Surveys in Geophysics

, 29:187 | Cite as

A Survey of Techniques for Predicting Earthquake Ground Motions for Engineering Purposes

Original Paper

Abstract

Over the past four or five decades many advances have been made in earthquake ground-motion prediction and a variety of procedures have been proposed. Some of these procedures are based on explicit physical models of the earthquake source, travel-path and recording site while others lack a strong physical basis and seek only to replicate observations. In addition, there are a number of hybrid methods that seek to combine benefits of different approaches. The various techniques proposed have their adherents and some of them are extensively used to estimate ground motions for engineering design purposes and in seismic hazard research. These methods all have their own advantages and limitations that are not often discussed by their proponents. The purposes of this article are to: summarise existing methods and the most important references, provide a family tree showing the connections between different methods and, most importantly, to discuss the advantages and disadvantages of each method.

Keywords

Earthquake Earthquake scenario Seismic hazard assessment Strong ground motion Ground-motion prediction 

1 Introduction

The accurate estimation of the characteristics of the ground shaking that occurs during damaging earthquakes is vital for efficient risk mitigation in terms of land-use planning and the engineering design of structures to adequately withstand these motions. This article has been provoked by a vast, and rapidly growing, literature on the development of various methods for ground-motion prediction. In total, this article surveys roughly two dozen methods proposed in the literature. Only about half are commonly in use today. Some techniques are still in development and others have never been widely used due to their limitations or lack of available tools, constraints on input parameters or data for their application.

Earthquake ground-motion estimation that transforms event parameters, e.g. magnitude and source location, to site parameters, either time-histories of ground motions or strong-motion parameters (e.g. peak ground acceleration, PGA, or response spectral displacement) is a vital component within seismic hazard assessment be it probabilistic or deterministic (scenario-based). Ground-motion characteristics of interest depend on the structure or effects being considered (e.g. McGuire 2004). At present, there are a number of methods being used within research and engineering practice for ground-motion estimation; however, it is difficult to understand how these different procedures relate to each another and to appreciate their strengths and weaknesses. Hence, the choice of which technique to use for a given task is not easy to make. The purpose of this article is to summarise the links between the different methods currently in use today and to discuss their advantages and disadvantages. The details of the methods will not be discussed here; these can be found within the articles cited. Only a brief description, list of required input parameters and possible outputs are given. The audience of this article includes students and researchers in engineering seismology but also seismic hazard analysts responsible for providing estimates for engineering projects and earthquake engineers seeking to understand limits on the predictions provided by hazard analyses. Numerous reviews of ground-motion simulation techniques have been published (e.g. Aki 1982; Shinozuka 1988; Anderson 1991; Erdik and Durukal 2003) but these have had different aims and scopes to this survey.

Only methods that can be used to estimate ground motions of engineering significance are examined here, i.e. those motions from earthquakes with moment magnitude Mw greater than 5 at source-to-site distances <100 km for periods between 0 and 4s (but extending to permanent displacements for some special studies). In addition, focus is given to the estimation of ground motions at flat rock sites since it is common to separate the hazard at the bedrock from the estimation of site response (e.g. Dowrick 1977) and because site response modelling is, itself, a vast topic (e.g. Heuze et al. 2004). Laboratory models, including foam models (e.g. Archuleta and Brune 1975), are not included because it is difficult to scale up to provide engineering predictions from such experiments.

Section 2 summarises the different procedures that have been proposed within a series of one-page tables (owing to the vast literature in this domain, only brief details can be given) and through a diagram showing the links between the methods. The problem of defining an earthquake scenario is discussed in Section 3. Section 4 is concerned with the testing of methods using observations. The article concludes with a discussion of how to select the most appropriate procedure for a given task.

2 Summaries of Different Procedures

As described by Ólafsson et al. (2001) there are basically two approaches to the construction of models for the prediction of earthquake ground motions: the mathematical approach, where a model is analytically based on physical principles, and the experimental one, where a mathematical model, which is not necessarily based on physical insight, is fitted to experimental data. In addition, there are hybrid approaches combining elements of both philosophies. Earthquakes are so complex that physical insight alone is currently not sufficient to obtain a reasonable model. Ólafsson et al. (2001) term those models that only rely on measured data ‘black-box’ models.

Figure 1 summarises the links between the different methods described in Tables 122. Each table briefly: (1) describes the method; (2) lists the required input parameters (bold for those parameters that are invariably used, italic for parameters that are occasionally considered and normal font for those parameters that are often implicitly, but not often explicitly, considered) and the outputs that can be reliably obtained; (3) lists a maximum of a dozen key references (preference is given to: the original source of the method, journal articles that significantly developed the approach and review articles) including studies that test the approach against observations; (4) lists the tools that are easily available to apply approach (public domain programs with good documentation help encourage uptake of a method 1); (5) gives the rough level of use of the technique in practice and in research; and finally (6) summarises the advantages and disadvantages/limitations of the method. The following sections introduce each of the four main types of methods.
Fig. 1

Summary of the approximate date when a method was developed on the x-axis, links to other approaches and the level of detail of the scenario modelled on the y-axis. Boxes indicate those methods that are often used in research and/or practice

Table 1

Method of representative accelerograms

Description of method

Records are chosen from databanks containing accelerograms that are appropriate for the considered site. Selection is often made considering the magnitude and distance (and occasionally other characteristics such as style-of-faulting) of the scenario event. Records with elastic response spectra that match a design spectrum are often preferred. After selection scaling of the amplitude (and occasionally the time scale) is often performed to corrected for differences to the design ground-motion parameters (e.g. PGA). A modern variant of this technique that is increasing in popularity is the minor adjustment of time-histories so that their response spectra better match the design spectrum

Input parameters

Outputs

Key references

Magnitude, distance, design response spectrum, seismotectonic regime, source depth, style-of-faulting

Scaled (modified) natural accelerogram reliable up to 1–4s for analogue or for digital (Akkar and Bommer 2006)

Guzman and Jennings (1976), Dowrick (1977), Campbell (1986), Joyner and Boore (1988), Shome et al. (1998), Bommer et al. (2000), Bommer and Ruggeri (2002), Bommer and Acevedo (2004), Baker and Cornell (2006), Watson-Lamprey and Abrahamson (2006), Beyer and Bommer (2007), Hancock et al. (2008)

Available tools

Used in research

Used in practice

Various websites (e.g. Ambraseys et al. 2004b) and CD ROMs (e.g. Ambraseys et al. 2004a) providing accelerograms; RSPMATCH2005 (Hancock et al. 2006); RASCAL (Silva and Lee 1987); WAVGEN (Mukherjee and Gupta 2002)

Often

Very often although they are rarely called ‘representative accelerograms’

Advantages

Disadvantages/limitations

Rapid; straightforward; many available records from Internet sites and CD ROM collections; can account for effects (e.g. near-field pulses) that are not well modelled by other methods; well established; since the ground motions have occurred in the past, they are physically possible; more easily understood and accepted by decision makers since based on observations; only requires standard scenario characteristics; includes ground-motion variability; can provide triaxial time-histories consistent with observed correlations between components

Still lack of near-source records from large events (hence difficult to know if observations are well representative of the true range of possible motions or sampling artifact); difficult to find records to match scenario characteristics in addition to magnitude and distance; small databanks for most regions (outside California and Japan); often implicit assumption is that host and target regions have similar characteristics (or that strong motions are not dependent on region); difficult to ascertain whether certain records are applicable elsewhere due to particular site or source effects; scaling can have significant impact on results of dynamic analyses

Table 2

Method of empirical ground-motion models (ground-motion prediction equations, GMPES)

Description of method

A databank of accelerograms and metadata from a region are collated and processed. Strong-motion intensity parameters (e.g. PGA) are computed for these accelerograms. Regression analysis is performed using a handful of source, path and site independent variables and the intensity parameter as the dependent variable. Less popular variants consist of the development of tables, graphs or neural nets for prediction purposes. The developed models are evaluated for a given scenario and the results are commonly weighted

Input parameters

Output parameters

Key references

Magnitude, distance, near-surface site characteristics, style-of-faulting, source depth, seismotectonic regime, gross source characteristics, deep geology

Strong-motion intensity parameters (e.g. PGA, PGV, PGD, response spectral ordinates, duration, other parameters)

Esteva and Rosenblueth (1964), Trifunac (1976), Joyner and Boore (1988), Abrahamson and Shedlock (1997), Anderson (1997b), Lee et al. (2000), Campbell (2002), Douglas (2003), Scherbaum et al. (2004), Bommer and Alarćon (2006), Power et al. (2008), Abrahamson et al. (2008)

Available tools

Used in research

Used in practice

Various websites (e.g. Ambraseys et al. 2004b) and CD ROMs (e.g. Ambraseys et al. 2004a) providing accelerograms; various spreadsheets and computer codes for evaluating models and for regression analysis; OpenSHA (Field et al. 2003)

Very often

Very often

Advantages

Disadvantages/limitations

Rapid; well established; can be simply and easily applied without having to set up lots of simulations (hence useful for regional PSHA); only requires standard scenario characteristics; more easily understood and accepted by decision makers since based on observations; easy to develop new GMPEs; includes ground-motion variability; can model different causes of variability (e.g. inter-event, inter-site and record-to-record variation)

Output is strong-motion parameter rather than time-history; strong-motion parameter is not always useful for sophisticated engineering analyses; still lack of near-source records from large events (hence difficult to know if observations are well representative of the true range of possible motions or sampling artifact); small databanks for most regions (outside California and Japan); often implicit assumption is that host and target regions have similar characteristics (or that strong motions are not dependent on region); applies to a generic (mainly unknown) situation so cannot account for site-specific conditions; never sure of having the correct functional form; observed data smoothed due to large scatter in observations; requires lots of records to derive models; at edges of dataspace predictions poorly constrained; physically basis of coefficients is not always clear; ground motions from small and large events scale differently with magnitude and distance hence difficult to use weak records to predict strong motions; debate over preference for global, regional or local models; large epistemic uncertainty, mainly due to limited data

Table 3

Methods based on macroseismic intensity-ground-motion correlations

Description of method

A databank of accelerograms and their associated macroseismic intensity (and possibly other metadata) from a region are collated and processed. Strong-motion intensity parameters (e.g. PGA) are computed for these accelerograms. Regression analysis is performed with macroseismic intensity (and possibly other parameters) as the independent variable(s) and the strong-motion parameter as the dependent variable. Assessed macroseismic site intensity is converted to a strong-motion intensity parameter using the previously derived correlation

Input parameters

Outputs

Key references

Macroseismic site intensity, seismotectonic regime, source depth, magnitude, distance

Strong-motion intensity parameters (e.g. PGA, PGV, PGD, response spectral ordinates, duration, other parameters)

Cancani (1904), Gutenberg and Richter (1942), Hershberger (1956), Ambraseys (1974), Trifunac and Brady (1975), Murphy and ÓBrien (1977), Campbell (1986), Wald et al. (1999), Atkinson and Sonley (2000), Sokolov and Wald (2002), Kaka and Atkinson (2004), Souriau (2006)

Available tools

Used in research

Used in practice

None known

Rarely

Occasionally

Advantages

Disadvantages/limitations

Rapid; straightforward; more easily understood and accepted by decision makers since based on observations; only requires standard scenario characteristics; includes ground-motion variability; historical earthquake catalogues often defined only in terms of macroseismic intensities hence less conversions required than other techniques; does not require strong-motion data if adopt data/model from another region; easier to apply ground-motion estimates for risk evaluation if vulnerability functions defined in terms of macroseismic intensity

Output is strong-motion parameter rather than time-history; strong-motion parameter not always useful for sophisticated engineering analyses; often implicit assumption is that host and target regions have similar characteristics (or that strong motions are not dependent on region); weak statistical dependence (lack of clear physical relationship) between ground-motion parameters and intensity; intensities in catalogues are subjective and can be associated with large inaccuracies; few reliable usable correlations between intensity and different strong-motion parameters because there are many intensity scales, intensity assessment can be country-dependent and lack of intensity data from close to accelerograph stations; many intensity relationships derived using isoseismal contours, which leads to positive bias in estimated motions; applies to a generic (mainly unknown) situation so cannot account for site-specific conditions; never sure of having the correct functional form; observed data smoothed due to large scatter in observations; requires lots of records to derive correlations; at edges of dataspace predictions poorly constrained; physically basis of coefficients not always clear; ground motions from small and large events scale differently with magnitude and distance hence difficult to use weak records to predict strong motions; debate over preference for global, regional or local models; large epistemic uncertainty, mainly due to limited data

2.1 Empirical Methods

The three methods described in this section are closely based on strong ground motion observations. Such empirical techniques are the most straightforward way to predict ground motions in future earthquakes and they are based on the assumption that shaking in future earthquakes will be similar to that observed in previous events. The development of these methods roughly coincided with the recording of the first strong-motion records in the 1930s but they continue to be improved. Empirical methods remain the most popular procedure for ground-motion prediction, especially in engineering practice. Tables 13 summarise the three main types of empirical methods.

