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

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 Imperial College Press

AN IMPROVED METHOD OF MATCHING RESPONSE SPECTRA 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 the form of acceleration time-series, and these will generally be required to be compatible with the elastic response spectra representing the design seismic actions at the site. The advantages of using real accelerograms matched to the target response spectrum using wavelets for this purpose are discussed. The program RspMatch, which performs spectral matching 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 displacement spectral ordinates as well as the spectrum of absolute acceleration, and additionally allows the matching to be performed simultaneously to a given spectrum at several damping 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 earthquake actions in the form of a response spectrum. In many situations, however, including the design of critical facilities, highly irregular buildings and base-isolated struc-tures, the simulation of structural response using a scaled elastic response spectrum is not considered appropriate to verify the earthquake resistance. In such cases, dynamic nonlinear analysis of the structure will be required and the seismic input then 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 the design seismic actions at the site. There are many different options for obtaining suites 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: j.bommer@ imperial.ac.uk.

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and Acevedo, 2004], the most widely used approaches being the use of artificial spectrum-compatible time-series, generated from white noise, and the use of scaled real accelerograms.

Artificial records constitute a convenient tool but their shortcomings, arising from their dissimilarity with real earthquake ground motions in terms of num-ber of cycles, phase content and duration, are widely recognised, and their use in nonlinear analyses is not recommended. These problems are avoided by using real strong-motion accelerograms, appropriately scaled to the target spectrum (at least in 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 large numbers of dynamic analyses in order to obtain stable estimates of the inelastic response of the structure. The required number of inelastic dynamic analyses can be significantly reduced if the real records are first matched to the target response spectrum, by eliminating the largest differences between the target spectrum and the spectral ordinates of individual accelerograms. This is clearly a compromise and in some sense the records become ‘artificial’ as a result, although the records can retain most (if not, in fact, all) of the characteristics of real earthquake records. The choice is essentially one of compromise between engineering pragmatism and seismological rigour, reducing the number of time-consuming structural analyses whilst 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 by adjusting 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 real ground motions and also have a close match to the target spectrum. However, adjusting the Fourier spectrum corrupts the velocity and displacement time-series and can result in motions with unrealistically high energy content [Naeim and Lew, 1995].

An alternative approach for spectral matching adjusts the time history in the time domain by adding wavelets to the acceleration time-series. Wavelet adjust-ment of recorded accelerograms has the same advantages as the Fourier adjustadjust-ment methods 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-stationary characteristics 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 that they do not cause a drift in the velocity or displacement time-series. RspMatch2005 also allows the records to be matched to a pseudo-acceleration spectrum rather than only the spectrum of absolute acceleration, and through improved convergence

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

2. Existing Wavelet Methods

There are several different methods of using wavelets to adjust accelerograms so that 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 continuous wavelet transform (CWT) to de-compose the original acceleration time-series into a number of time series with energy in non-overlapping frequency bands. An iterative procedure is used to scale each time history so that when they are added together they produce a spectrum-compatible ground motion. Although the approximate duration of the original accelerogram is retained using this type of adjustment pro-cedure, the adjusted accelerograms have visibly different amplitudes and frequency contents from the original accelerogram.

The method proposed by Lilhanand and Tseng [1987, 1988] employs wavelets but uses the response of elastic SDOF systems rather than the CWT. This enables accelerograms to be made spectrum compatible with smaller adjustments than the wavelet 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 given in Fig. 1. The essence of the methodology is as follows:

(1) Calculate the response of an elastic SDOF system under the action of the acceleration 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 amplitudes and 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 SDOF response under the action of the wavelet is the same as the time of the peak response to 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 result of adding the wavelet adjustment. This assumption is not always valid and this can lead 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 less than 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 by solving matrix, minimising the spectral misfits Scale and sum wavelets to create adjustment function Add adjustment to total adjustment function

Is this the last 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 frequency

range to be matched If required subdivide target

into frequency bands

Is last frequency range?

No

Yes Input in accelerogram

Calculate spectral response

Calculate amplitude and sign of spectral of misfit

Is misfit less than 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

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 by solving C matrix, minimising the spectral misfits Scale and sum wavelets to create adjustment function Add adjustment to total adjustment function

Is this the last 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 frequency

range to be matched If required subdivide target

into frequency bands

Is last frequency range?

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 equasolu-tions that account for the cross correlasolu-tion of each wavelet with each response to be matched. This can be expressed in matrix form:

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where C is a square matrix with elements that describe the amplitude of each SDOF response, at the time that the response needs to be adjusted, under the action of each wavelet, b is a vector of linear scale factors for each wavelet used in the adjustment, and δR is a vector of the required adjustment, the difference between the peak SDOF response of the unadjusted time-series and the required amplitude specified by the target spectra response for each period and damping level to be matched.

