AN IMPROVED METHOD OF MATCHING RESPONSE SPECTRA OF RECORDED EARTHQUAKE GROUND MOTION USING WAVELETS

April 28, 2006 13:47 WSPC/124-JEE 00273

Journal of Earthquake EngineeringVol. 10, Special Issue 1 (2006) 67–89c© Imperial College Press

AN IMPROVED METHOD OF MATCHING RESPONSESPECTRA OF RECORDED EARTHQUAKE GROUND

MOTION USING WAVELETS

JONATHAN HANCOCK†, JENNIE WATSON-LAMPREY‡,NORMAN A. ABRAHAMSON§, JULIAN J. BOMMER∗,†,

ALEXANDROS MARKATIS†, EMMA McCOY|| and RISHMILA MENDIS††Department of Civil and Environmental Engineering, Imperial College London, UK

‡University of California, Berkeley, California, USA§Pacific Gas and Electricity Company, San Francisco, USA||Department of Mathematics, Imperial College London, UK

Dynamic nonlinear analysis of structures requires the seismic input to be defined in theform of acceleration time-series, and these will generally be required to be compatiblewith the elastic response spectra representing the design seismic actions at the site. Theadvantages of using real accelerograms matched to the target response spectrum usingwavelets for this purpose are discussed. The program RspMatch, which performs spectralmatching using wavelets, is modified using new wavelets that obviate the need to sub-sequently apply a baseline correction. The new version of the program, RspMatch2005,enables the accelerograms to be matched to the pseudo-acceleration or displacementspectral ordinates as well as the spectrum of absolute acceleration, and additionallyallows the matching to be performed simultaneously to a given spectrum at severaldamping ratios.

Keywords: Dynamic analysis; accelerograms; wavelets; spectrum-compatible records;spectral matching; RspMatch.

1. Introduction

Seismic design of structures is invariably based on representation of the earthquakeactions in the form of a response spectrum. In many situations, however, includingthe design of critical facilities, highly irregular buildings and base-isolated struc-tures, the simulation of structural response using a scaled elastic response spectrumis not considered appropriate to verify the earthquake resistance. In such cases,dynamic nonlinear analysis of the structure will be required and the seismic inputthen needs to be defined in the form of acceleration time-series, which will gener-ally be required to be compatible with the elastic response spectra representing thedesign seismic actions at the site. There are many different options for obtainingsuites of accelerograms for use in engineering design and assessment [e.g. Bommer

∗Corresponding author: Tel.: +44-20-7594-5984, Fax: +44-20-7594-5934, E-mail: [email protected].

67

April 28, 2006 13:47 WSPC/124-JEE 00273

68 J. Hancock et al.

and Acevedo, 2004], the most widely used approaches being the use of artificialspectrum-compatible time-series, generated from white noise, and the use of scaledreal accelerograms.

Artificial records constitute a convenient tool but their shortcomings, arisingfrom their dissimilarity with real earthquake ground motions in terms of num-ber of cycles, phase content and duration, are widely recognised, and their use innonlinear analyses is not recommended. These problems are avoided by using realstrong-motion accelerograms, appropriately scaled to the target spectrum (at leastin the vicinity of the structure’s natural period of vibration), but the inherent vari-ability of real earthquake motions means that it will often be necessary to run largenumbers of dynamic analyses in order to obtain stable estimates of the inelasticresponse of the structure. The required number of inelastic dynamic analyses canbe significantly reduced if the real records are first matched to the target responsespectrum, by eliminating the largest differences between the target spectrum andthe spectral ordinates of individual accelerograms. This is clearly a compromise andin some sense the records become ‘artificial’ as a result, although the records canretain most (if not, in fact, all) of the characteristics of real earthquake records.The choice is essentially one of compromise between engineering pragmatism andseismological rigour, reducing the number of time-consuming structural analyseswhilst avoiding the use of completely artificial accelerograms generated from mod-ified white noise.

