An improved CWT-based algorithm for the generation of spectrum-compatible records

Abstract

The seismic design of most civil structures is usually accomplished using the response spectrum approach or simplified equivalent lateral force methods. However, some special tasks require the use of dynamic time history analyses. In the nuclear industry, for example, dynamic analyses are required in the design verification and seismic assessment of critical buildings and in the development of floor response spectra and free-field ground response spectra. The input motion for these analyses requires acceleration time series whose response spectrum matches a target design spectrum. This article revises the continuous wavelet transform (CWT) approach to generate spectrum-compatible records from the modification of acceleration time histories recorded in actual seismic events. The computational efficiency of the algorithm is increased greatly by performing the wavelet decomposition and details reconstruction via fast convolution using fast Fourier transforms. The new algorithm is evaluated using a typical design spectrum from the nuclear industry and different seed records.

Full text

O R I G I N A L R E S E A R C H Open Access
An improved CWT-based algorithm for the
generation of spectrum-compatible records
Luis A Montejo
1*
and Luis E Suarez
2
Abstract
The seismic design of most civil structures is usually accomplished using the response spectrum approach or
simplified equivalent lateral force methods. However, some special tasks require the use of dynamic time history
analyses. In the nuclear industry, for example, dynamic analyses are required in the design verification and seismic
assessment of critical buildings and in the development of floor response spectra and free-field ground response
spectra. The input motion for these analyses requires acceleration time series whose response spectrum matches a
target design spectrum. This article revises the continuous wavelet transform (CWT) approach to generate
spectrum-compatible records from the modification of acceleration time histories recorded in actual seismic events.
The computational efficiency of the algorithm is increased greatly by performing the wavelet decomposition and
details reconstruction via fast convolution using fast Fourier transforms. The new algorithm is evaluated using a
typical design spectrum from the nuclear industry and different seed records.
Keywords: Wavelet transform; Fast convolution; Fast Fourier transform; Artificial earthquake records; Seismic design
Introduction
Independent of the approach selected, either displace-
ment or force-based, most of the seismic designs of civil
structures are performed using variations of the re-
sponse spectrum method, where the ground motion is
represented as a design response spectrum that is repre-
sentative of the seismic hazard on the site. The (pseudo-
accelerations or relative displacements) design response
spectrum is used to estimate the force or deformation
demand imposed on the structure. The main goal of this
technique is to translate the ground motion into forces
acting on the building. Nevertheless, for some special tasks
or structures, dynamic time history analyses are required.
In the standard American Society of Civil Engineers
(ASCE) 7–10, for example, if the structure meets the
characteristics necessary to be considered as irregular,
dynamic analyses (either modal response spectrum or
time history) are required to verify the design of the
structure. In the nuclear industry, dynamic time history
analyses are required in the design, verification, and
seismic assessment of critical buildings. In addition, they
are needed for the development of floor (or in-structure)
response spectra and free-field ground response spectra.
When dynamic time history analyses are to be performed,
the seismic input needs to be defined as a time history of
accelerations. To comply with code requirements, the
acceleration series used should be compatible with the
design spectrum, i.e., the response spectrum of the ac-
celeration time history should match with the design
spectrum in average.
To select these input records, one can look at historic
records with their amplitudes scaled by a factor to optimize
the fitting over the design spectrum (e.g., Bommer
and Acevedo 2004). Alternatively, there are different
methodologies available to generate ‘synthetic’spectrum-
compatible records. When using the first option, i.e.,
amplitude scaling, a larger number of records are required
to obtain a reliable average of the system response due to
the natural scatter of the records. On the other hand, if
spectrum-compatible records are used, the number of
analyses required to obtain a reliable estimate of the
response is substantially reduced (e.g., Watson-Lamprey
and Abrahamson 2006; Heo et al. 2011; Hancock et al.
2008).
* Correspondence: luis.montejo@upr.edu
1
Department of Engineering Science and Materials, University of Puerto Rico
at Mayaguez, Mayaguez, 00680, Puerto Rico
Full list of author information is available at the end of the article
© Montejo and Suarez; licensee Springer. This is an open access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Montejo and Suarez International Journal of Advanced
Structural Engineering
2013
2013, 5:26
http://www.advancedstructeng.com/content/5/1/26

