Research Article  Open Access
Huiguo Chen, Ting Zhong, Guocui Liu, Junru Ren, "Improvement of and Parameter Identification for the Bimodal TimeVarying Modified KanaiTajimi Power Spectral Model", Shock and Vibration, vol. 2017, Article ID 7615863, 11 pages, 2017. https://doi.org/10.1155/2017/7615863
Improvement of and Parameter Identification for the Bimodal TimeVarying Modified KanaiTajimi Power Spectral Model
Abstract
Based on the KanaiTajimi power spectrum filtering method proposed by Du Xiuli et al., a genetic algorithm and a quadratic optimization identification technique are employed to improve the bimodal timevarying modified KanaiTajimi power spectral model and the parameter identification method proposed by Vlachos et al. Additionally, a method for modeling timevarying power spectrum parameters for ground motion is proposed. The 8244 Orion and ChiChi earthquake accelerograms are selected as examples for timevarying power spectral model parameter identification and ground motion simulations to verify the feasibility and effectiveness of the improved bimodal timevarying modified KanaiTajimi power spectral model. The results of this study provide important references for designing ground motion inputs for seismic analyses of major engineering structures.
1. Introduction
Severe ground motion is an extremely complex process. Affected by factors such as the seismic source mechanism, the travel path, and the field characteristics, observed ground motions demonstrate prominent nonstationary characteristics in both the time and the frequency domain. Among the known seismic characteristics that have significant impacts on the seismic response of a structure, in addition to its nonstationary strength characteristics, the impact of nonstationary seismic frequency characteristics on a structure’s nonlinear seismic response has become a focus in recent years. Research by Yeh and Wen [1] showed that nonstationary seismic frequencies have substantial impacts on rigid structures as their strength degrades; Cao et al. [2]. believed that concentrated seismic energy in the time domain had a considerably unfavorable effect on a structure. Research by Conte and Wang suggested that, due to the structural damage caused by ground motion, the structural period is prolonged; when the longperiod seismic component after a strong earthquake segment is close to a structure’s prolonged period, “instantaneous resonance” can occur, and the damage to the structure becomes more severe or even exceeds that due to a strong earthquake [3, 4]. Therefore, creating a method for modeling nonstationary ground motion in which the characteristics of the simulated ground motion are similar to those of the characteristic timefrequency distribution of actual ground motion has become a realistic and important topic in seismic research.
In 1965, Priestley proposed evolutionary stochastic process [5] theory in which a nonstationary process is converted to an integral of a stationary process and a deterministic modulation function, and then, the timevarying spectrum density of the nonstationary process is obtained from the spectrum density of the stationary process, and a method of describing the characteristics of the spectrum of a nonstationary process and a model of nonstationary ground motion is created. However, in this method, to simulate ground motion with a fully nonstationary frequency, a timevarying power spectral model is required. For many years, researchers worldwide conducted various studies of timevarying power spectral models for ground motion. For instance, in 1975, Kameda [6] employed a multifilter technique, recorded the power spectrum of an actual earthquake that varied over time and with frequency, and created a timevarying power spectral model; Sugito et al. [7] performed a statistical analysis of 118 actual earthquakes in Japan using the Kameda model and provided a typical timevarying power spectrum attenuation relationship. In 1988, Deodatis [8] analyzed the Niigata earthquake’s field liquefaction characteristics in Japan, modified the timevarying property of the KanaiTajimi model, and simulated ground motion with a fully nonstationary timefrequency based on the characteristics of timevarying power spectrum of the ground motion associated with the Niigata earthquake. As modern timefrequency signal analysis develops, the methods of describing timevarying power spectra continue to evolve. In 2010, Fan estimated the timevarying spectrum of an actual strong earthquake using harmonic wavelet transformation and performed a statistical analysis of the characteristics of the timevarying spectra of seismic waves at various sites [9]; he also created a modified KanaiTajimi nonstationary stochastic model with timevarying seismic waves by means of nonlinear fitting. In 2015, based on a power spectrum density model for stationary seismic processes, Liu et al. [10] developed a general evolutionary spectral model for fully nonstationary timefrequency seismic processes based on the Specification for Seismic Design of Hydraulic Construction in China; values for the parameters of this model were determined. In 2016, based on the theory that there is a time gap between when the P, S, and surface waves in a seismic wave reach the recording field and when the earthquake releases energy via component waves of various frequencies, Vlachos et al. [11] created a timevarying power spectral model for ground motion using a bimodal timevarying modified KanaiTajimi model. This model accurately described characteristics of fully nonstationary ground motion and was an attempt to model the timevarying power spectrum of actual ground motion. In this paper, based on work by Vlachos, a genetic algorithm is employed to perform identify optimal parameters for the bimodal timevarying modified KanaiTajimi power spectral model proposed by Vlachos. In combination with the filtering technique proposed by XiuLi and HouQun [12], a method is created to describe and model the nonstationary timefrequency characteristics of actual ground motion. Additionally, the method of simulating ground motion with a nonstationary timefrequency proposed by Z.J. Liu and L. Liu [13] is employed to verify the effectiveness of the bimodal timevarying modified KanaiTajimi power spectral model and parameter identification method.