A commonly used method to reduce the spectral mismatch of the individ-ual ground motions is to apply spectral matching in the frequency domain byadjusting the Fourier amplitude spectra [e.g. Rizzo et al., 1975; Silva and Lee,1987]. This is useful in that it generates accelerograms that are based on realground motions and also have a close match to the target spectrum. However,adjusting the Fourier spectrum corrupts the velocity and displacement time-seriesand can result in motions with unrealistically high energy content [Naeim andLew, 1995].

An alternative approach for spectral matching adjusts the time history in thetime domain by adding wavelets to the acceleration time-series. Wavelet adjust-ment of recorded accelerograms has the same advantages as the Fourier adjustmentmethods but leads to a more focused correction in the time domain thus intro-ducing less energy into the ground motion and also preserves the non-stationarycharacteristics of the original ground motion. This paper describes the work con-ducted to create an improved version of the program RspMatch, originally devel-oped by Abrahamson [1992] using the technique of Lilhanand and Tseng [1987,1988], which is named RspMatch2005. The wavelet adjustment techniques incor-porated to this new version of the software have the additional advantage thatthey do not cause a drift in the velocity or displacement time-series. RspMatch2005also allows the records to be matched to a pseudo-acceleration spectrum ratherthan only the spectrum of absolute acceleration, and through improved convergence

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Matching Response Spectra of Recorded Earthquake Ground Motion 69

properties allows the record to be matched to a given target spectrum with severaldifferent levels of damping simultaneously. This feature is particularly useful whenlong-period highly damped spectral displacements are relevant, such as in thedirect displacement-based design approaches [e.g. Kowalsky et al., 1995] and inthe analysis of buildings and bridges with base isolation or supplementary dampingdevices.

2. Existing Wavelet Methods

There are several different methods of using wavelets to adjust accelerograms sothat they have a closer match to a target response spectrum. Mukherjee and Gupta[2002a, 2002b] and Suarez and Montejo [2003, 2005] use wavelets and the continuouswavelet transform (CWT) to de-compose the original acceleration time-series into anumber of time series with energy in non-overlapping frequency bands. An iterativeprocedure is used to scale each time history so that when they are added togetherthey produce a spectrum-compatible ground motion. Although the approximateduration of the original accelerogram is retained using this type of adjustment pro-cedure, the adjusted accelerograms have visibly different amplitudes and frequencycontents from the original accelerogram.

The method proposed by Lilhanand and Tseng [1987, 1988] employs waveletsbut uses the response of elastic SDOF systems rather than the CWT. This enablesaccelerograms to be made spectrum compatible with smaller adjustments than thewavelet adjustment methodologies which use the CWT. The Lilhanand and Tseng[1987, 1988] methodology is adopted as the basis for this work.

A flowchart showing the original procedure as employed in RspMatch is givenin Fig. 1. The essence of the methodology is as follows:

(1) Calculate the response of an elastic SDOF system under the action of theacceleration time-series for each period and damping level to be matched.

(2) Compare the peak of each SDOF response with the target amplitude and deter-mine the mismatch.

(3) Add wavelets to the acceleration time-series with the appropriate amplitudesand phasing so that the peak of each response matches the target amplitude.One wavelet is used to match one SDOF response.

Each wavelet is applied to the time series so that the time of maximum SDOFresponse under the action of the wavelet is the same as the time of the peak responseto be adjusted from the unadjusted acceleration time-series. A fundamental assump-tion of the method is that the time of the peak response does not change as a resultof adding the wavelet adjustment. This assumption is not always valid and this canlead to diverging solutions, as is discussed in more detail later.

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Input in accelerogram

Calculate spectral response

Calculate amplitude and sign of spectral of misfit

Is misfit lessthan tolerance?

Subdivide target into subgroups, each with periods spread throughout the frequency range

Calculate spectral response at periods in subgroup

Calculate spectral misfit at periods in subgroup

Calculat rix for subgroup. This relates the amplitude of each wavelet to peak response at each period to be adjusted.

Conduct singular value decomposition of atrix

Find linear scale factor for each wavelet bysolving matrix, minimising the spectral misfits

Scale and sum wavelets to create adjustment function

Add adjustment to total adjustment function

Is this thelast subgroup?

