ORIGINAL ARTICLE

Empirical models for the prediction of ground motionduration for intraplate earthquakes

P. Anbazhagan & M. Neaz Sheikh & Ketan Bajaj &P. J. Mariya Dayana & H. Madhura & G. R. Reddy

Received: 18 June 2016 /Accepted: 10 February 2017 /Published online: 28 February 2017# Springer Science+Business Media Dordrecht 2017

Abstract Many empirical relationships for the earth-quake ground motion duration were developed forinterplate region, whereas only a very limited numberof empirical relationships exist for intraplate region.Also, the existing relationships were developed basedmostly on the scaled recorded interplate earthquakesto represent intraplate earthquakes. To the author’sknowledge, none of the existing relationships for theintraplate regions were developed using only the datafrom intraplate regions. Therefore, an attempt is madein this study to develop empirical predictive relation-ships of earthquake ground motion duration (i.e., sig-nificant and bracketed) with earthquake magnitude,hypocentral distance, and site conditions (i.e., rockand soil sites) using the data compiled from intraplateregions of Canada, Australia, Peninsular India, andthe central and southern parts of the USA. The com-piled earthquake ground motion data consists of 600records with moment magnitudes ranging from 3.0 to

6.5 and hypocentral distances ranging from 4 to1000 km. The non-linear mixed-effect (NLMEs) andlogistic regression techniques (to account for zeroduration) were used to fit predictive models to theduration data. The bracketed duration was found tobe decreased with an increase in the hypocentral dis-tance and increased with an increase in the magnitudeof the earthquake. The significant duration was foundto be increased with the increase in the magnitude andhypocentral distance of the earthquake. Both signifi-cant and bracketed durations were predicted higher inrock sites than in soil sites. The predictive relation-ships developed herein are compared with the existingrelationships for interplate and intraplate regions. Thedeveloped relationship for bracketed duration predictslower durations for rock and soil sites. However, thedeveloped relationship for a significant duration pre-dicts lower durations up to a certain distance andthereafter predicts higher durations compared to theexisting relationships.

Keywords Intraplate earthquakes . Bracketed duration .

Significant duration .Magnitude . Distance . Predictiverelationship

1 Introduction

Earthquakes can be broadly classified as interplateand intraplate earthquakes. Interplate earthquakesoccur on well-defined plate boundaries, where tec-tonic plates either move away from, or move

J Seismol (2017) 21:1001–1021DOI 10.1007/s10950-017-9648-2

P. Anbazhagan (*) :K. Bajaj : P. J. Mariya Dayana :H. MadhuraDepartment of Civil Engineering, Indian Institute of Science,Bangalore 560012, Indiae-mail: anbu@civil.iisc.ernet.in

M. Neaz SheikhSchool of Civil, Mining and Environmental Engineering,University of Wollongong, Wollongong, NSW 2500, Australia

G. R. ReddyBhabha Atomic Research Centre, Trombay, Mumbai 400085,India

towards, or slide past each other. On the other hand,intraplate earthquakes occur within the tectonic plateon fault zones. The differences between strongground motion characteristics of interplate and in-traplate earthquakes arise due to the source and patheffects. In general, intraplate earthquakes havehigher stress drop than interplate earthquakes. Interms of path effect, inelastic attenuation is generallyassumed to be greater in the more fragmentedinterplate regions (Atkinson 2004). Earthquakeground motion can be characterized by various pa-rameters (e.g., amplitude, frequency, and duration)reflecting a particular feature of the ground motion.Traditionally, the intensity of the earthquake groundmotion was considered as the scaling parameter forthe structural response analysis. However, intensityalone is not adequate to evaluate the damage poten-tial of earthquake ground motions (Fajfar et al.1990). In the current practice, earthquake groundmotions are usually characterized by the amplitudeof the shaking, such as peak ground acceleration andpeak spectral ordinates (response spectral accelera-tion, velocity, and displacement). However, theseparameters do not adequately consider damage ordegradation due to the hysteretic behavior of thestructure, which significantly depends on the dura-tion of strong ground motion. It has been shown thatduration is a meaningful predictor of performance,along with amplitude and frequency content param-eters for structural or geotechnical systems whoseperformance is measured by the damage that accu-mulates during shaking (Kempton and Stewart2006). Hence, for a complete characterization ofthe earthquake ground motion, duration of thestrong ground motion needs to be included. Howev-er, the influence of the ground motion duration onthe structural response may depend on the primaryground motion parameter used to characterize themotion. If two earthquakes of the same magnitudebut different durations are compared, the earthquakewith the longer duration will cause more damage tothe structure than the earthquake with shorterduration (Bommer and Martinez-Pereira 1999). Onthe other hand, if two earthquakes with the sameenergy content but different durations are compared,the earthquake with the shorter duration will causemore damage than the earthquake with longer dura-tion (Bommer and Martinez-Pereira 1999). It wasobserved that larger acceleration and spectral

parameters caused less damage due to shorter dura-tion of the shaking in the 1996 Parkfield earthquake(Housner 1975).

The influence of ground motion duration on theresponse of geotechnical structures is wellestablished. The duration of ground motion playsan important role in soil liquefaction and slope sta-bility. Liquefaction is a phenomenon that occurs insaturated cohesionless soil due to the generation ofpore water pressure under seismic cyclic loading.The generation of pore water pressure hence de-pends on the amplitude and the number of cyclesof the shaking or the ground motion duration. Thelateral spread displacement resulting from the soilliquefaction depends on the amplitude and durationof the shaking (Rauch and Martin 2000), with dis-placements increasing with duration. Seismic dis-placement of the soil mass along the slope alsodepends on the duration of shaking (e.g., Bray andRathje 1998).