There are several methodologies that have been pro-
posed for the generation of spectrum-compatible re-
cords, like the adjustment of the power spectrum of a
random process or the manipulation in the time or
frequency domain of historic records. The criteria for the
generation and evaluation of such records specified in
standard seismic provisions (e.g., ASCE 7; American
Society of Civil Engineers 2010) are rather slight when
compared to the requirements in the nuclear regula-
tory guide RG 1.208 (US-NRC 2007). For example,
RG 1.208 explicitly prohibits the use of synthetic
motions that are not based on seed-recorded time
histories so that only modified recorded ground mo-
tions can be used for site response analysis.
The available methodologies for obtaining spectrum-
compatible accelerograms based on the modification
of historic records can be classified into three groups:
(1) those based on matching in the frequency domain by
manipulation of the Fourier spectrum (e.g., Tsai 1972),
(2) those grounded on wavelet adjustments to the re-
corded accelerograms at specific times (e.g., Lilhanand
and Tseng 1988; Abrahamson 1992; Hancock et al. 2006;
Al Atik and Abrahamson 2010), and (3) those centered on
the manipulation of the wavelet coefficients obtained via
continuous wavelet transform (e.g., Mukherjee and Gupta
2002; Suarez and Montejo 2003, 2005). Although an ac-
ceptable level of matching can be obtained by means of
any of the available methodologies, the use of spectrum-
compatible records is a matter of discussion in the earth-
quake engineering community, as the characteristics of
the final compatible record may largely differ from those
observed in real records e.g., (Naeim and Lew 1995;
Bommer and Ruggeri 2002; Bommer and Acevedo
2004). Although these arguments may be valid, the dis-
cussion of this issue is beyond the scope of the paper. This
article revises the continuous wavelet transform approach
to generate spectrum-compatible records from the modifi-
cation of actual registered seismic events and propose an
alternative computation algorithm to reduce its computa-
tional cost.
Methods
CWT-based methodology for the generation of
spectrum-compatible records
For completeness, this section briefly describes the
methodology to manipulate historic earthquake records
to obtain spectrum-compatible accelerograms based on
the continuous wavelet transform (CWT). A compre-
hensive presentation of the CWT theory behind the
methodology and the properties of the wavelet function
used are available elsewhere (Suarez and Montejo
2005).
The CWT of a signal f(t) can be defined as the sum
over all times of the signal multiplied by scaled and
shifted versions of a wavelet function Ψas defined by
Equation 1
Cs;pðÞ¼
∫
þ∞
−∞ftðÞΨ
s;pdt ¼
∫
þ∞
−∞ftðÞ1
ffiffis
pΨt−p
s

dt
ð1Þ
The parameter sand pare used to scale and shift the
wavelet, respectively. The asterisk * indicates complex con-
jugation and the normalizing factor 1/√sensures that the
energy is the same for all values of s.Theresultofthe
transform is a matrix of wavelet coefficients C(s,p)thatcon-
tain information about the signal at the scale sand time
position p, that is, the CWT can be viewed as a two-
dimensional transform that maps an original time signal f(t)
into a time-scale domain. Since the scale scan be related to
frequency, the CWT is occasionally used in earthquake en-
gineering to obtain simultaneous time-frequency represen-
tations of earthquake records (e.g., Montejo and Kowalsky
2008). Once the wavelet coefficients are calculated, the sig-
nal can be reconstructed using Equation 2:
ftðÞ¼ 1
Kψ
∫
þ∞
s¼0
∫
∞
p¼−∞Cs;pðÞΨs;ptðÞdp
s2

