezaeian and Der Kiureghian, 2010]. F (ω, t) =

1−

ω ωf (t)

ω2 ωf (t)2

2

−

2iξf (t)ω ωf (t)

2iξg ω ωg 2iξg ω ω2 − ωg ωg 2

1− 1−

.

(24)

Hence, according to Eq. (3), its evolutionary power spectral density can be expressed as:

2 1 + 2ξg ωωg G (ω) =

2

2

2

2 . 2 2 1 − ω ω(t)2 + 2ξf (t) ωfω(t) 1 − ωωg 2 + 2ξg ωωg

ω ωf (t)

4

(25)

f

This is an improved version of the well-known modified Kanai-Tajimi model introduced by Clough and Penzien [1993] which will be called non-stationary modified Kanai-Tajimi model (NSMKT). Its IRF can be described as: h (t − τ , τ ) = Ae−(t−τ )ξg ωg cos

!" # 1 − ξ g2 ωg (t − τ )

+ Be−(t−τ )ξg ωg sin

%$−(t−τ )ξf (τ )ωf (τ )

+ Ce

$% & cos 1 − ξf (τ )2 ωf (τ ) (t − τ )

−(t−τ )ξf (τ )ωf (τ )

+ De

& 1 − ξg 2 ωg (t − τ )

$% sin

(26)

& 1 − ξf (τ ) ωf (τ ) (t − τ ) 2

Parameters A, B, C, and D depend on ξg , ωg , ξf , and wωf . The evolutionary power spectral density of the model is shown in Fig. 2a whose IRF is compared with that of Rezaeian and Der Kiureghian [2008] in Fig. 2b.


11

FIGURE 2 Comparison of the proposed double NSMKT model and Rezaeian and Der Kiureghian’s model [2010]: (a) Fourier Amplitudes, (b) The IRFs. Because of having two peaks in PSD of the IRF, one can easily see the combination of two sinusoidal components in the time-domain representation of the IRF. To make the model time variant, the parameters ξf and ωf are assumed to be linear functions of time: ωf (t) = ωf0 +

ωfn − ωf0 t, tn

ξf − ξf0 t ξf (t) = ξf0 + n tn

(27)

where tn is the record duration. Therefore, the proposed model’s IRF includes six parameters consisting two ground motion characteristics ωg and ξg as well as four filter parameters: ωf 0 , ωfn , ξf 0 , and ξfn where “0” and “f” subscripts represent initial and final values of each parameter, respectively. The model does not require simultaneous consideration of amplitude and frequency non-stationarities if they can be separated in Eq using the following relation: ⎫ ⎧ ⎬ ⎨ 1 t h (t − τ , τ , λ (τ )) w (τ ) dτ , f (t) = q (t) ⎭ ⎩ σ (t)

(28)

−∞

where q (t) and σ (t) represent the modulating function and the instantaneous standard deviation of the integral in the bracket, respectively.

3.1. Level Crossing of Proposed Model Since the nature of the proposed model is more complicated than previously introduced models, a try and error approach cannot simply be used here to find IRF parameters. Therefore, it is necessary to estimate the properties of the generated record as a function of six IRF parameters. Having tried various common time-frequency analysis methods such as STFT, wavelet transform, second order distributions like Wigner-Ville distribution and Hilbert Huang transform, the higher-order crossings turned out to be the most efficient way to trace the non-stationary characteristics of these signals.

