Ground Motion Process Research Articles

Time-domain dimension-reduction representation for stochastic ground motion utilizing filtered white noise

A method is proposed for characterizing and simulating both stationary and fully non-stationary stochastic ground motions. This method is based on discrete filtered white noise models, including single and double filtered, with the latter is introduced to suppress low-frequency components. Specifically, the proposed method expresses seismic ground motion as a linear combination of products involving orthogonal random variables and deterministic functions. Further, by defining high-dimensional orthogonal random variables as orthogonal functions of extremely low dimensional elementary random variables, efficient dimension-reduction (DR) of primitive ground motion process can be achieved. To illustrate this concept, three distinct categories of random orthogonal functions involving only one or two elementary random variables are examined, employing filtered white noise models to simulate ground motion acceleration processes, thereby demonstrating the accuracy and efficiency of the proposed method. Simultaneously, recommendations for employing the proposed method in simulations are provided based on an analysis of the impacts of various parameters on random ground motion processes. Case studies demonstrate the accuracy and robustness of the proposed method compared to Monte Carlo (MC) methods. Furthermore, case studies on fully non-stationary ground motion highlight the practical applicability of the proposed method in engineering.

Duration and damping for ground motion characterization in frequency‐domain damage estimation of SDOF structures

AbstractIn estimating the damage to a single‐degree‐of‐freedom (SDOF) structure due to multiple earthquake events, it may be convenient to use a frequency‐based approach, in which the anticipated ground motions are characterized through power spectral density functions (PSDFs) compatible with the specified design spectra for those motions. Since a spectrum‐compatible PSDF corresponds to a fictitious stationary ground motion process of a specified duration, the duration of this process should preferably be same as the length of the stationary segment in the actual ground acceleration process for a meaningful computation of the stochastic response. On assuming the strong‐motion segment of a given ground motion to represent its stationary segment, the nonstationarity of the strong‐motion segment is evaluated for the existing strong‐motion duration definitions. It is found that the Trifunac‐Brady definition is the most reasonable strong‐motion duration definition in respect of consistently giving the longest strong‐motion segment with least nonstationarity. Further, a spectrum‐compatible PSDF can be used for estimating the stochastic response of a linear system, provided the system has the same damping ratio as that of the design spectrum from which the PSDF has been computed. This causes errors in estimating the peak response of an equivalent linear system due to its damping ratio being higher than the initial damping ratio of the nonlinear system. To address this, an empirical expression is developed for the modified damping ratio of a linear system, which would lead to the correct peak response of the system on using the PSDF obtained from the 5%‐damping design spectrum.

A physics‐informed Bayesian framework for characterizing ground motion process in the presence of missing data

AbstractA Bayesian framework to characterize ground motions even in the presence of missing data is developed. This approach features the combination of seismological knowledge (a priori knowledge) with empirical observations (even incomplete) via Bayesian inference. At its core is a Bayesian neural network model that probabilistically learns temporal patterns from ground motion data. Uncertainties are accounted for throughout the framework. Performance of the approach has been quantitatively demonstrated via various missing data scenarios. This framework provides a general solution to dealing with missing data in ground motion records by providing various forms of representation of ground motions in a probabilistic manner, allowing it to be adopted for numerous engineering and seismological applications. Notably, it is compatible with the versatile Monte Carlo simulation scheme, such that stochastic dynamic analyses are still achievable even with missing data. Furthermore, it serves as a complementary approach to current stochastic ground‐motion models in data‐scarce regions under the growing interests of PBEE (performance‐based earthquake engineering), mitigating the data‐model dependence dilemma due to the paucity of data, and ultimately, as a fundamental solution to the limited data problem in data scarce regions.

