Physically compliant,conditionally simulated spatially variable seismic ground motions for performance‐based design

The performance‐based design of lifeline systems requires spatially variable seismic excitations at the structures' supports that are consistent with prescribed seismic ground motion characteristics and an appropriate spatial variability model—such motions can be obtained through conditional simulation. This work revisits the concept of conditional simulation and critically examines the conformity of the generated motions with the characteristics of the target random field and observations from data recorded at dense instrument arrays. Baseline adjustment processing techniques for recorded earthquake accelerograms are extended to fit the requirements of simulated and conditionally simulated spatially variable ground motions. Emphasis is placed on the use of causal vs acausal filtering in the data processing. Acceleration, velocity and displacement time histories are evaluated in two example applications of the approach. The first application deals with a prescribed synthetic time history that incorporates nonstationarity in the amplitude and frequency content of the motions and depends on earthquake magnitude, source–site distance and local soil conditions; this example results in zero residual displacements. The second application considers as prescribed time history a recording in the vicinity of a fault and yields nonzero residual displacements. It is shown that the conditionally simulated time histories preserve the characteristics of the prescribed ones and are consistent with the target random field. The results of this analysis suggest that the presented methodology provides a useful tool for the generation of spatially variable ground motions to be used in the performance‐based design of lifeline systems. Copyright © 2006 John Wiley & Sons, Ltd.     

Influence of irregular topography and random soil properties on coherency loss of spatial seismic ground motions

Coherency functions are used to describe the spatial variation of seismic ground motions at multiple supports of long span structures. Many coherency function models have been proposed based on theoretical derivation or measured spatial ground motion time histories at dense seismographic arrays. Most of them are suitable for modelling spatial ground motions on flat‐lying alluvial sites. It has been found that these coherency functions are not appropriate for modelling spatial variations of ground motions at sites with irregular topography (Struct. Saf. 1991; 10 (1):1–13). This paper investigates the influence of layered irregular sites and random soil properties on coherency functions of spatial ground motions on ground surface. Ground motion time histories at different locations on ground surface of the irregular site are generated based on the combined spectral representation method and one‐dimensional wave propagation theory. Random soil properties, including shear modulus, density and damping ratio of each layer, are assumed to follow normal distributions, and are modelled by the independent one‐dimensional random fields in the vertical direction. Monte‐Carlo simulations are employed to model the effect of random variations of soil properties on the simulated surface ground motion time histories. The coherency function is estimated from the simulated ground motion time histories. Numerical examples are presented to illustrate the proposed method. Numerical results show that coherency function directly relates to the spectral ratio of two local sites, and the influence of randomly varying soil properties at a canyon site on coherency functions of spatial surface ground motions cannot be neglected. Copyright © 2010 John Wiley & Sons, Ltd.     

A stochastic model and synthesis for near‐fault impulsive ground motions

The orientations of ground motions are paramount when the pulse‐like motions and their unfavorable seismic responses are considered. This paper addresses the stochastic modeling and synthesizing of near‐fault impulsive ground motions with forward directivity effect taking the orientation of the strongest pulses into account. First, a statistical parametric analysis of velocity time histories in the orientation of the strongest pulse with a specified magnitude and various fault distances is performed. A new stochastic model is established consisting of a velocity pulse model with random parameters and a stochastic approach to synthesize high‐frequency velocity time history. The high‐frequency velocity history is achieved by integrating a stochastic high‐frequency accelerogram, which is generated via the modified K‐T spectrum of residual acceleration histories and then modulated by the specific envelope function. Next, the associated parameters of pulse model, envelope function, and power spectral density are estimated by the least‐square fitting. Some chosen parameters in the stochastic model of near‐fault motions based on correlation analysis are regarded as random variables, which are validated to follow the normal or lognormal distribution. Moreover, the number theoretical method is suggested to select efficiently representative points, for generating artificial near‐fault impulsive ground motions with the feature of the strongest pulse, which can be used to the seismic response and reliability analysis of critical structures conveniently. Finally, the simulated ground motions demonstrate that the synthetic ground motions generated by the proposed stochastic model can represent the impulsive characteristic of near‐fault ground motions. Copyright © 2014 John Wiley & Sons, Ltd.     