12


The first- and second-order crossings rate of the signals which are called the zero crossing and the local maxima rate of the signal, are among the most informative indices of the non-stationary records. In contrast to those of a stationary signal, these parameters are changing with time and their dependence on the model parameters can be evaluated numerically. As it was previously presented, there is a solid relationship between the rate of zero crossing, positive minima/negative maxima with PSD of the stationary records. However, this relation cannot be used for non-stationary signals as it is based on Gaussian stationary signal assumption. Moreover, using an overall average PSD for these records and substituting that in the aforementioned relationship would lead to a constant rate which violates its non-stationary property. In order to modify the Rice’s relationship to compensate for the prescribed shortcomings, adjusted versions of the Eqs. (13) and (25) can be considered for the estimation of the crossings using an evolutionary (time-dependent) PSD. Nevertheless, the evolutionary PSD for these kinds of models cannot be easily determined since the convolution property of Fourier Transform does not hold. Instead, using slow-varying property of IRF, it can be assumed that the evolutionary power spectral density is equal to: S (ω, t) = S0 |H (ω, t)|2 ,

(29)

where S0 is the PSD of input stationary white noise w (τ ) and H (ω, t) is the Fourier transform of the impulse response function h (t − τ , τ ), which is the response of the system to the white noise at time τ with a time lag of (t − τ ). Since the modified Kanaii-Tajimi model is used to derive IRF, it follows: S (ω, t) =

1−

ω2 ωf (t)2

2

ω ωf (t)

+

4

1+

2

2ξf (t)ω ωf (t)

2ξg ω ωg

1−

2 ω2 ωg 2

2

+

2ξg ω ωg

2 S0 .

(30)

Thus, using Eqs. (13) and (23), the time-varying, zero-crossing and positive minima and negative maxima rate can be obtained as: 1 E ZC1+ (t) = 2π 1 E [PmNm (t)] = 2π

∞ 0∞ 0

∞ 2 0 ω S (ω, t) dω ∞ 0 S (ω, t) dω

ω4 S (ω, t) dω ω2 S (ω, t) dω

−

∞ 2 0 ω S (ω, t) dω ∞ 0 S (ω, t) dω

(31) .

(32)

It is should be noted that the uppermost frequency ω0 present in the discrete signal (Nyquist frequency) should be used as the upper bound in the shown integrations. To check the validity of proposed relations, 20,000 time domain simulations were carried out and the mean value of the zero-crossing and positive minima/negative maxima rates were estimated and compared with the values obtained from Eqs. (31) and (32). In these simulations, ωg and ξg are assumed to be constant and different values for each of the parameters ωf0 , ωfn , ξf0 , and ξfn are considered. Figures 3 and 4 compare the zero crossing and positive minima/negative maxima cumulative counts for the simulated record for two arbitrary sets of filter parameters with those obtained by analytic formulas described before, respectively.


13

FIGURE 3 Comparison of different methods in estimating the cumulative zero-crossings for simulated records using Rice’s Formula, Eq. (29), Kedem’s Formula, Eq. (32), and the proposed NSMKT model, Eq. (26): (a) fg = 2 Hz, ff0 = 9 Hz, ffn = 1(Hz), ξ f0 = 0.8, ξ fn = 0.1; (b) fg = 2 Hz, ff0 = 3 Hz, ffn = 7Hz, ξ f0 = 0.8, ξ fn = 0.1.

FIGURE 4 Comparison of different methods in estimating the positive minima and negative maxima for simulated records using Rice Formula, Eq. (30), Kedem Formula, Eq. (32). The proposed NSMKT model, Eq. (26): (a) fg = 2 Hz, ff0 = 8 Hz, ffn = 1Hz, ξ f0 = 0.9, ξ fn = 0.1; (b) fg = 2 Hz, ff0 = 8 Hz, ffn = 1 Hz, ξ f0 = 0.5, ξ fn = 0.5. Although the proposed Eqs. (31) and (34) are obtained by means of some simplifying assumptions, it can be observed that there is a good agreement between the result of realtime simulation and the estimated values of zero-crossings using the proposed relations. As it is shown, while Rice formula for estimating the zero crossing works properly, it fails to accurately predict the positive minima/negative maxima for the same records. This is due to the fact that Rice’s formula is based on Gaussian stationary continuous signals whereas the records simulated by the new method are not necessarily Gaussian or even stationary. However, since the assumption of stationarity in a small time range could be considered acceptable, thus the difficulties that arise from violation of such a property are ignorable. Using Gaussian assumption the coherence between signal and its time derivative in Eq. (14) is ignored which can be the source of minor deviation seen from the actual

