**2. Simulation method**

On the other hand, realistic acceleration time-histories should be employed in structural analysis to reduce the uncertainties in estimating the standard engineering parameters (Hutchings 1994), particularly for non-linear seismic behavior of structures. Thus, designers need to know the dynamic characteristics of predicted ground motion consistent with source rupture for a particular site to be able to adequately design an earthquake-resistant structure. Hall et al. (1995), Makris (1997), Chopra and Chintanapakdee (2001), Zhang and Iwan (2002) have experimentally as well as analytically studied the elastic and inelastic response of engineering structures subjected to actual near-fault records or simplified waveforms intend‐

During the past decades, much effort has been given in reliable simulation of strong ground motion from finite faults through methodologies that include theoretical or semi-empirical modeling of the parameters affecting shape, duration and frequency content of the strong motion records. Due to unavailability of strong recorded ground motion, simulation of ground motion has been carried out using the stochastic method proposed by Boore (2003). The ground motion spectrum has been generated by Atkinson and Boore model (1995). Even though the success of the point-source model has been pointed out repeatedly, it is also well known that it often breaks down, especially near the sources of large earthquakes. Recently, Beresnev and Atkinson (1997) have proposed a technique that overcomes the limitation posed by the hypothesis of a point source. Their technique is based on the original idea of Hartzell (1979) to model large events by the summation of smaller ones. In Beresnev and Atkinson (1997), the high-frequency seismic field near the epicentre of a large earthquake is modeled by subdivid‐ ing the fault plane into a certain number of sub-elements and summing their contributions, with appropriate time delays, at the observation point. Each element is treated as a point source. A stochastic model is used to calculate the ground motion contribution from each subelement, while the propagation effects are empirically modeled. Combining the stochastic method with the finite fault source model, Silva (1997), Beresnev and Atkinson (1998), Motazedian and Atkinson (2005) have proposed different methods, which could be effective

Two Californian earthquake events may be characterized as historical milestones related to near-source ground motions: the 1966 Parkfield and the 1971 San Fernando earthquakes. The 1966 Parkfield, California, event provided the now famous Station 2 (C02) record at a distance of only 80 m from the fault break (Housner and Trifunac 1967). Modern quantitative analysis of strong ground motion observations was started with this record. Aki (1968) and Haskell (1969) demonstrated that the observed transverse (i.e., fault-normal) displacement component of this ground motion record, which exhibited a simple impulsive form, was precisely what is expected for a right-lateral strike-slip rupture propagating from northwest to southeast. The 1971 San Fernando, California, earthquake provided the equally well-known Pacoima Dam (PCD) record. The strike-normal velocity component of this record also exhibited an impulsive character that several investigators attempted to model (e.g., Boore and Zoback 1974; Niazy 1975; Bouchon 1978). In addition, this record was the one that made earthquake engineers recognize the severe implications of the impulsive characteristics of near-source ground

ing to represent the typical ground motion pulses observed in near-field regions.

56 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

for simulating or predicting near-field ground motions.

motions on flexible structures.

A simple and powerful method for simulating ground motions is to combine parametric or functional descriptions of the ground motion's amplitude spectrum with a random phase spectrum modified such that the motion is distributed over a duration related to the earthquake magnitude and to the distance from the source. This method of simulating ground motions often goes by the name ''Stochastic method''. It is particularly useful for simulating the higherfrequency ground motions of most interest to engineers (generally, *f* > 1 *Hz*), and it is widely used to predict ground motions for regions of the world in which recordings of motion from potentially damaging earthquakes are not available. One of the essential characteristics of the method is that it distills what is known about the various factors affecting ground motions (source, path, and site) into simple functional forms.

#### **2.1. Stochastic finite-fault simulation method**

In this study, the Stochastic Method is used for simulating the strong ground motion. The method assumes that the far-field accelerations on an elastic half space are band-limited, finiteduration, white Gaussian noise, and that the source spectra are described by a single cornerfrequency model whose corner frequency depends on earthquake size (Mayeda and Malagnini 2009). The ground spectrum *Y (M0, R, f)* is conveniently broken into several simple functions – the Earthquake Source *(E)*; the Path *(P)*; the Site *(G)* and the instrument or type of motion *(I)*:

$$E\left(\boldsymbol{M}\_{0},\boldsymbol{R}\_{\prime}\boldsymbol{f}\right) = E\left(\boldsymbol{M}\_{0},\boldsymbol{f}\right)P\left(\boldsymbol{R},\boldsymbol{f}\right)G\left(\boldsymbol{f}\right)I\left(\boldsymbol{f}\right) \tag{1}$$

( ) ( ) ( ) 0 00 ,,, *a b SM f S M f S M f* = ´ (3)

Simulation of Near-Field Strong Ground Motions using Hybrid Method

http://dx.doi.org/10.5772/55682

By adopting the source spectrum model AB95 (Atkinson and Boore model 1995), the above equation for source spectrum is rewritten considering the seismic moment dependence of the

( ) 0 0 2 2

1 1

4 *sso R VF <sup>C</sup>* pr b*R*

Here, <*RΘΦ* > accounts for the radiation pattern (≈ 0.55); *V* represents the partition of total shear wave energy into horizontal components (= 0.707); *F* accounts the effect of free surface (≈ 2); *ρs* and *β<sup>s</sup>* are the density and shear wave velocity of the bedrock; *Ro* is a reference distance and usually taken as 1 *km*. The corner frequencies *fa* and *fb* are obtained from the seismic moment

