Empirical Relationships for Frequency
Content Parameters of Earthquake
Ground Motions
Ellen M. Rathje,a) M.EERI, Fadi Faraj,b) Stephanie Russell,c) and Jonathan
D. Bray,d) M.EERI
The frequency content of an earthquake ground motion is important because it affects the dynamic response of earth and structural systems. Four
scalar parameters that characterize the frequency content of strong ground
motions are (1) the mean period (Tm), (2) the average spectral period (Tavg),
(3) the smoothed spectral predominant period (To), and (4) the predominant
spectral period (Tp). Tm and Tavg distinguish the low frequency content of
ground motions, while To is affected most by the high frequency content. Tp
does not adequately describe the frequency content of a strong ground motion and is not recommended. Empirical relationships are developed that predict three parameters (Tm , Tavg , and To) as a function of earthquake magnitude, site-to-source distance, site conditions, and rupture directivity. The
relationships are developed from a large strong-motion database that includes
recorded motions from the recent earthquakes in Turkey and Taiwan. The new
relationships update those previously developed by the authors and others.
The results indicate that three site classes, which distinguish between rock,
shallow soil, and deep soil, provide a better prediction of the frequency content parameters and smaller standard error terms than conventional ‘‘rock’’
and ‘‘soil’’ site classes. Forward directivity significantly increases the frequency content parameters, particularly Tm and To , at distances less than 20
km. Each of the frequency content parameters can be predicted with reasonable accuracy, but Tm is the preferred because it best distinguishes the frequency content of strong ground motions. [DOI: 10.1193/1.1643356]
INTRODUCTION
The dynamic response of geotechnical and structural systems subjected to earthquake ground shaking is significantly affected by the frequency content of the input
earthquake ground motion. When the frequency content of an earthquake ground motion
closely matches the natural period of a geotechnical system (e.g., soil deposit, earth
dam) or structural system (e.g., building, bridge), the dynamic response is enhanced,
larger forces are exerted on the system, and significant damage may occur (Kramer
a)
Assistant Professor, Dept. of Civil Engineering, University of Texas, Austin, TX 78712
Graduate Research Assistant, Dept. of Civil Engineering, University of Texas, Austin, TX 78712
c)
Undergraduate Research Assistant, Dept. of Civil Engineering, University of Texas, Austin, TX 78712
d)
Professor, Dept. of Civil and Environmental Engineering, University of California, Berkeley, CA 94720
b)
119
Earthquake Spectra, Volume 20, No. 1, pages 119–144, February 2004; © 2004, Earthquake Engineering Research Institute
120
E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
1996, Chopra 2001). As a result, it is important to evaluate the frequency content of an
earthquake ground motion and consider its effect on the dynamic response of a structure.
Acceleration response spectra and Fourier Amplitude Spectra provide the most complete characterization of the frequency content of strong ground motions. These spectra
provide information regarding the distribution of motion across a range of frequencies.
Several attenuation relationships are available that predict the full acceleration response
spectrum of a strong ground motion (e.g., Abrahamson and Silva 1997, Boore et al.
1997, Sadigh et al. 1997), and theoretical seismological models (e.g., Brune 1970, 1971)
can be used to predict the Fourier Amplitude Spectrum. However, it is often useful to
characterize the frequency content of a strong ground motion with a single, scalar parameter. A scalar representation of frequency content allows the frequency content of
different strong ground motions to be compared quickly and easily (Stewart et al.
2001a). Additionally, a scalar frequency content parameter can be compared with the
natural period of a dynamic system to evaluate the possibility of resonance conditions or
an enhanced dynamic response (e.g., Nadim and Whitman 1983, Bray et al. 1998, Rathje
et al. 1998). Recently, seismic design procedures for landslides and earth fills have incorporated scalar frequency content parameters (e.g., Blake et al. 2002, Stewart et al.
2003). These examples illustrate the usefulness of scalar frequency content parameters
in engineering design and analysis.