ds
¼1
Kψ
∫
∞
0Ds;tðÞds ð2Þ
where K
ψ
is a constant that depends on the wavelet func-
tion selected for the analysis and the functions D(s,t)are
Ds;tðÞ¼
∫
∞
p¼−∞
1
s2Cs;pðÞΨs;ptðÞdp:ð3Þ
They are referred to as the detail functions. They have
a dominant frequency that depends on the type of wave-
let. In practice, a set of ndiscrete values of the continu-
ous scale sare used.
The selection of an appropriate wavelet function is
crucial for the effective implementation of a given appli-
cation. For strong motion data synthesis and analysis,
the impulse response wavelet has been successfully used
in the past. It is define as
ΨtðÞ¼e−ζΩtjjsin ΩtðÞ;ð4Þ
where ζand Ωare the parameters that define the shape
and central frequency of the wavelet, respectively. For
the applications presented here, values of ζ=0.05 and
Ω=πare used. The mathematical properties and ad-
vantages of this wavelet in the analysis of earthquakes
records are discussed in Suarez and Montejo (2005)
and Hancock et al. (2006).
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To obtain spectrum-compatible records one may proceed
as follows:
1. Perform the CWT, using Equations 1and 4,to
obtain the wavelet coefficients C(s,p)atselected
values of s. The values of sare defined based on
the periods or frequencies where a match with
the target spectrum is desired. For the impulse
response wavelet, the relation between sand the
predominant frequency (f) in hertz and period (T)
is given by Equation 5:
f¼Ω
2π
1
sT¼2π
Ωs:ð5Þ
In the applications presented in this work, a total of
n= 100 frequency values uniformly spaced over the
log frequency scale from 0.1 to 50 Hz is used as
required in Appendix F of the Nuclear Regulatory
Guide RG 1.208 (US-NRC 2007). Depending on the
application, the user may decide to modify these
values.
2. Once the wavelet coefficients C(s,p) are calculated,
the detail functions are constructed using
Equation 3.
3. Calculate the ratios between the target spectrum Sa
T
(f
j
) and the response spectrum of the historical
record Sa
R
(f
j
) at the frequencies defined in step 1:
Rf
j

