The cross-over frequency between pulse and high-frequency components of near-fault forward-directivity ground motions

Abstract

Due to the scarcity of near-fault records with forward-directivity effects, synthetic near-fault pulse-like ground motions through superimposing pulse models and high-frequency components are commonly used in the earthquake engineering. However, in existing studies, the cross-over frequency in generating high-frequency components is usually empirical and unclear. In this article, a hybrid decomposition and resynthesis method is developed to quantify the cross-over frequency, in which the wavelet decomposition and high-pass filter are, respectively, used to get the pulse and high-frequency components for near-fault records. Using the 30 near-fault pulse-like records, the distribution of cross-over frequencies is obtained. It is interestingly found that the cross-over frequency is inversely proportional to the pulse period.

Keywords

cross-over frequency high-frequency components hybrid method near-fault ground motions pulse

Introduction

A near-fault site, that is located within about 20 km from the fault rupture, may experience the near-fault effects. Comparing to far-field ground motions, the near-fault ground motions often possess some distinguishing features, such as the long-period velocity pulse and concentrated energy at the beginning of the records, permanent displacement and abundant high-frequency components (Abrahamson and Somerville, 1996; Bray and Rodriguez-Marek, 2004; Somerville et al., 1997). As a result, significant seismic demands may be imposed for civil structures when the near-fault ground motions attack (Bray and Rodriguez-Marek, 2004). Severe damages have been observed in historic earthquakes, including the 1994 Northridge, 1995 Kobe, 1999 Chi-Chi earthquakes and so on (Mavroeidis et al., 2004). Obviously, it is of significant necessity to investigate the near-fault effects on civil structures.

The near-fault pulse-like ground motions can be generally categorized into two classes, namely, forward-directivity motions and fling-step motions. The forward-directivity motions usually occur in the perpendicular direction to strike-slip faults when the fault rupture propagates towards the site with a velocity almost equal to the shear-wave velocity. The fling-step motions arise parallel to strike and dip directions for the strike-slip and dip-slip faults, respectively. There are significant differences between the forward-directivity motions and fling-step motions in the waveform. For example, the forward-directivity motions possess two-sided pulse(s) in its time-domain velocities and there are no permanent displacements, while the fling-step motions are characterized by a unidirectional large-amplitude velocity pulse and a monotonic step in the displacement time history (Bray and Rodriguez-Marek, 2004; Somerville et al., 1997; Xin et al., 2019). Herein, the near-fault forward-directivity ground motions are the main point of this article. In the following, the near-fault motions refer in particular to the near-fault forward-directivity motions for brevity.

For a long term, the scarcity of near-fault records makes it difficult to conduct the seismic analysis of civil structures under the near-fault ground motions. Thus, numerous studies have tried to use the simple mathematical models to characterize the pulse components, for example, the trigonometric functions (Makris, 1997), piece-wise triangular function (Alavi and Krawinkler, 2001), wavelet form (Mavroeidis and Papageorgiou, 2003; Yang and Zhou, 2015) and some other composite formulations (Fu and Menun, 2004; Menun and Fu, 2002). These simplified representations are quite effective and convenient in qualifying the pulse effects; however, they do not contain any high-frequency content portion. As demonstrated by Ghahari et al. (2006, 2010), the high-frequency components play an important and non-negligible role in seismic responses. With regard to the shortcomings, researchers generated the broadband near-fault ground motions by superposing the artificial or recorded high-frequency components to the pulse models in the seismic analysis. For instance, Mavroeidis and Papageorgious (2003) proposed a procedure for the generation of full near-fault ground motions by combining high-frequency synthetics from a specific barrier model with their proposed mathematical expression for the replication of long-period pulses. Halldórsson et al. (2010) also applied the specific barrier model and the mathematical model of Mavroeidis and Papageorgious (2003) to obtain the broadband near-fault time series. Boundary frequency of 1 Hz was taken during the superposition. The specific barrier model can provide the most complete, yet parsimonious, and self-consistent descriptions for the generation of the high frequencies of ground motions $(\geq 1 Hz)$ in the earthquake faulting processes (Halldórsson et al., 2010), and it is a stochastic source-based model belonging to physical earthquake models. However, stochastic source-based models usually have complex theoretical basis, because they describe the fault rupture process, wave propagation from the source to the site and site responses in a physical-based manner (Liu and Hong, 2013; Zerva, 1988). Besides, the stochastic source-based models are computationally demanding and heavily rely on the identifications of region-specific physical parameters for the source, path and site characteristics. The disadvantages seriously hinder their wide applications in the engineering practice.