The wavelet scale factors in the b vector are found using amplitude of the required 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 from the sum of the amplitudes of the wavelets at that time, aj(t), multiplied by their individual scale factors bj.

Adjustment(t) = j=Nw

j=1

bj· aj(t), (3)

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

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

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

The Lilhanand and Tseng [1987, 1988] method generally works well, but there are two main problems with the procedure. Firstly, the wavelet used corrupts the velocity and displacement time-series of the accelerograms, so a baseline correction is required after the wavelet adjustment, which can partially undo the spectral match. Secondly, the method is not always stable and diverges if the user attempts to 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 for the wavelet adjustment. Wavelets have many different functional forms, but in the interest of brevity, only those used in the RspMatch and RspMatch2005 programs are 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 a reverse impulse function:

aj(t) = −ωj 1− βj2 exp(−ωjβj(tj− t))j2− 1sinωj(tj− t) − 2βj  1− βj2cosωj(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 of the jth wavelet

ωj is the circular frequency of the jth wavelet

βj is the damping level (proportion of critical) of the jth oscillator ωj is the damped circular frequency ωj = ωj

 1− βj2.

Although this wavelet is very efficient in adjusting the response, it has the disadvantage that it corrupts the velocity and displacement time-history because the wavelet does not end with zero velocity or displacement (Fig. 2). To over-come this issue two new displacement compatible wavelets have been created for RspMatch2005.

3.2. Sinusoidal corrected wavelet

The sinusoidal corrected wavelet is a hybrid wavelet based the wavelet used by Suarez and Montejo [2003, 2005] that includes a sinusoidal correction to ensure zero final 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 reference origin of the wavelet. Unlike the other wavelets described above, this wavelet is only applied for a fixed number of cycles (Nc) specified by the user and automatically reduced by the program as required to ensure the whole wavelet is applied to the accelerogram.

The error in the final displacement from the Suarez and Montejo wavelet is obtained by double integration of the wavelet and applying the appropriate initial conditions: DispErrorj = 2      −2βj+ e−βjtdjωj(2βj+ tdjωj+ βj2tdj) cos(tdjωj) + e−βtdjω(1− βj2− βtdjωj) sin(tdjωj)  1 + β2j 2 ω2     , (6) where tdj is half the duration of the uncorrected jth wavelet equal to π

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Fi g. 2. A ccel er ati o n (upp e r ), v e lo ci ty (mi dd le ) a nd di sp la cem en t (lowe r ) time -se rie s o f re v e rse imp u lse w a v e le t (le ft ), si n u so id al cor rected w a v e le t (mi dd le ) a nd cor rected tap e re d c os in e w a v el et (righ t).

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

SinAmplitudej= −DispError · ω j ωj



+ 4tdj . (7)

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

3.3. Corrected tapered cosine wavelet

The corrected tapered cosine wavelet is an update of the wavelet used by Abrahamson [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[ωj(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  ωj . (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 particular frequency, the ψj should be selected such that the adjustment function at that frequency will also have a short duration. A tri-linear model for ψj(f ) is used in this 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 in Hz; the recommended values of the constants are f1 = 1 Hz, f2 = 4 Hz, z1 = 1.25 and z2= 0.25. The equation for the corrected tapered cosine wavelet is given by:

aj(t) = cos[ωj(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 1/4 acceleration 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 in RspMatch is that the time of the peak response is the same before and after each adjustment is applied. If this assumption was always valid the problem would be linear and could be solved exactly with a single iteration. Unfortunately, this is not the case and the problem is nonlinear, particularly if closely spaced spectral points and multiple damping levels are to be matched. Movement in the time of a response peak 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 by the cross correlation of different wavelet corrections. Put simply the wavelet added to correct one period and damping level can also affect the peak response of SDOF systems at other periods and damping levels. A general framework for describ-ing non-linear problems is presented by Tarantola [2005]. The specific methods developed by the authors for the solving this particular problem is described in the following sections.

Fig. 3. Illustration of sources of diverging response. Note that the response amplitude remains approximately unchanged at the time of the original peak response, but the response increases through 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 correlasolu-tion of the wavelets and the likelihood of a phase change in the peak response. The subdivision essentially sets 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 cross correlation of some wavelets is not taken into account. RspMatch2005 avoids this issue by using the full C matrix and obtains numerical stability by reducing the 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 very effective in providing numerical stability.

4.2. Preventing secondary peaks

Reducing the off-diagonal terms of the C matrix improves numerical stability, but it does not reduce the occurrence of secondary peaks. The new solution procedure checks for divergence using the maximum misfit, which is defined as the greatest misfit 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 this iteration at period T , and SAtarget(T ) is the target spectral acceleration.