¼
SaTfj

SaRfj
 ;j¼1;2;…;n:ð6Þ
4. Multiply each detail function D(s
j
,t)bythecorresponding
spectral ratio R(f
j
)andreconstructthesignalwith
Equation 2.
0
0.2
0.4
0.6
0.8
1
1.2
]g%[noitareleccalartceps
: target spectrum
: sc aled recods
10
-1
10
0
10
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
frequency [ Hz]
]g%[noitareleccalartceps
: target spectrum
: com patible recods
Figure 2 Target uniform hazard response spectra. Top: the target UHRS at the rock outcrop and the scaled response spectra for the three
seed records. Bottom: the target UHRS and the response spectra of the resulting compatible records.
0.5 11.5 22. 5 3
10
-2
10
0
10
2
number of data points in t he seed rec od (*10
4
)
estimated real time (hours)
: frequency domain
: time domain
Figure 1 Time savings using the proposed algorithm
implementation.
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5. Calculate and compare the response spectrum of the
reconstructed signal Sa
R
(f
j
) with the target spectrum
and repeat steps 3, 4, and 5 until a desired level of
matching is attained or a maximum number of
iterations are reached.
Improved efficiency using fast convolution via FFT
The CWT approach previously described for generating
spectrum-compatible records has been largely used in the
past, offering stable solutions and good matching with dif-
ferent target spectra (e.g., Velez 2007; Priestley et al. 2007;
Bahar and Taherpour 2008; Linzell and Nadakuditi 2011;
Montejo et al. 2012). However, a common complaint is the
amount of time that takes for the algorithm to run, espe-
cially when lengthy records are used as input. To improve
the processing time, a new algorithm is proposed in this
paper. It computes the wavelet decomposition, Equa-
tion 1, and constructs the detail functions, Equation 3,
using a fast convolution implementation by means of the
fast Fourier transform (FFT).
By looking at Equation 1, it can be said that the CWT
for a fixed scaled sis the convolution of the signal f(t)
with the wavelet function scaled by s. The matrix of
wavelet coefficients C(s,p) is formed by changing the
values of sand repeating the convolution operation. The
original implementation of the algorithm computed
these convolutions in the time domain. However, ac-
cording to the convolution theorem, convolution in the
time domain is equivalent to multiplication in the fre-
quency domain, that is, if we have the two functions f
and gin the time domain, its convolution can be calcu-
lated in the frequency domain using Equation 7:
f⊗g¼F−1Ff
fg
•Fg
fgfg ð7Þ
where ⊗denotes convolution, •denotes pointwise prod-
uct, and Fand F
−1
denote the forward and inverse Fourier
transform, respectively. Since we are dealing with discrete
signals, transformations between the time and frequency
domains call for the application of the discrete Fourier
-0.2
0
0.2
accel . [g]
-0.02
0
0.02
vel./g
-0.01
0
0.01
displ./g
0.2
AI/g
1
2
CAV/g
20 25 30 35 40 45 50 55 60
0.2
0.4
100*CSV/ g
2
time [s]
Figure 3 Characteristics of the seed motions corresponding to the TCU089 1999 Chi-Chi record. Top to bottom: time histories of acceleration,
velocity, displacement, AI, CAV, and CSV for the scaled (blue line) and compatible records (black line) corresponding to the TCU089 1999 Chi-Chi record.
Table 1 Summary of the selected seed records
NGA
number
Event Station Magnitude Mechanism R
rupt
(km)
Vs30
(m/s)
1521 1999 Chi-Chi,
Taiwan
TCU089 7.62 Reverse/
oblique
8.9 680
1508 1999 Chi-Chi,
Taiwan
TCU072 7.62 Reverse/
oblique
7 468
284 1980 Irpinia,
Italy
Auletta 6.90 Normal 9.6 1,000
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transforms (DFT). If the DFT is implemented using the
FFT algorithm, the convolution via the frequency domain
can be significantly faster than directly convolving the two
time domain signals. While the computational cost for
direct time domain convolution of two N-point sequences
is of the order O(N
2
), the cost for frequency domain con-
volution using the FFT algorithm is O(NlogN). Consid-
ering the typical length of the signals used in this
application and the number of convolution operations
required by the algorithm (one per each value of s), this
translates into large savings in the computational cost.
In the new implementation, the CWT is executed using
Equation 8:
Cs;pðÞ¼ftðÞ⊗Ψs;p¼IFFT FFT ftðÞ
fg
•FFT Ψs;p

ð8Þ
where FFT and IFFT are, respectively, the direct and in-
verse discrete Fourier Transforms calculated with the
FFT algorithm.
Note that in order to be able to compute the pointwise
product (•), the lengths of the FFT for the record and
for the wavelet must be the same. To obtain a faster and
most precise implementation, the lengths of the wavelet
and the record are set to the next power of 2 of the ex-
pected convolution length (2 * N−1) (Smith 2002).
Finally, notice from Equation 3 that the detail func-
tions can also be calculated using the FFT convolution
between the wavelet coefficients for a fixed sand the
wavelet function scaled by the same svalue, as shown in
Equation 9:
Ds;tðÞ¼
1
s2Cs;pðÞ⊗Ψs;ptðÞ
¼1
s2IFFT FFT Cs;pðÞfg•FFT Ψs;p