A modified method for simulating non-stationary multi-point earthquake ground motion

A spectral-representation-based algorithm is proposed to simulate non-stationary and stochastic processes with evolutionary power,according to a prescribed non-stationary cross-spectral density matrix. Non-stationary multi-point seismic ground motions at different locations on the ground surface are generated for use in engineering applications. First,a modified iterative procedure is used to generate uniformly modulated non-stationary ground motion time histories which are compatible with the prescribed power spectrum. Then,ground motion time histories are modeled as a non-stationary stochastic process with amplitude and frequency modulation. The characteristic frequency and damping ratio of the Clough-Penzien acceleration spectrum are considered as a function of time in order to study the frequency time variation. Finally,two numerical examples are presented to validate the efficiency of the proposed method,and the results show that this method can be effectively applied to the dynamic seismic analysis of long and large scale structures.     

Simulating earthquake ground motion at a site, for given intensity and uncertain source location

Following a companion article, ground motion acceleration time historiesduring earthquakes can be described as realizations of non-stationarystochastic processes with evolutionary frequency content and instantaneousintensity. The parameters characterizing those processes can be handled asuncertain variables with probabilistic distributions that depend on themagnitude of each seismic event and the corresponding source-to-sitedistance. Accordingly, the generation of finite samples of artificial groundmotion acceleration time histories for earthquakes of given intensities isformulated as a two-stage Monte Carlo simulation process. The first stageincludes the simulation of samples of sets of the parameters of thestochastic process models of earthquake ground motion. The second stageincludes the simulation of the time histories themselves, given theparameters of the associated stochastic process model. In order to accountfor the dependence of the probability distribution of the latter parameterson magnitude and source-to-site distance, the joint conditional probabilitydistribution of these variables must be obtained for a given value of theground motion intensity. This is achieved by resorting to Bayes Theoremabout the probabilities of alternate assumptions.Two options for the conditional simulation of ground motion time historiesare presented. The more refined option makes use of all the informationabout the conditional distribution of magnitude and distance for thepurpose of simulating values of the statistical parameters of the groundmotion stochastic process models. The second option considers allprobabilities concentrated at the most likely combination of magnitude anddistance for each of the seismic sources that contribute significantly to theseismic hazard at the site of interest.     

A stochastic ground motion model with separable temporal and spectral nonstationarities

A fully nonstationary stochastic model for strong earthquake ground motion is developed. The model employs filtering of a discretized white‐noise process. Nonstationarity is achieved by modulating the intensity and varying the filter properties in time. The formulation has the important advantage of separating the temporal and spectral nonstationary characteristics of the process, thereby allowing flexibility and ease in modeling and parameter estimation. The model is fitted to target ground motions by matching a set of statistical characteristics, including the mean‐square intensity, the cumulative mean number of zero‐level up‐crossings and a measure of the bandwidth, all expressed as functions of time. Post‐processing by a second filter assures zero residual velocity and displacement, and improves the match to response spectral ordinates for long periods. Copyright © 2008 John Wiley & Sons, Ltd.     

Influence of time‐varying frequency content in earthquake ground motions on seismic response of linear elastic systems

Earthquake ground motion records are nonstationary in both amplitude and frequency content. However, the latter nonstationarity is typically neglected mainly for the sake of mathematical simplicity. To study the stochastic effects of the time‐varying frequency content of earthquake ground motions on the seismic response of structural systems, a pair of closely related stochastic ground motion models is adopted here. The first model (referred to as ground motion model I) corresponds to a fully nonstationary stochastic earthquake ground motion model previously developed by the authors. The second model (referred to as ground motion model II) is nonstationary in amplitude only and is derived from the first model. Ground motion models I and II have the same mean‐square function and global frequency content but different features of time variation in the frequency content, in that no time variation of the frequency content exists in ground motion model II. New explicit closed‐form solutions are derived for the response of linear elastic SDOF and MDOF systems subjected to stochastic ground motion model II. New analytical solutions for the evolutionary cross‐correlation and cross‐PSD functions between the ground motion input and the structural response are also derived for linear systems subjected to ground motion model I. Comparative analytical results are presented to quantify the effects of the time‐varying frequency content of earthquake ground motions on the structural response of linear elastic systems. It is found that the time‐varying frequency content in the seismic input can have significant effects on the stochastic properties of system response. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling spatially correlated spectral accelerations at multiple periods using principal component analysis and geostatistics