14


simulated curves. Moreover, using the continuous formula for a discrete system may lead to leakage and aliasing phenomena. Although leakage problem cannot be compensated analytically, aliasing is taken into account via careful consideration of maximum useable frequency. Having examined the sources of errors, it is possible to considerably reduce them by using Kedem’s formula which is based on discrete formulation as the following: π ω 2k + S (ω) dω d E ZCk+1 1 −π cos (ω) sin 2 . = acos π ω 2k dt 2π t sin S (ω) dω −π

(33)

2

In which ZCk+1 is previously defined and t is the time increment ofthe record. Integrating + . For example, the above equation with respect to t gives the cumulative count of E ZCk+1 in order to calculate the number of zero crossings and positive minima/negative maxima of the signal, the following equations can be used:

E ZC

+

t = 0

t NegMax&PosMin = 0

1 acos 2π t

π

1 acos 2π t t

− 0

−π

π

1 acos 2π t

cos (ω) S (ω, τ ) dω π dτ . −π S (ω, τ ) dω

−π

(34)

2 cos (ω) sin ω2 S (ω, τ ) dω dt π ω 2 −π sin 2 S (ω, τ ) dω

π

−π

cos (ω) S (ω, τ ) dω π dt −π S (ω, τ ) dω

Figure 4 depicts the time domain simulations for positive minima/negative maxima cumulative curves compared to those obtained using Rice and Kedem’s formulas. It is worthy to notice that since the positive minima/negative maxima rate is highly affected by high frequencies, it is more sensitive to the formula used for estimation. There is still some difference between the exact values and the results obtained using Kedem’s formula especially at the beginning of the record, but it can be considered acceptable knowing the simplicity of the proposed method. It should be pointed out that instead of using predicted values, one can use time domain simulations to determine the crossings with higher computation time. This advantage of the closed-form solution over real-time evaluation will be appreciated later when it is necessary to estimate the crossing properties of the system for any specific model parameters. For the time being it can be concluded that Eqs. (31) and (34) can be used to estimate the first- and second-order crossings, respectively, for the new model proposed here.

4. Record Simulation Using the Proposed Model In this part a procedure to generate an ensemble of acceleration records whose time and frequency characteristics are in proper agreement with those of a target record is described. The target records that are used here are no pulse-like, near-field, earthquakes records with strike-slip type of mechanism and a rupture distance less than 10 km from the fault, and are selected from the PEER database. Table 2 in the Appendix presents the properties of the selected earthquake records.

15

30 95 145 147 148 149 160 165 189 233 235 240 251 252 253 254 255 256 257 258 259 261 262 263 264

NGA#

Parkfield Managua- Nicaragua-01 Coyote Lake Coyote Lake Coyote Lake Coyote Lake Imperial Valley-06 Imperial Valley-06 Imperial Valley-06 Mammoth Lakes-02 Mammoth Lakes-02 Mammoth Lakes-04 Mammoth Lakes-07 Mammoth Lakes-07 Mammoth Lakes-07 Mammoth Lakes-07 Mammoth Lakes-07 Mammoth Lakes-07 Mammoth Lakes-08 Mammoth Lakes-08 Mammoth Lakes-08 Mammoth Lakes-08 Mammoth Lakes-08 Mammoth Lakes-08 Mammoth Lakes-08

Event 1966 1972 1979 1979 1979 1979 1979 1979 1979 1980 1980 1980 1980 1980 1980 1980 1980 1980 1980 1980 1980 1980 1980 1980 1980

Year

TABLE 2 Properties of the database records

Cholame - Shandon Array #5 Managua- ESSO Coyote Lake Dam (SW Abut) Gilroy Array #2 Gilroy Array #3 Gilroy Array #4 Bonds Corner Chihuahua SAHOP Casa Flores Convict Creek Mammoth Lakes H. S. Convict Creek Fish & Game (FIS) Green Church Long Valley Fire Sta Mammoth Elem School USC Cash Baugh Ranch USC McGee Creek Inn Cashbaugh (CBR) Convict Lakes (CON) Fish & Game (FIS) Long Valley Fire Sta Mammoth Elem School USC Convict Lakes USC McGee Creek Inn