2.2 Black-box Methods

This section describes four methods (Tables 47) that can be classified as black-box approaches because they do not seek to accurately model the underlying physics of earthquake ground motion but simply to replicate certain characteristics of strong-motion records. They are generally characterised by simple formulations with a few input parameters that modify white noise so that it more closely matches earthquake shaking. These methods were generally developed in the 1960s and 1970s for engineering purposes to fill gaps in the small observational datasets then available. With the great increase in the quantity and quality of strong-motion data and the development of powerful techniques for physics-based ground-motion simulation, this family of prediction techniques has become less important although some of the procedures are still used in engineering practice.
Table 4

Methods based on stationary black-box simulations

Description of method

This type of method was developed to fill in gaps in early observational databanks, particularly, for large earthquakes. White noise (sum of cosines with random time delays) is modified by filtering in the frequency domain to obtain acceleration time-histories that conform to the observed main characteristics of earthquake ground motions

Input parameters

Outputs

Key references

Magnitude, distance, near-surface site characteristics, source depth, seismotectonic regime

Artificial acceleration time-histories reliable from 0 to about 2s

Housner (1947, 1955), Bycroft (1960), Housner and Jennings (1964), Jennings et al. (1968), Dowrick (1977)

Available tools

Used in research

Used in practice

None known

Very rarely

Very rarely

Advantages

Disadvantages/limitations

Rapid; straightforward; provides as many independent time-histories for a scenario as required; includes consideration of ground-motion variability; time-histories adequate for examining elastic response of lightly damped structures; well-suited for analytic solutions and Monte Carlo simulations of structural response; do not require knowledge of source, path and site

Do not generally involve rigorous considerations of the physics of the earthquakes; not appropriate for modelling smaller earthquake motions or for use in studies where the less intense but longer tails of accelerograms are thought to be significant, e.g. liquefaction studies; does not consider non-stationarity in time and frequency domains of earthquake ground motions; true ground-motion variability can be underestimated; frequency content not realistic; not accurate close to source where non-stationarity important; for generic scenario; too many cycles in ground motions; energy content of motions not realistic

Table 5

Methods based on non-stationary black-box simulations

Description of method

White noise is modified by filtering in the frequency domain and then it is multiplied by an envelope function in the time domain. Also this method can account for non-stationarity in frequency domain and a consideration of phase. Frequency content and envelope function developed using equations developed through regression analysis of observational data

Input parameters

Outputs

Key references

Magnitude, distance, near-surface site characteristics, style-of-faulting, source depth, seismotectonic regime

Artificial acceleration time-histories reliable from 0 to about 4s (e.g. Sabetta and Pugliese 1996)

Sabetta and Pugliese (1996), Montaldo et al. (2003), Pousse et al. (2006)

Available tools

Used in research

Used in practice

Program of Pousse et al. (2006)

Occasionally

Rarely

Advantages

Disadvantages/limitations

Rapid; straightforward; only requires a handful of input parameters; close link to observations; provides as many independent time-histories for a scenario as required; includes consideration of ground-motion variability; accounts for non-stationarity in time and frequency domains; do not require knowledge of source, path and site

Do not generally involve rigorous considerations of the physics of the earthquakes; require good databanks to constrain empirical parameters; true ground-motion variability can be underestimated

Table 6

Methods based on autoregressive/moving average (ARMA) simulations

Description of method

Parametric time-series models (ARMA models), where a random process is modelled by a recursive filter using random noise as input, are used. The parameters of the filter are determined from observed accelerations by using a suitable criterion for the goodness of fit

Input parameters

Outputs

Key references

Magnitude, distance, near-surface site characteristics, seismotectonic regime, source depth

Artificial acceleration time-histories reliable from 0 to about 2s

Jurkevics and Ulrych (1978), Nau et al. (1982), Ólafsson and Sigbjörnsson (1995), Ólafsson et al. (2001)

Available tools

Used in research

Used in practice

None known

Rarely

Very rarely

Advantages

Disadvantages/limitations

Rapid; nonparametric method to compute acceleration envelopes so does not rely on assumed envelope shape; provides as many independent time-histories for a scenario as required; includes consideration of ground-motion variability; well-suited for Monte Carlo simulations of structural response; ARMA models only need a handful of coefficients to give a good statistical fit to time histories; do not require knowledge of source, path and site

Do not generally involve rigorous considerations of the physics of the earthquakes; true ground-motion variability can be underestimated; not commonly used so poorly known; requires observational data to constrain input parameters; assumes that the strong-motion phase can be modelled as a locally stationary stochastic process; does not give reliable estimate outside range of data

Table 7

Methods based on spectrum-matching simulations

Description of method

This method was developed to provide acceleration time-histories whose elastic response spectra exactly match a target spectrum. White noise is modified by filtering in the frequency domain and then it is multiplied by an envelope function in the time domain so that the response spectrum matches the target within a specified tolerance. An iterative process is used

Input parameters

Outputs

Key references

Elastic response spectrum, duration of strong shaking

Artificial acceleration time-histories reliable from 0 to about 2s

Kaul (1978), Vanmarcke (1979), Naeim and Lew (1995)

Available tools

Used in research

Used in practice

SIMQKE (Vanmarcke and Gasparini 1976), various updates and numerous similar codes

Occasionally

Often

Advantages

Disadvantages/limitations

Rapid; straightforward; provides time-histories whose elastic response spectra exactly match design spectrum; only requires an elastic response spectrum as input; commonly used in past so well established; do not require knowledge of source, path and site; easy-to-use software freely available

Do not generally involve rigorous considerations of the physics of the earthquakes; true ground-motion variability can be underestimated; too many cycles in ground motions; energy content of motions not realistic; velocity and displacement time-histories not realistic

2.3 Physics-based Methods

Although this class of methods was simply called the ‘mathematical approach’ by Ólafsson et al. (2001) the recent advances in the physical comprehension of the dynamic phenomena of earthquakes and in the simulation technology means that we prefer the name ‘physics-based methods’. These techniques often consist of two stages: simulation of the generation of seismic waves (through fault rupture) and simulation of wave propagation. Due to this separation it is possible to couple the same source model with differing wave propagation approaches or different source models with the same wave propagation code (e.g. Aochi and Douglas 2006). In this survey emphasis is placed on wave propagation techniques.

Source models that have been used extensively for ground-motion prediction include theoretical works by: Haskell (1969), Brune (1970, 1971), Papageorgiou and Aki (1983), Gusev (1983), Joyner (1984), Zeng et al. (1994) and Herrero and Bernard (1994). Such insights are introduced into prescribed earthquake scenarios, called ‘kinematic’ source models. It is well known that the near-source ground motion is significantly affected by source parameters, such as the point of nucleation on the fault (hypocentre), rupture velocity, slip distribution over the fault and the shape of the slip function (e.g. Miyake et al. 2003; Mai and Beroza 2003; Tinti et al. 2005; Ruiz et al. 2007). This aspect is difficult to take into account in empirical methods. Recently it has become possible to introduce a complex source history numerically simulated by pseudo- or fully-dynamic modelling (e.g. Guatteri et al. 2003, 2004; Aochi and Douglas 2006; Ripperger et al. 2008) into the prediction procedure. Such dynamic simulations including complex source processes have been shown to successfully simulate previous large earthquakes, such as the 1992 Landers event (e.g. Olsen et al. 1997; Aochi and Fukuyama 2002). This is an interesting and on-going research topic but we do not review it in this article.

All of the physics-based deterministic methods convolve the source function with synthetic Green’s functions (the Earth’s response to a point-source double couple) to produce the motion at ground surface. Erdik and Durukal (2003) provide a detailed review of the physics behind ground-motion modelling and show examples of ground motions simulated using different methods. Tables 818 summarise the main types of physics-based procedures classified based on the method used to calculate the synthetic seismograms in the elastic medium for a given earthquake source. Most of these are based on theoretical concepts introduced in the 1970s and 1980s and intensively developed in the past decade when significant improvements in the understanding of earthquake sources and wave propagation (helped by the recording of near-source ground motions) were coupled with improvements in computer technology to develop powerful computational capabilities. Some of these methods are extensively used for research purposes and for engineering projects of high-importance although most of them are rarely used in general engineering practice due to their cost and complexity.
Table 8

Methods based on physics-based stochastic models

Description of method

A Fourier spectrum of ground motion is estimated using a stochastic model of the source spectrum that is transferred to the site by considering geometric decay and anelastic attenuation. The parameters that define the source spectrum and the geometric and anelastic attenuation are based on simple physical models of the earthquake process and wave propagation. These parameters are estimated by analysing many seismograms. After the Fourier spectrum at a site is estimated time-histories can be computed by adjusting and enveloping Gaussian white noise to give the desired spectrum and duration of shaking. Some authors develop equations like those developed from observational data (Table 2) based on thousands of simulations for various magnitudes and distances

Input parameters

Outputs

Key references

Source spectral amplitude, geometric decay rates, anelastic attenuation, local site amplification and attenuation, source spectral shape, source duration, path duration

Ground-motion time-histories reliable from 0 to about 2s

Hanks (1979), Hanks and McGuire (1981), Boore (1983), Silva et al. (1999), Atkinson and Somerville (1994), Boore (2003), Atkinson and Boore (2006)

Available tools

Used in research

Used in practice

SMSIM (Boore 2005), RASCAL (Silva and Lee 1987) and numerous similar codes

Often

Occasionally

Advantages

Disadvantages/limitations

Rapid; good predictions for short-period motions; useful for regions lacking observational data from damaging earthquakes because the parameters required can be estimated using data from standard seismological networks; input parameters have physical meaning hence link between physics and ground motions; realistic looking time-histories; acts as a link between engineering and seismological approaches

Long-period motions can be poorly estimated since generally only for S waves; does not generate three-component seismograms with physically-expected coherency; does not account for phase effects due to propagating rupture or wave propagation and, therefore, may not be reliable in near-source region; uncertainty in shape of source spectra for moderate and large events; variability only taken into account by the random generation of the phase; frequency content is stationary with time hence late-arriving surface waves and attenuated shear waves are not modelled; for generic scenario and not a specific source, path and site

Table 9

Methods based on physics-based extended stochastic models

Description of method

The fault rupture plane is modelled as an array of subfaults. Rupture initiates at the hypocentre and spreads along the fault plane. The radiation from each subfault is modelled as in the physics-based stochastic method (Table 8). Simulations from each subfault are summed at each considered observation point (after accounting for correct time delays at observation point). The size of the subfaults controls the overall spectral shape at medium frequencies. Some authors develop equations like those developed from observational data (Table 2) based on thousands of simulations for various magnitudes and distances

Input parameters

Outputs

Key references

Source spectral amplitude, fault location and size, rupture history, geometric decay rates, anelastic attenuation, local site amplification and attenuation, source spectral shape, source duration, path duration

Ground-motion time-histories reliable from 0 to about 4s

See Table 8, Beresnev and Atkinson (1998), Atkinson and Silva (2000), Motazedian and Atkinson (2005)

Available tools

Used in research

Used in practice

FINSIM (Beresnev and Atkinson 1998), EXSIM (Motazedian and Atkinson 2005)

Occasionally

Rarely

Advantages

Disadvantages/limitations

Rapid; good predictions for short-period motions; useful for regions lacking observational data from damaging earthquakes because most parameters required can be estimated using data from standard seismological networks; input parameters have physical meaning hence link between physics and ground motions; good predictions for near-source regions; realistic looking time-histories

Uncertainty in shape of source spectra for moderate and large events

Table 10

Method based on group-velocity dispersion curves

Description of method

The dispersive properties of earthquake waves propagating through low-velocity layers of the crust are used to model the phase characteristics of the simulated ground motion. Higher order modes of Love and Rayleigh-wave group velocity dispersion curves are used. This technique models time variations in frequency content as well as in amplitude due to surface wave dispersion. The stochastic nature of motion is captured by random phasing. The smooth Fourier amplitude spectrum and duration used to scale the ground motions are defined based on empirical ground-motion models or correlations with macroseismic intensity (Tables 2, 3)

Input parameters

Outputs

Key references

Magnitude (or epicentral macroseismic intensity), distance, velocity and density profile of site, style-of-faulting, source depth, seismotectonic regime

Ground-motion time-histories reliable from 0 to about 4s

Trifunac (1971, 1990), Wong and Trifunac (1978), Lee and Trifunac (1985, 1987)

