EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS
Earthquake Engng Struct. Dyn. 2006; 35:1145–1165
Published online 19 April 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.577
Inuence of long-period lter cut-o on elastic
spectral displacements
Sinan Akkar1; ∗; † and Julian J. Bommer2; ‡
1Department
of Civil Engineering; Earthquake Engineering Research Center; Middle East Technical University;
Ankara 06531; Turkey
2Department of Civil and Environmental Engineering; Imperial College London; SW7 2AZ; U.K.
SUMMARY
The eect of the long-period lter cut-o, Tc , on elastic spectral displacements is investigated using a strong ground-motion database from Europe and the Middle East. The relation between the
lter and oscillator responses is considered to observe the inuence of Tc for both analogue and digital records, and the variations with site classication, magnitude, lter order and viscous damping.
Robust statistics are derived using the re-processed European data to generalize the eects of the longperiod lter cut-o on maximum oscillator deformation demands as a function of these seismological
and structural features. Statistics with a 95% condence interval are derived to suggest usable period
ranges for spectral displacement computations as a function of Tc . The results indicate that the maximum period at which spectral displacements can be condently calculated depend strongly on the
site class, magnitude and lter order. The period range where reliable long-period information can
be extracted from digital accelerograms is twice that of analogue records. Copyright ? 2006 John
Wiley & Sons, Ltd.
KEY WORDS:
data processing; long-period lter cut-o; elastic oscillator displacement demands;
European ground-motion database
1. INTRODUCTION
A reliable ground-motion data set processed by a consistent scheme with a sound physical
basis is an a priori requisite for most research in earthquake engineering and engineering
seismology. This issue plays an essential role for displacement-based design and seismic
∗ Correspondence
to: Sinan Akkar, Department of Civil Engineering, Earthquake Engineering Research Center,
Middle East Technical University, Ankara 06531, Turkey.
† E-mail: [email protected]
‡ E-mail: [email protected]
Contract=grant sponsor: Royal Society
Contract=grant sponsor: Scientic and Technical Research Council of Turkey (TUBITAK)
Copyright ? 2006 John Wiley & Sons, Ltd.
Received 31 October 2005
Revised 15 February 2006
Accepted 24 February 2006
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S. AKKAR AND J. J. BOMMER
performance assessment procedures that are in continuous improvement for future seismic
design codes, as there is a need for extracting reliable long-period information from the
ground-motion records to model structural performance in terms of seismic displacement demands. In general, the methods that have been developed within the context of these procedures use elastic and inelastic displacement spectra to accomplish this objective [1]. Given
the displacement ductility demand () the equivalent linear methods make use of both elastic and inelastic spectral displacement information for a wide range of oscillator periods and
high viscous damping values to describe the most appropriate relationship between the actual
inelastic displacement response and the corresponding equivalent linear system (e.g. Reference [2]). The methods that directly estimate the inelastic spectral displacement from its elastic
counterpart also need the information revealed by the actual elastic and inelastic maximum
oscillator response to derive the regression relationships as functions of displacement ductility
(), initial oscillator period (To ) and normalized lateral strength ratio (i.e. R, elastic to yield
strength ratio of the oscillator) (e.g. References [3, 4]). Thus, regardless of the methodology or
its underlying theory, a well-processed ground-motion data set, including reliable information
about its usable period range, is crucial for such procedures in order to yield realistic results
for engineering use.
Of the various ground-motion processing schemes, ltering is the most popular procedure
among the data-processing agencies for removing the short- and long-period noise from the
raw ground-motion data to obtain processed waveforms. The ideal ltering should remove
the noise-contaminated part of the frequency content from the record and should not alter
the important features of the ground motion for a pre-determined frequency band. There is
abundant literature proposing dierent ltering schemes, particularly to remove the long-period
noise from the recorded acceleration time series in order to compute the ground velocity and
displacement, and response spectra (e.g. References [5–9]). Although the ultimate objective
is well-dened in ltering techniques, the choice of major lter parameters (i.e. the cut-o
periods to remove the short- and long-period noise and the lter order to control the rate
of roll-o) can strongly aect the computed velocity, displacements and response spectral
ordinates.
The choice of long-period lter cut-o period (Tc ) is the prime concern for displacement
spectra as its computation emphasizes long-period ground-motions [10]. Studies to dene a
suitable Tc value for removing the long-period noise date back to early 1970s. Trifunac and
co-workers [5, 11] suggested a Tc value of approximately 16 s to remove the long-period noise
from the raw analogue accelerograms. Later studies indicated that Tc can vary substantially
from record to record depending on the signal-to-noise ratio that can be determined from
the xed trace on the original lm. References [6, 12, 13] proposed noise spectra for analogue records depending on dierent sources of digitization errors; the reader is referred to
Reference [14] for detailed information about the sources of digitization errors in analogue
accelerograms. These studies showed variations among each other as they focused on the
particular features of dierent sensor–digitizer combinations. Moreover, the lack of digitized
xed trace information in most analogue records makes the use of similar procedures impractical to dene a Tc value for a given accelerogram. The rapid development in sensor and
data-acquisition technology has resulted in the deployment of digital accelerographs in seismically active regions around the world during the last three decades. Digital accelerographs
provide signicantly higher quality and broader dynamic range recordings, allowing one to
extract longer period information from the ground motions when compared to the analogue
Copyright ? 2006 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2006; 35:1145–1165
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records. However, long-period errors also exist in digital accelerograms [15, 16] that similarly
need to be removed by a proper choice of Tc .
The uncertainty and subjectivity in the estimation of Tc become important because of the
particular inuence of this lter parameter on the usable period range of spectral displacements
computed from the processed accelerograms. Most prediction equations derived for estimating
spectral ordinates dene the useful period range of spectrum as a fraction of long-period lter
cut-o Tc . This factor is devised for reducing the lter response eects (due to roll-o and
lter order) on the computed spectral values. Abrahamson and Silva [17] and Spudich et al.