2. Bimodal TimeVarying Modified KanaiTajimi Power Spectral Model and Improvement
2.1. Fundamental Theory of the Evolutionary Spectrum Method
According to evolutionary spectrum densitybased model of nonstationary stochastic processes proposed by Priestley [5], any nonstationary stochastic process with an average value of zero is defined as where is a complex stationary stochastic process with an orthogonal increment and is a slowly varying modulation function.
Assume that a stationary process satisfies
Then, the evolutionary spectrum function (the timevarying or progressive power spectrum) of is represented as where is the power spectrum of stationary process .
Based on the above evolutionary spectrum theory, a numerical simulation of a nonstationary stochastic process, , was initially proposed by Shinozuka et al. [14, 15] and was improved continuously until the following final ground motion simulation formula was reached:where is a random phase that is evenly distributed on .
Equations (3) and (4) show that the power spectrum density function, , is an important index for the characteristics of the timefrequency of ground motion.
2.2. Bimodal TimeVarying Modified KanaiTajimi Spectral Model
In a stochastic model of early stage ground motion, due to the difficulty of creating a fully timevarying power spectral model, researchers prefer to generate a stochastic process from the power spectrum of a stationary stochastic process and then to modify the nonstationary characteristics of this stochastic process to simulate ground motion. Based on this idea, Japanese researchers Kanai (1957) and Tajimi (1960) [16, 17] selected a soil layer on bedrock as a singledegreeoffreedom linear filter to investigate the effect of various types of soil layers on ground motion and proposed a wellknown ground motion power spectral model, the KanaiTajimi model, shown as follows: where is the power spectrum based on white noise excitation of the bedrock, is the characteristic frequency of the soil layer, and is the damping ratio of the singledegreeoffreedom linear filter.
This model suggests that an earthquake causes movement in the bedrock, which is then transmitted to the structure via the surface soil layer. The bedrock’s acceleration is treated as white noise whose average is zero and whose spectrum density is . The surface soil layer is treated as a filter. The KanaiTajimi power spectral model considers the effect of the surface soil layer’s characteristics on the characteristics of the seismic frequency spectrum and is physically interpretable. Therefore, this model is widely adopted in seismic engineering. Because the KanaiTajimi power spectral model is not timevarying in the frequency domain, the KanaiTajimi power spectrum is unable to directly model ground motion with fully nonstationary timefrequency characteristics. To make the conventional KanaiTajimi power spectrum capable of representing a timevarying seismic power spectrum, Dedatis et al. [8] proposed a timevarying modification of the KanaiTajimi power spectral model. Vlachos et al. [11] optimized the resulting timevarying modified KanaiTajimi power spectral model, superposed the conventional singlepeak KanaiTajimi equation, and proposed the following bimodal fully nonstationary timevarying KanaiTajimi spectral model: where is a timevarying power spectrum for frequency and time, is a filter, and , , and are the characteristic frequency, damping ratio, and modal participant factor, respectively, under the kth modal condition in the timevarying model.