The influence of ground motion duration on thestructural response is more pronounced, especiallyfor structures whose strength and stiffness degradeunder earthquake ground motion (Lumantarna et al.2010). An example of such structure is masonrystructure. Moreover, the duration of earthquakeground motion has great influence on the inelasticresponse of structures when cumulative parametersare used to measure the damage (Hancock andBommer 2007). Although the importance of groundmotion duration for the seismic design and seismicdamage assessment of structures are well document-ed, duration parameter has not yet been included inmost of the current design codes (Victor andFederico 2013).

2 Duration definitions

Several definitions for the duration of earthquakeground motion are available in the literature. But, themost commonly used are bracketed duration and signif-icant duration (Bommer and Martinez-Pereira 1999).Bracketed duration (Db) is defined as the time elapsedbetween first and last exceedance of a specified level ofacceleration. The change of threshold acceleration cansignificantly increase or decrease Db. In this study, thethreshold acceleration values of 0.05g (Bommer et al.2009) and 0.03g (Ambraseys and Sarma 1967) are

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considered. Figure 1 shows Db of the 2011 Virginiaearthquake for the considered threshold accelerationvalues. It can be seen that Db changes from 27.6 to11.8 s with the change of threshold acceleration from0.03 to 0.05g. It is very clear that bracketed durationparameters can be sensitive to the threshold accelera-tions and to small sub-events occurring towards the endof a recording (Kempton and Stewart 2006). It is be-cause of these and other reasons, other definitions ofduration are often preferred. Significant duration (Ds) isan energy-based approach. It is defined as the time

interval across which a specified amount of energy isdissipated. Energy is represented by the integral of thesquare of ground acceleration, or velocity, or displace-ment. The integration of acceleration is related to theArias intensity (AI) (Arias 1970), which is defined as

AI ¼ π2g

∫tr0 a2 tð Þdt ð1Þ

where a(t) is the acceleration time history, tr is the totalduration of accelerogram, and g is the acceleration dueto gravity. Husid plot is used to evaluate the Ds for a

Fig. 1 Bracketed duration of the 2011 Virginia earthquake. a Db , 0.03g. b Db , 0.05g

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given energy interval (Husid 1969). Two common mea-sures of significant duration are the time intervals be-tween 5–75 and 5–95% of AI, which are denoted asDs5

− 75 and Ds5 − 95 (Bommer et al. 2009). The Ds5 − 75

captures the energy from body waves, whereas the Ds5

− 95 captures the energy from the full wave trains. UnlikeDb which determines the strong motion duration basedon an absolute value, Ds considers the characteristics ofthe whole accelerogram and defines a continuous timeinterval in which the motion can be considered asstrong. Figure 2 shows the Ds for the 2011 Virginiaearthquake. TheDs5 − 75 has been observed to be slightlyover 40% ofDs5 − 95. It is possible to obtain non-zeroDs

for low-amplitude earthquake ground motion, whichmight not be significant for damage analyses. However,as usual practices are, the problem can be easily over-come whenever the damage analysis takes into accountboth the amplitude and the duration of the groundmotion.

A few predictive equations have been developed inthe past to determine the strongmotion duration. Inmostcases, the predictive equations were expressed in termsof magnitude, site to source distance/hypocentral dis-tance, and site condition (Simon 2011; Lee and Green2008, 2012, 2014). However, almost all of the predictiveequations were developed for interplate regions

(Trifunac and Brady 1975; Dobry et al. 1978;McGuire and Barnhard 1979; Kamiyama 1984;Abrahamson and Silva 1996; Kempton and Stewart2006; Bommer et al. 2009; Abrahamson and Silva2008), and only a few were developed for intraplateregions; moreover, these studies were based on thescaled recorded interplate earthquakes to represent in-traplate earthquakes, for e.g., Lee and Green (2008,2012, 2014). Hence, to the knowledge of the authors,none of the previous studies have developed predictiveequations for the earthquake ground motion duration,considering only the recorded earthquake events in theintraplate regions.

The objective of the study presented herein is thedevelopment of empirical predictive models correlatingthe most common duration definitions Db and Ds toearthquake magnitude (moment magnitude, M), hypo-central distance (R), and local site condition (S) (i.e.,rock site or soil site) for intraplate earthquakes, whichhas not yet been adequately developed and studied. Themodels are developed through non-linear mixed-effect(NLME) regression analyses on data comprising 700recorded horizontal ground motions from intraplate re-gions. Only the horizontal components of the groundmotion records have been considered. The data werecollected from 75 intraplate earthquake events. Lastly,

Fig. 2 Husid plot for significant duration (Ds)

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the predictive durations have been compared with thepreviously published studies for interplate and intraplateregions.

3 Strong motion database

The recorded earthquake ground motion data were col-lected from the intraplate regions on the east coast ofAustralia, southeast of Australia, southeast of Canada,north of Canada, Peninsular India, South Atlantic regionof the USA, midwestern region of the USA, southernregion of the USA, and the northeastern regions of theUSA. The ground motion time histories are collectedfrom different sources, including Next Generation ofAttenuation (NGA)-East database (NGA-East flat file)and engineering seismology toolbox. Most of the avail-able ground motion time histories were processed andfiltered using fourth-order Butterworth filter. The data-base having signal-to-noise ratio more than 4 is used forthe processing. For the unprocessed time histories,fourth-order Butterworth filter and quadratic baselinecorrection were applied. The procedure used in process-ing seismic data is similar to Boore (2005). The databasehad records from 75 earthquakes which included 1050ground motion records, with the 2014 Conway Springs,Kansas earthquake being the latest event included in thedatabase.

Earthquake magnitudes in the collected databasewere in terms of either moment magnitude (M) or bodywave magnitude (MB) or short-period body wave mag-nitude (MN). For the regression analysis, MB and MN

were converted to M using the equations developed in

Joshi and Ramancharla (2008) and Atkinson and Sonley(2005), respectively.