[18] dened this factor as 0.8 for 5-pole acausal (zero phase-shift) and causal Butterworth lter
response, respectively. The attempt was to have less than about a 10% eect on the 5% damped
oscillator response for 4- to 5-pole causal Butterworth lter. It has recently suggested that this
factor takes a value of approximately 0.5 for acausal Butterworth ltering (Dr W. J. Silva,
written communication, 2005). Ambraseys et al. [19] also considered 0:8Tc for the usable
bandwidth in deriving the most recent European prediction equations for acceleration spectral
ordinates. The ground-motion database maintained by PEER (Pacic Earthquake Engineering
Research Center) recommends the use of 0:8Tc as the usable bandwidth of processed groundmotions for causal 4-pole Butterworth lter. The California Strong-Motion Instrumentation
Program (CSMIP) suggested using its processed data right up to the long-period lter cut-o
for an acausal 4-pole Butterworth lter. Bommer and Elnashai [20] derived a displacementbased design spectrum by computing spectral displacement ordinates for vibration periods up
to 0:1 s below the lter cut-o period. While discussing the eects of acausal and causal lters
using particular high-quality digital accelerograms on spectral displacements, Boore and Akkar
[21] showed that the long-period lter cut-o can start inuencing the elastic and inelastic
displacement spectra at oscillator periods considerably shorter than Tc . Bazzurro et al. [22]
used 20 pairs of near-fault and 30 simulated ground motions and conrmed these ndings.
Using one analogue record Boore and Bommer [10] indicated that when low-order lters (i.e.
lters with gradual roll-o) are employed the usable spectral period range should be about one
half of the long-period lter cut-o. The common recommendation in these studies is the use of
acausal lters in order to reduce the inuence of Tc on the usable period range of the processed
records as this type of ltering results in zero phase-shift in the processed ground motions.
This paper describes the particular eects of Tc on the elastic oscillator displacements considering the relation between the lter and oscillator response. The distinction between analogue and digital records, and the inuence of magnitude, site classication, viscous damping
and lter order are investigated within the context of the study to observe the inuence of Tc on
oscillator displacement response. A European strong ground-motion database is re-processed
for this purpose to derive statistics on the usable period range of displacement spectra as a
function of long-period lter cut-o. Statistics with 95% condence level are computed to
guide researchers for determining spectral period ranges that would be least aected by Tc .
The results presented also provide insight regarding the maximum information that can be
retrieved from the actual European ground-motion data set for spectral displacement computations as a function of magnitude and site classication.
2. GROUND-MOTION DATA SET AND DATA-PROCESSING SCHEME
The European ground-motion databank is comprised of accelerograms from Europe and surrounding countries in the Middle East and North Africa. The databank has been compiled
Copyright ? 2006 John Wiley & Sons, Ltd.
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S. AKKAR AND J. J. BOMMER
numerous times for dierent objectives but the rst well-organized data processing was carried out to derive predictive equations for spectral acceleration using an elliptical lter with
Tc = 5 s for all records [23]. The second systematic processing of the data set was conveyed
in the same way by reducing the long-period lter cut-o to 4 s [24]. None of these studies
considered the frequency content or the noise in the ground-motion data while deciding on
the lter cut-os used. Bommer and Elnashai [20] re-processed all the records in the data set
to produce predictive equations for elastic spectral displacement ordinates. The ltering was
undertaken in an iterative manner, starting from Tc = 10 s and reducing it until the physical
appearance of the velocity and displacement traces was judged to be acceptable by visual
inspection. As indicated in the introductory section, Bommer and Elnashai [20] extracted the
spectral displacement information from each record for a spectral period range up to 0:1 s
below its lter cut-o period. Dening the usable period range up to almost the lter cut-o
resulted in low spectral displacement estimates, especially at long periods, due to the lter
cut-o and roll-o eects on the spectral displacement ordinates. Moreover, this procedure distorted the long-period information that could be retrieved from the data set since many of the
lter cut-o periods determined for removing the long-period noise were less than the corner
periods estimated from the theoretical source spectrum models (e.g. References [25, 26]). The
most recent processing of a European ground-motion data set was undertaken by Ambraseys
et al. [27] using an acausal Butterworth lter. The procedure used the available xed trace
information for analogue records to model the noise and determined Tc at the period where
signal-to-noise ratio is greater than 2. For digital records, Ambraseys et al. [27] used the
pre-event buer to model the noise and determined Tc for each record in the same way
as for analogue records. The long-period lter cut-o for analogue records with no xed
traces was determined using either the noise spectra proposed in Reference [16] or the procedure in Reference [28] that inspects the acceleration Fourier amplitude spectrum (FAS)
and determines Tc as the period where FAS no longer tends towards zero at long periods. The tendency of FAS towards large values at long periods is unjustiable, since source
theory indicates that the long-period FAS portion should decay in proportion to the reciprocal of frequency squared (e.g. References [26, 29, 30]). Although this procedure has the
most physical basis with respect to other schemes used, the use of pre-event memory to
model the noise in digital records will not provide the complete noise information as it lacks
the signal-generated noise [10]. Furthermore, the procedure does not include a comparison
between the selected Tc values and the corner periods estimated from theoretical source models
(e.g. References [26, 29, 30]) to observe whether the chosen lter cut-o removes a signicant
part of the actual signal (indicated by the plateau of the FAS of acceleration).
The majority of records from a recently compiled European ground-motion data set [19]
were re-processed in this study since the particular objective is to determine the inuence of Tc
on elastic oscillator displacements and to dene the usable period range for spectral
displacement ordinates by examining the inuence of the lter cut-o, lter order, site
classication, magnitude and viscous damping ratio. An acausal Butterworth lter was used in
the re-processing of the database. The total length of leading and trailing zero pads required
for acausal ltering was computed based on the recommendations by Converse and Brady [8].
The freely available record processing software (http:==quake.wr.usgs.gov= ∼boore=software
online.htm) developed by Dr David M Boore of the USGS was used in the computations.
This software applies a Butterworth lter in the time domain twice, forward and reverse,
resulting in a zero phase-shift that ensures the acausal lter behaviour. Bazzurro et al. [22]
Copyright ? 2006 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2006; 35:1145–1165
INFLUENCE OF LONG-PERIOD FILTER CUT-OFF
1149
noted dierences between this technique and the direct application of acausal Butterworth ltering in the frequency domain. The results presented should be considered within the context
of 2-pass time-domain ltering applied in this study. The processing scheme is summarized
below.
Step 1: Remove the mean from the raw accelerogram as an initial baseline correction by
subtracting the mean of pre-event portion from digital accelerograms and removing the mean
of whole accelerogram from analogue accelerograms. The record that is obtained after this
step is referred to as ‘mean removed’ in this study.