Station 6.19 6.24 5.74 5.74 5.74 5.74 6.53 6.53 6.53 5.69 5.69 5.7 4.73 4.73 4.73 4.73 4.73 4.73 4.8 4.8 4.8 4.8 4.8 4.8 4.8

Mag 9.6 4.1 6.1 9 7.4 5.7 2.7 7.3 9.6 9.5 9.1 5.3 5.6 5.3 6.3 8.5 9.4 4.7 9.7 6.8 5.7 5.9 8.6 6.7 4.4

Rrup (km) 289.6 288.8 597.1 270.8 349.9 221.8 223 274.5 338.6 338.5 370.8 338.5 338.5 338.5 338.5 338.5 338.5 338.5 338.5 370.8 338.5 338.5 338.5 370.8 338.5

Vs30 (m/s)

(Continued)

0.25 0.38 0.38 0.25 0.25 0.25 0.12 0.06 0.25 0.62 0.62 0.25 0.25 0.38 0.88 1 0.62 1.12 0.25 1 0.62 0.75 0.5 1 1.25

Low freq. (Hz)

16

269 317 320 321 448 461 547 558 585 727 901 1111 1165 1611 1612 1615 1617 1618 1631 2019 2020 2048 2049 2734

NGA#

Victoria- Mexico Westmorland Mammoth Lakes-10 Mammoth Lakes-11 Morgan Hill Morgan Hill Chalfant Valley-01 Chalfant Valley-02 Baja California Superstition Hills-02 Big Bear-01 Kobe- Japan Kocaeli- Turkey Duzce- Turkey Duzce- Turkey Duzce- Turkey Duzce- Turkey Duzce- Turkey Upland Gilroy Gilroy Yorba Linda Yorba Linda Chi-Chi- Taiwan-04

Event

TABLE 2 (Continued)

1980 1981 1983 1983 1984 1984 1986 1986 1987 1987 1992 1995 1999 1999 1999 1999 1999 1999 1990 2002 2002 2002 2002 1999

Year Victoria Hospital Sotano Salton Sea Wildlife Refuge Convict Creek Convict Creek Anderson Dam (Downstream) Halls Valley Zack Brothers Ranch Zack Brothers Ranch Cerro Prieto Superstition Mtn Camera Big Bear Lake - Civic Center Nishi-Akashi Izmit Lamont 1058 Lamont 1059 Lamont 1062 Lamont 375 Lamont 531 Pomona - 4th & Locust FF Gilroy - Gavilan Coll. Gilroy Array #3 Anaheim - Hwy 91 & Weir Cyn Rd Anaheim - Lakeview & Riverdale CHY074

Station 6.33 5.9 5.34 5.31 6.19 6.19 5.77 6.19 5.5 6.54 6.46 6.9 7.51 7.14 7.14 7.14 7.14 7.14 5.63 4.9 4.9 4.26 4.26 6.2

Mag 7.3 7.8 6.5 7.7 3.3 3.5 6.4 7.6 4.5 5.6 9.4 7.1 7.2 0.2 4.2 9.2 3.9 8 7.3 8.6 10.0 8.8 9.4 6.2

Rrup (km) 274.5 191.1 338.5 338.5 488.8 281.6 271.4 271.4 659.6 362.4 338.5 609 811 424.8 424.8 338 424.8 659.6 229.8 729.6 349.9 376.1 345.4 553.4

Vs30 (m/s)

0.25 0.1 0.19 0.5 0.12 0.25 0.14 0.12 0.12 0.38 0.12 0.12 0.12 0.07 0.07 0.06 0.19 0.07 0.5 0.39 0.39 0.39 0.39 0.31

Low freq. (Hz)