Load next subgroup

No Yes

No

Add total adjustment function to accelerogram and clear total adjustment function

Input in target spectrum

Yes

Save results to file

Load next frequencyrange to be matched

If required subdivide target into frequency bands

Is last frequencyrange?

No

Yes





Subdivide target into subgroups, each with periods spread throughout the frequency range



Calculate “C” matrix for subgroup. This relates the amplitude of each wavelet to peak response at each period to be adjusted.

Conduct singular value decomposition of C matrix

Find linear scale factor for each wavelet bysolving C matrix, minimising the spectral misfits


Add adjustment to total adjustment function


Load next subgroup

No Yes

No

Add total adjustment function to accelerogram and clear total adjustment function


Yes




Is last frequencyrange?

No

Yes

Fig. 1. Basic methodology of RspMatch program.

The amplitude of each wavelet used in the adjustment is determined by the solu-tion of a set of simultaneous equations that account for the cross correlation of eachwavelet with each response to be matched. This can be expressed in matrix form:

C · b = δR, (1)

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where C is a square matrix with elements that describe the amplitude of eachSDOF response, at the time that the response needs to be adjusted, under theaction of each wavelet, b is a vector of linear scale factors for each wavelet usedin the adjustment, and δR is a vector of the required adjustment, the differencebetween the peak SDOF response of the unadjusted time-series and the requiredamplitude specified by the target spectra response for each period and dampinglevel to be matched.

The wavelet scale factors in the b vector are found using amplitude of therequired adjustment and the inverse of the correlation matrix C:

b = C−1 · δR. (2)

The amplitude of the wavelet adjustment function at time t is determined fromthe sum of the amplitudes of the wavelets at that time, aj(t), multiplied by theirindividual scale factors bj .

Adjustment(t) =j=Nw∑j=1

bj · aj(t), (3)

where Nw is the total number of wavelets. The adjusted acceleration time-series isthe sum of the original time-series and the adjustment function.

Unfortunately, the correlation matrix C can be singular and of a size that tookconsiderable time to solve when the method was first proposed. These problemswere overcome by splitting the problem into smaller subgroups and conductingsingular value decomposition on the C matrix.

Users of RspMatch have known for many years that the original time-serieswill retain more of its original character if the adjustment is applied in stages overprogressively wider frequency bands. This is also true for RspMatch2005 and in theexample given below the frequency match has been applied in two stages: the firstmatches from 0.05 s (20Hz) to 1.0 s, and the second from 0.05 s to 5.0 s. For brevityonly the results of the final stage are presented.

The Lilhanand and Tseng [1987, 1988] method generally works well, but thereare two main problems with the procedure. Firstly, the wavelet used corrupts thevelocity and displacement time-series of the accelerograms, so a baseline correctionis required after the wavelet adjustment, which can partially undo the spectralmatch. Secondly, the method is not always stable and diverges if the user attemptsto match at closely spaced periods and multiple damping levels.

3. Wavelet Functional Form

One of the key features to the adjustment method is the functional form used forthe wavelet adjustment. Wavelets have many different functional forms, but in theinterest of brevity, only those used in the RspMatch and RspMatch2005 programsare described in this section.

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3.1. Wavelet used by Lilhanand and Tseng

The original wavelet used by Lilhanand and Tseng [1987, 1988] and RspMatch is areverse impulse function:

aj(t) =−ωj√1 − β2

j

exp(−ωjβj(tj − t))[(

2β2j − 1

)sin

(ω′

j(tj − t))

− 2βj

√1 − β2

j cos(ω′

j(tj − t))]

, (4)

where,

aj(t) is the amplitude of the jth wavelet at time t

tj is the time of the peak response of the jth oscillator under the action ofthe jth wavelet

ωj is the circular frequency of the jth waveletβj is the damping level (proportion of critical) of the jth oscillator

ω′j is the damped circular frequency ω′

j = ωj

√1 − β2

j .

Although this wavelet is very efficient in adjusting the response, it has thedisadvantage that it corrupts the velocity and displacement time-history becausethe wavelet does not end with zero velocity or displacement (Fig. 2). To over-come this issue two new displacement compatible wavelets have been created forRspMatch2005.