Step 2: If available, use digitized xed trace information for analogue records to compute (signal + noise)= noise ratio and determine Tc at the period where (signal + noise)= noise
ratio becomes greater than 3. Verify the selected Tc by reviewing the resulting velocity and
displacement waveforms computed from the ltered data. If physically unjustiable trends
still persist (i.e. velocity signicantly dierent than zero at the end of the record, long-period
uctuations running along the total record length, etc.) in these time series, lter the meanremoved record by choosing a smaller Tc value.
Step 3: For the rest of the records, use the guidance of acceleration FAS corner periods
estimated from two theoretical source models [25, 26] to select Tc in an iterative manner.
Step 3.1: Given the magnitude (M) of the events, compute the theoretical FAS corner
periods using the single-corner frequency model (JB88) [25] and two-corner frequency model
(AS00) [26] to have generic information on the theoretical long-period content of the records.
Step 3.2: Start with a Tc value that is longer than the theoretical FAS corner periods estimated from the above two models and gradually reduce it until the displacement waveforms
do not contain long-period uctuations that run along the total record length or any other
physically unjustiable variation such as ending with very large displacements. Try not to
choose a Tc value shorter than the FAS corner periods suggested by the theoretical models.
This would preserve long-period information as much as possible after ltering out the frequencies that are judged to be contaminated by noise. The lter cut-o can be shorter than
the theoretical FAS corner period if the long-period noise is judged to be excessive from the
visual inspection of ltered velocity and displacement data. However, the analyst should seriously consider the rejection of records for which extremely short cut-o periods are found to
be necessary. The frequency content of such records can be distorted severely due to excessive
ltering.
The choice of reference theoretical source spectra and the decision stages described in Steps
2 and 3 are still subjective and can vary from one analyst to another. Moreover, the operator
should be aware of the fundamental phase arrivals in seismograms to distinguish the longperiod noise uctuations described in Step 2 from those of wave trains such as surface wave
arrivals or long-period SH pulses. Some particular records from the Hector Mine and Denali
earthquakes constitute good examples to these complex waveforms [15, 31]. The inuence of
such particular waveforms cannot be captured on the theoretical source spectra. The prime
advantage of this procedure is to provide information about the frequency content of the ltered record in the context of chosen single- and double-corner source models that reect
the overall seismological features of the record given the magnitude. No instrument response
correction was applied to the records as there is no complete information about the characteristic features of all the accelerometers in the database and because the instrument correction
aects the very short-period information that is generally not a concern for displacement
response.
Copyright ? 2006 John Wiley & Sons, Ltd.
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S. AKKAR AND J. J. BOMMER
Figure 1 illustrates the processing scheme applied in this study using three (two analogue and one digital) accelerograms recorded on rock within relatively close distances of
the causative fault rupture. The rst row in Figure 1 shows the acceleration FAS for mean
removed and ltered data for dierent lter orders (i.e. n = 2; 4 and 8). (Clarication for the
abbreviations used in lter order within the context of time-domain ltering applied: n = 2; 4
and 8 corresponds to 2-pass, 2-, 4- and 8-pole Butterworth ltering, respectively.) The vertical
black dotted lines show the location of Tc chosen for each record. An increase in lter order
results in a faster decay in FAS values beyond Tc , which indicates faster roll-o in the lter
response. The vertical light-grey solid lines are the suggested FAS corner periods from the theoretical source models. The theoretical JB88 FAS corner period (TJB88 ) is the result of Brune’s
single-corner source spectrum model [29, 30] for a stress parameter (also referred to as stress
Mean removed
Tc=10s, n=2
Tc=10s, n=4
1
Tc=10s, n=8
Tc=10s
JB88, AS00 (Ta)
0.1
0.01
0.1
1
Period (s)
Mean removed
Tc=4s, n=2
Tc=4s, n=4
0.1
Tc=4s, n=8
Tc=4s
JB88, AS00 (Ta)
0.1
1
Period (s)
2
Ta
Digital
1
Mean removed
Tc=5s, n=2
0.1
Tc=5s, n=4
Tc=5s, n=8
Tc=5s
JB88, AS00 (Ta)
0.01
0.01
10
0.1
1
Period (s)
10
5
4
20
3
10
0
-10
-20
Mean removed
Filtered - Tc=10s
-30
Filtered - Tc=20s
Velocity (cm/s)
1
Velocity (cm/s)
Velocity (cm/s)
10
TJB88
Ta
1
0.01
0.01
10
TJB88
10 Analogue
Fourier Amplitude Spectrum (cm/s)
10
Fourier Amplitude Spectrum (cm/s)
Analogue
Umbria Marche, 1997 - M5.2
Borgo Cerreto Torre (Rock, d=11 km)
Umbria Marche, 1998 - M5.1
Cassignano (Rock, d=17 km)
Ta
100
TJB88
Fourier Amplitude Spectrum (cm/s)
Campano Lucano,1980 - M6.9
Sturno (Rock, d=14 km)
0
-1
Mean removed
Filtered - Tc=4s
40
50
0
-1
-2
Mean removed
Filtered - Tc=5s
-4
-5
5
60
1
-3
-2
30
2
10
time (s)
15
20
10
15
time (s)
20
25
time (s)
0.8
0.8
0
-10
0.4
Displacement (cm)
Displacement (cm)
Displacement (cm)
10
0.4
0.0
0.0
-0.4
-0.8
-20
-1.2
-0.4
30
40
50
time (s)
60
5
10
15
time (s)
20
10
15
20
time (s)
25
Figure 1. Data-processing scheme used in the study (time axis of the mean removed data is shifted to
conceal the leading zeros in the acausally ltered data for a better comparison between the time series).
Copyright ? 2006 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2006; 35:1145–1165
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drop) of 100 bar . The theoretical AS00 FAS corner period is computed from the double-corner
source spectrum [26] and represents the lower corner (Ta ) of the model that is related to the
nite-fault size. The second and third rows in Figure 1 show the velocity and displacement
time series computed from the mean removed and ltered data, respectively. The inuence
of long-period errors are more apparent in the mean removed displacement traces as these
tend to very large values that are not physically justiable for the given basic seismological
features of each record. The long-period lter cut-os chosen for the small magnitude events
(columns 2 and 3 in Figure 1) are longer than the theoretical FAS corner periods estimated
by the source models thus ensuring that the long-period waveforms suggested by the source
models are retained after processing. The lter cut-o for the large magnitude (M6.9) event
is in between the theoretical FAS corner periods, closer to the one estimated by JB88. The
reason for such a Tc choice is shown in the computed velocity and displacement time series.