Various scalar parameters have been used to characterize the frequency content of
strong ground motions. Rathje et al. (1998) described several frequency content parameters and developed predictive relationships for three parameters. In this paper the three
frequency content parameters identified by Rathje et al. (1998) are re-examined and a
fourth parameter is introduced. These frequency content parameters are the mean period
(Tm), the average spectral period (Tavg), the smoothed spectral predominant period (To),
and the predominant spectral period (Tp). The mean period (Tm) utilizes the Fourier Amplitude Spectrum, averaging periods (over a specified frequency range) weighted by the
Fourier amplitudes. The average spectral period (Tavg) utilizes the 5% damped acceleration response spectrum and averages periods (over a specified frequency range) weighted
by the spectral accelerations. The smoothed spectral predominant period (To) also utilizes the 5% damped acceleration response spectrum, but averages periods only over the
range of spectral amplification (i.e., spectral acceleration greater than 1.2 times the peak
ground acceleration). Finally, the predominant spectral period (Tp) utilizes the acceleration response spectrum and is simply defined as the period of the maximum spectral acceleration.
This paper updates the predictive relationships developed by Rathje et al. (1998) for
Tm and To for strong motions from shallow crustal events in active tectonic regions (e.g.,
western United States). Additionally, a relationship for Tavg is developed. The relationships developed in this study improve upon the previous study in several ways. The data
set used in this study is significantly larger, with 835 recordings used for Tm and Tavg and
1208 recordings used for To (previously, 306 records were used). The new model uses
three site classes that distinguish between rock, shallow soil, and deep soil, while the
previous study combined rock and shallow soil into a single site category. A recent development in earthquake engineering and engineering seismology is the recognition of
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
121
fault rupture directivity and its effect on strong ground motions (e.g., Somerville et al.
1997). Forward directivity enhances the long period components of motion, and hence,
affects the frequency content of strong ground motions. As a result, the new model accounts for the effects of forward directivity on the scalar frequency content parameters.
Finally, the more sophisticated random-effects model (e.g., Brillinger and Preisler 1984)
is used in the regression to develop the new relationships. This paper describes the process used to establish the functional form and regression coefficients for the predictive
relationships. The newly developed relationships are compared with previous relationships, and pertinent observations are discussed.
FREQUENCY CONTENT CHARACTERIZATION
Only a few previous studies have addressed predicting the frequency content of
strong ground motions, and most of these previous studies have chosen predominant
spectral period (Tp) as the parameter to characterize frequency content. Gutenberg and
Richter (1956) were the first to study the frequency content of strong ground motions,
considering the period ‘‘of the waves of maximum amplitude,’’ which is typically assumed to correspond to the predominant spectral period. Seed et al. (1969) suggested a
relationship between Tp and distance for different earthquake magnitudes based on a
limited data set. Idriss (1991) modified the Seed et al. (1969) relationships based on recorded strong motions from the 1989 Loma Prieta (M w⫽6.9) earthquake. Rathje et al.
(1998) performed a systematic, statistical empirical study of scalar frequency content
parameters for a relatively large database of recorded motions. In the 1998 study, 306
strong ground motions from 20 earthquake events in active plate-margin regions were
used to develop predictive relationships for Tp , as well as mean period (Tm) and
smoothed spectral predominant period (To). The Rathje et al. (1998) study concluded
that Tm and To better characterize the frequency content of strong ground motions than
Tp , and that these two parameters can be more accurately estimated. This paper reexamines the scalar frequency content parameters described in Rathje et al. (1998) and
introduces another parameter, the average spectral period (Tavg). These parameters are
defined and discussed below.
Tm is computed from the Fourier Amplitude Spectrum and is defined as
Tm⫽
兺C2i 共1/f i兲
i
兺C2i
for 0.25 Hz ⭐ f i ⭐ 20 Hz, with ⌬f ⭐ 0.05 Hz
(1)
i
where Ci are the Fourier amplitude coefficients, f i are the discrete fast Fourier transform
(FFT) frequencies between 0.25 and 20 Hz, and ⌬f is the frequency interval used in the
FFT computation. The discrete FFT frequencies are equally spaced in the frequency domain, but are not equally spaced when the reciprocal is used in Equation 1. As a result,
the frequency interval used in the FFT calculation can affect the Tm calculation, if unusually large frequency intervals are used. Using a theoretical model for the Fourier Amplitude Spectrum of an earthquake ground motion (Brune 1970, 1971), it was found that
a stable value of Tm (i.e., the same value of Tm is computed) was obtained for frequency
intervals smaller than 0.05 Hz. The frequency interval (⌬f ) is related to the time step
(⌬t) and number of points (N) in a time series by ⌬f⫽1/(N•⌬t). For common strongmotion recording time steps of 0.02 s, 0.01 s, and 0.005 s, the number of points corre-
122
E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
sponding to ⌬f⫽0.05 Hz is 1000, 2000, and 4000, respectively. To ensure a stable value
of Tm is calculated for recorded strong ground motions, motions should contain at least
the minimum number of points indicated above or should be augmented with zeroes to
attain these minimum values. A benefit to using the FAS to define a scalar frequency
content parameter is that the Fourier amplitude coefficients at each frequency are mutually independent. This is not the case for spectral acceleration.