3.2. Sinusoidal corrected wavelet

The sinusoidal corrected wavelet is a hybrid wavelet based the wavelet used bySuarez and Montejo [2003, 2005] that includes a sinusoidal correction to ensure zerofinal displacement. The equation describing the Suarez and Montejo wavelet is:

aj(t) = e−βjωj |t−tj+∆tj | sin(ωj(t − tj + ∆tj)), (5)

where ∆tj is the difference between time of peak response tj and the referenceorigin of the wavelet. Unlike the other wavelets described above, this wavelet is onlyapplied for a fixed number of cycles (Nc) specified by the user and automaticallyreduced by the program as required to ensure the whole wavelet is applied to theaccelerogram.

The error in the final displacement from the Suarez and Montejo wavelet isobtained by double integration of the wavelet and applying the appropriate initialconditions:

DispErrorj = 2

−2βj + e−βjtdjωj (2βj + tdjωj + β2

j tdj) cos(tdjωj)+ e−βtdjω(1 − β2

j − βtdjωj) sin(tdjωj)(1 + β2

j

)2ω2

, (6)

where tdj is half the duration of the uncorrected jth wavelet equal to πωj

∗ Nc.

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Fig

.2.

Acc

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ati

on

(uppe

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vel

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dis

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(lower

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rever

seim

puls

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avel

et(lef

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sinuso

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dw

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)and

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(rig

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.

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To overcome this problem a sinusoidal half cycle at the start and end of thewavelet is used to correct the displacement time-series (Fig. 2). The amplitude ofthe sinusoidal correction is given by:

SinAmplitudej =−DispError · ωj(

2πωj

)+ 4tdj

. (7)

In the unlikely event that there is insufficient space within the record to apply asinusoidal correction at the end of the wavelet, a polynomial baseline correction isapplied to the wavelet.

3.3. Corrected tapered cosine wavelet

The corrected tapered cosine wavelet is an update of the wavelet used byAbrahamson [1992] that includes an additional correction to ensure zero final dis-placement (Fig. 2). The equation describing the tapered cosine wave is given by:

aj(t) = cos[ω′j(t − tj + ∆tj)] exp[−|t − tj + ∆tj |ψj ], (8)

where ∆t for the tapered cosine wavelet is given by:

∆tj =tan−1

[√1−βj

βj

]ω′

j

. (9)

The frequency dependence of ψj should be consistent with the reference time-history. That is, if the reference time-history has a short duration at a particularfrequency, the ψj should be selected such that the adjustment function at thatfrequency will also have a short duration. A tri-linear model for ψj(f) is used inthis program:

ψ(f) =

z1 for fj < f1

z1 + (z2 − z1)(f − f1)(f2 − f1)

for f1 < fj < f2

z2 for fj > f2

, (10)

where f1, f2, z1 and z2 are constants and fj is the frequency of the jth wavelet inHz; the recommended values of the constants are f1 = 1 Hz, f2 = 4 Hz, z1 = 1.25and z2 = 0.25. The equation for the corrected tapered cosine wavelet is given by:

aj(t) = cos[ω′j(t − tj + ∆tj)] exp[−|t − tj + ∆tj |ψj

+ [c1(t − tj + ∆tj) + c2] exp[−|t − tj + ∆tj |5ψj]. (11)

The corrected tapered cosine wavelet is set so that it starts with an initial 1/4acceleration cycle to avoid long-period drift in the displacement time-series.

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4. Improved Solution Procedure

The fundamental assumption underlying the matrix solution method used inRspMatch is that the time of the peak response is the same before and after eachadjustment is applied. If this assumption was always valid the problem would belinear and could be solved exactly with a single iteration. Unfortunately, this is notthe case and the problem is nonlinear, particularly if closely spaced spectral pointsand multiple damping levels are to be matched. Movement in the time of a responsepeak is caused from either:

• A phase change in the response peak• A new “secondary” peak becoming critical

Both of these sources of divergence are illustrated in Fig. 3 and are caused bythe cross correlation of different wavelet corrections. Put simply the wavelet addedto correct one period and damping level can also affect the peak response of SDOFsystems at other periods and damping levels. A general framework for describ-ing non-linear problems is presented by Tarantola [2005]. The specific methodsdeveloped by the authors for the solving this particular problem is described in thefollowing sections.

