Synopsis

This online resource summarizes all empirical ground-motion prediction equations (GMPEs), to estimate earthquake peak ground acceleration (PGA) and elastic response spectral ordinates, published between 1964 and early 2021 (inclusive). This resource replaces: the Imperial College London reports of Douglas (2001a), Douglas (2002) and Douglas (2004a), which provide a summary of all GMPEs from 1964 until the end of 2003; the BRGM report of Douglas (2006), which summarizes all GMPEs from 2004 to 2006 (plus some earlier models); the report of Douglas (2008), concerning GMPEs published in 2007 and 2008 (plus some earlier models); and the report of Douglas (2011), which superseded all these reports and covered the period up to 2010. It is planned to continually update this website when new GMPEs are published or errors/omissions are discovered. In addition, this resource lists published GMPEs derived from simulations, although details are not given since the focus here is on empirical models. Studies that only present graphs are only listed, as are those non-parametric formulations that provide predictions for different combinations of distance and magnitude because these are more difficult to use for seismic hazard analysis than those which give a single formula. Equations for single earthquakes or for earthquakes of approximately the same size are excluded due to their limited usefulness. Those relations based on conversions from macroseismic intensity are only listed. Finally, conditional ground-motion models (e.g. Sung, Abrahamson, and Huang 2021), which provide predictions for a secondary intensity measure conditional on a primary measure, are excluded due to a lack of resources to identify and summarise these models.

This website summarizes, in total, the characteristics of 485 empirical GMPEs for the prediction of PGA and 316 empirical models for the prediction of elastic response spectral ordinates. In addition, 87 simulation-based models to estimate PGA and elastic response spectral ordinates are listed but no details are given. 52 complete stochastic models, 45 GMPEs derived in other ways, 39 non-parametric models and 18 backbone (G. M. Atkinson, Bommer, and Abrahamson 2014; Douglas 2018b) models are also listed. Finally, the table provided by Douglas (2012) is expanded and updated to include the general characteristics of empirical GMPEs for the prediction of: Arias intensity (34 models), cumulative absolute velocity (12 models), Fourier spectral amplitudes (19 models), maximum absolute unit elastic input energy (6 models), inelastic response spectral ordinates (6 models), Japanese Meterological Agency seismic intensity (5 models), macroseismic intensity (52 models, commonly called intensity prediction equations), mean period (6 models), peak ground velocity (147 models), peak ground displacement (37 models), relative significant duration (20 models) and vertical-to-horizontal response spectral ratio (13 models). This report will be updated roughly once every six months.

It should be noted that the size of this resource means that it may contain some errors or omissions. The boundaries between empirical, simulation-based and non-parametric ground-motion models are not always clear so I may classify a study differently than expected. No discussion of the merits, ranges of applicability or limitations of any of the relationships is included herein except those mentioned by the authors or inherent in the data used. This compendium is not a critical review of the models.

This compilation was made when I was employed at: Imperial College London, University of Iceland, BRGM and University of Strathclyde. I thank: my current and former employers for their support, many people for references, suggestions and encouragement while producing this resource, and the developers of LaTeXand associated packages, without whom this report would never have been written.

If required, you can cite this resource in the following way:

Douglas, J. (2022), Ground motion prediction equations 1964–2021, http://www.gmpe.org.uk.

Introduction

ESEE Report 01-1 ‘A comprehensive worldwide summary of strong-motion attenuation relationships for peak ground acceleration and spectral ordinates (1969 to 2000)’ (Douglas 2001a) was completed and released in January 2001. A report detailing errata of this report and additional studies was released in October 2002 (Douglas 2002). These two reports were used by Douglas (2003) as a basis for a review of previous ground-motion prediction equations (GMPEs). Following the release of these two reports, some further minor errors were found in the text and tables of the original two reports, and additional studies were found in the literature that were not included in ESEE 01-1 or the follow-on report. Also some new studies were published. Rather than produce another report listing errata and additions it was decided to produce a new report that included details on all the studies listed in the first two reports (with the corrections made) and also information on the additional studies. This report was published as a research report of Imperial College London at the beginning of 2004 (Douglas 2004a). At the end of 2006 a BRGM report was published (Douglas 2006) detailing studies published in 2004–2006 plus a few earlier models that had been missed in previous reports. Finally, at the end of 2008 another BRGM report was published (Douglas 2008) containing summaries of GMPEs from 2007 and 2008 and some additional earlier models that had been recently uncovered.

Because of the large number of new GMPEs published in 2009 and 2010 and the discovery of some additional earlier studies and various errors in the previous reports, it was decided to publish a new comprehensive report to replace the previous reports (Douglas 2001a, 2002, 2004a, 2006, 2008) containing all previous reports plus additional material rather than publish yet another addendum to the 2004 report. It was also decided that, for completeness and due to the lack of another comprehensive and public source for this information, to include a list of GMPEs developed using other methods than regression of strong-motion data, e.g. simulation-based models (e.g. Douglas and Aochi 2008). However, due to the complexity of briefly summarizing these models it was decided not to provide details but only references. This report was published as Douglas (2011).

In order to make the compendium easier to use and to update in the future it was decided to port the entire report to html using the LaTeXTeX 4ht package as well as add models from 2011 to 2021 and some older GMPEs that were recently found. Finally, GMPEs for intensity measures other than PGA and elastic response spectral ordinates are listed but details are not given (although some of these correspond to models for PGA and elastic spectral ordinates and hence they are summarized elsewhere in this compendium).

This report summarizes, in total, the characteristics of 485 empirical GMPEs for the prediction of peak ground acceleration (PGA) and 317 models for the prediction of elastic response spectral ordinates as well as 34 models for the prediction of Arias intensity, 12 models for cumulative absolute velocity, 19 models for Fourier spectral amplitudes, 6 models for maximum absolute unit elastic input energy, 6 models for inelastic response spectral ordinates, 5 models for Japanese Meterological Agency seismic intensity, 52 models¹ (intensity prediction equations) for macroseismic intensity, 6 models for mean period, 147 for peak ground velocity, 37 for peak ground displacement, 20 for relative significant duration and 13 models for vertical-to-horizontal response spectral ratio. With this many GMPEs available it is important to have criteria available for the selection of appropriate models for seismic hazard assessment in a given region — Cotton et al. (2006) and, more recently, Bommer et al. (2010) suggest selection requirements for the choice of models. For the selection of GMPEs routinely applicable to state-of-the-art hazard analyses of ground motions from shallow crustal earthquakes Bommer et al. (2010) summarize their criteria thus.

1. Model is derived for an inappropriate tectonic environment (such as subduction-zone earthquakes or volcanic regions).

2. Model not published in a Thomson Reuters ISI-listed peer-reviewed journal (although an exception can be made for an update to a model that did meet this criterion).

3. The dataset used to derive the model is not presented in an accessible format; the minimum requirement would be a table listing the earthquakes and their characteristics, together with the number of records from each event.

4. The model has been superseded by a more recent publication.

5. The model does not provide spectral predictions for an adequate range of response periods, chosen here to be from \(0\) to \(2\,\mathrm{s}\).

6. The functional form lacks either non-linear magnitude dependence or magnitude-dependent decay with distance.

7. The coefficients of the model were not determined with a method that accounts for inter-event and intra-event components of variability; in other words, models must be derived using one- or two-stage maximum likelihood approaches or the random effects approach.

8. Model uses inappropriate definitions for explanatory variables, such as \(M_L\) or \(r_{epi}\), or models site effects without consideration of \(V_{s,30}\).

9. The range of applicability of the model is too small to be useful for the extrapolations generally required in PSHA: \(M_{\min}>5\), \(M_{\max}<7\), \(R_{\max}<80\,\mathrm{km}\).

10. Model constrained with insufficiently large dataset: fewer than 10 earthquakes per unit of magnitude or fewer than 100 records per \(100\,\mathrm{km}\) of distance.

Similar criteria could be developed for other types of earthquakes (e.g. subduction). For example, the reader is referred to Stewart et al. (2015) for a discussion of the selection of GMPEs for hazard assessments for the three principal tectonic regimes. Application of such criteria would lead to a much reduced set of models. The aim of this report, however, is not to apply these, or any other, criteria but simply to summarize all models that have been published. Bommer et al. (2010) also note that: ‘[i]f one accepts the general approach presented in this paper, then it becomes inappropriate to develop and publish GMPEs that would subsequently be excluded from use in PSHA [probabilistic seismic hazard analysis] on the basis of not satisfying one or more of the requirements embodied in the criteria.’

Predictions of median ground motions from GMPEs show great dispersion (Douglas 2010a, 2010b, 2012) demonstrating the large epistemic uncertainties involved in the estimation of earthquake shaking. This uncertainty should be accounted for within seismic hazard assessments by, for example, logic trees (e.g. Bommer and Scherbaum 2008).

Other summaries and reviews of GMPEs

A number of reviews of GMPEs have been made in the past that provide a good summary of the methods used, the results obtained and the problems associated with such relations. Trifunac and Brady (1975a, 1976) provide a brief summary and comparison of published relations. McGuire (1976) lists numerous early relations. Idriss (1978) presents a comprehensive review of published attenuation relations up until 1978, including a number which are not easily available elsewhere. Hays (1980) presents a good summary of ground-motion estimation procedures up to 1980. Boore and Joyner (1982) provide a review of attenuation studies published in 1981 and they comment on empirical prediction of strong ground motion in general. Campbell (1985) contains a full survey of attenuation equations up until 1985. Joyner and Boore (1988) give an excellent analysis of ground motion prediction methodology in general, and attenuation relations in particular; Joyner and Boore (1996) update this by including more recent studies. N. N. Ambraseys and Bommer (1995) provide an overview of relations that are used for seismic design in Europe although they do not provide details about methods used. Recent reviews include those by Campbell (2003c, 2003a) and Bozorgnia and Campbell (2004a), which provide the coefficients for a number of commonly-used equations for peak ground acceleration and spectral ordinates, and Douglas (2003). Bommer (2006) discusses some pressing problems in the field of empirical ground-motion estimation. The International Institute of Seismology and Earthquake Engineering provides a useful online resource
http://iisee.kenken.go.jp/eqflow/reference/Start.htm summarising a number of GMPEs (particularly those from Japan) and providing coefficients and an Excel spreadsheet for their evaluation. A recent discussion of current and future trends in ground-motion prediction is provided by Douglas and Edwards (2016).

Summaries and reviews of published ground-motion models for the estimation of strong-motion parameters other than PGA and elastic response spectral ordinates are available². For example: Bommer and Martı́nez-Pereira (1999), Alarcón (2007) and Bommer, Stafford, and Alarcón (2009) review predictive equations for strong-motion duration; Tromans (2004) summarizes equations for the prediction of PGV and displacement (PGD); Bommer and Alarcón (2006) provide a more recent review of GMPEs for PGV; Hancock and Bommer (2005) discuss available equations for estimating number of effective cycles; Stafford, Berrill, and Pettinga (2009) briefly review GMPEs for Arias intensity; Rathje et al. (2004) summarize the few equations published for the prediction of frequency-content parameters (e.g. predominant frequency); and Cua et al. (2010) review various intensity prediction equations.

GMPEs summarised here

Equations for single earthquakes (e.g. Bozorgnia, Niazi, and Campbell 1995) or for earthquakes of approximately the same size (e.g. Seed et al. 1976; K. Sadigh, Youngs, and Power 1978) are excluded because they lack a magnitude-scaling term and, hence, are of limited use. Also excluded are those originally developed to yield the magnitude of an earthquake (e.g. Espinosa 1980), i.e. the regression is performed the other way round, which should not be used for the prediction of ground motion at a site. The model of Kim and Shin (2017) is not included because it is based on the ratio of the magnitude of the mainshock to an aftershock rather than the magnitude directly. The model of J. X. Zhao and Gerstenberger (2010) is not summarised since it uses recorded motions to estimate motions at sites without observations, within a rapid-response system. Models such as that by Olszewska (2006) and Golik and Mendecki (2012), who use ’source energy logarithms’ to characterize mining-induced events, have been excluded because such a characterization of event size is rare in standard seismic hazard assessments. Similarly, equations derived using data from nuclear tests, such as those reported by Mickey (1971; Hays 1980), are not included. Finally, conditional ground-motion models (e.g. Sung, Abrahamson, and Huang 2021), which provide predictions for a secondary intensity measure conditional on a primary measure, are excluded due to a lack of resources to identify and summarise these models.

Those based on simulated ground motions from stochastic source models (e.g G. M. Atkinson and Boore 1990) and other types of simulations (e.g. Megawati, Pan, and Koketsu 2005), those derived using the hybrid empirical technique (e.g Campbell 2003b; Douglas, Bungum, and Scherbaum 2006), those relations based on intensity measurements (e.g. Battis 1981) and backbone models (G. M. Atkinson, Bommer, and Abrahamson 2014; Douglas 2018b) are listed in Chapter 6 but no details are given because the focus here is on empirical models derived from ground-motion data. Studies using simulation techniques other than the classic stochastic method and which do not provide a closed-form GMPE (e.g. Medel-Vera and Ji 2016) are not listed as they are often difficult to use. Studies which provide graphs to give predictions (e.g. Schnabel and Seed 1973) are only listed and not summarized as are those non-parametric formulations that give predictions for different combinations of distance and magnitude (e.g. Anderson 1997), both of which are generally more difficult to use for seismic hazard analysis than those which report a single formula. For similar reasons, models derived using neural networks (e.g. Güllü and Erçelebi 2007) are only listed.

GMPEs for the prediction of PGA are summarized in Chapters 2 and 3 and those for spectral ordinates are summarized in Chapters 4 and 5. Chapter 6 lists other ground-motion models that are not detailed in the previous chapters. The final chapter (Chapter 7) provides the general characteristics of GMPEs for intensity measures other than PGA and elastic spectral ordinates. All the studies that present the same GMPE are mentioned at the top of the section and in the tables of general characteristics (Illustrations [tab:pga] & [tab:speccomp]). The information contained within each section, and within tables, is the sum of information contained within each of the publications, i.e. not all the information may be from a single source. Note that GMPEs are ordered in chronological order both in the section titles and the order of the sections. Therefore, a well-known model presented in a journal article may not be listed where expected since it had previously been published in a conference proceedings or technical report. To find a given model it is recommended to examine the table of content carefully or apply a keyword search to the PDF. Some models (e.g. Abrahamson and Silva 1997) provide GMPEs for spectral accelerations up to high frequencies (e.g. \(100\,\mathrm{Hz}\)) but do not explicitly state that these equations can be used for the prediction of PGA. Therefore, they are only listed in the chapters dealing with GMPEs for the prediction of spectral ordinates (Chapters 4 and 5) and their coefficients are not given. This should be considered when searching for a particular model.

To make it easier to understand the functional form of each GMPE the equations are given with variable names replacing actual coefficients and the derived coefficients and the standard deviation, \(\sigma\), are given separately (for PGA equations). These coefficients are given only for completeness and if an equation is to be used then the original reference should be consulted. If a coefficient is assumed before the analysis is performed then the number is included directly in the formula.

Obviously all the details from each publication cannot be included in this report because of lack of space but the most important details of the methods and data used are retained. The style is telegraphic and hence phrases such as ‘Note that …’ should be read ‘The authors [of the original model] note that …’. The number of records within each site and source mechanism category are given if this information was reported by the authors of the study. Sometimes these totals were found by counting the numbers in each category using the tables listing the data used and, therefore, they may be inaccurate.

This report contains details of all studies for PGA and response spectra that could be found in the literature (journals, conference proceedings, technical reports and some Ph.D. theses) although some may have been inadvertently missed³. Some of the studies included here have not been seen but are reported in other publications and hence the information given here may not be complete or correct. Since this resource has been written in many distinct periods over almost two decades (2000–2021), the amount of information given for each model varies, as does the style.

In the equations unless otherwise stated, \(D\), \(d\), \(R\), \(r\), \(X\), \(\Delta\) or similar are distance and \(M\) or similar is magnitude and all other independent variables are stated. PGA is peak ground acceleration, PGV is peak ground velocity and PSV is relative pseudo-velocity.

In Tables [tab:pga], [tab:speccomp] and [tab:gmpes] the gross characteristics of the data used and equation obtained are only given for the main equation in each study. The reader should refer to the section on a particular publication or the original reference for information on other equations derived in the study.

In earlier reports the name ‘attenuation relation(ships)’ is used for the models reported. The current de facto standard is to refer to such models as ‘ground motion prediction equations’ (GMPEs) and, therefore, this terminology is adopted here. However, as discussed by Boore and Atkinson (2007 Appendix A) there is some debate over the best name for these models (e.g. ‘ground-motion model’ or ‘ground motion estimation equations’) and some people disagree with the use of the word ‘prediction’ in this context.

No discussion of the merits, ranges of applicability or limitations of any of the relationships is included herein except those mentioned by the authors or inherent in the data used. This report is not a critical review of the models. The ground-motion models are generally reported in the form given in the original references. The boundaries between empirical, simulation-based and non-parametric ground-motion models are not always clear so I may classify a study differently than expected. Note that the size of this report means that it may contain some errors or omissions — the reader is encouraged to consult the original reference if a model is to be used.

Summary of published GMPEs for PGA

Esteva and Rosenblueth (1964)

• Ground-motion model is: \[a=c\exp(\alpha M) R^{-\beta}\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(c=2000\), \(\alpha=0.8\) and \(\beta=2\) (\(\sigma\) is not given).

Kanai (1966)

• Ground-motion model is: \[\begin{aligned} a&=&\frac{a_1}{\sqrt{T_G}} 10^{a_2 M-P \log_{10} R+Q}\\ P&=&a_3+a_4/R\\ Q&=&a_5+a_6/R\end{aligned}\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(a_1=5\), \(a_2=0.61\), \(a_3=1.66\), \(a_4=3.60\), \(a_5=0.167\) and \(a_6=-1.83\) (\(\sigma\) is not given).

• \(T_G\) is the fundamental period of the site.

Milne and Davenport (1969)

• Ground-motion model is: \[A=\frac{a_1 \mathrm{e}^{a_2 M}}{a_3 \mathrm{e}^{a_4 M} +\Delta^2}\] where \(A\) is in percentage of \(\,\mathrm{g}\), \(a_1=0.69\), \(a_2=1.64\), \(a_3=1.1\) and \(a_4=1.10\) (\(\sigma\) not given).

• Use data from Esteva and Rosenblueth (1964).

Esteva (1970)

• Ground-motion model is: \[a=c_1 \mathrm{e}^{c_2 M} (R+c_3)^{-c_4}\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(c_1=1230\), \(c_2=0.8\), \(c_3=25\), \(c_4=2\) and \(\sigma=1.02\) (in terms of natural logarithm).

• Records from soils comparable to stiff clay or compact conglomerate.

• Records from earthquakes of moderate duration.

Denham and Small (1971)

• Ground-motion model is: \[\log Y=b_1 + b_2 M+b_3 \log R\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1=-0.2\), \(b_2=0.2\) and \(b_3=-1.1\) (\(\sigma\) not given).

• Records from near dam on recent unconsolidated lake sediments which are \(\geq 50\,\mathrm{m}\) thick.

• Note need for more points and large uncertainty in \(b_1\), \(b_2\) and \(b_3\).

Davenport (1972)

• Ground-motion model is: \[A=\alpha \mathrm{e}^{\beta m}R^{-\gamma}\] where \(A\) is in \(\,\mathrm{g}\), \(\alpha=0.279\), \(\beta=0.80\), \(\gamma=1.64\) and \(\sigma=0.74\) (in terms of natural logarithms).

Denham, Small, and Everingham (1973)

• Ground-motion model is: \[\log Y_a=a_1+a_2 M_L+b_3 \log R\] where \(Y_a\) is in \(\,\mathrm{cm/s^2}\), \(a_1=2.91\), \(a_2=0.32\) and \(a_3=-1.45\) (\(\sigma\) is not given).

• Use records from Yonki station (20 records) which is on \(50\,\mathrm{m}\) of recent alluvium and from Paguna station (5 records) which is on unconsolidated volcanic rock.

• Question validity of combining data at the two sites because of differences in geological foundations.

• Note large standard errors associated with coefficients preclude accurate predictions of ground motions.

• Also derive equation for Yonki station separately.

Donovan (1973)

• Ground-motion model is: \[y=b_1 \mathrm{e}^{b_2 M} (R+25)^{-b_3}\] where \(y\) is in \(\,\mathrm{gal}\), \(b_1=1080\), \(b_2=0.5\), \(b_3=1.32\) and \(\sigma=0.71\). \(25\) adopted from Esteva (1970).

• 214 (\(32\%\)) records from San Fernando (9/2/1971) earthquake and \(53\%\) of records with PGA less than \(0.5\,\mathrm{m/s^2}\).

• Considers portions of data and finds magnitude dependence increases with increasing distance from source and more small accelerations increase magnitude dependence. Thus magnitude and distance cannot be considered independent variables.

Esteva and Villaverde (1973) & Esteva (1974)

• Ground-motion model is: \[Y_c=b_1 \mathrm{e}^{b_2 M} (R+b_4)^{-b_3}\] where \(Y_c\) is in \(\,\mathrm{cm/s^2}\), \(b_1=5600\), \(b_2=0.8\), \(b_3=2\), \(b_4=40\) and \(\sigma=0.64\) (in terms of natural logarithm).

Katayama (1974)

• Ground-motion model is: \[\log A=c_1+c_2 \log(R+c_3)+c_4 M\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=2.308\), \(c_2=-1.637\), \(c_3=30\) and \(c_4=0.411\) (\(\sigma\) not reported⁴). Also derive equation using \(r_{epi}\): \(c_1=0.982\), \(c_2=-0.129\), \(c_3=0\) and \(c_4=0.466\) (\(\sigma\) not reported⁵).

McGuire (1974) & McGuire (1977)

• Ground-motion model is: \[E[v]=a10^{bM}(R+25)^{-c}\] where \(E\) indicates expectation, \(v\) is in \(\,\mathrm{gal}\), \(a=472\), \(b=0.278\), \(c=1.301\).

• Excludes records for which significant soil amplification established but makes no distinction between rock and soil sites.

• Focal depths between \(9\) and \(70\,\mathrm{km}\) with most about \(10\,\mathrm{km}\). Most records from earthquakes with magnitudes about \(6.5\) and most distances less than \(50\,\mathrm{km}\). Uses records from 21 different sites.

• Notes that physical laws governing ground motion near the source are different than those governing motion at greater distances therefore excludes records with epicentral distance or distance to fault rupture smaller than one-half of estimated length of rupture.

• Examines correlation among the records but find negligible effect.

Orphal and Lahoud (1974)

• Ground-motion model is: \[A=\lambda 10^{\alpha M} R^{\beta}\] where \(A\) is in \(\,\mathrm{g}\), \(\lambda=6.6\times 10^{-2}\), \(\alpha=0.40\), \(\beta=-1.39\) and \(\sigma=1.99\) (this is multiplication factor).

• Use 113 records with distances between \(15\) to \(350\,\mathrm{km}\) from San Fernando earthquake to find distance dependence, \(\beta\).

• Use 27 records of Wiggins (1964) from El Centro and Ferndale areas, with magnitudes between \(4.1\) and \(7.0\) and distances between \(17\) and \(94\,\mathrm{km}\) (assuming focal depth of \(15\,\mathrm{km}\)), to compute magnitude dependent terms assuming distance dependence is same as for San Fernando.

Ahorner and Rosenhauer (1975)

• Ground-motion model is: \[A=c_1 \exp(c_2 M) (R+c_3)^{-c_4}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=1230\), \(c_2=0.8\), \(c_3=13\) and \(c_4=-2\) (\(\sigma\) is not reported).

N. N. Ambraseys (1975), N. Ambraseys (1975) & N. N. Ambraseys (1978a)

• Ground-motion model is: \[\log Y=b_1 +b_2 M_L+b_3 \log R\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(b_1=0.46\), \(b_2=0.63\), \(b_3=-1.10\) and \(\sigma=0.32\)⁶

• N. N. Ambraseys and Bommer (1995) state that uses earthquakes with maximum focal depth of \(15\,\mathrm{km}\).

Shah and Movassate (1975)

• Ground-motion model is: \[A=c_1 \exp(c_2 M) (R+c_3)^{-c_4}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=5000\), \(c_2=0.8\), \(c_3=40\) and \(c_4=-2\) (\(\sigma\) is not reported).

Trifunac and Brady (1975a), Trifunac (1976a) & Trifunac and Brady (1976)

• Ground-motion model is: \[\begin{aligned} \log_{10} a_{\max}&=&M+\log_{10} A_0 (R) -\log_{10} a_0(M,p,s,v)\\ \log_{10} a_0(M,p,s,v)&=&\left\{ \begin{array}{r} ap+bM+c+ds+ev+fM^2-f(M-M_{\max})^2\\ \quad \mbox{for}\quad M\geq M_{\max}\\ ap+bM+c+ds+ev+fM^2\\ \quad \mbox{for}\quad M_{\max}\geq M \geq M_{\min}\\ ap+bM_{\min}+c+ds+ev+fM^2_{\min}\\ \quad \mbox{for}\quad M\leq M_{\min}\\ \end{array} \right.\end{aligned}\] where \(a_{\max}\) is in \(\,\mathrm{cm/s^2}\), \(\log_{10} A_0 (R)\) is an empirically determined attenuation function from Richter (1958) used for calculation of \(M_L\), \(p\) is confidence level and \(v\) is component direction (\(v=0\) for horizontal and \(1\) for vertical). Coefficients are: \(a=-0.898\), \(b=-1.789\), \(c=6.217\), \(d=0.060\), \(e=0.331\), \(f=0.186\), \(M_{\min}=4.80\) and \(M_{\max}=7.50\) (\(\log_{10} A_0(R)\) not given here due to lack of space).

• Use three site categories:

  1. Alluvium or other low velocity ‘soft’ deposits: \(63\%\) of records.

  2. ‘Intermediate’ type rock: \(23\%\) of records.

  3. Solid ‘hard’ basement rock: \(8\%\) of records.

• Exclude records from tall buildings.

• Do not use data from other regions because attenuation varies with geological province and magnitude determination is different in other countries.

• Records baseline and instrument corrected. Accelerations thought to be accurate between \(0.07\) and \(25\,\mathrm{Hz}\) or between \(0.125\) and \(25\,\mathrm{Hz}\) for San Fernando records.

• Most records (\(71\%\)) from earthquakes with magnitudes between \(6.0\)–\(6.9\), \(22\%\) are from \(5.0\)–\(5.9\), \(3\%\) are from \(4.0\)–\(4.9\) and \(3\%\) are from \(7.0\)–\(7.7\) (note barely adequate data from these two magnitude ranges). \(63\%\) of data from San Fernando earthquake.

• Note that for large earthquakes, i.e. long faults, \(\log_{10} A_0(R)\) would have a tendency to flatten out for small epicentral distances and for low magnitude shocks curve would probably have a large negative slope. Due to lack of data \(\lesssim 20\,\mathrm{km}\) this is impossible to check.

• Note difficulty in incorporating anelastic attenuation because representative frequency content of peak amplitudes change with distance and because relative contribution of digitization noise varies with frequency and distance.

• Note that \(\log_{10} A_0(R)\) may be unreliable for epicentral distances less than \(10\,\mathrm{km}\) because of lack of data.

• Change of slope in \(\log_{10} A_0(R)\) at \(R=75\,\mathrm{km}\) because for greater distances main contribution to strong shaking from surface waves, which are attenuated less rapidly (\(\sim 1/R^{1/2}\)) than near-field and intermediate-field (\(\sim 1/R^{2-4}\)), or far-field body waves (\(\sim 1/R\)).

• Note lack of data to reliably characterise \(\log_{10} a_0(M,p,s,v)\) over a sufficiently broad range of their arguments. Also note high proportion of San Fernando data may bias results.

• Firstly partition data into four magnitude dependent groups: \(4.0\)–\(4.9\), \(5.0\)–\(5.9\), \(6.0\)–\(6.9\) and \(7.0\)–\(7.9\). Subdivide each group into three site condition subgroups (for \(s=0\), \(1\) and \(2\)). Divide each subgroup into two component categories (for \(v=0\) and \(1\)). Calculate \(\log_{10} a_0(M,p,s,v)=M+\log_{10} A_0(R)-\log_{10} a_{\max}\) within each of the \(24\) parts. Arrange each set of \(n\) \(\log_{10} a_0\) values into decreasing order with increasing \(n\). Then \(m\)th data point (where \(m\) equals integer part of \(pn\)) is estimate for upper bound of \(\log_{10} a_0\) for \(p \%\) confidence level. Then fit results using least squares to find \(a\), …\(f\).

• Check number of PGA values less than confidence level for \(p=0.1\), …, \(0.9\) to verify adequacy of bound. Find simplifying assumptions are acceptable for derivation of approximate bounds.

Blume (1977)

• Ground-motion model is: \[a=b_1 \mathrm{e}^{b_2 M_L} (R+25)^{-b_3}\] where \(a\) is in \(\,\mathrm{gal}\), for \(M_L \leq 6\frac{1}{2}\) \(b_1=0.318\times 29^{1.14\bar{b}}\), \(b_2=1.03\), \(b_3=1.14 \bar{b}\) and \(\sigma=0.930\) (in terms of natural logarithm) and for \(M_L>6\frac{1}{2}\) \(b_1=26.0 \times 29^{1.22\bar{b}}\), \(b_2=0.432\), \(b_3=1.22\bar{b}\) and \(\sigma=0.592\) (in terms of natural logarithm).

• Assumes all earthquakes have focal depth of \(8\,\mathrm{km}\).

• Makes no distinction for site conditions in first stage where uses only earthquake records.

• Studies effects of PGA cutoff (no cutoff, \(0.01\), \(0.02\) and \(0.05\,\mathrm{m/s^2}\)), distance cutoff (no cutoff and \(<150\,\mathrm{km}\)) and magnitude cutoff (all, \(\geq 5\frac{1}{2}\), \(\geq 6\), \(\geq 6 \frac{1}{2}\), \(\geq 6 \frac{3}{4}\) and \(\leq 6\frac{1}{2}\)).

• Selects \(6\frac{1}{2}\) as optimum magnitude cutoff but uses all data to derive equation for \(M_L \leq 6\frac{1}{2}\) because not much difference and dispersion is slightly lower (in terms of \(\pm 1\) standard deviation have \(2.53\) and \(2.61\)).

• In second stage uses only records from underground nuclear explosions, consistent with natural earthquake records, to derive site factor.

• Uses 1911 alluvium and 802 rock records and derive PGA ratio of alluvium to rock assuming their PGAs equal at \(4\,\mathrm{km}\).

• Finds site impedance \(\rho V_s\), where \(\rho\) is density and \(V_s\) is shear-wave velocity under site, is best measure of site condition. Use \(2000\,\mathrm{fps}\) (\(600\,\mathrm{m/s}\)) as shear-wave velocity of alluvium stations.

• Multiplies equation (after taking logarithms) by \(\bar{b}=\frac{1}{2} \log_{10} (\rho V_s)\) and normalise to \(4\,\mathrm{km}\).

• Notes may not be a good model for other regions.

Gürpinar (1977)

• Ground-motion model is: \[\ln y=\ln a_1+a_2 \ln M+a_3\ln R\] where \(y\) is in \(\,\mathrm{cm/s^2}\); \(a_1=0.15\), \(a_2=4.84\) and \(a_3=-0.68\) (\(\sigma\) not reported) for site class I; \(a_1=9.80\), \(a_2=3.57\) and \(a_3=-1.12\) (\(\sigma\) not reported) for site class II; and \(a_1=0.0022\), \(a_2=8.31\) and \(a_3=-1.31\) (\(\sigma\) not reported) for site class III.

• Use 3 site classes (based on Caltech classification):

  1. Soft alluvium. 128 components.

  2. Stiff soil. 68 components.

  3. Hard rock. 26 components.

• Only uses data from \(20<r_{epi}<70\,\mathrm{km}\) because of large dispersion in data for closer distances.

• Uses F-values to test goodness of fit.

Milne (1977)

• Ground-motion model is: \[\mathrm{ACC}=a_1 \mathrm{e}^{a_2M} R^{a_3}\] where \(\mathrm{ACC}\) is in \(\,\mathrm{g}\), \(a_1=0.04\), \(a_2=1.00\) and \(a_3=-1.4\).

Saeki, Katayama, and Iwasaki (1977)

• Ground-motion model is: \[\log A=c_1+c_2M+c_3 \log(\Delta)\] where \(A\) is in \(\,\mathrm{cm/s^2}\), for class I: \(c_1=1.455\), \(c_2=0.207\) and \(c_3=-0.598\); for class II: \(c_1=1.121\), \(c_2=0.330\) and \(c_3=-0.806\); for class III: \(c_1=1.507\),

  \(c_2=0.254\) and \(c_3=-0.757\); for class IV: \(c_1=0.811\), \(c_2=0.430\) and \(c_3=-0.977\); and for all sites: \(c_1=1.265\), \(c_2=0.302\) and \(c_3=-0.800\). \(\sigma\) not reported.