The frequency content parameters Tp , To , and Tavg are based on the 5% damped acceleration response spectrum. Tp is simply defined as the period of the maximum spectral acceleration. To is computed as
共T 兲
冉SPGA
冊
T⫽
S 共T 兲
冊
兺 ln冉
PGA
兺Ti•ln
a
i
i
o
a
for Ti with
i
Sa
⭓ 1.2, ⌬ log Ti ⭐ 0.02
PGA
(2)
i
where Ti are the discrete periods in the acceleration response spectrum equally spaced
on a log axis, Sa(Ti) are the spectral accelerations at periods Ti , and PGA is the peak
ground acceleration. Only periods where the spectral acceleration is greater than 1.2
•PGA are used in the computation in Equation 2. In essence, To attempts to define the
peak in the response spectrum by smoothing the spectral accelerations over the range
where Sa is greater than 1.2•PGA. Consequently, To is similar to Tp . However, To represents an improvement over Tp because it smoothes the response spectrum to find its
peak. In the previous study (Rathje et al. 1998), Tp displayed the largest variability because it represents only one point in the response spectrum. To can provide similar information regarding the response spectrum but can be predicted with more certainty.
Based in its definition, To is most affected by the high to moderate frequency content of
strong motions and may be best suited for structures that are sensitive to motions in this
frequency range (e.g., nuclear reactors). Tavg is computed as
Tavg
共T 兲
冉SPGA
冊
⫽
S 共T 兲
冊
兺冉
PGA
兺Ti•
a
i
i
a
i
2
2
for 0.05 s ⭐ Ti ⭐ 4 s, ⌬Ti ⭐ 0.05 s
(3)
i
where Ti are the discrete periods in the acceleration response spectrum equally spaced
on an arithmetic axis, Sa(Ti) are the spectral accelerations at periods Ti , and PGA is the
peak ground acceleration. As defined, Tavg is similar to Tm , except that the periods are
equally spaced on an arithmetic axis and spectral acceleration is used in lieu of Fourier
amplitudes.
It is important to note how the period spacing in an acceleration response spectrum
affects scalar frequency content parameters. When periods are spaced equally on an
arithmetic axis, the spectral accelerations at long periods are not independent of one another. These spectral accelerations are not independent because a response spectrum represents the response of single-degree-of-freedom (SDOF) oscillators and the frequency
bandwidth of the SDOF response depends on the natural frequency of the system (e.g.,
Chopra 2001). For lower frequency (longer period) systems, the response bandwidth is
larger, and therefore the response of closely spaced periods (e.g., 1.0 s versus 1.1 s) is a
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
123
Figure 1. (a) Normalized acceleration response spectra (Sa /PGA) and (b) Fourier Amplitude
Spectra for the TCU048, TCU047, and TCU129 motions recorded during the 1999 Chi-Chi
earthquake.
function of almost the same frequency bandwidth of the motion. Consequently, when
averaging spectral accelerations at periods equally spaced on an arithmetic axis, the low
frequency content of the motion is weighted more heavily than the high frequency content. This consequence is advantageous if one wishes to emphasize the low frequency
content of a strong motion.
Figure 1 shows the normalized acceleration response spectra (Sa /PGA), Fourier Amplitude Spectra, and associated frequency content parameters for three strong motions
recorded during the 1999 Chi-Chi, Taiwan, (M w⫽7.6) earthquake. These motions are the
TCU048 (shallow soil, rupture distance 14.4 km), the TCU047 (shallow soil, rupture
distance 33 km), and the TCU129 (deep soil, rupture distance 1.2 km) recordings. The
124
E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
acceleration response spectra (Figure 1a) indicate that TCU048 and TCU047 have similar spectral shapes in the period range of 0.01 to 1.0 s, but TCU129 drops off quickly at
periods greater than 0.35 s. Additionally, TCU048 demonstrates the largest spectral accelerations at long periods. The Fourier Amplitude Spectra reveal similar trends (Figure
1b); TCU129 contains the most high frequency energy, while TCU048 contains the most
long period energy. However, the enhanced high frequency energy in TCU129 is more
easily identified in the Fourier Amplitude Spectrum.