Fig. 3. Illustration of sources of diverging response. Note that the response amplitude remainsapproximately unchanged at the time of the original peak response, but the response increasesthrough both a phase shift and the emergence of a secondary peak.

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4.1. Reduction of off-diagonal C matrix elements

RspMatch subdivides the C matrix into smaller sub-matrices, which increases solu-tion speed but more importantly reduces the cross correlation of the wavelets andthe likelihood of a phase change in the peak response. The subdivision essentiallysets some of the off-diagonal terms to zero, which is beneficial as it provides numer-ical stability but is at the expense of the accuracy of the solution because the crosscorrelation of some wavelets is not taken into account. RspMatch2005 avoids thisissue by using the full C matrix and obtains numerical stability by reducing theoff-diagonal terms by a constant factor. From conducting trials with different off-diagonal reduction factors, it is found that a reduction factor of about 0.7 is veryeffective in providing numerical stability.

4.2. Preventing secondary peaks

Reducing the off-diagonal terms of the C matrix improves numerical stability, butit does not reduce the occurrence of secondary peaks. The new solution procedurechecks for divergence using the maximum misfit, which is defined as the greatestmisfit calculated from all of the periods to be matched; misfit at spectral period T

is defined as:

Misfit(T ) =∣∣∣∣SA(T ) − SAtarget(T )

SAtarget(T )

∣∣∣∣ ∗ 100, (12)

where SA(T ) is the spectral acceleration of the adjusted ground motion at thisiteration at period T , and SAtarget(T ) is the target spectral acceleration.

The new solution procedure prevents divergence from secondary peaks by addinga new wavelet at the period, damping level and time of the new secondary peak.The adjustment function is recalculated using the secondary wavelet. An exampleof the application of this method is shown in Fig. 4.

4.3. Reduction of correction amplitude

The solution can sometimes still diverge, even with the correction of secondarypeaks. This occurs when the cross correlation of the responses causes a phase changeof the response peak. Figure 5 shows one such case; here the original response hasa good match with the target, but wavelets introduced to adjust the responseat other periods cause a phase change and a divergence. This issue is overcomeby increasing the importance of the SDOF with the diverging response. This isachieved by reducing the amplitude of the adjustment correction, δR, by 30% for allthe points to be matched except the point causing divergence. Although artificiallyreducing the amplitude of the adjustment function increases the numerical stability,it results in a greater number of iterations being required to obtain the requiredspectral match.

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Fig. 4. Prevention of diverging response with an additional wavelet adjustment.

Fig. 5. Prevention of diverging response with reduction of correlated wavelet targets.

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4.4. Pseudo spectral acceleration

Spectral displacements are required for very long-period structures, tunnels, baseisolated structures and for direct displacement-based design. Nonlinear static(pushover) methods of analysis may require elastic spectral displacements for peri-ods up to about 5 seconds and damping levels from 5% of critical up to about30% in most cases. As spectral displacements are directly related to the pseudospectral acceleration (PSA), not the absolute spectral acceleration, PSA should beused when matching spectral displacements. PSA can be calculated directly fromthe spectral displacement, SD for any period T :

PSA = SD

(2π

T

)2

. (13)

RspMatch2005 uses pseudo-spectral accelerations for this reason. The differencebetween the pseudo and absolute spectral accelerations only becomes significantfor damping levels above about 20%.

4.5. New solution algorithm

A flowchart detailing the new solution algorithm is shown in Fig. 6. Although thenew algorithm prevents the solution from diverging it does not guarantee that thesolution will converge to within the requested tolerance. For cases with multipledamping levels and closely spaced spectral points it may be necessary to accept amore relaxed tolerance criteria than for cases where only a single damping level isto be matched. A balance needs to be maintained between the goodness-of-fit tothe response spectra and the degree of adjustment made to the accelerogram.