• Use 4 site classes:

  1. 29 records

  2. 74 records

  3. 127 records

  4. 68 records

N. N. Ambraseys (1978b)

• Ground-motion model is: \[\bar{a}=a_1\bar{R}^{a_2}\exp(a_3\bar{M})\] where \(\bar{a}\) is in \(\,\mathrm{cm/s^2}\), \(a_1=1.31\), \(a_2=-0.92\) and \(a_3=1.455\) (\(\sigma\) is not given).

• Uses data from former USSR, former Yugoslavia, Portugal, Italy, Iran, Greece and Pakistan.

• Peak ground accelerations have either been taken from true-to-scale accelerograms or have been supplied by local networks. Records have not been high- or low-pass filtered because it was found not to work with short records.

• Believes body-wave or local magnitude are the appropriate magnitude scales because interested in the high-frequency range of spectra, which are seen and sampled by strong-motion instruments, and most engineering structures have high natural frequencies.

• Most of the magnitudes were recalculated using P-waves of periods of not more than \(1.2\,\mathrm{s}\) because it was found that the magnitude was dependent on the period of the P-waves used for its determination.

• Groups data into intervals of \(0.5\) magnitude units by \(10\,\mathrm{km}\) in which the mean and standard deviations of the PGAs is calculated. This grouping minimises distance and magnitude-dependent effects. Notes that the number of observations is barely sufficient to allow a statistical treatment of the data and hence only test general trend. Notes that scatter is significant and decreases with increasing magnitude.

Donovan and Bornstein (1978)

• Ground-motion model is: \[\begin{aligned} y&=&b_1 \mathrm{e}^{b_2 M} (R+25)^{-b_3}\\ \mbox{where }\quad b_1&=&c_1 R^{-c_2}\\ b_2&=&d_1+d_2 \log R\\ b_3&=&e_1+e_2 \log R\end{aligned}\] where \(y\) is in \(\,\mathrm{gal}\), \(c_1=2,154,000\), \(c_2=2.10\), \(d_1=0.046\), \(d_2=0.445\), \(e_1=2.515\), \(e_2=-0.486\), for \(y=0.01\,\mathrm{g}\) \(\sigma=0.5\), for \(y=0.05\,\mathrm{g}\) \(\sigma=0.48\), for \(y=0.10\,\mathrm{g}\) \(\sigma=0.46\) and for \(y=0.15\,\mathrm{g}\) \(\sigma=0.41\) (in terms of natural logarithm).

  Use \(25\) because assume energy centre of Californian earthquakes to be at depth \(5\,\mathrm{km}\).

• Consider two site conditions but do not model:

  1. Rock: (21 records)

  2. Stiff soil: (38 records)

• \(32\%\) of records from San Fernando (9/2/1971) but verifies that relationship is not significantly biased by this data.

• Most records within \(50\,\mathrm{km}\) and most from earthquakes with magnitudes of about \(6.5\).

• Recognises that magnitude and distance are not independent variables.

• Find \(b_1\), \(b_2\) and \(b_3\) by dividing data according to distance and computing \(b\) parameters for each set using least squares. Find a distinct trend with little scatter.

Faccioli (1978)

• Ground-motion model is: \[y=a 10^{b M} (R+25)^{-c}\] where \(y\) is in \(\,\mathrm{gal}\), \(a=108.60\), \(b=0.265\), \(c=0.808\) and \(\sigma=0.236\) (in terms of logarithm to base \(10\)).

• Records from sites underlain by cohesive or cohesionless soils with shear-wave velocities less than about \(100\,\mathrm{m/s}\) and/or standard penetration resistance \(N\leq10\) in uppermost \(10\,\mathrm{m}\) with layers of considerably stiffer materials either immediately below or at depths not exceeding a few tens of metres.

• Focal depths between \(9\) and \(100\,\mathrm{km}\).

• Free-field accelerograms, to minimize soil-structure interaction.

• Excludes records with \(\mathrm{PGA}<0.4\,\mathrm{m/s^2}\).

• 21 Japanese records processed with frequency cutoffs of bandpass filter, for baseline correction, adjusted so as to account for length and mean sampling rate of records and response characteristics of SMAC-2. 4 of remaining 7 records processed in same way.

Goto et al. (1978)

• Ground-motion model is: \[\log A=c_1+c_2M+c_3 \log(\Delta+30)\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=2.610\), \(c_2=0.160\) and \(c_3=-0.752\) (\(\sigma\) not reported⁷).

• Data from alluvial sites

• All PGAs \(>50\,\mathrm{gal}\).

R. K. McGuire (1978b)

• Ground-motion model is: \[\ln x=b_1+b_2 M +b_3 \ln R +b_4 Y_s\] where \(x\) is in \(\,\mathrm{cm/s^2}\), \(b_1=3.40\), \(b_2=0.89\), \(b_3=-1.17\), \(b_4=-0.20\) and \(\sigma=0.62\).

• Uses two site categories:

  1. Rock: sedimentary or basement rock or soil less than \(10\,\mathrm{m}\) thick, 11 records.

  2. Soil: alluvium or other soft material greater than \(10\,\mathrm{m}\) thick, 59 records.

• Uses records from basement of buildings or from ‘free-field’. Uses no more than seven records from same earthquake and no more than nine from a single site to minimize underestimation of calculated variance. Retains records which give a large distance and magnitude range.

• Notes that near-field ground motion governed by different physical laws than intermediate and far field so excludes near-field data, for example El Centro (19/5/1940) and Cholame-2, from Parkfield earthquake (28/6/1966)

• Considers a distance dependent site term but not statistically significant. Also uses a magnitude dependent site term and although it was statistically significant it did not reduce the scatter and also since largest magnitude for a rock site is \(6.5\), result may be biased.

A. Patwardhan, K. Sadigh, I.M. Idriss, R. Youngs (1978) reported in Idriss (1978)

• Ground-motion model is: \[\ln y=\ln A+B M_s +E \ln [R+d\exp(fM_s)]\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(d=0.864\) and \(f=0.463\) and for path A (rock): \(A=157\) (for median), \(A=186\) (for mean), \(B=1.04\) and \(E=-1.90\), for path A (stiff soil): \(A=191\) (for median), \(A=224\) (for mean), \(B=0.823\) and \(E=-1.56\) and for path B (stiff soil): \(A=284\) (for median), \(A=363\) (for mean), \(B=0.587\) and \(E=-1.05\) (\(\sigma\) not given).

• Separate equations for two types of path:

  1. Shallow focus earthquakes (California, Japan, Nicaragua and India), 63 records.

  2. Subduction (Benioff) zone earthquakes (Japan and South America), 23 earthquakes, \(5.3\leq M_s \leq 7.8\), 32 records.

• Use two site categories for path A earthquakes for which derive separate equations:

  1. Rock: 21 records.

  2. Stiff soil: 42 records.

  Use only stiff soil records for deriving subduction zone equation.

• Most earthquakes for path A have \(5\leq M_s \leq 6.7\).

• All data corrected. PGA for corrected Japanese and South American records much higher than uncorrected PGA.

Cornell, Banon, and Shakal (1979)

• Ground-motion model is: \[\ln A_p=a+b M_L+c \ln (R+25)\] where \(A_p\) is in \(\,\mathrm{cm/s^2}\), \(a=6.74\), \(b=0.859\), \(c=-1.80\) and \(\sigma=0.57\).

• No more than 7 records from one earthquake to avoid biasing results.

• Records from basements of buildings or free-field.

Faccioli (1979)

• Ground-motion model is: \[\log y=b_1+b_2M+b_3 \log(R+25)\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(b_1=0.44\), \(b_2=0.33\), \(b_3=-2.66\) and \(\sigma=0.12\).

• Uses data from three sedimentary rock sites (Somplago, San Rocco and Robic) because aim of study to provide zoning criteria as free as possible from influence of local conditions.

• Compares predictions and observations and find close fit, possibly because of restricted distance range.

• Note that use of simple functional form and \(r_{hypo}\) acceptable approximation because of short rupture lengths.

Faccioli and Agalbato (1979)

• Ground-motion model is: \[\log y=b_1+b_2 M+b_3 \log(R+\alpha)\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(b_1=1.59\pm 0.69\), \(b_2=0.25\pm 0.03\), \(b_3=-0.79\pm 0.12\), \(\alpha=0\) and \(\sigma=0.25\) for horizontal PGA and \(b_1=1.38\pm 1.89\), \(b_2=0.24\pm 0.09\), \(b_3=-0.78\pm 0.25\) and \(\sigma=0.25\) for vertical PGA.

• Use two site classes:

  1. Includes alluvium and moraine deposits of varying thicknesses and characteristics.

  2. Includes limestone, dolomite, flysch and cemented conglomerates, even if heavily fractured, overlain by not more than \(4\)–\(5\,\mathrm{m}\) of alluvium.

  Use published and unpublished material for classification.

• Focal depths between \(6\) and \(11\,\mathrm{km}\).

• Use data from Friuli 1976 mainshock and subsequent earthquakes from four networks including temporary stations (ENEL, CNEN, IZIIS and CEA/DSN). Data from ENEL, CNEN and IZIIS from RFT-250 and SMA-1 instruments and data from CEA/DSN from short-period seismographs. Some records not available in digital form so used reported PGAs.

• Almost all records from free-field stations.

• 58 PGAs from \(r_{hypo}\leq 20\,\mathrm{km}\).

• \(13\,\mathrm{cm/s^2}\leq PGA \leq 515\,\mathrm{cm/s}\) with \(93\%\) above \(30\,\mathrm{cm/s^2}\).

• Best-recorded earthquake (mainshock) contributed 24 PGAs.

• One station contributed 17 PGAs.

• Also regresses just using data from mainshock.

• \(\alpha\) is either \(0\) or \(25\) in regression. Prefer results with \(\alpha=0\) because smaller standard errors in \(b_3\).

• Statistical tests show \(b_2\) and \(b_3\) are significantly different than \(0\).

• Also present coefficients for rock-like stations only and soil stations only. Find that effect of selection by site class does not greatly affect coefficients.

• Process a smaller set of records available in digitized form (76 horizontal components) using high-pass filter (cut-off and roll-off of \(0.4\)–\(0.8\,\mathrm{Hz}\)) based on digitization noise. Note difficulty in standard processing due to high-frequency content and short durations. Use sampling rate of \(100\,\mathrm{Hz}\). Find that corrected horizontal PGAs are on average \(6\%\) lower than uncorrected PGAs and \(15\%\) show difference larger than \(10\%\). For vertical PGAs average difference is \(12\%\). Develop equations based on this subset (for horizontal PGA \(b_1=1.51\pm 0.77\), \(b_2=0.24\pm 0.04\), \(b_3=0.70 \pm 0.21\) and \(\sigma=0.24\)). Note similarity to results for uncorrected PGAs.

• Also derive equation using only 39 PGAs from \(r_{hypo}\leq 20\,\mathrm{km}\) and note weak magnitude and distance dependence. Compare to data from shallow soil sites at Forgaria-Cornino and Breginj and note that local site conditions can significantly modify bedrock motions even at close distances.

Aptikaev and Kopnichev (1980)

• Ground-motion model is: \[\log A_e=a_1 M+a_2 \log R+a_3\] where \(A_e\) is in \(\,\mathrm{cm/s^2}\), for \(A_e \geq 160\,\mathrm{cm/s^2}\) \(a_1=0.28\), \(a_2=-0.8\) and \(a_3=1.70\) and for \(A_e< 160\,\mathrm{cm/s^2}\) \(a_1=0.80\), \(a_2=-2.3\) and \(a_3=0.80\) (\(\sigma\) not given).

• As a rule, PGA corresponds to S-wave.

• Use five source mechanism categories (about 70 records, 59 earthquakes from W. N. America including Hawaii, Guatemala, Nicaragua, Chile, Peru, Argentina, Italy, Greece, Romania, central Asia, India and Japan):

  1. Contraction faulting (uplift and thrust), about 16 earthquakes.

  2. Contraction faulting with strike-slip component, about 6 earthquakes.

  3. Strike-slip, about 17 earthquakes.

  4. Strike-slip with dip-slip component, about 6 earthquakes.

  5. Dip-slip, about 9 earthquakes.

• Use these approximately 70 records to derive ratios of mean measured, \(A_0\), to predicted PGA, \(A_e\), \(\log(A_0/A_e)\), and for ratios of mean horizontal to vertical PGA, \(\log A_h/A_v\), for each type of faulting. Use every earthquake with equal weight independent of number of records for each earthquake.

• Results are:

  	Category 1	Category 2	Category 3	Category 4	Category 5
  \(\log A_0/A_e\)	\(0.35\pm 0.13\) (16)	\(0.11\pm 0.17\) (5)	\(0.22\pm 0.08\) (17)	\(0.06\pm 0.13\) (6)	\(-0.06 \pm 0.20\) (9)
  \(\log A_h/A_v\)	\(0.32\pm 0.13\) (12)	\(0.32\pm 0.08\) (5)	\(0.27\pm 0.07\) (12)	\(0.18\pm 0.10\) (5)	\(0.17 \pm 0.11\) (5)

  where \(\pm\) gives \(0.7\) confidence intervals and number in brackets is number of earthquakes used.

• Also calculate mean envelope increasing speed for P-wave amplitudes, \(A\), obtained at teleseismic distances: \(n=\mathrm{d} \ln A/ \mathrm{d} t\), where \(t\) is time for P-wave arrival and try to relate to ratios for each type of faulting.

Blume (1980)

• Ground-motion model is: \[a=b_1 \mathrm{e}^{b_2 M} (R+k)^{-b_3}\] where \(a\) is in \(\,\mathrm{gal}\), for method using distance partitioning \(b_1=18.4\), \(b_2=0.941\), \(b_3=1.27\) and \(k=25\) and for ordinary one-stage method \(b_1=102\), \(b_2=0.970\), \(b_3=1.68\) and \(k=25\) (\(\sigma\) not given).

• Does not use PGA cutoff because PGA is, by itself, a poor index of damage in most cases.

• Mean magnitude is \(5.4\) and mean distance is \(84.4\,\mathrm{km}\).

• Notes problem of regression leverage for some attenuation studies. Lots of data in fairly narrow distance band, e.g. records from San Fernando earthquake, can dominate regression and lead to biased coefficients.

• Divides data into ten distance bands (A-J) which are \(10\,\mathrm{km}\) wide up to \(60\,\mathrm{km}\) and then \(60\)-\(99.9\,\mathrm{km}\), \(100\)–\(139.9\,\mathrm{km}\), \(140\)–\(199.9\,\mathrm{km}\) and \(\geq 200\,\mathrm{km}\). Fits \(\log_{10} a=b M-c\) to data in each band and fits Ground-motion model to selected point set in \(M\), \(R\) and \(a\).

• Also fits equation using all data using normal least squares.

• Adds 52 records (\(3.2\leq M \leq 6.5\), \(5\leq R \leq 15\,\mathrm{km}\)) and repeats; finds little change.

Iwasaki, Kawashima, and Saeki (1980)

• Ground-motion model is: \[\mathrm{PGA}=a_1 10^{a_2 M} (\Delta+10)^{a_3}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{gal}\), for type I sites \(a_1=46.0\), \(a_2=0.208\) and \(a_3=-0.686\) , for type II sites \(a_1=24.5\), \(a_2=0.333\) and \(a_3=-0.924\), for type III sites \(a_1=59.0\), \(a_2=0.261\) and \(a_3=-0.886\), for type IV sites \(a_1=12.8\), \(a_2=0.432\), \(a_3=-1.125\) and for all sites \(a_1=34.1\), \(a_2=0.308\) and \(a_3=-0.925\) (\(\sigma\) not given).

• Use four site categories:

  1. Tertiary or older rock (defined as bedrock) or diluvium with depth to bedrock, \(H<10\,\mathrm{m}\), 29 records.

  2. Diluvium with \(H\geq 10\,\mathrm{m}\) or alluvium with \(H<10\,\mathrm{m}\), 74 records.

  3. Alluvium with \(H<25\,\mathrm{m}\) including soft layer (sand layer vulnerable to liquefaction or extremely soft cohesive soil layer) with thickness \(<5\,\mathrm{m}\), 130 records.

  4. Other than above, usually soft alluvium or reclaimed land, 68 records.

• Select earthquakes with Richter magnitude \(\geq 5.0\), hypocentral depth \(\leq 60\,\mathrm{km}\) and which include at least one record with PGA \(\geq 50\,\mathrm{gals}\) (\(0.5\,\mathrm{m/s^2}\)). Exclude records with PGA \(<10\,\mathrm{gals}\) (\(0.1\,\mathrm{m/s^2}\)).

• All records for \(M\geq 7.0\) are from distance \(>60\,\mathrm{km}\).

• Do regression separately for each soil category and also for combined data.

Matuschka (1980)

• Ground-motion model is: \[Y_c=b_1 \mathrm{e}^{b_2 M} (R+b_4)^{-b_3}\] Coefficients unknown.

Ohsaki, Watabe, and Tohdo (1980)

• Ground-motion model is: \[A=10^{a_1 M-a_2 \log x +a_3}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), for horizontal PGA \(a_1=0.440\), \(a_2=1.381\) and \(a_3=1.04\) and for vertical PGA \(a_1=0.485\), \(a_2=1.85\) and \(a_3=1.38\) (\(\sigma\) not given).

• All records from free-field bedrock sites.

TERA Corporation (1980)

• Ground-motion model is: \[\mathrm{PGA}=a \exp(b M)[R+c_1 \exp(c_2 M)]^{-d}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for constrained model \(a=0.0782\), \(b=1.10\), \(c_1=0.343\), \(c_2=0.629\), \(d=1.75\) and \(\sigma=0.457\) (in terms of natural logarithm).

• Similar to Campbell (1981) (see Section 2.36) but different data.

Campbell (1981)

• Ground-motion model is: \[\mathrm{PGA}=a \exp(b M)[R+c_1 \exp(c_2 M)]^{-d}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for unconstrained model \(a=0.0159\), \(b=0.868\), \(c_1=0.0606\), \(c_2=0.700\), \(d=1.09\) and \(\sigma=0.372\) (on natural logarithm) and for constrained model \(a=0.0185\), \(b=1.28\), \(c_1=0.147\), \(c_2=0.732\), \(d=1.75\) and \(\sigma=0.384\) (in terms of natural logarithm).

  Uses this functional form because capable of modelling possible nonlinear distance scaling in near field and because distance at which transition from near field to far field occurs probably proportional to fault rupture zone size.

• Considers six site classifications but does not model:

  1. Recent alluvium: Holocene Age soil with rock \(\geq 10\,\mathrm{m}\) deep, 71 records.

  2. Pleistocene deposits: Pleistocene Age soil with rock \(\geq 10\,\mathrm{m}\) deep, 22 records.

  3. Soft rock: Sedimentary rock, soft volcanics, and soft metasedimentary rock, 14 records.

  4. Hard rock: Crystalline rock, hard volcanics, and hard metasedimentary rock, 9 records.

  5. Shallow soil deposits: Holocene or Pleistocene Age soil \(<10\,\mathrm{m}\) deep overlying soft or hard rock, 17 records. Not used in analysis.

  6. Soft soil deposits: extremely soft or loose Holocene Age soils, e.g. beach sand or recent floodplain, lake, swamp, estuarine, and delta deposits, 1 record. Not used in analysis.

• Notes that data from areas outside western USA may be substantially different than those from western USA due to tectonics and recording practices but far outweighed by important contribution these data can make to understanding of near-source ground motion.

• Notes use of only near-source data has made differences in anelastic attenuation negligible to inherent scatter from other factors.

• Selects data from shallow tectonic plate boundaries generally similar to western N. America, deep subduction events excluded because of differences in travel paths and stress conditions.

• Selects data from instruments with similar dynamic characteristics as those used in USA to avoid bias, therefore excludes data from SMAC accelerographs in Japan.

• Selects data which meet these criteria:

  1. Epicentres known with an accuracy of \(5\,\mathrm{km}\) or less, or accurate estimate of closest distance to fault rupture surface known.

  2. Magnitudes accurate to within \(0.3\) units.

  3. Distances were within \(20\), \(30\), and \(50\,\mathrm{km}\) for magnitudes less than \(4.75\) between \(4.75\) and \(6.25\) and greater than \(6.25\) respectively. Only uses data from earthquakes with magnitude \(\geq 5.0\) because of greatest concern for most design applications.

  4. Hypocentres or rupture zones within \(25\,\mathrm{km}\) of ground surface.

  5. PGA\(\geq 0.2\,\mathrm{m/s^2}\) for one component, accelerographs triggered early enough to capture strong phase of shaking.

  6. Accelerograms either free-field, on abutments of dams or bridges, in lowest basement of buildings, or on ground level of structures without basements. Excluded Pacoima Dam record, from San Fernando (9/2/1971) earthquake due to topographic, high-frequency resonance due to large gradation in wave propagation velocities and amplification due to E-W response of dam.

• Well distributed data, correlation between magnitude and distance only \(6\%\).

• Uses PGA from digitised, unprocessed accelerograms or from original accelerograms because fully processed PGAs are generally smaller due to the \(0.02\,\mathrm{s}\) decimation and frequency band-limited filtering of records.

• Uses mean of two horizontal components because more stable peak acceleration parameter than either single components taken separately or both components taken together.

• Magnitude scale chosen to be generally consistent with \(M_w\). Division point between using \(M_L\) and \(M_s\) varied between \(5.5\) and \(6.5\); finds magnitudes quite insensitive to choice.

• Notes \(r_{rup}\) is a statistically superior distance measure than epicentral or hypocentral and is physically consistent and meaningful definition of distance for earthquakes having extensive rupture zones.

• Does not use all data from San Fernando earthquake to minimize bias due to large number of records.

• Uses seven different weighting schemes, to control influence of well-recorded earthquakes (e.g. San Fernando and Imperial Valley earthquakes). Giving each record or each earthquake equal weight not reasonable representation of data. Uses nine distance dependent bins and weights each record by a relative weighting factor \(1/n_{i,j}\), where \(n_{i,j}\) is total number of recordings from \(i\)th earthquake in \(j\)th interval.

• Finds unconstrained coefficients and all coefficients statistically significant at \(99\%\).

• Finds coefficients with \(d\) constrained to \(1.75\) (representative of far-field attenuation of PGA) and \(c_2=b/d\), which means PGA is independent of magnitude at the fault rupture surface. All coefficients statistically significant at \(99\%\). Notes similarity between two models.

• Plots normalised weighted residuals against distance, magnitude⁸

  and predicted acceleration. Finds that residuals uncorrelated, at \(99\%\), with these variables.

• Normal probability plots, observed distribution of normalised weighted residuals and Kolmogorov-Smirnov test, at \(90\%\), confirms that PGA can be accepted as being lognormally distributed.

• Finds effects of site geology, building size, instrument location and mechanism to be extensively interrelated so selects only records from free-field or small structures.

• Analyses all selected data, find sites of classes E and F significantly higher PGA , at \(90\%\) level, so removes records from E and F.

• Finds differences in PGA from other site categories to be negligible but notes that it cannot be extended to PGV, PGD, spectral ordinates or smaller magnitudes or further distances.

• Distribution with mechanism is: 69 from strike-slip, 40 from reverse, 5 from normal and 2 records from oblique. Finds that reverse fault PGAs are systematically higher, significant at \(90\%\), than those from other fault types although size of bias is due to presence of data from outside N. America.

• Considers soil (A and B) records from small buildings (115 components) and in free-field and those obtained in lowest basement of large buildings (40 components). Finds PGA significantly lower, at \(90\%\) level, in large buildings.

• Finds topographic effects for 13 components used in final analysis (and for 11 components from shallow soil stations) to be significantly higher, at \(90\%\), although states size of it may not be reliable due to small number of records.

• Removes Imperial Valley records and repeats analysis. Finds that saturation of PGA with distance is not strongly dependent on this single set of records. Also repeats analysis constraining \(c_2=0\), i.e. magnitude independent saturation, and also constraining \(c_1=c_2=0\), i.e. no distance saturation, finds variance when no distance saturation is significantly higher, at \(95\%\), than when there is saturation modelled.

• Finds that magnitude saturation effects in modelling near-source behaviour of PGA is important and \(c_2\) is significantly greater than zero at levels of confidence exceeding \(99\%\). Also variance is reduced when \(c_2\neq0\) although not at \(90\%\) or above.

• Repeats analysis using distance to surface projection of fault, finds reduced magnitude saturation but similar magnitude scaling of PGA for larger events.

Chiaruttini and Siro (1981)

• Ground-motion model is: \[\log a =b_0 + b_{AN} X_{AN} + b_{AB} X_{AB} + b_M M_L +b_d \log d\] where \(a\) is in \(\,\mathrm{g}/100\), \(b_0=0.04\), \(b_{AN}=0.24\), \(b_{AB}=0.23\), \(b_M=0.41\) and \(b_d=-0.99\) (\(\sigma\) not given).

• Use three site categories for Friuli records, although note that information is rather superficial:

  • Alluvium with depth \(>20\,\mathrm{m}\), 36 records.

  • Rock-like: hard rock or stiff soil, 24⁹ records.

  • Alluvium-like with depth \(\leq 20\,\mathrm{m}\): includes sites for which thickness of deposit is reported to be very small which accounts for a few metres of weathering of underlying bedrock, 60 records.

  Alpide belt records divided into two categories: rock-like (25 records) and alluvium-like (40 records).

• Use data from free-field instruments or from instruments in basements of small structures and divide data into three regions: those from 1976 Friuli shocks (120 records) \(\Rightarrow X_{AN}=X_{AB}=0\), those from 1972 Ancona swarm (40 records) \(\Rightarrow X_{AN}=1\) & \(X_{AB}=0\) and those from Alpide Belt (Azores to Pakistan excluding those from Friuli and Ancona) (64 records) \(\Rightarrow X_{AN}=0\) & \(X_{AB}=1\). Exclude records with PGA \(<0.15\,\mathrm{m/s^2}\) to avoid possible bias at low acceleration values.

• Assume average focal depth of \(6\,\mathrm{km}\).

• Note some PGA values derived from velocity records which are retained because compatible with other data. No instrument corrections applied to Friuli records because correction does not substantially alter PGA.

• Use \(M_L\) because determined at short distances and allows homogenous determination from lowest values up to saturation at \(M_L=7.0\) and it is determined at frequencies of nearly \(1\,\mathrm{Hz}\), close to accelerographic band.

• Perform regression on PGAs from each of the three regions and each soil types considered within that region.

• Group rock-like (R) and thick alluvium (ThA) records together for Friuli. Find \(b_d\) for Friuli equations derived for thin alluvium-like and rock and thick alluvium not significantly different but \(b_M\) is significantly different, at \(95\%\) level. Repeat analysis using only Tolmezzo records because of large scatter in residuals but decide it is in thA category.

• For Alpide belt equations find \(b_M\) is almost the same for Rl and Al records and the difference in \(b_d\) is less than standard error, thus repeat analysis using a dummy variable \(X_{Al}\) which equals \(0\) for Rl and \(1\) for Al records.

Goto, Kameda, and Sugito (1981)

• Ground-motion model is: \[\log A=b_0+b_1M+b_2\log(\Delta+b_3)\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(b_0=2.305\), \(b_1=0.178\), \(b_2=-0.666\) and \(b_3=30\) (\(\sigma\) not reported¹⁰).

• Use \(N\)-value profiles from standard penetration tests (SPTs) to characterise sites. Use data from alluvial and diluvial sites. Exclude data from rock and very soft soils. Define \(S_n\) as a weighting function for SPT profile to characterise softness of surface layers. Plot residuals from model against \(S_n\) and find correlation. Derive site correction factors for model. Find coefficient of variation decreases after applying correction.

• Use 346 uncorrected components (magnitudes from \(5\) to about \(7.8\) and \(r_{epi}\) from about \(7\) to \(500\,\mathrm{km}\)) to derive preliminary model without site term: \(\bar{A}=b_0 10^{b_1M}/(r_{epi}+30)^{b_2}\). Derive models using different data selections: all data, \(M<6.6\), \(M\geq 6.6\), \(r_{epi}\leq 119\,\mathrm{km}\), \(r_{epi}>119\,\mathrm{km}\), \(M\)-\(r_{epi}\) region where expected PGA (from model using all data) \(\geq 39\,\mathrm{cm/s^2}\) or expected PGA \(<39\,\mathrm{cm/s^2}\) (these selected divide data into two equal halves). Examine scaling of the various models in 3D plots. Based on this analysis, conclude that model depends on \(M\)-\(r_{epi}\) range used for data selection. Because of engineering interest in PGA\(>10\,\mathrm{gal}\) believe model should be derived using \(M\)-\(r_{epi}\) region defined by expected PGA.

• 18 records from 1978 Off Miyagi earthquakes and 6 records from 1978 Izu-oshima-kinkai earthquake.

• Strong correlation between \(M\) and \(r_{epi}\) with almost all data from \(M>7\) being from \(r_{epi}>100\,\mathrm{km}\).

• Try different \(b_3\) values but find influence on coefficient of variation minimal so fix to \(30\,\mathrm{km}\).

• For final model use corrected accelerograms (PGAs generally 10% to 30% higher than uncorrected values). Most data from SMAC instruments.

• Plot residuals w.r.t. \(r_{epi}\) and \(M\) and find no trends.

Joyner and Boore (1981)

• Ground-motion model is: \[\begin{aligned} \log y&=&\alpha +\beta \mathbf{M}-\log r +b r\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(\alpha=-1.02\), \(\beta=0.249\), \(b=-0.00255\), \(h=7.3\) and \(\sigma=0.26\).

• Use two site categories (not all records have category):

  1. Rock: sites described as granite, diorite, gneiss, chert, greywacke, limestone, sandstone or siltstone and sites with soil material less than \(4\) to \(5\,\mathrm{m}\) thick overlying rock, 29 records. Indicate caution in applying equations for \(\mathbf{M}>6.0\) due to limited records.

  2. Soil: sites described as alluvium, sand, gravel, clay, silt, mud, fill or glacial outwash except where soil less than \(4\) to \(5\,\mathrm{m}\) thick, 96 records.

• Restrict data to western North American shallow earthquakes, depth less than \(20\,\mathrm{km}\), with \(\mathbf{M}>5.0\). Most records from earthquakes with magnitudes less than \(6.6\).

• Exclude records from base of buildings three or more storeys high and from abutments of dams.

• Exclude records associated with distances which had an uncertainty greater than \(5\,\mathrm{km}\).

• Exclude records from distances greater than or equal to the shortest distance to an instrument which did not trigger.

• Six earthquakes recorded at only one station so not included in second stage regression.

• Include quadratic dependence term, \(\gamma \mathbf{M}^2\), but not significant at \(90\%\) level so omitted.

• Include site term, \(cS\), but not significant so omitted.

• Examine residuals against distance for difference magnitude ranges, no obvious differences in trends are apparent among the different magnitude classes.

• Consider a magnitude dependent \(h=h_1 \exp (h_2 [\mathbf{M}-6.0])\) but reduction in variance not significant. Also prefer magnitude independent \(h\) because requires fewer parameters.

• Examine effect of removing records from different earthquakes from data.

• Examine effect of different \(h\) on residuals and \(b\). Note coupling between \(h\) and \(b\).

• Note coincidence of anelastic coefficient, \(b\), and measured \(Q\) values. Also note similarity between \(h\) and proportions of depth of seismogenic zone in California.

Bolt and Abrahamson (1982)

• Ground-motion model is: \[y=a\{(x+d)^2+1\}^c \mathrm{e}^{-b (x+d)}\] where \(y\) is in \(\,\mathrm{g}\), for \(5\leq \mathbf{M}<6\) \(a=1.2\), \(b=0.066\), \(c=0.033\), \(d=23\) and standard error for one observation of \(0.06\,\mathrm{g}\), for \(6 \leq \mathbf{M}<7\) \(a=1.2\), \(b=0.044\), \(c=0.042\), \(d=25\) and standard error for one observation of \(0.10\,\mathrm{g}\), for \(7 \leq \mathbf{M}\leq 7.7\) \(a=0.24\) \(b=0.022\), \(c=0.10\), \(d=15\) and standard error for one observation of \(0.05\,\mathrm{g}\) and for \(6\leq \mathbf{M}\leq 7.7\) \(a=1.6\), \(b=0.026\), \(c=-0.19\), \(d=8.5\) and standard error for one observation of \(0.09\,\mathrm{g}\).

• Use data of Joyner and Boore (1981).

• Form of equation chosen to satisfy plausible physical assumptions but near-field behaviour is not determined from overwhelming contributions of far-field data.