M ¼ 0:85�MB þ 1:03 3:5≤MB≤6:8 ð2Þ

M ¼ 1:03�MN – 0:61 1:0≤MN ≤6:0 ð3ÞOnly the horizontal components of the earthquake

ground motion record were considered in develop-ing the equations. Vertical components of the earth-quake ground motion data were omitted from the1050 ground motion records. Hence, the recordedearthquake ground motion data considered in thisstudy were reduced to 700 records. The epicentraldistance of the records varies from 4.036 to1768.53 km. For all of the ground motion records,epicentral distances were calculated using the lati-tudes and longitudes of the recording station andearthquake epicenter. The focal depth of the groundmotion records varies from 0.64 to 30.8 km.Figure 3a shows the distribution of M and R of the700 recorded data points. The moment magnitudesM of the ground motion records range from 3.0 to6.5. It can be also seen from Fig. 3a that very fewrecordings are available for M > 5 and R < 100. Thiscan be improved by scaling the strong motion datafrom the interplate region, but this will be addressedin the future work. Figure 3b shows the distributionof database considered in this study with respect todifferent region. About 26% of the ground motionrecords are within R < 100 km and 30% are withinR = 100–500 km, and only 44% are within

Fig. 3 a Moment magnitude (M) versus hypocentral distance (R) of the earthquake records used in this study. b Distribution of data fordifferent region for soil and rock sites

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R > 500 km. Around 14% of the database, which isrecorded for R > 1000 km, is discarded as it ishaving insignificant peak ground acceleration.Hence, for deriving the regression coefficients, only600 ground motion components have been used.

The ground motions were classified as either“rock” or “soil” based on the site conditions at therespective seismograph stations. The site classifica-tion scheme used for the data is adapted fromMcGuire et al. (2001), which are based on the thirdletter of the Geomatrix three-letter site classificationsystem. Site categories A and B were considered torepresent rock sites, and site categories C, D, and Ewere considered to represent soil sites. The respec-tive stations were classified based on the shear wavevelocity data. Only those data was considered forwhich proper information regarding the site class orshear wave velocity is available. Only the groundmotion recorded considered in this study for whichmeasure Vs is available; the strong motion recordswith calculated Vs based on proxies or other methodare discarded from the study. It is noted that hypo-central distance (R) of the earthquake ground motionrecords was calculated based on epicentral distanceand the depth of the earthquake. Among the 700earthquake ground motion records, 509 records wereon rock site and 191 records were on soil site.

4 Regression analyses

As mentioned in Sect. 1, very few equations havebeen published, at least in recent years, for theprediction of ground motion duration. Hence, thereis relatively little guidance available on suitablefunctional forms. It is noted that development ofpredictive relationships for ground motion parame-ters including PGA and spectral parameters for dif-ferent periods at the rock and soil sites for intraplateearthquakes still remains a challenge (Anbazhaganet al. 2014, 2015). A NLME modeling is a powerfultechnique for the regression analysis of dataconsisting of multiple groups. The duration dataconsists of groups of ground motion representingground motions from the same earthquake. TheNLME regression modeling is a maximum likeli-hood method, which is based on Gaussian (normal)distribution and is mainly used for the analysis ofgrouped data (i.e., datasets comprising subsets)

(Pinheiro and Bates 2000). The traditional regres-sion techniques (e.g., least squares method) analyzethe entire data at once. However, the dataset iscomposed of motions from different earthquakes,with the number of motions for each earthquakevarying significantly. The resulting regression is un-duly influenced by the earthquake having the largestnumber of ground motions. Also, NLME regressionanalyses quantify both interearthquake andintraearthquake uncertainties. The model producedby the NLME regression technique has unbiasedfittings for each subset having a different numberof duration recordings.

This regression technique inherently assumes thedata and errors to be normally distributed. There-fore, quantile-quantile plot (or Q-Q plot) was usedto check this assumption (i.e., if the data points plotapproximately as a straight line on a normal Q-Qplot, it indicates that the data is normally distribut-ed). The NLME regression analyses were performedusing the statistical analysis program R (Program-R2015).

The functional form of the predictive relation-ships in this study was adapted from those proposedin Abrahamson and Silva (1996) and Kempton andStewart (2006), which were based on the seismicsource duration relationships (Hanks 1979;McGuire and Hanks 1980) and seismic source mod-el of Brune (1970, 1971). It is noted that althoughthe functional form may be based upon physicalreasoning, the final model is entirely empirical andcare should be taken when inferring physical attri-butes from regression coefficients. The authors haveadopted the philosophy that while it may be possibleto identify more complex expressions with statisti-cally significant terms, the resulting reduction invariance is very minor. In this study, earthquakeground motion duration is expressed as a functionof M (magnitude), R (distance), and S (site condi-tion) for both bracketed and significant durations. Inorder to ensure that the derived empirical models arerobust, residuals were checked and the statisticalsignificance of all the coefficients was inspected.The standard deviations for Ds5 − 95and Ds5 − 75 werefound to be smaller than those for Db , 0.03g andDb , 0.05g. This is attributed to the fact that bracketeddurations are inherently far less predictable than thesignificant durations (Bommer et al. 2009). Variousfunctional forms were considered in this study, and

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statistical significance of the functional forms isdetermined based on the recorded data. The detailregarding the selection of functional form is givenbelow.