Available tools

Used in research

Used in practice

SYNACC (Wong and Trifunac 1978)

Rarely

Very rarely

Advantages

Disadvantages/limitations

Rapid; accounts for non-stationary of time-histories; can be used to generate strain, curvatures and rotation (torsion and rocking) components of motion consistent with translation components; accounts for detailed site characteristics; includes some variability in ground motions; combines aspects of empirical and physics-based techniques; does not require detailed source description; seismograms have realistic appearance

Medium structure limited to stratified layers; requires detailed velocity and density profile for site; no large-scale validation exercise conducted; not widely used and therefore not widely accepted by community; approach is strictly only valid for surface waves; for generic source; mainly based on observations at deep alluvium sites

Table 11

Semi-analytical methods

Description of method

Solve the elastodynamic equation, complying with the boundary conditions of the free surface, continuity of wave field across each interface and bonded motion at infinity, for a layered homogeneous and isotropic elastic medium over a half-space with an earthquake point source buried inside. The solution is usually derived using the generalized reflection and transmission matrix method, which excludes the growing exponential terms. The solution is computed in the frequency domain and then converted to the time domain. This easily allows the introduction of frequency-dependent attenuation parameters (e.g. quality factor) independently for P and S waves

Input parameters

Outputs

Key references

Source location, velocity and density profiles of layered medium, source time function and mechanism, quality factor of medium

Ground-motion time-histories reliable for a frequency range defined by number of discrete frequencies or wavenumbers

Aki and Larner (1970), Kennett and Kerry (1979), Bouchon (1981), Apsel and Luco (1983), Luco and Apsel (1983), Koketsu (1985), Takeo (1985), Zeng and Anderson (1995), Wang (1999), Aki and Richards (2002), Bouchon and Sánchez- Sesma (2007), Chen (2007)

Available tools

Used in research

Used in practice

Many authors freely provide their codes on demand; COMPSYN (Spudich and Xu 2003)

Often

Often

Advantages

Disadvantages/limitations

Numerically accurate over wide range of frequencies; useful for inverse problems; seismograms have realistic appearance; more rapid than typical FDM; more accurate than typical FDM; stable technique for layers of thicknesses from ms to kms; valid for a wide range of frequencies; can account for material attenuation; widely used in different fields of seismology; can provide static deformation field; can give theoretical Green’s function for a unit source so for arbitrary source (finite source with complex source time function) synthetic waveforms can be generated through convolution

Medium structure often limited to stratified elastic layers; time consuming to calculate motions at many points

Table 12

Finite difference methods (FDM)

Description of method

Directly solve the differential equation of elastic or (viscoelastic) wave propagation in a medium. The volume is discretised, usually by equally-spaced grids, but some intelligent ways of using unstructured grids have also been proposed. Finite fault sources are usually (except when dynamically modelling the rupture process along the fault plane) treated as a series of point sources in the form of double couple forces or stress gluts corresponding to a seismic moment. As for other pure numerical methods, anelastic attenuation can be approximated as a damping factor in the elastic medium but more realistically it is necessary to solve the visco-elastic equations. To simulate an unbounded medium, such as the Earth, some absorbing boundary conditions should be introduced at the edges of the model space so as to avoid artificial wave reflections. Both these aspects are still research topics

Input parameters

Outputs

Key references

Source location, time function and mechanism, velocity and density profiles of layered medium, quality factor of medium

Ground-motion time-histories reliable for low frequencies in heterogeneous model corresponding to grid spacing (normally one wavelength needs 5–10 spatial grid points)

Boore (1973), Virieux and Madariaga (1982), Frankel and Clayton (1986), Levander (1988), Graves (1996), Olsen et al. (1997), Pitarka et al. (1998), Aoi and Fujiwara (1999), Day and Bradley (2001), Oprsal and Zahradnik (2002), Olsen et al. (2006), Komatitsch and Martin (2007), Moczo et al. (2007b)

Available tools

Used in research

Used in practice

Many authors freely provide their codes on demand, e.g. http://geo.mff.cuni.cz/∼io/

Often

Occasionally

Advantages

Disadvantages/limitations

Can treat any heterogeneous medium; can allow volumetric visualization of wave propagation without increasing number of numerical calculations; rapid computer development in 1990s means that large calculations are easy for practical applications; most efficient of all purely numerical methods; complex geometry more easy to model; can also treat any anisotropy and/or anelastic media

Not better than semi-analytical methods with respect to numerical accuracy; numerical dispersion; shows best performance for structured grids; not good at treating sharp interfaces with strong contrasts (e.g. internal layering and topography); gridding does not always correspond to material interfaces, which means that elastic properties attributed to each grid point is usually an average value thereby limiting the accuracy of the method in heterogeneous media

Table 13

Finite element methods (FEM)

Description of method

Solve the variational, or weak form, of the equations of wave propagation with low-order polynomial bases in the framework of unstructured elements. This leads to a linear system of equations in matrix form. Normally the tensors are not diagonal and therefore the unknown solution vectors have to be numerically inverted from these equations

Input parameters

Outputs

Key references

Source location, time function and mechanism, velocity and density profiles of layered medium, mesh, quality factor of medium

Ground-motion time-histories reliable for a frequency defined by element spacing

Lysmer and Drake (1972), Bao et al. (1998), Ma et al. (2007), Moczo et al. (2007a)

Available tools

Used in research

Used in practice

Mostly commercial codes

Rarely

Rarely

Advantages

Disadvantages/limitations

Can treat any heterogeneous medium; can allow volumetric visualization of wave propagation without increasing number of numerical calculations; complex geometry more easy to model; parallelization of computer codes possible; meshing can be made consistent with material interfaces, which improves accuracy of method (see Table 12)

Numerical dispersion; very numerically expensive; parallelization usually difficult because of domain participation and matrix; complicated meshing is a big task that must be completed before application of FEM code

Table 14

Spectral element methods (SEM)

Description of method

Solve the variational, or weak form, of the equations of wave propagation with high-order basic functions for unstructured elements. It is an integrated formulation of classical FEM (Table 13). This approach is becoming popular for the simulation of ground motions from large earthquakes and for motions affected by basin structures

Input parameters

Outputs

Key references

Source location, time function and mechanism; velocity and density profiles of layered medium; mesh, quality factor of medium

Ground-motion time-histories reliable for a frequency defined by element spacing and order of basic functions

Faccioli et al. (1997), Komatitsch and Vilotte (1998), Komatitsch and Tromp (1999), Komatitsch et al. (2004), Chaljub et al. (2007a)

Available tools

Used in research

Used in practice

SPECFEM3D (Chen et al. 2008)

Occasionally

Very rarely

Advantages

Disadvantages/limitations

See Table 13; compared to FEM calculation is faster thanks to diagonal matrix; can use larger elements thanks to higher-order basic functions compared to FEM

Much more numerically expensive then FDM but less expensive than FEM; simple structured elements generally preferred

Table 15

Methods based on modal summation

Description of method

For a wave field in a limited area only consisting of wave-trains propagating away from the source, the surface-wave formulation is adequate. Lateral heterogeneity can also be treated as coupling of local modes

Input parameters

Outputs

Key references

Source location, time function and mechanism, velocity and density profiles of layered medium, quality factor of medium

Ground-motion time-histories reliable for low frequencies in heterogeneous model defined by used mode frequencies

Woodhouse (1974), Swanger and Boore (1978), Panza (1985), Panza and Suhadolc (1987), Florsch et al. (1991), Douglas et al. (2004), Maupin (2007)

Available tools

Used in research

Used in practice

Some authors freely provide their codes on demand

Occasionally

Rarely

Advantages

Disadvantages/limitations

Useful when surface waves dominate, e.g. at long periods and moderate distances; widely used for teleseismic studies so efficient programs exist; the dispersion parameters and eigenfunctions need only be computed once for time-domain synthesis for any type and depth of source, at any azimuth and any distance; time-domain synthesis simple and rapid; useful for interpretation of relative importance of source depth and site response; easy to extend point source solutions to extended sources; number of layers not a practical limitation; useful for inverse problems

Only reliable when epicentral distance is greater than focal depth; only gives an approximation (of unknown accuracy) of the total motion; not suitable when no surface layers

Table 16

Lattice particle method

Description of method

Instead of solving differential equation in continuous medium simulate physical interaction between particles on a discrete lattice. Depending on the physical description and numerical discretisation this method is also known as: lattice solid model, discrete element method or distinct element method

Input parameters

Outputs

Key references

Source location, time function and mechanism, velocity and density profiles of layered medium, mesh, quality factor of medium

Ground-motion time-histories reliable for low frequencies in heterogeneous model corresponding to a large number of elements

Mora and Place (1994), Place and Mora (1999), Dalguer et al. (2003), Shi and Brune (2005)

Available tools

Used in research

Used in practice

None known

Very rarely

Very rarely

Advantages

Disadvantages/limitations

Applicable for complex hydro-dynamical problems that cannot be described as a system of continuous mediums; accurate for compressive waves

Complex calculation; less accurate for shear waves; numerically expensive

Table 17

Finite volume method

Description of method

Transform the differential equation into a conservative formulation inside a discrete volume. This leads to an integral equation different from those of FEM and SEM; however, for certain simple cases the method corresponds to FDM or FEM

Input parameters

Outputs

Key references

Source location, time function and mechanism, velocity and density profiles of layered medium, mesh, quality factor of medium

Ground-motion time-histories reliable for a frequency defined by element spacing

Dormy and Tarantola (1995), LeVeque (2002), Käser and Iske (2005)

Available tools

Used in research

Used in practice

None known

Very rarely

Very rarely

Advantages

Disadvantages/limitations

Can correctly treat the material interfaces; suitable for unstructured meshes; can be more accurate than FDM

Higher-order approximation numerically costly; numerical efforts much heavier than FDM

Table 18

Methods based on ray theory

Description of method

Green’s functions are calculated to describe the effect of wave propagation from source to site considering the direct and reflected rays. The overall time-history is produced by summing the rays, which arrive at different times. The amplitude and time relationships between these arrivals change with distance. Overall duration related to crustal structure and focal depth. Maximum distance for realistic wave propagation modelling depends on the number of rays

Input parameters

Outputs

Key references

Source location, time function and mechanism, velocity and density profiles of layered medium, quality factor of medium

Ground-motion time-histories reliable for low frequencies depending on heterogeneities

Heaton and Helmberger (1977), Atkinson and Somerville (1994)

Available tools

Used in research

Used in practice

Some authors freely provide their codes on demand; ISOSYN (Spudich and Xu 2003)

Often

Rarely

Advantages

Disadvantages/limitations

Economical, especially for high frequencies where the contribution of surface waves is small; arrival of different phases accurately modelled; attenuation function derived from focal depth and crustal structure and therefore more appropriate when empirical attenuation information lacking; provides insight through analysis of crustal conditions controlling details of observed ground motions and also the effects of focal depth on attenuation

Not efficient when many layers; cannot easily account for attenuation; time-histories not realistic because scattering not included; low frequencies better predicted than high frequencies

2.4 Hybrid Methods

To benefit from the advantages of two (or more) different approaches and to overcome some of their disadvantages a number of hybrid methods have been proposed. These are summarised in Tables 1922. These techniques were developed later than the other three families of procedures, which are the bases of these methods. Since their development, mainly in the 1980s and 1990s, they have been increasingly used, especially for research purposes. Their uptake in engineering practice has been limited until now, although they seem to be gaining in popularity due to the engineering requirement for broadband time-histories, e.g. for soil–structure interaction analyses.
Table 19

Methods based on empirical Green’s functions (EGF) (classic)

Description of method

Observed ground motion(s) recorded at a site (e.g. from aftershock(s) of a mainshock that is to be modelled) are collected and are used as EGF(s). EGF(s) should have same focal mechanism(s) as modelled earthquake. The modelled fault is divided into subfaults whose sizes equal the rupture area of the event(s) contributing the EGF(s). Fault rupture is simulated and the EGFs are used as the ground motion from each subfault. Therefore the simulated ground motion at a site is the weighted (moment scaling of small events and correction for radiation pattern) time-delayed (to model rupture propagation) sum of the EGFs

Input parameters

Outputs

Key references

Recorded accelerogram(s) of small event(s) (1–3 magnitude units smaller than modelled event) in the source region of the modelled earthquake, basic fault model, source-to-site distances

Ground-motion time-histories reliable from 0 to 1–10s, depending on quality of EGF(s)

Hartzell (1978), Kanamori (1979), Hadley and Helmberger (1980), Dan et al. (1990), Irikura and Kamae (1994), Tumarkin and Archuleta (1994), Frankel (1995), Kamae et al. (1998), Pavic et al. (2000)