• Apply nonlinear regression on \(y\) not on \(\log y\) to give more weight to near-field values.

• Split data into four magnitude dependent groups: \(5 \leq \mathbf{M} < 6\), \(6 \leq \mathbf{M} < 7\), \(7 \leq \mathbf{M} \leq 7.7\) and \(6 \leq \mathbf{M} \leq 7.7\).

• Use form of equation and regression technique of Joyner and Boore (1981), after removing 25 points from closer than \(8\,\mathrm{km}\) and find very similar coefficients to Joyner and Boore (1981). Conclude from this experiment and their derived coefficients for the four magnitude groups that using their form of equation predicted near-field accelerations are not governed by far-field data.

• Find no evidence of systematic increase in PGA near the source as a function of magnitude and that the large scatter prevents attaching significance to differences in near-field PGA which are predicted using their attenuation relations for different magnitude ranges.

Joyner and Boore (1982b) & Joyner and Boore (1988)

• Ground-motion model is: \[\begin{aligned} \log y&=&\alpha+\beta(M-6)+\gamma(M-6)^2-p \log r+b r+c S\\ r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(\beta=0.23\), \(\gamma=0\), \(p=1\), \(b=-0.0027\), \(c=0\), \(h=8.0\) and \(\sigma=0.28\) and for randomly oriented component \(\alpha=0.43\) and for larger component \(\alpha=0.49\).

• Use same data and method as Joyner and Boore (1981), see Section 2.39, for PGA.

• Use data from shallow earthquakes, defined as those for which fault rupture lies mainly above a depth of \(20\,\mathrm{km}\).

PML (1982)

• Ground-motion model is: \[\ln (a)=C_1+C_2 M+C_3 \ln [R+C_4 \exp(C_5 M)]\] where \(a\) is in \(\,\mathrm{g}\), \(C_1=-1.17\), \(C_2=0.587\), \(C_3=-1.26\), \(C_4=2.13\), \(C_5=0.25\) and \(\sigma=0.543\).

• Use data from Italy (6 records, 6 earthquakes), USA (18 records, 8 earthquakes), Greece (13 records, 9 earthquakes), Iran (3 records, 3 earthquakes), Pakistan (3 records, 1 earthquake), Yugoslavia (3 records, 1 earthquake), USSR (1 record, 1 earthquake), Nicaragua (1 record, 1 earthquake), India (1 record, 1 earthquake) and Atlantic Ocean (1 record, 1 earthquake).

• Develop for use in .

Schenk (1982)

• Ground-motion model is: \[\log A_{\mathrm{mean}}=a M -b\log R +c\] where \(A_{\mathrm{mean}}\) is in \(\,\mathrm{cm/s^2}\), \(a=1.1143\), \(b=1.576\) and \(c=2.371\) (\(\sigma\) not given).

• Fits equation by eye because least squares method is often strictly dependent on marginal observations, particularly for little pronounced dependence.

Brillinger and Preisler (1984)

• Ground-motion model is: \[A^{1/3}=a_1+a_2 M+a_3 \ln (d^2+a_4^2)\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=0.432 (0.072)\), \(a_2=0.110 (0.012)\), \(a_3=-0.0947 (0.0101)\), \(a_4=6.35 (3.24)\), \(\sigma_1=0.0351 (0.0096)\) (inter-event) and \(\sigma_2=0.0759 (0.0042)\) (intra-event), where numbers in brackets are the standard errors of the coefficients.

• Use exploratory data analysis (EDA) and alternating conditional expectations (ACE) techniques.

• Firstly sought to determine functions \(\theta(A)\), \(\phi (M)\) and \(\psi (d)\) so that \(\theta (A) \doteq \phi(M)+\psi(d)\), i.e. an approximately additive relationship. Prefer additivity because of linearity, ease of interpolation and interpretation and departures from fit are more easily detected.

• Use ACE procedure to find model. For set of data, with response \(y_i\) and predictors \(w_i\) and \(x_i\) find functions to minimize: \(\sum_{i=1}^n [\theta(y_i)-\phi(w_i)-\psi(x_i)]^2\) subject to \(\sum \phi (w_i)=0\), \(\sum \psi (x_i)=0\), \(\sum \theta (y_i)=0\) and \(\sum \theta (y_i)^2=n\). Search amongst unrestricted curves or unrestricted monotonic curves. Use EDA to select specific functional forms from the estimates of \(\theta\), \(\phi\) and \(\psi\) at each data point.

• Do not use weighting because does not seem reasonable from statistical or seismological points of view.

• Do not want any individual earthquake, e.g. one with many records, overly influencing results.

• Note that because each earthquake has its own source characteristics its records are intercorrelated. Therefore use ‘random effects model’ which accounts for perculiarities of individual earthquakes and correlation between records from same event.

• On physical grounds, restrict \(\theta\), \(\phi\) and \(\psi\) to be monotonic and find optimal transformation of magnitude is approximately linear, optimal transformation of distance is logarithmic and cube root is optimal for acceleration transformation.

• Note that need correlations between coefficients, which are provided, to attach uncertainties to estimated PGAs.

• Provide method of linearization to give \(95\%\) confidence interval for acceleration estimates.

• Also provide a graphical procedure for estimating accelerations that does not rely on an assumed functional form.

• Examine residual plots (not shown) and found a candidate for an outlying observation (the record from the Hollister 1974 earthquake of \(0.011\,\mathrm{g}\) at \(17.0\,\mathrm{km}\)).

• Find that assumption of normality after transformation seems reasonable.

Campbell (1984) & K.W. Campbell (1988) reported in Joyner and Boore (1988)

• Ground-motion model is: \[\begin{aligned} \ln y&=&a+b M+d \ln [r+h_1 \exp (h_2 M)]+s\\ \mbox{where }s&=&e_1 K_1+e_2 K_2+e_3 K_3+e_4 K_4+e_5 K_5+e_6(K_4+K_5)\tanh(e_7 r)\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(a=-2.817\), \(b=0.702\), \(d=-1.20\), \(h_1=0.0921\), \(h_2=0.584\), \(e_1=0.32\), \(e_2=0.52\), \(e_3=0.41\), \(e_4=-0.85\), \(e_5=-1.14\), \(e_6=0.87\), \(e_7=0.068\) and \(\sigma=0.30\).¹¹

• Uses two site categories:

  1. Soils \(\leq 10\,\mathrm{m}\) deep.

  2. Other.

• Uses three embedment categories:

  1. Basements of buildings \(3\)–\(9\) storeys.

  2. Basements of buildings \(\geq 10\) storeys.

  3. Other.

• Selects data using these criteria:

  1. Largest horizontal component of peak acceleration was \(\geq 0.02\,\mathrm{g}\) [\(\geq 0.2\,\mathrm{m/s^2}\)].

  2. Accelerograph triggered early enough to record strongest phase of shaking.

  3. Magnitude of earthquake was \(\geq 5.0\).

  4. Closest distance to seismogenic rupture was \(<30\) or \(<50\,\mathrm{km}\), depending on whether magnitude of earthquake was \(<6.25\) or \(>6.25\).

  5. Shallowest extent of seismogenic rupture was \(\leq 25\,\mathrm{km}\).

  6. Recording site located on unconsolidated deposits.

• Excludes records from abutments or toes of dams.

• Derives two equations: unconstrained (coefficients given above) and constrained which includes a anelastic decay term \(kr\) which allows equation to be used for predictions outside near-source zone (assumes \(k=-0.0059\) for regression, a value appropriate for region of interest should be chosen).

• Uses two source mechanism categories:

  1. Strike-slip.

  2. Reverse.

• Uses two directivity categories:

  1. Rupture toward site.

  2. Other.

Joyner and Fumal (1984), Joyner and Fumal (1985) & Joyner and Boore (1988)

• Ground-motion model is: \[\begin{aligned} \log y&=&c_0+c_1 (\mathbf{M}-6) +c_2(\mathbf{M}-6)^2 +c_3 \log r +c_4 r +S\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\\ \mbox{and: }S&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&\mbox{rock site}\\ c_6 \log \frac{V}{V_0}&\mbox{soil site}\\ \end{array} \right.\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), coefficients \(c_0\) to \(c_4\), \(h\) and \(\sigma\) are from Joyner and Boore (1981) and \(c_6\) and \(V_0\) are not significant at \(90\%\) level so do not report them.

• Use data of Joyner and Boore (1981).

• Continuous site classification for soil sites in terms of shear-wave velocity, \(V\), to depth of one quarter wavelength of waves of period of concern. \(V\) measured down to depths of at least \(30\,\mathrm{m}\) and then extrapolated using geological data. \(V\) known for 33 stations.

• Soil amplification factor based on energy conservation along ray tubes, which is a body wave argument and may not hold for long periods for which surface waves could be important. Does not predict resonance effects.

• Regress residuals, \(R_{ij}\), w.r.t. motion predicted for rock sites on \(\log R_{ij}=P_i + c_6 V_j\), where \(j\) corresponds to \(j\)th station and \(i\) to \(i\)th earthquake. Decouples site effects variation from earthquake-to-earthquake variation. Find unique intercept by requiring average site effect term calculated using shear-wave velocity to be same as that calculated using rock/soil classification.

• No significant, at \(90\%\), correlation between residuals and \(V\) for PGA.

• Repeat regression on residuals using \(V\) and depth to underlying rock (defined as either shear-wave velocity \(>750\,\mathrm{m/s}\) or \(>1500\,\mathrm{m/s}\)). Find no correlation.

Kawashima, Aizawa, and Takahashi (1984) & Kawashima, Aizawa, and Takahashi (1986)

• Ground-motion model is: \[X(M,\Delta,\mathrm{GC}_i)=a(\mathrm{GC}_i) 10^{b(\mathrm{GC}_i) M} (\Delta+30)^{c}\] where \(X(M,\Delta,\mathrm{GC}_i)\) is in \(\,\mathrm{gal}\), \(c=-1.218\), for group 1 sites \(a(\mathrm{GC}_1)=987.4\), \(b(\mathrm{GC}_1)=0.216\) and \(\sigma=0.216\), for group 2 sites \(a(\mathrm{GC}_2)=232.5\), \(b(\mathrm{GC}_2)=0.313\) and \(\sigma=0.224\) and for group 3 sites \(a(\mathrm{GC}_3)=403.8\), \(b(\mathrm{GC}_3)=0.265\) and \(\sigma=0.197\).

• Use three site categories:

  1. Tertiary or older rock (defined as bedrock) or diluvium with \(H<10\,\mathrm{m}\) or fundamental period \(T_G<0.2\,\mathrm{s}\).

  2. Diluvium with \(H\geq 10\,\mathrm{m}\), alluvium with \(H<10\,\mathrm{m}\) or alluvium with \(H<25\,\mathrm{m}\) including soft layer with thickness \(<5\,\mathrm{m}\) or fundamental period \(0.2<T_G<0.6\,\mathrm{s}\).

  3. Other than above, normally soft alluvium or reclaimed land.

• Only includes free-field records with \(M_{\mathrm{JMA}}\geq 5.0\) and focal depths \(D_p<60\,\mathrm{km}\). Excludes records from structures with first floor or basement.

• Records instrument corrected, because Japanese instruments substantially suppress high frequencies, considering accuracy of digitization for frequencies between \(\frac{1}{3}\) and \(12\,\mathrm{Hz}\).

• Note that \(M_{\mathrm{JMA}}\) and \(\Delta\) not necessarily most suitable parameters to represent magnitude and distance but only ones for all records in set.

• Note lack of near-field data for large magnitude earthquakes, approximately \(\frac{3}{4}\) of records from \(M_{\mathrm{JMA}}<7.0\).

• Use \(30\,\mathrm{km}\) in distance dependence term because focal depth of earthquakes with magnitudes between \(7.5\) and \(8.0\) are between \(30\) and \(100\,\mathrm{km}\) so \(30\) is approximately half the fault length.

• Try equation: \(\log X= f_1+f_2M+f_3 \log(\Delta+30)+f_4 D_p +f_5 M \log (\Delta+30)+f_6 M D_p+f_7 D_p \log (\Delta+30)+f_8 M^2+f_9\{\log(\Delta+30)\}^2+f_{10} D_p^2\) where \(f_i\) are coefficients to be found considering each soil category separately. Apply multiple regression analysis to 36 combinations of retained coefficients, \(f_i\), and compute multiple correlation coefficient, \(R\), and adjusted multiple correlation coefficient, \(R^*\). Find that inclusion of more than three coefficients does not give significant increase in \(R^*\), and can lead to unrealistic results. Conclude due to insufficient data.

• Consider \(a\), \(b\) and \(c\) dependent and independent of soil type and examine correlation coefficient, \(R\), and adjusted correlation coefficient, \(R^*\). Find that \(c\) is not strongly dependent on soil type.

• Find match between normal distribution and histograms of residuals.

McCann Jr. and Echezwia (1984)

• Four Ground-motion models: \[\begin{aligned} \log_{10} Y&=&a+b M+d \log_{10} [(R^2+h^2)^{1/2}] \mbox{\qquad Model I}\\ \log_{10} Y&=&a+b M+d \log_{10} [R+c_1 \exp (c_2 M)] \hfill \mbox{\qquad Model II}\\ \log_{10} Y&=&a+b M+d \log_{10} \left[ \frac{c_1}{R^2} + \frac{c_2}{R} \right] +e R \mbox{\qquad Model III} \\ \log_{10} Y&=&a+b M+d \log_{10} [R+25] \mbox{\qquad Model IV}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), for model I \(a=-1.320\), \(b=0.262\), \(d=-0.913\), \(h=3.852\) and \(\sigma=0.158\), for model II \(a=-1.115\), \(b=0.341\), \(c_1=1.000\), \(c_2=0.333\), \(d=-1.270\) and \(\sigma=0.154\), for model III \(a=-2.000\), \(b=0.270\), \(c_1=0.968\), \(c_2=0.312\), \(d=0.160\), \(e=-0.0105\) and \(\sigma=0.175\) and for model IV \(a=1.009\), \(b=0.222\), \(d=-1.915\) and \(\sigma=0.174\).

• Note \(25\) in Model IV should not be assumed but should be found by regression.

• Note tectonics and travel paths may be different between N. American and foreign records but consider additional information in near field more relevant.

• Selection procedure composite of Campbell (1981) and Joyner and Boore (1981). Exclude data from buildings with more than two storeys.

• Weighted least squares, based on distance, applied to control influence of well recorded events (such as San Fernando and Imperial Valley). Similar to Campbell (1981)

• Test assumption that logarithm of residuals are normally distributed. Cannot disprove assumption.

• Variability between models not more than \(\pm 20\%\) at distances \(>10\,\mathrm{km}\) but for distances \(<1\,\mathrm{km}\) up to \(\pm 50\%\).

Schenk (1984)

• Ground-motion model is: \[\log A_{\mathrm{mean}}=aM-b \log R+c\] where \(A_{\mathrm{mean}}\) is in \(\,\mathrm{cm/s^2}\), \(a=0.37\), \(b=1.58\) and \(c=2.35\) (\(\sigma\) not given).

• Considers two site conditions but does not model:

  1. Solid

  2. Soft

• Fits equation by eye.

• States applicable approximately for: \(R_{\mathrm{lower}} \leq R \leq R_{\mathrm{upper}}\) where \(\log R_{\mathrm{lower}} \doteq 0.1 M+0.5\) and \(\log R_{\mathrm{upper}} \doteq 0.35 M +0.4\), due to distribution of data.

• Notes great variability in recorded ground motions up to \(R=30\,\mathrm{km}\) due to great influence of different site conditions.

• Notes for \(M\leq 4\) source can be assumed spherical but for \(M>4\) elongated (extended) shape of focus should be taken into account.

Xu, Shen, and Hong (1984)

• Ground-motion model is: \[\mathrm{PGA}=a_1 \exp (a_2 M) (R+a_3)^{-a_4}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(a_1=0.1548\), \(a_2=0.5442\), \(a_3=8\) and \(a_4=1.002\) (\(\sigma\) not given).

• All records from aftershocks of 1975 Haicheng earthquake and from 1976 Tangshan earthquake and aftershocks.

• Most records from earthquakes with magnitude less than \(5.8\) and from distances \(<30\,\mathrm{km}\).

• Exclude records with PGA \(<0.5\,\mathrm{m/s^2}\) to avoid too much contribution from far field.

• Due to small number of records simple regression technique justified.

• States valid for \(4\leq M\leq6.5\) and \(R\leq 100\,\mathrm{km}\).

• Also use 158 records from western N. America to see whether significantly different than N. Chinese data. Derive equations using both western N. American and N. Chinese data and just western N. American data and find that predicted PGAs are similar, within uncertainty.

• Insufficient data to find physically realistic anelastic term.

Brillinger and Preisler (1985)

• Ground-motion model is: \[\begin{aligned} \log A&=&a_1+a_2 M-\log r+a_3 r\\ \mbox{where} \quad r^2&=&d^2+a_4^2\end{aligned}\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=-1.229 (0.196)\), \(a_2=0.277 (0.034)\), \(a_3=-0.00231 (0.00062)\), \(a_4=6.650 (2.612)\), \(\sigma_1=0.1223 (0.0305)\) (inter-event) and \(\sigma=0.2284 (0.0127)\) (intra-event), where numbers in brackets are the standard errors of the coefficients.

• Provide algorithm for random effects regression.

• Note that the functional form adopted in Brillinger and Preisler (1984) is strictly empirical and hence repeat analysis using functional form of Joyner and Boore (1981), which is based on physical reasoning.

• Note that need correlations between coefficients, which are provided, to attach uncertainties to estimated PGAs.

Kawashima, Aizawa, and Takahashi (1985)

• Use very similar data to Kawashima, Aizawa, and Takahashi (1984); do not use some records because missing due to recording and digitizing processes. Use equation and method (although do not check all 36 combinations of forms of equation) used by Kawashima, Aizawa, and Takahashi (1984), see section 2.47.

• \(X(M,\Delta,\mathrm{GC}_i)\) is in \(\,\mathrm{gal}\). Coefficients are: \(c=-1.190\) and for ground group 1 \(a=117.0\) and \(b=0.268\) and for ground group 2 \(a=88.19\) and \(b=0.297\) and for group ground 3 \(a=13.49\) and \(b=0.402\) with \(\sigma=0.253\).

Makropoulos and Burton (1985) & Makropoulos (1978)

• Ground-motion model is: \[A=b_1 \exp(b_2 M) (R+h)^{-b_3}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(b_1=2164\), \(b_2=0.7\pm 0.03\), \(h=20\) and \(b_3=-1.8\pm 0.02\) (\(\sigma\) is not reported).

• Derived by averaging (at \(M=7.5\)) eight previous models: Donovan (1973), Orphal and Lahoud (1974), Esteva (1974), Katayama (1974) and Trifunac (1976a).

• Check predictions against eight Greek accelerograms and find agreement.

K.-Z. Peng, Wu, and Song (1985)

• Ground-motion model is: \[\log_{10} a =A+B M+C \log_{10} R +DR\] where \(a\) is in \(\,\mathrm{cm/s^2}\), for N.E. China \(A=-0.474\), \(B=0.613\), \(C=-0.873\) and \(D=-0.00206\) (\(\sigma\) not given) and for S.W. China \(A=0.437\), \(B=0.454\), \(C=-0.739\) and \(D=-0.00279\) (\(\sigma\) not given).

• Consider two site conditions for NE records but do not model:

  1. Rock: 28 records.

  2. Soil: 45 records.

• Consider all records to be free-field.

• Note that Chinese surface-wave magnitude, \(M\), is different than \(M_s\) and may differ by \(0.5\) or more. Use \(m_b\) or \(M_s\) and find larger residuals.

• Most records from \(M\leq 5.8\).

• Note isoseismals are not elongated for these earthquakes so use of another distance measure will not change results by much.

• Also derives equation for SW China (\(3.7 \leq M \leq 7.2\), \(6.0 \leq R \leq 428.0\,\mathrm{km}\) all but one record \(\leq 106.0\,\mathrm{km}\) , 36 records from 23 earthquakes) and note difference between results from NE China although use less data.

• Note that some scatter may be due to radiation pattern.

• Note that data is from limited distance range so need more data to confirm results.

K. Peng et al. (1985)

• Ground-motion model is: \[\begin{aligned} \log A_m&=&a_1+a_2M-\log R-a_3 R\\ R&=&\sqrt{d^2+h^2}\end{aligned}\] where \(A_m\) is \(\,\mathrm{g}\), \(a_1=-1.49\), \(a_2=0.31\), \(a_3=0.0248\), \(h=9.4\,\mathrm{km}\) and \(\sigma=0.32\) (for horizontal components) and \(a_1=-1.92\), \(a_2=0.29\), \(a_3=0.0146\), \(h=6.7\,\mathrm{km}\) and \(\sigma=0.36\) (for vertical components).

• Data from experimental strong-motion array consisting of 12 Kinemetrics PDR-1 instruments deployed in the epicentral area of the \(M_s=7.8\) Tangshan earthquake of 28th July 1976. Provide details of site geology at each station; most stations are on soil.

• Records from earthquakes recorded by only one station were excluded from analysis.

• Note that equations are preliminary and more refined equations await further studies of magnitudes and distances used in analysis.

• Note that high anelastic attenuation coefficient may be due to biases introduced by the distribution in magnitude-distance space and also because of errors in magnitude and distances used.

PML (1985)

• Ground-motion model is: \[\ln (a)=C_1+C_2 M+C_3 \ln [R+C_4 \exp(C_5 M)]+C_6 F\] where \(a\) is in \(\,\mathrm{g}\), \(C_1=-0.855\), \(C_2=0.46\), \(C_3=-1.27\), \(C_4=0.73\), \(C_5=0.35\), \(C_6=0.22\) and \(\sigma=0.49\).

• Use data from Italy (47 records, 9 earthquakes), USA (128 records, 18 earthquakes), Greece (11 records, 8 earthquakes), Iran (2 records, 2 earthquakes), Yugoslavia (7 records, 2 earthquake), Nicaragua (1 record, 1 earthquake), New Zealand (3 records, 3 earthquakes), China (2 records, 2 earthquakes) and Canada (2 records, 1 earthquake).

• Develop for use in .

• Select earthquakes with \(M_s<7\) and \(R\leq 40\,\mathrm{km}\).

• Focal depths \(<40\,\mathrm{km}\).

• Use two source mechanism categories (40 records have no source mechanism given):

  1. Strike-slip and normal, 85 records.

  2. Thrust, 78 records.

• Also derive equation not considering source mechanism, i.e. \(C_6=0\).

McCue (1986)

• Ground-motion model is: \[A=a_1 (\mathrm{e}^{a_2 M_L})(d_h)^{a_3}\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=0.00205\), \(a_2=1.72\) and \(a_3=-1.58\) (\(\sigma\) not given).

C.B. Crouse (1987) reported in Joyner and Boore (1988)

• Ground-motion model is: \[\ln y=a+bM_s+cM_s^2+d \ln (r+1)+kr\] where \(y\) is in \(\,\mathrm{gal}\), \(a=2.48456\), \(b=0.73377\), \(c=-0.01509\), \(d=-0.50558\), \(k=-0.00935\) and \(\sigma=0.58082\).

• Records from deep soil sites (generally greater than \(60\,\mathrm{m}\) in thickness).

• Data from shallow crustal earthquakes.

Krinitzsky, Chang, and Nuttli (1987) & Krinitzsky, Chang, and Nuttli (1988)

• Ground-motion model is (for shallow earthquakes): \[\log A=a_1+a_2M-\log r+a_3 r\] where A is in \(\,\mathrm{cm/s^2}\), \(a_1=1.23\) (for hard sites), \(a_1=1.41\) (for soft sites), \(a_2=0.385\) and \(a_3=-0.00255\) (\(\sigma\) is not given).

  Ground-motion model is (for subduction zone earthquakes): \[\log A=b_1+b_2 M-\log \sqrt{r^2+100^2}+b_3 r\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(b_1=2.08\) (for hard sites), \(b_1=2.32\) (for soft sites), \(b_2=0.35\) and \(b_3=-0.0025\) (\(\sigma\) is not given).

• Use four site categories:

  1. Rock

  2. Stiff soil

  3. Deep cohesionless soil (\(\geq 16\,\mathrm{m}\))

  4. Soft to medium stiff clay (\(\geq 16\,\mathrm{m}\))

  Categories 1 and 2 are combined into a hard (H) class and 3 and 4 are combined into a soft (S) class. This boundary established using field evidence at a shear-wave velocity of \(400\,\mathrm{m/s}\) and at an SPT N count of 60.

• Use data from ground floors and basements of small or low structures (under 3 stories) because believe that small structures have little effect on recorded ground motions.

• Separate earthquakes into shallow (\(h\leq 19\,\mathrm{km}\)) and subduction (\(h\geq 20\,\mathrm{km}\)) because noted that ground motions have different characteristics.

• Use epicentral distance for Japanese data because practical means of representing deep subduction earthquakes with distant and imprecise fault locations.

• Do not use rupture distance or distance to surface projection of rupture because believe unlikely that stress drop and peak motions will occur with equal strength along the fault length and also because for most records fault locations are not reliably determinable.

• Note that there is a paucity of data but believe that the few high peak values observed (e.g. Pacoima Dam and Morgan Hill) cannot be dismissed without the possibility that interpretations will be affected dangerously.

• For subduction equations, use records from Japanese SMAC instruments that have not been instrument corrected, even though SMAC instruments show reduced sensitivity above \(10\,\mathrm{Hz}\), because ground motions \(>10\,\mathrm{Hz}\) are not significant in subduction earthquakes. Do not use records from SMAC instruments for shallow earthquakes because high frequency motions may be significant.

• Examine differences between ground motions in extensional (strike-slip and normal faulting) and compressional (reverse) regimes for shallow earthquakes but do not model. Find that the extensional ground motions seem to be higher than compressional motions, which suggest is because rupture propagation comes closer to ground surface in extensional faults than in compressional faults.

• Group records into \(1\,M\) unit intervals and plot ground motions against distance. When data is numerous enough the data points are encompassed in boxes (either one, two or three) that have a range equal to the distribution of data. The positions of the calculated values within the boxes were used as guides for shaping appropriate curves. Initially curves developed for \(M=6.5\) were there is most data and then these were extended to smaller and larger magnitudes.

Sabetta and Pugliese (1987)

• Ground-motion model is: \[\log y=a+bM-\log (R^2+h^2)^{1/2}+eS\] where \(y\) is in \(\,\mathrm{g}\) and for distance to surface projection of fault \(a=-1.562\), \(b=0.306\), \(e=0.169\), \(h=5.8\) and \(\sigma=0.173\).

• Use two site categories:

  1. Stiff and deep soil: limestone, sandstone, siltstone, marl, shale and conglomerates (\(V_s>800\,\mathrm{m/s}\)) or depth of soil, \(H\), \(>20\,\mathrm{m}\), 74 records.

  2. Shallow soil: depth of soil, \(H\), \(5\leq H\leq 20\,\mathrm{m}\), 21 records.

• Select records which satisfy these criteria:

  1. Reliable identification of the triggering earthquake.

  2. Magnitude greater than \(4.5\) recorded by at least two stations.

  3. Epicentres determined with accuracy of \(5\,\mathrm{km}\) or less.

  4. Magnitudes accurate to within \(0.3\) units.

  5. Accelerograms from free-field. Most are from small electric transformer cabins, 4 from one- or two-storey buildings with basements and 5 from near abutments of dams.

• Depths between \(5.0\) and \(16.0\,\mathrm{km}\) with mean \(8.5\,\mathrm{km}\).

• Focal mechanisms are: normal and oblique (7 earthquakes, 48 records), thrust (9 earthquakes, 43 records) and strike-slip (1 earthquake, 4 records).

• Notes lack of records at short distances from large earthquakes.

• Records baseline-, instrument-corrected and filtered with cutoff frequencies determined by visual inspection in order to maximise signal to noise ratio within band. Cutoff frequencies ranged from \(0.2\) to \(0.4\,\mathrm{Hz}\) and from \(25\) to \(35\,\mathrm{Hz}\). This correction routine thought to provide reliable estimates of PGA so uncorrected PGA do not need to be used.

• For well separated multiple shocks, to which magnitude and focal parameters refer, use only first shock.

• Magnitude scale assures a linear relationship between logarithm of PGA and magnitude and avoids saturation effects of \(M_L\).

• Distance to surface projection of fault rupture thought to be a more physically consistent definition of distance for earthquakes having extensive rupture zones and is easier to predict for future earthquakes. Also reduces correlation between magnitude and distance.

• Use Exploratory Data Analysis using the ACE procedure to find transformation functions of distance, magnitude and PGA.

• Include anelastic attenuation term but it is positive and not significant.

• Include magnitude dependent \(h\) equal to \(h_1\exp(h_2M)\) but find \(h_2\) not significantly different than zero. Note distribution of data makes test not definitive.

• Find geometric attenuation coefficient, \(c\), is close to \(-1\) and highly correlated with \(h\) so constrain to \(-1\) so less coefficients to estimate.

• Consider deep soil sites as separate category but find difference between them and stiff sites is not significant.

• Also use two-stage method but coefficients and variance did not change significantly with respect to those obtained using one-stage method, due to uniform distribution of recordings among earthquakes.

• Find no significant trends in residuals, at \(99\%\) level and also no support for magnitude dependent shape for attenuation curves.

• Exclude records from different seismotectonic and geological regions and repeat analysis. Find that predicted PGA are similar.

• Plot residuals from records at distances \(15\,\mathrm{km}\) or less against magnitude; find no support for magnitude dependence of residuals.

• Note some records are affected by strong azimuthal effects, but do not model them because they require more coefficients to be estimated, direction of azimuthal effect different from region to region and azimuthal effects have not been used in other relationships.

K. Sadigh (1987) reported in Joyner and Boore (1988)

• Ground-motion model is: \[\ln y=a+b \mathbf{M}+c_1(8.5-\mathbf{M})^{c_2}+d\ln[r+h_1 \exp(h_2 \mathbf{M})]\] where \(y\) is in \(\,\mathrm{g}\). For strike-slip earthquakes: \(b=1.1\), \(c_1=0\), \(c_2=2.5\), for PGA at soil sites \(a=-2.611\) and \(d=-1.75\), for \(\mathbf{M}<6.5\) \(h_1=0.8217\), \(h_2=0.4814\) and for \(\mathbf{M}\geq 6.5\) \(h_1=0.3157\) and \(h_2=0.6286\), for PGA at rock sites \(a=-1.406\) and \(d=-2.05\), for \(\mathbf{M}<6.5\) \(h_1=1.353\) and \(h_2=0.406\) and for \(\mathbf{M}\geq 6.5\) \(h_1=0.579\) and \(h_2=0.537\). For reverse-slip increase predicted values by \(20\%\). For \(\mathbf{M}<6.5\) \(\sigma=1.26-0.14\mathbf{M}\) and for \(\mathbf{M} \geq 6.5\) \(\sigma=0.35\).

• Uses two site categories:

  1. Soil

  2. Rock

• Use two source mechanism categories:

  1. Strike-slip

  2. Reverse-slip

• Supplement data with significant recordings of earthquakes with focal depths \(<20\,\mathrm{km}\) from other parts of world.

• Different equations for \(\mathbf{M}<6.5\) and \(\mathbf{M}\geq 6.5\).

Singh et al. (1987)

• Ground-motion model is: \[\log y_{\mathrm{max}}=\alpha M_s -c \log R +\beta\] where \(y_{\mathrm{max}}\) is in \(\,\mathrm{cm/s^2}\), \(\alpha=0.429\), \(c=2.976\), \(\beta=5.396\) and \(\sigma=0.15\).

  More complicated functional form unwarranted due to limited distance range.

• Depths between \(15\) and \(20\,\mathrm{km}\).

• Only use data from a single firm site (Ciudad Universitaria), on a surface layer of lava flow or volcanic tuff.

• Only records from coastal earthquakes.

• Residuals plotted against distance, no trends seen.

• Give amplification factor for lake bed sites (\(25\) to \(80\,\mathrm{m}\) deposit of highly compressible, high water content clay underlain by resistant sands), but note based on only a few sites so not likely to be representative of entire lake bed.

Algermissen, Hansen, and Thenhaus (1988)

• Ground-motion model is: \[\ln (A)=a_1+a_2M_s+a_3 \ln(R)+a_4R\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=-1.987\), \(a_2=0.604\), \(a_3=-0.9082\), \(a_4=-0.00385\) and \(\sigma=0.68\).

Annaka and Nozawa (1988)

• Ground-motion model is: \[\log A =C_m M+C_h H -C_d \log (R+A\exp BM )+C_o\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(A\) and \(B\) so PGA becomes independent of magnitude at fault rupture, \(H\) is depth of point on fault plane when \(R\) becomes closest distance to fault plane, \(C_m=0.627\), \(C_h=0.00671\), \(C_d=2.212\), \(C_o=1.711\) and \(\sigma=0.211\).