The new solution procedure prevents divergence from secondary peaks by adding a new wavelet at the period, damping level and time of the new secondary peak. The adjustment function is recalculated using the secondary wavelet. An example of 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 secondary peaks. This occurs when the cross correlation of the responses causes a phase change of the response peak. Figure 5 shows one such case; here the original response has a good match with the target, but wavelets introduced to adjust the response at other periods cause a phase change and a divergence. This issue is overcome by increasing the importance of the SDOF with the diverging response. This is achieved by reducing the amplitude of the adjustment correction, δR, by 30% for all the points to be matched except the point causing divergence. Although artificially reducing the amplitude of the adjustment function increases the numerical stability, it results in a greater number of iterations being required to obtain the required spectral match.

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

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4.4. Pseudo spectral acceleration

Spectral displacements are required for very long-period structures, tunnels, base isolated 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 about 30% in most cases. As spectral displacements are directly related to the pseudo spectral acceleration (PSA), not the absolute spectral acceleration, PSA should be used when matching spectral displacements. PSA can be calculated directly from the spectral displacement, SD for any period T :

P SA = SD  T 2 . (13)

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

4.5. New solution algorithm

A flowchart detailing the new solution algorithm is shown in Fig. 6. Although the new algorithm prevents the solution from diverging it does not guarantee that the solution will converge to within the requested tolerance. For cases with multiple damping levels and closely spaced spectral points it may be necessary to accept a more relaxed tolerance criteria than for cases where only a single damping level is to be matched. A balance needs to be maintained between the goodness-of-fit to the 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 ensure compatibility with earlier versions of the code; however, this is not recommended with 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. The first half of this section shows the results of matching to the target acceleration and displacement spectra at a 5% damping level. The second half of this section shows that RspMatch2005 is capable of producing a ground motion that simultaneously matches the 5, 10, 20 and 30% damping levels, provided a relaxation of the solution tolerance is accepted.

5.1. Selection of seed accelerogram

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

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Calculate spectral response

Calculate amplitude and sign of spectral of misfit

Is misfit less than 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

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

Conduct singular value decomposition of matrix

Find linear scale factor for each wavelet by solving C matrix, minimising the spectral misfits Scale and sum wavelets to create adjustment function

Temporally add adjustment to total adjustment function and check response

Is this the last subgroup? Load next subgroup

No

Yes No

Add total adjustment function to accelerogram

Is the solution converging?

Add wavelet to adjust secondary peak

Is the mismatch peak within a half cycle of an

existing matched point? Reduce amplitude of elements in that of diverging spectral point Yes No Yes No Yes

Save results to file Load next frequency

range to be matched If required subdivide target

into frequency bands

Is this the last frequency range?

No

Yes Input in accelerogram

Calculate spectral response

Calculate amplitude and sign of spectral of misfit

Is misfit less than 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

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

Conduct singular value decomposition of C matrix

Find linear scale factor for each wavelet by solving C matrix, minimising the spectral misfits Scale and sum wavelets to create adjustment function

Temporally add adjustment to total adjustment function and check response

Is this the last subgroup? Load next subgroup

No

Yes No

Add total adjustment function to accelerogram

Is the solution converging?

Add wavelet to adjust secondary peak

Is the mismatch peak within a half cycle of an

existing matched point? Reduce amplitude of elements in δR except that of diverging spectral point Yes No Yes No

Input in target spectrum

Yes

Save results to file Load next frequency

range to be matched If required subdivide target

into frequency bands

Is this the last frequency range?

No

Yes

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before applying the wavelets adjustments. The issues related to selecting and scaling accelerograms are beyond the scope of this paper but the reader is referred to the following papers for guidance — and different perspectives — on this issue: Watson-Lamprey and Abrahamson [2006a], Bommer and Acevedo [2004], Naeim et al. [2004], Malhotra [2003]. The accelerograms in this paper have been selected in accordance with the recommendations of Bommer and Acevedo [2004] who show that distance has little influence on spectral shape and so they recommend a narrow search 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 that spectral matched accelerograms with scale factors of over a factor of 10 could be used without causing a bias in nonlinear response.

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

∆SAnRMS=     1 Np Np  i=1 PSA 0(Ti) PGA0 PSAs(Ti) PGAs 2 , (14)

where Np is the number of periods, P SA0(T i) is the pseudo spectral acceleration from the record at period Ti, P SAs(Ti) is the target pseudo spectral acceleration at the same period; P GA0 and P GAs are the peak ground acceleration of the accelerogram and the zero-period anchor point of the target spectrum.