These plots indicate that a cut-o (Tc = 20 s) that is larger than the estimated FAS corner
periods from both source models yields very large ground-displacement amplitudes suggesting
a shorter lter cut-o for removing the long-period noise. The iterative procedure described
in Step 3.2 resulted in a Tc value of 10 s for this record that yields tolerable variations in
ground displacements. Note that the long-period noise eect for the M5.2 digital record (third
column) is also clear from the long-period drift in the mean removed velocity waveform.
Figures 2(a) and (b) display M vs Tc scatter diagrams for the re-processed European
ground-motion database for analogue and digital records, respectively, for dierent site classes.
The shapes on the right-hand y-axes show the log-normal Tc distribution for analogue and
digital records and suggest relatively longer Tc values for digital records. Figure 2(a) presents
a total of 220 rock, 314 sti and 164 soft site analogue recordings. The scatter diagram in
Figure 2(b) shows the M vs Tc variations for 130 rock, 126 sti and 114 soft site digital
records. These numbers show the dominance of analogue records in the database, which
constitute 65% of the total. Rock sites have shear wave velocities (Vs;30 ) greater than 750 m= s
in the uppermost 30 m whereas sti and soft soil site classes have shear wave velocities of
360 m= s¡Vs;30 6750 m= s and 180 m= s¡Vs;30 6360 m= s, respectively.
Figure 2. Long-period lter cut-o (Tc ) vs magnitude (M) variation for the European ground-motion
data set in terms of: (a) analogue; and (b) digital records.
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S. AKKAR AND J. J. BOMMER
These scatter diagrams also show the theoretical FAS corner periods computed from the
source models considered. The JB88 single-corner period takes values that lie between the FAS
corner periods Ta and Tb estimated from the two-corner AS00 model. The corner period Ta
controls the nite-fault size whereas Tb controls the sub-fault size that reects the assumption
that large fault ruptures can be modelled by a series of smaller earthquakes [32]. Considering
the uncertainty in the FAS corner period estimations in these models, which are based on
empirical ground-motion data, lter cut-o values closer to the Tb bound can be interpreted
as representing the removal of an integral part of the signal while ltering out the longperiod noise. The information revealed from Figure 2 suggests that the ltering process has
removed a considerable amount of information from some of the large magnitude, rock and
sti site analogue records. These records should be treated carefully and should not be used
for calculated spectral ordinates at very long spectral periods as they would yield articially
reduced spectral displacements.
In general, the long-period lter cut-os presented in this study are longer than those
reported by some recent studies conducted on dierent European ground-motion data sets
[19, 24, 27]. The main disagreement between the Tc values suggested in each study is due
to dierent criteria employed in the proposed methodologies. For example, the lter cut-os
suggested by Ambraseys et al. [19, 27] were chosen by considering all three components
(i.e. two horizontal and one vertical) that resulted conservative Tc values for cases in which
the vertical components had lower amplitudes. Moreover, Ambraseys et al. [27] indicated
that the use of the noise spectra proposed in Reference [16] might result in signicantly
conservative estimations for automatically or semi-automatically digitized analogue records.
The long-period lter cut-os presented here are believed to convey physically reasonable
information and indicate that the corresponding velocity and displacement traces are visually
acceptable based on the iterative criteria established in Steps 2 and 3.
3. INFLUENCE OF LONG-PERIOD FILTER CUT-OFF ON ELASTIC
DISPLACEMENT RESPONSE
Figure 3 shows the 5% damped elastic oscillator displacement response history of the records
illustrated in Figure 1 at dierent vibration periods. The rst row presents the normalized acceleration FAS of the mean removed and ltered ground motions with respect to
their maxima. The period axes of the FAS plots are normalized by the lter cut-o period
(i.e. T=Tc ) to dene the relative location of oscillator period with respect to the chosen lter
cut-o. In this respect, the vertical grey dashed lines point to the locations of oscillator periods with respect to the lter cut-os for each case analysed. The spectrum plots also contain
the theoretical zero-phase lter response curves to describe the inuence of ltering in the
frequency domain. The interaction between the ground motion and ltering is clearly visualized when the theoretical lter response is evaluated together with the complex variations of
long-period spectral decay for each processed record relative to the mean removed record. The
second row in Figure 3 shows the corresponding oscillator displacement response histories of
mean removed and ltered records. The mean removed record brings out the distinct eects
of lter response and its interaction with the oscillator response in a comparative manner. The
plots in the rst column, which pertain to an M6.9 event recorded on an analogue instrument, show the results for a vibration period that is 0.7 times the selected lter cut-o period
Copyright ? 2006 John Wiley & Sons, Ltd.
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Figure 3. Elastic oscillator response of ltered and mean removed data at dierent T=Tc values.
(i.e. T=Tc = 0:7). The ltered data yields approximately 25% smaller absolute maximum oscillator displacement with respect to the mean removed data, indicating that the lter cut-o
inuence on oscillator response has started at a period much shorter than T=Tc = 0:7. This
observation is consistent with the information revealed by the corresponding FAS plot that
displays a fast decay in the long-period content of the ltered data with respect to the mean
removed data for T=Tc ¿0:5. Another important observation from this case is the dierence
in time intervals at which the mean removed and ltered data reach their absolute maxima
(locations of absolute maxima are shown by circles in each plot). The ltered data reach
its absolute maximum response slightly later than the mean removed data, which reects the
phase shift due to oscillator response as lter response is phase distortionless. The second
and third columns in this gure show that the oscillator response of the ltered record attains
larger values with respect to the mean removed record at periods signicantly shorter than
the lter cut-o. The discrepancies between the mean removed and ltered FAS plots in the
middle column where the theoretical lter response is at are also noted. These dierences can
be attributed to the application of ltering in the time domain that might alter the waveform
components at the frequency bands away from the lter cut-o. The dierence between the
mean removed and ltered traces, in terms of absolute maximum displacement, is 16% for the
analogue record (second column) whereas it is approximately 6% for the digital data (third
column). The analogue M5.1 record is a typical S-wave triggered record as can be perceived
from the relatively large oscillator response amplitude just at the beginning of the time series.
