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Abstract

The non-stationary properties of the strong ground motions are considered to have dominant influence on the dynamic response of structures. In this article, a new method is proposed to study the instantaneous frequency by constructing analytic signals with two orthogonal horizontal components of ground motion vectors directly. Comparing with the traditional method of constructing analytic signals based on Hilbert transforms, the new method is digestible and simple in calculation, and it can also avoid some associated limits of Hilbert transform. In this article, instantaneous frequency obtained from the new method is simulated with an exponential decay model, and its attenuation rules which can be used for synthesizing design ground motions are investigated. By comparing the attenuation rules between vector and unidirectional ground motions, the difference between new method and traditional method are investigated. The results show that the new method of constructing analytic signals is feasible, and the exponential decay model is effective to describe the change of instantaneous frequency with time.

Keywords

Earthquake engineering ground motion non-stationary property analytic signal instantaneous frequency

Introduction

The conventional properties of amplitude, frequency, and duration represent the basic characteristics of ground motions. And they are important factors to specify and select strong ground motions for structure dynamic analysis and seismic design. While they are not enough to describe all the characteristics of ground motions, there are other nonnegligible properties that need more attention. The non-stationary properties in amplitude and frequency contents are important. Herein “non-stationary” means the property of signal is changing with time. In both time and frequency domains, ground motions are typical non-stationary signals. The non-stationary property has been widely studied,¹ methods to generate non-stationary ground motions,^2,3 especially non-stationary ground motion in two orthogonal horizontal directions,^4,5 have gained increasing attention recently. The non-stationary frequency content is harder to describe than amplitude content; therefore, some methods are put forward, such as Fourier transform, physical spectrum,⁶ Hilbert transform and Hilbert-Huang transform,^7,8 wavelet transform,^9,10 zero-crossing rate,¹¹ and univariate phase spectrum model.¹²

Armstrong found that the frequency modulation was able to reduce disturbances in radio signals, which drew the attention of people to the properties of frequency.¹³ The concept of instantaneous frequency (IF) was initially put forward based on the frequency modulation theory of communication and was further developed benefited from research results of analytic signals. Because of its clear physical meaning and simplicity, Dong¹⁴ used IF to describe non-stationary frequency content of ground motions and achieved satisfactory results.

Both ground motion itself and its structures dynamic response are three-dimensional in nature. It is obvious that researches based on one-dimensional ground motion cannot fully describe the whole characteristics nor reveal the nature of seismic response on structures, it is necessary to consider ground motion as a vector in research.

The traditional method to obtain IF consists of the following processes (1) seismic signal is processed through Hilbert transform to get its orthogonal complex signal; (2) an analytic signal is constructed by the original and its complex signal; and (3) a complex plane is built where IF can be obtained as the derivative of instantaneous phase. However, the calculations of Hilbert transform are complicated and are limited with restrictions, such as the limits related to the Bedrosian¹⁵ theorem and the Nuttall theorem.¹⁶ This article presents a new method to obtain IF, that is, using two orthogonal horizontal components of ground motion to construct vector signal, the derivative of instantaneous phase in real plane is IF. An exponential decay function is used to simulate IF based on optimization algorithm; then the corresponding model parameters are simulated with an attenuation model. Finally, the attenuation rules of IF model parameters between vector and unidirectional ground motion are compared, and the new method and traditional method to obtain IF are compared.

IF of signals

Definition of IF

Before introducing the process of obtaining IF of ground motion vector, the definition and limits of traditional frequency must be introduced to make the concept of IF clearly. Frequency is only physically meaningful for monocomponent signals with invariant frequency, or frequency fluctuates within a narrow range.¹⁷ For multicomponent signal, Fourier transformation can be used to decompose it into sum of harmonic signals with particular frequencies. However, there are some limits that Fourier transformation is only suitable for stationary signals. In addition, the frequency obtained this way only represents global information rather than local properties of signals; therefore, a new concept generalized from constant frequency is necessary.