4.1 Determination of functional form

Various authors such as Abrahamson and Silva(1996), Kempton and Stewart (2006), and Lee andGreen (2012, 2014) gave different functional formfor bracketed and significant durations. These func-tional forms were derived either based on physicalreasoning or statistical significance. However, in thisstudy, both ways are considered to determine thebest suitable functional form for duration corre-sponding to magnitude and distance scaling. The

functional form of the predictive model is developedthrough “trial-and-error” approach, where the modelselection is based on the standard error/deviation(i.e., the smallest) and the correct representation ofthe underlying physics. The functional forms forground motion duration model (GMDM) for signif-icant duration used are

log Dð Þ ¼ C1 þ C2 M−M*� �þ C3log Rð Þþ C4 þ C5 M−M*� �þ C6log Rð Þ� �

S FF 1ð Þ

log Dð Þ ¼ C1 þ C2exp M−M*� �þ C3R

þ C4 þ C5 M−M*� �þ C6R� �

S FF 2ð Þ

log Dð Þ ¼ log C1 þ C2exp M−M*� �þ C3Rþ C4 þ C5 M−M*� �þ C6R� �

S� �

FF 3ð Þ

where D is the significant duration Ds , 5 − 75 orDs , 5

− 95; M is the moment magnitude; M∗ is the hingemagnitude; R is the hypocentral distance; S is the site

condition , where S=0 for rock sites and S=1 for soilsites; and C1 ,C2 ,C3 ,C4 ,C5 , and C6 are the regres-sion coefficients.

Fig. 4 Typical plot of residuals for FF1for significant duration. a Interevent residuals with magnitude. b Intraevent residuals with logarithmof distance. Difference between the two lines in a shows the bias in the model

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The regression coefficients corresponding to theabove functional forms are derived, and bias for thesethree models is calculated. For determining the bias ineach model, the residuals are divided into intraevent andinterevent residuals. To accomplish that, a mixed-effectregression proposed byAbrahamson and Youngs (1992)and further extended by Jayaram and Baker (2010) isconducted by considering spatial correlation. The wholealgorithm used is explained in Bajaj and Anbazhagan(2016). The following equation has been used to per-form the mixed-effect regression.

Ri; j� �

k ¼ ck þ ηið Þk þ ϵi; j� �

k ð4Þ

where ck is the mean offset (or bias) of the datarelative to kth GMDM, ηi represents the event termfor event i, and εi , j is the intraevent residuals forrecording j in event i. The event term ηi representsthe mean offset of data for event i from the predictionprovided by the GMDM median after adjusting theoffset ck. In other words, the intraevent residual, ϵi , j,is the residual after accounting for the intereventresidual, ηi. The event term would be helpful inproviding a convenient mechanism for testing theability of a GMDM to provide the magnitude scaling

of a database. Interevent (η) and intraevent (ϵ) termsare assumed to have zero mean and τ and σ, respec-tively, as standard deviation. Hence, τ refers to theevent-to-event variability, and on the other hand, σrefers to the variability in a single event. A typicalexample for showing bias in a model is shown inFig. 4. A similar study has been carried out with allthe three models given as equations FF1 to FF3 fordifferent durations. In Fig. 4, it can be seen that biasin model 1 is 0.05, and the residual line matches withthe zero line. For Ds , 5 − 95, the bias values for FF2and FF3, respectively, are 0.1 and 0.18, and for FF3,residual line does not match with the zero line. Sim-ilar observations are seen for Ds , 5 − 95. Hence, FF1 isfurther used for deriving the ground motion durationmodel. However, as far as bracketed duration is ofconcern, the functional form available in the litera-ture is used, as this predicts less standard deviation.

A similar procedure is used for determining thereference magnitude (M∗) in case of both significantand bracketed duration. The M∗ varies from 3 to 6with an interval of 0.1, and it has been found thatM∗ equals to 4. The value of ck is very less ascompared to other magnitude. Hence, M∗= 4 is se-lected as a reference magnitude. The final functional

Fig. 5 Q-Q plots of the non-zero duration data and the resultant errors. a ln(Db , 0.03g ). b ln(Db , 0.05g ). c Db , 0.03g . d Db , 0.05g

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form used in determining significant duration is asfollows:

log Dð Þ ¼ C1 þ C2 M−4ð Þ þ C3log Rð Þþ C4 þ C5 M−4ð Þ þ C6log Rð Þf gS

ð5Þ

Additionally, as both the components of the groundmotion are available, an attempt has beenmade to derivethe regression coefficients using the geometric meanand maximum of both the components also by takingboth the components.

4.2 Bracketed duration

The Db has been determined from 600 ground motionrecords for the absolute acceleration of 0.03g (Db , 0.03g)

and 0.05g (Db , 0.05g). The entire bracketed durationdataset does not follow a normal distribution. Also, thepresence of zero-duration data precludes the data frombeing log-normally distributed (Lee and Green 2012).Moreover, the zero duration does not correlate well tothe independent variables in the regression analyses. Inorder to bypass these issues, the zero and non-zeroduration data were treated separately, with the non-zero duration data reasonably following a log-normaldistribution. The zero-duration data, however, needed tobe incorporated in the predictive model. Otherwise, themodel would be biased towards longer durations. Thisbecomes important for Db , 0.05g as it has a significantnumber of zero-duration records.

The corresponding expected value of duration can beexpressed using the total probability theorem, for a

Fig. 6 Residuals for Db , 0.03g versus amoment magnitude and b hypocentral distance. Residuals for Db , 0.05g versus c moment magnitudeand d hypocentral distance

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given set of momentmagnitude, local site condition, andhypocentral distance,

E D½ � ¼ E DjD > 0½ � � p D > 0ð Þ þ E DjD ¼ 0½ ��p D ¼ 0ð Þ

ð6Þ

where E[ ] and p( ) represent the expected value andprobability, respectively. Since the value of E[D|D=0]would be zero, the second term in Eq. (6) automaticallybecomes zero for all values of the input variables.Therefore, the expected value for D becomes equal tothe quantity of the expected value for a given non-zeroduration (i.e., E[D|D>0]) times the probability of non-zero duration occurrence (i.e., p(D>0)). For modelingthe expected value of duration, the NLME regressiontechnique is employed, and for modeling the probabilityof non-zero duration occurrence, a logistic regressionmethod is used. The probability of non-zero duration

occurrence can be easily obtained by subtracting theprobability of zero duration occurrences from 1. After-wards, this probability model is applied as a weightingfunction to the NLME regression result.