Comparing to the stochastic source-based models, the stochastic site-based models are preferred by design engineers, since the stochastic site-based models describe the ground motions for a specific site by fitting to a recorded motion with readily available and simple knowledge of earthquake and site characteristics, such as the magnitude, distance and shear-wave velocity of the soil (Huang and Wang, 2015a, 2015b, 2017; Rezaeian and Der Kiureghian, 2008, 2012). Moreover, the stochastic site-based models are time-efficient. Therefore, the stochastic site-based models have been commonly used to generate the high-frequency contents of near-fault ground motions in the last decades. For instance, Tian et al. (2007) presented an engineering-oriented approach to generate the near-fault ground motions, in which the Clough–Penzien spectrum larger than empirically 1.0 Hz and time–frequency transformation method are jointly employed to get the high-frequency components, just as the far-field synthesis process. Fu and Menun (2004) simulated fault-normal near-fault ground motions for a specified seismic environment by superimposing a stochastic trigonometric-series model with a pulse model. Yang and Zhou (2015) regarded the modified Gabor wavelet as the pulse and the Kanai–Tajimi spectrum as the foundation of high-frequency components; besides, the high-pass filter with a fitting value of 1.18 rad/s is used to filter the low-frequency components in the spectrum transformation to avoid the overflow phenomenon. Dabaghi and Der Kiureghian (2018) performed an interesting study in generating horizontal pairs of the synthetic near-fault ground motion for a specific earthquake source and site, in which the modified pulse model and modulated, filtered white-noise process with a time-varying filter are combined to ensure the pulse-like characteristics and non-stationaries in both temporal and spectral domains. Li et al. (2016) used the Butterworth filtering to decompose the original near-fault records into pulse part and background part, then simply replaced the pulse part with pulse models and finally obtained the artificial ground motions. From the above literatures, it can be seen that the stochastic site-based models in combination with deterministic pulse have become an important strategy in generating the artificial near-fault ground motions. It is noted that different stochastic site-based models can generate the high-frequency components with different period ranges. For example, the spectral representation may fail to match the response spectrum in the long-period ranges (typically greater than 2 s) (Tian et al., 2007). Rezaeian and Der Kiureghian (2008) used a second filter to improve a white-noise-based model in the match to response spectral ordinates for long period as long as 5–10 s.

However, it can also be observed from the above references that there always exists a cut-off frequency for the high-frequency components whether using the stochastic source-based models or site-based models, for example, 1 Hz or 1.18 rad/s. The cut-off frequency is usually considered as the cross-over frequency between the pulse and residual components, which is currently unclear and empirical. Virtually, the cross-over frequency plays a very important role in the synthesis process. When the frequency is inappropriately valued, the quality of artificial near-fault ground motions will be significantly reduced. As shown in Figure 1, if the frequency is too large, some frequency bands may be lost; if the frequency is too small instead, excessive low-frequency parts will be superimposed. Inspired by the artificial synthesis methods, this article presents a hybrid decomposition and resynthesis method to get the distribution characteristics of the cross-over frequency from real records. The results are expected to be useful in generating near-fault ground motions.

Hybrid method

The objective of this article is determining the optimal cross-over frequency between the pulse and high-frequency components from the real records. The hybrid method developed in this article contains three parts: decomposition process, resynthesis process and determining the optimal cross-over frequency.

Decomposing process

As mentioned in section ‘Introduction’, the concept ‘cross-over frequency’ comes from the artificial synthesis methods of near-fault ground motions. Therefore, the first step is to decompose the real records into pulse and high-frequency components following the inverse procedure of artificially generating near-fault ground motions.

Pulse parts

The pulse parts from a given near-fault ground motion can be identified and extracted by the wavelet-based signal processing technique, as outlined in the following steps (Baker, 2007):

1. Perform the continuous wavelet transformation (CWT) for the velocity time history and decompose it into a series of wavelet packet coefficients in the time–frequency domains, which is

C_{s, l} = \int_{- \infty}^{\infty} V (t) Φ_{s, l} (t) dt

(1)

with

Φ_{s, l} (t) = \frac{1}{\sqrt{s}} Φ (\frac{t - l}{s})

where $Φ (•)$ is the mother wavelet function, which is chosen as the Daubechies wavelet of order 4 because it approximates the general two-sided shape of velocity pulses; s and l are the scale parameter and location parameter, respectively; and $V (t)$ is the velocity time history of the record.