Available tools

Used in research

Used in practice

None known

Often

Rarely

Advantages

Disadvantages/limitations

Computation is rapid; EGFs already contain all the information about the path and local site effects; does not explicitly compute the wave path or site effects (since captured within the time-histories from the small earthquake); simulated motions are closely based on observations; ground motions look realistic

Only possible where appropriate records of small events from the source area recorded at sites of interest are available (rare for source areas of future large earthquakes); EGF(s) must have same focal mechanism(s) as modelled earthquake; many (poorly constrained) degrees of freedom therefore large epistemic uncertainties in results; strictly only for site(s) with available EGF(s); signal-to-noise ratio of Green’s function limits long-period estimation; event should be able to be considered as a point source; difficult to match the source characteristics since the stress drops of small and large earthquakes may be different; valid up to the corner frequency of EGF(s); debate over correct method to sum the EGFs; results can have strong dependence on choice of EGF(s); does not account for nonlinear site effects (not a problem if predicting at rock sites)

Table 20

Methods based on empirical Green’s functions (stochastic)

Description of method

As in the classic EGF method (Table 19) observed ground motion(s) recorded at a site (e.g. from aftershock(s) of a mainshock that is to be modelled) are collected and are used as EGF(s). These are stochastically summed (using a probability density of time delays) so that the simulated ground motions are, on average, in exact agreement with current knowledge on earthquake scaling relations

Input parameters

Outputs

Key references

Recorded accelerogram(s) of small event(s) (1–3 magnitude units smaller than modelled event) in the source region of the modelled earthquake, magnitude, stress drop source-to-site distance

Ground-motion time-histories reliable from 0 to 1–10s, depending on quality of EGF(s)

See Table 19, Joyner and Boore (1986), Wennerberg (1990), Ordaz et al. (1995), Kohrs-Sansorny et al. (2005)

Available tools

Used in research

Used in practice

None known

Often

Rarely

Advantages

Disadvantages/limitations

Rapid; far fewer degrees-of-freedom than classic EGF approach; simulates a multitude of rupture processes; variability in simulated ground motions; see Table 19

Source-to-site distance must be greater than source dimensions therefore not for near-source region since assumes point source and hence does not model directivity; see Table 19

Table 21

Hybrid stochastic-empirical method

Description of method

A stochastic model (Table 8) is constructed for a target region (e.g. from existing literature). Stochastic models are estimated for existing empirical ground-motion models (for different host regions) for response spectra by finding models that lead to the minimum misfit between predicted response spectra from empirical and stochastic models. Response spectra are predicted for various magnitudes and distances (and other independent variables) by the empirical ground-motion models and then are multiplied by the ratio between the response spectrum predicted by the stochastic models for the target and host regions. These response spectral ordinates are then regressed to develop hybrid stochastic-empirical ground-motion models for the target region

Input parameters

Outputs

Key references

Magnitude, distance, near-surface site characteristics, style-of-faulting, seismotectonic regimes of host and target regions, source depth, gross source characteristics, deep geology, Source spectral amplitude, geometric decay rates, anelastic attenuation, local site amplification and attenuation, source spectral shape, source duration, path duration

Strong-motion intensity amplitude parameters (e.g. PGA, PGV, PGD and response spectral ordinates)

See Tables 2 and 8, Atkinson (2001), Campbell (2003), Tavakoli and Pezeshk (2005), Douglas et al. (2006), Scherbaum et al. (2006), Campbell (2007)

Available tools

Used in research

Used in practice

CHEEP (Douglas et al. 2006)

Occasionally

Rarely

Advantages

Disadvantages/limitations

See Tables 2, 8

See Tables 2 and 8; difficult to assess true variability of derived models; not yet validated by observations

Table 22

Hybrid numerical methods

Description of method

High frequencies from one method and low frequencies from another method to get hybrid synthetic ground motions (after used matched filters to combine the two approaches) that are then used to simulate motions from large earthquakes. This approach is taken since smaller scale heterogeneity in the Earth (source, propagation path and site) is difficult to deterministically identify and our knowledge in each method is limited. Those who propose EGF or stochastic methods (e.g. Tables 8, 9, 19 and 20) to generate high frequencies assume relatively simple earthquake source description, whereas those who use semi-analytical or numerical methods (see Tables 1113) up to high frequencies adopt complex descriptions of the earthquake source, which have been greatly developed in the past decade. There are numerous combinations proposed in the literature

Input parameters

Outputs

Key references

See tables for the two methods comprising the hybrid approach

See tables for the two methods comprising the hybrid approach

Berge et al. (1998), Kamae et al. (1998), Pitarka et al. (2000), Hartzell et al. (2002), Mai and Beroza (2003), Gallovič and Brokešoá (2007), Hisada (2008)

Available tools

Used in research

Used in practice

No ready-to-use code is known to exist

Occasionally

Occasionally

Advantages

Disadvantages/limitations

Practical for a wide range of frequencies; reduces computation time considerably; works for near-source region; can handle complex propagation media because crustal phases and surface waves evaluated with complete Green’s functions; can statistically adjust the frequency content of ground motion to that desired; see tables for the two methods comprising the hybrid approach

Combination of two sets of simulation results is not always easy; not evident how to obtain triaxial time-histories with correct correlation between components; not evident that velocity and displacement time-histories are realistic, especially in the time domain, due to the lack of causality of phase; see tables for the two methods comprising the hybrid approach

3 Earthquake Scenario

Before predicting the earthquake ground motions that could occur at a site it is necessary to define an earthquake scenario or scenarios, i.e. earthquake(s) that need(s) to be considered in the design (or risk assessment) process for the site. The methods proposed in the literature to define these scenarios (e.g. Dowrick 1977; Hays 1980; Reiter 1990; Anderson 1997a; Bazzurro and Cornell 1999; Bommer et al. 2000) are not discussed here. In this section the focus is on the level of detail required to define a scenario for different ground-motion prediction techniques, which have varying degrees of freedom. In general, physics-based (generally complex) methods require more parameters to be defined than empirical (generally simple) techniques. As the number of degrees of freedom increases sophisticated prediction techniques can model more specific earthquake scenarios, but it becomes difficult to constrain the input parameters. The various methods consider different aspects of the ground-motion generation process to be important and set (either explicitly or implicitly) different parameters to default values. However, even for methods where a characteristic can be varied it is often set to a standard value due to a lack of knowledge. In fact, when there is a lack of knowledge (epistemic uncertainty) the input parameters should be varied within a physically realistic range rather than fixed to default values. Care must be taken to make sure that parameters defining a scenario are internally consistent. For example, asperity size and asperity slip contrast of earthquake ruptures are generally inversely correlated (e.g. Bommer et al. 2004).

The basic parameters required to define a scenario for almost all methods are magnitude and source-to-site distance (note that, as stated in Section 1, hazard is generally initially computed for a rock site and hence site effects are not considered here). In addition, other gross source characteristics, such as the style-of-faulting mechanism, are increasingly being considered. An often implicit general input variable for simple techniques is ‘seismotectonic regime’, which is explicitly accounted for in more complex approaches through source and path modelling. In this article, we assume that kinematic source models (where the rupture process is a fixed input) are used for ground-motion simulations. Dynamic source modelling (where the rupture process is simulated by considering stress conditions) is a step up in complexity from kinematic models and it remains mainly a research topic that is very rarely used for generating time-histories for engineering design purposes. Dynamic rupture simulations have the advantage over kinematic source models in proposing various possible rupture scenarios of different magnitudes for a given seismotectonic situation (e.g. Anderson et al. 2003; Aochi et al. 2006). However, it is still difficult to tune the model parameters for practical engineering purposes (e.g. Aochi and Douglas 2006) (see Section 2.3 for a discussion of dynamic source models).

Many factors (often divided into source, path and site effects) have been observed to influence earthquake ground motions, e.g.: earthquake magnitude (or in some approaches epicentral macroseismic intensity), faulting mechanism, source depth, fault geometry, stress drop and direction of rupture (directivity); source-to-site distance, crustal structure, geology along wave paths, radiation pattern and directionality; and site geology, topography, soil–structure interaction and nonlinear soil behaviour. The combination of these different, often inter-related, effects leads to dispersion in ground motions. The varying detail of the scenarios (i.e. not accounting for some factors while modelling others) used for the different techniques consequently leads to dispersion in the predictions. The unmodelled effects, which can be important, are ignored and consequently predictions from some simple techniques (e.g. empirical ground-motion models) contain a bias due to the (unknown) distribution of records used to construct the model with respect to these variables (e.g. Douglas 2007). There is more explicit control in simulation-based procedures. Concerning empirical ground-motion models McGuire (2004) says that ‘only variables that are known and can be specified before an earthquake should be included in the predictive equation. Using what are actually random properties of an earthquake source (properties that might be known after an earthquake) in the ground motion estimation artificially reduces the apparent scatter, requires more complex analysis, and may introduce errors because of the added complexity.’

In empirical methods the associated parameters that cannot yet be estimated before the earthquake, e.g. stress drop and details of the fault rupture, are, since observed ground motions are used, by definition, within the range of possibilities. Varying numbers of these parameters need to be chosen when using simulation techniques, which can be difficult. On the other hand, only a limited and unknown subset of these parameters are sampled by empirical methods since not all possible earthquakes have been recorded. In addition, due to the limited number of strong-motion records from a given region possible regional dependence of these parameters cannot usually be accounted for by empirical procedures since records from a variety of areas are combined in order to obtain a sufficiently large dataset.

Various prediction methods account for possible regional dependence (e.g. Douglas 2007) in different ways. Methods based on observed ground motions implicitly hope that the strong-motion records capture the complete regional dependence and that the range of possible motions is not underestimated. However, due to limited databanks it is not often possible to only use records from small regions of interest; data from other areas usually need to be imported. Physics-based methods explicitly model regional dependence through the choice of input parameters, some of which, e.g. crustal structure, can be estimated from geological information or velocimetric (weak-motion) data, while others, e.g. stress parameters, can only be confidently estimated based on observed strong-motion data from the region. If not available for a specific region parameters must be imported from other regions or a range of possible values assumed.

Although this article does not discuss site effects nor their modelling, it is important that the choice of which technique to use for a task is made considering the potential use of the ground-motion predictions on rock for input to a site response analysis. For example, predictions from empirical methods are for rock sites whose characteristics (e.g. velocity and density profiles and near-surface attenuation) are limited by the observational database available and therefore the definition of rock cannot, usually, be explicitly defined by the user; however, approximate adjustments to unify predictions at different rock sites can be made (e.g. Cotton et al. 2006). In addition, the characteristics of the rock sites within observational databases are generally poorly known (e.g. Cotton et al. 2006) and therefore the rock associated with the prediction is ill-defined. In contrast, physics-based techniques generally allow the user to explicitly define the characteristics of the rock site and therefore more control is available. The numerical resolution of each method puts limits on the velocities and thicknesses of the sufficiently layers that can be treated. Black-box approaches generally neglect site effects; when they do not the parameters for controlling the type of site to use are, as in empirical techniques, constrained based on (limited) observational databases.

4 Testing of Methods

Predicted ground motions should be compared to observations for the considered site, in terms of amplitude, frequency content, duration, energy content and more difficult to characterise aspects, such as the ‘look’ of the time-histories. This verification of the predictions is required so that the ground-motion estimates can be used with confidence in engineering and risk analyses. Such comparisons take the form of either point comparisons for past earthquakes (e.g. Aochi and Madariaga 2003), visually checking a handful of predictions and observations in a non-systematic way, or more general routine validation exercises, where hundreds of predictions and observations are statistically compared to confirm that the predictions are not significantly biased and do not display too great a scatter (a perfect fit between predictions and observations is not expected, or generally possible, when making such general comparisons) (e.g. Atkinson and Somerville 1994; Silva et al. 1999; Douglas et al. 2004). In a general comparison it is also useful to check the correlation coefficients between various strong-motion parameters (e.g. PGA and relative significant duration, RSD) to verify that they match the correlations commonly observed (Aochi and Douglas 2006).

For those techniques that are based on matching a set of strong-motion intensity parameters, such as the elastic response spectral ordinates, it is important that the fit to non-matched parameters is used to verify that they are physically realistic, i.e. to check the internal consistency of the approach. For example, black-box techniques that generate time-histories to match a target elastic response spectrum can lead to time-histories with unrealistic displacement demand and energy content (Naeim and Lew 1995).

A potentially useful approach, although one that is rarely employed, is to use a construction set of data to calibrate a method and then an independent validation set of data to test the predictions. Using such a two-stage procedure will demonstrate that any free parameters tuned during the first step do not need further modifications for other situations. Such a demonstration is important when there is a trade-off between parameters whereby various choices can lead to similar predicted ground motions for a given scenario.