17

The proposed model is a LTV system described by Eq. (28) using NSMKT type of IRF defined in Eq. (26) as h (t − τ , τ , λ (τ )) and a piecewise modulating function, q (t), introduced by Amin and Ang [1966]. The considered modulating function includes 6 parameters T0 , T1 , T2 , α1 , α2 andα3 , while the IRF itself has the variables ωg , ξg , ωf 0 , ωfn , ξf 0 , ξfn already described in Sec. 3. Therefore, the proposed model has 12 unknown parameters which should be determined for proper simulation of the target record. Assuming independent amplitude and frequency non-stationarity for the model, the procedure of parameter identification can be divided into two parts for modulating function and IRF parameters, respectively.

4.1. The Optimal Modulating Function and IRF Parameters Employing the Arias intensity of the target record, and performing extensive sensitivity analyses, the portion between 1% and 99% of the maximum intensity is considered as the effective duration of the record. The upper and lower limits of the Arias intensity are chosen in way that the insignificant zero-padding and noise of the beginning and ending parts of the record do not interfere in the evaluation of the crossing statistics. The best modulating function parameters are obtained using curve-fitting modules and optimizing the difference between the effective target record intensity and calculated intensity from the assumed modulating function. Figure 5 shows a target record and its optimized modulating function obtained using the prescribed procedure. The IRF parameters are determined according to a scheme that does not depend on amplitude non-stationarity. For the proposed model, the crossing statistics of the acceleration record will be used to determine the optimized parameters. Considering an objective function based on zero-crossing and positive minima/negative maxima of the signal as the following, the parameters of the IRF would be obtained such that the objective function become minimum: ' 2 ' 2 PMNMSim − PMNMTarget + ZCSim − ZCTarget , F ωg , ξg , ωf 0 , ωfn , ξf 0 , ξfn = (35)

FIGURE 5 Piecewise modulating function obtained using curve-fitting procedure.

18


where PMNM and ZC are the cumulative count of positive minima/negative maxima and zero-crossing of the signal, respectively. The subscripts “Sim” and “Target” refer to the simulated and target records, accordingly. This is a strictly nonlinear optimization problem that includes stochastic functions PMNMSim and ZCSim and should be repeatedly evaluated for any assumed set of six parameters. Instead of real-time simulation, in this study the analytical expressions given by Eqs. (31) and (34) are used to estimate the ZC and PMNM rates, respectively. With the aid of analytical formulas, the gradient based methods can be used to find the optimized parameters in a very short time. It is clear that the functions used in Eq. (35) are complicated and any optimization procedure (except heuristic methods) needs good initial values for the procedure to optimally locate the global minima for the problem. As an acceptable initial values, the parameters of the corresponding stationary synthetic record are used which are obtained from nonlinear fitting of modified Kanaii-Tajimi’s PSD to the PSD of target record. Then, the feasible range for each parameter is searched to get the best fit between the target and the simulated record in terms of different level crossings.After obtaining all 12 parameters of the model and simulating the records using Eq. (28), a high-pass filter (HPF) should be applied to the output signals in order to eliminate the unwanted low-frequency components. The high-pass frequency used in this study has the following transfer function: 2 H (ω) =

ω ωc

1−

2

ω ωc 2

− 2iξc

, ω ωc

(36)

where ωc and ξc are called HPF corner frequency and damping ratio. The HPF parameters are determined in a way to minimize the difference between the simulated and the target record’s low-frequency content. Figures 6–8 illustrate the efficiency of the proposed procedure in simulating the no-pulse-like near-field records. These figures display the response parameters of the simulated and target records for comparison purposes. Figure 8 shows the comparison between spectral acceleration of four target records and their respective simulated ones.