2. Identify the maximum coefficient and return the scale parameter and location parameter

{s 1, l 1} = o [max (c_{si, li})]

(2)

where $o [•]$ is the operator returning the scale and location parameter of coefficient [•].

3. Conduct nine times of loop computation and identify the other nine largest wavelet coefficients for the residual components, namely, $c_{si, li}$ and ${s 1, l 1}, i = 2, 3, \dots, 10$ . These coefficients are located in a time window of $\pm (1 / 2)$ the width of the first wavelet $(s 1)$ .

4. Reconstruct the velocity time history only using the 10 wavelet coefficients

V_{p} (t) = \sum_{i = 1}^{10} c_{si, li} Φ_{si, li} (t)

(3)

where $V_{p} (t)$ is the pulse parts.

The extracted pulse can be considered as the best result of various pulse models for the site recorded from the point view of synthesis ground motions.

High-frequency components

The high-frequency components for the given near-fault ground motion can be easily extracted using an ideal high-pass filter, namely

X (f) = FFT [A (t)]

(4)

Y (f) = H (f) \cdot X (f)

(5)

A_{H} (t) = IFFT [Y (f)]

(6)

with

H (f) = {\begin{matrix} 1, f_{c} < | f | \\ 0, | f | \leq f_{c} \end{matrix}

where the high-pass frequency, denoted by $f_{c}$ , is exactly the cross-over frequency between the pulse and high-frequency components; $FFT [•]$ and $IFFT [•]$ are the discrete Fourier transform and inverse Fourier transform operator, respectively; $A (t)$ is the acceleration time history of the record; and $A_{H} (t)$ is the high-frequency acceleration time history.

The extracted high-frequency components can also be regarded as the best result, no matter for the stochastic source-based or site-based models.

Example

In Figure 2, a record is taken as an example, and the decomposing results are plotted, including the original record, pulse and high-frequency components. It can be seen that varying $f_{c}$ can significantly affect the high-frequency components.

Resynthesis process

Superimpose the pulse and high-pass high-frequency components in section ‘Decomposing process’, and the ‘fake’ recorded near-fault ground motions will be generated. The process can be represented by

A_{fake} (t) = A_{P} (t) + A_{H} (t)

(7)

where $A_{P} (t)$ is the acceleration of pulse, derived from $V_{p} (t)$ . It can be easily known that the ‘fake’ recorded near-fault ground motion is related to $f_{c}$ .

Determination of the optimal cross-over frequency

To determine the optimal cross-over frequency, comparisons are made between the recorded and ‘fake’ near-fault ground motions, namely $A (t)$ and $A_{fake} (t)$ . The elastic response spectrum is taken as the reference standard because it is highly sensitive for the frequency components (Ghahari et al., 2010). The optimal cross-over frequency for arbitrary near-fault record can be obtained by

EP (T) = M [A (t)]; E P_{fake} (T) = M [A_{fake} (t)]

(8)

D = \int_{0 s}^{6 s} {| EP (T) - E P_{fake} (T) |}^{2} dT

(9)

f_{c} = \arg min (D)

(10)

under the boundary condition

1 / T_{p} < f_{c} \leq 4 Hz

(11)

where $M [•]$ is the operator of calculating acceleration response spectrum $EP (T)$ , T is the period (s), Tp is the pulse period derived from Baker’s procedure (Baker, 2007). In this way, the pulse of records can be definitely removed during the decomposition process. After that, the linear regression between $EP (T)$ and $E P_{fake} (T)$ is made to judge whether the two response spectra match or not. The coefficient of determination (Pearson correlation coefficient) has to be larger than 90% (Gomes et al., 2006).

As shown in Figure 3, the high-pass frequency $f_{c} = 0.35 Hz$ is the optimal cross-over frequency for the record in section ‘Decomposing process’. It can be seen that a good agreement between the ‘fake’ and original record has been achieved. Moreover, the coefficient of determination is 98.44%.

Distribution of optimal cross-over frequencies

In this section, 30 records of pulse-like near-fault ground motions are compiled into a database, as listed in Tables 1 and 2. All records are taken from stations fell in 20 km of the fault rupture with moment magnitude greater than 6.5 or 15 km greater than 5.5. The digital data for these records were collected from the Pacific Earthquake Engineering Research (PEER) Center database (http://peer.berkeley.edu/ngawest2/). No scaling is applied to the ground motions in order to preserve the pulse-like inherent nature of the as-recorded ground motions. To highlight the forward-directivity effect, the near-fault components of each station have been rotated to fault-normal orientations.

Table 1.

The earthquake events.