Regional seismic risk assessments and quantification of portfolio losses often require simulation of spatially distributed ground motions at multiple intensity measures. For a given earthquake, distributed ground motions are characterized by spatial correlation and correlation between different intensity measures, known as cross‐correlation. This study proposes a new spatial cross‐correlation model for within‐event spectral acceleration residuals that uses a combination of principal component analysis (PCA) and geostatistics. Records from 45 earthquakes are used to investigate earthquake‐to‐earthquake trends in application of PCA to spectral acceleration residuals. Based on the findings, PCA is used to determine coefficients that linearly transform cross‐correlated residuals to independent principal components. Nested semivariogram models are then fit to empirical semivariograms to quantify the spatial correlation of principal components. The resultant PCA spatial cross‐correlation model is shown to be accurate and computationally efficient. A step‐by‐step procedure and an example are presented to illustrate the use of the predictive model for rapid simulation of spatially cross‐correlated spectral accelerations at multiple periods.     

Seismic ground motion simulations from a class of spatial variability models

An improved version of the spectral representation method (References 12–14) for the generation of simulations of random processes and random fields by means of the Fast Fourier Transform algorithm is developed. The new formulation produces simulations that are ergodic in the mean without any restricting assumptions at the origin of the spectra and leads to a faster convergence rate of the stochastic characteristics of the simulations to those of the random field. As an example of the methodology, simulations in space and time based on a spatial variability model for the seismic ground motions are generated.     

Ground motion selection for simulation‐based seismic hazard and structural reliability assessment

This paper examines four methods by which ground motions can be selected for dynamic seismic response analyses of engineered systems when the underlying seismic hazard is quantified via ground motion simulation rather than empirical ground motion prediction equations. Even with simulation‐based seismic hazard, a ground motion selection process is still required in order to extract a small number of time series from the much larger set developed as part of the hazard calculation. Four specific methods are presented for ground motion selection from simulation‐based seismic hazard analyses, and pros and cons of each are discussed via a simple and reproducible illustrative example. One of the four methods (method 1 ‘direct analysis’) provides a ‘benchmark’ result (i.e., using all simulated ground motions), enabling the consistency of the other three more efficient selection methods to be addressed. Method 2 (‘stratified sampling’) is a relatively simple way to achieve a significant reduction in the number of ground motions required through selecting subsets of ground motions binned based on an intensity measure, IM. Method 3 (‘simple multiple stripes’) has the benefit of being consistent with conventional seismic assessment practice using as‐recorded ground motions, but both methods 2 and 3 are strongly dependent on the efficiency of the conditioning IM to predict the seismic responses of interest. Method 4 (‘generalized conditional intensity measure‐based selection’) is consistent with ‘advanced’ selection methods used for as‐recorded ground motions and selects subsets of ground motions based on multiple IMs, thus overcoming this limitation in methods 2 and 3. Copyright © 2015 John Wiley & Sons, Ltd.     

全非平稳地震作用的结构随机反应与可靠度分析

在平稳地震动过程的Clough-Penzien功率谱基础上,采用林家浩非均匀调制函数建立全非平稳地震动过程的演变功率谱。根据我国现行的《建筑抗震设计规范》进行全非平稳地震动演变功率谱的参数识别研究。应用非平稳随机过程模拟的谱表示-随机函数方法,生成建筑结构抗震设计所用地震动的代表性样本集合。通过代表性样本集合的二阶统计值及地震反应谱与目标值的拟合比较,验证本文方法的有效性。最后结合概率密度演化方法,进行以层间位移角为控制准则的结构随机地震反应分析与抗震可靠度计算。     

Simulations of non‐stationary frequency content and its importance to seismic assessment of structures

To verify the importance of the non‐stationary frequency characteristic of seismic ground motion, a joint time–frequency analysis technique of time signals, called chirplet‐based signal approximation, is developed to extract the non‐stationary frequency information from the recorded data. The chirplet‐based signal approximation is clear in concept, similar to Fourier Transform in mathematical expressions but with different base functions. Case studies show that the chirplet‐based signal approximation can represent the joint time–frequency variation of seismic ground motion quite well. Both the random models of uniform modulating process and evolutionary process are employed to generate artificial seismic waves. The joint time–frequency modulating function in the random model of evolutionary process is determined by chirplet‐based signal approximation. Finally, non‐linear response analysis of a SODF system and a frame structure is performed based on the generated artificial seismic waves. The results show that the non‐stationary frequency characteristic of seismic ground motion can significantly change the non‐linear response characteristics of structures, particularly when a structure goes into collapse phase under seismic action. It is concluded that non‐stationary frequency characteristic of seismic ground motion should be considered for the assessment of seismic capacity of structures. Copyright © 2002 John Wiley & Sons, Ltd.     