4.3. Discussion on the Results The proposed model in this study is used to regenerate the target records with a predefined evolutionary spectrum shape. The EPSD of the proposed model possess only two dominant frequencies, for which the determination of the optimum values requires higher order crossings of the target signals. Having assumed a fixed EPSD function, the model tries to find the parameters which can simulate the target EPSD with acceptable accuracy. Knowing the complicated nature of the ground motion physics, it is apparent that this simplified spectrum will not exactly match the spectrum of the considered ground motions. However, it is desired to obtain the model parameters which can capture the general trend of the target records’ evolutionary spectrum. In this procedure, it is necessary to make a compromise between the representations of the general or “approximate level” features of records versus their local or “detailed level” features, respectively. While being concerned about the total shape of the spectral responses, it is observed that the simulated records’ mean is close to those of the target records. For example, in the case of Kobe 1995 Nishi-Akashi record (Fig. 8c), the local strong harmonics with periods around 0.3 s, 0.5 s, and 2.5 s are underestimated in the spectrum of simulated records. However, the comparison of the power spectrum of the simulated and the target records (Fig. 12) reveals that


19

FIGURE 6 The simulation results for the Mammoth Lakes 05/31/80, Long Valley Fire Station. the simulated PSD is in good agreement with those of original records at the prescribed periods. This phenomenon is the results of smoothing and simplification inherent in the model as discussed above. The results of the simulation for the Mammoth Lakes May 30, 1980 earthquake (Fig. 6a) show an acceptable agreement between the simulated and the target spectral responses except for the period range of (0.4–2.0 s). The investigation of the resulted power spectrum (Fig. 6d) indicates that both of (0.4 s and 2.0 s) corresponds to two prominent harmonics of the original signal. Even though the local maxima are fairly captured in the mean simulated spectra, because of the limitation of the predefined EPSD, the region between these two values are underestimated. On the other hand, many researchers such as Yeh and Wen [1990], Alamilla et al. [2001], Chang et al. [1982] did not provide any information with regard to the standard deviation of the spectral responses of their models. Although Saragoni and Hart [1974] presented the deviation of the results of Arias intensity and zero-crossing from mean, but they did not provide any information about the spectral responses. The results of Rofooei et al. [2001] indicates a coefficient of variation (C.O.V) of 100% for the peak SPA and up to 25% for the PGA values. The results of Aimiri et al. [2014] indicates a less than 50% C.O.V. The results of Conte and Peng [1996] for Capitola 1989 earthquake shows that the maximum C.O.V is almost 100% for large periods and 17% for the PGA. The results of Polhemus and Cakmak [1981] indicate less than 15% C.O.V for PGA simulation which this

20


FIGURE 7 The simulation results for the Duzce 11/12/99, (Lamont Station 1058).

value grows as period increases. Table 3 shows the average value of the C.O.V in different regions of the SPA spectra. Even though the obtained C.O.V is not as high as some other studies, it can be said that it is in the range of the other models cited here. However, since the response of a Gaussian zero-mean input is also Gaussian [Lutes and Sarkani, 2004], the concept of firstcrossing time of symmetric barrier given by Vanmarcke [1975] could be used for the present case if the conditions of slow-varying characteristics of the generated ground motions are maintained. Vanmarcke defined the cumulative distribution function of the peak absolute response over a time duration Td (denoted here as RTd = max |R (t)|) as: Td

$

FRτ (r) = 1 − exp −

2 &

s 2

⎡

%

⎤ Td 1 − exp − π2 δe s ⎢ ⎥ 2

exp ⎣−ν ⎦,r > 0 s exp 2 − 1

(37)

where s=

" r = r/ λ0 σR

(38)


21

FIGURE 8 The simulation results compared to target records for (a) Yorbalinda 2002, Anaheim - Hwy, (b) Kobe 1995, Nishi-Akashi, (c) Baja Calif. Eq. 1987, Cerro Prieto, (d) Chalfant Valley 1986, Zack Brothers Ranch.