2.3. Improvement of the Bimodal TimeVarying Modified KanaiTajimi Spectral Model
In (6), the selection of the filter, , is critical. The reason is that there are two essential limitations to the application of the KanaiTajimi power spectral model. The first one is on the process energy. Research by JinRen et al. [18] showed that, to satisfy the requirement of limited seismic process energy, the process variance of the ground motion acceleration and the variances of the first and secondorder derivative processes should be limited. However, based on the rules for the convergence of general integrals, the variance of the acceleration process in the KanaiTajimi power spectral model is unbounded. To satisfy the condition of the process energy being limited, the KanaiTajimi power spectral model should be filtered and modified. The second limitation is that the energy must be zero when the frequency is zero. In the KanaiTajimi power spectral model, when approaches zero, the ground velocity and displacement become infinite, which is obviously unrealistic. Therefore, to solve the problem of nonzero energy when the frequency is zero frequency in the conventional KanaiTajimi power spectral model and to address longperiod noise pollution in the modified simulation signal, the KanaiTajimi power spectral model should be filtered and modified. However, improvements to the KanaiTajimi spectrum in the past only addressed a single aspect, which focused on the acceleration’s boundary integral or derivative process; few considered both. When creating a timevarying KanaiTajimi spectral model, Vlachos et al. [11] employed a fourthorder Butterworth highpass filter. Because of the Butterworth highpass filter’s characteristics, its higherorder characteristics may result in pseudofiltering behavior and prolong harmful lowamplitude zeroaverage oscillations in the signal’s pulse response. Therefore, adding a highpass filter improves the boundary integral process but does not improve the derivative process. However, adding a lowpass filter improves the boundary derivative process but has no effect on the integral process. Additionally, when only the order of the lowpass filter is increased, the power spectrum of the integral process still has a singular point [18]. To resolve this dilemma, XiuLi and HouQun [12] investigated the simplified seismic acceleration Fourier spectrum proposed by MingZheng, which considers the propagation path and the effect of local field conditions [19], and proposed a method for filtering the KanaiTajimi spectrum using concatenated lowpass and highpass filters. This method effectively combines the characteristics of lowpass and highpass filters, satisfies the high and lowfrequency requirements of the power spectrum, and is an ideal method for improving the KanaiTajimi spectral model of ground motion. To guarantee accuracy in the KanaiTajimi power spectrum identification process, this paper improves the bimodal timevarying modified KanaiTajimi model created by Vlachos et al. [11] by replacing the filter in the model with the filter model proposed by XiuLi and HouQun [12]. The filter is as follows:where is a spectrum parameter that reflects the bedrock’s characteristics and is the low corner frequency.
and have clear physical interpretations. Mijatovic (2013) [20] proposed that be 0.03 second and be as follows:where is the radius of an equivalent disc and km/s. MingZheng (1986) [19] provided a statistical relationship between and the earthquake’s magnitude. A simple way to determine is where is the duration of the fault crack. Ying (1987) [21] identified a statistical relationship between and an earthquake’s magnitude, : where . Here, is equivalent to 90% of the duration of energy at the epicenter.
The improved bimodal timevarying modified KanaiTajimi model is as follows:
3. Model Parameter Identification
3.1. Introduction of the Genetic Algorithm
When creating a nonstationary timefrequency model of ground motion, the key is to identify the parameters of the resulting timevarying power spectral model via numerical analysis to obtain a timevarying seismic power spectrum. The bimodal timevarying modified KanaiTajimi model has six unknown parameters, . Identifying these six parameters is a typical multiparameter and nonlinear optimization problem. Among current parameter identification methods and numerical optimization methods, the genetic algorithm is based on the theory of biological evolution; it simulates a biological evolutionary process by producing a nextgeneration solution via operations such as replication, intersection, and mutation and gradually eliminates solutions with low fitness values to obtain a solution with a high fitness value. This method was initially proposed in 1975 by Professor Holland of the University of Michigan in the United States [22]. Since then, it has been widely adopted in various fields, such as optimization identification, machine learning, artificial intelligence control, image processing, and pattern recognition. The basic idea underlying its application in optimization identification is to treat a solution to an actual problem as a representation of a biological individual and to treat the process of solving an optimization problem as the genetic process of a species obtaining an optimal biological individual, where the representation of the optimal biological individual is the optimal solution. Because this is essentially a stochastic probability search algorithm and has the characteristics of an adaptive global search, it is ideal for solving multiparameter and nonlinear planning problems. Therefore, this paper introduces this method as the primary means of identifying the parameters of the improved bimodal modified KanaiTajimi model.