<sup>0</sup> log 2.41 0.533 *<sup>a</sup>*

<sup>0</sup> log 1.431 0.188 *<sup>b</sup>*

is given by the multiplication of the geometrical spreading and *Q* functions:

The path effects are represented by simple functions that describe the geometric spreading function, attenuation (intrinsic and scattering attenuation), and the general increase of duration with distance due to wave propagation and scattering. The simplified path effect, *P*,

( ) ( ) exp

<sup>=</sup> í ý ï ï î þ

*Rf P ZR*

*Q*

*QfC* ì ü ï ï p

e

ï ï - = + í ý

*a b*

 e

QF < > <sup>=</sup> (5)

*f M* = - (6)

*f M* = - (7)

(4)

59

(8)

*f f f f*

ï é ù é ùï + + ï ê ú ê úï î þ ëû ëû

ì ü

<sup>1</sup> ,

above factors *Sa* in terms of corner frequencies *fa* and *fb*:

*E M f CM*

where, *C* is a constant given by

using the following relations

The Source duration is evaluated as 0.5/*fa.*

where, *M0* is the seismic moment, *R* is the shortest distance from the fault to the site and *f* is the frequency. Atkinson and Boore model (1995) is used to obtain the ground motion spectrum. The process of strong ground motion simulation is depicted as a flowchart in Figure 1.

**Figure 1.** Flow chart showing the structure of the FOTRAN program for Atkinson and Boore model (1995)

The source spectrum, *E*, is obtained by the following equations specifying both the shape and the amplitude as a function of the earthquake size:

$$E\left(M\_{0\prime},f\right) = \mathbb{C}\,M\_0\,\mathbb{S}\left(M\_{0\prime},f\right) \tag{2}$$

Simulation of Near-Field Strong Ground Motions using Hybrid Method http://dx.doi.org/10.5772/55682 59

$$\mathcal{S}\left(M\_{0'}f\right) = \mathcal{S}\_a\left(M\_{0'}f\right) \times \mathcal{S}\_b\left(M\_{0'}f\right) \tag{3}$$

By adopting the source spectrum model AB95 (Atkinson and Boore model 1995), the above equation for source spectrum is rewritten considering the seismic moment dependence of the above factors *Sa* in terms of corner frequencies *fa* and *fb*:

$$E\left(M\_{0'},f\right) = \text{CM}\_0\left\{\frac{1-x}{1+\left[\bigwedge\_{a}^{f}f\_a\right]^2} + \frac{x}{1+\left[\bigwedge\_{f\_b}^{f}f\_b\right]^2}\right\}\tag{4}$$

where, *C* is a constant given by

method is that it distills what is known about the various factors affecting ground motions

In this study, the Stochastic Method is used for simulating the strong ground motion. The method assumes that the far-field accelerations on an elastic half space are band-limited, finiteduration, white Gaussian noise, and that the source spectra are described by a single cornerfrequency model whose corner frequency depends on earthquake size (Mayeda and Malagnini 2009). The ground spectrum *Y (M0, R, f)* is conveniently broken into several simple functions – the Earthquake Source *(E)*; the Path *(P)*; the Site *(G)* and the instrument or type of motion *(I)*:

where, *M0* is the seismic moment, *R* is the shortest distance from the fault to the site and *f* is the frequency. Atkinson and Boore model (1995) is used to obtain the ground motion spectrum. The process of strong ground motion simulation is depicted as a flowchart in Figure 1.

**Figure 1.** Flow chart showing the structure of the FOTRAN program for Atkinson and Boore model (1995)

the amplitude as a function of the earthquake size:

The source spectrum, *E*, is obtained by the following equations specifying both the shape and

() () 0 00 *E M f CM S M f* , , = (2)

( ) ( ) ( ) ()() 0 0 *YM Rf EM f PRf G f I f* ,, , , = (1)

(source, path, and site) into simple functional forms.

58 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

**2.1. Stochastic finite-fault simulation method**

$$C = \frac{ VF}{4\pi\rho\_s\beta\_s R\_o} \tag{5}$$

Here, <*RΘΦ* > accounts for the radiation pattern (≈ 0.55); *V* represents the partition of total shear wave energy into horizontal components (= 0.707); *F* accounts the effect of free surface (≈ 2); *ρs* and *β<sup>s</sup>* are the density and shear wave velocity of the bedrock; *Ro* is a reference distance and usually taken as 1 *km*. The corner frequencies *fa* and *fb* are obtained from the seismic moment using the following relations

$$1\log f\_a = 2.41 - 0.533M\_0 \tag{6}$$

$$
\log f\_b = 1.431 - 0.188M\_0 \tag{7}
$$

The Source duration is evaluated as 0.5/*fa.*

The path effects are represented by simple functions that describe the geometric spreading function, attenuation (intrinsic and scattering attenuation), and the general increase of duration with distance due to wave propagation and scattering. The simplified path effect, *P*, is given by the multiplication of the geometrical spreading and *Q* functions:

$$P = Z\{R\}\exp\left\{\frac{-\pi Rf}{\mathcal{Q}\{f\}\mathcal{C}\_Q}\right\} \tag{8}$$

The relation between distance and geometrical spreading function, *Z(R)*, is given by the following function

$$Z\left(R\right) = \frac{1}{70} \sqrt{\frac{130}{R}}\tag{9}$$

Boore suggested the values of ε and η to be 0.2 and 0.05, respectively and *fTgm* = 2 based on Saragoni and Hart (1974). The windowed WGN is converted to frequency domain and normalized by its root mean square amplitude. The entire process of obtaining WGN is shown in Figure 2. Then, the ground motion spectrum, shown in Figure 2, is multiplied with the normalized windowed noise to get the Fourier amplitude spectrum as shown in Figure 2.

and *Q(f)* is the frequency dependent quality factor which is given by the following equation

where, *I* = (-1)0.5. n = 0, 1, 2 for ground displacement, velocity and acceleration, respectively.