Simulation of non-stationary and non-Gaussian random processes by 3rd-order Spectral Representation Method: Theory and POD implementation

This paper introduces the 3rd-order Spectral Representation Method for simulation of non-stationary and non-Gaussian stochastic processes. The proposed method extends the classical 2nd-order Spectral Representation Method to expand the stochastic process from an evolutionary bispectrum and an evolutionary power spectrum, thus matching the process completely up to third-order. A Proper Orthogonal Decomposition (POD) approach is further proposed to enable an efficient FFT-based implementation that reduces computational cost significantly. Two examples are presented, including the simulation of a fully non-stationary seismic ground motion process, highlighting the accuracy and efficacy of the proposed method.

Power spectral models of stationary earthquake-induced ground motion process considering site characteristics

In this article, several spectral models describing the stationary stochastic process of earthquake ground motion are explored and compared. The Hu-Zhou spectrum, which is regarded as an improved model of Kanai-Tajimi spectrum, is concerned specially. It is proved that the earthquake-induced ground acceleration process described by the Hu-Zhou spectrum is a twice filtered white noise process in essence, and two filters for modifying low-frequency components and moderate- and high-frequency components respectively are investigated. Total 1946 strong earthquake records in different sites are employed to determine the parameters of spectral models, including the Kanai-Tajimi spectrum, the Clough-Penzien spectrum and the Hu-Zhou spectrum. The results show that the Hu-Zhou spectrum fits well with the actual observed ground motions over the whole frequency range, it is not only distinct in physical meaning and concise in mathematical expression, but also reasonable in practice.

Effect of residual displacement and sequence of events on Design Force Ratio spectrum

In the areas of high seismicity, where several medium-to-strong events may occur before the design event, the cumulative damage due to earthquake events expected during the design life of the structure should be kept within safe limits. This may be achieved by raising the design force level corresponding to the largest seismic event anticipated during the design life of the structure by a factor known as Design Force Ratio (DFR). This study is an attempt to generalize the existing DFR spectrum by modifying how the damage during an event is computed and how the power spectral density function (PSDF) of the ground motion process for that event is estimated. As the existing models do not estimate incremental damage during an event of a sequence, the modified Park-Ang damage model is proposed to be modified further such that its estimates for the individual events add up to the cumulative damage index for the entire sequence as well as the effects of residual displacements are accounted for. The proposed model can estimate additional damage due to an event (in a multiple-event situation) without requiring the excitation time-history. The generalized procedure for DFR spectrum is based on using this model and the concept of spectrum-compatible PSDF. A numerical study on a hypothetical seismic environment shows that the DFR spectrum is very sensitive to the directions of the residual displacements and that all residual displacements should be assumed in the same direction for a conservative estimate of the DFR spectrum. It is also shown that in that case, the event with maximum damage potential should be assumed to occur as the last event in the sequence in order to maximize the cumulative damage.

Probability density evolution analysis of stochastic nonlinear structure under non-stationary ground motions

In the present paper, a framework of dimension-reduction modeling method is developed for a dual stochastic dynamic structural system of spectrum-compatible non-stationary stochastic ground motion processes and stochastic structures. With the aid of the proposed method, the random variables used to describe the stochastic characteristics of the non-stationary ground motion processes and the structural parameters are respectively represented by the one-elementary-random-variable functions, contributing to an entire stochastic dynamic structural system readily described by merely two elementary random variables. Owing to the fact that the number of the elementary random variables needed is extremely small, the set of the representative points associated with the elementary random variables can thus be selected by the widely-used number theoretical method, and then the probability density evolution method can be conveniently combined to conduct the dynamic response analysis and dynamic reliability evaluation of nonlinear stochastic structures. In the numerical examples, the probability density evolution analysis of an eight-storey reinforced concrete frame structure with random parameters subjected to spectrum-compatible non-stationary stochastic ground motion processes is investigated as a case study. Numerical results fully demonstrated the effectiveness and robustness of the proposed method.