• Focal depths \(<100\,\mathrm{km}\).

• Convert records from sites with \(V_s<300\,\mathrm{m/s}\) into records from sites with \(V_s>300\,\mathrm{m/s}\) using 1-D wave propagation theory.

• Introduce term \(C_h H\) because it raises multiple correlation coefficient for PGA.

• Note equations apply for site where \(300 \leq V_s \leq 600\,\mathrm{m/s}\).

Fukushima, Tanaka, and Kataoka (1988) & Fukushima and Tanaka (1990)

• Ground-motion model is: \[\log A =a M -\log (R+c 10^{a M}) -b R +d\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a=0.41\), \(b=0.0034\), \(c=0.032\), \(d=1.30\) and \(\sigma=0.21\).

• Use four site categories for some Japanese stations (302 Japanese records not classified):

  1. Rock: 41 records

  2. Hard: ground above Tertiary period or thickness of diluvial deposit above bedrock \(<10\,\mathrm{m}\), 44 records.

  3. Medium: thickness of diluvial deposit above bedrock \(>10\,\mathrm{m}\), or thickness of alluvial deposit above bedrock \(<10\,\mathrm{m}\), or thickness of alluvial deposit \(<25\,\mathrm{m}\) and thickness of soft deposit is \(<5\,\mathrm{m}\), 66 records.

  4. Soft soil: other soft ground such as reclaimed land, 33 records.

• Use 1100 mean PGA values from 43 Japanese earthquakes (\(6.0\leq M_{\mathrm{JMA}} \leq 7.9\), focal depths \(\leq 30\,\mathrm{km}\)) recorded at many stations to investigate one and two-stage methods. Fits \(\log A=c-b\log X\) (where \(X\) is hypocentral distance) for each earthquake and computes mean of \(b\), \(\bar{b}\). Also fits \(\log A =a M -b^{*} \log X +c\) using one-stage method. Find that \(\bar{b}>b^{*}\) and shows that this is because magnitude and distance are strongly correlated (\(0.53\)) in data set. Find two-stage method of Joyner and Boore (1981) very effective to overcome this correlation and use it to find similar distance coefficient to \(\bar{b}\). Find similar effect of correlation on distance coefficient for two other models: \(\log A=a M-b \log (\Delta +30)+c\) and \(\log A=aM-\log X -b X +c\), where \(\Delta\) is epicentral distance.

• Japanese data selection criteria: focal depth \(<30\,\mathrm{km}\), \(M_{\mathrm{JMA}}>5.0\) and predicted PGA \(\geq 0.1\,\mathrm{m/s^2}\). US data selection criteria: \(d_r \leq 50\,\mathrm{km}\), use data from Campbell (1981).

• Because \(a\) affects distance and magnitude dependence, which are calculated during first and second steps respectively use an iterative technique to find coefficients. Allow different magnitude scaling for US and Japanese data.

• For Japanese data apply station corrections before last step in iteration to convert PGAs from different soil conditions to standard soil condition using residuals from analysis.

• Two simple numerical experiments performed. Firstly a two sets of artificial acceleration data was generated using random numbers based on attenuation relations, one with high distance decay and which contains data for short distance and one with lower distance decay, higher constant and no short distance data. Find that the overall equation from regression analysis has a smaller distance decay coefficient than individual coefficients for each line. Secondly find the same result for the magnitude dependent coefficient based on similar artificial data.

• Exclude Japanese data observed at long distances where average acceleration level was predicted (by using an attenuation relation derived for the Japanese data) to be less than the trigger level (assume to be about \(0.05\,\mathrm{m/s^2}\)) plus one standard deviation (assume to be \(0.3\)), i.e. \(0.1\,\mathrm{m/s^2}\), to avoid biasing results and giving a lower attenuation rate.

• Use the Japanese data and same functional form and method of Joyner and Boore (1981) to find an attenuation relation; find the anelastic coefficient is similar so conclude attenuation rate for Japan is almost equal to W. USA.

• Find difference in constant, \(d\), between Japanese and W. USA PGA values.

• Plot residuals against distance and magnitude and find no bias or singularity.

Gaull (1988)

• Ground-motion model is: \[\log \mathrm{PGA} =[(a_1\log R+a_2)/a_3] (M_L-a_4)-a_5\log R -a_6 R +a_7\] where \(\mathrm{PGA}\) is in \(\,\mathrm{m/s^2}\), \(a_1=5\), \(a_2=3\), \(a_3=20\), \(a_4=6\), \(a_5=0.77\), \(a_6=0.0045\) and \(a_7=1.2\) (\(\sigma\) not given).

• Considers three site categories but does not model:

  1. Rock: 6 records

  2. Alluvium: 5 records

  3. Average site: 10 records

• Most records from earthquakes with magnitudes about \(3\) and most from distances below about \(20\,\mathrm{km}\).

• Band pass filter records to get PGA associated with waves with periods between \(0.1\) and \(0.5\,\mathrm{s}\) because high frequency PGA from uncorrected records not of engineering significance.

• Adds 4 near source (\(5\leq R \leq 10\,\mathrm{km}\)) records from US, Indian and New Zealand earthquakes with magnitudes between \(6.3\) and \(6.7\) to supplement high magnitude range.

• Add some PGA points estimated from intensities associated with 14/10/1968 \(M_L=6.9\) Meckering earthquake in Western Australia.

• Plot 6 records from one well recorded event with \(M_L=4.5\) and fit an attenuation curve of form \(\log \mathrm{PGA}=b_1 -b_2 \log R-b_3 R\) by eye. Plot PGA of all records with \(2\leq R\leq 20\,\mathrm{km}\) against magnitude, fit an equation by eye. Use these two curves to normalise all PGA values to \(M_L=4.5\) and \(R=5\,\mathrm{km}\) from which estimates attenuation relation.

McCue, Gibson, and Wesson (1988)

• Ground-motion model is: \[A=a (\exp(bM)) \left( \frac{R}{R_0}+c \right)^{-d}\] where \(A\) is in \(\,\mathrm{g}\), \(\ln a=-5.75\), \(b=1.72\), \(c=0\), \(d=1.69\) and \(R_0=1\) (\(\sigma\) not given).

• Few records from free-field, most are in dams or special structures.,

• Because only 62 records, set \(R_0=1\) and \(c=0\).

• Most records from earthquakes with \(M_L\) between \(1.5\) and \(2.0\).

• Maximum PGA in set \(3.05\,\mathrm{m/s^2}\).

• Nonuniform distribution of focal distances. One quarter of records from same hypocentral distance. Therefore plot PGA of these records against magnitude (\(1.2\lesssim M_L \lesssim 4.3\) most less than \(2.1\)) to find \(b\). Then plot \(bM-\ln A\) against \(\ln (R/R_0)\) for all records to find \(a\) and \(d\).

• Notes limited data.

Petrovski and Marcellini (1988)

• Ground-motion model is: \[\ln (a)=b_1'+b_2 M+b_3 \ln (R+c)\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(b_1'=6.4830\), \(b_2=0.5438\), \(b_3=-1.3330\), \(c=20\,\mathrm{km}\) and \(\sigma=0.6718\) (for horizontal PGA) and \(b_1=5.6440\), \(b_2=0.5889\), \(b_3=-1.3290\), \(c=20\,\mathrm{km}\) and \(\sigma=0.6690\) (for vertical PGA) (also give coefficients for other choices of \(c\)).

• Data from ‘moderate’ soil conditions.

• Data mainly from SMA-1s but 17 from RFT-250s.

• Data from northern Greece (5 records, 4 stations, 3 earthquakes), northern Italy (45 records, 18 stations, 20 earthquakes) and former Yugoslavia (70 records, 42 stations, 23 earthquakes).

• Data from free-field or in basements of structures.

• Select records from earthquakes with \(3\leq M \leq 7\). Most earthquakes with \(M\leq 5.5\). 4 earthquakes (4 records) with \(M\leq 3.5\), 20 (27 records) with \(3.5<M\leq 4.5\), 13 (25 records) with \(4.5<M\leq 5.5\), 8 (50 records) with \(5.5<M\leq 6.5\) and 1 (14 records) with \(M>6.5\).

• Select records from earthquakes with \(h\leq 40\,\mathrm{km}\). Most earthquakes with \(h\leq 10\,\mathrm{km}\). 6 earthquakes with \(h\leq 5\,\mathrm{km}\), 30 with \(5<h\leq 10\,\mathrm{km}\), 5 with \(10<h\leq 20\,\mathrm{km}\), 4 with \(20<h \leq 30\,\mathrm{km}\) and 1 with \(h>30\).

• Select records that satisfied predetermined processing criteria so that their amplitude would be such as to give negligible errors after processing.

• Select records to avoid concentration of records w.r.t. certain sites, magnitudes, hypocentral distances or earthquakes. Most well-recorded earthquakes is 15/4/1979 Montenegro earthquake with 14 records.

• Try values of \(c\) between \(0\) and \(40\,\mathrm{km}\). Find standard deviation does not vary much for different choices.

• Test assumption of the log-normal probability distribution of data using graph in a coordinate system for log-normal distribution of probability, by \(\chi^2\) test and by the Kolmogorov-Smirnov test (not shown). Find assumption is acceptable.

PML (1988)

• Ground-motion model is: \[\ln y=C_0+C_1 M_s+C_2 \ln R\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(C_0=4.75\), \(C_1=0.52\), \(C_2=-1.00\) and \(\sigma=0.53\) for hard ground and horizontal; \(C_0=3.77\), \(C_1=0.70\), \(C_2=-1.24\) and \(\sigma=0.64\) for hard ground and vertical; \(C_0=4.37\), \(C_1=0.57\), \(C_2=-0.91\) and \(\sigma=0.52\) for medium ground and horizontal; \(C_0=3.20\), \(C_1=0.75\), \(C_2=-1.07\) and \(\sigma=0.55\) for medium ground and vertical; \(C_0=4.68\), \(C_1=0.55\), \(C_2=-0.99\) and \(\sigma=0.54\) for soft ground and horizontal; and \(C_0=3.53\), \(C_1=0.71\), \(C_2=-1.10\) and \(\sigma=0.66\) for soft ground and vertical.

• Use 3 site classes (and develop independent models for each):

  1. Equivalent to rock. 57 records.

  2. Soil depth \(<20\,\mathrm{m}\) or ‘shallow’. Includes sites described as ‘medium’.53 records.

  3. Soil depth \(>20\,\mathrm{m}\) or ‘deep’. Includes sites described as ‘soft’ or ‘alluvium’. 53 records.

  Notes that difficult to classify due to lack of information for some stations (particularly for soft and medium sites).

• Extends models of PML (1982, 1985) to spectral ordinates.

• Develop for use in the UK.

• Use earthquakes with focal depths \(\leq 30\,\mathrm{km}\).

• Prioritize data away from well-defined plate-boundaries, e.g. central and eastern North America (Miramichi, Nahanni, New Madrid), mainland China (e.g. Tangshan) and the Alpide Belt (Ancona, Friuli, Irpinia, Cephalonia, Koyna, Montenegro, Imotski, Gazli, Tabas), over data from California. When Californian data is required use near-field records so that model not affected by specific attenuation of California. California data was required for soft ground model because of lack of near-field data from other areas.

• Vast majority of data from \(M_w 4\) to \(7\) and from \(\leq 100\,\mathrm{km}\).

• Because of limited information on stress field and diversity of focal mechanisms in the UK do not consider style of faulting when selecting data.

• Correct data for instrument response and apply elliptic filter for majority of data (some data already processed using similar schemes). Note that differences below \(0.04\,\mathrm{s}\) between records corrected in different ways are minor.

• Use PGAs from processed data, which note may be slightly lower than those from uncorrected records.

Tong and Katayama (1988)

• Ground-motion model is: \[\log \bar{A}=\alpha M-\beta \log (\Delta+10)+\gamma T +\delta\] where \(\bar{A}\) is in \(\,\mathrm{gal}\), \(T\) is predominant period of site, \(\alpha=0.509\), \(\beta=2.32\), \(\gamma=0.039\) and \(\delta=2.33\) (\(\sigma\) not given).

• Correlation coefficient between magnitude and distance is \(0.84\), so magnitude and distance cannot be considered independent, so attenuation rate, \(\beta\), is difficult to find.

• First step fit \(\log \bar{A}=-\beta_i \log (\Delta+10)+\delta_i\) to each earthquake. Define reliability parameter, \(\psi_i=N_i R_i^2\), where \(N_i\) is degrees of freedom for \(i\) earthquake and \(R_i\) is correlation coefficient. Plot \(\psi_i\) against \(\beta_i\) and find attenuation rate scattered, between \(-6\) and \(9\), for \(\psi_i<1\) (Group B) and for \(\psi_1>1\) attenuation rate converges (Group U).

• Group B includes earthquakes with focal depths \(>388\,\mathrm{km}\), earthquakes with small magnitudes and records from distances \(\approx 100\,\mathrm{km}\), earthquakes with records from great distances where spread of distances is small, earthquakes recorded by only 3 stations and earthquakes with abnormal records. Exclude these records.

• Apply multiple regression on Group U to find \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) simultaneously. Also fix \(\beta=\sum \psi_i \beta_i/\sum \psi_i\) and find \(\alpha\), \(\gamma\) and \(\delta\). Find different coefficients but similar correlation coefficient. Conclude due to strong correlation between \(M\) and \(\Delta\) so many regression planes exist with same correlation coefficient.

• Perform Principal Component Analysis (PCA) on \(\log A\), \(M\), \(\log (\Delta+10)\), \(T\) and \(\log \bar{A}/A\) and find that equation found by fixing \(\beta\) is not affected by ill-effect of correlation between \(M\) and \(\Delta\).

• Omit \(T\) from regression and find little effect in estimation.

Yamabe and Kanai (1988)

• Ground-motion model is: \[\begin{aligned} \log_{10} a&=&\beta-\nu \log_{10} x\\ \mbox{where }\beta&=&b_1 +b_2 M\\ \mbox{and: }\nu&=&c_1+c_2 M\end{aligned}\] where \(a\) is in \(\,\mathrm{gal}\), \(b_1=-3.64\), \(b_2=1.29\), \(c_1=-0.99\) and \(c_2=0.38\) (\(\sigma\) not given).

• Focal depths between \(0\) and \(130\,\mathrm{km}\).

• Regress recorded PGA of each earthquake, \(i\), on \(\log_{10} a=\beta_i -\nu_i \log_{10} x\), to find \(\beta_i\) and \(\nu_i\). Then find \(b_1\) and \(b_2\) from \(\beta=b_1+b_2 M\) and \(c_1\) and \(c_2\) from \(\nu=c_1+c_2 M\).

• Also consider \(\nu=d_1 \beta\).

• Find \(\beta\) and \(\nu\) from 6 earthquakes (magnitudes between \(5.4\) and \(6.1\)) from Tokyo-Yokohama area are much higher than for other earthquakes, so ignore them. Conclude that this is due to effect of buildings on ground motion.

Youngs, Day, and Stevens (1988)

• Ground-motion model is: \[\ln (a_{\mathrm{max}})=C_1+C_2M_w-C_3 \ln[R+C_4 \exp (C_5 M_w)]+B Z_t\] where \(a_{\mathrm{max}}\) is in \(\,\mathrm{g}\), \(C_1=19.16\), \(C_2=1.045\), \(C_3=4.738\), \(C_4=205.5\), \(C_5=0.0968\), \(B=0.54\) and \(\sigma=1.55-0.125 M_w\).

• Use only rock records to derive equation but use some (389 records) for other parts of study. Classification using published shear-wave velocities for some sites.

• Exclude data from very soft lake deposits such as those in Mexico City because may represent site with special amplification characteristics.

• Data from subduction zones of Alaska, Chile, Peru, Japan, Mexico and Solomon Islands.

• Use two basic types of earthquake:

  1. Interface earthquakes: low angle, thrust faulting shocks occurring on plate interfaces.

  2. Intraslab earthquakes: high angle, predominately normal faulting shocks occurring within down going plate.

  Classification by focal mechanisms or focal depths (consider earthquakes with depths \(>50\,\mathrm{km}\) to be intraslab). Note that possible misclassification of some intraslab shocks as interface events because intraslab earthquakes do occur at depths \(<50\,\mathrm{km}\).

• Plots PGA from different magnitude earthquakes against distance; find near-field distance saturation.

• Originally include anelastic decay term \(-C_6 R\) but \(C_6\) was negative (and hence nonphysical) so remove.

• Plot residuals from original PGA equation (using rock and soil data) against \(M_w\) and \(R\); find no trend with distance but reduction in variance with increasing \(M_w\). Assume standard deviation is a linear function of \(M_w\) and find coefficients using combined rock and soil data (because differences in variance estimation from rock and soil are not significant).

• Use derived equation connecting standard deviation and \(M_w\) for weighted (weights inversely proportional to variance defined by equation) nonlinear regression in all analyses.

• Plot residuals from original PGA equation; find that hypothesis that coefficients of equations for interface and intraslab earthquakes are the same can be rejected (using likelihood ratio test for nonlinear regression models) at \(0.05\) percentile level for both soil and rock. Try including a term proportional to depth of rupture into equation (because intraslab deeper than interface events) but find no significant reduction in standard error. Introduce \(B Z_t\) term into equation; find \(B\) is significant at \(0.05\) percentile level. Try including rupture type dependence into other coefficients but produces no further decrease in variance so reject.

• Use only data from sites with multiple recordings of both interface and intraslab earthquakes and include dummy variables, one for each site, to remove differences due to systematic site effects. Fix \(C_1\) to \(C_5\) to values from entire set and find individual site terms and \(B\); find \(B\) is very similar to that from unconstrained regression.

• Examine residuals for evidence of systematic differences between ground motion from different subduction zones; find no statistically significant differences in PGA among different subduction zones.

• Use geometric mean of two horizontal components to remove effect of component-to-component correlations that affect validity of statistical tests assuming individual components of motion represent independent measurements of ground motion. Results indicate no significant difference between estimates of variance about median relationships obtained using geometric mean and using both components as independent data points.

• Extend to \(M_w>8\) using finite difference simulations of faulting and wave propagation modelled using ray theory. Method and results not reported here.

Abrahamson and Litehiser (1989)

• Ground-motion model is: \[\log_{10} a=\alpha +\beta M -\bar{c} \log_{10} [r+\exp(h_2 M)] +F\phi+E b r\] where \(F=1\) for reverse or reverse oblique events and \(0\) otherwise and \(E=1\) for interplate events and \(0\) otherwise, \(a\) is in \(\,\mathrm{g}\), for horizontal PGA \(\alpha=-0.62\), \(\beta=0.177\), \(\bar{c}=0.982\), \(h_2=0.284\), \(\phi=0.132\), \(b=-0.0008\) and \(\sigma=0.277\) and for vertical PGA \(\alpha=-1.15\), \(\beta=0.245\), \(\bar{c}=1.096\), \(h_2=0.256\), \(\phi=0.096\), \(b=-0.0011\) and \(\sigma=0.296\).

• Consider three site classifications, based on Joyner and Boore (1981):

  1. Rock: corresponds to C, D & E categories of Campbell (1981), 159 records.

  2. Soil: corresponds to A,B & F categories of Campbell (1981), 324 records.

  3. Unclassified: 102 records.

  Use to examine possible dependence in residuals not in regression because of many unclassified stations.

• Data based on Campbell (1981).

• Fault mechanisms are: strike-slip (256 records from 28 earthquakes), normal (14 records from 7 earthquakes), normal oblique (42 records from 12 earthquakes), reverse (224 records from 21 earthquakes) and reverse oblique (49 records from 8 earthquakes). Grouped into normal-strike-slip and reverse events. Weakly correlated with magnitude (\(0.23\)), distance (\(0.18\)) and tectonic environment (\(0.03\)).

• Tectonic environments are: interplate (555 records from 66 earthquakes) and intraplate (30 records from 10 earthquakes) measurements. Weakly correlated with magnitude (\(-0.26\)), distance (\(-0.17\)) and fault mechanism (\(0.03\)).

• Depths less than \(25\,\mathrm{km}\).

• Use array average (37 instruments are in array) from 10 earthquakes recorded at array in Taiwan.

• Most records from distances less than \(100\,\mathrm{km}\) and magnitude distribution is reasonably uniform but correlation between magnitude and distance of \(0.52\).

• Try two-stage technique and model (modified to include fault mechanism and tectonic environment parameters) of Joyner and Boore (1981), find inadmissable positive anelastic coefficient, so do not use it.

• Use a hybrid regression technique based on Joyner and Boore (1981) and Campbell (1981). A method to cope with highly correlated magnitude and distance is required. First step: fit data to \(f_2(r)=\bar{c} \log_{10}(r+h)\) and have separate constants for each earthquake (like in two-stage method of Joyner and Boore (1981)). Next holding \(\bar{c}\) constant find \(\alpha\), \(\beta\), \(b\) and \(h_2\) from fitting \(h=\exp(h_2 M)\). Weighting based on Campbell (1981) is used.

• Form of \(h\) chosen using nonparametric function, \(H(M)\), which partitions earthquakes into \(0.5\) unit bins. Plot \(H(M)\) against magnitude. Find that \(H(M)=h_1 \exp (h_2 M)\) is controlled by Mexico (19/9/1985) earthquake and \(h_1\) and \(h_2\) are highly correlated, \(0.99\), although does given lower total variance. Choose \(H(M)=\exp (h_2 M)\) because Mexico earthquake does not control fit and all parameters are well-determined, magnitude dependent \(h\) significant at \(90\%\).

• Try removing records from single-recorded earthquakes and from shallow or soft soil but effect on predictions and variance small (\(<10\%\)).

• Plot weighted residuals within \(10\,\mathrm{km}\) no significant, at \(90\%\), trends are present.

• Find no significant effects on vertical PGA due to site classification.

Campbell (1989)

• Ground-motion model is: \[\ln \mathrm{PHA} = a+b M_L - 1.0 \ln [R+c_1]\] where \(\mathrm{PHA}\) is in \(\,\mathrm{g}\), \(a=-2.501\), \(b=0.623\), \(c_1=7.28\) and \(\sigma=0.506\).

• Selects records from deep soil (\(>10\,\mathrm{m}\)). Excludes data from shallow soil (\(\leq 10\,\mathrm{m}\)) and rock sites and those in basements of buildings or associated with large structures, such as dams and buildings taller than two storeys. Selects records with epicentral distances \(\leq 20\,\mathrm{km}\) for \(M_L<4.75\) and distances \(\leq 30\,\mathrm{km}\) for \(M_L\geq 4.75\) to minimize regional differences in anelastic attenuation and potential biases associated with nontriggering instruments and unreported PGAs.

• Focal depths, \(H\), between \(1.8\) and \(24.3\,\mathrm{km}\) with mean of \(8.5\,\mathrm{km}\).

• PGAs scaled from either actual or uncorrected accelerograms in order to avoid potential bias due to correction.

• Uses weighted nonlinear least squares technique of Campbell (1981).

• Tries two other forms of equation: \(\ln \mathrm{PHA}=a+b M_L-1.0 \ln [R+c_1] +e_1 H\) and \(\ln \mathrm{PHA}=a+b M_L -1.0 \ln [R+c_1]+e_2 \ln H\) for epicentral and hypocentral distance. Allows saturation of PGA for short distances but finds nonsignificant coefficients, at \(90\%\). Also tries distance decay coefficient other than \(-1.0\) but finds instability in analysis.

• Examines normalised weighted residuals against focal depth, \(M_L\) and distance. Finds that although residuals seem to be dependent on focal depth there are probably errors in focal depth estimation for deep earthquakes in the study so the dependence may not be real. Finds residuals not dependent on magnitude or distance.

• Uses 171 records (\(0.9\leq R\leq 28.1\,\mathrm{km}\)) from 75 earthquakes (\(2.5 \leq M_L \leq 5.0\), \(0.7 \leq H \leq 24.3\,\mathrm{km}\)) excluded from original analysis because they were on shallow soil, rock and/or not free-field, to examine importance of site geology and building size. Considers difference between PGA from records grouped according to instrument location, building size, embedment, and site geology and the predicted PGA using the attenuation equation to find site factors, \(S\). Groups with nonsignificant, at \(90\%\), values of \(S\) are grouped together. Finds two categories: embedded alluvial sites from all building sizes (38 records) and shallow-soil (depth of soil \(\leq 10\,\mathrm{m}\)) sites (35 records) to have statistically significant site factors.

• Performs regression analysis on all records (irrespective of site geology or building size) from Oroville (172 records from 32 earthquakes) and Imperial Valley (71 records from 42 earthquakes) to find individual sites that have significant influence on prediction of PGA (by using individual site coefficients for each station). Finds equations predict similar PGA to those predicted by original equation. Finds significant differences between PGA recorded at different stations in the two regions some related to surface geology but for some finds no reason.

• Uses 27 records (\(0.2 \leq R \leq 25.0\,\mathrm{km}\)) from 19 earthquakes (\(2.5 \leq M_{bLG} \leq 4.8\), \(0.1 \leq H \leq 9\,\mathrm{km}\)) from E. N. America to examine whether they are significantly different than those from W. N. America. Finds residuals significantly, at \(99\%\) level, higher than zero and concludes that it is mainly due to site effects because most are on shallow soils or other site factors influence ground motion. Correcting the recorded PGAs using site factors the difference in PGA between E. N. America and W. N. America is no longer significant although notes may not hold for all of E. N. America.

Huo (1989)

• Ground-motion model is: \[\log Y=c_1+c_2M+c_4\log[R+c_5\exp(c_6M)]\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(c_1=0.207\), \(c_2=0.808\), \(c_4=-2.026\), \(c_5=0.183\), \(c_6=0.703\) and \(\sigma=0.247\) (for rock sites) and \(c_1=0.716\), \(c_2=0.647\), \(c_4=-1.706\), \(c_5=0.187\), \(c_6=0.703\) and \(\sigma=0.251\) (for soil sites).

• Use 2 site classes and derive separate models:

Ordaz, Jara, and Singh (1989)

• Ground-motion model is unknown.

Alfaro, Kiremidjian, and White (1990)

• Ground-motion model for near field is: \[\log(A)=a_1+a_2 M_s-\log (r^2+a_3^2)^{\frac{1}{2}}\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=-1.116\), \(a_2=0.312\), \(a_3=7.9\) and \(\sigma=0.21\).

  Ground-motion model for far field is: \[\log(A)=b_1+b_2M_s+b_3 \log (r^2+b_4^2)^{\frac{1}{2}}\] where \(A\) is in \(\,\mathrm{g}\), \(b_1=-1.638\), \(b_2=0.438\), \(b_3=-1.181\), \(b_4=70.0\) and \(\sigma=0.21\).

• Separate crustal and subduction data because of differences in travel path and stress conditions:

  1. Near field

  2. Far field, 20 records from San Salvador, 20 earthquakes, \(4.2\leq M_s \leq 7.2\), depths between \(36\) and \(94\,\mathrm{km}\), \(31 \leq r \leq 298\,\mathrm{km}\).

Ambraseys (1990)

• Ground-motion model is: \[\begin{aligned} \log y&=&\alpha +\beta M_w-\log r +b r\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(\alpha=-1.101\), \(\beta=0.2615\), \(b=-0.00255\), \(h=7.2\) and \(\sigma=0.25\).

• Uses data and method of Joyner and Boore (1981) but re-evaluates \(M_w\) for all earthquakes. Finds some large changes, e.g. Santa Barbara changes from \(M_w=5.1\) to \(M_w=5.85\). Uses \(M_L\) for 2 earthquakes (\(M_L=5.2\), \(6.2\)).

• Find effect of uncertainty in \(M_w\) causes less than \(10\%\) change in \(\sigma\).

• Also calculates equation using \(M_s\) instead of \(M_w\).

• Finds assumption \(M_s=M_w\) introduces bias, particularly for small magnitude shocks, on unsafe side, and this can be significant in cases where there is a preponderance of small earthquakes in set.

Campbell (1990)

• Ground-motion model is: \[\begin{aligned} \ln(Y)&=&a+bM+d \ln [R+c_1 \exp (c_2 M)]+eF+f_1\tanh[f_2(M+f_3)]\\ &&{}+g_1 \tanh(g_2 D)+h_1K_1+h_2K_2+h_3 K_3\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-2.245\), \(b=1.09\), \(c_1=0.361\), \(c_2=0.576\), \(d=-1.89\), \(e=0.218\), \(f_1=0\), \(f_2=0\), \(f_3=0\), \(g_1=0\), \(g_2=0\), \(h_1=-0.137\), \(h_2=-0.403\) and \(h_3=0\). \(\sigma=0.517\) for \(M\leq 6.1\) and \(\sigma=0.387\) for \(M\geq 6.2\). Also given is \(\sigma=0.450\) for \(M\geq 4.7\).

• Records from firm soil and soft rock sites. Characterises site conditions by depth to basement rock (sediment depth) in \(\,\mathrm{km}\), \(D\).

• Records from different size buildings. \(K_1=1\) for embedded buildings 3–11 storeys, \(K_2=1\) for embedded buildings with \(>\)11 storeys and \(K_3=1\) for non-embedded buildings \(>\)2 storeys in height. \(K_1=K_2=K_3=0\) otherwise.

• Uses two fault mechanisms:

  1. Strike-slip

  2. Reverse

Dahle, Bungum, and Kvamme (1990) & Dahle, Bugum, and Kvamme (1990)

• Ground-motion model is: \[\begin{aligned} \ln A&=&c_1+c_2 M+c_4 R +\ln G(R,R_0)\\ \mbox{where } G(R,R_0)&=&R^{-1} \quad \mbox{for} \quad R \leq R_0\\ \mbox{and: } G(R,R_0)&=&R_0^{-1}\left(\frac{R_0}{R}\right)^{5/6} \quad \mbox{for} \quad R>R_0\end{aligned}\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(c_1=-1.471\), \(c_2=0.849\), \(c_4=-0.00418\) and \(\sigma=0.83\).

• Use records from rock sites (presumably with hard rock or firm ground conditions).

• Assume intraplate refers to area that are tectonically stable and geologically more uniform than plate boundary areas. Select records from several ‘reasonably’ intraplate areas (eastern N. America, China, Australia, and some parts of Europe), due to lack of data.

• Select records which are available unprocessed and with sufficient information on natural frequency and damping of instrument.

• Use \(M_s\), when available, because reasonably unbiased with respect to source dimensions and there is globally consistent calculation method.

• Most (\(72\%\)) records from earthquakes with \(M\leq5.5\). Tangshan and Friuli sequence comprise a large subset. Correlation coefficient between magnitude and distance is \(0.31\).

• Instrument correct records and elliptical filter with pass band \(0.25\) to \(25.0\,\mathrm{Hz}\).

• If depth unknown assume \(15\,\mathrm{km}\).

• Choose \(R_0=100\,\mathrm{km}\) although depends on crustal structure and focal depth. It is distance at which spherical spreading for S waves overtaken by cylindrical spreading for Lg waves.

• PGA attenuation relation is pseudo-acceleration equation for \(0.025\,\mathrm{s}\) period and \(5\%\) damping.

• Plot residuals against magnitude and distance.

• Note ‘first order’ results, because data from several geological regions and use limited data base.

Jacob et al. (1990)

• Ground-motion model is: \[A=10^{(a_1+a_2M+a_3 \log d+a_4 d)}\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=-1.43\), \(a_2=0.31\), \(a_3=-0.62\) and \(a_4=-0.0026\) (\(\sigma\) not given).

• Note equation only for hard rock sites.

• Equation from a composite of two separate regressions: one using data from 6 earthquakes, \(4.7\leq M\leq 6.4\) and \(d\) primarily between \(40\) and \(820\,\mathrm{km}\) and one using the same data supplemented with data from 2 earthquakes with \(M=1.8\) and \(M=3.2\) and \(d\leq 20\,\mathrm{km}\) to extend results to smaller \(M\) and \(d\). Give no details of this composite regression.

• Note regressions are preliminary and should be tested against more data.

• Note careful assessment of uncertainties is required.

Sen (1990)

• Ground-motion model is: \[\ln \mathrm{PGA}=a+bM+c \ln (r+h)+\phi F\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(a=1.375\), \(b=1.672\), \(c=-1.928\) and \(\phi=0.213\) (\(h\) not given). Standard deviation is composed of two parts, inter-site \(\tau=0.261\) and intra-site \(\sigma=0.653\). \(F=1\) for thrust mechanism and \(0\) otherwise.

• Computes theoretical radiation pattern and finds a linear trend between residuals and radiation pattern but does not model.

Sigbjörnsson (1990)

• Ground-motion model is: \[a_{\mathrm{peak}}=\alpha_0 \exp(\alpha_1M)\exp(-\alpha_2R)R^{-\alpha}P\] where \(P=1\).