The concept of IF was initially put forward in the frequency modulation theory by Carson and Fry¹⁸ in 1937. They argued that IF is the rate of phase angle changing with time. IF can describe instantaneous properties and has advantages in the study of non-stationary signals. Balth¹⁹ proposed the definition of IF, based on a harmonic oscillation written in the form

s (t) = a (t) \cdot \cos [\int_{0}^{t} 2 π f (t) dt + φ] = a (t) \cdot \cos [ϕ (t)]

(1)

where $a (t)$ is the envelope, $ϕ$ is initial phase, and $ϕ (t) = \int_{0}^{t} 2 π f (t) dt + φ$ is instantaneous phase. Then IF $f (t)$ is given by

f (t) = \frac{1}{2 π} \cdot \frac{d ϕ (t)}{dt}

(2)

Constructing analytic signal using Hilbert transform

Most signals are not harmonic signals. To obtain the IF of a real non-harmonic signal, the first step is to construct an analytic signal, whose real and imaginary part represent orthogonal information in the complex plane.

Gabor²⁰ proposed a practical method of constructing an analytic signal and laid the foundation for further study. That is, first creating a complex signal from the real one using the Hilbert transform and then combining the two orthogonal signals into an analytic signal. For a real signal $x (t)$ , its complex signal generated by Hilbert transform is

y (t) = \frac{1}{π} PV \int_{- \infty}^{\infty} \frac{x (τ)}{t - τ} d τ

(3)

where PV is the Cauchy principal value. Then a corresponding analytic signal $z (t)$ is constructed

z (t) = x (t) + iy (t)

(4)

With Euler’s formulation, $z (t)$ can be expressed in the form of complex exponential

z (t) = A (t) e^{i θ (t)}

(5)

A (t) = \sqrt{x^{2} (t) + y^{2} (t)}

(6)

θ (t) = \tan^{- 1} (\frac{y (t)}{x (t)})

(7)

In the complex plane constructed by an analytic signal $z (t)$ , $A (t)$ is the instantaneous amplitude, and $θ (t)$ is the instantaneous phase.

The classical definition of the IF $f (t)$ is given as

f (t) = \frac{1}{2 π} \cdot \frac{d}{dt} [\arg z (t)]

(8)

In addition, IF was also expressed as

f (t) = \frac{1}{2 π} \cdot \frac{d θ (t)}{dt}

(9)

Constructing analytic signal using ground motion components

One important step to obtain IF with the aforementioned method is Hilbert Transform. However, it has some disadvantages and is complex in comprehension and computation.

To avoid the aforementioned problems, this article presents a new method which is free from Hilbert Transform. The new model proposes to construct an analytic signal with two orthogonal components of the seismic vector, instead of the original seismic signal and its Hilbert transformation. Equations (4)–(9) are still available in this new case, where $x (t)$ and $y (t)$ represent two orthogonal horizontal time histories from the same ground motion record, that is, two components of horizontal vector $z (t)$ . A real complex plane is constructed naturally and makes parameters physically meaningful.

As Figure 1 shows, $x (t)$ and $y (t)$ represent the projections of vector $z (t)$ on two axes, $θ (t)$ represents instantaneous phase, and $A (t)$ represents instantaneous amplitude. Meanwhile, IF can be deduced from equation (9).

Figure 1.

Envelope and instantaneous phase of signal.

Modeling of IF

IF was modeled by an exponential decay function by Dong¹⁴

f (t) = {\begin{matrix} f_{0} t < t_{0} \\ f_{0} e^{- c_{f} (t - t_{0})} t ⩾ t_{0} \end{matrix}

(10)

θ (t) = \int_{0}^{t} f (t) dt = {\begin{matrix} f_{0} t_{0} t < t_{0} \\ f_{0} t_{0} + \frac{f_{0}}{c_{f}} (1 - e^{- c_{f} (t - t_{0})}) t ⩾ t_{0} \end{matrix}