The record selected is the 1989 (Mw 6.9) Loma Prieta earthquake recorded 71 km from the fault rupture at Diamond Heights (record 00794T in the NGA database). The code allows for scaling of the accelerogram to either PGA or a selected scale factor. Here we have chosen to linearly scale by a factor of 3.2 so that the difference between the record and target PSA and SD is minimised. For engineering projects where high-frequency ground motion is of importance it is appropriate to scale to PGA or a high-frequency spectral acceleration in order to preserve 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 propro-posed by Tromans and 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

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the formulae derived by Bommer et al. [2000], which was subsequently adopted by Eurocode 8 [CEN 2002]: η =  10 5 + β, (15)

where, η is the linear scale factor between the 5% damped response spectrum and the required spectrum at β damping at intermediate periods. Equation (15) is a simplification and is used herein only for illustrative purposes; the scaling of the 5%-damped spectral ordinates for higher damping levels has recently been shown to be a function of the strong-motion duration [Bommer and Mendis, 2005; Mendis and 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 possible but 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. Akkar and Bommer, 2006). The average spectral misfit between 0.05 s and 5 s period for the 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 is defined as: AverageMisfit = 1 Np Np  i=1 

PSAo(Ti)− PSAs(Ti) PSAs(Ti)



 ∗ 100. (16) Note that precisely the same average misfit is calculated if the PSA terms in Eq. (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 spectral matching.

Examination of the acceleration, velocity and displacement time-series before and after the wavelet adjustment shows that the characteristics of the original records have been retained (Fig. 8). Checks of the build up of Arias intensity also demonstrate that the energy distribution within the record is similar to the original ground motion and that the total energy content has been changed by less than about 5% by the wavelet adjustment (Fig. 9).

5.4. Matching multiple damping levels

The ability of RspMatch2005 to adjust accelerograms to fit multiple damping levels is investigated by running the program four times, fitting the accelerogram used in the previous section to increasing numbers of damping levels (Fig. 10). Figure 10 shows that matching to the 5% damping spectra alone does not ensure a good match 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 dampdamp-ing

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Fig. 7. 5% spectral acceleration (upper) and displacement (lower) of the target response (dashed black 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 tolerance is accepted when matching increased numbers of damping levels. Examination of the acceleration, velocity and displacement time-series before and after the wavelet adjustment 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 energy distribution within the record is similar to the original ground motion and that the total energy content has been changed by less than about 10% by the wavelet adjustment (Fig. 12).

6. Discussion

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

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Fi g. 8. C o m p ar is on of the a ccel er ati o n, v e lo ci ty and d is pl acem en t ti m e-se ri es of the o ri gi nal li n ear ly-sc al ed gr ound m o ti on (gr ey li ne )a n d a d ju st e d gr ound m o ti on (bl a c k line ).

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

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

The option of adjusting real strong-motion recordings to achieve a match to the target spectrum renders the use of artificial spectrum-compatible signals generated from white noise redundant. The choices that remains then are to use natural accelerograms scaled to achieve an approximate match to the target spectrum over 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 option reduces the variability of the inelastic response, which is particularly beneficial as the number of accelerograms required to predict the response to a given confidence level depends on the standard deviation of the response. This means that inelastic response can be predicted with greater confidence and fewer analyses using accelero-grams matched to the elastic response spectrum. Studies by Carballo [2000] and Watson-Lamprey and Abrahamson [2006b] suggest that matched accelerograms can reduce the standard deviation of the inelastic response by a factor of 2 compared to linearly scaled accelerograms. This reduces the number of accelerograms to estimate the 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 (top row); 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 spectral displacement (right column).

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Fi g. 11. A ccel er ati o n, v e lo ci ty and d is pl acem en t ti m e-se ri es fr om or ig in al li b ear ly scal e d g ro und m o ti on and that a dj us ted to m atc h the 5 , 10, 20 a n d3 0 % d a m p in gl e v e ls fr o m 0 .0 5t o5s e c o n d s p e ri o d .

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

Damping Level

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

Original 14.8 12.7 8.7 6.6 10.7

Matched 5% 1.0 6.1 10.6 12.3 7.5

Matched 5 and 10% 2.8 2.3 6.3 9.2 5.2

Matched 5, 10 and 20% 4.7 2.8 2.1 3.6 3.3

Matched 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 adjusted to 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 of conserving the characteristics of real ground motion, using spectrally-matched accelerograms has the advantage that the variability in spectral amplitude is greatly reduced. If the target response spectrum has been obtained from probabilistic seis-mic hazard analysis (PSHA), then the ground-motion variability will already be incorporated into the ordinates of the target spectrum; using scaled natural records can thus mean double counting of this aleatory variability.

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

Acknowledgements

We would like to express our thanks to Luis Montejo for sending us his thesis and journal 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 David Boore on the possible methods of measuring the difference between recorded and target spectral shape. The paper has benefited from the thorough reviews of Miguel Castro and two anonymous reviewers, for which we are most grateful. The work of the first and seventh authors is supported by doctoral training grants from the EPSRC and Marie Curie Fellowships.

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