Probability density evolution analysis of a shear-wall structure under stochastic ground motions by shaking table test

In this study, the accurate dynamic response analysis and dynamic reliability assessment upon a 12-storey shear-wall structure induced by stochastic seismic excitations are implemented through shaking table test combined with the probability density evolution method. To generate the stochastic seismic excitations, this study decomposes the ground motion process as a superposition of two contributions: the first one is a fully non-stationary counterpart modeled by a known evolutionary power spectrum using the dimension-reduction spectral representation; the second one is a corrective term adjusting the first counterpart in order to make it spectrum compatible. Through this treatment, the dimension-reduction simulation of the non-stationary stochastic ground motion processes satisfying the provision imposed by certain seismic codes is accomplished. Making the generated stochastic ground motion processes as the input stimulus of the shaking table test, the stochastic dynamic response analysis and the dynamic reliability assessment of the structural components as well as the whole shear-wall structure can then be carried out adopting the probability density evolution method. The typical studies sufficiently demonstrate the superiority of the proposed method and provide a powerful reference for the reliability-based seismic optimization design with regard to the cast-in-place high-rise shear-wall structures.

Evolution of Secondary Deformations Captured by Satellite Radar Interferometry: Case Study of an Abandoned Coal Basin in SW Poland

The issue of monitoring surface motions in post-mining areas in Europe is important due to the fact that a significant number of post-mining areas lie in highly-urbanized and densely-populated regions. Examples can be found in: Belgium, the Czech Republic, France, Germany, the Netherlands, Spain, the United Kingdom, as well as the subject of this study, the Polish Walbrzych Hard Coal Basin. Studies of abandoned coal fields show that surface deformations in post-mining areas occur even several dozen years after the end of underground coal extraction, posing a threat to new development of these areas. In the case of the Walbrzych area, fragmentary, geodetic measurements indicate activity of the surface in the post-mining period (from 1995 onward). In this work, we aimed at determining the evolution of surface deformations in time during the first 15 years after the end of mining, i.e., the 1995–2010 period using ERS 1/2 and Envisat satellite radar data. Satellite radar data from European Space Agency missions are the only source of information on historical surface movements and provide spatial coverage of the entirety of the coal fields. In addition, we attempted to analyze the relationship of the ground deformations with hydrogeological changes and geological and mining data. Three distinct stages of ground movements were identified in the study. The ground motions (LOS (Line Of Sight)) determined with the PSInSAR (Persistent Scatterer Interferometry) method indicate uplift of the surface of up to +8 mm/a in the first period (until 2002). The extent and rate of this motion was congruent with the process of underground water table restoration in separate water basins associated with three neighboring coal fields. In the second period, after the stabilization of the underground water table, the surface remained active, as indicated by local subsidence (up to −5 mm/a) and uplift (up to +5 mm/a) zones. We hypothesize that this surface activity is the result of ground reaction disturbed by long-term shallow and deep mining. The third stage is characterized by gradual stabilization and decreasing deformations of the surface. The results accentuate the complexity of ground motion processes in post-mining areas, the advantages of the satellite radar technique for historical studies, and provide information for authorities responsible for new development of such areas, e.g., regarding potential flood zones caused by restoration of groundwater table in subsided areas.

Spectral representation-based dimension reduction for simulating multivariate non-stationary ground motions

A framework of spectral representation-based dimension reduction for simulating multivariate non-stationary stochastic ground motion processes is addressed in this paper. By means of introducing random functions serving as constraints correlating with the orthogonal random variables in the original spectral representation scheme, the high-dimensional randomness degree involved in the multivariate stochastic processes can thus be reduced substantially. To this aim, three random function forms considering the combination of trigonometric functions and orthogonal polynomials are constructed for simulation purpose. Accordingly, the accurate representation of the original stochastic processes is realized with merely three elementary random variables, overcoming the principal challenge of numerous random variables faced by the Monte Carlo simulation method. Also, the consistency of the statistics between the sample functions of stochastic ground motions and strong motion records is established with update of the ground motion model parameters in stochastic simulation techniques. Numerical investigations involving the comparisons with the Monte Carlo simulation method and the validation based on the strong motion records are presented to demonstrate the superiority and effectiveness of the proposed methodology in practical engineering applications.