空间相关的多点地震动合成(Ⅱ)合成实例

本文应用已提出的自功率谱、相干函数、视速度模型生成了空间相关的多点地震动时程。采用分段合成、乘强度包络函数的方法近似地考虑了地震动强度和频率尬发的非平稳性、生成的地震动符合空间相关性、传播性、随机性和非平稳性,可用于长结构多点输入地震反应分析。     

Energy‐compatible and spectrum‐compatible (ECSC) ground motion simulation using wavelet packets

A stochastic ground‐motion simulation and modification technique is developed to generate energy‐compatible and spectrum‐compatible (ECSC) synthetic motions through wavelet packet characterization and modification in both frequency and time domains. The ECSC method significantly advances traditional spectral matching approaches, because it generates ground motions that not only match the target spectral accelerations, but also match Arias intensity build‐up and significant durations. The great similarity between the ECSC simulated motions and the actual recorded motions is demonstrated through one‐to‐one comparison of a variety of intensity measures. Extensive numerical simulations were also performed to validate the performance of the ECSC ground motions through nonlinear analyses of elasto‐plastic oscillators. The ECSC method can be easily implemented in the generalized conditional intensity measure framework by directly simulating a set of motions following a targeted distribution of multiple intensity measures. Therefore, the ECSC method has great potential to be used in performance‐based earthquake design and analysis. Copyright © 2017 John Wiley & Sons, Ltd.     

Seismic data analysis using local time‐frequency decomposition

Many natural phenomena, including geologic events and geophysical data, are fundamentally nonstationary ‐ exhibiting statistical variation that changes in space and time. Time‐frequency characterization is useful for analysing such data, seismic traces in particular. We present a novel time‐frequency decomposition, which aims at depicting the nonstationary character of seismic data. The proposed decomposition uses a Fourier basis to match the target signal using regularized least‐squares inversion. The decomposition is invertible, which makes it suitable for analysing nonstationary data. The proposed method can provide more flexible time‐frequency representation than the classical S transform. Results of applying the method to both synthetic and field data examples demonstrate that the local time‐frequency decomposition can characterize nonstationary variation of seismic data and be used in practical applications, such as seismic ground‐roll noise attenuation and multicomponent data registration.     

一种改进的近断层脉冲型地震动模拟方法

本文基于小波包技术的随机地震动模拟方法,提出一种改进的参数化随机近断层脉冲型地震动模拟方法。然后,通过识别和提取近断层脉冲型地震动数据库中脉冲型地震动的特征参数,建立了基于震源、传播路径和场地特征等参数的脉冲模型参数预测方程。最后,通过模拟实际记录和误差分析检验了改进的模拟方法的有效性。结果表明:应用改进的模拟方法得到的地震动时程无论在波形、频率特性还是峰值上均与实际记录具有较好的一致性。改进的模拟方法在保留地震动时频非平稳性的基础上,能够有效地提高近断层脉冲型地震动的模拟效果,并且能够很好地体现脉冲型地震动的主要特征。     

Simulation of synthetic ground motions for specified earthquake and site characteristics

A method for generating a suite of synthetic ground motion time‐histories for specified earthquake and site characteristics defining a design scenario is presented. The method employs a parameterized stochastic model that is based on a modulated, filtered white‐noise process. The model parameters characterize the evolving intensity, predominant frequency, and bandwidth of the acceleration time‐history, and can be identified by matching the statistics of the model to the statistics of a target‐recorded accelerogram. Sample ‘observations’ of the parameters are obtained by fitting the model to a subset of the NGA database for far‐field strong ground motion records on firm ground. Using this sample, predictive equations are developed for the model parameters in terms of the faulting mechanism, earthquake magnitude, source‐to‐site distance, and the site shear‐wave velocity. For any specified set of these earthquake and site characteristics, sets of the model parameters are generated, which are in turn used in the stochastic model to generate the ensemble of synthetic ground motions. The resulting synthetic acceleration as well as corresponding velocity and displacement time‐histories capture the main features of real earthquake ground motions, including the intensity, duration, spectral content, and peak values. Furthermore, the statistics of their resulting elastic response spectra closely agree with both the median and the variability of response spectra of recorded ground motions, as reflected in the existing prediction equations based on the NGA database. The proposed method can be used in seismic design and analysis in conjunction with or instead of recorded ground motions. Copyright © 2010 John Wiley & Sons, Ltd.     