• Notes that data are very limited and any definite conclusions should, therefore, be avoided.

• Does not give coefficients, only predictions.

Tsai, Brady, and Cluff (1990)

• Ground-motion model is: \[\ln y=C_0+C_1M+C_2 (8.5-M)^{2.5}+C_3\ln [D+C_4 \exp(C_5M)]\] where \(y\) is in \(\,\mathrm{g}\), \(C_3=-2.1\), \(C_4=0.616\), \(C_5=0.524\) and for \(M\geq 6.5\) \(C_0=-1.092\), \(C_1=1.10\), \(C_2=0\) and \(\sigma=0.36\) and for \(M<6.5\) \(C_0=-0.442\), \(C_1=1.0\), \(C_2=0\) and \(\sigma=1.27-0.14M\).

• All records from rock or rock-like sites.

• Separate equation for \(M<6.5\) and \(M\geq 6.5\).

• Use only shallow crustal thrust earthquakes.

• Use another database of rock and soil site records and simulated acceleration time histories to find conversion factors to predict strike-slip and oblique ground motions from the thrust equation given above. For strike-slip conversion factor is \(0.83\) and for oblique conversion factor is \(0.91\).

• Standard deviation, \(\sigma\), for \(M\geq 6.5\) from regression whereas \(\sigma\) for \(M<6.5\) from previous results. Confirm magnitude dependence of standard deviation using 803 recordings from 124 earthquakes, \(3.8\leq M_w \leq 7.4\), \(D<100\,\mathrm{km}\).

Ambraseys and Bommer (1991) & N. N. Ambraseys and Bommer (1992)

• Ground-motion model is: \[\begin{aligned} \log a&=&\alpha +\beta M -\log r +b r\\ \mbox{where } r&=&(d^2+h_0^2)^{1/2}\\ \mbox{or: } r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), for horizontal PGA \(\alpha=-1.09\), \(\beta=0.238\), \(b=-0.00050\), \(h=6.0\) and \(\sigma=0.28\) and for vertical PGA \(\alpha=-1.34\), \(\beta=0.230\), \(b=0\), \(h=6.0\) and \(\sigma=0.27\). When use focal depth explicitly: for horizontal PGA \(\alpha=-0.87\), \(\beta=0.217\), \(b=-0.00117\) and \(\sigma=0.26\) and for vertical PGA \(\alpha=-1.10\), \(\beta=0.200\), \(b=-0.00015\) and \(\sigma=0.26\).

• Consider two site classifications (without regard to depths of deposits) but do not model:

  1. Rock

  2. Alluvium

• Select records which have: \(M_s\geq 4.0\) and standard deviation of \(M_s\) known and reliable estimates of source-site distance and focal depth, \(h\leq25\,\mathrm{km}\), regardless of local soil conditions from free-field and bases of small buildings. No reliable data or outliers excluded. Records from instruments at further distances from the source than the closest non-triggered instrument were non-excluded because of non-homogeneous and irregularly spaced networks and different and unknown trigger levels.

• Most data, about \(70\%\), with distances less than \(40\,\mathrm{km}\). Note strong bias towards smaller values of magnitude and PGA.

• PGA read from analogue and digitised data, with different levels of processing. Differences due to different processing usually below \(5\%\), but some may be larger.

• Errors in distances for small shocks may be large.

• Prefer one-stage technique because second step of two-stage method would ignore records from singly-recorded earthquakes which compose over half the events, also find more realistic, \(b\), and \(h_0\) using one-stage method. Do not use weighting because involves assumptions which are difficult to verify.

• Find inadmissable and positive \(b\) for vertical PGA so remove and repeat.

• Remove records from distances less than or equal to half their focal depth and also less than or equal to their focal depth, find that \(h_0\) is governed by near-field data.

• Use focal depth explicitly, by replacing \(r=(d^2+h_0^2)^{1/2}\) by \(r=(d^2+h^2)^{1/2}\). Find lower standard deviation and that it is very significant.

• Repeat analysis on subsets of records grouped by focal depth. Find no correlation between \(h_0\) and focal depth of subset. Use \(h_0\) equal to mean focal depth in each subset and find similar results to when focal depth used explicitly.

• Repeat analysis with geometric attenuation coefficient equal to \(-0.83\), corresponding to the Airy phase, as opposed to \(-1.0\).

• Find small dependence of horizontal PGA on site classification, note due to level of information available.

Crouse (1991)

• Ground-motion model is: \[\ln \mathrm{PGA}=p_1+p_2M+p_4 \ln [R+p_5 \exp(p_6M)]+p_7h\] where \(\mathrm{PGA}\) is in \(\,\mathrm{gal}\), using all PGA values \(p_1=6.36\), \(p_2=1.76\), \(p_4=-2.73\), \(p_5=1.58\), \(p_6=0.608\), \(p_7=0.00916\) and \(\sigma=0.773\).

• Use data from stiff soil sites (depth of soil \(<25\,\mathrm{m}\)).

• Include data from any zones with strong seismic coupling, such as the younger subduction zones (S.W. Japan, Alaska, C. America (Mexico), C. Chile, Peru and northern Honshu and Kuril subduction zones in Japan) unless compelling reasons to exclude data. Do this because lack of data from Cascadia. Most (\(>70\%\)) are from Japan.

• Focal depths, \(h\), between \(0\) and \(238\,\mathrm{km}\).

• Compare Japanese and Cascadia PGA values for earthquakes with similar magnitude and depths and find similar.

• Do not exclude data from buildings or which triggered on S-wave. Note could mean some PGAs are underestimated.

• Plot ground motion amplitude (PGA and also some maximum displacements from seismograms) against distance for a number of large magnitude shocks (including some data from rock sites which not included in set for regression). Find that rate of attenuation becomes smaller for shorter distances and process is magnitude dependent. Also plot Japanese PGA data, from earthquakes with \(h\leq50\,\mathrm{km}\), split into three distance groups (between \(50\) and \(75\,\mathrm{km}\), between \(100\) and \(150\,\mathrm{km}\) and between \(250\) and \(300\,\mathrm{km}\)) find as distance increases magnitude scaling becomes larger and possible saturation in PGA for large magnitudes. Fit \(\ln \mathrm{PGA}=p_1+p_2 \ln (R+C)\) to some PGA values from large magnitude shocks for \(C=0\) and \(C>0\), find lower standard deviation for \(C>0\).

• Fit \(\ln \mathrm{PGA}=a+bM\) and \(\ln \mathrm{PGA}=a+bM+cM^2\) to Japanese data split into the three distance groups (mentioned above); find \(b\) increases with increasing distance range but both equations fit data equally well.

• Constrain \(p_4\) to negative value and \(p_5\) and \(p_6\) to positive values.

• Include quadratic magnitude term, \(p_3 M^2\), but find equal to zero.

• Plot residuals against \(M\); find uniformly distributed and evidence for smaller residuals for larger \(M\).

• Plot residuals against \(R\)¹² and find decreasing residuals for increasing \(R\).

• Give equation using only those records available in digital form (235 records).

Garcı̀a-Fernàndez and Canas (1991) & Garcia-Fernandez and Canas (1995)

• Ground-motion model is: \[\ln \mathrm{PGA}=\ln C_0+C_1 M-0.5 \ln r -\gamma r\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), for Iberian Peninsula \(\ln C_0=-5.13\), \(C_1=2.12\) and \(\gamma=0.0039\), for NE region \(\ln C_0=-4.74\), \(C_1=2.07\) and \(\gamma=0.0110\) and for SSE region \(\ln C_0=-5.30\), \(C_1=2.21\) and \(\gamma=0.0175\) (\(\sigma\) is not given).

• Derive equations for two regions:

  1. South south-east part of the Iberian peninsula, from the Guadalquivir basin to the Mediterranean Sea, including the Betic Cordillera, 140 records from 5 stations.

  2. North-east part of the Iberian peninsula, including the Pyrenees, the Catalan Coastal Ranges, the Celtiberian chain and the Ebro basin, 107 records from 3 stations.

• Use vertical-component short-period analogue records of Lg-waves (which are believed to have the largest amplitudes for the period range \(0.1\) to \(1s\)) from regional earthquakes in Iberian Peninsula.

• Processing procedure is: digitise seismogram using irregular sampling rate to get better sampling at peaks and ‘kinks’, select baseline, apply cubic spline interpolation and compare original and digitised seismograms. Next the Fourier amplitude spectrum is computed and the instrument amplitude response is removed.

• Estimate PGA using the maximum value of pseudo-absolute acceleration obtained from Fourier amplitude spectra. Derived equations are for characteristic frequency of \(5\,\mathrm{Hz}\).

• Compare estimated PGAs with observed PGAs from five earthquakes and find good agreement.

• Use \(5\,\mathrm{Hz}\) \(\gamma\) values from Garcia-Fernandez and Canas (1992) and Vives and Canas (1992).

Geomatrix Consultants (1991), Sadigh et al. (1993) & Sadigh et al. (1997)

• Ground-motion model is: \[\ln\mathrm{PGA}=C_1+C_2M+C_3\ln\left(r_{\mathrm{rup}}+C_4\mathrm{e}^{C_5M}\right)+C_6Z_T\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for horizontal PGA, rock sites and strike-slip faulting \(C_3=0\) and \(C_4=-2.100\), for \(M\leq 6.5\) \(C_1=-0.624\), \(C_2=1.0\), \(C_5=1.29649\) and \(C_6=0.250\) and for \(M>6.5\), \(C_1=-1.274\), \(C_2=1.1\), \(C_5=-0.48451\) and \(C_6=0.524\). For reverse and thrust earthquakes multiply strike-slip prediction by \(1.2\). \(\sigma=1.39-0.14M\) for \(M<7.21\) and \(\sigma=0.38\) for \(M\geq 7.21\). For horizontal PGA and deep soil \(C_2=1.0\), \(C_3=1.70\) and \(C_6=0\), for strike-slip faulting \(C_1=-2.17\) and for reverse or thrust faulting \(C_1=-1.92\), for \(M\leq 6.5\) \(C_4=2.1863\) and \(C_5=0.32\) and for \(M>6.5\) \(C_4=0.3825\) and \(C_5=0.5882\). \(\sigma=1.52-0.16M\) for \(M\leq 7\) and \(\sigma=0.40\) for \(M=7\).

  For vertical PGA, rock sites and strike-slip faulting \(C_3=0\) and \(C_4=-2.300\), for \(M\leq 6.5\) \(C_1=-0.430\), \(C_2=1.0\), \(C_5=1.2726\) and \(C_6=0.228\) and for \(M>6.5\), \(C_1=-1.080\), \(C_2=1.1\), \(C_5=-0.3524\) and \(C_6=0.478\). For reverse and thrust earthquakes multiply strike-slip prediction by \(1.1\) and for oblique faulting multiply by \(1.048\). \(\sigma=0.48\) for \(M\geq 6.5\), \(\sigma=3.08-0.40M\) for \(6<M<6.5\) and \(\sigma=0.68\) for \(M\leq 6\).

• Use two site categories (for horizontal motion):

  1. Rock: bedrock within about a metre of surface. Note that many such sites are soft rock with \(V_s \leq 750\,\mathrm{m/s}\) and a strong velocity gradient because of near-surface weathering and fracturing, 274 records.

  2. Deep soil: greater than \(20\,\mathrm{m}\) of soil over bedrock. Exclude data from very soft soil sites such as those from San Francisco bay mud, 690 records.

  Vertical equations only for rock sites.

• Crustal earthquakes defined as those that occur on faults within upper \(20\) to \(25\,\mathrm{km}\) of continental crust.

• Use source mechanism: RV=reverse (26+2) \(\Rightarrow Z_T=1\) and SS=strike-slip (and some normal) (89+0) \(\Rightarrow Z_T=0\). Classified as RV if rake\(>45^{\circ}\) and SS if rake\(<45^{\circ}\). Find peak motions from small number of normal faulting earthquakes not to be significantly different than peak motions from strike-slip events so were including in SS category.

• Records from instruments in instrument shelters near ground surface or in ground floor of small, light structures.

• 4 foreign records (1 from Gazli and 3 from Tabas) supplement Californian records.

• Separate equations for \(M_w <6.5\) and \(M_w \geq 6.5\) to account for near-field saturation effects and for rock and deep soil sites.

Huo and Hu (1991)

• Ground-motion model is (case II): \[\log y =C_1+C_2 M-C_4 \log [R+C_5 \exp (C_6 M)]\] where \(y\) is in \(\,\mathrm{gal}\), \(C_5=0.231\) and \(C_6=0.626\), for rock \(C_1=0.894\), \(C_2=0.563\), \(C_4=1.523\) and \(\sigma=0.220\) and for soil \(C_1=1.135\), \(C_2=0.462\), \(C_4=1.322\) and \(\sigma=0.243\) (these coefficients are from regression assuming \(M\) and \(R\) are without error).

• Use two site categories:

  1. Rock

  2. Soil

• Supplement western USA data in large magnitude range with 25 records from 2 foreign earthquakes with magnitudes \(7.2\) and \(7.3\).

• Note that there are uncertainties associated with magnitude and distance and these should be considered in derivation of attenuation relations.

• Develop method, based on weighted consistent least-square regression, which minimizes residual error of all random variables not just residuals between predicted and measured ground motion. Method considers ground motion, magnitude and distance to be random variables and also enables inverse of attenuation equation to be used directly.

• Note prediction for \(R>100\,\mathrm{km}\) may be incorrect due to lack of anelastic attenuation term.

• Use both horizontal components to maintain their actual randomness.

• Note most data from moderate magnitude earthquakes and from intermediate distances therefore result possibly unreliable outside this range.

• Use weighted analysis so region of data space with many records are not overemphasized. Use \(M\)-\(R\) subdivisions of data space: for magnitude \(M<5.5\), \(5.5\leq M \leq 5.9\), \(6.0 \leq M \leq 6.4\), \(6.5 \leq M \leq 6.9\), \(7.0 \leq M \leq 7.5\) and \(M>7.5\) and for distance \(R<3\), \(3 \leq R \leq 9.9\), \(10 \leq R \leq 29.9\), \(30 \leq R \leq 59.9\), \(60 \leq R \leq 99.9\), \(100 \leq R \leq 300\) and \(R>300\,\mathrm{km}\). Assign equal weight to each subdivision, and any data point in subdivision \(i\) containing \(n_i\) data has weight \(1/n_i\) and then normalise.

• To find \(C_5\) and \(C_6\) use 316 records from 7 earthquakes (\(5.6 \leq M \leq 7.2\)) to fit \(\log Y=\sum^m_{i=1} C_{2,i} E_i -C_4 \log [r+\sum^m_{i=1} R_{0,i} E_i]\), where \(E_i=1\) for \(i\)th earthquake and \(0\) otherwise. Then fit \(R_0=C_5 \exp (C_6 M)\) to results.

• Also try equations: \(\log y=C_1+C_2 M -C_4 \log [R+C_5]\) (case I) and \(\log y=C_1 +C_2 M-C_3M^2-C_4 \log [R+C_5 \exp (C_6 M)]\) (case III) for \(M\leq M_c\), where impose condition \(C_3 =(C_2-C_4C_6/\ln 10)/(2M_c)\) so ground motion is completely saturated at \(M=M_c\) (assume \(M_c=8.0\)).

• Find equations for rock and soil separately and for both combined.

I.M. Idriss (1991) reported in Idriss (1993)

• Ground-motion model is: \[\ln(Y)=[\alpha_0+\exp(\alpha_1+\alpha_2 M)]+[\beta_0-\exp(\beta_1+\beta_2 M)]\ln (R+20)+a F\] where \(Y\) is in \(\,\mathrm{g}\), \(a=0.2\), for \(M\leq 6\) \(\alpha_0=-0.150\), \(\alpha_1=2.261\), \(\alpha_2=-0.083\), \(\beta_0=0\), \(\beta_1=1.602\), \(\beta_2=-0.142\) and \(\sigma=1.39-0.14M\) and for \(M>6\) \(\alpha_0=-0.050\), \(\alpha_1=3.477\), \(\alpha_2=-0.284\), \(\beta_0=0\), \(\beta_1=2.475\), \(\beta_2=-0.286\) and for \(M<7\frac{1}{4}\) \(\sigma=1.39-0.14M\) and for \(M\geq 7\frac{1}{4}\) \(\sigma=0.38\).

• Records from rock sites.

• Uses three fault mechanisms:

  1. Strike slip

  2. Oblique

  3. Reverse

• Separate equations for \(M\leq 6\) and \(M>6\).

• Examines residuals for PGA. Finds average residual almost zero over entire distance range; trend reasonable up to about \(60\,\mathrm{km}\) but beyond \(60\,\mathrm{km}\) relationship would underestimate recorded PGA.

• Finds standard deviation to be linear function of magnitude.

Loh et al. (1991)

• Ground-motion model is: \[a=b_1 \mathrm{e}^{b_2 M} (R+b_4)^{-b_3}\] where \(a\) is in \(\,\mathrm{g}\), \(b_1=1.128\), \(b_2=0.728\), \(b_3=1.743\), \(b_4=32\,\mathrm{km}\) and \(\sigma=0.563\) (in terms of \(\ln\)).

• Use only data from rock sites.

• Focal depths, \(h\), between \(0.2\) and \(97.4\,\mathrm{km}\). Most records from \(h<30\,\mathrm{km}\).

• Also derive equations for PGA using \(\log_{10}(a)=b_1+b_2 M+b_3 \log \sqrt{R^2+b_5^2}\) and \(a=b_1 \mathrm{e}^{b_2 M} (R+b_4 \mathrm{e}^{b_5 M})^{-b_3}\) in order to have diversity in the characterisation of ground motion.

• Use \(r_{hypo}\) because no clear fault ruptures identified for Taiwanese earthquakes.

• All data from SMA-1s.

• PGAs between \(7.3\) and \(360.2\,\mathrm{cm/s^2}\).

Matuschka and Davis (1991)

• Exact functional form unknown but based on those of Campbell (1981), Fukushima and Tanaka (1990) and Abrahamson and Litehiser (1989).

• Use three site classes.

• Develop separate equations for each site class. Only possible for two classes. Therefore, modify equation derived for site class C to obtain coefficients for other two classes.

• Digitization sampling rate of records used is \(50\,\mathrm{Hz}\). Most data low-pass filtered at \(24.5\,\mathrm{Hz}\).

• Most data high-pass filtered with cut-offs above \(0.25\,\mathrm{Hz}\).

• Due to limited data, advise caution when using model.

Niazi and Bozorgnia (1991)

• Ground-motion model is: \[\ln Y=a+b M+d \ln [R+c_1 \mathrm{e}^{c_2 M}]\] where \(Y\) is in \(\,\mathrm{g}\), for horizontal PGA \(a=-5.503\), \(b=0.936\), \(c_1=0.407\), \(c_2=0.455\), \(d=-0.816\) and \(\sigma=0.461\) and for vertical PGA \(a=-5.960\), \(b=0.989\), \(c_1=0.013\), \(c_2=0.741\), \(d=-1.005\) and \(\sigma=0.551\).

• All records from array so essentially identical site conditions and travel paths.

• All records from free-field instruments mounted on \(4\mathrm{inch}\) (\(10\,\mathrm{cm}\)) thick concrete base mats, approximately \(2\) by \(3\,\mathrm{feet}\) (\(60\) by \(90\,\mathrm{cm}\)) across.

• Select earthquakes to cover a broad range of magnitude, distance and azimuth and ensuring thorough coverage of the array. Criteria for selection is: at least 25 stations recorded shock, focal depth \(<30\,\mathrm{km}\), hypocentral distance \(<50\,\mathrm{km}\) except for two large earthquakes from beyond \(50\,\mathrm{km}\) to constrain distance dependence.

• Focal depths between \(0.2\) and \(27.2\,\mathrm{km}\) with all but one \(\leq 13.9\,\mathrm{km}\).

• Azimuths between \(60^{\circ}\) and \(230^{\circ}\).

• Most records (\(78\%\)) have magnitudes between \(5.9\) and \(6.5\). Note magnitude and distance are not independent (correlation coefficient is \(0.6\)).

• Records have sampling interval of \(0.01\,\mathrm{s}\). Processed using trapezoidal band passed filter with corner frequencies \(0.07\), \(0.10\), \(25.0\) and \(30.6\,\mathrm{Hz}\).

• Not enough information to use distance to rupture zone.

• Source mechanisms of earthquakes are: 4 normal, 2 reverse, 1 reverse oblique and 1 normal oblique with 4 unknown. Do not model source mechanism dependence because of 4 unknown mechanisms.

• Use weighted regression, give equal weight to recordings from each earthquake within each of 10 distance bins (\(<2.5\), \(2.5\)–\(5.0\), \(5.0\)–\(7.5\), \(7.5\)–\(10.0\), \(10.0\)–\(14.1\), \(14.1\)–\(20.0\), \(20\)–\(28.3\), \(28.3\)–\(40.0\), \(40.0\)–\(56.6\) and \(56.6\)–\(130\,\mathrm{km}\)). Do this so earthquakes with smaller number of recordings are not overwhelmed by those with a larger coverage and also to give additional weight to shocks recorded over multiple distance bins. Apply two-stage regression, because of high correlation between magnitude and distance, excluding 3 earthquakes (\(M=3.6\), \(5.0\), \(7.8\)) with 162 records from first stage to reduce correlation between \(M\) and \(R\) to \(0.1\). Also do one-stage regression although do not give coefficients.

• Use mean horizontal component because reduces uncertainty in prediction.

• Examine coefficient of variation for each earthquake using median and normalized standard deviation of recordings in inner ring of array. Find evidence for magnitude dependent uncertainty (large magnitude shocks show less uncertainty). Find that main contribution to scatter is inter-event variations again by examining coefficient of variation; although note may be because using dense array data.

• Examine mean residuals of observations from each earthquake. Find evidence for higher than predicted vertical PGA from reverse faulting earthquakes and lower than predicted vertical PGA from normal faulting earthquakes, although due to lack of information for 4 earthquakes note that difficult to draw any conclusions.

• Examine mean residuals of observations from each station in inner ring. Find mean residuals are relatively small compared with standard deviation of regression so variation between stations is less than variation between earthquakes. Find for some stations some large residuals.

Rogers et al. (1991)

• Ground-motion model is: \[\log a_p=a_1+0.36M-0.002R+a_2 \log R+a_3 S_1+a_4 S_1 \log R+a_5 S_5+a_6 S_5 \log R+a_7 S_6 \log R\] where \(a_p\) is in \(\,\mathrm{g}\), \(a_1=-1.62\), \(a_2=-1.01\), \(a_3=0.246\), \(a_4=0.212\), \(a_5=0.59\), \(a_6=-0.29\), \(a_7=0.21\) and \(\sigma=0.29\).

• Use six local site classifications:

  1. Holocene

  2. Pleistocene soil

  3. Soft rock

  4. Hard rock

  5. Shallow (\(<10\,\mathrm{m}\) depth) soil

  6. Soft soil (e.g. bay mud)

• Data from about 800 different stations.

• Note that inclusion of subduction-zone events in analysis may affect results with unmodelled behaviour, particularly with regard to distance scaling although believe use of \(r_{rup}\) partially mitigates this problem.

• Firstly compute an equation does not include site coefficients. Conduct regression analysis on site-condition subsets of the residuals using \(M\) or \(\log R\) as dependent variable. Find several regressions are not statistically significant at the \(5\%\) level and/or the predicted effects are small at the independent variable extremes. Find strongest effects and most significant results are for shallow soil sites and soft soil sites although because of the high correlation between \(M\) and \(\log R\) in the set used it is difficult to construct unbiased models.

• Use a stochastic random-vibration approach to find theoretical equations for estimating PGA that include the effect of local site conditions as distance-dependent terms. Using the results from this analysis construct equation based on the observed PGAs. Try including terms for \(S_1\), \(S_2\), \(S_5\), \(S_6\) and corresponding \(\log R\) terms for each site type but iterate to retain only the significant terms.

• Fix magnitude scaling (\(0.36M\)) and anelastic attenuation (\(0.002R\)). Do not try to optimise the fit other than using fixed values similar to those given by the stochastic analysis.

• Note that anelastic coefficient may be too low but it produces an acceptable geometric spreading term.

• Note that because Moho critical reflections can increase amplitudes beyond about \(50\,\mathrm{km}\) the effects of anelastic or geometric attenuation may be masked.

• Allowing all the coefficients in the equation to be free produces a smaller magnitude scaling coefficient, a smaller geometric spreading coefficient, and a non-significant anelastic attenuation term.

• Note that data from \(S_5\) and \(S_6\) are sparse.

• Compare estimated PGAs with data from within small magnitude ranges. Find that PGAs from Morgan Hill earthquake are overestimated, which believe is due to the unilateral rupture of this earthquake masking the effect of the local site conditions.

Stamatovska and Petrovski (1991)

• Ground-motion model is: \[\mathrm{Acc}=b_1 \exp(b_2M)(R_h+c)^{b_3}\] \(\mathrm{Acc}\) is in \(\,\mathrm{cm/s^2}\), \(b_1=534.355\), \(b_2=0.46087\), \(b_3=-1.14459\), \(c=25\) and \(\sigma_{\ln \mathrm{Acc}}=0.72936\).

• Data from 141 different sites, which are considered to have average soil conditions.

• Data from Yugoslavia (23 earthquakes), Italy (45 earthquakes), northern Greece (3 earthquakes), Romania (1 earthquake), Mexico (1 earthquake) and the USA (5 earthquakes). Select earthquakes to have range of magnitudes and focal depths.

• Data processed using standard procedure.

• Conduct Pearson \(\chi^2\) and Kolmogorov-Smirnov tests to test acceptability of log-normal assumption using a \(5\%\) significance level. Conclude that assumption is justified.

• Note the strong influence of the data used on results and the need to improve it.

Abrahamson and Youngs (1992)

• Ground-motion model is: \[\ln y=a+b M+d \ln (r+c)+e F\] where \(a=0.0586\), \(b=0.696\), \(c=12.0\), \(d=-1.858\), \(e=0.205\), \(\sigma=0.399\) (intra-event) and \(\tau=0.201\) (inter-event) (units of \(y\) are not given but probably \(\,\mathrm{g}\)).

• \(F\) is fault type (details not given).

• Develop new algorithm for one-stage maximum-likelihood regression, which is more robust than previous algorithms.

N. N. Ambraseys, Bommer, and Sarma (1992)

• Ground-motion model is: \[\begin{aligned} \log (a)&=&c_1+c_2 M+c_3 r+c_4 \log r\\ r&=&(d^2+h_0^2)^{\frac{1}{2}}\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), \(c_1=-1.038\), \(c_2=0.220\), \(c_3=-0.00149\), \(c_4=-0.895\), \(h_0=5.7\) and \(\sigma=0.260\).

• Investigate equations of PML (1982) and PML (1985) using criteria:

  1. Is the chosen data set of earthquake strong-motion records suitable to represent the seismic environment?

  2. Are the associated seismological and geophysical parameters used in these reports reliable and consistent?

  3. Is the methodology used to derive attenuation laws and design spectra from the data set reliable?

• Investigate effect of different Ground-motion model, one and two-stage regression technique, record selection technique and recalculation of associated parameters. Find these choice cause large differences in predictions.

• Coefficients given above are for PML (1985) data with recalculated magnitudes and distances and addition of extra records from some earthquakes.

J. Huo and Hu (1992)

• Ground-motion model is¹³: \[\ln Y=a_1+a_2M+a_3\ln[R+a_4\exp(a_5M)]\] where \(Y\) is in \(\,\mathrm{gal}\), \(a_1=0.1497\), \(a_2=1.9088\), \(a_3=-2.049\), \(a_4=0.1818\) and \(a_5=0.7072\) (\(\sigma\) is unknown).

• Use macroseismic intensities and strong-motion data to derive model. Details unknown.

• There is another model by these authors Huo and Hu (1991) (see Section 2.89) to which this model is probably similar.

Kamiyama, O’Rourke, and Flores-Berrones (1992) & Kamiyama (1995)

• Ground-motion model is (note that there is a typographical error in Kamiyama, O’Rourke, and Flores-Berrones (1992; Kamiyama 1995) because \(r_t\) has been replaced by \(r_c\) in equations): \[\begin{aligned} \log_{10} a_{\mathrm{max}}&=&-1.64 R_0+b_1 R_1+b_2 R_2 +c_a +\sum_{i=1}^{N-1} A_i S_i\\ R_0&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&r \leq r_t\\ \log_{10}r-\log_{10}r_c&r >r_t\\ \end{array} \right.\\ R_1&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&r \leq r_t\\ 1&r >r_t\\ \end{array} \right.\\ R_2&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&r \leq r_t\\ M&r >r_t\\ \end{array} \right.\end{aligned}\] where \(S_i=1\) for \(i\) station, \(S_0=0\) otherwise, \(a_{\mathrm{max}}\) is in \(\,\mathrm{cm/s^2}\), \(b_1=-1.164\), \(b_2=0.358\), \(c_a=2.91\), \(r_c=5.3\,\mathrm{km}\) and \(\sigma=0.247\) (\(A_i\) given in publications but not reported here due to lack of space).

• Instrument correct records and filter with pass band between \(0.24\) and \(11\,\mathrm{Hz}\).

• Model individual soil conditions at each site as amplification factors, \(\mathrm{AMP}_i\), as described by Kamiyama and Yanagisawa (1986).

• Most records are from hypocentral distances between \(30\) and \(200\,\mathrm{km}\).

• Focal depths between \(0\) and \(130\,\mathrm{km}\).

• Models peak ground accelerations independent of magnitude and distance in a fault zone, \(r_t\), where \(r_t=r_c 10^{(b_1+b_2M)/1.64}\).

• Constrain decay with distance in far field to \(-1.64\) using results from other studies to avoid problems due to correlation between \(M\) and \(\log_{10} r\).

• Use trial and error method to find \(r_c\) so that resulting values of \(r_t\) are consistent with empirical estimates of fault length from past studies.

• Also give expression using shortest distance to fault plane (rupture distance), \(R\), by replacing the expression for \(r\leq r_c\) and \(r>r_c\) by one expression given by replacing \(r\), hypocentral distance, by \(R+r_c\) in expression for \(r>r_c\). This gives PGA independent of magnitude at distance \(R=0\,\mathrm{km}\).

• Note that use of \(r_{hypo}\) is not necessarily best choice but use it due to simplicity.

• Check residual plots; find no trends so conclude adequate from statistical point of view.

Sigbjörnsson and Baldvinsson (1992)

• Ground-motion model is: \[\begin{aligned} \log A&=&\alpha +\beta M -\log R +b R\\ \mbox{with: }R&=&\sqrt{d^2+h^2}\end{aligned}\] where \(A\) is in \(\,\mathrm{g}\), for average horizontal PGA and \(4<M<6\) \(\alpha=-1.98\), \(\beta=0.365\), \(b=-0.0039\) and \(\sigma=0.30\), for larger horizontal PGA and \(4<M<6\) \(\alpha=-1.72\), \(\beta=0.327\), \(b=-0.0043\) and \(\sigma=0.30\) and for both horizontal PGAs and \(2<M<6\) \(\alpha=-2.28\), \(\beta=0.386\), \(b=0\) and \(\sigma=0.29\).

• Find that Icelandic data does not fit other published relations.

• Find equation using only records with \(M\geq 4.0\), \(h\) equal to focal depth and both the horizontal components.

• Find equation using only records with \(M\geq 4.0\), \(h\) equal to focal depth and larger horizontal component.

• Also repeated with all data. Anelastic coefficient constrained to zero because otherwise positive.

• Also done with \(h\) free.

• Note that large earthquakes have \(h \approx 10\,\mathrm{km}\) while small events have \(h \approx 5\,\mathrm{km}\).

Silva and Abrahamson (1992)

• Ground-motion model is: \[\ln \mathrm{pga}=c_1+1.2 M+c_3\ln(r+20)+0.25 F\] where \(\mathrm{pga}\) is in \(\,\mathrm{g}\), \(c_1=-3.27\), \(c_3=-1.79\) and \(\sigma_{total}=0.46\) for deep soil and \(c_1=-3.56\), \(c_3=-1.67\) and \(\sigma_{total}=0.46\) for rock/shallow soil.

• Originally use five site classes (chosen based on site response analyses using broad categories and generic site profiles):

  1. Rock. 66 records

  2. Shallow soil (\(<250\,\mathrm{ft}\). 6 records.)

  3. Intermediate depth soil (\(250\)–\(1000\,\mathrm{ft}\)). 2 records.

  4. Deep soil (\(>1000\,\mathrm{ft}\)). 51 records.

  5. Alluvium of unknown depth. 10 records.

  but insufficient records in shallow and intermediate classes to evaluate separately so combine rock and shallow classes and intermediate, deep and unknown depth categories to leave two classes: \(<250\,\mathrm{ft}\) and \(>250\,\mathrm{ft}\).