The attenuation or diminution operator *D(f)* accounts for the path-independent loss of high frequency in the ground motions. A simple multiplicative filter can account for the diminution of the high frequency motions. Here, *fmax* is 10 *Hz*. The diminution factor

The particular type of ground motion resulting from the simulation is controlled by the filter *I(f)*. If ground motion is desired, then

A time domain simulation is carried out to get the actual Fourier amplitude spectrum. A White Gaussian Noise (WGN) is produced

Simulation of Near-Field Strong Ground Motions using Hybrid Method

http://dx.doi.org/10.5772/55682

61

Boore suggested the values of ε and η to be 0.2 and 0.05, respectively and *fTgm* = 2 based on Saragoni and Hart (1974). The windowed WGN is converted to frequency domain and normalized by its root mean square amplitude. The entire process of obtaining WGN is shown in Figure 2. Then, the ground motion spectrum, shown in Figure 2, is multiplied with the normalized

The path duration function of 0.05R is calculated from Atkinson and Boore (1995).

 1 130 70

0.8 *Qf f* 100 (10)

*<sup>R</sup>* (9)

is calculated based on the following equation

(11)

and windowed off using a windowing function given below

 

(14)

windowed noise to get the Fourier amplitude spectrum as shown in Figure 2.

Figure 2. Basis of procedure for simulating ground motions using the stochastic method **Figure 2.** Basis of procedure for simulating ground motions using the stochastic method

**2.2. Analytical model proposed by Mavroeidis and Papageorgiou (2003)**

For near-field strong ground motions, most of the elastic energy arrives coherently in a single, intense, relatively long period pulse at the beginning of record, representing the cumulative effect of almost all the seismic radiation from the fault. The phenomenon is even more pronounced when the direction of slip on the fault plane points toward the site as well. The Mavroeidis and Papageorgiou model (2003) adequately describes the impulsive character of near-faults ground motions both qualitatively and quantitatively. In addition, it can be used to analytically reproduce empirical observations that are based on available near-source records. The input parameters of the model have an unambiguous physical meaning. The proposed analytical model has been calibrated using a large number of actual near-field ground motion records. It successfully simulates the entire set of available near-fault displace‐ ment, velocity, and (in many cases) acceleration time histories, as well as the corresponding deformation, velocity, and acceleration response spectra. An "objective" definition of the pulse duration is given based on model input parameters. In addition, Mavroeidis and Papageorgiou (2003) investigate the scaling characteristics of the model parameters with earthquake magnitude. Also, Mavroeidis et al. (2004) derive the Fourier transform of the analytical model and identify the parameters that have the most significant effect on the spectral characteristics of the model. Finally, a simplified (adequate for engineering purposes) method is proposed for the synthesis of near-fault ground motions. The pulse duration (or period), the pulse

0.5 <sup>8</sup>

 ;,, exp *<sup>b</sup> W t t att ctt* 

*b*

ln 1 ln 1

 

(13)

<sup>1</sup> <sup>10</sup> *<sup>f</sup> D f*

*Z R*

 <sup>2</sup> *<sup>n</sup> I f* (12)

> 

*Tgm gm*

exp

*a ie*

where,

 

*b c b tf T* 

and *Q(f)* is the frequency dependent quality factor which is given by the following equation

$$Q(f) = 100 \, f^{0.8} \tag{10}$$

The path duration function of 0.05R is calculated from Atkinson and Boore (1995).

The attenuation or diminution operator *D(f)* accounts for the path-independent loss of high frequency in the ground motions. A simple multiplicative filter can account for the diminution of the high frequency motions. Here, *fmax* is 10 *Hz*. The diminution factor is calculated based on the following equation

$$D\left(f\right) = \left\{1 + \left(\bigvee\_{1\mid 0} \right)^{8}\right\}^{-0.5} \tag{11}$$

The particular type of ground motion resulting from the simulation is controlled by the filter *I(f)*. If ground motion is desired, then

$$I = -\left(2\pi f\right)^{\mu} \tag{12}$$

where, *I* = (-1)0.5. n = 0, 1, 2 for ground displacement, velocity and acceleration, respectively.

A time domain simulation is carried out to get the actual Fourier amplitude spectrum. A White Gaussian Noise (WGN) is produced and windowed off using a windowing function given below

$$\mathcal{W}\left(t; \varepsilon, \eta, t\_{\eta}\right) = a\left(t/t\_{\eta}\right)^{b} \exp\left\{-c\left(t/t\_{\eta}\right)\right\} \tag{13}$$

where,

$$\begin{aligned} a &= \left\{ \exp\{i\} / e \right\}^b \\ b &= -\left( \varepsilon \ln \eta \right) \Big/ \left[ 1 + \varepsilon \left( \ln \varepsilon - 1 \right) \right] \\ c &= b / \varepsilon \\ t\_\eta &= f\_{T\_{\text{Sym}}} \times T\_{\text{Sym}} \end{aligned} \tag{14}$$

Boore suggested the values of ε and η to be 0.2 and 0.05, respectively and *fTgm* = 2 based on Saragoni and Hart (1974). The windowed WGN is converted to frequency domain and normalized by its root mean square amplitude. The entire process of obtaining WGN is shown in Figure 2. Then, the ground motion spectrum, shown in Figure 2, is multiplied with the normalized windowed noise to get the Fourier amplitude spectrum as shown in Figure 2. ln 1 ln 1 *Tgm gm b c b tf T* (14) Boore suggested the values of ε and η to be 0.2 and 0.05, respectively and *fTgm* = 2 based on Saragoni and Hart (1974). The windowed WGN is converted to frequency domain and normalized by its root mean square amplitude. The entire process of obtaining WGN is shown in Figure 2. Then, the ground motion spectrum, shown in Figure 2, is multiplied with the normalized

and *Q(f)* is the frequency dependent quality factor which is given by the following equation

where, *I* = (-1)0.5. n = 0, 1, 2 for ground displacement, velocity and acceleration, respectively.