3.2. Model Parameter Identification and Fitting
Research by Vlachos et al. [11] suggested that, in the bimodal modified KanaiTajimi model, the spectrum strength factors, and , have no direct relationship with the spectral energy of each mode; their values only represent linear combinations of the modes. Therefore, the following ratio is defined to replace the absolute values: According to (12), when is fixed, that is, when , the number of parameters to be identified in the model is reduced to five; that is, . To satisfy the requirements of the global optimization method in a genetic algorithm, the least squaresbased model parameter identification fitness function is as follows: where is the timevarying power spectrum of the actual ground motion, which is obtained via signal timefrequency analysis, and is the timevarying power spectral model of the ground motion represented by (11).
MATLAB’s genetic algorithm toolbox (Mijatovic [20]) is employed to identify and to obtain the values of parameters in an instantaneous frequency curve model at each time point of the ground motion’s timevarying power spectrum. Among the identified parameters, are defined as damping ratios of the top soil layer on the bedrock, which are normally constants. To simplify the analysis, the values of parameters are set to the mean value of identified at each time : Next the five original unknown parameters, , are reduced to three, which are . , which are determined by (12) and (14), and are treated as known parameters, and a genetic algorithm is employed to perform quadratic optimization of to identify the values of the remaining parameters in the model at each time point ; they are recorded as . These identified parameters are a series of scattered points at time point .
The resulting scattered point parameter values should be translated to mathematical expressions via modeling before they are used in ground motion simulations. Therefore, based on characteristics of the actual parameters identified for the ground motion timevarying power spectral model, this paper employs a Gaussian distribution function to fit the identified and , whereas is fitted using smooth splines. The resulting expressions for and are as follows: where and are the parameter values for the corresponding values of .
Figure 1 is a diagram of the parameter identification process for the bimodal modified KanaiTajimi model. This diagram shows that, of the six model parameters, only is defined directly; the other parameters are identified by the genetic algorithm. A quadratic optimization identification technique is employed during the identification process. First, a genetic algorithm is employed to identify each parameter, and the results are averaged to obtain values of the damping ratio parameters, , for the cover soil layer. In quadratic optimization, values of , are used to reidentify the values of the scattered points at each time point . Then, based on the identified parameters, a timevarying equation relating is obtained via fitting.
3.3. Example of Model Parameter Identification
The 8244 Orion (1971, San Fernando) and ChiChi 1999, station CHY102, NS component accelerograms seismic wave records are selected as examples for model parameter identification; their acceleration timehistory curves are shown in Figure 2. Based on a study by XiuLi and HouQun [12], the values of the filter parameters are as follows: is always 0.03 seconds and is 0.6214 for the 8244 Orion record and is 0.2844 for the ChiChi record. Shorttime Fourier transformation and the Hanning window smoothing technique are combined to obtain timevarying power spectra for the acceleration timehistories of the two ground motions; they are shown in Figure 3.
(a) 8244 Orion
(b) ChiChi
(a) 8244 Orion
(b) ChiChi
Figure 4 shows the initial values of from the genetic algorithm. The values for the 8244 Orion and ChiChi seismic waves, , calculated using (14) are as follows: 8244 Orion ChiChi
(a) 8244 Orion
(b) ChiChi
Based on the mean of and genetic algorithmbased quadratic optimization, the identified values of the parameters in the model at each time point are obtained; they are shown in Figures 5–7. The diagrams also include the curve fitted after parameter identification. Tables 1 and 2 list the final values identified for the model parameters and their fitted timevarying expressions. The timevarying power spectra are obtained after modeling by inverse calculation based on the fitting parameters and the timevarying equation, as shown in Figure 8. Comparison of Figures 3 and 8 shows that the timevarying power spectral model effectively reflects the major nonstationary timefrequency characteristics of the original ground motion.