The entire model for bracketed duration estimationcan be split into two parts, one part is the non-zerodurationmodel that is developed through NLME regres-sion technique (which follows log-normal distributionas shown in Fig. 5) and the other part is a weightingfunction that represents the probability of non-zero du-ration occurrence for a given earthquake magnitude,hypocentral distance, and site condition which is devel-oped using logistic regressions.

The natural logarithm of bracketed duration is usedbecause the bracketed duration data is assumed to belog-normally distributed. The determination of the func-tional was explained above; hence, based on the statis-tical significance in the data and the relative magnitude

Fig. 7 Predicted bracketed duration by the combinedmodel forMw = 4.5, 5.5, and 6.0Db , 0.03g for a rock site and b soil site andDb , 0.05g forc rock site and d soil site

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of the resulting standard deviations, the model gave thebest fit of the data,

ln E DbjDb > 0½ �ð Þ ¼ C1 þ C2 M−4ð Þ þ C3log Rð Þþ C4 þ C5 M−4ð Þ þ C6log Rð Þf gS

ð7Þ

where Db is the bracketed duration – Db , 0 .03g

orDb , 0.05g ;M is the moment magnitude; R is the hy-pocentral distance; S is the site condition , whereS=0 for rock sites and S=1 for soil sites; and C1 ,C2 ,C3 ,C4 ,C5 , and C6 are the regression coefficients.

It is noted that if duration obtained fromEq. (7) is lessthan zero, then zero should be used as the predictedduration. The results from the regression analysis alongwith the associated 95% confidence interval are reportedin Table 1. The given regression coefficients are derivedby taking geometric mean, maximum of both the hori-zontal component and by considering both the compo-nents. As it is known, there are inherent assumptions innon-linear mixed-effect modeling that the errors arenormally distributed. Therefore, normal Q-Q plots ofthe errors (i.e., theoretical quantiles of the standardnormal versus the standardized errors) are produced asshown in Fig. 5. As may be observed from this figure,theQ-Q plots are relatively straight lines, demonstrating

that the distributional assumptions inherent to the re-gression analyses are valid. The residuals (i.e., predicted− actual value) for the recorded ground motions forintraplate are shown in Fig. 6 as a function of distanceand magnitude. With the exception of only a few datapoints, the residuals are within model ± standard devia-tion range in natural log, indicating that the modelpredictions are in good agreement with the data. Also,the residuals do not show any trend with magnitude ordistance, suggesting that the model is appropriate. InFig. 6 and Table 1, it can be concluded that use of boththe components significantly reduced the standard devi-ation. Hence, for comparison purpose, regression coef-ficient corresponding to both the components is used.

Logistic regressions were implemented separately foreach site condition, as a function ofM and R to estimatethe probability of the occurrence of a zero-durationmotion. The logistic function is defined as

p Dbracketed ¼ 0jM ;Rð Þ ¼ eβ1þβ2Mþβ3R

1þ eβ1þβ2Mþβ3Rð8Þ

where p(Dbracketed = 0|M,R) is the probability of zeroduration for a given M and R and β1 ,β2 , and β3 arethe regression coefficients determined from logisticregression and are separate for both site conditions.Then, the probability of the duration being greater

Fig. 8 Comparison of normal Q-Q plots for a Ds5 − 95 and b Ds5 − 75

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than zero is one minus the probability of that it iszero duration.

p Dbracketed > 0jM ;Rð Þ ¼ 1−p Dbracketed ¼ 0jM ;Rð Þ

¼ 1

1þ eβ1þβ2Mþβ3Rð9Þ

The results of the logistic regression are shownin Table 2. Equation (5) is used in conjunctionwith Eq. (7), which is used as a weighting func-tion. The combined model proposed for bracketeddurations that account for zero duration is givenby

E Dbð Þ ¼ exp C1 þ C2 M−4ð Þ þ C3Rþ C4 þ C5 M−4ð Þ þ C6Rf gSf g � p Dbracketed > 0jM ;Rð Þ≥0 ð10Þ

Using Eq. (10) in conjunction with the regressioncoefficients mentioned in Tables 1 and 2, the bracketeddurations predicted for intraplate regions are shown inFig. 7, as functions of hypocentral distance R for M4.5,M5.5, and M6.0 for both rock and soil sites. It has beenobserved that with the increase in R, the bracketeddurations decrease but increase with increasing magni-tude. This trend holds true for Db , 0.03g, Db , 0.05g, andalso for rock and soil sites. It is very clear from thegraphs that the duration varies significantly with mag-nitude, particularly for distances less than around 50 km,where an increase in one magnitude unit results in afourfold increase in duration. For rock sites,Db , 0.03g haslonger durations than Db , 0.05g, and similar trend hasbeen observed for the soil sites. In comparing soil androckmotions, soil motions seemingly have considerablyshorter durations than rock motions for both Db , 0.03g

and Db , 0.05g. However, in case of Db , 0.03g, for lowermagnitude (i.e.,Mw<4.5) and less hypocentral distance,duration at soil sites is observed to be more as comparedto the rock sites. This may be due to high upper crustalamplification for lower distances for intraplate region. In

the case ofDb , 0.05g, for larger hypocentral distance, andhighmagnitude, duration at soil sites is more as compareto rock sites. This trend may be attributed to relativelyrich, high-frequency content of intraplate motions andlow attenuation for larger distance. As the motionspropagate up through the soil layers, high frequenciesin intraplate motions filter out (Lee and Green 2012).