The attenuation or diminution operator *D(f)* accounts for the path-independent loss of high frequency in the ground motions. A simple multiplicative filter can account for the diminution of the high frequency motions. Here, *fmax* is 10 *Hz*. The diminution factor

The particular type of ground motion resulting from the simulation is controlled by the filter *I(f)*. If ground motion is desired, then

A time domain simulation is carried out to get the actual Fourier amplitude spectrum. A White Gaussian Noise (WGN) is produced

The path duration function of 0.05R is calculated from Atkinson and Boore (1995).

 1 130 70

0.8 *Qf f* 100 (10)

*<sup>R</sup>* (9)

is calculated based on the following equation

(11)

and windowed off using a windowing function given below

 

0.5 <sup>8</sup>

 ;,, exp *<sup>b</sup> W t t att ctt* 

(13)

<sup>1</sup> <sup>10</sup> *<sup>f</sup> D f*

*Z R*

 <sup>2</sup> *<sup>n</sup> I f* (12)

> 

*b*

exp

*a ie*

where,

The relation between distance and geometrical spreading function, *Z(R)*, is given by the

*<sup>R</sup>* <sup>=</sup> (9)

( ) 0.8 *Qf f* = 100 (10)

(12)

(11)

(14)

( ) 1 130 70

The path duration function of 0.05R is calculated from Atkinson and Boore (1995).

( )

e h

*b c b*

=

=

h

{ ( ) }

eh

*Tgm gm*

exp

*tf T*

= ´

e

*a ie*

*<sup>f</sup> D f*

and *Q(f)* is the frequency dependent quality factor which is given by the following equation

The attenuation or diminution operator *D(f)* accounts for the path-independent loss of high frequency in the ground motions. A simple multiplicative filter can account for the diminution of the high frequency motions. Here, *fmax* is 10 *Hz*. The diminution factor is calculated based on

<sup>1</sup> <sup>10</sup>


The particular type of ground motion resulting from the simulation is controlled by the filter

(2 ) *<sup>n</sup> I f* = p

where, *I* = (-1)0.5. n = 0, 1, 2 for ground displacement, velocity and acceleration, respectively.

A time domain simulation is carried out to get the actual Fourier amplitude spectrum. A White Gaussian Noise (WGN) is produced and windowed off using a windowing function given

> ( ;,, ) ( ) exp{ ( )} *<sup>b</sup> W t t att ctt* hh

> > ( ) ( )

=- + - é ù

*b*

ln 1 ln 1

 e e

ë û

 h

= - (13)

0.5 <sup>8</sup>

*Z R*

60 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

following function

the following equation

below

where,

*I(f)*. If ground motion is desired, then

Figure 2. Basis of procedure for simulating ground motions using the stochastic method **Figure 2.** Basis of procedure for simulating ground motions using the stochastic method

windowed noise to get the Fourier amplitude spectrum as shown in Figure 2.

#### **2.2. Analytical model proposed by Mavroeidis and Papageorgiou (2003)**

For near-field strong ground motions, most of the elastic energy arrives coherently in a single, intense, relatively long period pulse at the beginning of record, representing the cumulative effect of almost all the seismic radiation from the fault. The phenomenon is even more pronounced when the direction of slip on the fault plane points toward the site as well. The Mavroeidis and Papageorgiou model (2003) adequately describes the impulsive character of near-faults ground motions both qualitatively and quantitatively. In addition, it can be used to analytically reproduce empirical observations that are based on available near-source records. The input parameters of the model have an unambiguous physical meaning. The proposed analytical model has been calibrated using a large number of actual near-field ground motion records. It successfully simulates the entire set of available near-fault displace‐ ment, velocity, and (in many cases) acceleration time histories, as well as the corresponding deformation, velocity, and acceleration response spectra. An "objective" definition of the pulse duration is given based on model input parameters. In addition, Mavroeidis and Papageorgiou (2003) investigate the scaling characteristics of the model parameters with earthquake magnitude. Also, Mavroeidis et al. (2004) derive the Fourier transform of the analytical model and identify the parameters that have the most significant effect on the spectral characteristics of the model. Finally, a simplified (adequate for engineering purposes) method is proposed for the synthesis of near-fault ground motions. The pulse duration (or period), the pulse amplitude, as well as the number and phase of half cycles are the key parameters that define the waveform characteristics of near-fault velocity pulses. Therefore, an analytical model with four parameters in principle should suffice to describe the entire set of velocity pulses generated due to forward directivity or permanent translation effects. Seismologists have used "wavelets" (also referred to as "signals," "signatures," or "pulses"), particularly in fields such as seismic filtering, wavelet processing, wave-propagation modelling, and trace inversion (Hubral and Tygel 1989). Although, various wavelets have been proposed in the literature, only a limited number of them are popular and frequently used in practice. In this study, the analytical wavelet signal proposed by Mavroeidis and Papageorgiou (2003) is chosen and expressed by

$$f(t) = A \frac{1}{2} \left| 1 + \cos\left(\frac{2\pi f\_p}{\mathcal{Y}} t\right) \right| \cos\left(2\pi f\_p t + \nu\right) \tag{15}$$