(a)
(b)
(a)
(b)
(a) 8244 Orion
(b) ChiChi
(a) 8244 Orion
(b) ChiChi
4. Ground Motion Simulation Example Verification
In Section 2.3, the parameters in bimodal timevarying modified KanaiTajimi model, , are obtained via genetic algorithmbased optimization. However, the most direct and effective method of verifying the effectiveness of the bimodal timevarying modified KanaiTajimi model and the identified parameters in describing the nonstationary timefrequency characteristics of the ground motion is to perform an inverse calculation, analyze the ground motion based on the identified parameters, and compare the actual and simulated ground motions. Therefore, in this section, the effectiveness of model and its parameters is analyzed using the timevarying power spectrumbased stochastic process simulation method proposed by Z.J. Liu and L. Liu [13].
4.1. TimeVarying Power SpectrumBased Stochastic Process Simulation Method for Ground Motion
Z.J. Liu and L. Liu [13] introduced a random functionbased expression for standard orthogonal random variable and created a sample function set that could generate a stochastic process via a discrete set of representative points of a basic random variable. When this method is applied to simulating nonstationary timefrequency ground motion, the expression is where is the timehistory of the simulated ground motion, is the ground motion’s timevarying power spectrum, and and are standard orthogonal random variables that satisfy the basic condition shown inwhere represents the mathematical expectation and is the Kronecker delta.
In an actual stochastic process simulation, the random variables and are defined as functions of a basic random variable, ; that is, the following random functions are constructed: where the basic random variable is evenly distributed on . It is easy to verify that the standard orthogonal random variables and constructed in (20) satisfy the basic condition required by (18).
4.2. Simulation Verification of Ground Motion
Based on the parameters identified for the bimodal timevarying modified KanaiTajimi models of the 8244 Orion and ChiChi earthquakes obtained in Section 2.3, timevarying power spectra, , for the two earthquakes are calculated using (11). On this basis as well as the method of representing nonstationary stochastic process spectra defined in (18)–(20), the simulated timehistories of the two earthquakes shown in Figures 9(a) and 9(b) are obtained. Figure 10 shows comparisons of the strength envelope functions and the normalized cumulative energies, respectively, of the simulated and original ground motions. These figures show that the simulated and original ground motions are generally consistent in terms of their nonstationary characteristics in the time domain. To compare the nonstationary characteristics of the simulated and original ground motions in the frequency domain, Figure 11 shows a comparison of the cumulative zerocrossing rates for the simulated and original ground motions. These graphs show that the cumulative zerocrossing rates of the simulated and original ground motions are generally consistent, which means that the improved bimodal timevarying modified KanaiTajimi model accurately reflects the nonstationary frequency characteristics of the original ground motion.
(a) 8244 Orion
(b) ChiChi
(a) 8244 Orion
(b) ChiChi
(a) 8244 Orion
(b) ChiChi
5. Conclusions
In this paper, the KanaiTajimi power spectrum filtering method proposed by Du Xiuli et al. is employed to improve the bimodal timevarying modified KanaiTajimi power spectral model proposed by Vlachos et al. A genetic algorithm is employed to identify the optimal model parameters and to establish a timevarying power spectrum parameterization model of the ground motion. As examples of the parameter identification process, timevarying power spectral models of the 8244 Orion and ChiChi ground motions and simulation verification of these ground motions show that the improved bimodal timevarying modified KanaiTajimi power spectral model accurately describes the nonstationary timefrequency characteristics of ground motion. But the proposed bimodal model is unable to reproduce accelerogram with the strong nonstationarity in frequency content and must be improved in the future. These research findings provide important references for the design of ground motion inputs for seismic analyses of major engineering structures.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant no. 51478068.
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Copyright © 2017 Huiguo Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.