The ability of using different sub-groups has been left in the program to ensurecompatibility with earlier versions of the code; however, this is not recommendedwith use of the new algorithms that employ off-diagonal reduction.

5. Examples of New Matching Procedure

To illustrate the new matching procedure a spectrum-matched accelerogram is pro-duced for the scenario of a stiff soil site at 10 km from an Ms 7 earthquake. Thefirst half of this section shows the results of matching to the target acceleration anddisplacement spectra at a 5% damping level. The second half of this section showsthat RspMatch2005 is capable of producing a ground motion that simultaneouslymatches the 5, 10, 20 and 30% damping levels, provided a relaxation of the solutiontolerance is accepted.

5.1. Selection of seed accelerogram

The first step in the process is to select a suite of real accelerograms that maybe linearly scaled to obtain an approximate match with the spectral ordinates

April 28, 2006 13:47 WSPC/124-JEE 00273





Subdivide target into subgroups, each withperiods spread throughout the frequency range



Calculate “C” matrix for subgroup. Applying off-diagonal reduction.

Conduct singular value decomposition of matrix



Temporally add adjustment to total adjustment functionand check response


Load next subgroup

No

Yes

No

Add total adjustment function to accelerogram

Is the solutionconverging?

Add wavelet to adjustsecondary peak

Is the mismatch peak

within a half cycle of an existing matched

point?

Reduce amplitude ofelements in that of divergingspectral point

Yes

No

Yes

No

Yes




Is this the lastfrequency range?

No

Yes





Subdivide target into subgroups, each withperiods spread throughout the frequency range



Calculate “ atrix for subgroup. Applying off-diagonal reduction.

Conduct singular value decomposition of C matrix



Temporally add adjustment to total adjustment functionand check response


Load next subgroup

No

Yes

No

Add total adjustment function to accelerogram

Is the solutionconverging?

Add wavelet to adjustsecondary peak

Is the mismatch peak

within a half cycle of an existing matched

point?

Reduce amplitude ofelements in δR exceptthat of divergingspectral point

Yes

No

Yes

No


Yes




Is this the lastfrequency range?

No

Yes

Fig. 6. RspMatch2005 methodology including new solution algorithms (bold text).

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before applying the wavelets adjustments. The issues related to selecting and scalingaccelerograms are beyond the scope of this paper but the reader is referred tothe following papers for guidance — and different perspectives — on this issue:Watson-Lamprey and Abrahamson [2006a], Bommer and Acevedo [2004], Naeimet al. [2004], Malhotra [2003]. The accelerograms in this paper have been selectedin accordance with the recommendations of Bommer and Acevedo [2004] who showthat distance has little influence on spectral shape and so they recommend a narrowsearch window in terms of magnitude but allow broad limits in terms of distance.Some engineers might consider the scale factors used in this paper are quite large;however, a recent study by Watson-Lamprey and Abrahamson [2006b] found thatspectral matched accelerograms with scale factors of over a factor of 10 could beused without causing a bias in nonlinear response.

For the illustrative example shown herein, a seed accelerogram has been selectedfrom the 3551 records of the PEER NGA dataset [PEER 2005]. Initial selection isconducted based on an approximate match to the earthquake magnitude and thespectral shape using the RMS of the difference in normalised spectral accelera-tion (∆SAnRMS ), Equation (14). Other methods of matching spectral shapes arepossible: for example, the shape could be normalised to a high-frequency spectralacceleration rather than PGA, or to the log of the normalised spectral acceleration.

∆SAnRMS =

√√√√ 1Np

Np∑i=1

(PSA0(Ti)

PGA0− PSAs(Ti)

PGAs

)2

, (14)

where Np is the number of periods, PSA0(T i) is the pseudo spectral accelerationfrom the record at period Ti, PSAs(Ti) is the target pseudo spectral accelerationat the same period; PGA0 and PGAs are the peak ground acceleration of theaccelerogram and the zero-period anchor point of the target spectrum.