The step-like changes in the M5.2 digital record indicate that this record is aected by low
digitizer resolution. Such records are mostly from small magnitude events and their spectral quantities may show variations from large to small values at dierent oscillator periods
Copyright ? 2006 John Wiley & Sons, Ltd.
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S. AKKAR AND J. J. BOMMER
depending on the duration of initial portion lost due to trigger on S-wave or the level of digitizer resolution [33]. These non-standard problems cannot be resolved by ltering procedures,
although their application will tend to at least partially conceal them. Application of tapering
techniques is one procedure for handling the late triggered data, which should be applied with
caution in order not to alter the essential seismological features of the data. In this study, no
tapering was applied for late triggered data in order not to distort the data prior to ltering. Instead, these plots are presented to show how ltering may aect such cases Accelerograms that
exhibit recording problems are common to most strong ground-motion databases (e.g. S-wave
triggered pre-1999 analogue records in California [21]). The scarcity of empirical groundmotion data discourages analysts from simply discarding these records as they can be used up
to a certain period range for many engineering and strong ground-motion related studies [33].
The results discussed above show that the relation between the lter and oscillator response
is complex and the eects of ltering on elastic spectral displacements are signicant after
a certain level of T=Tc depending on the particular features of the ground motion. One way
of generalizing the overall eects of Tc on ground motions is to provide basic statistics
that describe the central tendency and dispersion information considering the relationships
between the lter cut-o and important oscillator and ground-motion features. The statistics
are provided for analogue and digital records by combining the inuence of the lter cut-o on
the usable period range with the possible contributions from site class, lter order, magnitude,
and viscous damping ratio. The displacement spectrum computed from the ltered ground
motion is normalized by the spectral displacement of the mean removed data (designated as
Sd;proc =Sd;unproc in the plots) to clearly study the lter cut-o eects on the processed records and
to have a general perspective on the suciency of mean removed data in spectral displacement
calculations for a certain period interval. A ratio of Sd;proc =Sd;unproc equal to 1 indicates that the
processed data are not inuenced by the lter response at that particular period of vibration.
If this ratio takes a value less than 1, the interpretation is that the ltering has started to
inuence the spectral displacement by removing some part of the signal in the vicinity of that
particular vibration period. Values of Sd;proc =Sd;unproc greater than 1 suggest the complex and
rigorous time-domain ltering modications in the frequency band of the record that contribute
to the spectral amplitude. Within the observations of this study such cases are conspicuous
amongst the low-quality data. The period axis of the displacement spectrum is normalized by
the lter cut-o period (T=Tc ) to extract general conclusions from the statistics presented about
the usable period range of ltered data in terms of Tc . As this ratio takes values closer to
1, the oscillator period approaches to the long-period lter cut-o selected for that particular
record.
3.1. Inuence of Tc on the usable period range in terms of magnitude
The long-period information carried by strong ground-motion records is mostly dominated by
the earthquake magnitude and site class. The magnitude is related to the source kinematics
that determine the fundamental waveform features. Figure 4 presents central tendency and
dispersion statistics to highlight the contribution of magnitude variation to the Tc inuence on
the useful period range of spectral displacements. The plots show the median and log-normal
standard deviation variations of 5% damped Sd;proc =Sd;unproc vs T=Tc for dierent magnitude
bins (i.e. 56M66 and M¿6). The lter order (n) was chosen as 4 (i.e. 2-pass, 4-pole)
while processing the data. The displacement spectra were computed for 0:16T=Tc 62. The
Copyright ? 2006 John Wiley & Sons, Ltd.
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INFLUENCE OF LONG-PERIOD FILTER CUT-OFF
Stiff Soil Records
Stiff Soil Records
1
Std. Dev. - Sd,proc/Sd,unproc
Median -Sd,proc/Sd,unproc
1
5 < M < 6 - Analogue
M > 6 - Analogue
5 < M < 6 - Digital
M > 6 - Digital
0.1
0.01
0.1
0.1
1
T/Tc
0.1
1
T/Tc
Figure 4. Inuence of magnitude on the usable period range of spectral displacements.
extension of T=Tc for values greater than 1 is for illustrative purposes rather than any physical
meaning and only completes the general view of the variation of spectral displacements under
the inuence of Tc . The same T=Tc limits are used while presenting the majority of the statistical results discussed in the paper to preserve the uniformity in graphical illustrations. The
results are presented for sti site records but the general conclusions derived from these plots
apply to the other site classes. The central tendency (median) plots indicate that an increase
in magnitude, on average, results in a longer usable period range for spectral displacement
computations for both digital and analogue accelerograms. The log-normal standard deviations
of large magnitude events are also fairly low with respect to the small magnitude events suggesting that lter cut-o eects are rather limited for ground motions recorded from higher
magnitude earthquakes. The rich long-period frequency content of large magnitude events is
the main reason for the reduced inuence of Tc on the ltered ground-motion data.
Another observation from these plots is the superior performance of digital records with
respect to the analogue records in terms of spectral displacement information. Regardless of
the magnitude bins, digital records show a more stable Sd;proc =Sd;unproc variation for longer T=Tc
values suggesting that the lter cut-o starts inuencing the computed spectral displacements
at relatively longer periods with respect to the analogue records. The stable Sd;proc =Sd;unproc
variation in digital records is the result of broadband frequency response of the instrument
that yields richer long-period information in the recorded data; analogue recordings tend to
have greater noise in the long period range. The superior behaviour of digital records with
respect to analogue records in terms of usable period range is also noted in almost all of the
discussions presented in the following sections.
3.2. Inuence of Tc on the usable period range in terms of site class
The plots in Figure 5 show the lter cut-o inuence on the usable period range of elastic
spectral displacements for dierent site classes. A fourth-order (i.e. 2-pass, 4-pole) acausal
Copyright ? 2006 John Wiley & Sons, Ltd.