:ð9Þ
Figure 1 shows the computational savings obtained
using the proposed algorithm and assuming that the rec-
ord is decomposed at 100 scale (frequency) values. It is
seen that for a relative short record of 2,000 data points
(e.g., 10 s sampled at 200 Hz), the estimated ‘real time’
that will take the new implementation of the algorithm
to run is about 42 s, while the old implementation will
take approximately 37 min. As the length of the seed
record increases, the differences become exponentially
larger (notice the logarithmic scale used in the y-axis).
The estimated real times in Figure 1 were computed
based on a laptop PC i7 at 2.90 GHz, 8 GB RAM, and
64-bit OS.
-0.2
0
0.2
accel. [g]
-0.02
0
0.02
vel./g
-0.02
0
0.02
displ. /g
0.1
0.2
AI/g
1
CAV/ g
20 25 30 35 40 45 50 55 60
0
0.2
100*CSV/ g
2
time [s]
Figure 4 Characteristics of the seed motions corresponding to the TCU072 1999 Chi-Chi record. Top to bottom: time histories of acceleration,
velocity, displacement, AI, CAV, and CSV for the scaled (blue line) and compatible records (black line) corresponding to the TCU072 1999 Chi-Chi record.
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Results and discussion
The new implementation of the algorithm is evaluated
using a design spectrum typical of the US nuclear indus-
try. The target spectrum is extracted from Section 5 of
RG 1.208 (US-NRC 2007) and represents a typical uni-
form hazard response spectrum (UHRS) at the outcrop
rock (Figure 2). In practice, synthetic records compatible
with this spectrum need to be generated to perform site
analyses and develop a site-specific design spectrum.
Three different seed records to be used as input for
the algorithm were selected from the PEER NGA data-
base (PEER 2013) based on the similarity of the response
spectrum shape of the records with the shape of the tar-
get spectrum. Table 1 summarizes the main characteris-
tics for each of the records selected. Since the selection
is based on spectral shape similarities, the amplitude of
the records is scaled to provide a best fit. The scaled re-
sponse spectra for the three records are displayed and
compared with the target spectrum in Figure 2 (top).
Figure 2 (bottom) shows the response spectra for 5%
damping of the resulting compatible records along with
the target spectrum. The 90% and 130% design spectra
are also displayed (dashed lines) as these are the spectral
amplitude limits specified in RG 1.208. To prevent the
spectra in large frequency windows from falling below
the spectrum, RG 1.208 also requires that no more than
nine adjacent frequency points fall below the target
spectrum. It is seen that the compatible records gener-
ated satisfy these requirements.
Figures 3,4,5 display the time histories of acceleration,
velocity, displacement, Arias intensity (AI), cumulative ab-
solute velocity (CAV), and cumulative squared velocity
(CSV) for the linearly scaled and compatible records gen-
erated. It is seen that the characteristics of the seed mo-
tions are, in general, retained in the compatible records.
Conclusions
An improved algorithm for faster generation of spectrum-
compatible records was developed. The algorithm is based
on the modification of the CWT coefficients of a historic
record. The efficiency of the algorithm is improved by per-
forming the required convolution operations in the fre-
quency domain via fast Fourier transforms. It was shown
that the algorithm can be used to obtain compatible
records that satisfy current US NRC requirements. A
Matlab implementation of the algorithm is available upon
request from the corresponding author.
-0.2
0
0.2
accel. [g]
-0.02
0
0.02
vel./g
-0.01
0
0.01
0.02
displ. /g
0.1
0.2
AI/g
0
1
CAV/g
0 5 10 15 20 25 30
0
0.2
100*CSV/g
2
time [s]
Figure 5 Characteristics of the seed motions corresponding to the Auletta 1980 Irpinia record. Top to bottom: time histories of acceleration,
velocity, displacement, AI, CAV, and CSV for the scaled (blue line) and compatible records (black line) corresponding to the Auletta 1980 Irpinia record.
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Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors contributed equally to this research work.
Acknowledgements
This work was performed at the University of Puerto Rico at Mayaguez under
award NRC-HQ-12-G-38-0018 from the US Nuclear Regulatory Commission.
The statements, findings, conclusions, and recommendations are those of
the authors and do not necessarily reflect the view of the US Nuclear
Regulatory Commission.
Author details
1
Department of Engineering Science and Materials, University of Puerto Rico
at Mayaguez, Mayaguez, 00680, Puerto Rico.
2
Department of Civil
Engineering and Surveying, University of Puerto Rico at Mayaguez,
Mayaguez, 00680, Puerto Rico.
Received: 13 May 2013 Accepted: 8 October 2013
Published:
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Cite this article as: Montejo and Suarez: An improved CWT-based
algorithm for the generation of spectrum-compatible records.
International Journal of Advanced Structural Engineering
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