Fast and slow nonlinear elastic response of disparate rocks and the influence of moisture

We study nonlinear elastic phenomena in rocks at the laboratory scale, with the goal of characterizing and understanding observations at crustal scales, for instance, during strong ground motion and earthquake slip processes. A dynamic perturbation of microstrain amplitude in rocks results in a transient elastic softening followed by a log(t)-type relaxation back to the initial unperturbed elastic modulus as soon as the excitation is removed. Here we use Dynamic Acousto-Elastic Testing (DAET) to investigate the relaxation behavior over 7 orders of magnitude in time (from 10-4s to more than 103s). We find that relaxation starts for all samples between 10-3 and 10-2s. Some samples then exhibit a nearly perfect log(t)-relaxation, implying that no characteristic time can be extracted, while some other samples show a preferential relaxation around 0.1s/1s. Such features appear insensitive to the amplitude of the dynamic perturbation and to the moisture content within the sample. The full nonlinear elastic response (fast dynamics) is also extracted at all amplitudes and moisture content. Adsorption of water on the grains strongly increases the elastic softening during the dynamic perturbation and the non-classical nonlinear features, whereas the classical features seem rather unaffected.

Ground seismic response analysis based on the probability density evolution method

Based on an overview of pertinent literature, the work presented in this paper focuses on the simulation of non-stationary ground motion processes and the stochastic seismic response analysis of site-specific soils. Using the random process orthogonal expansion method and the concept of random functions, initial earthquake ground motion is expressed as a random process based on only one random variable. Then, by discretizing the random acceleration process, we obtained 628 non-stationary earthquake acceleration time history samples with assigned probabilities. By comparing second-order statistical quantities of the random acceleration process between the sampled ensembles and an initial random targeting process, we demonstrate the advantages and effectiveness of this hybrid approach. The first part of this process also laid a foundation for subsequent stochastic seismic response analysis. Then, during the next part of the procedure, the calculation accuracy of the probability density evolution method (PDEM) was verified for a single-degree-of-freedom system with random excitation. The site-specific stochastic seismic response of the ground was then analyzed using the PDEM with the previously mentioned 628 acceleration time history samples. Abundant information was obtained on the probability of the ground seismic response, and the amplification effect on the target soil was verified from a random viewpoint. According to the performance-based method and PDEM, the reliability of the Shanghai soft engineering site was determined to be 0.7959. Furthermore, this new research introduces the use of PDEM techniques for modern geotechnical engineering problems.

Ground motion and rupture process of the 2004 Mid Niigata Prefecture earthquake obtained from strong motion data of K-NET and KiK-net

Abstract The 2004 Mid Niigata Prefecture earthquake (37.289°N, 138.870°E, 13.1 km, MJMA 6.8; JMA), also known as the 2004 Niigata Prefecture Chuetsu earthquake, was a thrust type earthquake that occurred on October 23, 2004 at 17:56 (JST). Strong ground motions of PGA 800-1700 cm/s2 and PGV 60-130 cm/s were observed at stations located immediately above the source region. We deduced the rupture process of this earthquake with a multi-time-window linear waveform inversion procedure. We used near-fault strong ground motion data observed at nine K-NET and KiK-net stations within 50 km from the epicenter. In order to obtain appropriate Green’s functions for the waveform inversion, we constructed two velocity structure models for stations on the hanging wall and one structure model for stations on the footwall. The estimated total slip distribution contains three asperities: (a) around the hypocenter, (b) in the upper-middle section of the fault plane, and (c) southwest of the hypocenter. The maximum slip is 3.8 m at the hypocenter and the total seismic moment is 1.2 × 1019 Nm, which corresponds to Mw=6.7. The moment rate functions in asperities (a) and (c) have a short rise time, while those in asperity (b) have a longer rise time.