• Use two faulting mechanisms:

  1. Strike-slip

  2. Reverse or oblique

• Process data by: 1) interpolation of uncorrected unevenly sampled records to 400 samples per second; 2) frequency domain low-pass filtering using a causal five-pole Butterworth filter with corner frequencies selected based on visual examination of Fourier amplitude spectrum; 3) removal of instrument response; 4) decimation to 100 or 200 samples per second depending on low-pass filter corner frequencies; and 5) application of time-domain baseline correction, using polynomials of degrees zero to ten depending on integrated displacements, and final high-pass filter chosen based on integrated displacements that is flat at corner frequency and falls off proportional to frequency on either side, which is applied in the time domain twice (forward and backwards) to result in zero phase shift.

• Note that due to limited magnitude range of data, magnitude dependence is not well constrained nor is dependency on mechanism. Hence these coefficients are fixed based on previous studies.

• Plot residuals w.r.t. distance. Find slight increase at \(70\)–\(100\,\mathrm{km}\). To test if due to Moho bounce repeat regression assuming functional form that is flat between \(70\) and \(90\,\mathrm{km}\) but this produced a smaller likelihood. Conclude that data does not support significant flattening at \(<100\,\mathrm{km}\).

• Note that model is preliminary.

Taylor Castillo et al. (1992)

• Ground-motion model is: \[\ln (A)=a_1+a_2 M_s+a_3 \ln(R)+a_4 R\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(a_1=0.339\), \(a_2=0.455\), \(a_3=-0.67\), \(a_4=-0.00207\) and \(\sigma=0.61\).

Tento, Franceschina, and Marcellini (1992)

• Ground-motion model is: \[\begin{aligned} \ln \mathrm{PGA}&=&b_1+b_2M+b_3R-\ln R\\ \mbox{where }R&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{gal}\), \(b_1=4.73\), \(b_2=0.52\), \(b_3=-0.00216\), \(h\) is mean focal depth of group into which each earthquake is classified and \(\sigma=0.67\).

• Most records from distances between \(10\,\mathrm{km}\) and \(40\,\mathrm{km}\).

• Correction technique based on uniform Caltech correction procedure. Most (125) were automatically digitised, rest were manually digitised. Roll-on and cutoff frequencies of Ormsby filter were selected by adopting a record dependent criteria. Cutoff frequencies range between \(0.13\,\mathrm{Hz}\) and \(1.18\,\mathrm{Hz}\) with a median of \(0.38\,\mathrm{Hz}\).

• Records included from analysis were from free-field stations. Excluded those not complete (e.g. started during strong-motion phase). Excluded those with epicentral distances greater than that of first nontriggered station.

• Note relatively small influence of form of equation adopted although two step method seems preferable.

• Note correction procedure plays a relevant role in analysis.

• Note using \(d\) instead of \(R\) causes greater scatter in data.

• Note moderate underestimation for low magnitude in near field and for high magnitude in far field.

Theodulidis and Papazachos (1992)

• Ground-motion model is: \[\ln Y=C_1+C_2 M+C_3 \ln (R+R_0)+C_4 S\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(C_1=3.88\), \(C_2=1.12\), \(C_3=-1.65\), \(R_0=15\), \(C_4=0.41\) and \(\sigma=0.71\).

• Use two site categories (mean opinion of seven specialists who classified sites into three categories: soft alluvium, crystalline rock and intermediate):

  1. Rock: 34+4 records. Japanese sites have diluvium with depth to bedrock \(H<10\,\mathrm{m}\). Alaskan sites have \(\mathrm{PGV}/\mathrm{PGA} \approx 66\pm 7\,\mathrm{cms^{-1}g^{-1}}\).

  2. Alluvium: 71+12 records. Japanese sites have diluvium \(H>10\,\mathrm{m}\) or alluvium \(H<10\,\mathrm{m}\), and alluvium with \(H<25\,\mathrm{m}\) as well as soft layers with thickness \(<5\,\mathrm{m}\). Alaskan sites have \(\mathrm{PGV}/\mathrm{PGA} > 66\pm 7\,\mathrm{cms^{-1}g^{-1}}\).

• \(70\%\) of records from ground level or basement of buildings with two storeys or less. Rest from buildings with up to eight storeys.

• Some (16) Greek records manually digitized and baseline corrected, some (22) Greek records manually digitized and filtered and rest of the Greek records automatically digitized and filtered.

• Due to lack of data for \(7.0<M_s<7.5\) include shallow subduction data from other regions with similar seismotectonic environments (Japan and Alaska) using criteria i) depth \(<35\,\mathrm{km}\), ii) \(M_w\) or \(M_{\mathrm{JMA}}\) between \(7.0\) and \(7.5\), iii) instruments triggered before S-wave, iv) free-field recording, v) surface geology known at station. Note \(M_s\), \(M_w\) and \(M_{\mathrm{JMA}}\) are equivalent between \(6.0\) and \(8.0\).

• Focal depths between \(0\,\mathrm{km}\) (\(13\,\mathrm{km}\)) and \(18\,\mathrm{km}\) (\(31\,\mathrm{km}\)).

• Most data from \(M_s<5.5\) and from \(R<50\,\mathrm{km}\).

• Use four step regression procedure. First step use only Greek data from \(M_s>6.0\) (\(9 \leq R \leq 128\,\mathrm{km}\), 14 records) for which distances are more reliable (use both hypocentral and epicentral distance find epicentral distance gives smaller standard deviation) to find geometrical coefficient \(C_{31}\) and \(R_0\) ignoring soil conditions. Next find constant (\(C_{12}\)), magnitude (\(C_{22}\)) and soil (\(C_{42}\)) coefficients using all data. Next recalculate geometrical (\(C_{33}\)) coefficient using only Greek data with \(M_s>6.0\). Finally find constant (\(C_{14}\)), magnitude (\(C_{24}\)) and soil (\(C_{44}\)) coefficients using all the data; final coefficients are \(C_{14}\), \(C_{24}\), \(C_{33}\) and \(C_{44}\).

• Plot residuals against \(M_s\) and \(R\) and find no apparent trends. Find residuals (binned into \(0.2\) intervals) fit normal distribution.

Abrahamson and Silva (1993)

• Ground-motion model is: \[\begin{aligned} \ln \mbox{pga}_{rock}&=&\theta_1+\theta_2 M+\theta_3\ln [r+\exp(\theta_4+\theta_5M)]+\theta_{11}F_1\\ \ln \mbox{pga}_{soil}&=&\theta_6+\theta_7 M+\theta_8\ln [r+\exp(\theta_9+\theta_{10})]+\theta_{11}F_1\end{aligned}\] where \(\mbox{pga}\) is in \(\,\mathrm{g}\), \(\theta_1=-4.364\), \(\theta_2=1.016\), \(\theta_3=-1.285\), \(\theta_4=-3.34\), \(\theta_5=0.79\), \(\theta_6=-8.698\), \(\theta_7=1.654\), \(\theta_8=-1.166\), \(\theta_9=-6.80\), \(\theta_{10}=1.40\), \(\theta_{11}=0.17\), \(\sigma=0.44\), \(\tau=0.00\) (sic) and \(\sigma_{total}=0.44\).

• Originally use five site classes (chosen based on site response analyses using broad categories and generic site profiles):

  1. Rock. 78 records

  2. Shallow soil (\(<250\,\mathrm{ft}\). 25 records.)

  3. Intermediate depth soil (\(250\)–\(1000\,\mathrm{ft}\)). 5 records.

  4. Deep soil (\(>1000\,\mathrm{ft}\)). 62 records.

  5. Alluvium of unknown depth. 31 records.

  but insufficient records in shallow and intermediate classes to evaluate separately so combine rock and shallow classes and intermediate, deep and unknown depth categories to leave two classes: \(<250\,\mathrm{ft}\) and \(>250\,\mathrm{ft}\).

• Use two faulting mechanisms:

  1. Strike-slip or normal

  2. Reverse

• Based on Silva and Abrahamson (1992) (see Section 2.101.

• Only use Nahanni records for spectral ordinates and not PGA because more representative of eastern US rock than western US rock.

Boore, Joyner, and Fumal (1993), Boore, Joyner, and Fumal (1997) & Boore (2005)

• Ground-motion model is: \[\begin{aligned} \log Y&=&b_1+b_2 (\mathbf{M}-6)+b_3 (\mathbf{M}-6)^2+b_4 r+b_5\log r+b_6 G_B+b_7 G_C\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), for randomly-oriented horizontal component (or geometrical mean) \(b_1=-0.105\), \(b_2=0.229\), \(b_3=0\), \(b_4=0\), \(b_5=-0.778\), \(b_6=0.162\), \(b_7=0.251\), \(h=5.57\) and \(\sigma=0.230\) (for geometrical mean \(\sigma=0.208\)) and for larger horizontal component \(b_1=-0.038\), \(b_2=0.216\), \(b_3=0\), \(b_4=0\), \(b_5=-0.777\), \(b_6=0.158\), \(b_7=0.254\), \(h=5.48\) and \(\sigma=0.205\).

• Due to an error in Equation (3) of Boore, Joyner, and Fumal (1994a) and Equation (6) of Boore, Joyner, and Fumal (1997) \(\sigma_c\) reported in Boore, Joyner, and Fumal (1994a, 1997) are too large by a factor of \(\sqrt{2}\). Therefore correct values of standard deviations are: \(\sigma_f=0.431\), \(\sigma_c=0.160\), \(\sigma_r=0.460\), \(\sigma_s=0.184\) and \(\sigma_{\ln Y}=0.495\).

• Use three site categories:

  1. \(V_{s,30} >750\,\mathrm{m/s}\), some categorised using measured shear-wave velocity, most estimated \(\Rightarrow G_B=0, G_C=0\), 48 records

  2. \(360<V_{s,30} \leq 750\,\mathrm{m/s}\), some categorised using measured shear-wave velocity, most estimated \(\Rightarrow G_B=1, G_C=0\), 118 records.

  3. \(180<V_{s,30} \leq 360\,\mathrm{m/s}\),some categorised using measured shear-wave velocity, most estimated \(\Rightarrow G_B=0, G_C=1\), 105 records.

  where \(V_{s,30}\) is average shear-wave velocity to \(30\,\mathrm{m}\).

• Define shallow earthquakes as those for which fault rupture lies mainly above a depth of \(20\,\mathrm{km}\).

• Peak acceleration scaled directly from accelerograms, in order to avoid bias from sparsely sampled older data.

• Do not use data from structures three storeys or higher, from dam abutments or from base of bridge columns. Do not use data from more than one station with the same site condition within a circle of radius \(1\,\mathrm{km}\) (note that this is a somewhat arbitrary choice).

• Exclude records triggered by S wave.

• Do not use data beyond cutoff distance which is defined as equal to lesser of distance to the first record triggered by S wave and closest distance to an operational nontriggered instrument.

• Note that little data beyond \(80\,\mathrm{km}\).

• Due to positive values of \(b_4\) when \(b_5=-1\), set \(b_4\) to zero and let \(b_5\) vary.

Campbell (1993)

• Ground-motion model is: \[\begin{aligned} \ln (Y)&=&\beta_0+a_1 M+ \beta_1 \tanh[a_2(M-4.7)]-\ln (R^2+[a_3\exp(a_1M)]^2)^{1/2}\\ &&{}-(\beta_4+\beta_5M)R+a_4 F+[\beta_2+a_5\ln(R)]S+\beta_3\tanh(a_6D)\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(\beta_0=-3.15\), \(\beta_1=0\), \(\beta_2=0\), \(\beta_3=0\), \(\beta_4=0.0150\), \(\beta_5=-0.000995\), \(a_1=0.683\), \(a_2=0.647\), \(a_3=0.0586\), \(a_4=0.27\), \(a_5=-0.105\), \(a_6=0.620\) and \(\sigma=0.50\).

• Uses two site categories:

  1. Quaternary deposits (soil).

  2. Tertiary or older sedimentary, metamorphic, and igneous deposits (rock).

  Also includes depth to basement rock (\(\,\mathrm{km}\)), \(D\).

• Uses two fault mechanisms:

  1. Strike-slip.

  2. Reverse, reverse-oblique, thrust, and thrust-oblique.

  Recommends use \(F=0.5\) for normal or unknown mechanisms.

• Gives estimates of average minimum depths to top of seismogenic rupture zone.

• Uses stochastic simulation model to find anelastic coefficients \(\beta_4\) and \(\beta_5\) because uses only near-source records.

• Uses weighted nonlinear regression method based on Campbell (1981) to control dominance of well-recorded earthquakes.

Dowrick and Sritharan (1993)

• Ground-motion model is: \[\begin{aligned} \log y&=&\alpha +\beta \mathbf{M}-\log r +b r\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] Coefficients are unknown.

• Data from earthquakes occurring between 1987 and 1991.

Gitterman, Zaslavsky, and Shapira (1993)

• Ground-motion model is: \[\log Y=a+bM-\log \sqrt{r^2+h^2}-c r\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-5.026\), \(b=0.989\), \(h=2.7\) and \(c=0.00443\) (\(\sigma\) not reported).

• Some data from velocity sensors have been used, after differentiation, to increase amount of data at moderate and long distances.

McVerry et al. (1993) & McVerry, Dowrick, and Zhao (1995)

• Ground-motion model is (Type A): \[\log_{10} \mathrm{PGA}=a+bM_w-cr-d\log_{10}r\] where \(PGA\) is in \(\,\mathrm{g}\), \(a=-1.434\pm0.339\), \(b=0.209\pm0.036\), \(c=0.00297\pm0.00093\), \(d=-0.449\pm0.186\) and \(\sigma=0.276\).

• Find that ground motions in previous earthquakes were significantly higher than the motions predicted by equations derived from W. N. America data.

• Only include records from earthquakes for which \(M_w\) is known because of poor correlation between \(M_L\) and \(M_w\) in New Zealand.

• Focal depths, \(h_e\leq 122\,\mathrm{km}\).

• 140 records from reverse faulting earthquakes.

• Divide records into crustal and deep earthquakes.

• Only use records for which reliable event information is available, regardless of their distances with respect to untriggered instruments.

• Only use records which triggered on the P-wave.

• Also derive separate equations for shallow, upper crustal earthquakes (\(h_e\leq 20\,\mathrm{km}\), 102 records, \(5.1\leq M_w \leq 7.3\), \(13\leq r \leq 274\,\mathrm{km}\)) and crustal earthquakes (\(h_e\leq 50\,\mathrm{km}\), 169 records, \(5.1\leq M_w \leq 7.3\), \(13\leq r \leq 274\,\mathrm{km}\)).

• Also try equations of form: \(\log_{10} \mathrm{PGA}=a+bM_w-d\log_{10}r\) (Type B) and \(\log_{10} \mathrm{PGA}=a+bM_w-cr-\log_{10}r\) (Type C) because of large standard errors and highly correlated estimates for some of the coefficients (particularly \(c\) and \(d\)). Find Type B usually gives much reduced standard errors for \(d\) than Type A model and have lowest correlation between coefficients, but are sceptical of extrapolating to distance ranges shorter and longer than the range of data. Type C usually has similar standard deviations to Type A. Find that usually all three models give similar predictions over distance range of most of the data, but sometimes considerably different values at other distances.

• Derive separate equations for reverse faulting earthquakes only and usually find similar results to the combined equations.

• Find deep earthquakes produce significantly higher PGAs than shallow earthquakes for similar \(r\).

Midorikawa (1993a)

• Ground-motion model is: \[\log y=c_1M-\log(d+c_2 10^{c_1M})+c_3d+c_4\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(c_1=0.42\), \(c_2=0.025\), \(c_3=-0.0033\) and \(c_4=1.22\) (\(\sigma\) unknown).

Quijada et al. (1993)

• Ground-motion model is unknown.

• Used by Tanner and Shedlock (2004).

Singh et al. (1993)

• Ground-motion model is: \[\begin{aligned} \log (A)&=&a_1+a_2M+a_3\log [G(R_0)]+a_4R_0\\ \mbox{where }R_0^2&=&R^2+(\mathrm{e}^{a_5 M})^2\\ G(R_0)&=&R_0 \mbox{ for: } R_0 \leq 100\,\mathrm{km}\\ \mbox{and: }G(R_0)&=&\sqrt{(100 R_0)} \mbox{ for: } R_0>100\,\mathrm{km}\end{aligned}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a_1=2.74\), \(a_2=0.212\), \(a_3=-0.99\), \(a_4=-0.000943\), \(a_5=0.47\) and \(\sigma=0.26\).

• Use same data as Taylor Castillo et al. (1992).

• Employ several different regression techniques.

• Select equation found by Bayesian method (given above) for hazard study.

Steinberg et al. (1993)

• Ground-motion model is: \[\log(A_{\max})=a_1 M+a_2 \log(D+a_3)+a_4\] where \(A_{\max}\) is in \(\,\mathrm{cm/s^2}\), \(a_1=0.54\), \(a_2=-1.5\), \(a_3=10\) and \(a_4=1.25\) (\(\sigma\) not reported).

Sun and Peng (1993)

• Ground-motion model is: \[\ln A = a+b M -c \ln (R+h) +d T_s\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a=7.7\), \(b=0.49\), \(c=1.45\), \(d=0.19\), \(h=25.0\) and \(\sigma=0.46\).

• Model soil using its fundamental period of the overburden soil, \(T_s\). Thickness of deposit defined as depth to rock base, defined either as \(V_s > 800\,\mathrm{m/s}\) or when ratio of shear-wave velocity in \(i\)th layer to shear-wave velocity in \(i-1\)th layer is greater than 2 (only calculate period to \(100\,\mathrm{m}\) because only have important effect on structure). For outcropping rock, \(T_s=0.05\,\mathrm{s}\).

• Eight distance intervals used for weighting, five \(10\,\mathrm{km}\) wide up to \(50\,\mathrm{km}\), \(50\)–\(69.9\,\mathrm{km}\), \(70\)–\(99.9\,\mathrm{km}\) and \(100\)–\(200\,\mathrm{km}\). Within each interval each earthquake received equal weight, inversely proportional to number of records from that earthquake in interval.

• Use resolve accelerations in direction, \(\theta\), which gives largest value. Find scatter is lower than for larger horizontal component.

• Many (27) earthquakes only have one record associated with them and 60 records are from San Fernando.

Ambraseys and Srbulov (1994)

• Ground-motion model is: \[\begin{aligned} \log a&=&b_1+b_2M_s+b_3r+b_4\log r\\ \mbox{where }r&=&(d^2+h_0^2)^{0.5}\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), \(b_1=-1.58\), \(b_2=0.260\), \(b_3=-0.00346\), \(b_4=-0.625\), \(h_0=4\) and \(\sigma=0.26\).

• Do not consider effect of site geology but expect it to be statistically insignificant for PGA.

• Focal depths, \(h<25\,\mathrm{km}\). Mean focal depth is \(10\pm 4\,\mathrm{km}\).

• Mean magnitude of earthquakes considered is \(6.0\pm 0.7\).

• Most records from \(d<100\,\mathrm{km}\).

• Only use records with \(\mathrm{PGA}>0.01\,\mathrm{g}\).

• Records mainly from SMA-1s located at ground floor or in basements of buildings and structures and free-field sites regardless of topography.

• Records from thrust earthquakes (\(46\%\) of total), normal earthquakes (\(26\%\)) and strike-slip earthquakes (\(28\%\)).

• Baseline correct and low-pass filter records. Select cut-offs from visual examination of Fourier amplitude spectrum of uncorrected time-histories and choose cut-off below which the Fourier amplitude spectrum showed an unrealistic energy increase due to digitization noise and instrument distortions.

• Find (from reprocessing about 300 records) that with very few exceptions differences in PGAs arising from different methods of processing are not significant, remaining below \(3\%\).

• Also derive equation which includes focal depth explicitly.

Boore, Joyner, and Fumal (1994a) & Boore, Joyner, and Fumal (1997)

• Based on Boore, Joyner, and Fumal (1993) see Section 2.106

• Ground-motion model is: \[\begin{aligned} \log Y&=&b_1+b_2 (\mathbf{M}-6)+b_3 (\mathbf{M}-6)^2+b_4 r+b_5\log r+b_V (\log V_S -\log V_A)\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1\) to \(b_5\), \(h\) and \(\sigma\) are same as for Boore, Joyner, and Fumal (1993) (see Section 2.106) and for randomly oriented component \(b_V=-0.371\) and \(V_A=1400\) and for larger horizontal component \(b_V=-0.364\) and \(V_A=1390\).

• Model site effect as a continuous function of average shear-wave velocity to \(30\,\mathrm{m}\) deep, \(V_S\).

• Coefficients \(b_1\), \(b_2\), \(b_3\),\(b_4\) and \(b_5\) from Boore, Joyner, and Fumal (1993).

• Find no basis for different magnitude scaling at different distances.

• Find evidence for magnitude dependent uncertainty.

• Find evidence for amplitude dependent uncertainty.

• Find marginal statistical significance for a difference between strike-slip (defined as those with a rake angle within \(30^{\circ}\) of horizontal) and reverse-slip motions but do not model it. Modelled in Boore, Joyner, and Fumal (1994b) (by replacing \(b_1\) by \(b_{SS}G_{SS}+b_{RS}G_{RS}\) where \(G_{SS}=1\) for strike-slip shocks and \(0\) otherwise and \(G_{RS}=1\) for reverse-slip shocks and \(0\) otherwise) and reported in Boore, Joyner, and Fumal (1997). Coefficients for randomly oriented horizontal component are: \(b_{SS}=-0.136\) and \(b_{RS}=-0.051\)¹⁴.

• Analysis done using one and two-stage maximum likelihood methods; note that results are very similar.

• Earthquakes with magnitudes below \(6.0\) are poorly represented.

• Note that few Class A records.

• Note that \(V_S\) does not model all the effects of site because it does not model effect of the thickness of attenuating material on motion.

• Note that ideally would like to model site in terms of average shear-wave velocity to one-quarter wavelength.

• Note lack measurements from distances greater than \(100\,\mathrm{km}\) so that weak-motion data from seismographic stations maybe should be used.

• Note that use of cutoff distances independent of geology or azimuth may be over strict but it is simple and objective. Note that methods based on data from nontriggered stations or using seismogram data may be better.

El Hassan (1994)

• Ground-motion model is: \[\log a=C_1+C_2M+C_3 \log(R+C_4)\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(C_1=8.65\), \(C_2=0.71\), \(C_3=-1.6\), \(C_4=40\) and \(\sigma=0.6\).

• May not be an empirical GMPE but derived through a intensity-PGA relations.

Fat-Helbary and Ohta (1994)

• Ground-motion model is: \[\ln A=a_1 M+a_2 \ln \Delta +a_3\] where \(A\) is in \(\,\mathrm{gal}\), \(a_1=1.812\), \(a_2=-0.796\), \(a_3=-3.616\) and \(\sigma=0.558\).

• Use velocity data recorded at GRW. Differentiate to acceleration. Use frequency range \(1\) to \(6\,\mathrm{Hz}\).

Fukushima, Gariel, and Tanaka (1994) & Fukushima, Gariel, and Tanaka (1995)

• Ground-motion model is: \[\log Y=aM+bX-\log X+\sum \delta_i c_i\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(\delta_i=1\) at \(i\)th receiver and \(0\) otherwise, for horizontal PGA \(a=0.918\) and \(b=-0.00846\) (\(\sigma\) not given) and for vertical PGA \(a=0.865\) and \(b=-0.00741\) (\(\sigma\) not given). \(c_i\) given in paper but are not reported here due to lack of space.

• Data from three vertical arrays in Japan so predictions at surface and at different depths down to \(950\,\mathrm{m}\).

• Different definition of \(M_{\mathrm{JMA}}\) for focal depths \(>60\,\mathrm{km}\) so exclude such data. Focal depths between \(2\) and \(60\,\mathrm{km}\).

• Exclude data from earthquakes \(M<5.0\) because errors are larger for smaller events.

• Exclude data for which predicted, using a previous attenuation relation, \(\mathrm{PGV}<0.1\,\mathrm{cm/s}\) in order to find precise attenuation rate.

• Most data from earthquakes with \(M\leq 6.0\) and most from \(X\leq 100\,\mathrm{km}\).

• Records low-pass filtered with cutoff frequency \(25\,\mathrm{Hz}\) for records from 2 sites and \(30\,\mathrm{Hz}\) for records from 1 site.

• Use two-stage method because positive correlation between \(M\) and \(X\). Also apply one step; find it is biased and two-stage method is most effective method to correct bias.

• Check residuals (not shown) against \(M\) and \(X\) find no remarkable bias.

Lawson and Krawinkler (1994)

• Ground-motion model is: \[\log Y=a+b(M-6)+c(M-6)^2+d \sqrt{R^2+h^2}+e\log \sqrt{R^2+h^2}+fS_B+gS_C\]

• Use three site categories:

  1. Firm to hard rock: granite, igneous rocks, sandstones and shales with close to widely spaced fractures, \(750\leq V_{s,30}\leq 1400\,\mathrm{m/s}\) \(\Rightarrow S_B=0\), \(S_C=0\).

  2. Gravelly soils and soft to firm rocks: soft igneous rocks, sandstones and shales, gravels and soils with \(>20\%\) gravel, \(360\leq V_{s,30}\leq 750\,\mathrm{m/s}\) \(\Rightarrow S_B=1\) , \(S_C=0\).

  3. Stiff clays and sandy soils: loose to very dense sands, silt loams and sandy clays, and medium stiff to hard clay and silty clays (\(N>5\,\mathrm{blows/ft}\)), \(180\leq V_{s,30}\leq360\,\mathrm{m/s}\) \(\Rightarrow S_B=0\), \(S_C=1\).

• For shallow (fault rupture within \(20\,\mathrm{km}\) of earth surface) crustal earthquakes.

• Use free-field records. Records not significantly contaminated by structural feedback, excludes records from structures with \(>\)2 stories.

• Chooses Ground-motion model because of simplicity. Note that other possible forms of equation may have significant effect on results, but including more terms complicates relationships without reducing variability.

• Do not give coefficients only predictions.

Lungu et al. (1994)

• Ground-motion model is: \[\ln \mathrm{PGA}=c_1+c_2M_w+c_3 \ln R+c_4 h\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(c_1=-2.122\), \(c_2=1.885\), \(c_3=-1.011\), \(c_4=-0.012\) and \(\sigma=0.502\).

• Focal depth, \(h\), between \(79\) and \(131\,\mathrm{km}\).

• Consider to separate areas of \(90^{\circ}\) to investigate variation with respect to azimuth; find azimuthal dependence.

• Find individual attenuation equations for three earthquakes. Note faster attenuation for smaller magnitude and faster attenuation for deeper events.

Musson, Marrow, and Winter (1994)

• Ground-motion model is (model 1): \[\ln A=a+b M-\ln (R)+d R\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a=2.11\), \(b=1.23\) and \(d=-0.014\).

  Ground-motion model is (model 2): \[\begin{aligned} \ln A&=&c_1+c_2 M+c_4 R +\ln G(R,R_0)\\ \mbox{where } G(R,R_0)&=&R^{-1} \quad \mbox{for} \quad R \leq R_0\\ \mbox{and: } G(R,R_0)&=&R_0^{-1}\frac{R_0}{R}^{5/6} \quad \mbox{for} \quad R>R_0\end{aligned}\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(c_1\) and \(c_2\) are from Dahle, Bungum, and Kvamme (1990), \(c_4=-0.0148\) and \(\sigma\) is recommended as \(0.65\) (although this is from an earlier study and is not calculated in regression).

• Use data from Canada (Saguenay earthquake and Nahanni sequence) and Belgium (Roermond earthquake).

• Focal depths, \(h\), between \(1\) and \(30\,\mathrm{km}\) with average \(14.4\,\mathrm{km}\).

• Assume peak ground acceleration equals pseudo-acceleration at \(30\,\mathrm{Hz}\) due to few unclipped horizontal records and because instrument response of instruments means records unreliable above \(30\,\mathrm{Hz}\). Use only digital records for \(30\,\mathrm{Hz}\) model.

• Note poorness of data due to data and other data being widely separated thus preventing a comparison between the two sets. Also means straightforward regression methods would be inadequate as there would be little control on shape of curves derived.

• Note earlier models over predict data.

• Use two-stage least squares method to give model 1. First stage fit only /Belgian data to find \(b\), in second stage use this value of \(b\) and use all data to find \(a\) and \(d\).

• Do not recommend model 1 for general use because too influenced by limitations of data to be considered reliable. Canadian data probably insufficient to anchor curves at small \(R\)/large \(M\) and extremely high Saguenay earthquake records carry undue weight.

• Use model of Dahle, Bungum, and Kvamme (1990) to get model 2. Fix \(c_1\) and \(c_2\) to those of Dahle, Bungum, and Kvamme (1990) and find \(c_4\). Prefer this model.

Radu et al. (1994), Lungu, Coman, and Moldoveanu (1995) & Lungu et al. (1996)

• Ground-motion model is: \[\ln \mathrm{PGA}=c_1+c_2 M+c_3 \ln R+c_4 h\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(c_1=5.432\), \(c_2=1.035\), \(c_3=-1.358\), \(c_4=-0.0072\) and \(\sigma=0.397\).

• Sites have different soil conditions, some medium and stiff sites and some very soft soil sites.

• Use some records from Moldova and Bulgaria.

• Focal depths, \(h\), between \(91\) and \(133\,\mathrm{km}\).

• Records from free-field or from basements of buildings.

• Originally include data from a shallower (focal depth \(79\,\mathrm{km}\)), smaller magnitude (\(M_L=6.1\), \(M_w=6.3\)) earthquake with shorter return period than other three earthquakes, but exclude in final analysis.

• Originally do attenuation analysis for two orthogonal directions N45E (which is in direction of fault plane) and N35E (which is normal to fault plane). From this define 3 \(90^{\circ}\) circular sectors based roughly on tectonic regions, and calculate attenuation relations for each of these sectors as well as for all data. Find azimuthal dependence.

• Remove 1 to 3 anomalous records per sector.

• Remove the only record from the 4/3/1977 earthquake, because it has a strong influence on results, and repeat analysis using model \(\ln \mathrm{PGA}=b_1+b_2M+b_3\ln R\), find lower predicted PGA.

• Find slower attenuation in direction of fault plane compared with normal to fault plane.

• Find faster attenuation and larger standard deviation (by finding attenuation equations for two different earthquakes) for deeper focus and larger magnitude shocks.

Ramazi and Schenk (1994)

• Ground-motion model is: \[\begin{aligned} a_h&=&a_1 (a_2+d+H)^{a_5} \exp(a_6 M_s)\\ H&=&|d-a_3|^{a_4}\end{aligned}\] where for horizontal peak acceleration \(a_h\) is in \(\,\mathrm{cm/s^2}\), \(a_1=4000\), \(a_2=20\), \(a_3=16\) and \(a_4=0.63\) for soil sites \(a_5=-2.02\) and \(a_6=0.8\) and for rock sites \(a_5=-2.11\) and \(a_6=0.79\) (\(\sigma\) not given). For vertical peak acceleration on soil sites \(a_v\) is in \(\,\mathrm{cm/s^2}\) \(a_1\) to \(a_3\) are same as horizontal and \(a_4=0.48\), \(a_5=-1.75\) and \(a_6=0.53\) (\(\sigma\) not given).

• Use two site categories (from original of four) for which derive two separate equations:

  1. Rock: mainly category (2) a) loose igneous rocks (tuffs), friable sedimentary rocks, foliated metamorphic rock and rocks which have been loosened by weathering, b) conglomerate beds, compacted sand and gravel and stiff clay (argillite) beds where soil thickness \(>60\,\mathrm{m}\) from bed rock. 29 records.

  2. Soil: mainly category (4) a) soft and wet deposits resulting from high level of water table, b) gravel and sand beds with weak cementation and/or uncementated unindurated clay (clay stone) where soil thickness \(>10\,\mathrm{m}\) from bed rock. 54 records.

• Focal depths between \(10\) and \(69\,\mathrm{km}\).

• Find equations using hypocentral distance but find that poor fit for Rudbar (Manjil) earthquake (\(M_s=7.7\)) which conclude due to use of hypocentral rather than rupture distance.

• Find equations using rupture distance¹⁵ for Rudbar (Manjil) earthquake and hypocentral distances for other earthquakes. Coefficients given above. They conclude that it is important that equations are derived using rupture distance rather than hypocentral distance because most destructive earthquakes rupture surface in Iran.

• Do not know physical meaning of \(H\) term but find that it causes curves to fit data better.

Xiang and Gao (1994)

• Ground-motion model is: \[A_p=a \mathrm{e}^{b M_s} (R+\Delta)^c\] where \(A_p\) is in \(\,\mathrm{cm/s^2}\) and for combined Yunnan and W. N. American data \(a=1291.07\), \(b=0.5275\), \(c=-1.5785\), \(\Delta=15\) and \(\sigma=0.5203\) (in terms of natural logarithm).

