An improvement over Fourier transform to enhance its performance for frequency content evaluation of seismic signals

Highlights

• An improvement over Fourier transform were proposed for frequency content evaluation.

• Strong correlation with structural responses proves the accuracy of proposed method.

• Two simple equations were proposed to modify conventional Fourier spectrum.

Abstract

The Fourier analysis is conventional technique for approximating the frequency content of earthquake ground motions. These approaches assume that earthquake excitation is a stationary process. Time frequency analysis techniques such as wavelet transform consider an earthquake to be a non-stationary process that shows a variation in frequency content over time, but these techniques are not easy to apply and most engineers are not familiar with their sophisticated mathematics. This paper introduces a new method for investigating of frequency content of earthquake records which is more accurate than Fourier spectrum but yet easy to apply for structural design and seismological studies. The efficiency of this new approach is investigated by analyzing a variety of single and multiple degree of freedom systems against near, mid and far field earthquake components. The results demonstrate the efficiency of the proposed method for a range of systems having different natural fundamental periods.

Introduction

The frequency content of ground motion is an important parameter in dynamic analysis and a key factor in the intensification of structural damage caused by resonance. Earthquakes are non-stationary processes, which makes measurement of their frequency content challenging. The Fourier transform is usually implemented for frequency analysis, but does not consider variation in the frequency content during earthquake excitation; thus, it might not provide adequate information about characteristic frequencies. A number of studies have been conducted to quantify frequency content and determine its relation to seismic assessment and the severity of damage to structures. The development of mathematical techniques for investigating frequency content has produced practical applications such as damage detection, structural health monitoring and artificial records generation.

There are many studies in the scientific literature on the properties and applications of Fourier transform in earthquake engineering. Fourier transform is conventionally used to study frequency content of ground acceleration in earthquake engineering. Some of researchers have developed more specific applications of Fourier transform in seismology. For instance, inverse discrete Fourier transform was used for simulation of horizontal components of earthquake ground motion [1]. Fractional Fourier transform has been employed to denoise seismic data [2]. The issue of resolution in time frequency analysis was examined using short time Fourier transform [3]. Short time Fourier transform along some other techniques has been used to investigate landslide process [4]. In addition, to reduce computational effort for dynamic time history analysis, earthquake accelerograms were simplified using inverse Fourier transform [5].

This study proposes a novel method for frequency content evaluation of seismic signals that shows stronger correlation with the maximum responses of structures in comparison with conventional Fourier spectrum. The accuracy of this claim has been proven by analyzing a wide variety of single and multi-degree freedom systems. Furthermore, simple formulas were developed to convert conventional Fourier spectrum to the proposed one.

However, the Fourier transform of a non-stationary signal is the average of each spectral amplitude calculated for the total length of the signal, avoiding local spectrum information and then losing the time varying behavior [6].

A variety of engineering fields in which non-stationary signals are frequently encountered (such as wind, ocean and earthquake engineering) require tools which allow time and frequency localization beyond that of customary Fourier analysis [7]. Many researchers have investigated seismic loads using time frequency analysis techniques. The short-time Fourier transform, Winger-Ville, Choi-Williams and reduced interference distributions were applied to earthquakes recorded in Mexico City to compare time and frequency resolution of results [8].

Wavelet transform has been used to assess the energy content of seismic sequences to evaluate the contribution of each frequency to the total energy induced by ground motion [9], [10].

The Mexican hat wavelet was used for time frequency analysis with the assumption that earthquake ground motions “are produced by a sequence of simple penny-shaped ruptures at different locations along a fault line” [11]. The energy spectrum was developed using wavelet analysis for time frequency localization of energy induced by ground motion [12]. Wavelet transform also has been used to de-noise seismic signals [13].

Also, Hilbert-Huang transform was applied to investigate non-stationary characteristics of ground motions recorded in Mexico City [14]. The large pulse of near field events has been extracted using wavelet transforms to find the effect of pulse-like ground motions to calculate the inelastic demand of ductility level of structures [15]. In this area, wavelet analysis has been used to quantitatively classify near-fault ground motions which contain a strong velocity pulse [16]. Fourier transform and wavelet packet analysis were employed to examine time frequency characteristics of blasting seismic waves [17].

In the field of damage detection, it is possible to mention to monitoring the variation of mean instantaneous frequency using wavelet transforms of the nonlinear lateral displacement response of structures to assess structural damage caused by earthquakes [18]. The S-transform can be used for time frequency analysis of nonlinear response time histories so that the mode shapes of a structure can be evaluated over time [6]. The Hilbert-Huang transform is used to estimate a damage detection index for detecting structural damage to steel structures due to strong ground motion [19]. In general, the wavelet transform of the nonlinear dynamic response of structures has widespread applications in damage detection [20], [21], [22], [23].

Time frequency analysis techniques are also applicable for generating synthetic records. The Gabor transformation has been used to generate site-specific synthetic time-histories [24]. With the wavelet transform, it is possible to use compressed earthquake records (instead of entire records) to train neural networks for generating artificial earthquake records. This enhances the efficiency of the model and can generate artificial records for any given design spectrum [25]. Also, a stochastic model has been developed to synthesize ground motion signals using discrete wavelet transform [26]. The Morlet wavelet has been used for spectral matching of earthquake records to a target spectrum [27]. Wavelet analysis can also be applied to construct signals that can be used in the design of civil engineering structures. These simplified signals contain the prominent features of the data distribution recorded from pulse-like earthquakes [28].

For tsunami early warning, it is possible to detect the arrival of very long period seismic waves before seismic S-waves reach the station using wavelet analysis [29]. Wavelet analysis has been applied to calculate the total energy for a frequency range of 0.33 to 16.25 Hz. Note that the maximum total energy for non-tsunamigenic earthquakes is more than twice that of tsunamigenic events and can be used as a threshold for tsunami warnings [30].

All of aforementioned studies deals with frequency content of different types of signals and the widespread attention of researchers to the topic of frequency content and the development of various mathematical methods to measure it, illustrate the importance of this issue. The many practical applications of time frequency analysis methods, some of which are mentioned here, additionally emphasize the importance of the subject.