This problem is easily resolved by limiting the time interval of the signal as follows

$$-\frac{\gamma}{2f\_p} \le t \le \frac{\gamma}{2f\_p} \tag{16}$$

zero crossings increases); and *t0* specifies the epoch of the envelope's peak. The analytical expressions for the ground acceleration and displacement time histories, compatible with the

0 0

Simulation of Near-Field Strong Ground Motions using Hybrid Method

0 0

(21)

pg

n

 pg  pg

g

 pg

 g

> g

g

2 2 2

*p p p*

g

*otherwisw*

, 1

 g

> g

g

http://dx.doi.org/10.5772/55682

g ü ï ï ï ï ï ï ï ï ý ï

þ

(19)

63

(20)

(22)

(23)

ground velocity given by equation (18), are

*p*

*p*

sin cos 2

sin 2 1 cos

 n

( ( ) ) ( )

2 1

( )

*<sup>f</sup> ftt t t*

g

g

0

( )

 g

+ + >+

previously. We define the normalized time variable as

0 2

*p p*

*f f*

p

<sup>1</sup> sin , 4 2 <sup>1</sup>

<sup>1</sup> sin , 4 2 <sup>1</sup>

*<sup>A</sup> Ctt f f*

n pg

n pg

<sup>ï</sup> <sup>=</sup> <sup>í</sup> - + <-

sin 2 sin

 n

 g

0 2

*p p*

2

gp

( ) ( )

*<sup>A</sup> d t Ctt*

g

*p*

p

2 1

g g

g

g


( ) ( )

rewriting equations (19) and (20), as

p

g

 g

( )

n

2

g

*d t t*

p

p

p 2

g

( ) ( )

n pg

n pg

1 1 sin , <sup>4</sup> <sup>1</sup>

( ) ( )

1 1 sin , <sup>4</sup> <sup>1</sup>

0, *<sup>p</sup>*

p

> g

( ) ( )

0 0

p

( ) ( )

*a t <sup>f</sup> f f ftt t t*

0 0

 n

g

g

1 2 1

*p*

0 0

*p*

 g

p

*<sup>f</sup> tt ftt A f t t t with*

p

<sup>ì</sup> é ù æ ö <sup>ü</sup> <sup>ï</sup> ê ú ç ÷ - é ù - ++ <sup>ï</sup> ë û <sup>ï</sup> ê ú <sup>ï</sup> è ø ïï- ê ú - ££ + > ïï <sup>=</sup> <sup>í</sup> ê ú é ù æ ö <sup>ý</sup> <sup>ï</sup> ê ú é ù -+ + - ê ú ç ÷ <sup>ï</sup> ë û ç ÷ <sup>ï</sup> ê ú ê ú è ø <sup>ï</sup> ë û ë û <sup>ï</sup> <sup>ï</sup> ïî ïþ

g

 n

( )

n

, 1 <sup>4</sup> 2 1 2 2 <sup>1</sup> sin

<sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup>

A parametric study in terms of *ν* and *γ* of the normalized (with respect to *fp* and *A*) acceleration, velocity, and displacement pulses can be performed based on the equations presented

Then, the normalized acceleration and displacement time histories can be expressed by

( ) <sup>0</sup> 2 *<sup>p</sup> t ftt* = p

( ) ( ) sin cos sin 1 cos , ( ) ( ) <sup>1</sup>

n

<sup>ì</sup> <sup>ü</sup> <sup>ï</sup> é ù æö æö - + <sup>ï</sup> <sup>ï</sup> ê ú + + ç÷ ç÷ + + + - ££ > <sup>ï</sup> <sup>ï</sup> ë û - + èø èø <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> = = <sup>í</sup> - <- <sup>ý</sup> <sup>ï</sup> - <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> + > <sup>ï</sup> <sup>ï</sup> - ïî ïþ

 n

1 1 11 1 sin sin sin , <sup>1</sup>

 gg

 gg

*t t t t with*

ì ü é ù æ ö æ ö æ ö ï ï - ê ú ç ÷ + + + + - ££ > ç ÷ ç ÷ = = í ý ê ú ë û è ø è ø è ø ï ï î þ

*Af otherwisw*

ng

4 21 2 1

 gg

gg

*t*

 pg

 pg

*t t a t <sup>t</sup> <sup>t</sup> t with*

*p p p p*

*<sup>A</sup> C t t t with <sup>f</sup> f f <sup>f</sup> t t*

<sup>ì</sup> é ù é ù - <sup>ï</sup> ê ú - ++ ê ú - ++ <sup>ï</sup> ê ú - ê ú ë û <sup>ï</sup> + - ££ + > <sup>ï</sup> é ù <sup>+</sup> <sup>ï</sup> ê ú - + <sup>ï</sup> <sup>+</sup> ë û ë û <sup>ï</sup>

 g

( )

( )

0,

p

p

ï ï

> p

î

*a t*

( ) ( ) ( )