One problem faced by all validation analysis is access to all the required independent parameters, such as local site conditions, in order that the comparisons are fair. If a full set of independent variables is not available then assumptions need to be made, which can lead to uncertainty in the comparisons. For example, Boore (2001), when comparing observations from the Chi-Chi earthquake to shaking predicted by various empirical ground-motion models, had to make assumptions on site classes due to poor site information for Taiwanese stations. These assumptions led to a lack of precision in the level of over-prediction of the ground motions.

Until recently most comparisons between observations and predictions were visual or based on simple measures of goodness-of-fit, such as: the mean bias and the overall standard deviation sometimes computed using a maximum-likelihood approach (Spudich et al. 1999). Scherbaum et al. (2004) develop a statistical technique for ranking various empirical ground-motion models by their ability to predict a set of observed ground motions. Such a method could be modified for use with other types of predictions. However, the technique of Scherbaum et al. (2004) relies on estimates of the scatter in observed motions, which are difficult to assess for techniques based on ground-motion simulation, and the criteria used to rank the models would probably require modification if applied to other prediction techniques. Assessment of the uncertainty in simulations requires considering all sources of dispersion—modelling (differences between the actual physical process and the simulation), random (detailed aspects of the source and wave propagation that cannot be modelled deterministically at present) and parametric (uncertainty in source parameters for future earthquakes) (Abrahamson et al. 1990). The approach developed by Abrahamson et al. (1990) to split total uncertainty into these different components means that the relative importance of different source parameters can be assessed and hence aids in the physical interpretation of ground-motion uncertainty.

In addition to this consideration of different types of uncertainty, work has been undertaken to consider the ability of a simulation technique to provide adequate predictions not just for a single strong-motion intensity parameter but many. Anderson (2004) proposes a quantitative measure of the goodness-of-fit between synthetic and observed accelerograms using ten different criteria that measure various aspects of the motions, for numerous frequency bands. This approach could be optimised to require less computation by adopting a series of strong-motion parameters that are poorly correlated (orthogonal), and hence measure different aspects of ground motions, e.g. amplitude characterised by PGA and duration characterised by RSD. A goodness-of-fit approach based on the time-frequency representation of seismograms, as opposed to strong-motion intensity parameters as in the method of Anderson (2004), is proposed by Kristeková et al. (2006) to compare ground motions simulated using different computer codes and techniques. Since it has only recently been introduced this procedure has yet to become common but it has the promise to be a useful objective strategy for the validation of simulation techniques by comparing predicted and observed motions and also by internal comparisons between methods. Some comprehensive comparisons of the results from numerical simulations have been made in the framework of recent research projects and workshops (e.g. Day et al. 2005; Chaljub et al. 2007b).

If what is required from a method is a set of ground motions that include the possible variability in shaking at a site from a given event then it is important to use a method that introduces some randomness into the process (e.g. Pousse et al. 2006) to account for random and parametric uncertainties. For example, results from physically based simulation techniques will not reproduce the full range of possible motions unless a stochastic element is introduced into the prediction, through the source or path. However, if what is required from a technique is the ability to give the closest prediction to an observation then this stochastic element is not necessarily required.

5 Synthesis and Conclusions

Dowrick (1977) notes that ‘[a]s with other aspects of design the degree of detail entered into selecting dynamic input [i.e. ground-motion estimates] will depend on the size and vulnerability of the project’. This is commonly applied in practice where simple methods (GMPEs, representative accelerograms or black-box methods) are applied for lower importance and less complex projects whereas physics-based techniques are used for high importance and complex situations (although invariably in combination with simpler methods). Methods providing time-histories are necessary for studies requiring non-linear engineering analyses, which are becoming increasingly common. Dowrick (1977) believes that ‘because there are still so many imponderables in this topic only the simpler methods will be warranted in most cases’. However, due to the significant improvements in techniques, knowledge, experience and computing power this view from the 1970s is now less valid. Simple empirical ground-motion estimates have the advantage of being more defensible and are more easily accepted by decision makers due to their close connection to observations. Simulations are particularly important in regions with limited (or non-existent) observational databanks and also for site-specific studies, where the importance of different assumptions on the input parameters can be studied. However, reliable simulations require good knowledge of the propagation media and they are often computationally expensive.

One area where physics-based forward modelling breaks down is in the simulation of high-frequency ground motions where the lack of detail in source (e.g. heterogeneities of the rupture process) and path (e.g. scattering) models means high frequencies are poorly predicted. Hanks and McGuire (1981) state that ‘[e]vidently, a realistic characterization of high-frequency strong ground motion will require one or more stochastic parameters that can account for phase incoherence.’ In contrast, Aki (2003) believes that ‘[a]ll these new results suggest that we may not need to consider frequencies higher than about 10 Hz in Strong Motion Seismology. Thus, it may be a viable goal for strong motion seismologists to use entirely deterministic modeling, at least for path and site effects, before the end of the twenty-first century.’

The associated uncertainties within ground-motion prediction remain high despite many decades of research and increasingly sophisticated techniques. The unchanging level of aleatory uncertainties within empirical ground-motion estimation equations over the past thirty years are an obvious example of this (e.g. Douglas 2003). However, estimates from simulation methods are similarly affected by large (and often unknown) uncertainties. These large uncertainties oblige earthquake engineers to design structures with large factors of safety that may not be required.

The selection of the optimum method for ground-motion estimation depends on what data are available for assessing the earthquake scenario, resources available and experience of the group. Currently the choice of method used for a particular study is generally controlled by the experience and preferences of the worker and the tools and software available to them rather than it being necessarily selected based on what is most appropriate for the project.

There are still a number of questions concerning ground-motion prediction that need to be answered. These include the following—possible regional dependence of ground motions (e.g. Douglas 2007), the effect of rupture complexity on near-source ground motion (e.g. Aochi and Madariaga 2003), the spatial variability of shaking (e.g. Goda and Hong 2008) and the determination of upper bounds on ground motions (e.g. Strasser et al. 2008). All these questions are difficult to answer at present due to the lack of near-source strong-motion data from large earthquakes in many regions (little near-source data exists outside the western USA, Japan and Taiwan). Therefore, there is a requirement to install, keep operational and improve, e.g. in terms of spatial density (Trifunac 2007), strong-motion networks in various parts of the world. In addition, the co-location of accelerometers and high-sample-rate instruments using global navigation satellite systems (e.g. the Global Positioning System, GPS) could help improve the prediction of long-period ground motions (e.g. Wang et al. 2007).

In addition to the general questions mentioned above, more specific questions related to ground-motion prediction can be posed, such as: what is the most appropriate method to use for varying quality and quantity of input data and for different seismotectonic environments? how can the best use be made of the available data? how can the uncertainties associated with a given method be properly accounted for? how can the duration of shaking be correctly modelled? These types of questions are rarely explicitly investigated in articles addressing ground-motion prediction. In addition, more detailed quantitative comparisons of simulations from different methods for the same scenario should be conducted through benchmarks.

Over time the preferred techniques will tend to move to the top of Fig. 1 (more physically based approaches requiring greater numbers of input parameters) (e.g. Field et al. 2003) since knowledge of faults, travel paths and sites will become sufficient to constrain input parameters. Such predictions will be site-specific as opposed to the generic estimations commonly used at present. Due to the relatively high cost and difficulty of ground investigations, detailed knowledge of the ground subsurface is likely to continue to be insufficient for fully numerical simulations for high-frequency ground motions, which require data on 3D velocity variations at a scale of tens of metres. In the distant future when vast observational strong-motion databanks exist including records from many well-studied sites and earthquakes, more sophisticated versions of the simplest empirical technique, that of representative accelerograms, could be used where selections are made not just using a handful of scenario parameters but many, in order to select ground motions from scenarios close to that expected for a study area.

Footnotes

  1. 1.

    Some of the programs for ground-motion prediction are available for download from the ORFEUS Seismological Software Library http://www.orfeus-eu.org/Software/softwarelib.html

Notes

Acknowledgements

The design of the diagram in this article has benefited from advice contained in the book by Tufte (2006). Some of the work presented in this article was funded by the ANR project ‘Quantitative Seismic Hazard Assessment’ (QSHA). The rest was funded by internal BRGM research projects. We thank the rest of the BRGM Seismic Risks unit for numerous discussions on the topics discussed in this article. Finally, we thank two anonymous reviewers for their careful and detailed reviews, which led to significant improvements to this article.