(11)

where instantaneous phase $θ (t)$ is the summation (integral) of IF, $f_{0}$ represents the initial frequency, $c_{f}$ is the decay coefficient. $f_{0}$ and $c_{f}$ are identified by the optimization algorithm with $θ (t)$ as the target parameter. $t_{0}$ is the start time of the strong part of ground motion records, there are more than one ways to get $t_{0}$ , a common method is to use the time at which 0.5% of the final Arias intensity for the record has been accumulated as $t_{0}$ .²¹ This article uses piecewise intensity envelope model E(t) modified by Peng and Li¹² to get $t_{0}$

E (t) = {\begin{matrix} 0 t < t_{0} \\ I_{0} {(\frac{t - t_{0}}{t_{1} - t_{0}})}^{2} t_{0} < t < t_{1} \\ I_{0} t_{1} < t < t_{2} \\ I_{0} e^{- c (t - t_{2})} t ⩾ t_{2} \end{matrix}

(12)

In this model, $t_{0}$ , $t_{1}$ , and $t_{2}$ represent different time, and $I_{0}$ and c are characteristic parameters.

Results and discussion

Ground motions used

It is a basic step to collect many seismic records and screen them properly for research. Parameters obtained based on enough records can achieve statistical significance and can be applied to structural design as well as structural analysis. This article collected seismic records from the following database institutions: NGA (PEER),²² ISESD (Europe),²³ NIED (Japan),²⁴ and GeoNet (New Zealand).²⁵ Most of the records were collected from NGA and NIED, and only a small part of them came from ISESD and GeoNet.

To make analysis results statistically significant, only corrected records with complete time history were used. And every record is composed of one vertical component and two orthogonal horizontal components. Moreover, records with magnitudes below than 4.0 and peak ground acceleration (PGA) less than 0.2 m/s² are not used. After screening, 4904 seismic records are adopted for following analysis. Since the distributions by magnitude and epicentral distance of those records are uneven, it is inappropriate if they were analyzed with equal weight.²⁶ Therefore, they are classified into several categories by magnitude and epicentral distance (Figure 2). After classification, they are weighted according to the size of their categories in order to improve the confidence of the statistical results.

Figure 2.

Distributions of magnitude and epicentral distance: (a) distribution of magnitude and (b) distribution of epicentral distance.

Fitting results of IF model

Figure 3 shows the time history of vector envelope of record TCU031. Besides its vector envelope, time histories of two orthogonal acceleration components, that is, TCU031-E and TCU031-N, are also represented as references.

Figure 3.

Time history of vector envelope.

Figure 4 shows the estimated results of instantaneous phase and IF. While fitting, amplitude is used as weight to ensure the fitting accuracy of strong motion. The instantaneous phase model derives from the exponential attenuation IF model (equation (10)) which fits the actual instantaneous phase precisely, it indicates that the IF model can simulate the changing rules of the ground motion IF vector accurately. Although the more complicated model can improve the fitting accuracy, this exponential decay model is an appropriate choice considering its practicability, simplicity as well as high fitting accuracy.

Figure 4.

Estimation results of instantaneous phase and instantaneous frequency: (a) estimation result of IP and (b) estimation result of IF.

Attenuation rules

Previous researches on the attenuation rules of ground motion parameters showed that they are dominated by magnitude M, epicentral distance R, and site characteristic. PGA and PGV, respectively, represent PGA and peak ground velocity. In addition, the PGV/PGA ratio is used to represent the site characteristic.¹⁴ After reviewing attenuation models of other ground motion parameters, the following model is used

\lg Y = c_{1} M + c_{2} R + c_{3} \lg (R + 10) + c_{4} (\frac{PGV}{PGA}) + c_{5} + ε

(13)

where Y represents the IF parameter ( $f_{0}$ or $c_{f}$ ), $c_{1}$ to $c_{5}$ are regression coefficients. Regression algorithm is used to determine those parameters. It is found that this model has higher fitting accuracy among others, and regression algorithm is used for it.

Unidirectional non-vector records are always used in previous researches, so besides the IF attenuation rules of vector ground motions, the rules of non-vector ground motion also needed to be explored as references. To remain the information of two ground motion components comprehensively and keep the nature of non-vector, a special “unidirectional ground motion,” whose partial parameters are geometric average of parameters obtained from two components, is constructed. The regression coefficients of both vector and unidirectional ground motions are given in Table 1.