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S. AKKAR AND J. J. BOMMER
Rock - Analogue - n = 4 (5% Elastic)
Rock - Digital - n = 4 (5% Elastic)
1
Sd,proc/Sd,unproc
Sd,proc/Sd,unproc
1
Median
16 and 84 percentiles
Filter Response
0.1
0.1
Median
16 and 84 percentiles
Filter Response
0.1
1
1
0.1
T/Tc
T/Tc
Stiff - Analogue - n = 4 (5% Elastic)
Stiff - Digital - n = 4 (5% Elastic)
1
Sd,proc/Sd,unproc
Sd,proc/Sd,unproc
1
Median
16 and 84 percentiles
Filter Response
Median
16 and 84 percentiles
Filter Response
0.1
0.1
0.1
1
1
0.1
T/Tc
T/Tc
Soft - Analogue - n = 4 (5% Elastic)
Soft - Digital - n= 4 (5% Elastic)
1
Sd,proc/Sd,unproc
Sd,proc/Sd,unproc
1
Median
16 and 84 percentiles
Filter Response
Median
16 and 84 percentiles
Filter Response
0.1
0.1
0.1
1
T/Tc
0.1
1
T/Tc
Figure 5. Inuence of site conditions on the usable spectral period range in
terms of analogue and digital records.
Copyright ? 2006 John Wiley & Sons, Ltd.
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1157
Butterworth lter was used in the ltering process and 5% damping was used in the spectral
displacement calculations. The rst row presents the Sd;proc =Sd;unproc vs T=Tc plots for all rock
site recordings whereas the second and third rows display the same information for sti and
soft soil sites, respectively. The results pertaining to analogue and digital records are displayed
in the left- and right-hand panels, respectively. Each plot in Figure 5 also shows the median,
16 and 84% percentile curves. The theoretical lter response curves are also presented in these
plots. These curves should be considered as a rst-order approximation to the inuence of
Tc on the usable period range as spectral response amplitudes are inuenced from a range of
ground-motion frequencies whereas lter response is computed for a particular cut-o period.
However, they emphasize the complicated interaction between the lter and elastic oscillator
response associated with the limitations during the recording phase and contributions from
complex seismological features.
The ground-motion data recorded on soft soil site class is the least aected group from
the lter cut-o for both analogue and digital recordings. This site class yields a more stable Sd;proc =Sd;unproc variation for longer T=Tc ratios manifested by a less dispersive behaviour
when compared to the variations in rock and sti soil records. The soft soil site recordings are rich in long-period motions that shift the lter cut-o eect towards longer periods with respect to the other site classes. In all plots the dispersive behaviour increases as
the oscillator periods take values closer to the lter cut-o, which is more discernible for
analogue records. The level of dispersion increases for ground motions recorded on stier
sites.
Figure 6 shows a more comparative view for the Tc eect on spectral displacements in
terms of site classes by plotting the median and log-normal standard deviation variation of
Sd;proc =Sd;unproc against T=Tc . The statistics presented are derived form 5% damped displacement
spectra. The median values for analogue and digital ground motions recorded on dierent site
classes are presented in the rst row. The second row shows the log-normal standard deviation
(dispersion) in the same format. These plots complete the observations highlighted in this
section and Section 3.1. The digital records yield substantially lower dispersion and longer
usable period range with respect to the analogue records for all site classes. On average, the
soft site recordings are aected to a lesser degree by the lter cut-o as the oscillator periods
get closer to Tc . They also yield smaller dispersion about the central tendency of Sd;proc =Sd;unproc
for T=Tc 60:4 in analogue and T=Tc 60:7 in digital records, suggesting that the inuence of
Tc is smaller for this group with respect to the other site classes. Combining the information
revealed by the central tendency and dispersion statistics, the unprocessed (mean removed)
soft soil records would yield fairly similar spectral displacement values as the processed data
for T 60:4Tc for analogue and for T 60:7Tc for digital records. Although the median curves
for rock and sti records follow a very similar pattern to the ones presented for soft soil,
the larger dispersion associated with these site classes suggests that greater caution should be
applied in using their unprocessed data for spectral displacement computations, particularly
for analogue records.
3.3. Inuence of Tc on the usable period range in terms of lter order
The lter order (n) controls the decay in the roll-o of the lter response between the
pass-band and stop-band. The increase in lter order will result in a faster decay in the
roll-o, which results a more abrupt change in the lter response in the vicinity of
Copyright ? 2006 John Wiley & Sons, Ltd.
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S. AKKAR AND J. J. BOMMER
Digital - Elastic 5%, n=4
Median - Sd,proc/Sd,unproc
Median - Sd,proc/Sd,unproc
Analogue - Elastic 5%, n=4
1
Rock
Stiff
Soft
0.1
1
0.1
1
Digital - Elastic 5%, n=4
Analogue - Elastic 5%, n=4
1
Std.Dev. - Sd,proc/Sd,unproc
1
Std.Dev. - Sd,proc/Sd,unproc
1
T/Tc
T/Tc
0.1
0.01
0.001
0.1
1
T/Tc
0.1
0.01
0.001
0.1
1
T/Tc
Figure 6. Median and dispersion statistics for usable spectral period in terms of Tc for analogue and
digital records from rock, sti and soft soil sites.
long-period lter cut-o. Thus, higher lter orders would mimic the ideal lter behaviour
but they may cause distortions in the ltered time-series due to the ringing eect that occurs at the edges of the lter response [34]. For acausal ltering, an increase in lter order
is associated with an increase in the length of zero pads in the ltered data [8]. Figure 7
presents the median and log-normal standard deviation of 5% damped Sd;proc =Sd;unproc vs T=Tc
plots for analogue and digital soft soil recordings in terms of dierent lter orders; the format of this gure is similar to Figure 6. The central tendency (median) plots show that
the increase in lter order from n = 2 to 4 would increase the usable period range in spectral displacement computations signicantly. The dispersion is also reduced when a lter
order of 4 instead of 2 is used for digital records. The scope for reducing the dispersion
in the usable period range with increasing the lter order is substantially limited for analogue records, suggesting once again that the long-period information that can be retrieved
from these records is relatively limited compared to the digital ground-motion data. The
plots in Figure 7 convey that a lter order of 4 is optimum for acausal ltering and lter
Copyright ? 2006 John Wiley & Sons, Ltd.