1 σ˙ ν= R = π σR π

λ2 . λ0

(39)

In which s is the normalized barrier level and ν is the mean % zero-crossing rate of the oscilλ2

lator’s response. Defining δe = δ 1.2 , the parameter δ = 1 − λ0 λ1 2 is the shape factor of the response’s power spectrum calculated using the first 3 spectral moments of the response power spectrum λ0 , λ1 , λ2 . For such a case, the mean and standard deviation of the response spectrum can be obtained using: R¯ τ = pσR σRτ = qσR

(40)

where p and q are the peak factors dependent on the first three spectral moments and the duration Td . For the case of 10 ≤ νTd ≤ 1000 and 0.11 ≤ δ ≤ 1 one can have (from Der Kiureghian, 1980):

22


TABLE 3 The sample C.O.V value of the response spectrum of the ground motions simulated using the proposed model Coefficient of variation Record Name Mammoth Lakes 05/31/80, Long Valley Fire Station Duzce 11/12/99, (Lamont Station 1058) Yorbalinda, 09/03/2002, Anaheim-HWY 91 & Weir Cyn RD, 090 Baja Calif EQ, 02/07/87, 03:45:14, Cerro Prieto, 251 Kobe 01/16/95 2046, Nishi-Akashi, 000 (Cue Chalfant Valley 07/21/86 14:42, Zack Brothers Ranch

p=

Figure

PGA

0–1 s

1–5 s

5–10 s

6

0.37

0.44

0.54

0.48

7

0.17

0.21

0.36

0.40

8a

0.14

0.30

0.31

0.28

8b

0.13

0.28

0.32

0.28

8c

0.22

0.27

0.29

0.35

8d

0.21

0.30

0.39

0.34

" 0.5772 2 ln (ve Td ) + √ 2 ln (ve Td )

(41)

1.2 5.4 q= √ − 2 ln (ve Td ) 13 + (2 ln (ve Td ))3.2 and * ve =

1.6δ 0.45 − 0.38 v δ < 0.69 . v δ ≥ 0.69

(42)

It can be shown that as δ grows, the p/q factor which is an indication of the C.O.V. of the response spectrum, decreases. It is known that, δ is larger for wide-band processes and since the proposed model has multiple dominant frequencies, its bandwidth is expected to be bigger. Therefore, in comparison to the studies that have used single dominant frequencies such as Rezaeian and Der Kiureghian [2008], the resulted response spectrum is expected to have lower standard deviation. It is also noteworthy that ratio p/q decreases as νTd becomes larger, and the C.O.V for the proposed model increases as period grows. Since νTd generally builds up as the frequency of the oscillator increases, these two observations are in agreement with each other. In order to get a perspective of the efficiency of the proposed model in accurately simulating the records, the numerical relative error between the target and simulated records’ mean spectral displacement (SD) computed for different period ranges within 0.0–10 s. The statistical distribution of relative SD errors for all of 106 target records is generated as a discrete cumulative distribution function shown in Figure 9. This figure illustrates the distribution function as a measure of proposed method’s efficiency. Since the frequency content of the accelerograms is affected by seismological parameters of the ground motion, its dependency of the record simulation efficiency is of main


23

FIGURE 9 The cumulative distribution function for the mean SD error for different period ranges. concern. Figure 11 depicts the general trend of the SD relative error with respect to the moment magnitude (Mw), Shear Velocity (Vs30 ) and Rupture Distance (Rrup ). Each figure includes two curves for low-frequency and high-frequency range of the response spectrum. According to the Fig. 11, the simulation of the records with greater moment magnitudes is less successful in the larger period region while it is more effective in the low period region. However, it seems that soil type and rupture distance are not important factors on the performances of the method for higher periods of the spectrum. Nevertheless, the proposed model does better for lower range of periods when target records are recorded on stiffer soil types. The overall goodness of simulation measures indicate that the model has performed fairly well considering the fact that only the time domain characteristics of the records have been utilized. It should be noted that the proposed method only for 20% of the whole database, which includes 53 pairs of records, leads to a mean spectral error less than 20%. One should observe that the SD errors are estimated for periods as large as 10 s. This is a region where low-frequency content dominates the results and the stochastic methods cannot generally do well in synthesizing the earthquake records. The SD relative estimation error for 0–5 s and 0–1 s period ranges are also shown in the same figure. It is clear that the simulation method yields better results for high-frequency region of the response spectra. Figure 10 shows the cumulative distribution of relative error for simulated and target record’s zero-crossing. The simulation results indicate very good agreement between the simulated and target records zero-crossing and positive minima/negative maxima characteristics. Given only two constraints, i.e., zero-crossing and positive minima/negative maxima, to determine the optimal model parameters, multiple sets of answers may be obtained. In other words, the initial values used for the optimization process may not lead to the actual optimized parameters. The introduced method here does not involve any complicated transformation and is easy to relate its results to the record’s time-domain characteristics. It is clear that there is a relationship between a signal’s spectrum and its crossing statistics. It can be easily proved that the zero-crossing of the signal is related to its dominant frequency or the so-called “central frequency.” The time-variance of the model parameters enables it to efficiently capture the frequency evolution of the signal and its compatibility to the target signal according to the response spectrum. The proposed method’s ability to capture three