4.3 Significant duration

It has been determined from 600 ground motion re-cords for two common measures, which are Ds5 − 75

andDs5 − 95. In assessing the normal distribution of thesignificant duration dataset, it was found that the datafollowed a log-normal distribution. Also, adding 1 s tothe durations optimized the overall log-normality ofthe duration data. As shown in Fig. 8, log(Dsig + 1)more closely follows a normal distribution thanlog(Dsig). This optimization of the normality is neces-sary because normal distribution of data and errors isinherently assumed in the theoretical formulation ofthe NLME regression. Accordingly, the NLME

Fig. 9 Q-Q plot of the standard errors. a Ds5 − 95. b Ds5 − 75

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regression analyses were performed on log(Dsig+1)and log(Dsig). Both the equations will be statisticallytested, and the coefficient that corresponds to both isgiven in Table 1. Numerous regression analyses were

performed using various functional forms for the pre-dictive relationships. The model gave the best fit forthe data (i.e., smallest standard deviation) as ex-plained above.

log Dsig

� � ¼ C1 þ C2 M−4ð Þ þ C3log Rð Þ þ C4 þ C5 M−4ð Þ þ C6log Rð Þf gS½ � ð11Þ

where Dsig is the significant duration –Ds5 − 75 orDs5

− 95;M is the momentmagnitude; R is the hypocentraldistance; S is the site condition , where S=0 for rock

sites and S=1 for soil sites; and C1 ,C2 ,C3 ,C4 ,C5 ,and C6 are the regression coefficients.

For determining the coefficients corresponding tolog(Dsig+1), the proposed model given by Eq. (11)was rewritten from its original form by taking the

exponential and subtracting 1 from both sides of the

original equation, i.e., Dsig ¼ 10 log Dsigþ1ð Þ½ �−1. The re-sults from the NLME regression analyses of the signif-icant duration dataset for both log(Dsig+1) and log(Dsig)are given in Table 1. The regressions of both the Ds5 − 75

and Ds5 − 95 data were performed in two stages. For thefirst stage, the significant duration data were regressedusing Eq. (11) as mentioned above. However, several

Fig. 10 Residuals for Ds5 − 95

and Ds5 − 75 with respect to ahypocentral distance and bmagnitude

J Seismol (2017) 21:1001–1021 1013

regression coefficients were determined to have no sta-tistical significance, as per the p value for the likelihoodratio test (Pinheiro and Bates 2000). Therefore, a secondround of regressions was performed, in which the sta-tistically insignificant regression coefficients were re-moved. The statistically insignificant coefficients aresimply listed as “zero” in Table 1, instead of rewritingthe regression equation for both Ds5 − 75 and Ds5 − 95.This allows only one model to be used for both Ds5

− 75 and Ds5 − 95 predictive relationships. Figure 9 showsthe Q-Q plots of the standard errors from the regressionanalyses. As shown in the figure, the errors follownormal distributions, consistent with the assumptionsinherent to NLME modeling. Similarly, the residualsfor the recorded groundmotions for intraplate are shownin Fig. 10 as a function of distance and magnitude. It isevident from Fig. 10 that barring a few data points, theresiduals are within model ± one standard deviation

range in natural log, representing that the model predic-tions are in general in accordance with the data. Inaddition, the residuals do not form a trend either withmagnitude and or with distance, which is suggestive ofthe model being fit. However, it can also be observedfrom Fig. 10 that residual for log(Dsig+1) is less ascompared to log (Dsig); hence, it is recommended thatcoefficients corresponding to log (Dsig+1) need to beused for calculating the duration for intraplate region.

Using Eq. (11) and the regression coefficientslisted in Table 1, Ds5 − 75 and Ds5 − 95 for intraplateregions are plotted in Fig. 11 as functions of hypo-central distance for M4.5, M5.5, and M6.5 for rockand soil sites. The significant durations for intraplateregion increase with increasing hypocentral distanceand increasing magnitude, which is in harmony withthe findings of Abrahamson and Silva (1996) andKempton and Stewart (2006) for interplate regions

Fig. 11 Ds5 − 95 for a rock site and b soil site and Ds5 − 75 for c rock site and d soil site

1014 J Seismol (2017) 21:1001–1021

and also with Lee and Green (2014) for intraplateregions. Dissimilar to the trends observed forinterplate regions, the significant durations for intra-plate regions for rock sites tend to be a little longerthan those for soil sites, except for the small earth-quake magnitude (i.e., M4.5) and Ds5 − 75 measure ofsignificant duration. Another observation fromFig. 11 is that the soil sites have a higher durationfor shorter distance compared to the wreck sites. Thiscould be because of high-frequency content for intra-plate region being significantly filter out as theground motions propagate though the soil stratum.The same trend was observed for bracketed durationsas well. The reason may be the same as mentionedearlier in the paper. The durations for rock sites forDs5 − 95 are consistently longer than rock durationsforDs5 − 75. Similar observation is also noticed for thesoil sites.

5 Comparison with existing relationships

The bracketed duration relation proposed herein is com-pared with the relationships proposed by Koutrakis et al.(2002) and Lee and Green (2012). Koutrakis et al.(2002) derived their relation using Greek strong motiondata, which comprised of 93 seismic events and 141accelerograms covering magnitudes between 4.5 to 6.9and distances from 1 to 128 km. They used multi-linearregression analysis technique (i.e., least squares method)as their regression methodology. The distance consid-ered by them is epicentral distance, since, for most of thecases, fault rupture was not adequately determined.Moreover, due to the uncertainty in the focal depthdetermination, the application of the hypocentral dis-tance was avoided. Also, they did not consider the soilcondition parameter at the recording site as one of theindependent variables. In contrast to Koutrakis et al.