Simulation of non-stationary ground motion processes (II)

Simulation of non-stationary ground motion processes (II)

Simulation of non-stationary ground motion processes (I)

This paper presents a spectral representation method for simulation of non-stationary ground motion processes on the basis of Priestley’s evolutionary spectral theory. Following this method, sample processes can be generated using a cosine series formula. It is shown that, these sample processes accurately reflect the prescribed characteristics of the evolutionary power spectral density function when the number of the terms in the cosine series is large enough; and the ensemble expected value and the ensemble autocorrelation function approach the corresponding target functions, respectively, as the sample size increases; and these sample processes are asymptotically normal as the number of the terms in the series tends to infinity. Finally, a few special cases of the formula are discussed, one of which is non-stationary white noise process, and other one is reduced to the formula for simulation of stationary stochastic processes.

Erratum to: Ground motion and rupture process of the 2003 Tokachi-oki earthquake obtained from strong motion data of K-NET and KiK-net

Erratum to: Ground motion and rupture process of the 2003 Tokachi-oki earthquake obtained from strong motion data of K-NET and KiK-net

Ground motion and rupture process of the 2003 Tokachi-oki earthquake obtained from strong motion data of K-NET and KiK-net

A great earthquake, named the 2003 Tokachi-oki earthquake, occurred in the southern Kuril subduction zone on 26th September 2003,4:50 JST (41.7797°N, 144.0795°E, 42 km depth; Japan Meteorological Agency). Its ground motion was recoreded at 655 stations of the nationwide strong motion networks, K-NET and KiK-net. A maximum peak ground acceleration of 988 cm/s2 (gal) was observed at station HKD100 and amplitudes greater than 200 cm/s2 were observed over a wide area of eastern Hokkaido. We used a multi-line linear waveform inversion method to estimate the rupture process from the strong motion data of supplied by 15 stations. We assumed a fault plane model of 140 km × 160 km with strike and dip angles of our fault model are N246°E and 18°, respectively, placed on the estimated upper boundary of the subducting Pacific Plate. The estimated total slip distribution consisted of three major slip areas; (a) around the hypocenter, (b) the northwest part of the fault with the maximum slip of 5.9 m, and (c) the northeast edge of the fault plane. The major asperity (b) was composed of two large slip areas with different slip rate functions: the duration of moment release in the sourtheast part is longer than 15 sec, but in contrast most of the seismic moment of the northwest part was released in a short period of less than 10 sec. Our estimation of the total seismic moment was 2.9 × 1021 N·m which corresponded to Mw = 8.2.

Differences in ground motion and fault rupture process between the surface and buried rupture earthquakes

We have studied differences in ground motion and fault rupture characteristics between surface rupture and buried rupture earthquakes. We found that the ground motion generated by buried rupture in the period range around 1 second is on average 1.5 times larger than the average empirical relationship. In contrast, ground motion from earthquakes that rupture the surface is 1.5 times smaller in the same period range. This phenomenon is considered to be caused by differences in fault rupture process between the two types of earthquakes. To examine possible reasons of the above effect we analyzed source slip distribution data derived from waveform inversions, and divided them into two groups: surface rupture and buried rupture earthquakes. It was found that the large slip areas (asperities) of surface rupture earthquakes are concentrated in the depth range shallower than about 5 km. In contrast, large slip areas of buried rupture earthquakes are spread over the depth deeper than 5 km. We also found that the total rupture area of buried rupture earthquakes is 1.5 times smaller than that of surface rupture earthquakes having the same seismic moment, and that deep asperities have about 3 times larger effective stress drops and 2 times higher slip velocities than shallow asperities. These observations are verified by numerical simulations using stochastic Green’s function method.

Wavelet‐based characterization of design ground motions

AbstractWith 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.

WAVELET-BASED STOCHASTIC SEISMIC RESPONSE OF A DUFFING OSCILLATOR

This paper presents formulation for the non-stationary response of a Duffing oscillator under a seismic excitation process. The excitation process is assumed to be characterized through wavelet coefficients and the non-linear system is replaced by a stochastic equivalent linear system with time-varying parameters. An example ground motion process has been used to show that the proposed approach gives accurate response estimates in case of mildly non-linear systems.