24


FIGURE 10 The cumulative distribution function for the mean zero-crossing error.

FIGURE 11 Dependence of simulation error on the seismological parameters of the earthquake.

different frequency regions are compared against stationary MKT and also Rezaeian and DerKiureghian’s model in Fig. 12 for Kobe 1995, Nishi Akashi earthquake. Figure 13 displays the superiority of the proposed method over the stationary version, considering the relative error of the computed displacement response spectrum.


25

FIGURE 12 The Fourier Amplitudes of the simulated records using three methods: Rezaeian and Der Kiureghian, Modified Kanaii-Tajimi, and NSMKT for Kobe 1995, Nishi Akashi.

FIGURE 13 The empirical distribution for the mean relative SD error for non-stationary and stationary modified Kanaii-Tajimi models.

It should be emphasized that even though the classic modified Kanaii-Tajimi method has some advantages over other stationary methods in synthesizing recorded accelerograms, it cannot capture the frequency content of the near-field, no pulse-like records which have a time-varying PSD as well as multiple dominant frequencies. Despite having 12 unknown parameters in the proposed model, a minimization procedure could easily lead to the best result if appropriate initial values are used. The proper initial values as it was explained could be determined using the stationary parameters obtained from regression analysis. Compared to the similar works based on wavelet or Hilbert-Huang methods, this method focuses on a feasible and time-efficient method to simulated earthquake records

26


with minimum number of parameters possible, without the need to use complicated mathematical transformation. In other words, even though an insight can be obtained from the non-stationary characteristics of a recorded ground motion using wavelet or even Hilbert Huang transforms, they still are not suitable to be used for the purpose of scenariobased record simulation in which model parameters are related to different seismological characteristics of the event.

5. Conclusion In this study a new method has been introduced that can capture the non-stationary features of near-field, no-pulse-like records of strike slip mechanism and resynthesize them. The proposed procedure for record simulation is mainly based on time-varying characteristics of the target signals which are zero-crossing and positive minima/negative maxima. The main advantage of the introduced method is its time domain detection of optimal parameters instead of frequency region. The model involves two main parts for capturing the non-stationary characteristics of the signal from amplitude and frequency points of view in a separate manner. Each part consists of six parameters by which the best approximation of the records can be achieved. In order to determine the model parameters, the Arias intensity of the target record is fitted with the Arias intensity of the model obtained through calculation of the piecewise modulating function opted here. To determine the frequency-related parameters of the model, the crossing behavior of the signal including its zero-crossings as well as the positive minima/negative maxima count is used as a measure of instantaneous dominant frequency and band-width indicators. An objective function has been defined according to the goodness of fit between the actual records crossing curves and the simulated ones. In order to efficiently calculate the crossing counts of the signal resulted from a set of model parameters, the analytic formulas for the output of the model crossings is developed and used instead of lengthy process of actual real-time simulation of the records. The results obtained for 53 pairs of near-field records indicates the significance of considering double dominant frequency in time instead of one, as well as careful assessment of the time-variant characteristics of the frequency content.

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