References

  1. Abrahamson NA, Shedlock KM (1997) Overview. Seismological Research Letters 68(1):9–23Google Scholar
  2. Abrahamson NA, Somerville PG, Cornell CA (1990) Uncertainty in numerical strong motion predictions. In: Proceedings of the fourth U.S. national conference on earthquake engineering, vol 1, pp 407–416Google Scholar
  3. Abrahamson N, Atkinson G, Boore D, Bozorgnia Y, Campbell K, Chiou B, Idriss IM, Silva W, Youngs R (2008) Comparisons of the NGA ground-motion relations. Earthq Spectra 24(1):45–66. doi:10.1193/1.2924363 CrossRefGoogle Scholar
  4. Aki K (1982) Strong motion prediction using mathematical modeling techniques. Bulletin of the Seismological Society of America 72(6):S29–S41Google Scholar
  5. Aki K (2003) A perspective on the history of strong motion seismology. Physics of the Earth and Planetary Interiors 137:5–11CrossRefGoogle Scholar
  6. Aki K, Larner KL (1970) Surface motion of a layered medium having an irregular interface due to incident plane SH waves. Journal of Geophysical Research 75(5):933–954CrossRefGoogle Scholar
  7. Aki K, Richards PG (2002) Quantitative Seismology. University Science Books, Sausalito, California, USAGoogle Scholar
  8. Akkar S, Bommer JJ (2006) Influence of long-period filter cut-off on elastic spectral displacements. Earthquake Engineering and Structural Dynamics 35(9):1145–1165CrossRefGoogle Scholar
  9. Ambraseys NN (1974) The correlation of intensity with ground motion. In: Advancements in Engineering Seismology in Europe, TriesteGoogle Scholar
  10. Ambraseys NN, Douglas J, Sigbjörnsson R, Berge-Thierry C, Suhadolc P, Costa G, Smit PM (2004a) Dissemination of European strong-motion data, vol 2. In: Proceedings of thirteenth world conference on earthquake engineering, paper no. 32Google Scholar
  11. Ambraseys NN, Smit P, Douglas J, Margaris B, Sigbjörnsson R, Ólafsson S, Suhadolc P, Costa G (2004b) Internet site for European strong-motion data. Bollettino di Geofisica Teorica ed Applicata 45(3):113–129Google Scholar
  12. Anderson JG (1991) Strong motion seismology. Rev Geophys 29(Part 2):700–720Google Scholar
  13. Anderson JG (1997a) Benefits of scenario ground motion maps. Engineering Geology 48(1–2):43–57CrossRefGoogle Scholar
  14. Anderson JG (1997b) Nonparametric description of peak acceleration above a subduction thrust. Seismological Research Letters 68(1):86–93Google Scholar
  15. Anderson JG (2004) Quantitative measure of the goodness-of-fit of synthetic seismograms. In: Proceedings of thirteenth world conference on earthquake engineering, paper no. 243Google Scholar
  16. Anderson G, Aagaard BT, Hudnut K (2003) Fault interactions and large complex earthquakes in the Los Angeles area. Science 302(5652):1946–1949, doi:10.1126/science.1090747 CrossRefGoogle Scholar
  17. Aochi H, Douglas J (2006) Testing the validity of simulated strong ground motion from the dynamic rupture of a finite fault, by using empirical equations. Bulletin of Earthquake Engineering 4(3):211–229, doi:10.1007/s10518-006-0001-3 CrossRefGoogle Scholar
  18. Aochi H, Fukuyama E (2002) Three-dimensional nonplanar simulation of the 1992 Landers earthquake. J Geophys Res 107(B2). doi:10.1029/2000JB000061
  19. Aochi H, Madariaga R (2003) The 1999 Izmit, Turkey, earthquake: Nonplanar fault structure, dynamic rupture process, and strong ground motion. Bulletin of the Seismological Society of America 93(3):1249–1266CrossRefGoogle Scholar
  20. Aochi H, Cushing M, Scotti O, Berge-Thierry C (2006) Estimating rupture scenario likelihood based on dynamic rupture simulations: The example of the segmented Middle Durance fault, southeastern France. Geophysical Journal International 165(2):436–446, doi:10.1111/j.1365-246X.2006.02842.x CrossRefGoogle Scholar
  21. Aoi S, Fujiwara H (1999) 3D finite-difference method using discontinuous grids. Bulletin of the Seismological Society of America 89(4):918–930Google Scholar
  22. Apsel RJ, Luco JE (1983) On the Green’s functions for a layered half-space. Part II. Bulletin of the Seismological Society of America 73(4):931–951Google Scholar
  23. Archuleta RJ, Brune JN (1975) Surface strong motion associated with a stick-slip event in a foam rubber model of earthquakes. Bulletin of the Seismological Society of America 65(5):1059–1071Google Scholar
  24. Atkinson GM (2001) An alternative to stochastic ground-motion relations for use in seismic hazard analysis in eastern North America. Seismological Research Letters 72:299–306Google Scholar
  25. Atkinson GM, Boore DM (2006) Earthquake ground-motion prediction equations for eastern North America. Bulletin of the Seismological Society of America 96(6):2181–2205. doi:10.1785/0120050245 CrossRefGoogle Scholar
  26. Atkinson GM, Silva W (2000) Stochastic modeling of California ground motion. Bulletin of the Seismological Society of America 90(2):255–274CrossRefGoogle Scholar
  27. Atkinson GM, Somerville PG (1994) Calibration of time history simulation methods. Bulletin of the Seismological Society of America 84(2):400–414Google Scholar
  28. Atkinson GM, Sonley E (2000) Empirical relationships between modified Mercalli intensity and response spectra. Bulletin of the Seismological Society of America 90(2):537–544CrossRefGoogle Scholar
  29. Baker JW, Cornell CA (2006) Spectral shape, epsilon and record selection. Earthquake Engineering and Structural Dynamics 35(9):1077–1095, doi:10.1002/eqe.571 CrossRefGoogle Scholar
  30. Bao HS, Bielak J, Ghattas O, Kallivokas LF, O’Hallaron DR, Shewchuk JR, Xu JF (1998) Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers. Computer Methods in Applied Mechanics and Engineering 152(1–2):85–102CrossRefGoogle Scholar
  31. Bazzurro P, Cornell CA (1999) Disaggregation of seismic hazard. Bulletin of the Seismological Society of America 89(2):501–520Google Scholar
  32. Beresnev IA, Atkinson GM (1998) FINSIM: A FORTRAN program for simulating stochastic acceleration time histories from finite faults. Seismological Research Letters 69:27–32Google Scholar
  33. Berge C, Gariel JC, Bernard P (1998) A very broad-band stochastic source model used for near source strong motion prediction. Geophysical Research Letters 25(7):1063–1066CrossRefGoogle Scholar
  34. Beyer K, Bommer JJ (2007) Selection and scaling of real accelerograms for bi-directional loading: A review of current practice and code provisions. Journal of Earthquake Engineering 11(S1):13–45, doi:10.1080/13632460701280013 Google Scholar
  35. Bommer JJ, Acevedo AB (2004) The use of real earthquake accelerograms as input to dynamic analysis. Journal of Earthquake Engineering 8(Special issue 1):43–91CrossRefGoogle Scholar
  36. Bommer JJ, Alarcón JE (2006) The prediction and use of peak ground velocity. Journal of Earthquake Engineering 10(1):1–31CrossRefGoogle Scholar
  37. Bommer JJ, Ruggeri C (2002) The specification of acceleration time-histories in seismic design codes. European Earthquake Engineering 16(1):3–17Google Scholar
  38. Bommer JJ, Scott SG, Sarma SK (2000) Hazard-consistent earthquake scenarios. Soil Dynamics and Earthquake Engineering 19(4):219–231CrossRefGoogle Scholar
  39. Bommer JJ, Abrahamson NA, Strasser FO, Pecker A, Bard PY, Bungum H, Cotton F, Fäh D, Sabetta F, Scherbaum F, Studer J (2004) The challenge of defining upper bounds on earthquake ground motions. Seismological Research Letters 75(1):82–95Google Scholar
  40. Boore DM (1973) The effect of simple topography on seismic waves: Implications for the accelerations recorded at Pacoima Dam, San Fernando valley, California. Bulletin of the Seismological Society of America 63(5):1603–1609Google Scholar
  41. Boore DM (1983) Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra. Bulletin of the Seismological Society of America 73(6):1865–1894Google Scholar
  42. Boore DM (2001) Comparisons of ground motions from the 1999 Chi-Chi earthquake with empirical predictions largely based on data from California. Bulletin of the Seismological Society of America 91(5):1212–1217CrossRefGoogle Scholar
  43. Boore DM (2003) Simulation of ground motion using the stochastic method. Pure and Applied Geophysics 160(3–4):635–676, doi:10.1007/PL00012553 CrossRefGoogle Scholar
  44. Boore DM (2005) SMSIM—Fortran programs for simulating ground motions from earthquakes: Version 2.3—A revision of OFR 96-80-A. Open-File Report 00-509, United States Geological Survey, modified version, describing the program as of 15 August 2005 (Version 2.30)Google Scholar
  45. Bouchon M (1981) A simple method to calculate Green’s functions for elastic layered media. Bulletin of the Seismological Society of America 71(4):959–971Google Scholar
  46. Bouchon M, Sánchez-Sesma FJ (2007) Boundary integral equations and boundary elements methods in elastodynamics. In: Advances in Geophysics: advances in wave propagation in heterogeneous Earth, vol 48, Chap 3. Academic Press, London, UK, pp 157–189Google Scholar
  47. Brune JN (1970) Tectonic stress and the spectra of seismic shear waves from earthquakes. Journal of Geophysical Research 75(26):4997–5009CrossRefGoogle Scholar
  48. Brune JN (1971) Correction. Journal of Geophysical Research 76(20):5002CrossRefGoogle Scholar
  49. Bycroft GN (1960) White noise representation of earthquake. Journal of The Engineering Mechanics Division, ASCE 86(EM2):1–16Google Scholar
  50. Campbell KW (1986) An empirical estimate of near-source ground motion for a major, m b = 6.8, earthquake in the eastern United States. Bulletin of the Seismological Society of America 76(1):1–17Google Scholar
  51. Campbell KW (2002) A contemporary guide to strong-motion attenuation relations. In: Lee WHK, Kanamori H, Jennings PC, Kisslinger C (eds) International handbook of earthquake and engineering seismology, Chap 60. Academic Press, LondonGoogle Scholar
  52. Campbell KW (2003) Prediction of strong ground motion using the hybrid empirical method and its use in the development of ground-motion (attenuation) relations in eastern North America. Bulletin of the Seismological Society of America 93(3):1012–1033CrossRefGoogle Scholar
  53. Campbell KW (2007) Validation and update of hybrid empirical ground motion (attenuation) relations for the CEUS. Tech. rep., ABS Consulting, Inc. (EQECAT), Beaverton, USA, Award number: 05HQGR0032Google Scholar
  54. Cancani A (1904) Sur l’emploi d’une double échelle sismique des intensités, empirique et absolue. Gerlands Beitr z Geophys 2:281–283, not seen. Cited in Gutenberg and Richter (1942)Google Scholar
  55. Chaljub E, Komatitsch D, Vilotte JP, Capdeville Y, Valette B, Festa G (2007a) Spectral-element analysis in seismology. In: Advances in geophysics: advances in wave propagation in heterogeneous earth, vol 48, chap 7. Academic Press, London, UK, pp 365–419Google Scholar
  56. Chaljub E, Tsuno S, Bard PY, Cornou C (2007b) Analyse des résultats d’un benchmark numérique de prédiction du mouvement sismique dans la vallée de Grenoble. In: 7ème Colloque National AFPS 2007, in FrenchGoogle Scholar
  57. Chen XF (2007) Generation and propagation of seismic SH waves in multi-layered media with irregular interfaces. In: Advances in geophysics: advances in wave propagation in heterogeneous earth, vol 48, chap 4. Academic Press, London, UK, pp 191–264Google Scholar
  58. Chen M, Hjörleifsdóttir V, Kientz S, Komatitsch D, Liu Q, Maggi A, Savage B, Strand L, Tape C, Tromp J (2008) SPECFEM 3D: User manual version 1.4.3. Tech. rep., Computational Infrastructure for Geodynamics (CIG), California Institute of Technology (USA); University of Pau (France). URL: http://www.gps.caltech.edu/jtromp/research/downloads.html
  59. Cotton F, Scherbaum F, Bommer JJ, Bungum H (2006) Criteria for selecting and adjusting ground-motion models for specific target regions: Application to central Europe and rock sites. Journal of Seismology 10(2):137–156, DOI: 10.1007/s10950-005-9006-7 CrossRefGoogle Scholar
  60. Dalguer LA, Irikura K, Riera JD (2003) Simulation of tensile crack generation by three-dimensional dynamic shear rupture propagation during an earthquake. J Geophys Res 108(B3), article 2144Google Scholar
  61. Dan K, Watanabe T, Tanaka T, Sato R (1990) Stability of earthquake ground motion synthesized by using different small-event records as empirical Green’s functions. Bulletin of the Seismological Society of America 80(6):1433–1455Google Scholar
  62. Day SM, Bradley CR (2001) Memory-efficient simulation of anelastic wave propagation. Bulletin of the Seismological Society of America 91(3):520–531CrossRefGoogle Scholar
  63. Day SM, Bielak J, Dreger D, Graves R, Larsen S, Olsen KB, Pitarka A (2005) Tests of 3D elastodynamic codes. Final report for Lifelines Project 1A03. Pacific Earthquake Engineering Research Center, University of California, Berkeley, USAGoogle Scholar
  64. Dormy E, Tarantola A (1995) Numerical simulation of elastic wave propagation using a finite volume method. Journal of Geophysical Research 100(B2):2123–2133CrossRefGoogle Scholar
  65. Douglas J (2003) Earthquake ground motion estimation using strong-motion records: A review of equations for the estimation of peak ground acceleration and response spectral ordinates. Earth-Science Reviews 61(1–2):43–104CrossRefGoogle Scholar
  66. Douglas J (2007) On the regional dependence of earthquake response spectra. ISET Journal of Earthquake Technology 44(1):71–99Google Scholar