*d t*

*A f*

*p*

*p*

p

g

The period of the harmonic oscillation should be smaller than the period of the envelope represented by the elevated cosine function in order to produce physically acceptable signals; that is,

$$\frac{1}{f\_p} < \frac{\gamma}{f\_p} \Rightarrow \gamma > 1\tag{17}$$

The combination of equations (15) to (17) yields the formulation of the proposed analytical model for the near-fault ground velocity pulses:

$$\nu\_{\nu}(t) = \begin{bmatrix} A \frac{1}{2} \left[ 1 + \cos\left(\frac{2\pi f\_p}{\nu} (t - t\_0) \right) \right] \cos\left(2\pi f\_p \left(t - t\_0 \right) + \nu\right)\_{\prime} \\ 0 & \text{, otherwise} \end{bmatrix} \tag{18}$$

where, parameter *A* controls the amplitude of the signal; *fp* is the frequency of the amplitudemodulated harmonic (or the prevailing frequency of the signal); *ν* is the phase of the amplitudemodulated harmonic (i.e., *ν* =0 and *ν* = ± *π* / 2 define symmetric and antisymmetric signals, respectively); *γ* is a parameter that defines the oscillatory character (i.e., zero crossings) of the signal (i.e., for small *γ* the signal approaches a deltalike pulse; as *γ* increases, the number of zero crossings increases); and *t0* specifies the epoch of the envelope's peak. The analytical expressions for the ground acceleration and displacement time histories, compatible with the ground velocity given by equation (18), are

amplitude, as well as the number and phase of half cycles are the key parameters that define the waveform characteristics of near-fault velocity pulses. Therefore, an analytical model with four parameters in principle should suffice to describe the entire set of velocity pulses generated due to forward directivity or permanent translation effects. Seismologists have used "wavelets" (also referred to as "signals," "signatures," or "pulses"), particularly in fields such as seismic filtering, wavelet processing, wave-propagation modelling, and trace inversion (Hubral and Tygel 1989). Although, various wavelets have been proposed in the literature, only a limited number of them are popular and frequently used in practice. In this study, the analytical wavelet signal proposed by Mavroeidis and Papageorgiou (2003) is chosen and

> ( ) ( ) <sup>1</sup> <sup>2</sup> 1 cos cos 2

g

2 2 *p p t f f*

<sup>1</sup> <sup>1</sup>

( ) ( 0 0 ) ( ( ) ) <sup>1</sup> <sup>2</sup> 1 cos cos 2 , <sup>2</sup>

ì ü é ù æ ö ï ï ê ú + - ç ÷ - + <sup>=</sup> í ý ê ú ë û è ø ï ï î þ

*p*

p

g

*<sup>f</sup> <sup>A</sup> tt ftt <sup>t</sup>*

*p p f f* g < Þ> g

model for the near-fault ground velocity pulses:

n

0 ,

 g

The period of the harmonic oscillation should be smaller than the period of the envelope represented by the elevated cosine function in order to produce physically acceptable signals;

The combination of equations (15) to (17) yields the formulation of the proposed analytical

*otherwise*

where, parameter *A* controls the amplitude of the signal; *fp* is the frequency of the amplitudemodulated harmonic (or the prevailing frequency of the signal); *ν* is the phase of the amplitudemodulated harmonic (i.e., *ν* =0 and *ν* = ± *π* / 2 define symmetric and antisymmetric signals, respectively); *γ* is a parameter that defines the oscillatory character (i.e., zero crossings) of the signal (i.e., for small *γ* the signal approaches a deltalike pulse; as *γ* increases, the number of

*p*

p

*<sup>f</sup> ft A t ft* p

This problem is easily resolved by limiting the time interval of the signal as follows

g

é ù æ ö = + ê ú ç ÷ <sup>+</sup> ç ÷ ê ú ë û è ø

*p*

*p*

 n


(17)

 n (15)

(18)

p

2

62 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

expressed by

that is,

$$a(t) = \begin{bmatrix} -A\pi f\_p \\ -\frac{A\pi f\_p}{\mathcal{I}} \begin{bmatrix} \sin\left(\frac{2\pi f\_p}{\mathcal{I}}(t-t\_0)\right) \cos\left[2\pi f\_p \left(t-t\_0\right) + \nu\right] + \\ \gamma \sin\left[2\pi f\_p \left(t-t\_0\right) + \nu\right] \left[1 + \cos\left(\frac{2\pi f\_p}{\mathcal{I}}(t-t\_0)\right)\right] \end{bmatrix} \begin{bmatrix} t\_0 - \frac{\mathcal{I}}{2f\_p} \le t \le t\_0 + \frac{\mathcal{I}}{2f\_p} \text{ with } \gamma > 1 \\\\ \text{otherwise} \end{bmatrix} \tag{19}$$

$$d(t) = \begin{bmatrix} \frac{A}{4\pi f\_p} \left[ \sin\left(2\pi f\_p (t - t\_0) + \nu\right) + \frac{1}{2} \frac{\gamma}{1 - \gamma} \sin\left[\frac{2\pi f\_p (1 - \gamma)}{\gamma} (t - t\_0) + \nu\right] + \nu \right] \\\\ \frac{A}{2\pi f\_p} \left[ \frac{1}{2} \frac{\gamma}{1 - \gamma} \gamma \sin\left[\frac{2\pi f\_p (1 + \gamma)}{\gamma} (t - t\_0) + \nu\right] \right] \\\\ d(t) = \frac{A}{4\pi f\_p} \frac{1}{\gamma} \frac{1}{\left(1 - \gamma^2\right)} \sin\left(\nu - \pi\right) + C\_r \ t < t\_0 - \frac{\gamma}{2f\_p} \\\\ \frac{A}{4\pi f\_p} \frac{1}{\left(1 - \gamma^2\right)} \sin\left(\nu + \pi\right) + C\_r \ t > t\_0 + \frac{\gamma}{2f\_p} \\\\ \end{bmatrix} \tag{20}$$