The record selected is the 1989 (Mw 6.9) Loma Prieta earthquake recorded71 km from the fault rupture at Diamond Heights (record 00794T in the NGAdatabase). The code allows for scaling of the accelerogram to either PGA or aselected scale factor. Here we have chosen to linearly scale by a factor of 3.2 sothat the difference between the record and target PSA and SD is minimised. Forengineering projects where high-frequency ground motion is of importance it isappropriate to scale to PGA or a high-frequency spectral acceleration in order topreserve the high-frequency characteristics of the ground motion.

5.2. Target spectra

The median 5% damped target spectra is generated according to the method pro-posed by Bommer et al. [2000] using the peak ground motions proposed by Tromansand Bommer [2002] notwithstanding the limitations associated with these formula-tions arising from their derivation using analogue recordings [Boore and Bommer,2005]. The target spectra for damping levels other than 5% are obtained using

April 28, 2006 13:47 WSPC/124-JEE 00273


the formulae derived by Bommer et al. [2000], which was subsequently adopted byEurocode 8 [CEN 2002]:

η =√

105 + β

, (15)

where, η is the linear scale factor between the 5% damped response spectrum andthe required spectrum at β damping at intermediate periods. Equation (15) is asimplification and is used herein only for illustrative purposes; the scaling of the5%-damped spectral ordinates for higher damping levels has recently been shownto be a function of the strong-motion duration [Bommer and Mendis, 2005; Mendisand Bommer, 2006].

5.3. Matching 5% damped spectrum

The ground motion acceleration is adjusted so that it matches the target spec-trum between 0.05 s and 5 s period (Fig. 7). Matching to longer periods is possiblebut not conducted because the filter frequency of the seed accelerogram is 5 s;indeed the useable frequency range will be less than the filter frequency (e.g. Akkarand Bommer, 2006). The average spectral misfit between 0.05 s and 5 s period forthe 5% damping level has improved from 15% in the linearly scaled record to 1%after wavelet adjustment with RspMatch2005. The average of the spectral misfit isdefined as:

AverageMisfit =1

Np

Np∑i=1

∣∣∣∣PSAo(T i) − PSAs(T i)PSAs(T i)

∣∣∣∣ ∗ 100. (16)

Note that precisely the same average misfit is calculated if the PSA terms inEq. (16) are replaced with SD. When comparing results the misfit must be cal-culated at closely-spaced periods, not just those used to conduct the spectralmatching.

Examination of the acceleration, velocity and displacement time-series beforeand after the wavelet adjustment shows that the characteristics of the originalrecords have been retained (Fig. 8). Checks of the build up of Arias intensity alsodemonstrate that the energy distribution within the record is similar to the originalground motion and that the total energy content has been changed by less thanabout 5% by the wavelet adjustment (Fig. 9).

5.4. Matching multiple damping levels

The ability of RspMatch2005 to adjust accelerograms to fit multiple damping levelsis investigated by running the program four times, fitting the accelerogram used inthe previous section to increasing numbers of damping levels (Fig. 10). Figure 10shows that matching to the 5% damping spectra alone does not ensure a goodmatch at other damping levels. Although the match is not exact with four damp-ing levels, the program has reduced the average spectral misfit at all damping

April 28, 2006 13:47 WSPC/124-JEE 00273


Fig. 7. 5% spectral acceleration (upper) and displacement (lower) of the target response (dashedblack line), original linearly scaled ground motion (solid grey line) and adjusted ground motion(solid black line).

levels by a factor of about 3 (Table 1). This demonstrates RspMatch2005’s abil-ity to match multiple damping levels, provided a reduced convergence toleranceis accepted when matching increased numbers of damping levels. Examination ofthe acceleration, velocity and displacement time-series before and after the waveletadjustment shows that the characteristics of the original record have been retained(Fig. 11). Checks of the build up of Arias intensity also demonstrate that the energydistribution within the record is similar to the original ground motion and that thetotal energy content has been changed by less than about 10% by the waveletadjustment (Fig. 12).

6. Discussion

An improved method is presented for the wavelet adjustment of recorded groundmotions to achieve a match between the target design spectrum and the response

April 28, 2006 13:47 WSPC/124-JEE 00273


Fig

.8.