• All records from basement rock.

• Most Yunnan data from main and aftershocks of Luquan and Luncang-Gengma earthquakes.

• Records from Lancang-Gengma sequence corrected.

• Most Yunnan records with \(3 \leq M_s \leq 5\) and \(10 \leq R \leq 40\,\mathrm{km}\).

• To overcome difficulty due to shortage of large magnitude records and sample heterogeneous distribution in near and far fields use W. N. America data, because intensity attenuation is similar.

• Fit curves to Yunnan and Yunnan with W. N. American data. Find curve for combined data has lower variance and fit to observation data for large magnitudes is better (by plotting predicted and observed PGA).

Aman, Singh, and Singh (1995)

• Ground-motion model is: \[\log(a^{1/M})=b_1-b_3 \log(R)\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(b_1=0.433\), \(b_3=0.073\) and \(\sigma=0.037\).

• Data from three earthquakes with \(M_B\) of \(5.7\), one of \(M_B\) of \(5.8\) and the other \(M_B\) of \(7.2\).

• Compare predicted and observed ground motions for 20/10/1991 Uttarkashi earthquake (\(M 6.1\)) and find good fit.

N. N. Ambraseys (1995)

• Ground-motion model is: \[\begin{aligned} \log a&=&A+B M_s+C r +D \log r\\ \mbox{where }r^2&=&d^2+h_0^2\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), for \(4.0\leq M \leq 7.4\): for horizontal PGA not including focal depth \(A=-1.43\), \(B=0.245\), \(C=-0.0010\), \(D=-0.786\), \(h_0=2.7\) and \(\sigma=0.24\), for vertical PGA not including focal depth \(A=-1.72\), \(B=0.243\), \(C=-0.00174\), \(D=-0.750\), \(h_0=1.9\) and \(\sigma=0.24\), for horizontal PGA including focal depth \(A=-1.06\), \(B=0.245\), \(C=-0.00045\), \(D=-1.016\), \(h_0=h\) and \(\sigma=0.25\) and for vertical PGA including focal depth \(A=-1.33\), \(B=0.248\), \(C=-0.00110\), \(D=-1.000\), \(h_0=h\) and \(\sigma=0.25\).

• Reviews and re-evaluates distances, focal depths, magnitudes and PGAs because data from variety of sources with different accuracy and reliability. For \(M_s>6.0\) distances have acceptable accuracy but for \(M_s<6.0\) distance, depths and magnitudes are poorly known. Errors in locations for \(M_s<6.0\) still large with no foreseeable means of improving them. Use of \(r_{epi}\) for \(M_s<6.0\) justified because difference between \(r_{jb}\) and \(r_{epi}\) for small earthquakes is not larger than uncertainty in epicentre. Check and redetermine station locations; find large differences in excess of \(15\,\mathrm{km}\) for some stations.

• Focal depths poorly determined. Revises 180 depths using S-start times (time between P and S-wave arrival).

• Focal depths \(h<26\,\mathrm{km}\); most (60%+) between \(4\) and \(14\,\mathrm{km}\).

• Does not use \(M_L\) because no \(M_L\) values for Algeria, Iran, Pakistan, Turkey and former USSR and unreliable for other regions. Does not use magnitude calculated from strong-motion records because magnitude calculation requires point source approximation to be valid. Conversion from \(M_L\) to \(M_s\) should not be done because of uncertainty in conversion which should be retained.

• Notes that \(M_s\) results in nonlinear scaling on PGA with \(M_w\) due to nonlinear relationship between \(\log M_0\) and \(M_s\).

• Uses PGAs in four forms: maximum values from accelerograms read by others (34%), from corrected records (30%), scaled directly from accelerograms (13%) and from digitised plots (23%). Notes potential bias in using both corrected and uncorrected PGAs but neglects it because small difference (\(\lesssim 4\%\) for those checked). Excludes PGAs near trigger level because processing errors can be large. Some unfiltered digital records which require additional processing to simulate SMA-1 could be associated with larger differences (\(\lesssim 10\%\)).

• Excludes records from basements and ground floors of structures with more than 3 levels. Retains the few records from dam abutments and tunnel portals.

• Excludes records generated by close small magnitude earthquakes triggered by S-wave.

• Does not exclude records obtained at distances greater than shortest distance to an operational but not triggered instrument because of non-constant or unknown trigger levels and possible malfunctions of instruments.

• Uses weighted regression of Joyner and Boore (1988) for second stage.

• Splits data into five magnitude dependent subsets: \(2.0 \leq M_s \leq 7.3\) (1260 records from 619 shocks), \(3.0 \leq M_s \leq 7.3\) (1189 records from 561 shocks), \(4.0 \leq M_s \leq 7.3\) (830 records from 334 shocks), , \(5.0 \leq M_s \leq 7.3\) (434 records from 107 shocks), and \(3.0 \leq M_s \leq 6.0\) (976 records from 524 shocks). Calculates coefficients for each subset. Finds only small differences \(\pm 15\%\) over distance range \(1\)–\(200\,\mathrm{km}\) between predictions and uncertainties. Concludes results stable. Prefers results from subset with \(4.0 \leq M_s \leq 7.3\).

• Finds it difficult to obtain some vertical accelerations due to low ground motion so ignores data from \(>100\,\mathrm{km}\) with PGA \(<1\%\mathrm{g}\) (\(0.1\,\mathrm{m/s^2}\)).

• Repeats regression using \(r^2=d^2+h^2\). Finds depth important.

• Calculates using one-stage method; finds very similar results for \(10 < d <100\,\mathrm{km}\).

• Considers magnitude dependent function: \(\log a = b_1+b_2 M_s +b_3 r +b_4 [r+b_5 \exp (b_6 M_s)]\). Finds \(b_5\) is zero so drops \(b_3\) and repeats. Finds \(b_5\) close to zero so magnitude dependent function not valid for this dataset.

• Local shear-wave velocity, \(V_s\), profiles known for 44 stations (268 records from 132 earthquakes between \(2.5\) and \(7.2\)) although only 14 from \(>40\,\mathrm{km}\) so barely sufficient to derive equation. Use 145 records from 50 earthquakes with \(M_s>4.0\) to fit \(\log a=A+B M_s +Cr +D \log r +E \log V_{s30}\), where \(V_{s30}\) is average shear-wave velocity to reference depth of \(30\,\mathrm{m}\). Finds \(C\) positive so constrain to zero. Find no reduction in standard deviation.

• Uses residuals from main equation to find \(E\). Notes that should not be used because of small number of records. Considers different choices of reference depth; finds using between \(5\) and \(10\,\mathrm{m}\) leads to higher predicted amplifications. Notes better to use \(V_{s30}\) because no need for subjective selection of categories.

Dahle et al. (1995)

• Ground-motion model is: \[\begin{aligned} \ln A&=&c_1+c_2 M_w+c_3 \ln R+c_4 R+c_5 S\\ \mbox{with: }R&=&\sqrt{r^2+r_h^2}\end{aligned}\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(c_1=-1.579\), \(c_2=0.554\), \(c_3=-0.560\), \(c_4=-0.0032\), \(c_5=0.326\), \(r_h=6\) and \(\sigma=0.3535\)

• Use records from Costa Rica, Mexico, Nicaragua and El Salvador. Only Mexican earthquakes with \(M_w \geq 6.5\) were used.

• Use two site categories:

  1. Rock: 92 records

  2. Soil: 88 records

• Use a Bayesian one-stage regression method (Ordaz, Singh, and Arciniega 1994) to yield physically possible coefficients.

• Consider tectonic type: subduction or shallow crustal but do not model.

• Find no significant difference between Guerrero (Mexico) and other data.

• Find no significant difference between subduction and shallow crustal data.

V. W. Lee, Trifunac, Todorovska, et al. (1995)

• Ground-motion models are (if define site in terms of local geological site classification): \[\log a_{\max}=M+\mathrm{Att}(\Delta/L,M,T)+b_1M+b_2s+b_3v+b_4+b_5M^2+\sum_i b_6^i S_L^i+b_{70} r R+b_{71} (1-r) R\] or (if define site in terms of depth of sediment): \[\log a_{\max}=M+\mathrm{Att}(\Delta/L,M,T)+b_1M+b_2h+b_3v+b_4+b_5M^2+\sum_i b_6^i S_L^i+b_{70} r R+b_{71} (1-r) R\] where: \[\begin{aligned} \mathrm{Att}(\Delta,M,T)&=&\left\{ \begin{array}{r} b_0\log_{10} \Delta \quad \mbox{for}\quad R\leq R_{\max}\\ b_0 \log_{10} \Delta_{\max}-(R-R_{\max})/200 \quad \mbox{for}\quad R>R_{\max}\\ \end{array} \right.\\ \Delta&=&S\left(\ln\frac{R^2+H^2+S^2}{R^2+H^2+S_0^2}\right)^{-1/2}\\ \Delta_{\max}&=&\Delta(R_{\max},H,S)\\ R_{\max}&=&\frac{1}{2}(-\beta+\sqrt{\beta^2-4H^2})\end{aligned}\] \(S_0\) is correlation radius of source function and can be approximated by \(S_0\sim \beta T/2\) (for PGA assume \(T\approx 0.1\,\mathrm{s}\) so use \(S_0=0.1\,\mathrm{km}\)), \(\beta\) is shear-wave velocity in source region, \(T\) is period, \(S\) is ‘source dimension’ approximated by \(S=0.2\) for \(M<3\) and \(S=-25.34+8.51M\) for \(3 \leq M \leq 7.25\), \(L\) is rupture length of earthquake approximated by \(L=0.01 \times 10^{0.5M}\) \(\,\mathrm{km}\) and \(v\) is component direction (\(v=0\) for horizontal \(1\) for vertical). Different \(b_0\), \(b_{70}\) and \(b_{71}\) are calculated for five different path categories. Coefficients are not reported here due to lack of space.

• Use four types of site parameter:

  • Local geological site classification (defined for all records):

    1. Sites on sediments.

    2. Intermediate sites.

    3. Sites on basement rock.

  • Depth of sediments from surface to geological basement rock beneath site, \(h\) (defined for 1675 records out of 1926).

  • Local soil type parameter describes average soil stiffness in top \(100\)–\(200\,\mathrm{m}\) (defined for 1456 records out of 1926):

    1. ‘Rock’ soil sites \(\Rightarrow\) \(S_L^1=1\), \(S_L^2=0\) and \(S_L^3=0\). Characterises soil up to depth of less than \(10\,\mathrm{m}\).

    2. Stiff soil sites \(\Rightarrow\) \(S_L^1=1\), \(S_L^2=0\) and \(S_L^3=0\) (shear-wave velocities \(<800\,\mathrm{m/s}\) up to depth of \(75\)–\(100\,\mathrm{m}\)).

    3. Deep soil sites \(\Rightarrow\) \(S_L^2=1\), \(S_L^1=0\) and \(S_L^3=0\). (shear-wave velocities \(<800\,\mathrm{m/s}\) up to depth of \(150\)–\(200\,\mathrm{m}\)).

    4. Deep cohesionless soil sites \(\Rightarrow\) \(S_L^3=1\), \(S_L^1=0\) and \(S_L^2=0\) (only use for one site with 10 records).

  • Average soil velocity in top \(30\,\mathrm{m}\), \(v_L\) (if unavailable then use soil velocity parameter, \(s_T\)) (defined for 1572 records out of 1926):

    1. \(v_L>750\,\mathrm{m/s}\).

    2. \(360\,\mathrm{m/s}<v_L\leq 750\,\mathrm{m/s}\).

    3. \(180\,\mathrm{m/s}<v_L\leq 360\,\mathrm{m/s}\).

    4. \(v_L\leq 180\,\mathrm{m/s}\).

• Only include records for which significant subset of site parameters (\(s\), \(h\), \(s_L\), \(v_L\)) exist.

• Almost all earthquakes have focal depths \(H \leq 15\,\mathrm{km}\); all focal depths \(H\leq 43\,\mathrm{km}\).

• Use records from 138 aftershocks of Imperial Valley earthquake (15/10/1979), which contribute most of \(M\leq 3\) records.

• Use records from 109 earthquakes with \(M\leq 3\).

• Use free-field records.

• Characterise path by two methods:

  • Fraction of wave path travelled through geological basement rock measured at surface, from epicentre to station, \(0\leq r \leq 1\).

  • Generalised path type classification:

    1. Sediments to sediments.

    2. Rock-to-sediments, vertically.

    3. Rock-to-sediments, horizontally.

    4. Rock-to-rock.

    5. Rock-to-rock through sediments, vertically.

    6. Rock-to-sediments through rock and sediments, vertically.

    7. Rock-to-sediments though rock and sediments, horizontally.

    8. Rock-to-rock through sediments, horizontally.

    Due to lack of data combine path types 2 and 6 in new category 2’, combine path types 3 and 7 in new category 3’, combine path types 4, 5 and 8 in new category 4’ (when \(r\neq 1\)) and combine 4, 5 and 8 in new category 5’ (when \(r=1\)).

• Plot PGA against magnitude and distance to get surface by interpolation. Plot without smoothing and with light and intense smoothing. Find for small magnitude (\(M\approx 3\)–\(4\)) earthquakes attenuation is faster than for large magnitude (\(M\approx 6\)–\(7\)) earthquakes.

• Use a multi-step residue regression method. First fit \(\log a_{\max}=M+\mathrm{Att}(\Delta,M,T)+b_1M+b_2s+b_3v+b_4+b_5 M^2\) (or \(\log a_{\max}=M+\mathrm{Att}(\Delta,M,T)+b_1M+b_2h+b_3v+b_4+b_5 M^2\)) and calculate residuals \(\epsilon=\log a_{\max}-\log \hat{a}_{\max}\) where \(a_{\max}\) is estimated PGA and \(\hat{a}_{\max}\) is recorded PGA. Fit \(\epsilon=b_7^{(-1)}S_L^{(-1)}+b_7^{(0)}S_L^{(0)}+b_7^{(1)}S_L^{(1)}+b_7^{(2)}S_L^{(2)}+b_7^{(3)}S_L^{(3)}\) where \(S_L^{(i)}=1\) if \(s_L=i\) and \(S_L^{(i)}=0\) otherwise. Find significant dependence. Try including \(v_L\) both as a continuous and discrete parameter in model but not significant at \(5\%\) significance level. Next calculate residuals from last stage and fit \(\epsilon=b_0' \log_{10} (\Delta/L)+b_4'+b_{60}rR+b_{61}(1-r)R\) for each of the five path type groups (1’ to 5’). Lastly combine all the individual results together into final equation.

• Note that \(b_{70}\) and \(b_{71}\) can only be applied for \(R\lesssim 100\,\mathrm{km}\) where data is currently available. For \(R\gtrsim 100\,\mathrm{km}\) the predominant wave type changes to surface waves and so \(b_{70}\) and \(b_{71}\) do not apply.

Lungu et al. (1995)

• Study almost identical to Radu et al. (1994), see Section 2.124, but different coefficients given: \(c_1=3.672\), \(c_2=1.318\), \(c_3=-1.349\), \(c_4=-0.0093\) and \(\sigma=0.395\).

Molas and Yamazaki (1995)

• Ground-motion model is: \[\log y=b_0+b_1M+b_2r+b_3\log r+b_4 h+c_i\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(b_0=0.206\), \(b_1=0.477\), \(b_2=-0.00144\), \(b_3=-1\), \(b_4=0.00311\), \(\sigma=0.276\) and \(c_i\) is site coefficient for site \(i\) (use 76 sites), given in paper but are not reported here due to lack of space.

• Records from accelerometers on small foundations detached from structures; thus consider as free-field.

• Exclude records with one horizontal component with \(\mathrm{PGA}<1\,\mathrm{cm/s^2}[0.01\,\mathrm{m/s^2}]\) because weaker records not reliable due to resolution (\(\pm 0.03\,\mathrm{cm/s^2}[0.0003\,\mathrm{m/s^2}]\)) of instruments.

• Exclude earthquakes with focal depths equal to \(0\,\mathrm{km}\) or greater than \(200\,\mathrm{km}\), due to lack of such data. Depths (depth of point on fault plane closest to site), \(h\), between about \(1\,\mathrm{km}\) to \(200\,\mathrm{km}\).

• Apply a low-cut filter with cosine-shaped transition from \(0.01\) to \(0.05\,\mathrm{Hz}\).

• Positive correlation between magnitude and distance so use two-stage method.

• Note different definition for \(M_{\mathrm{JMA}}\) for focal depths \(>60\,\mathrm{km}\).

• Firstly do preliminary analysis with \(b_4=0\) and no site coefficients; find \(b_2\) is positive so constrain to \(0\) but find \(b_3<-1.0\) so constrain \(b_3\) to \(-1.0\) and unconstrain \(b_2\). Find linear dependence in residuals on \(h\) especially for \(h<100\,\mathrm{km}\). Find significant improvement in coefficient of determination, \(R^2\), using terms \(b_4h\) and \(c\).

• Find singularity in matrices if apply two-stage method, due to number of coefficients, so propose a iterative partial regression method.

• Also separate data into five depth ranges (A: \(h=0.1\) to \(30\,\mathrm{km}\), 553 records from 111 earthquakes; B: \(h=30\) to \(60\,\mathrm{km}\), 778 records from 136 earthquakes; C: \(h=60\) to \(90\,\mathrm{km}\), 526 records from 94 earthquakes; D: \(h=90\) to \(120\,\mathrm{km}\), 229 records from 31 earthquakes; E: \(h=120\) to \(200\,\mathrm{km}\), 112 records from 19 earthquakes) and find attenuation equations for each range. Note results from D & E may not be reliable due to small number of records. Find similar results from each group and all data together.

• Find weak correlation in station coefficients with soil categories, as defined in Iwasaki, Kawashima, and Saeki (1980), but note large scatter.

Sarma and Free (1995)

• Ground-motion model is: \[\begin{aligned} \log(a_h)&=&C_1+C_2M+C_3M^2+C_4\log(R)+C_5R+C_6S\\ \mbox{where }R&=&\sqrt{d^2+h_0^2}\end{aligned}\] where \(a_h\) is in \(\,\mathrm{g}\), \(C_1=-3.4360\), \(C_2=0.8532\), \(C_3=-0.0192\), \(C_4=-0.9011\), \(C_5=-0.0020\), \(C_6=-0.0316\), \(h_0=4.24\) and \(\sigma=0.424\).

• Use two site categories:

  1. Rock

  2. Soil

• Use one-stage method because of the predominance of earthquakes with single recordings in the set.

• Note that it is very important to choose a functional form based as much as possible on physical grounds because the data is sparse or non-existent for important ranges of distance and magnitude.

• Carefully verify all the distances in set.

• Use focal depths from (in order of preference): special reports (such as aftershock monitoring), local agencies and ISC and NEIS determinations. Focal depths \(<30\,\mathrm{km}\).

• Do not use \(M_L\) or \(m_b\) because of a variety of reasons. One of which is the saturation of \(M_L\) and \(m_b\) at higher magnitudes (\(M_L, m_b>6\)).

• If more than one estimate of \(M_w\) made then use average of different estimates.

• Use PGAs from: a) digital or digitised analogue records which have been baseline corrected and filtered, b) data listings of various agencies and c) other literature. Difference between PGA from different sources is found to be small.

• Also derive equations assuming \(C_3=0\) (using rock and soil records and only soil records) and \(C_3=0\), \(C_4=-1\) and \(C_6=0\) (using only rock records).

• Include records from Nahanni region and find similar results.

• Also derive equations for Australia (115 records from 86 earthquakes, \(2.4\leq M_w \leq 6.1\), \(1\leq d_e \leq 188\,\mathrm{km}\)) and N. E. China (Tangshan) (193 records from 64 earthquakes, \(3.5\leq M_w \leq 7.5\), \(2\leq d_e \leq 199\,\mathrm{km}\)) . Find considerable difference in estimated PGAs using the equations for the three different regions.

N. N. Ambraseys, Simpson, and Bommer (1996) & Simpson (1996)

• Ground-motion model is: \[\begin{aligned} \log y&=&C'_1+C_2M+C_4\log r+C_A S_A+C_S S_S\\ \mbox{where }r&=&\sqrt{d^2+h_0^2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(C'_1=-1.48\), \(C_2=0.266\), \(C_4=-0.922\), \(C_A=0.117\), \(C_S=0.124\), \(h_0=3.5\) and \(\sigma=0.25\).

• Use four site conditions but retain three (because only three records from very soft (L) soil which combine with soft (S) soil category):

  1. Rock: \(V_s>750\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=0\), 106 records.

  2. Stiff soil: \(360<V_s\leq750\,\mathrm{m/s}\), \(\Rightarrow S_A=1, S_S=0\), 226 records.

  3. Soft soil: \(180<V_s\leq360\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=1\), 81 records.

  4. Very soft soil: \(V_s\leq180\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=1\), 3 records.

• Lower limit of \(M_s=4.0\) because smaller earthquakes are generally not of engineering significance.

• Focal depths less than \(30\,\mathrm{km}\), \(81\%\) between \(5\) and \(15\,\mathrm{km}\).

• Note for some records distances have uncertainty of about \(10\,\mathrm{km}\).

• Most records from distances less than about \(40\,\mathrm{km}\).

• For some small events need to estimate \(M_s\) from other magnitude scales.

• Most records from free-field stations although some from basements or ground floors of relatively small structures, and tunnel portals. Do not exclude records from instruments beyond cutoff distance because of limited knowledge about triggered level.

• All uncorrected records plotted, checked and corrected for spurious points and baseline shifts.

• Uniform correction procedure was applied for all records. For short records (\(<5\,\mathrm{s}\)) a parabolic adjustment was made, for long records (\(>10\,\mathrm{s}\)) filtering was performed with pass band \(0.20\) to \(25\,\mathrm{Hz}\) and for intermediate records both parabolic and filtering performed and the most realistic record was chosen. Instrument correction not applied due to limited knowledge of instrument characteristics.

• Also analyze using one-stage method, note results comparable.

N. N. Ambraseys and Simpson (1996) & Simpson (1996)

• Based on N. N. Ambraseys, Simpson, and Bommer (1996), see Section 2.134.

• Coefficients are: \(C'_1=-1.74\), \(C_2=0.273\), \(C_4=-0.954\), \(C_A=0.076\), \(C_S=0.058\), \(h_0=4.7\) and \(\sigma=0.26\).

Aydan, Sedaki, and Yarar (1996) & Aydan (2001)

• Ground-motion model is: \[a_{\max}=a_1 [\exp(a_2 M_s) \exp (a_3 R)-a_4]\] where \(a_{\max}\) is in \(\,\mathrm{gal}\), \(a_1=2.8\), \(a_2=0.9\), \(a_3=-0.025\) and \(a_4=1\) (\(\sigma\) is not given).

• Most records from \(r_{hypo}>20\,\mathrm{km}\).

• Note that data from Turkey is limited and hence equation may be refined as amount of data increases.

• Also give equation to estimate ratio of vertical PGA (\(a_v\)) to horizontal PGA (\(a_h\)): \(a_v/a_h=0.217+0.046M_s\) (\(\sigma\) is not given).

Bommer et al. (1996)

• Ground-motion model is: \[\ln (A)=a+bM+d\ln(R)+qh\] where \(h\) is focal depth, \(A\) is in \(\,\mathrm{g}\), \(a=-1.47\), \(b=0.608\), \(d=-1.181\), \(q=0.0089\) and \(\sigma=0.54\).

• Only use subduction earthquakes.

• Do not recommend equation used for hazard analysis, since derive it only for investigating equations of Climent et al. (1994).

Crouse and McGuire (1996)

• Ground-motion model is: \[\ln Y=a+b M+d \ln (R+c_1 \exp\{c_2 M\})+eF\] where \(Y\) is in \(\,\mathrm{g}\), for site category B: \(a=-2.342699\), \(b=1.091713\), \(c_1=0.413033\), \(c_2=0.623255\), \(d=-1.751631\), \(e=0.087940\) and \(\sigma=0.427787\) and for site category C: \(a=-2.353903\), \(b=0.838847\), \(c_1=0.305134\), \(c_2=0.640249\), \(d=-1.310188\), \(e=-0.051707\) and \(\sigma=0.416739\).

• Use four site categories, \(\bar{V}_s\) is shear-wave velocity in upper \(100\,\mathrm{ft}\) (\(30\,\mathrm{m}\)):

  1. Rock: \(\bar{V}_s \geq 2500\,\mathrm{fps}\) (\(\bar{V}_s \geq 750\,\mathrm{m/s}\)), 33 records

  2. Soft rock or stiff soil: \(1200 \leq \bar{V}_s \leq 2500\,\mathrm{fps}\) (\(360 \leq \bar{V}_s < 750\,\mathrm{m/s}\)), 88 records

  3. Medium stiff soil: \(600 \leq \bar{V}_s < 1200\,\mathrm{fps}\) (\(180 \leq \bar{V}_s < 360\,\mathrm{m/s}\)), 101 records

  4. Soft clay: \(\bar{V}_s <600\,\mathrm{fps}\) (\(\bar{V}_s <180\,\mathrm{m/s}\)), 16 records

• Use two source mechanisms: reverse (R): \(\Rightarrow F=1\), 81 records and strike-slip (S) \(\Rightarrow F=0\), 157 records. Most (77) reverse records from \(M_s\leq 6.7\).

• Most (231) records from small building (up to 3 storeys in height) or from instrument shelters to reduce effect of soil-structure interaction. 6 records from 6 storey buildings and 1 record from a 4 storey building, included because lack of data in site or distance range of these records. Structures thought not to appreciably affect intermediate or long period and at large distances short period ground motion more greatly diminished than long period so less effect on predictions.

• Exclude records from Eureka-Ferndale area in N. California because may be associated with subduction source, which is a different tectonic regime than rest of data. Also excluded Mammoth Lake records because active volcanic region, atypical of rest of California.

• Include one record from Tarzana Cedar Hills although exclude a different record from this station due to possible topographic effects.

• Most records between \(6\leq Ms \leq 7.25\) and \(10\leq R \leq 80\,\mathrm{km}\).

• Apply weighted regression separately for site category B and C. Data space split into 4 magnitude (\(6.0\)–\(6.25\), \(6.25\)–\(6.75\), \(6.75\)–\(7.25\), \(7.25+\)) and 5 distance intervals (\(\leq 10\,\mathrm{km}\), \(10\)–\(20\,\mathrm{km}\), \(20\)–\(40\,\mathrm{km}\), \(40\)–\(80\,\mathrm{km}\), \(80\,\mathrm{km}+\)). Each recording within bin given same total weight.

• So that \(Y\) is increasing function of \(M\) and decreasing function of \(R\) for all positive \(M\) and \(R\) apply constraints. Define \(g=b/d\) and \(h=-(g+c_2)\), then rewrite equation \(\ln Y=a+d \{gM+\ln [R+c_1 \exp(c_2 M)]\}+eF\) and apply constraints \(g\leq 0\), \(d \leq 0\), \(c \geq 0\), \(c_2 \geq 0\) and \(h\geq 0\).

• Check plots of residuals (not shown in paper), find uniform distribution.

• Find \(e\) not significantly different than \(0\) and inconsistency in results between different soil classes make it difficult to attach any significance to fault type.

• Lack of records for A and D site categories. Find scale factors \(k_1=0.998638\) and \(k_2=1.200678\) so that \(Y_A=k_1 Y_B\) and \(Y_D=k_2 Y_C\), where \(Y_S\) is predicted ground motion for site class \(S\). Find no obvious dependence of \(k_1\) or \(k_2\) on acceleration from examining residuals. Find \(k_1\) and \(k_2\) not significantly different than \(1\).

• Note limited data for \(R<10\,\mathrm{km}\), advise caution for this range.

• Note equation developed to estimate site-amplification factors not for seismic hazard analysis.

Free (1996) & Free, Ambraseys, and Sarma (1998)

• Ground-motion model is: \[\begin{aligned} \log (Y)&=&C_1+C_2 \mathbf{M}+C_3 \mathbf{M}^2+C_4 \log(R)+C_5(R) +C_6(S)\\ R&=&\sqrt{d^2+h_0^2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), for \(\mathbf{M}>1.5\) using acceleration and velocity records, for horizontal PGA \(C_1=-4.2318\), \(C_2=1.1962\), \(C_3=-0.0651\), \(C_4=-1\), \(C_5=-0.0019\), \(C_6=0.261\), \(h_0=2.9\) and \(\sigma=0.432\) and for vertical PGA \(C_1=-4.1800\), \(C_2=1.0189\), \(C_3=-0.0404\), \(C_4=-1\), \(C_5=-0.0019\), \(C_6=0.163\), \(h_0=2.7\) and \(\sigma=0.415\).

• Use two site categories:

  1. Rock, H: 470 records, V: 395 records.

  2. Soil, H: 88 records, V: 83 records.

  Note that not most accurate approach but due to lack of site information consider this technique makes most consistent use of available information.

• Select data using these criteria:

  1. Epicentre and recording station must be within the stable continental region boundaries defined by Johnston and others (1994) because a) such regions form end of spectrum of regions described by ‘intraplate’ and hence allows differences with interplate regions to be seen, b) they are clearly delineated regions and c) intraplate oceanic crust is excluded.

  2. Minimum magnitude level \(\mathbf{M}=1.5\).

  3. Use records from dam abutments and downstream free-field sites but excludes records from crests, slopes, toes, galleries, or basements.

  4. Use records from acceleration and velocity instruments.

  5. Specify no minimum PGA.

  6. Specify no maximum source distance. Do not exclude records from distances greater than shortest distance to a non-triggered station.

• Data from Australia, N.W. Europe, Peninsular India and E. N. America.

• Focal depths, \(2\leq h \leq 28\,\mathrm{km}\).

• Most records from \(\mathbf{M}<4.0\).

• Visually inspect all records including integrated velocities and displacements, identify and remove traces dominated by noise, identify and correct transient errors (spikes, ramps, linear sections, back time steps and clipped peaks), identify scaling errors, identify and remove multiple event records. Linear baseline correct and elliptically filter with cut-off \(0.25\) to \(0.5\,\mathrm{Hz}\) (determine frequency by visual inspection of adjusted record) and \(33\) to \(100\,\mathrm{Hz}\) (generally pre-determined by Nyquist frequency).

• Large proportion of records from velocity time histories which differentiate to acceleration. Test time domain method (central difference technique) and frequency domain method; find very similar results. Use time domain method.

• Distribution with respect to magnitude did not allow two-stage regression technique.

• In many analyses distribution of data with respect to distance did not allow simultaneous determination of coefficients \(C_4\) and \(C_5\), for these cases constrain \(C_4\) to \(-1\).

• Test effect of minimum magnitude cut-off for two cut-offs \(\mathbf{M}=1.5\) and \(\mathbf{M}=3.5\). Find if include data from \(\mathbf{M}<3.5\) then there is substantial over prediction of amplitudes for \(d<10\,\mathrm{km}\) for large magnitudes unless include \(C_3\) term. \(C_3\) effectively accounts for large number of records from small magnitudes and so predictions using the different magnitude cut-offs are very similar over broad range of \(\mathbf{M}\) and \(d\).

• Try including focal depth, \(h\), explicitly by replacing \(h_0\) with \(h\) because \(h_0\) determined for whole set (which is dominated by small shocks at shallow depths) may not be appropriate for large earthquakes. Find improved fit at small distances but it does not result in overall improvement in fit (\(\sigma\) increases); this increase thought due to large errors in focal depth determination.

• Find larger standard deviations than those found in previous studies which note may be due to intrinsic differences between regional subsets within whole set. Repeat analysis separately for Australia (for horizontal and vertical), N. America (for horizontal and vertical) and N.W. Europe (horizontal); find reduced standard deviations (although still large), \(C_5\) varies significantly between 3 regions.

• Repeat analysis excluding velocity records.

• Also repeat analysis using only rock records.

Inan et al. (1996)

• Ground-motion model is: \[\log \mathrm{PGA}=aM+b\log R+c\] where \(\mathrm{PGA}\) is in an unknown unit but it is probably in \(\,\mathrm{gal}\), \(a=0.65\), \(b=-0.9\) and \(c=-0.44\) (\(\sigma\) not reported).

Ohno et al. (1996)

• Ground-motion model is: \[\log S(T)=a(T) M-\log X_{eq} -b(T) X_{eq} +c(T)+q \Delta{s(T)}\] where \(S(0.02)\) is in \(\,\mathrm{gal}\), \(a(0.02)=0.318\), \(b(0.02)=0.00164\) and \(c(0.02)=1.597\) (\(\Delta{s(0.02)}\) and \(\sigma\) only given in graphs).