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INFLUENCE OF LONG-PERIOD FILTER CUT-OFF
Soft Soil - Digital
Median - Sd,proc/Sd,unproc
Median - Sd,proc/Sd,unproc
Soft Soil - Analogue
1
n=2
n=4
n=8
0.1
1
0.1
1
1
T/Tc
Std. Dev - Sd,proc/Sd,unproc
Std. Dev - Sd,proc/Sd,unproc
T/Tc
0.1
0.01
0.1
0.1
0.01
1
T/Tc
1
0.1
T/Tc
Figure 7. Inuence of lter order on the usable range of spectral displacements.
orders higher than 4 would not improve the usable period range in spectral displacement
computations.
3.4. Inuence of Tc on the usable period range in terms of viscous damping
The elastic spectral displacements for 5, 10 and 20% viscous damping were computed using the re-processed European ground-motion databank in order to observe the contribution
of damping on the usable period range of spectral displacements. Similar statistics were
derived as discussed in the preceding sections. Figure 8 displays the analogue and digital
rock site median and log-normal standard deviations of Sd;proc =Sd;unproc vs T=Tc plots for the
above damping ratios in a format similar to Figures 6 and 7. The comparative statistics
suggest that the variations in the viscous damping have negligible inuence on the usable
period range when compared to the contributions of magnitude, site class, lter order and
the issue of analogue vs digital recording. However, the increase in damping results in a
slight increase in dispersion for both analogue and digital records particularly for small T=Tc
ratios.
Copyright ? 2006 John Wiley & Sons, Ltd.
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S. AKKAR AND J. J. BOMMER
Rock - Digital - n=4
Median - Sd,proc/Sd,unproc
Median - Sd,proc/Sd,unproc
Rock - Analogue - n=4
1
= 5%
= 10%
= 20%
0.1
1
1
0.1
T/Tc
1
1
Std. Dev. - Sd,proc/Sd,unproc
Std. Dev. - Sd,proc/Sd,unproc
1
T/Tc
0.1
0.01
0.1
1
T/Tc
0.1
0.01
0.1
1
T/Tc
Figure 8. Inuence of damping on the usable spectral period range.
4. USABLE PERIOD RANGE FOR SPECTRAL DISPLACEMENTS IN TERMS
OF LONG-PERIOD FILTER CUT-OFF
In essence the process of applying high-pass lters to accelerograms is the determination
of the maximum usable response period for the particular record. Once determined, almost
identical results, in terms of elastic response spectral ordinates, could be obtained using the
mean removed record provided the upper period limit is not exceeded. As has been made
clear in the previous section, this upper limit period will invariably be shorter than the lter
cut-o Tc . The probability of an observed proportion of X = Sd;proc =Sd;unproc being within a
certain acceptable interval was employed to assess the usable period range for spectral displacement computations. The ‘estimation of proportion’ technique, which accepts a proportion
of occurrences of an event in Bernoulli sequence [35], was used. Given a certain criterion, this
technique estimates the probability of occurrence (P) of an event associated with a condence
interval assuming the data are normally distributed. Based on the preceding observations, this
study selected X = Sd;proc =Sd;unproc to be between 0.9 and 1.1 as the acceptable interval criterion to determine a usable period range for spectral displacement computations as a function
Copyright ? 2006 John Wiley & Sons, Ltd.
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INFLUENCE OF LONG-PERIOD FILTER CUT-OFF
1161
of long-period lter cut-o. This decision on the acceptable interval criterion is arbitrary but
it ensures exclusion of data that experiences severe recording problems as well as excessive
ltering that would yield articially small displacements.
The probability of occurrences for 0:9¡X ¡1:1 for a condence band of 95% is plotted
in Figure 9 as a function of T=Tc . The rst row in this plot displays the calculated probabilities for rock site analogue and digital records, respectively. The second and third rows
present the same information for sti and soft soil site recordings, respectively. Considering the
lter order eects discussed in the previous section, the statistics were derived using the data
processed by a fourth-order acausal Butterworth lter, which seems to be an optimal lter
order to increase the usable period range for displacement spectrum. The plots are presented
for 5% viscous damping but they also apply for higher viscous damping values as damping
seems not to have a very signicant eect on the inuence of long-period lter cut-o on
spectral displacement computations (Section 3.4). The probability of occurrences presented
in Figure 9 is consistent with the statistical results discussed in Section 3. They show that
digital records would be aected less from long-period lter cut-o for all site classes and
the usable period range is longer for spectral displacements computed from soft soil site
records.
Two probability levels are selected to determine the usable period range for spectral displacement calculations as a function of Tc . The rst criterion is conservative and determines
the usable period range for a 90% probability of occurrence for the pre-determined interval criterion. In other words, for a given database, 90% of the ltered data will be within
0:9¡X ¡1:1 with 95% condence. This probability level dictates 0:3Tc , 0:35Tc and 0:45Tc as
the usable upper-bound period ranges for rock, sti and soft soil analogue records, respectively. For digital records, the usable upper-bound period ranges dened by this probability
level are 0:65Tc for rock and sti soil records and 0:7Tc for soft site ground motions. The
second alternative presented for determining the usable period range is based on 50% of
probability of occurrence that ensures half of the data will satisfy 0:9¡X ¡1:1 criterion with
95% condence. This choice is more tolerant than the rst one but given the strict acceptance
criterion used for computing the probability of occurrences it would still yield reasonably
reliable period ranges in terms of Tc for spectral displacement computations. The application
of this alternative yields an upper-bound usable period range of 0:65Tc for rock and sti
site analogue records, and is 0:7Tc for soft site analogue records. For digital records, the
usable upper-bound period ranges for spectral computations become 0:8Tc , 0:9Tc and 0:97Tc
for rock, sti and soft soil ground motions, respectively. It should be noted that the 0:97Tc
upper bound computed for soft soil ground motions is unexpectedly high and it should be
used with caution in the calculation of spectral displacement ordinates from the processed
data.
The alternative usable period ranges presented within the context of this study consider
the contributions of many parameters to long-period lter cut-o that essentially aect the
spectral displacement calculations. Their implementation to strong ground-motion databases
will depend on the quality of the records in the database. The application of the second
alternative to databases with high-quality accelerograms would yield condent spectral displacement information for a wide range of response periods, whereas the implementation
of former, and more conservative, option would result in a rather limited usable period
range for databases that involve signicant uncertainty in terms of recorded ground
motions.
Copyright ? 2006 John Wiley & Sons, Ltd.