A parametric study in terms of *ν* and *γ* of the normalized (with respect to *fp* and *A*) acceleration, velocity, and displacement pulses can be performed based on the equations presented previously. We define the normalized time variable as

$$
\overline{t} = 2\pi f\_p \left( t - t\_0 \right) \tag{21}
$$

Then, the normalized acceleration and displacement time histories can be expressed by rewriting equations (19) and (20), as

$$\overline{a}\left(\overline{\tau}\right) = \frac{a(t)}{Af\_p} = \left| -\frac{\pi}{\gamma} \left[ \sin\left(\frac{\overline{\tau}}{\gamma}\right) \cos\left(\overline{\tau} + \nu\right) + \gamma \sin\left(\overline{\tau} + \nu\right) \left(1 + \cos\left(\frac{\overline{\tau}}{\gamma}\right)\right) \right], -\pi\gamma \le \overline{\tau} \le \pi\gamma \text{ with } \gamma > 1\right.\tag{22}$$

$$\text{where}$$

$$\overline{A}\left(\overline{\tau}\right) = \frac{d(t)}{Af\_p}\left| -\frac{1}{4f\_f} \frac{1}{\gamma} \sin\left(\frac{\gamma - 1}{\gamma}\overline{\tau} + \nu\right) + \frac{1}{2} \frac{\gamma}{\gamma - 1} \sin\left(\frac{\gamma + 1}{\gamma}\overline{\tau} + \nu\right)\right|, -\pi\gamma \le \overline{\tau} \le \pi\gamma \text{ with } \gamma > 1\right.\tag{23}$$

$$\begin{aligned} \overline{A}\left(\overline{\tau}\right) &= \frac{d(t)}{Af\_p}\left| -\frac{1}{4\pi} \frac{1}{\left(1 - \gamma^2\right)} \sin\left(\nu - \pi\gamma\right), \ \overline{\tau} < -\pi\gamma\\ \frac{1}{4\pi} \frac{1}{\left(1 - \gamma^2\right)} \sin\left(\nu + \pi\gamma\right), \ \overline{\tau} &> \pi \end{aligned}\right.\tag{23}$$

Assuming that the duration of the pulse is independent of the source–station distance for stations located within ~10 km from the causative fault, the pulse period and the moment magnitude are related through the following empirical relationship obtained by least-squares fit analysis:

$$
\log T\_p = -2.2 + 0.4M\_W \tag{24}
$$

coincides with the phase of the Fourier transform of the synthetic time history generated

3. Calculate the Fourier transform of the synthetic acceleration time histories generated in steps 1 and 2.

2. For the selected fault–station geometry, generate the synthetic acceleration time histories for the moment magnitude, *MW*,

Simulation of Near-Field Strong Ground Motions using Hybrid Method

http://dx.doi.org/10.5772/55682

65

4. Subtract the Fourier amplitude spectrum of the synthetic time history generated in step 1 from the Fourier amplitude

5. Construct a synthetic acceleration time history so that (a) its Fourier amplitude spectrum is the difference of the Fourier amplitude spectra calculated in step 4; and (b) its phase coincides with the phase of the Fourier transform of the synthetic time

6. Superimpose the time histories generated in steps 1 and 5. The near-source pulse is shifted in time so that the peak of its

The Zagros region is one of the most seismically active regions in Iran. The Tombak LNG terminal is located along the Persian Gulf northern coast, south of the Zagros Mountains, which mark the deforming zone separating Arabia (Arabian plate) and Central Iran (Eurasian plate) (Figure 3(a)). Location of Tombak area is presented in Figure 3(b). The massive LNG storage tanks exist in this terminal. These tanks have high importance from engineering and economical point of view so seismic loads should be considered in their analysis and design. The relevant codes of LNG storage containers emphasize that a comprehensive seismic hazard

West of the Makran coast, where oceanic crust is subducting beneath Eurasia, the collision of the Arabian shield with Iran has uplifted the Zagros Mountains. The Zagros Mountains belt represents the early stage of a continental collision between the Arabian plate and the central Iran continental blocks. The Zagros Mountains are a seismically active region. Seismicity is restricted to the region between the Main Zagros Thrust and the Persian Gulf. Strong earthquakes are thought to occur on blind active thrust faults, which do not reach the surface. Fault plane solutions of these earthquakes indicate displacement mainly on low to high-angle reverse faults at depth of 6-12 km in the uppermost part of the basement. Most of the earthquakes for the region have generally M = 5.0 to 6.5, and have originated on sources beneath the decollement (Berberian 1995). Subduction on the main Zagros thrust has now ceased and it is seismically inactive (Ni and Barazangi 1986) except for the northern Zagros, where the surface trace of the thrust has been reactivated as right-slip main recent fault. The Zagros active fold-thrust belt lies on the north-eastern margin of the Arabian plate, on Precambrian (Pan-African) basement. It is composed of Cambrian to Neogene's folded series and is the result of five major tectonic events (Berberian and King 1981; Berberian 1983). The Zagros fold-thrust belt is composed of five units. The folds are parallel to the thrust faults. The axial part of the folds, striking NW·SE, appears as broad asymmetrical folds with axial planes dipping to the NE and North. Their north-eastern limbs gently dip (20°) to the NW whereas their south-western limbs are steeper (40°) to the SE reaching 60 to 80°down slope and in some cases are nearly vertical, overturned or thrusted. The Main Zagros Thrust Fault (MZTF) indicates a fundamental change in sedimentary and structural evolution and seismicity. It marks the geosuture between the two colliding plates of the Eurasia and the Arabia. The global zone taken into account lies between 32°N

**6.** Superimpose the time histories generated in steps 1 and 5. The near-source pulse is shifted in time so that the peak of its envelope coincides with the time that the rupture front passes

The Zagros region is one of the most seismically active regions in Iran. The Tombak LNG terminal is located along the Persian Gulf northern coast, south of the Zagros Mountains, which mark the deforming zone separating Arabia (Arabian plate) and Central Iran (Eurasian plate) (Figure 3(a)). Location of Tombak area is presented in Figure 3(b). The massive LNG storage tanks exist in this terminal. These tanks have high importance from engineering and econom‐ ical point of view so seismic loads should be considered in their analysis and design. The relevant codes of LNG storage containers emphasize that a comprehensive seismic hazard investigation should be conducted for regional seismicity and earthquake events of known

investigation should be conducted for regional seismicity and earthquake events of known near-fault.