Com

pari

son

ofth

eacc

eler

ati

on,vel

oci

tyand

dis

pla

cem

ent

tim

e-se

ries

of

the

ori

gin

allinea

rly-s

cale

dgro

und

moti

on

(gre

yline)

and

adju

sted

gro

und

moti

on

(bla

ckline)

.

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Fig. 9. Build up of Arias intensity from the original linearly-scaled accelerogram (grey line) andadjusted ground motion (black line).

spectra of the accelerograms. New wavelets have been developed that have zerofinal velocity and displacement, ensuring that records do not require a baselinecorrection after wavelet adjustment. The procedure is applied using pseudo-spectralacceleration so that spectral displacements can be matched. This method enablesrecords to be adjusted so that they match the target response spectrum at moredamping levels than previously possible, although the goodness-of-fit to the targetspectrum reduces as the number of target damping levels increases.

The option of adjusting real strong-motion recordings to achieve a match to thetarget spectrum renders the use of artificial spectrum-compatible signals generatedfrom white noise redundant. The choices that remains then are to use naturalaccelerograms scaled to achieve an approximate match to the target spectrumover a specified period range or to adjust the records using the wavelets tech-nique to achieve a close match with the target spectral ordinates. The latter optionreduces the variability of the inelastic response, which is particularly beneficial asthe number of accelerograms required to predict the response to a given confidencelevel depends on the standard deviation of the response. This means that inelasticresponse can be predicted with greater confidence and fewer analyses using accelero-grams matched to the elastic response spectrum. Studies by Carballo [2000] andWatson-Lamprey and Abrahamson [2006b] suggest that matched accelerograms canreduce the standard deviation of the inelastic response by a factor of 2 compared tolinearly scaled accelerograms. This reduces the number of accelerograms to estimatethe inelastic response to a given confidence level by a factor of about 4.

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Fig. 10. Spectral matching for different damping levels. Matched to 5% damped spectrum (toprow); 5 and 10% damped spectra (second row); 5, 10 and 20% damped spectra (third row); 5, 10,20 and 30% damped spectra (bottom row). Pseudo spectral acceleration (left column) and spectraldisplacement (right column).

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Table 1. Average spectral misfit between 0.05 and 5 seconds period for adjustmentconducted at different damping levels.

Damping Level

Damping level matched 5% 10% 20% 30% All

Original 14.8 12.7 8.7 6.6 10.7Matched 5% 1.0 6.1 10.6 12.3 7.5Matched 5 and 10% 2.8 2.3 6.3 9.2 5.2Matched 5, 10 and 20% 4.7 2.8 2.1 3.6 3.3Matched 5, 10, 20 and 30% 5.0 3.2 2.5 2.5 3.3

Fig. 12. Arias Intensity from original linearly scaled ground motion (grey line) and that adjustedto match the 5, 10, 20 and 30% damping levels from 0.05 to 5 seconds period (black line).

Although using scaled natural accelerograms may be preferable in terms ofconserving the characteristics of real ground motion, using spectrally-matchedaccelerograms has the advantage that the variability in spectral amplitude is greatlyreduced. If the target response spectrum has been obtained from probabilistic seis-mic hazard analysis (PSHA), then the ground-motion variability will already beincorporated into the ordinates of the target spectrum; using scaled natural recordscan thus mean double counting of this aleatory variability.

The program is available on request from the corresponding author, providedtogether with a user manual.

Acknowledgements

We would like to express our thanks to Luis Montejo for sending us his thesis andjournal publications and Luis Suarez for interesting discussions on this subject. We

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would also like to acknowledge fruitful discussions with John Douglas and DavidBoore on the possible methods of measuring the difference between recorded andtarget spectral shape. The paper has benefited from the thorough reviews of MiguelCastro and two anonymous reviewers, for which we are most grateful. The workof the first and seventh authors is supported by doctoral training grants from theEPSRC and Marie Curie Fellowships.

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Documents

AN IMPROVED METHOD OF MATCHING RESPONSE SPECTRA OF RECORDED EARTHQUAKE GROUND MOTION USING WAVELETS