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S. AKKAR AND J. J. BOMMER
Rock - Digital (Elastic 5%)
Rock - Analog (Elastic 5%)
1.0
1.0
n=4
0.8
P(0.9 < X < 1.1)
0.8
P(0.9 < X < 1.1)
n=4
0.6
0.4
0.2
0.6
0.4
0.2
95% Confidence
95% Confidence
X = Sd,proc/Sd,uproc
X = Sd,proc/Sd,uproc
0.0
0.0
0.1
1
1
0.1
T/Tc
T/Tc
Stiff - Analog (Elastic 5%)
1.0
Stiff - Digital (Elastic 5%)
1.0
n=4
0.8
P(0.9 < X < 1.1)
0.8
P(0.9 < X < 1.1)
n=4
0.6
0.4
0.2
0.6
0.4
0.2
95% Confidence
95% Confidence
X = Sd,proc/Sd,uproc
X = Sd,proc/Sd,uproc
0.0
0.0
0.1
1
1
0.1
T/Tc
T/Tc
Soft- Analogue (Elastic 5%)
1.0
Soft- Analogue (Elastic 5%)
1.0
n=4
0.8
P(0.9 < X < 1.1)
P(0.9 < X < 1.1)
0.8
0.6
0.4
0.2
n=4
0.6
0.4
0.2
95% Confidence
X = Sd,proc/Sd,uproc
X = Sd,proc/Sd,uproc
0.0
0.1
1
T/Tc
95% Confidence
0.0
0.1
1
T/Tc
Figure 9. Usable period range for elastic spectral displacements as a function of long-period lter cut-o.
Copyright ? 2006 John Wiley & Sons, Ltd.
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INFLUENCE OF LONG-PERIOD FILTER CUT-OFF
1163
5. SUMMARY AND CONCLUSIONS
Filters are widely used to process strong-motion accelerograms to prevent the long-period
noise from exerting an inuence on calculated parameters such as response spectral ordinates.
For high-quality digital recordings it is noted that there is an option to simply apply baseline
corrections instead of ltering; this may not produce very reliable displacement time-histories
but it can result nonetheless in usable response spectral ordinates up to periods of 10 or
even 20 s [36, 37]. This paper, however, has focused specically on the use of lters as the
standard tool for processing of accelerograms, and particular for the analogue recordings that
still dominate the global databank.
The inuence of long-period lter cut-o on elastic oscillator displacements is investigated
by re-processing a European strong-motion databank. The processing scheme uses the guidance
of theoretical single- and double-corner source spectra to provide insight about the frequency
content of the ltered data and to prevent excessive ltering as much as possible, while
removing the long-period noise from the data. An acausal Butterworth lter was used since
previous studies have shown that lter response with no phase distortion results in a processed
data that is less susceptible to the lter cut-o eects. Using particular case studies, the relationship between the lter and oscillator response is explored and dened by both the central
tendency and dispersion statistics. Regardless of the site conditions, the usable period range
for displacement spectra is signicantly longer for digital records than for analogue records:
on average, the spectral displacement information that can be extracted from digital records
is twice that from analogue records. The higher inuence of lter cut-os for analogue
records at shorter periods might be due to the inherently existing higher noise levels with
respect to their digital counterparts. This fact is believed to make the inuence of long-period
lter cut-os more prominent for analogue records at shorter vibration periods.
An increase in earthquake magnitude results in a longer usable period-band for displacement
spectra computations. The ground motions recorded on rock and sti soil site classes are
inuenced more by the long-period lter cut-o and their usable period range for spectral
displacement calculations is substantially lower with respect to the soft soil recordings relative
to the selected Tc . The rich long-period content of large magnitude and soft soil site records
is believed to be the primary reason for these conclusions. The results of this study also show
that a lter order of 4 is optimal for acausal Butterworth ltering in order to have a longer
usable period range for spectral displacements. Based on these observations, on average, the
unprocessed digital ground motions recorded on soft sites can be used for spectral displacement
computations for periods less than 0:7Tc . If the records are analogue, this period bound reduces
to just 0:4Tc .
Using the statistics obtained from a re-processed European ground-motion database, the usable period range for spectral displacements is computed as a fraction of the long-period lter
cut-o in a probabilistic manner. The recommended usable period ranges takes into account
the interaction between the lter and oscillator response and also considers the contributions
of magnitude, site class and lter order to the inuence of long-period cut-o on spectral
displacements. Two criteria are selected for determining the usable period ranges. The conservative criterion recommends the upper-bound usable period range for analogue rock, sti and
soft soil site records as 0:3Tc , 0:35Tc and 0:45Tc , respectively, given the processed to mean
removed spectral displacement ratios fall between 0.9 and 1.1 range with a 90% probability at
95% condence level. This upper-bound period range increases to 0:65Tc for rock and sti soil
Copyright ? 2006 John Wiley & Sons, Ltd.
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S. AKKAR AND J. J. BOMMER
site digital records whereas it is 0:7Tc for soft soil sites and digitally recorded ground motions.
The more tolerant criterion denes the upper-bound usable period range as 0:65Tc for analogue
rock and sti site records and 0:7Tc for analogue records from soft soil sites, ensuring that the
processed to mean removed spectral displacement ratios fall in the 0.9 and 1.1 range with a
50% probability at 95% condence level. The tolerant criterion yields 0:8Tc , 0:9Tc and 0:97Tc
as the usable upper-bound range for rock, sti and soft soil site digital records. Bearing on
the results presented in this study, which are driven using a large European data set, most,
if not all, existing equations derived from European analogue strong-motion recording are
likely to be biased at longer response periods. Therefore, the consideration of the proposed
limitations is believed to result in more trustworthy elastic displacement spectrum estimations
in particular at the long period range for Europe and surrounding countries.
ACKNOWLEDGEMENTS
A great part of this work was conducted during the academic visit of rst author to Imperial College
London. The nancial support for this academic visit was provided by the Royal Society and the
Scientic and Technical Research Council of Turkey (TUBITAK). Dr David M. Boore kindly responded
to many enquiries about the use of his program during the conduct of this study. Dr Boore also made
valuable suggestions for some of the ltering terminology used in the text. Mr Onder
Ozen
has helped
in data processing during the initial phase of the study. The constructive critiques of Professor Roberto
Paolucci and Dr John Douglas have signicantly increased the technical quality of the paper and its
presentation.
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