No.	Earthquake	Date	Magnitude	Fault mechanism	Slip
1	Coyote Lake	3 September 1979	5.7	Strike slip	336
2	Imperial Valley	15 October 1979	6.5	Strike slip	143
3	Superstition Hills (B)	24 November 1987	6.6	Strike slip	127
4	Kobe	17 January 1995	6.9	Strike slip	50
5	Kocaeli	17 August 1999	7.4	Strike slip	90
6	Duzce	12 November 1999	7.1	Strike slip	90
7	Parkfield	28 September 2004	6.0	Strike slip	138
8	Landers	28 June 1992	7.3	Strike slip	351
9	Northridge	17 January 1984	6.7	Reverse	122

Table 2.

Near-fault ground motion records used in this article.

No.	RSN	Earthquake	Station	R_jb (km)	V ₃₀ (m/s)	Tp_Baker (s)
1	147	1	Gilroy Array #2	8.47	270.84	1.07
2	148	1	Gilroy Array #3	6.75	349.85	1.17
3	149	1	Gilroy Array #4	4.79	221.78	1.05
4	159	2	Agrarias	0	242.05	2.37
5	161	2	Brawley Airport	8.54	208.71	4.21
6	170	2	EC County Centre FF	7.31	192.05	4.00
7	171	2	El Centro-Meloland Geot. Array	0.07	264.57	2.86
8	173	2	El Centro Array #10	8.6	202.85	4.11
9	179	2	El Centro Array #4	4.9	208.91	4.89
10	180	2	El Centro Array #5	1.76	205.63	4.38
11	181	2	El Centro Array #6	0	203.22	3.91
12	182	2	El Centro Array #7	0.56	210.51	3.69
13	184	2	El Centro Differential Array	5.09	202.26	4.88
14	185	2	Holtville Post Office	5.35	202.89	4.73
15	722	3	Kornbloom Road (temp)	18.48	266.01	0.69
16	723	3	Parachute Test Site	0.95	348.69	1.00
17	1106	4	KJMA	0.94	312	0.80
18	1114	4	Port Island (0 m)	3.31	198	2.47
19	1119	4	Takarazuka	0	312	1.78
20	1120	4	Takatori	1.46	256	0.89
21	1176	5	Yarimca	1.38	297	8.75
22	1602	6	Bolu	12.02	293.57	6.99
23	4098	7	Parkfield-Cholame 1E	1.66	326.64	1.25
24	4102	7	Parkfield-Cholame 3 W	2.55	230.57	1.06
25	4115	7	Parkfield-Fault Zone 12	0.88	265.21	1.24
26	879	8	Lucerne	2.19	1369	4.38
27	982	9	Jensen Filter Plant Administrative Building	0	373.07	3.18
28	983	9	Jensen Filter Plant Generator Building	0	525.79	3.18
29	1004	9	LA – Sepulveda VA Hospital	0	380.06	0.94
30	1085	9	Sylmar – Converter Sta East	0	370.52	1.82

RSN: record sequence number.

Following the procedures in section ‘Hybrid method’, the optimal cross-over frequency can be obtained for each record. Figure 4(a) shows the distribution of cross-over frequencies for the records. It can be seen that the cross-over frequency varies from different records. Using the rough 1 Hz or else was not rational in the previous studies. Moreover, it is interestingly found that the optimum cross-over frequencies are inversely proportional to the pulse periods, as shown in Figure 4(b), which can be regressed by equation (12). The frequency may be more suitable in the generation of high-frequency components

f_{c} = 1.423 / Tp + 0.04 \approx 1.423 / Tp

(12)

Conclusion

In the earthquake engineering, the stochastic source-based or site-based models with a pulse model are the most common methods in generating the artificial near-fault ground motions. However, the cross-over frequency between pulse and high-frequency components is currently unclear, which may lead to a false conclusion in the seismic analysis. In this article, a hybrid method is developed to quantify the cross-over frequency for given near-fault forward-directivity records, in which the decomposition and resynthesis procedures are applied to reconstruct the ‘fake’ records. By comparing the response spectra of original records, the quality of the ‘fake’ records is measured and the cross-over frequency is further determined. Taking as-recorded 30 near-fault pulse-like ground motions as research object, the distribution of cross-over frequencies is characterized. It is found that the optimum cross-over frequencies are inversely proportional to the pulse periods. The conclusion may have a better effect when generating the near-fault ground motions.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the Major Research Plan of China National Railway Ministry of China under Grant number P2018G007 and the China Railway Corporation Science and Technology Research and Development Plan under Grant number 2014-Major-1.

ORCID iD

Lifeng Xin

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