Fig. 12 Comparison of Db , 0.05g

between this study and Lee andGreen (2012) model for intraplateregions for a rock site and b soilsite

J Seismol (2017) 21:1001–1021 1015

(2002), the Lee and Green (2012) relationship wasdeveloped for stable continental regions of North Amer-ica using the NLME regression method on the data fromthe NGA database. The dataset considered by themcomprised of 28 recorded motions and 592 scaled mo-tions. Zero duration was also considered by conductinglogistic regression to model the probability of its occur-rence. However, their relation is only applicable for athreshold acceleration of 0.05g. This limits the compar-ison of Lee and Green’s (2012) relation to be made onlyto the Db , 0.05g measure of bracketed duration of thepresent study. On the other hand, Db , 0.03g measure ofbracketed duration has been compared with the relation-ship proposed by Koutrakis et al. (2002). The compar-isons are hampered by the use of different parameterdefinitions in the model proposed by Koutrakis et al.(2002); therefore, adjustments are made for magnitudescales using the conversion of Ambraseys and Free(1997) (for converting MS values to Mw). Figures 11

and 12 show the comparison of bracketed durationrelationships. Considerable differences exist betweenthe predictions from this study and from Lee andGreen (2012) and Koutrakis et al. (2002). This is likelydue to Koutrakis et al. (2002) not considering the dif-ference between the rock and soil sites in their analysis;not fully accounting for zero-duration motions; andusing least squares method for regression analysis,wherein the results are disproportionately influencedby the number of records in an earthquake event. Thisconclusion is further supported by the fact that similarwere reported in Bommer et al. (2009). Bommer et al.(2009) compared the empirical relationships for strongmotion duration with those in Koutrakis et al. (2002).Also, the dissimilarity might be because of differencesin the database used by Lee and Green (2012) and thisstudy. Lee and Green (2012) database have almost allthe records scaled from active crustal regions. The pres-ent study used only the recorded ground motion data

Fig. 13 Comparison of Db , 0.05g

between this study and Koutrakiset al. (2002) model for Greece fora rock site and b soil site. Thedistance on x axis is the epicentraldistance from the fault and variesfrom 0.1 to 128 km. The depth ofthe rupture for M4.5 is 10.37 km,for M5.5 is 16.72 km, and forM6.5 is 27 km

1016 J Seismol (2017) 21:1001–1021

from intraplate region. Overall, Lee and Green (2012)relationship predict significantly higher bracketed dura-tions than the model developed in this study. It is alsonoted that Db , 0.03g of this study and Koutrakis et al.(2002) prediction for M6.5 and M5.5 for rock site arerelatively close (Fig. 13).

Significant duration relationships proposed hereinfor two measures (Ds5 − 75 and Ds5 − 95) are comparedwith relationships proposed by Kempton and Stewart

(2006) and Lee and Green (2014). Both of theserelationships were developed using NLME regres-sion analyses. The relationship proposed byKempton and Stewart (2006) is for active crustalregions (e.g., western North America (WNA)),whereas the relationship proposed by Lee andGreen (2014) is for stable continental regions.Kempton and Stewart’s “base” model was developedusing data from 1557 recordings from 73 shallow

Fig. 14 Comparison ofDs5 − 95 between this study and Kempton and Stewart (2006) and Lee and Green (2014) for a rock site and b soil site

Fig. 15 Comparison ofDs5 − 75 between this study and Kempton and Stewart (2006) and Lee and Green (2014) for a rock site and b soil site