  67. Douglas J, Suhadolc P, Costa G (2004) On the incorporation of the effect of crustal structure into empirical strong ground motion estimation. Bulletin of Earthquake Engineering 2(1):75–99CrossRefGoogle Scholar
  68. Douglas J, Bungum H, Scherbaum F (2006) Ground-motion prediction equations for southern Spain and southern Norway obtained using the composite model perspective. Journal of Earthquake Engineering 10(1):33–72CrossRefGoogle Scholar
  69. Dowrick DJ (1977) Earthquake resistant design—a manual for engineers and architects. Wiley, LondonGoogle Scholar
  70. Erdik M, Durukal E (2003) Simulation modeling of strong ground motion. In: Earthquake engineering handbook, chap 6, CRC Press LLC, Boca Raton, FL, USAGoogle Scholar
  71. Esteva L, Rosenblueth E (1964) Espectros de temblores a distancias moderadas y grandes. Boletin Sociedad Mexicana de Ingenieria Sesmica 2:1–18, in SpanishGoogle Scholar
  72. Faccioli E, Maggio F, Paolucci R, Quarteroni A (1997) 2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method. Journal of Seismology 1(3):237–251CrossRefGoogle Scholar
  73. Field EH, Jordan TH, Cornell CA (2003) OpenSHA: A developing community-modeling environment for seismic hazard analysis. Seismological Research Letters 74(4):406–419Google Scholar
  74. Florsch N, Fäh D, Suhadolc P, Panza GF (1991) Complete synthetic seismograms for high-frequency multimode SH-waves. Pure and Applied Geophysics 136:529–560CrossRefGoogle Scholar
  75. Frankel A (1995) Simulating strong motions of large earthquakes using recordings of small earthquakes: The Loma Prieta mainshock as a test case. Bulletin of the Seismological Society of America 85(4):1144–1160Google Scholar
  76. Frankel A, Clayton RW (1986) Finite-difference simulations of seismic scattering—Implications for the propagation of short-period seismic-waves in the crust and models of crustal heterogeneity. Journal of Geophysical Research 91(B6):6465–6489CrossRefGoogle Scholar
  77. Gallovič F, Brokešová J (2007) Hybrid k-squared source model for strong ground motion simulations: Introduction. Physics of the Earth and Planetary Interiors 160(1):34–50, DOI: 10.1016/j.pepi.2006.09.002 CrossRefGoogle Scholar
  78. Goda K, Hong HP (2008) Spatial correlation of peak ground motions and response spectra. Bulletin of the Seismological Society of America 98(1):354–365, doi:10.1785/0120070078 CrossRefGoogle Scholar
  79. Graves RWJ (1996) Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences. Bulletin of the Seismological Society of America 86(4):1091–1106Google Scholar
  80. Guatteri M, Mai PM, Beroza GC, Boatwright J (2003) Strong ground motion prediction from stochastic-dynamic source models. Bulletin of the Seismological Society of America 93(1):301–313, DOI: 10.1785/0120020006 CrossRefGoogle Scholar
  81. Guatteri M, Mai PM, Beroza GC (2004) A pseudo-dynamic approximation to dynamic rupture models for strong ground motion prediction. Bulletin of the Seismological Society of America 94(6):2051–2063, DOI: 10.1785/0120040037 CrossRefGoogle Scholar
  82. Gusev AA (1983) Descriptive statistical model of earthquake source radiation and its application to an estimation of short-period strong motion. Geophysical Journal of the Royal Astronomical Society 74:787–808Google Scholar
  83. Gutenberg G, Richter CF (1942) Earthquake magnitude, intensity, energy, and acceleration. Bulletin of the Seismological Society of America 32(3):163–191Google Scholar
  84. Guzman RA, Jennings PC (1976) Design spectra for nuclear power plants. Journal of The Power Division, ASCE 102(2):165–178Google Scholar
  85. Hadley DM, Helmberger DV (1980) Simulation of strong ground motions. Bulletin of the Seismological Society of America 70(2):617–630Google Scholar
  86. Hancock J, Watson-Lamprey J, Abrahamson NA, Bommer JJ, Markatis A, McCoy E, Mendis R (2006) An improved method of matching response spectra of recorded earthquake ground motion using wavelets. Journal of Earthquake Engineering 10(Special issue 1):67–89CrossRefGoogle Scholar
  87. Hancock J, Bommer JJ, Stafford PJ (2008) Numbers of scaled and matched accelerograms required for inelastic dynamic analyses. Earthquake Engineering and Structural Dynamics DOI: 10.1002/eqe.827, in press
  88. Hanks TC (1979) b values and ω−γ seismic source models: Implications for tectonic stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion. Journal of Geophysical Research 84(B5):2235–2242CrossRefGoogle Scholar
  89. Hanks TC, McGuire RK (1981) The character of high-frequency strong ground motion. Bulletin of the Seismological Society of America 71(6):2071–2095Google Scholar
  90. Hartzell SH (1978) Earthquake aftershocks as Green’s functions. Geophysical Research Letters 5(1):1–4CrossRefGoogle Scholar
  91. Hartzell S, Leeds A, Frankel A, Williams RA, Odum J, Stephenson W, Silva S (2002) Simulation of broadband ground motion including nonlinear soil effects for a magnitude 6.5 earthquake on the Seattle fault, Seattle, Washington. Bulletin of the Seismological Society of America 92(2):831–853CrossRefGoogle Scholar
  92. Haskell NA (1969) Elastic displacements in the near-field of a propagating fault. Bulletin of the Seismological Society of America 59(2):865–908Google Scholar
  93. Hays WW (1980) Procedures for estimating earthquake ground motions. Geological Survey Professional Paper 1114, US Geological SurveyGoogle Scholar
  94. Heaton TH, Helmberger DV (1977) A study of the strong ground motion of the Borrego Mountain, California, earthquake. Bulletin of the Seismological Society of America 67(2):315–330Google Scholar
  95. Herrero A, Bernard P (1994) A kinematic self-similar rupture process for earthquakes. Bulletin of the Seismological Society of America 84(4):1216–1228Google Scholar
  96. Hershberger J (1956) A comparison of earthquake accelerations with intensity ratings. Bulletin of the Seismological Society of America 46(4):317–320Google Scholar
  97. Heuze F, Archuleta R, Bonilla F, Day S, Doroudian M, Elgamal A, Gonzales S, Hoehler M, Lai T, Lavallee D, Lawrence B, Liu PC, Martin A, Matesic L, Minster B, Mellors R, Oglesby D, Park S, Riemer M, Steidl J, Vernon F, Vucetic M, Wagoner J, Yang Z (2004) Estimating site-specific strong earthquake motions. Soil Dynamics and Earthquake Engineering 24(3):199–223, DOI: 10.1016/j.soildyn.2003.11.002 CrossRefGoogle Scholar
  98. Hisada Y (2008) Broadband strong motion simulation in layered half-space using stochastic Green’s function technique. Journal of Seismology 12(2):265–279, DOI:10.1007/s10950-008-9090-6 CrossRefGoogle Scholar
  99. Housner GW (1947) Characteristics of strong-motion earthquakes. Bulletin of the Seismological Society of America 37(1):19–31Google Scholar
  100. Housner GW (1955) Properties of strong-ground motion earthquakes. Bulletin of the Seismological Society of America 45(3):197–218Google Scholar
  101. Housner GW, Jennings PC (1964) Generation of artificial earthquakes. Journal of The Engineering Mechanics Division, ASCE 90:113–150Google Scholar
  102. Irikura K, Kamae K (1994) Estimation of strong ground motion in broad-frequency band based on a seismic source scaling model and an empirical Green’s function technique. Annali di Geofisica XXXVII(6):1721–1743Google Scholar
  103. Jennings PC, Housner GW, Tsai NC (1968) Simulated earthquake motions. Tech. rep., Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, California, USAGoogle Scholar
  104. Joyner WB (1984) A scaling law for the spectra of large earthquakes. Bulletin of the Seismological Society of America 74(4):1167–1188Google Scholar
  105. Joyner WB, Boore DM (1986) On simulating large earthquake by Green’s function addition of smaller earthquakes. In: Das S, Boatwright J, Scholtz CH (eds) Earthquake source mechanics, Maurice Ewing Series 6, vol 37. American Geophysical Union, Washington, D.C., USAGoogle Scholar
  106. Joyner WB, Boore DM (1988) Measurement, characterization, and prediction of strong ground motion. In: Proceedings of earthquake engineering & soil dynamics II, Geotechnical division, ASCE, pp 43–102Google Scholar
  107. Jurkevics A, Ulrych TJ (1978) Representing and simulating strong ground motion. Bulletin of the Seismological Society of America 68(3):781–801Google Scholar
  108. Kaka SI, Atkinson GM (2004) Relationships between instrumental ground-motion parameters and modified Mercalli intensity in eastern North America. Bulletin of the Seismological Society of America 94(5):1728–1736CrossRefGoogle Scholar
  109. Kamae K, Irikura K, Pitarka A (1998) A technique for simulating strong ground motion using hybrid Green’s functions. Bulletin of the Seismological Society of America 88(2):357–367Google Scholar
  110. Kanamori H (1979) A semi-empirical approach to prediction of long-period ground motions from great earthquakes. Bulletin of the Seismological Society of America 69(6):1645–1670Google Scholar
  111. Käser M, Iske A (2005) ADER schemes on adaptive triangular meshes for scalar conservations laws. Journal of Computational Physics 205(2):486–508CrossRefGoogle Scholar
  112. Kaul MK (1978) Spectrum-consistent time-history generation. Journal of The Engineering Mechanics Division, ASCE 104(ME4):781–788Google Scholar
  113. Kennett BLN, Kerry NJ (1979) Seismic waves in a stratified half-space. Geophysical Journal of the Royal Astronomical Society 57:557–583Google Scholar
  114. Kohrs-Sansorny C, Courboulex F, Bour M, Deschamps A (2005) A two-stage method for ground-motion simulation using stochastic summation of small earthquakes. Bulletin of the Seismological Society of America 95(4):1387–1400, DOI: 10.1785/0120040211 CrossRefGoogle Scholar
  115. Koketsu K (1985) The extended reflectivity method for synthetic near-field seismograms. Journal of the Physics of the Earth 33:121–131Google Scholar
  116. Komatitsch D, Martin R (2007) An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation. Geophysics 72(5):SM155–SM167CrossRefGoogle Scholar
  117. Komatitsch D, Tromp J (1999) Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophysical Journal International 139(3):806–822CrossRefGoogle Scholar
  118. Komatitsch D, Vilotte JP (1998) The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures. Bulletin of the Seismological Society of America 88(2):368–392Google Scholar
  119. Komatitsch D, Liu Q, Tromp J, Süss P, Stidham C, Shaw JH (2004) Simulations of ground motion in the Los Angeles basin based upon the spectral-element method. Bulletin of the Seismological Society of America 94(1):187–206CrossRefGoogle Scholar
  120. Kristeková M, Kristek J, Moczo P, Day SM (2006) Misfit criteria for quantitative comparison of seismograms. Bulletin of the Seismological Society of America 96(5):1836–1850, DOI: 10.1785/0120060012 CrossRefGoogle Scholar
  121. Lee VW, Trifunac MD (1985) Torsional accelerograms. Soil Dynamics and Earthquake Engineering 4(3):132–139CrossRefGoogle Scholar
  122. Lee VW, Trifunac MD (1987) Rocking strong earthquake accelerations. Soil Dynamics and Earthquake Engineering 6(2):75–89CrossRefGoogle Scholar
  123. Lee Y, Anderson JG, Zeng Y (2000) Evaluation of empirical ground-motion relations in southern California. Bulletin of the Seismological Society of America 90(6B):S136–S148CrossRefGoogle Scholar
  124. Levander AR (1988) Fourth-order finite-difference P-SV seismograms. Geophysics 53(11):1425–1436CrossRefGoogle Scholar
  125. LeVeque RJ (2002) Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UKGoogle Scholar
  126. Luco JE, Apsel RJ (1983) On the Green’s functions for a layered half-space. Part I. Bulletin of the Seismological Society of America 73(4):909–929Google Scholar
  127. Lysmer J, Drake LA (1972) A finite element method for seismology. In: Bolt BA (eds) Methods in Computational Physics. Academic Press Inc., New York, USAGoogle Scholar
  128. Ma S, Archuleta RJ, Page MT (2007) Effects of large-scale surface topography on ground motions as demonstrated by a study of the San Gabriel Mountains, Los Angeles, California. Bulletin of the Seismological Society of America 97(6):2066–2079, DOI: 10.1785/0120070040 CrossRefGoogle Scholar
  129. Mai PM, Beroza GC (2003) A hybrid method for calculating near-source, broadband seismograms: Application to strong motion prediction. Physics of the Earth and Planetary Interiors 137(1–4):183–199, DOI: 10.1016/S0031-9201(03)00014-1 Google Scholar
  130. Maupin V (2007) Introduction to mode coupling methods for surface waves. In: Advances in geophysics: advances in wave propagation in heterogeneous earth, vol 48, chap 2. Academic Press, London, UK, pp 127–155Google Scholar
  131. McGuire RK (2004) Seismic Hazard and Risk Analysis. Earthquake Engineering Research Institute (EERI), Oakland, California, USAGoogle Scholar
  132. Miyake H, Iwata T, Irikura K (2003) Source characterization for broadband ground-motion simulation: Kinematic heterogeneous source model and strong motion generation area. Bulletin of the Seismological Society of America 93(6):2531–2545, DOI: 10.1785/0120020183 CrossRefGoogle Scholar
  133. Moczo P, Kristek J, Galis M, Pazak P, Balazovjech M (2007a) The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion. Acta Physica Slovaca 57(2):177–406Google Scholar