Modification of stochastic ground motion models for matching target intensity measures

Stochastic ground motion models produce synthetic time‐histories by modulating a white noise sequence through functions that address spectral and temporal properties of the excitation. The resultant ground motions can be then used in simulation‐based seismic risk assessment applications. This is established by relating the parameters of the aforementioned functions to earthquake and site characteristics through predictive relationships. An important concern related to the use of these models is the fact that through current approaches in selecting these predictive relationships, compatibility to the seismic hazard is not guaranteed. This work offers a computationally efficient framework for the modification of stochastic ground motion models to match target intensity measures (IMs) for a specific site and structure of interest. This is set as an optimization problem with a dual objective. The first objective minimizes the discrepancy between the target IMs and the predictions established through the stochastic ground motion model for a chosen earthquake scenario. The second objective constraints the deviation from the model characteristics suggested by existing predictive relationships, guaranteeing that the resultant ground motions not only match the target IMs but are also compatible with regional trends. A framework leveraging kriging surrogate modeling is formulated for performing the resultant multi‐objective optimization, and different computational aspects related to this optimization are discussed in detail. The illustrative implementation shows that the proposed framework can provide ground motions with high compatibility to target IMs with small only deviation from existing predictive relationships and discusses approaches for selecting a final compromise between these two competing objectives.     

Investigations into critical earthquake load models within deterministic and probabilistic frameworks

This paper deals with the determination of critical earthquake load models for linear structures subjected to single‐point seismic inputs. The primary objective of this study is to examine the realism in critical excitations and critical responses vis a vis the framework adopted for the study and constraints that these excitations are taken to satisfy. Two alternative approaches are investigated. In the first approach, the critical earthquake is expressed in terms of a Fourier series that is modulated by an enveloping function that imparts transient nature to the inputs. The Fourier coefficients are taken to be deterministic and are constrained to satisfy specified upper and lower bounds. Estimates on these bounds, for a given site, are obtained by analysing past earthquake records from the same site or similar sites. The unknown Fourier coefficients are determined such that the response of a given structure is maximized subjected to these bounds and additional constraints on intensity, peak ground acceleration, peak ground velocity and peak ground displacement. In the second approach, the critical earthquake is modelled as a partially specified non‐stationary Gaussian random process which is defined in terms of a stationary random process of unknown power spectral density (psd) function modulated by a deterministic envelope function. The input is constrained to possess specified variance and average zero crossing rate. Additionally, a new constraint in terms of entropy rate representing the expected level of disorder in the excitation is also imposed. The unknown psd function of the stationary part of the input is determined so that the response of a given structure is maximized. The optimization problem in both these approaches is solved by using sequential quadratic programming method. The procedures developed are illustrated by considering the seismic response of a tall chimney and an earth dam. It is concluded that the imposition of lower and upper bounds on Fourier coefficients in the first approach and constraints on amount of disorder in the second approach are crucial in arriving at realistic critical excitations. Copyright © 2002 John Wiley & Sons, Ltd.     

Wavelet‐based characterization of design ground motions

With the recent emergence of wavelet‐based procedures for stochastic analyses of linear and non‐linear structural systems subjected to earthquake ground motions, it has become necessary that seismic ground motion processes are characterized through statistical functionals of wavelet coefficients. While direct characterization in terms of earthquake and site parameters may have to wait for a few more years due to the complexity of the problem, this study attempts such characterization through commonly available Fourier and response spectra for design earthquake motions. Two approaches have been proposed for obtaining the spectrum‐compatible wavelet functionals, one for input Fourier spectrum and another for input response spectrum, such that the total number of input data points are 30–35% of those required for a time‐history analysis. The proposed methods provide for simulating ‘desired non‐stationary characteristics’ consistent with those in a recorded accelerogram. Numerical studies have been performed to illustrate the proposed approaches. Further, the wavelet functionals compatible with a USNRC spectrum in the case of 35 recorded motions of similar strong motion durations have been used to obtain the strength reduction factor spectra for elasto‐plastic oscillators and to show that about ±20% variation may be assumed from mean to 5 and 95% confidence levels due to uncertainty in the non‐stationary characteristics of the ground motion process. Copyright © 2002 John Wiley & Sons, Ltd.