Table 1.

Regression coefficients of instantaneous frequency parameters.

	Y	c1	c2	c3	c4	c5
Vector	f₀	−0.0511	−4.19 × 10^–5	0.00	−0.018	1.269
Vector	c_f	−0.0048	−6.39 × 10^–4	−0.153	2.758	−1.585
Unidirectional	f₀	−0.0495	−5.39 × 10^–5	0.00	−0.125	1.274
Unidirectional	c_f	−0.0208	−1.59 × 10^–4	−0.173	2.918	−1.497

Regression coefficients are substituted into attenuation model equation (13) to ion attenuation rules, Figure 5 shows the results of attenuation rules of $f_{0}$ and $c_{f}$ for ground motion vectors and their variation trend with M, R, and PGV/PGA. Both $f_{0}$ and $c_{f}$ decrease as M and R increase, and increase as PGV/PGA increases. The downward trend of $c_{f}$ with the increase of M appears small or even negligible. With the increase of R, the downward trend of $f_{0}$ with R is always gentle and small, while $c_{f}$ decreases sharply when R is small, and the decrease rate slows down as R increases.

Figure 5.

Attenuation rules of f₀ and c_f: (a) attenuation rules of f₀ and (b) attenuation rules of c_f.

Comparison of vector and unidirectional cases

Figure 6 shows the comparison results on attenuation rules of IF model parameters $f_{0}$ and $c_{f}$ between vector and unidirectional ground motions when M = 6, and Figure 7 shows the case when PGV/PGA = 0.1. The shapes of surfaces between vector and unidirectional ground motion are overall similar, which indicates their attenuation rules of IF are also similar.

Figure 6.

Comparison of attenuation rules of f₀ and c_f when M = 6: (a) attenuation rules of f₀ and (b) attenuation rules of c_f.

Figure 7.

Comparison of attenuation rules of f₀ and c_f when PGV/PGA = 0.1: (a) attenuation rules of f₀ and (b) attenuation rules of c_f.

For $f_{0}$ , it decreases as M, R, and PGV/PGA increase. The change rules of the two types of ground motions with M and R are very similar, while as PGV/PGA increases, the downward trend of vector ground motions is smaller than that of the unidirectional ones.

For $c_{f}$ , it decreases as M and R increase but increases as PGV/PGA increases. With the increase of M and PGV/PGA, the downward trend is almost same in the two cases, while that in vector case are slightly smaller. When R is small, $c_{f}$ of unidirectional ground motions is smaller than that of the vector, while the situation reverses with the increase of R. And the downward trend of vector case is slightly larger with the increase of R.

Conclusion

This article has studied the frequency non-stationary property of horizontal ground motion vectors by investigating IF. A new method of constructing analytic signal is proposed to obtain IF, that is, two orthogonal horizontal seismic signal from the same seismic record are used to construct analytic signal. Then the IF of analytic signal is mathematically represented as an exponential decay model which is validated by instantaneous phases. For the next step, the attenuation rules of IF parameters are regressed, the fitting results show that both the IF model and the attenuation rules model have accurate simulation. The initial frequency $f_{0}$ decreases as M, R, and PGV/PGA increase; the decay coefficient $c_{f}$ also decreases as M and R increase and as PGV/PGA decreases. R and PGV/PGA are bigger influencing factors than M for $f_{0}$ and $c_{f}$ . The IF attenuation rules of vector ground motions are similar to the unidirectional counterpart, which indicates that the new method is effective and feasible for constructing analytic signals.

Footnotes

Handling Editor: Wen-Hsiang Hsieh

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 National Natural Science Foundation of China (grant no.: 10802104).

ORCID iD

Ying Hu

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Non-stationary property in frequency content of horizontal ground motion vectors

Abstract

Keywords

Introduction

IF of signals

Definition of IF

Constructing analytic signal using Hilbert transform

Constructing analytic signal using ground motion components

Modeling of IF

Results and discussion

Ground motions used

Fitting results of IF model

Attenuation rules

Comparison of vector and unidirectional cases

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References