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.
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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 M w 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 1–22. 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 Footnote 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.
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
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 1–3 summarise the three main types of empirical methods.
2.2 Black-box Methods
This section describes four methods (Tables 4–7) 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.
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 8–18 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.
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 19–22. 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.
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.
Notes
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
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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.
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Douglas, J., Aochi, H. A Survey of Techniques for Predicting Earthquake Ground Motions for Engineering Purposes. Surv Geophys 29, 187–220 (2008). https://doi.org/10.1007/s10712-008-9046-y
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DOI: https://doi.org/10.1007/s10712-008-9046-y