(a) (b)

envelope coincides with the time that the rupture front passes in front of the station.

**3. The seismotectonic and seismicity of Tombak region** 

Figure 3. Location map: (a) Zagros folded zone, and (b) Tombak area

**Figure 3.** Location map: (a) Zagros folded zone, and (b) Tombak area

and 26°N in latitude and 50°E and 58°E in longitude.

**4. Estimation of the model parameters** 

**3. The seismotectonic and seismicity of Tombak region**

spectrum of the synthetic time history produced in step 2.

specified previously, using the specific barrier model.

in step 2.

near-fault.

in front of the station.

history generated in step 2.

In this section, we propose a very simplified methodology for generating realistic synthetic ground motions that are adequate for engineering analysis and design. We exploit the simple analytical model introduced in the present work to describe the coherent (long-period) component of motion and the stochastic (or engineering) approach to synthesize the incoherent (high-frequency) seismic radiation (for a review of the stochastic approach of ground motion synthesis; see Boore (1983) and Shinozuka (1988)). For the latter component of motion, due to the proximity of the point of observation to the source, it is necessary to use a source model that provides guidance as how to distribute the available seismic moment of the simulated event on the fault plane. Such a source model is the specific barrier model of Papageorgiou and Aki (1983). According to this model, an earthquake is visualized as a sequence of equalsize sub-events uniformly distributed on a rectangular fault plane. At the present time, the proposed mathematical model along with its scaling laws can take into account (with confi‐ dence) only for the forward directivity effect. Even though the analytical expression can replicate near-fault ground motion records that manifest the permanent-translation effect as well, the limited number of recordings with permanent translation does not permit the derivation of appropriate scaling laws. Therefore, the proposed analytical model should be utilized with caution for the generation of synthetic long-period ground motions that intend to incorporate the permanent translation effect. In these cases, the permanent offsets of the synthetic displacement time histories should be compatible with the tectonic environment and earthquake magnitude of the simulated event. The proposed methodology is written in MATLAB with the following steps:


4. Subtract the Fourier amplitude spectrum of the synthetic time history generated in step 1 from the Fourier amplitude

5. Construct a synthetic acceleration time history so that (a) its Fourier amplitude spectrum is the difference of the Fourier

northern coast, south of the Zagros Mountains, which mark the deforming zone separating Arabia (Arabian plate) and Central Iran (Eurasian plate) (Figure 3(a)). Location of Tombak area is presented in Figure 3(b). The massive LNG storage tanks exist in this terminal. These tanks have high importance from engineering and economical point of view so seismic loads should be considered in their analysis and design. The relevant codes of LNG storage containers emphasize that a comprehensive seismic hazard

uplifted the Zagros Mountains. The Zagros Mountains belt represents the early stage of a continental collision between the Arabian plate and the central Iran continental blocks. The Zagros Mountains are a seismically active region. Seismicity is restricted to the region between the Main Zagros Thrust and the Persian Gulf. Strong earthquakes are thought to occur on blind active thrust faults, which do not reach the surface. Fault plane solutions of these earthquakes indicate displacement mainly on low to high-angle reverse faults at depth of 6-12 km in the uppermost part of the basement. Most of the earthquakes for the region have generally M = 5.0 to 6.5, and have originated on sources beneath the decollement (Berberian 1995). Subduction on the main Zagros thrust has now ceased and it is seismically inactive (Ni and Barazangi 1986) except for the northern Zagros, where the surface trace of the thrust has been reactivated as right-slip main recent fault. The Zagros active fold-thrust belt lies on the north-eastern margin of the Arabian plate, on Precambrian (Pan-African) basement. It is composed of Cambrian to Neogene's folded series and is the result of five major tectonic events (Berberian and King 1981; Berberian 1983). The Zagros fold-thrust belt is composed of five units. The folds are parallel to the thrust faults. The axial part of the folds, striking NW·SE, appears as broad asymmetrical folds with axial planes dipping to the NE and North. Their north-eastern limbs gently dip (20°) to the NW whereas their south-western limbs are steeper (40°) to the SE reaching 60 to 80°down slope and in some cases are nearly vertical, overturned or thrusted. The Main Zagros Thrust Fault (MZTF) indicates a fundamental change in sedimentary and structural evolution and seismicity. It marks the geosuture between the two colliding plates of the Eurasia and the Arabia. The global zone taken into account lies between 32°N

coincides with the phase of the Fourier transform of the synthetic time history generated in step 2. 2. For the selected fault–station geometry, generate the synthetic acceleration time histories for the moment magnitude, *MW*,

**6.** Superimpose the time histories generated in steps 1 and 5. The near-source pulse is shifted in time so that the peak of its envelope coincides with the time that the rupture front passes in front of the station. specified previously, using the specific barrier model. 3. Calculate the Fourier transform of the synthetic acceleration time histories generated in steps 1 and 2.