• Use two site conditions:

  1. Pre-Quaternary: Rock (sandstone, siltstone, shale, granite, mudstone, etc.); thickness of surface soil overlying rock is less than \(10\,\mathrm{m}\); shallow soil or thin alluvium, 160 records. S-wave velocities \(>600\,\mathrm{m/s}\).

  2. Quaternary: Soil (alluvium, clay, sand, silt, loam, gravel, etc.), 336 records. S-wave velocities \(\leq 600\,\mathrm{m/s}\).

  Exclude records from very soft soil such as bay mud or artificial fill because few such records and ground motions may be strongly affected by soil nonlinearity.

• Use equivalent hypocentral distance, \(X_{eq}\), because strong motion in near-source region affected from points other than nearest point on fault plane.

• Use portion of record after initial S-wave arrival.

• Approximates PGA by spectral acceleration for period of \(0.02\,\mathrm{s}\) and \(5\%\) damping.

• Plot the amplitude factors from first stage against \(M_w\); find well represented by linear function.

Romeo, Tranfaglia, and Castenetto (1996)

• Ground-motion model is: \[\log \mathrm{PHA} = a_1+a_2M_w-\log (d^2+h^2)^{1/2} + a_3 S\] where \(\mathrm{PHA}\) is in \(\,\mathrm{g}\), \(a_1=-1.870 \pm 0.182\), \(a_2=0.366 \pm 0.032\), \(a_3=0.168 \pm 0.045\), \(h=6\,\mathrm{km}\) and \(\sigma=0.173\) for \(r_{jb}\) and \(a_1=-2.238 \pm 0.200\), \(a_2=0.438 \pm 0.035\), \(a_3=0.195 \pm 0.049\), \(h=5\,\mathrm{km}\) and \(\sigma=0.190\) for \(r_{epi}\).

• Use two site categories:

  1. Rock or stiff soils and deep alluvium.

  2. All other sites.

• Use data and functional form of Sabetta and Pugliese (1987) but use \(M_w\) instead of magnitudes used by Sabetta and Pugliese (1987).

Sarma and Srbulov (1996)

• Ground-motion model is: \[\begin{aligned} \log(A_p/\mathrm{g})&=&b_1+b_2M_s+b_3 \log r+b_4 r\\ \mbox{where } r&=&(d^2+h_0^2)^{0.5}\end{aligned}\] where \(A_p\) is in \(\,\mathrm{g}\), using both horizontal components \(b_1=-1.617\), \(b_2=0.248\), \(b_3=-0.5402\), \(b_4=-0.00392\), \(h_0=3.2\) and \(\sigma=0.26\) and for larger horizontal component \(b_1=-1.507\), \(b_2=0.240\), \(b_3=-0.542\), \(b_4=-0.00397\), \(h_0=3.0\) and \(\sigma=0.26\).

• Consider two soil categories but do not model:

  1. Rock

  2. Soil

  Classify sites without regard to depth and shear-wave velocity of deposits.

• Most records from W. USA but many from Europe and Middle East.

• Focal depths between \(2\) and \(29\,\mathrm{km}\).

• Records from instruments on ground floor or in basements of buildings and structures up to 3 storeys and at free-field sites, regardless of topography.

• Records baseline corrected and low-pass filtered using elliptic filter.

Singh, Aman, and Prasad (1996)

• Ground-motion model is: \[\log_{10} \mathrm{AGM}=b_1+0.31M-b_3 \log R\] where \(\mathrm{AGM}\) is in \(\,\mathrm{cm/s^2}\), \(b_1=1.14\) and \(b_3=0.615\) (\(\sigma\) is not given). Note there are typographical errors in the abstract.

• Data from three earthquakes with \(m_b=5.7\), one with \(m_b=5.8\) and one with \(m_b=7.2\).

• Adopt magnitude scaling coefficient (\(0.31\)) from Boore (1983).

Spudich et al. (1996) & Spudich et al. (1997)

• Ground-motion model is: \[\begin{aligned} \log_{10}Y&=&b_1+b_2(M-6)+b_3(M-6)^2+b_4R+b_5\log_{10}R+b_6\Gamma\\ \mbox{where }R&=&\sqrt{r_{jb}^2+h^2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1=0.156\), \(b_2=0.229\), \(b_3=0\), \(b_4=0\), \(b_5=-0.945\), \(b_6=0.077\), \(h=5.57\), \(\sigma=\sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2}\) where \(\sigma_1=0.216\), \(\sigma_2=0\), for randomly orientated component \(\sigma_3=0.094\) and for geometric mean \(\sigma_3=0\).

• Use two site categories (following classification of Joyner and Boore (1981)):

  1. Rock: 35 records

  2. Soil: 93 records

• Applicable for extensional regimes, i.e. those regions where lithosphere is expanding areally.

• Reject records from structures of more than two storeys or from deeply embedded basements or those which triggered on S wave.

• Include records from those instruments beyond cutoff distance, i.e. beyond first instrument which did not trigger.

• Correction technique based on uniform correction and processing. Determine passband for filtering based on visual inspection of Fourier amplitude spectra and doubly-integrated displacements. Apply instrument correction.

• Not enough data to be able to find all coefficients so use \(b_2\) and \(b_3\) from Boore, Joyner, and Fumal (1994a)

• Note that should only be used in distance range \(0\) to \(70\,\mathrm{km}\) because further away ground motions tend to be over predicted.

Stamatovska and Petrovski (1996)

• Ground-motion model is: \[\begin{aligned} \mathrm{Acc}&=&\exp(b)\exp(b_M) (R_h+C)^{b_R}\\ \mbox{where } R_h^2&=&(R_e/\rho)^2+h^2\\ \mbox{and } \rho&=&\sqrt{\frac{1+t g^2 \alpha}{a^{-2}+t g^2 \alpha}}\end{aligned}\] where \(\mathrm{Acc}\) is in \(\,\mathrm{cm/s^2}\), \(\alpha\) is the azimuth of the site with respect to energy propagation pattern, \(b=3.49556\), \(b_M=1.35431\), \(C=30\), \(b_R=-1.58527\), \(a=1.2\) and \(\sigma=0.48884\) (definitions of \(t\) and \(g\) are not given).

• Correct PGAs for local site effects so that PGAs used correspond to a site with a shear-wave velocity of \(700\,\mathrm{m/s}\). Do not state how this is performed.

• Most records from SMA-1s.

• Not all records from free-field.

• Records from strong intermediate depth earthquakes in Vrancea region.

• Focal depths, \(89.1 \leq h \leq 131\,\mathrm{km}\).

• For each of the four earthquakes, calculate coefficents in equation \(\ln \mathrm{Acc}=b_0+b_1 \ln (R_e/\rho)\), the main direction of energy propagation and the relation between the semi-axes of the ellipse in two orthogonal directions (\(a:b\)).

• Also calculate coefficents in equation \(\ln \mathrm{Acc}=b+b_M M+ b_R \ln (R_h+C)\) for different azimuth by normalising the values of \(R_e/\rho\) by the azimuth. Give coefficients for Bucharest, Valeni and Cerna Voda.

• Note that uncertainty is high and suggest this is because of distribution of data with respect to \(M\), \(R_e\) and \(h\), the use of data processed in different ways, soil-structure interaction and the use of an approximate correction method for local site effects.

Ansal (1997)

• Ground-motion model is: \[\log A_p=a_1 M+a_2 R+a_3 \log R+a_4\] where \(A_p\) is in \(\,\mathrm{gal}\), \(a_1=0.329\), \(a_2=-0.00327\), \(a_3=-0.792\) and \(a_4=1.177\) (\(\sigma\) is not known).

Campbell (1997), Campbell (2000), Campbell (2001) & Campbell and Bozorgnia (1994)

Ground-motion model (horizontal component) is: \[\begin{aligned} \ln A_H&=&a_1+a_2 M+a_3 \ln \sqrt{R_{\mathrm{SEIS}}^2+[a_4 \exp(a_5 M)]^2}\\ &&{}+[a_6+a_7\ln R_{\mathrm{SEIS}}+a_8M]F+[a_9+a_{10}\ln R_{\mathrm{SEIS}}]S_{\mathrm{SR}}\\ &&{}+[a_{11}+a_{12}\ln R_{\mathrm{SEIS}}]S_{\mathrm{HR}}+f_A(D)\\ f_A(D)&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&D \geq 1\,\mathrm{km}\\ \{[a_{11}+a_{12}\ln (R_{\mathrm{SEIS}})]-[a_9+a_{10} \ln (R_{\mathrm{SEIS}})]S_{\mathrm{SR}}\}(1-D)(1-S_{\mathrm{HR}})&D<1\,\mathrm{km}\\ \end{array} \right.\end{aligned}\] where \(A_H\) is in \(\,\mathrm{g}\), \(a_1=-3.512\), \(a_2=0.904\), \(a_3=-1.328\), \(a_4=0.149\), \(a_5=0.647\), \(a_6=1.125\), \(a_7=-0.112\), \(a_8=-0.0957\), \(a_9=0.440\), \(a_{10}=-0.171\), \(a_{11}=0.405\), \(a_{12}=-0.222\), \(\sigma=0.55\) for \(A_H<0.068\,\mathrm{g}\), \(\sigma=0.173-0.140\ln (A_H)\) for \(0.068\,\mathrm{g}\leq A_H \leq 0.21\,\mathrm{g}\) and \(\sigma=0.39\) for \(A_H>0.21\,\mathrm{g}\) (when expressed in terms of acceleration) and \(\sigma=0.889-0.0691M\) for \(M<7.4\) and \(\sigma=0.38\) for \(M \geq 7.4\) (when expressed in terms of magnitude).

Ground-motion model (vertical component) is: \[\begin{aligned} \ln A_V&=&\ln A_H + b_1+b_2 M + b_3 \ln [R_{\mathrm{SEIS}}+b_4 \exp (b_5 M)]\\ &&{}+b_6 \ln[R_{\mathrm{SEIS}}+b_7 \exp(b_8 M)]+b_9 F\end{aligned}\] where \(A_V\) is in \(\,\mathrm{g}\), \(b_1=-1.58\), \(b_2=-0.10\), \(b_3=-1.5\), \(b_4=0.079\), \(b_5=0.661\), \(b_6=1.89\), \(b_7=0.361\), \(b_8=0.576\), \(b_9=-0.11\) and \(\sigma_V=\sqrt{\sigma^2+0.36^2}\) (where \(\sigma\) is standard deviation for horizontal PGA prediction).

Uses three site categories:

A

Hard rock: primarily Cretaceous and older sedimentary deposits, metamorphic rock, crystalline rock and hard volcanic deposits (e.g. basalt).

Soft rock: primarily Tertiary sedimentary deposits and soft volcanic deposits (e.g. ash deposits).

Alluvium or firm soil: firm or stiff Quaternary deposits with depths greater than \(10\,\mathrm{m}\).

Also includes sediment depth (\(D\)) as a variable.

Restricts to near-source distances to minimize influence of regional differences in crustal attenuation and to avoid complex propagation effects that have been observed at longer distances.

Excludes recordings from basement of buildings greater than two storeys on soil and soft rock, greater than five storeys on hard rock, toe and base of dams and base of bridge columns. Excludes recordings from shallow and soft soil because previous analyses showed such sites have accelerations significantly higher than those on deep, firm alluvium. Include records from dam abutments because comprise a significant number of rock recordings and due to stiff foundations are expected to be only minimally affected by dam. Some of these could be strongly affected by local topography.

Includes earthquakes only if they had seismogenic rupture within shallow crust (depths less than about \(25\,\mathrm{km}\)). Includes several large, shallow subduction interface earthquakes because previous studies found similar near-source ground motions to shallow crustal earthquakes.

Includes only earthquakes with \(M\) about \(5\) or larger to emphasize those ground motions of greatest engineering interest and limit analysis to more reliable, well-studied earthquakes.

Notes that distance to seismogenic rupture is a better measure than distance to rupture or distance to surface projection because top layer of crust is non-seismogenic and will not contribute to ground motion. Give estimates for average depth to top of seismogenic rupture for hypothetical earthquakes.

Considers different focal mechanisms: reverse (H:6, V:5), thrust (H:9, V:6), reverse-oblique (H:4, V:2) and thrust-oblique (0), total (H:19, V:13) \(\Rightarrow F=1\) (H:278 records, V:116 records) (reverse have a dip angle greater than or equal to \(45^{\circ}\)), strike-slip (H:27, V:13) \(\Rightarrow F=0\) (H:367 records, V:109 records) (strike-slip have an absolute value of rake less than or equal to \(22.5^{\circ}\) from the horizontal as measured along fault plane). There is only one normal faulting earthquakes in set of records (contributing four horizontal records) so difference is not modelled although \(F=0.5\) given as first approximation (later revised to \(F=0\)).

Mostly W. USA with 20 records from Nicaragua(1) Mexico (5), Iran (8), Uzbekistan (1), Chile (3), Armenia (1) and Turkey (1).

Does regression firstly with all data. Selects distance threshold for each value of magnitude, style of faulting and local site condition such that the 16th percentile estimate of \(A_H\) was equal to \(0.02\,\mathrm{g}\) (which corresponds to a vertical trigger of about \(0.01\,\mathrm{g}\)). Repeats regression repeated only with those records within these distance thresholds. Avoids bias due to non-triggering instruments.

Finds dispersion (uncertainty) to be dependent on magnitude and PGA, models as linear functions. Finds better fit for PGA dependency.

Munson and Thurber (1997)

• Ground-motion model is: \[\begin{aligned} \log_{10} \mathrm{PGA}&=&b_0+b_1(M-6)+b_2r-\log_{10} r+b_4S\\ \mbox{where }r&=&\sqrt{d^2+h^2}\end{aligned}\] \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(b_0=0.518\), \(b_1=0.387\), \(b_2=-0.00256\), \(b_4=0.335\), \(h=11.29\) and \(\sigma=0.237\).

• Use two site categories:

  1. Lava: 38 records

  2. Ash: \(60 \lesssim V_s \lesssim 200\,\mathrm{m/s}\), 13 records

• Depths between \(4\) and \(14\,\mathrm{km}\) with average \(9.6\,\mathrm{km}\) (standard deviation \(2.3\,\mathrm{km}\)). Limit of \(15\,\mathrm{km}\) chosen to differentiate between large tectonic earthquakes and deeper mantle events.

• Attenuation greater than for western USA due to highly fractured volcanic pile.

• Peak acceleration measured directly from accelerograms. Check against one from corrected records, small difference.

• Excludes records triggered on S-wave and those beyond cutoff distance (the distance to first nontriggered instrument).

• Does weighted and unweighted least squares analysis; find some differences.

Pancha and Taber (1997)

• Ground-motion model is: \[\begin{aligned} \log y&=&\alpha +\beta \mathbf{M}-\log r +b r\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] Coefficients are unknown.

• Also develop model using functional form of Molas and Yamazaki (1995).

• All data from rock sites.

• Data from seismographs of New Zealand National Seismograph Network and temporary deployments on East Cape of the North Island, the Marlborough region of the South Island and the central volcanic zone of the North Island.

• Most data from more than \(100\,\mathrm{km}\) from the source.

Rhoades (1997)

• Ground-motion model is: \[\begin{aligned} \log_{10} a&=&\alpha+\beta M-\log_{10} r+\gamma r\\ \mbox{where } r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), \(\alpha=-1.237 \pm 0.254\), \(\beta=0.278 \pm 0.043\), \(\gamma=-0.00220 \pm 0.00042\), \(h=6.565 \pm 0.547\), \(\tau^2=0.00645 \pm 0.00382\) and \(\sigma^2=0.0527 \pm 0.00525\) (where \(\tau^2\) is the inter-earthquake variance and \(\sigma^2\) is the intra-earthquake variance and \(\pm\) signifies the standard error of the estimate.

• Notes that errors in magnitude determination are one element that contributes to the between-earthquake component of variance and could thus cause apparent differences between earthquakes, even if none existed.

• Develops a method to explicitly include consideration of magnitude uncertainties in a random earthquake effects model so that the between-earthquake component of variance can be split into the part that is due only to magnitude uncertainty (and is therefore of no physical consequence) and the part for which a physical explanation may be sought.

• Applies method to data of Joyner and Boore (1981). Assume two classes of magnitude estimates: those with estimates of \(M_w\), which assumes to be associated with a standard error of \(0.1\), and those for which \(M_L\) was used as a surrogate for \(M_w\), which assumes to be associated with a standard error of \(0.3\). Find that the inter-earthquake variance is much lower than that computed assuming that the magnitudes are exact but that other coefficients are similar. Believes that the high inter-earthquake variance derived using the exact magnitudes model is largely explained by the large uncertainties in the magnitude estimates using \(M_L\).

Schmidt, Dahle, and Bungum (1997)

• Ground-motion model is: \[\begin{aligned} \ln A&=&c_1+c_2 M+c_3 \ln r+c_4 r+c_5 S_1 +c_6 S_2\\ \mbox{where }r&=&\sqrt{R^2+6^2}\end{aligned}\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(c_1=-1.589\), \(c_2=0.561\), \(c_3=-0.569\), \(c_4=-0.003\), \(c_5=0.173\), \(c_6=0.279\) and \(\sigma=0.80\) (for all earthquakes), \(c_1=-1.725\), \(c_2=0.687\), \(c_3=-0.742\), \(c_4=-0.003\), \(c_5=0.173\), \(c_6=0.279\) and \(\sigma=0.83\) (for shallow crustal earthquakes) and \(c_1=-0.915\), \(c_2=0.543\), \(c_3=-0.692\), \(c_4=-0.003\), \(c_5=0.173\), \(c_6=0.279\) and \(\sigma=0.74\) (for subduction zone earthquakes).

• Use three site categories:

  1. Rock, 54 records.

  2. Hard soil, 63 records.

  3. Soft soil, 83 records.

• Most records from SMA-1s with 6 records from SSA-2.

• Use PSA at \(40\,\mathrm{Hz}\) (\(0.025\,\mathrm{s}\)) as peak ground acceleration.

• Records instrument corrected and bandpass filtered with cut-offs of \(0.2\) and \(20\,\mathrm{Hz}\).

• Use data from shallow crustal earthquakes (133 records) and subduction zone earthquakes (67 records).

• Perform regression on combined shallow crustal and subduction zone records, on just the shallow crustal records using \(r_{hypo}\) and using \(r_{epi}\) and on just subduction zone records.

• Note that distribution w.r.t. distance improves in the near field when epicentral distance is used but only possible to use \(r_{epi}\) for shallow crustal earthquakes because for subduction zone earthquakes hypocentral distance is much greater than epicentral distance so should use \(r_{hypo}\) instead.

• For \(4 \leq M \leq 6\) distribution w.r.t. epicentral distance is quite good but for \(M>6\) no records from \(d_e<40\,\mathrm{km}\).

• Use a two step procedure. Firstly use entire set and both horizontal components and compute two soil terms (one for hard and one for soft soil). In second step use soil terms to correct motions for rock conditions and then repeat regression.

• Use Bayesian analysis (Ordaz, Singh, and Arciniega 1994) so that derived coefficients comply with physics of wave propagation because include a priori information on the coefficients to avoid physically unrealistic values. Choose initial values of coefficients based on theory and previous results

• Cannot find coefficient in \(r\) by regression so adopt \(6\,\mathrm{km}\) from previous study.

• Examine residuals w.r.t. distance and magnitude and find no trends.

Youngs et al. (1997)

Ground-motion model for soil is: \[\begin{aligned} \ln \mathrm{PGA}&=&C_1^*+C_2 \mathbf{M} +C_3^* \ln \left[ r_{\mathrm{rup}}+\mathrm{e}^{C_4^*-\frac{C_2}{C_3^*} \mathbf{M}}\right] +C_5 Z_t+C_9H+C_{10}Z_{ss}\\ \mbox{with: }C_1^*&=&C_1+C_6Z_r\\ C_3^*&=&C_3+C_7Z_r\\ C_4^*&=&C_4+C_8Z_r\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(C_1=-0.6687\), \(C_2=1.438\), \(C_3=-2.329\), \(C_4=\ln(1.097)\), \(C_5=0.3643\), \(C_9=0.00648\) and \(\sigma=1.45-0.1\mathbf{M}\) (other coefficients in equation not needed for prediction on deep soil and are not given in paper).

Ground-motion model for rock is: \[\begin{aligned} \ln \mathrm{PGA}&=&C_1^*+C_2 \mathbf{M} +C_3^* \ln \left[ r_{\mathrm{rup}}+\mathrm{e}^{C_4^*-\frac{C_2}{C_3^*}\mathbf{M}}\right] +C_5 Z_{ss}+C_8Z_t+C_9H\\ \mbox{with: }C_1^*&=&C_1+C_3C_4-C_3^*C_4^*\\ C_3^*&=&C_3+C_6Z_{ss}\\ C_4^*&=&C_4+C_7Z_{ss}\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(C_1=0.2418\), \(C_2=1.414\), \(C_3=-2.552\), \(C_4=\ln(1.7818)\), \(C_8=0.3846\), \(C_9=0.00607\) and \(\sigma=1.45-0.1\mathbf{M}\) (other coefficients in equation not needed for prediction on rock and are not given in paper).

Use different models to force rock and soil accelerations to same level in near field.

Use three site categories to do regression but only report results for rock and deep soil:

A

Rock: Consists of at most about a metre of soil over weathered rock, 96 records.

Deep soil: Depth to bedrock is greater than \(20\,\mathrm{m}\), 284 records.

Shallow soil: Depth to bedrock is less than \(20\,\mathrm{m}\) and a significant velocity contrast may exist within \(30\,\mathrm{m}\) of surface, 96 records.

Use free-field recordings, i.e. instruments in basement or ground-floor of buildings less than four storeys in height. Data excluded if quality of time history poor or if portion of main shaking not recorded.

Consider tectonic type: interface (assumed to be thrust) (98 records) \(\Rightarrow Z_t=0\), intraslab (assumed to be normal) (66 records) \(\Rightarrow Z_t=1\)

Focal depths, \(H\), between \(10\) and \(229\,\mathrm{km}\)

Not enough data to perform individual regression on each subset so do joint regression analysis.

Both effect of depth and tectonic type significant.

Large differences between rock and deep soil.

Note differences between shallow crustal and interface earthquake primarily for very large earthquakes.

Assume uncertainty to be linear function of magnitude.

Zhao, Dowrick, and McVerry (1997)

• Ground-motion model (Model 1) is: \[\log_{10} \mathrm{PGA}= A_1 M_w + A_2 \log_{10} \sqrt{r^2+d^2} + A_3 h_c+A_4+A_5 \delta_R+A_6 \delta_A+A_7 \delta_I\] where \(\mathrm{PGA}\) is in \(\,\mathrm{m/s^2}\), \(\delta_R=1\) for crustal reverse \(0\) otherwise, \(\delta_A=1\) for rock \(0\) otherwise, \(\delta_I=1\) for interface \(0\) otherwise, \(A_1=0.298\), \(A_2=-1.56\), \(A_3=0.00619\), \(A_4=-0.365\), \(A_5=0.107\), \(A_6=-0.186\), \(A_7=-0.124\), \(d=19\) and \(\sigma=0.230\).

• Models also given for soil sites only (Model 2), unspecified site (Model 3), focal mechanism and tectonic type unknown (Model 4) and only magnitude, depth and distance known (Model 5)

• Records from ground or base of buildings. 33 from buildings with more than 3 storeys; find no significant differences.

• Retain two site categories:

  1. Rock: Topographic effects expected, very thin soil layer (\(\leq 3\,\mathrm{m}\)) overlying rock or rock outcrop.

  2. Soil: everything else

• Use depth to centroid of rupture, \(h_c\), \(4\leq h_c \leq 149\). Only nine are deeper than \(50\,\mathrm{km}\). Exclude records from deep events which travelled through mantle.

• Consider tectonic type: C=crustal (24+17 records), I=interface (7+0 records) and S=slab (20+0 records)

• Consider source mechanism: N=normal (15+1 records), R=reverse (22+5 records) and S=strike-slip (12+11 records). Classify mixed mechanisms by ratio of components \(\geq 1.0\).

• For only five records difference between the distance to rupture surface and the distance to centroid could be more than 10%.

• 66 foreign near-source records (\(d_r \leq 10\,\mathrm{km}\)) from 17 crustal earthquakes supplement NZ data. Mainly from western North America including 17 from Imperial Valley and 12 from Northridge.

• Exclude one station’s records (Atene A) due to possible topographical effects.

• Exclude records which could have been affected by different attenuation properties in the volcanic region.

• Note regional difference between Fiordland and volcanic region and rest of country but do model.

• Retain coefficients if significant at \(\alpha=0.05\).

• Anelastic term not significant.

Baag et al. (1998)

• Ground-motion model is: \[\begin{aligned} \ln \mathrm{PGA}&=&a_1+a_2M+a_3\ln R+a_4R\\ \mbox{where } \quad R&=&\sqrt{R_{\mathrm{epi}}^2+a_5^2}\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(a_1=0.4\), \(a_2=1.2\), \(a_3=-0.76\), \(a_4=-0.0094\) and \(a_5=10\) (\(\sigma\) not given).

• This article has not been seen. The model presented may not be a fully empirical model.

Bouhadad et al. (1998)

• Ground-motion model is: \[A=c \exp(\alpha M)[R^k+a]^{-\beta-\gamma R}\]

• Coefficients not given, only predictions.

Costa, Suhadolc, and Panza (1998)

• Ground-motion model is: \[\log(A)=a+bM+c\log(r)\] where \(A\) is in \(\,\mathrm{g}\), \(a=-1.879\), \(b=0.431\) and \(c=-1.908\) (for vertical components) and \(a=-2.114\), \(b=0.480\) and \(c=-1.693\) (for horizontal components).

• All records from digital instruments.

• Try including a term \(d\log(M)\) but tests show that \(d\) is negligible with respect to \(a\), \(b\) and \(c\).

Manic (1998)

• Ground-motion model is: \[\begin{aligned} \log (A)&=&c_1+c_2 M+c_3 \log(D)+c_4 D+c_5 S\\ D&=&(R^2+d_0^2)^{1/2}\end{aligned}\] where \(A\) is in \(\,\mathrm{g}\), \(c_1=-1.664\), \(c_2=0.333\), \(c_3=-1.093\), \(c_4=0\), \(c_5=0.236\), \(d_0=6.6\) and \(\sigma=0.254\).

• Uses four site categories (following N. N. Ambraseys, Simpson, and Bommer (1996)) but only two have data within them:

  1. Rock (R): \(v_s>750\,\mathrm{m/s}\), 92 records.

  2. Stiff soil (A): \(360< v_s \leq 750\,\mathrm{m/s}\), 184 records.

  where \(v_s\) is average shear-wave velocity in upper \(30\,\mathrm{m}\).

• Uses both horizontal components to get a more reliable set of data.

• Tries using \(M_L\) rather than \(M_s\), epicentral distance rather than hypocentral distance and constraining anelastic decay coefficient, \(c_4\), to zero. Chooses combination which gives minimum \(\sigma\).

Reyes (1998)

• Ground-motion model is: \[\ln \mathrm{Sa}=\alpha_1+\alpha_2(M-6)+\alpha_3(M-6)^2+\alpha_4\ln R+\alpha_5R\] where \(\mathrm{Sa}\) is in \(\,\mathrm{cm/s^2}\), \(\alpha_1=5.8929\), \(\alpha_2=1.2457\), \(\alpha_3=-9.7565\times 10^{-2}\), \(\alpha_4=-0.50\), \(\alpha_5=-6.3159\times 10^{-3}\) and \(\sigma=0.420\).

• Use data from one station, University City (CU) in Mexico City, a relatively firm site.

Rinaldis et al. (1998)

• Ground-motion model is: \[\ln Y=C_{14}+C_{22}M+C_{31}\ln(R+15)+C_{43}S+C_{54}F\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(C_{14}=5.57\), \(C_{22}=0.82\), \(C_{31}=-1.59\), \(C_{43}=-0.14\), \(C_{54}=-0.18\) and \(\sigma=0.68\). Assume \(15\,\mathrm{km}\) inside \(\ln (R+\ldots)\) from Theodulidis and Papazachos (1992).

• Use two site categories:

  1. Rock: includes stiff sites.

  2. Alluvium: includes both shallow and deep soil sites.

• Use two source mechanism categories:

  1. Thrust and strike-slip earthquakes.

  2. Normal earthquakes.

• Use epicentral distance because in Italy and Greece the surface geology does not show any evident faulting, consequently it is impossible to use a fault distance definition.

• Good distribution and coverage of data with respect to site category and source mechanism.

• Consider six strong-motion records (three Italian and three Greek) with different associated distances, magnitudes and record length and apply the different processing techniques of ENEA-ENEL and ITSAK to check if data from two databanks can be merged. Digitise six records using same equipment. ITSAK technique: subtract the reference trace (either fixed trace or trace from clock) from uncorrected accelerogram and select band-pass filter based on either Fourier amplitude spectra of acceleration components or selected using a different technique. ENEA-ENEL technique: subtract the reference trace from uncorrected accelerogram and select band-pass filter by comparing Fourier amplitude spectra of acceleration components with that of fixed trace. Find small differences in PGA, PGV, PGD so can merge Italian and Greek data into one databank.

• Use four step regression procedure, similar to that Theodulidis and Papazachos (1992) use. First step use only data with \(M\geq 6.0\) (\(7 \leq R \leq 138\,\mathrm{km}\)) for which distances are more accurate to find geometrical coefficient \(C_{31}\). Next find constant (\(C_{12}\)) and magnitude (\(C_{22}\)) coefficients using all data. Next find constant (\(C_{13}\)) and soil (\(C_{43}\)) coefficients using all data. Finally find constant (\(C_{14}\)) and source mechanism (\(C_{54}\)) coefficients using data with \(M \geq 6.0\) for which focal mechanism is better constrained; final coefficients are \(C_{14}\), \(C_{22}\), \(C_{31}\), \(C_{43}\) and \(C_{54}\). Investigate influence of distance on \(C_{54}\) by subdividing data in final step into three categories with respect to distance (\(7\leq R \leq 140\,\mathrm{km}\), \(7 \leq R \leq100\,\mathrm{km}\) and \(7\leq R \leq 70\,\mathrm{km}\)).

• Equation intended as first attempt to obtain attenuation relations from combined databanks and site characteristics and fault rupture properties could and should be taken into account.

Sadigh and Egan (1998)

• Based on Sadigh et al. (1997), see Section 2.88.

• Ground-motion model is: \[\ln \mathrm{PGA}= C_1+C_2 M+C_3 \ln [r_{\mathrm{rup}} + \exp(C_4+C_5M)]\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for \(M<6.5\) \(C_4=1.29649\) and \(C_5=0.25\) and for \(M\geq 6.5\) \(C_4=-0.48451\) and \(C_5=0.524\). For rock sites: \(C_3=-2.100\), for strike-slip mechanism and \(M<6.5\) \(C_1=-0.949\) and \(C_2=1.05\), for strike-slip mechanism and \(M\geq 6.5\) \(C_1=-1.274\) and \(C_2=1.10\), for reverse-slip and \(M<6.5\) \(C_1=0.276\) and \(C_2=0.90\) and for reverse-slip and \(M\geq 6.5\) \(C_1=-1.024\) and \(C_2=1.10\). For soil sites: \(C_3=-1.75\), for strike-slip mechanism and \(M<6.5\) \(C_1=-1.1100\) and \(C_2=0.875\), for strike-slip mechanism and \(M\geq 6.5\) \(C_1=-1.3830\) and \(C_2=0.917\), for reverse-slip mechanism and \(M<6.5\) \(C_1=-0.0895\) and \(C_2=0.750\) and for reverse-slip mechanism and \(M\geq 6.5\) \(C_1=-1.175\) and \(C_2=0.917\) (\(\sigma\) not given).

• Use two site categories:

  1. Rock: bedrock within about a metre of surface. Note that many such sites are soft rock with \(V_s \leq 750\,\mathrm{m/s}\) and a strong velocity gradient because of near-surface weathering and fracturing, 274 records.

  2. Deep soil: greater than \(20\,\mathrm{m}\) of soil over bedrock. Exclude data from very soft soil sites such as those from San Francisco bay mud, 690 records.

• Define crustal earthquakes as those that occur on faults within upper \(20\) to \(25\,\mathrm{km}\) of continental crust.

• Consider source mechanism: RV=reverse (26+2) and SS=strike-slip (and some normal) (89+0). Classified as RV if rake\(>45^{\circ}\) and SS if rake\(<45^{\circ}\). Find peak motions from small number of normal faulting earthquakes not to be significantly different than peak motions from strike-slip events so include in SS category.

• Separate equations for \(M_w <6.5\) and \(M_w \geq 6.5\) to account for near-field saturation effects, for rock and deep soil sites and reverse and strike-slip earthquakes.

• Records from instruments in instrument shelters near ground surface or in ground floor of small, light structures.

• 4 foreign records (1 from Gazli and 3 from Tabas) supplement Californian records.