  134. Moczo P, Robertsson JOA, Eisner L (2007b) The finite-difference time-domain method for modeling of seismic wave propagation. In: Advances in geophysics: advances in wave propagation in heterogeneous Earth, vol 48, chap 8. Academic Press, London, UK, pp 421–516Google Scholar
  135. Montaldo V, Kiremidjian AS, Thráinsson H, Zonno G (2003) Simulation of the Fourier phase spectrum for the generation of synthetic accelerograms. Journal of Earthquake Engineering 7(3):427–445CrossRefGoogle Scholar
  136. Mora P, Place D (1994) Simulation of the frictional stick-slip instability. Pure and Applied Geophysics 143(1–3):61–87CrossRefGoogle Scholar
  137. Motazedian D, Atkinson GM (2005) Stochastic finite-fault modeling based on a dynamic corner frequency. Bulletin of the Seismological Society of America 95(3):995–1010, DOI: 10.1785/0120030207 CrossRefGoogle Scholar
  138. Mukherjee S, Gupta VK (2002) Wavelet-based generation of spectrum-compatible time-histories. Soil Dynamics and Earthquake Engineering 22(9–12):799–804CrossRefGoogle Scholar
  139. Murphy JR, O’Brien LJ (1977) The correlation of peak ground acceleration amplitude with seismic intensity and other physical parameters. Bulletin of the Seismological Society of America 67(3):877–915Google Scholar
  140. Naeim F, Lew M (1995) On the use of design spectrum compatible time histories. Earthquake Spectra 11(1):111–127CrossRefGoogle Scholar
  141. Nau RF, Oliver RM, Pister KS (1982) Simulating and analyzing artificial nonstationary earthquake ground motions. Bulletin of the Seismological Society of America 72(2):615–636Google Scholar
  142. Ólafsson S, Sigbjörnsson R (1995) Application of ARMA models to estimate earthquake ground motion and structural response. Earthquake Engineering and Structural Dynamics 24(7):951–966CrossRefGoogle Scholar
  143. Ólafsson S, Remseth S, Sigbjörnsson R (2001) Stochastic models for simulation of strong ground motion in Iceland. Earthquake Engineering and Structural Dynamics 30(9):1305–1331CrossRefGoogle Scholar
  144. Olsen K, Madariaga R, Archuleta RJ (1997) Three-dimensional dynamic simulation of the 1992 Landers earthquake. Science 278:834–838CrossRefGoogle Scholar
  145. Olsen KB, Day SM, Minster JB, Cui Y, Chourasia A, Faerman M, Moore R, Maechling P, Jordan T (2006) Strong shaking in Los Angeles expected from southern San Andreas earthquake. Geophys Res Lett 33(L07305). doi:10.1029/2005GL025472
  146. Oprsal I, Zahradnik J (2002) Three-dimensional finite difference method and hybrid modeling of earthquake ground motion. J Geophys Res 107(B8). doi:10.1029/2000JB000082
  147. Ordaz M, Arboleda J, Singh SK (1995) A scheme of random summation of an empirical Green’s function to estimate ground motions from future large earthquakes. Bulletin of the Seismological Society of America 85(6):1635–1647Google Scholar
  148. Panza GF (1985) Synthetic seismograms: The Rayleigh waves modal summation. Journal of Geophysics 58:125–145Google Scholar
  149. Panza GF, Suhadolc P (1987) Complete strong motion synthetics. In: Bolt BA (eds) Seismic strong motion synthetics. Academic Press, Orlando, pp. 153–204Google Scholar
  150. Papageorgiou AS, Aki K (1983) A specific barrier model for the quantitative description of inhomogeneous faulting and the prediction of strong ground motion. Part I. Description of the model. Bulletin of the Seismological Society of America 73(3):693–702Google Scholar
  151. Pavic R, Koller MG, Bard PY, Lacave-Lachet C (2000) Ground motion prediction with the empirical Green’s function technique: an assessment of uncertainties and confidence level. Journal of Seismology 4(1):59–77CrossRefGoogle Scholar
  152. Pitarka A, Irikura K, Iwata T, Sekiguchi H (1998) Three-dimensional simulation of the near-fault ground motion for the 1995 Hyogo-ken Nanbu (Kobe), Japan, earthquake. Bulletin of the Seismological Society of America 88(2):428–440Google Scholar
  153. Pitarka A, Somerville P, Fukushima Y, Uetake T, Irikura K (2000) Simulation of near-fault strong-ground motion using hybrid Green’s functions. Bulletin of the Seismological Society of America 90(3):566–586CrossRefGoogle Scholar
  154. Place D, Mora P (1999) The lattice solid model to simulate the physics of rocks and earthquakes: Incorporation of friction. Journal of Computational Physics 150(2):332–372CrossRefGoogle Scholar
  155. Pousse G, Bonilla LF, Cotton F, Margerin L (2006) Non stationary stochastic simulation of strong ground motion time histories including natural variability: Application to the K-net Japanese database. Bulletin of the Seismological Society of America 96(6):2103–2117, DOI: 10.1785/0120050134 CrossRefGoogle Scholar
  156. Power M, Chiou B, Abrahamson N, Bozorgnia Y, Shantz T, Roblee C (2008) An overview of the NGA project. Earthquake Spectra 24(1):3–21, DOI: 10.1193/1.2894833 CrossRefGoogle Scholar
  157. Reiter L (1990) Earthquake Hazard Analysis: Issues and Insights. Columbia University Press, New YorkGoogle Scholar
  158. Ripperger J, Mai PM, Ampuero JP (2008) Variability of near-field ground motion from dynamic earthquake rupture simulations. Bulletin of the Seismological Society of America 98(3):1207–1228, DOI: 10.1785/0120070076 CrossRefGoogle Scholar
  159. Ruiz J, Baumont D, Bernard P, , Berge-Thierry C (2007) New approach in the kinematic k 2 source model for generating physical slip velocity functions. Geophysical Journal International 171(2):739–754, DOI: 10.1111/j.1365-246X.2007.03503.x CrossRefGoogle Scholar
  160. Sabetta F, Pugliese A (1996) Estimation of response spectra and simulation of nonstationary earthquake ground motions. Bulletin of the Seismological Society of America 86(2):337–352Google Scholar
  161. Scherbaum F, Cotton F, Smit P (2004) On the use of response spectral-reference data for the selection and ranking of ground-motion models for seismic-hazard analysis in regions of moderate seismicity: The case of rock motion. Bulletin of the Seismological Society of America 94(6):2164–2185, DOI: 10.1785/0120030147 CrossRefGoogle Scholar
  162. Scherbaum F, Cotton F, Staedtke H (2006) The estimation of minimum-misfit stochastic models from empirical ground-motion prediction equations. Bulletin of the Seismological Society of America 96(2):427–445, DOI: 10.1785/0120050015 CrossRefGoogle Scholar
  163. Shi B, Brune JN (2005) Characteristics of near-fault ground motions by dynamic thrust faulting: Two-dimensional lattice particle approaches. Bulletin of the Seismological Society of America 95(6):2525–2533, DOI: 10.1785/0120040227 CrossRefGoogle Scholar
  164. Shinozuka M (1988) Engineering modeling of ground motion. In: Proceedings of ninth world conference on earthquake engineering, vol VIII, pp 51–62Google Scholar
  165. Shome N, Cornell CA, Bazzurro P, Carballo JE (1998) Earthquakes, records and nonlinear responses. Earthquake Spectra 14(3):469–500CrossRefGoogle Scholar
  166. Silva WJ, Lee K (1987) State-of-the-art for assessing earthquake hazards in the United States; report 24: WES RASCAL code for synthesizing earthquake ground motions. Miscellaneous Paper S-73-1, US Army Corps of EngineersGoogle Scholar
  167. Silva W, Gregor N, Darragh B (1999) Near fault ground motions. Tech. rep., Pacific Engineering and Analysis, El Cerrito, USA, PG & E PEER—Task 5.AGoogle Scholar
  168. Sokolov V, Wald DJ (2002) Instrumental intensity distribution for the Hector Mine, California, and the Chi-Chi, Taiwan, earthquakes: Comparison of two methods. Bulletin of the Seismological Society of America 92(6):2145–2162CrossRefGoogle Scholar
  169. Souriau A (2006) Quantifying felt events: A joint analysis of intensities, accelerations and dominant frequencies. Journal of Seismology 10(1):23–38, DOI: 10.1007/s10950-006-2843-1 CrossRefGoogle Scholar
  170. Spudich P, Xu L (2003) Software for calculating earthquake ground motions from finite faults in vertically varying media. In: IASPEI handbook of earthquake and engineering seismology, chap 85.14. Academic Press, Amsterdam, The Netherlands, pp 1633–1634Google Scholar
  171. Spudich P, Joyner WB, Lindh AG, Boore DM, Margaris BM, Fletcher JB (1999) SEA99: A revised ground motion prediction relation for use in extensional tectonic regimes. Bulletin of the Seismological Society of America 89(5):1156–1170Google Scholar
  172. Strasser FO, Bommer JJ, Abrahamson NA (2008) Truncation of the distribution of ground-motion residuals. Journal of Seismology 12(1):79–105, DOI: 10.1007/s10950-007-9073-z CrossRefGoogle Scholar
  173. Swanger HJ, Boore DM (1978) Simulation of strong-motion displacements using surface-wave modal superposition. Bulletin of the Seismological Society of America 68(4):907–922Google Scholar
  174. Takeo M (1985) Near-field synthetic seismograms taking into account the effects of anelasticity: The effects of anelastic attenuation on seismograms caused by a sedimentary layer. Meteorology & Geophysics 36(4):245–257CrossRefGoogle Scholar
  175. Tavakoli B, Pezeshk S (2005) Empirical-stochastic ground-motion prediction for eastern North America. Bulletin of the Seismological Society of America 95(6):2283–2296, DOI: 10.1785/0120050030 CrossRefGoogle Scholar
  176. Tinti E, Fukuyama E, Piatanesi A, Cocco M (2005) A kinematic source-time function compatible with earthquake dynamics. Bulletin of the Seismological Society of America 95(4):1211–1223, DOI: 10.1785/0120040177 CrossRefGoogle Scholar
  177. Trifunac MD (1971) A method for synthesizing realistic strong ground motion. Bulletin of the Seismological Society of America 61(6):1739–1753Google Scholar
  178. Trifunac MD (1976) Preliminary analysis of the peaks of strong earthquake ground motion – dependence of peaks on earthquake magnitude, epicentral distance, and recording site conditions. Bulletin of the Seismological Society of America 66(1):189–219Google Scholar
  179. Trifunac MD (1990) Curvograms of strong ground motion. Journal of The Engineering Mechanics Division, ASCE 116:1426–32CrossRefGoogle Scholar
  180. Trifunac MD (2007) Recording strong earthquake motion—instruments, recording strategies and data processing. Tech. Rep. CE 07-03. Department of Civil Engineering, University of Southern CaliforniaGoogle Scholar
  181. Trifunac MD, Brady AG (1975) On the correlation of seismic intensity scales with the peaks of recorded strong ground motion. Bulletin of the Seismological Society of America 65(1):139–162Google Scholar
  182. Tufte ER (2006) Beautiful Evidence. Graphics Press, Cheshire, Connecticut, USAGoogle Scholar
  183. Tumarkin A, Archuleta R (1994) Empirical ground motion prediction. Annali di Geofisica XXXVII(6):1691–1720Google Scholar
  184. Vanmarcke EH (1979) Representation of earthquake ground motion: Scaled accelerograms and equivalent response spectra. State-of-the-Art for Assessing Earthquake Hazards in the United States 14, Miscellaneous Paper S-73-1, U.S. Army Corps of Engineers, Vicksburg, Mississippi, USAGoogle Scholar
  185. Vanmarcke EH, Gasparini DA (1976) Simulated earthquake motions compatible with prescribed response spectra. Tech. Rep. R76-4. Dept. of Civil Engineering, Massachusetts Inst. of Technology, Cambridge, USAGoogle Scholar
  186. Virieux J, Madariaga R (1982) Dynamic faulting studied by a finite difference method. Bulletin of the Seismological Society of America 72(2):345–369Google Scholar
  187. Wald DJ, Quitoriano V, Heaton TH, Kanamori H (1999) Relationships between peak ground acceleration, peak ground velocity, and modified Mercalli intensity in California. Earthquake Spectra 15(3):557–564CrossRefGoogle Scholar
  188. Wang R (1999) A simple orthonormalization method for stable and efficient computation of Green’s functions. Bulletin of the Seismological Society of America 89(3):733–741Google Scholar
  189. Wang GQ, Boore DM, Tang G, Zhou X (2007) Comparisons of ground motions from collocated and closely-spaced 1-sample-per-second Global Positioning System (GPS) and accelerograph recordings of the 2003, M6.5 San Simeon, California, earthquake in the Parkfield Region. Bull Seismol Soc Am 97(1B):76–90. doi:10.1785/0120060053 Google Scholar
  190. Watson-Lamprey J, Abrahamson N (2006) Selection of ground motion time series and limits on scaling. Soil Dynamics and Earthquake Engineering 26(5):477–482CrossRefGoogle Scholar
  191. Wennerberg L (1990) Stochastic summation of empirical Green’s functions. Bulletin of the Seismological Society of America 80(6):1418–1432Google Scholar
  192. Wong HL, Trifunac MD (1978) Synthesizing realistic ground motion accelerograms. Tech. Rep. CE 78-07. Department of Civil Engineering, University of Southern CaliforniaGoogle Scholar
  193. Woodhouse JH (1974) Surface waves in a laterally varying layered structure. Geophysical Journal of the Royal Astronomical Society 37:461–490Google Scholar
  194. Zeng Y, Anderson JG (1995) A method for direct computation of the differential seismogram with respect to the velocity change in a layered elastic solid. Bulletin of the Seismological Society of America 85(1):300–307Google Scholar
  195. Zeng T, Anderson JG, Yu G (1994) A composite source model for computing realistic synthetic strong ground motions. Geophysical Research Letters 21(8):725–728CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.ARN/RISBRGMOrleans Cedex 2France

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