J Seismol (2017) 21:1001–1021 1017

Tab

le1

Coefficientsof

thepredictiv

eduratio

nsdevelopedin

thisstudy

C1

C2

C3

C4

C5

C6

τσ

σtotal

Db,0.03g

Geo-m

ean

−1.43±0.52

1.54

±0.12

−0.005

±0.0008

1.27

±1.41

−0.64±0.37

0.007±0.008

0.62

0.52

0.81

Maxim

um−1

.26±0.44

1.56

±0.12

−0.005

±0.0008

1.02

±1.30

−0.63±0.37

0.008±0.012

0.61

0.48

0.78

Both

−1.14±0.33

1.62

±0.09

−0.005

±0.0008

1.36

±0.88

−0.67±0.25

0.009±0.004

0.51

0.44

0.68

Db,0.05g

Geo-m

ean

−1.43±0.67

1.44

±0.15

−0.005

±0.0008

1.36

±1.64

−0.66±0.43

0.007±0.006

0.73

0.56

0.92

Maxim

um−1

.25±0.57

1.37

±0.15

−0.005

±0.0008

1.03

±1.48

−0.49±0.42

0.011±0.012

0.72

0.54

0.90

Both

−1.60±0.43

1.21

±0.11

−0.005

±0.0008

1.15

±1.03

−0.44±0.29

0.009±0.006

0.68

0.51

0.85

Ds5

−95

Geo-m

ean

−0.176

±0.11

0.04

±0.02

0.724±0.032

0.78

±.32

0.007±0.05

−0.29±0.11

0.15

0.28

0.32

Maxim

um−0

.110

±0.13

0.03

±0.03

0.694±0.042

0.05

±.36

0.005±0.07

−0.013

±0.12

0.24

0.32

0.40

Both

−0.061

±0.08

0.08

±0.02

0.712±0.032

0.82

±.32

0.005±0.05

−0.32±0.11

0.17

0.22

0.28

Ds5

−95+1

Geo-m

ean

0.001±0.09

0.052±0.02

0.662±0.02

0.71

±0.29

0.008±0.05

−0.26±0.11

0.13

0.27

0.30

Maxim

um−0

.036

±0.11

0.098±0.02

0.692±0.03

0.52

±0.29

0.004±0.05

−0.21±0.11

0.13

0.29

0.32

Both

−0.026

±0.11

0.121±0.02

0.601±0.03

0.56

±0.29

0.005±0.05

−0.25±0.11

0.16

0.19

0.25

Ds5

−75

Geo-m

ean

−1.146

±0.12

0.006±0.02

0.982±0.03

1.06

±0.29

0.00

±0.00

−0.38±0.09

0.24

0.30

0.39

Maxim

um−1

.001

±0.11

0.033±0.02

0.966±0.03

0.51

±0.31

0.00

±0.00

−0.21±0.11

0.20

0.26

0.34

Both

−0.975

±0.11

0.043±0.02

0.955±0.03

0.91

±0.29

0.00

±0.00

−0.34±0.09

0.19

0.26

0.33

Ds5

−75+1

Geo-m

ean

−1.146

±0.12

0.006±0.02

0.982±0.03

1.06

±0.29

0.00

±0.00

−0.38±0.09

0.24

0.30

0.39

Maxim

um−0

.516

±0.11

0.075±0.02

0.791±0.03

0.57

±0.31

0.00

±0.00

−0.23±0.11

0.17

0.19

0.26

Both

−0.528

±0.08

0.081±0.02

0.797±0.03

0.58

±0.29

0.00

±0.00

−0.28±0.11

0.19

0.19

0.27

The

plus

andminus

values

define

the95%

confidence

intervalforeach

parameter

1018 J Seismol (2017) 21:1001–1021

crustal earthquakes, which covered a magnituderange of ≈5–7.6 and closest hypocentral distancerange of ≈0–200 km. They expanded their base mod-el by adding several terms that incorporate near-faultdirectivity and deep basin effects. The local site con-ditions were represented via the average shear wavevelocity in the upper 30 m (Vs30) of a profile, insteadof using site classification. On the other hand, Leeand Green (2014) developed their relation from 620horizontal motions for stable continental regions(e.g., central and eastern North America (CENA)),which comprised of 28 recorded motions and 592scaled WNA motions for CENA conditions. Themoment magnitude for these motions range from4.5 to 7.6, and hypocentral distances range from 0.1to 199.1 km. In Figs. 14 and 15, the new model forsignificant durations is shown for various combina-tions of magnitude and distance and compared withthose from Kempton and Stewart (2006), Lee andGreen (2014), and Boore and Thompson (2015).The predictions of this study are lower until a certainhypocentral distance and after the predictions be-come higher compared to the previous studies. Thisis true for all the magnitudes of earthquake consid-ered in the study for rock sites and is clearly evidentin the case of small-magnitude earthquake event(M6.5). However, for the case of small-magnitudeearthquake event (M4.5), this trend is not visiblydistinct and one could argue that the prediction isroughly similar. However, in case of soil sites forsmall-magnitude earthquake event (M4.5), reversecan be seen from Figs. 14 and 15. Figure 14a showsthe comparison of estimated significant durationfrom proposed predictive relation with Boore andThompson (2015) path duration for rock and veryhard rock sites of stable continental regions. It is seenin Fig. 14a that for rock sites, predicted durations areslightly higher than Boore and Thompson (2015).This difference may be due to type of duration,

Boore and Thompson (2015) given path duration,

which is effective duration D095

� �minus source dura-

tion. Hence, path duration given by Boore andThompson (2015) is lower than significant durationproposed in this study.

6 Conclusions

In this study, new predictive equations for the brack-eted duration (Db , 0.03g and Db , 0.05g) and significantduration (Ds5 − 75 and Ds5 − 95) have been developedfor the intraplate regions based on the recordedintraplate earthquake ground motions. None of theexisting relationships for ground motion duration, tothe best of the author’s knowledge, were developedusing only recorded intraplate motions. The bestfunctional form is developed based on dividing theresidual into intraevent and interevent residuals. Themodels presented in this study are based on mixed-effect regression on the residuals determined fromthe functional forms available in the literature. Themodels include magnitude, hypocentral distance,and site condition. The database used consists of75 recorded intraplate earthquakes, 600 horizontalground motion records with moment magnitudesranging from 3.0 to 6.5 and hypocentral distancefrom 4 to 1000 km. Similar to the findings of pre-vious studies, the bracketed durations were predictedto decrease with increasing distance but predicted toincrease significantly with increasing magnitude.When compared with the existing relationships, thenew models predicted noticeably lower bracketeddurations for both rock and soil sites. Again, thesignificant durations were also predicted to followtrends proposed by previous studies; i.e., significantdurations increase with increasing hypocentral dis-tance and magnitude. In contrast to trends observedfor interplate regions, the significant durations forintraplate regions for rock sites were found to be alittle higher than those for soil sites, except for thesmall earthquake magnitude (i.e., M4.5) and Ds5 − 75

measure of significant duration. While comparingsignificant duration relation proposed herein withthe existing relations, the models presented in thisstudy predicted lower durations for both rock andsoil sites initially, but after a certain hypocentraldistance predicted higher durations. Quantitatively,

Table 2 Coefficients of the logistic regression for weightingfunctions

Site β1 β2 β3

Db , 0.03g Rock 2.45 −0.70 0.004

Soil 3.12 −0.69 0.018

Db , 0.05g Rock 4.27 −0.72 0.003

Soil 4.25 −0.68 0.008

J Seismol (2017) 21:1001–1021 1019

the relationships proposed herein and the ones pub-lished previously differ significantly, while qualita-tively, they are similar in some respects.

Acknowledgements First author would like to thank “Board ofResearch in Nuclear Sciences (BRNS),” Department of AtomicEnergy (DAE), Government of India, for funding the project titled“Seismic site classification for Indian shallow soil deposits” (Ref.No. Sanction No. 2012/36/33-BRNS-1656 dated October 10,2012).

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