Considering the scalar frequency content parameters, the values of Tp are very similar for all three motions (0.11 s for TCU048, 0.15 s for TCU047, 0.16 s for TCU129)
and do not distinguish the differences in the spectral shapes over the high to moderate
frequencies. The To values (0.35 s for TCU048, 0.33 s for TCU047, 0.17 s for TCU129)
can better distinguish the shape of TCU129 from the shapes of TCU048 and TCU047
because To considers the spectral accelerations throughout the range of spectral acceleration amplification (i.e., where Sa⬎1.2•PGA). However, To cannot distinguish between
TCU048 and TCU047, which differ significantly at long periods, because the normalized
spectral accelerations for TCU048 are below 1.2 at periods beyond 1.0 s. The values of
Tavg and Tm better differentiate between all three motions, with TCU129 displaying the
smallest values, TCU048 displaying the largest values, and TCU047 falling in between
(Figure 1).
There are many advantages to using a Fourier Amplitude Spectrum to define a scalar
frequency content parameter rather than using an acceleration response spectrum. The
Fourier amplitudes represent the amplitudes of harmonic waves that make up an
acceleration-time history and each Fourier amplitude coefficient is mutually exclusive.
An acceleration response spectrum is a collection of maximum responses of damped
SDOF oscillators. As a result, the acceleration response spectrum is not a direct representation of an acceleration-time history. Each spectral acceleration is affected by a
bandwidth of frequencies in the ground motion, which makes spectral accelerations at
long periods more similar to one another. Additionally, the shape of the Fourier Amplitude Spectrum for strong ground motions provides more information regarding the long
period content of a motion because the Fourier amplitude coefficients do not decline as
quickly as the spectral acceleration values at long periods (i.e., T⬎1.0 s). As a result, Tm
best characterizes the frequency content of a strong ground motion over moderate to
long periods. However, engineers who feel more comfortable with response spectra versus Fourier Amplitude Spectra may find Tavg to be most useful to characterize the moderate to long period content of a strong ground motion. To is most affected by the low to
moderate period content of ground motions, and may be most useful if these periods are
of engineering interest. Finally, Tp does not adequately describe the frequency content of
an earthquake ground motion and should not be used.
STRONG MOTION DATA SET
The strong motion recordings used in this study were processed by Dr. Walt Silva of
Pacific Engineering and Analysis and are available from the Pacific Earthquake Engineering Research Center strong motion database (http://peer.berkeley.edu/smcat). The
data set used in this study is significantly larger than the data set used in the Rathje et al.
(1998) study. Strong motion recordings from recent earthquakes provided a significant
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
125
amount of new data, including over 300 motions from the 1999 Chi-Chi (M w⫽7.6)
earthquake. Further, additional motions from previous earthquakes were included in this
study due to new information regarding site classification. As a result, 1208 motions
from 71 events ranging in magnitude from 4.7 to 7.6 were available for this study. An
additional constraint was imposed on motions for the Tm and Tavg data sets based on the
usable frequency band of the record, as indicated by the low-pass and high-pass filter
frequencies applied during processing. Motions were discarded if the high-pass filter
was greater than 0.3 Hz or the low-pass filter was less than 10 Hz, because filters at
these frequencies significantly affect the wave amplitudes within the frequency range
used in the Tm and Tavg calculation (i.e., 0.25 to 20 Hz). This constraint mainly affected
motions from smaller magnitude earthquakes because these motions have less long period energy and often require high-pass filters greater than 0.3 Hz to eliminate noise. As
a result, 835 motions from 44 events ranging in magnitude from 4.9 to 7.6 were used to
develop the predictive relationship for Tm and Tavg . Table 1 lists the events and number
of recordings per event included in the data sets for this study. It should be noted that
different tectonic settings (e.g., stable versus active tectonic regions) were not considered because all of the motions came from reverse and strike-slip events in active tectonic regions.
Figure 2 provides a comparison of the magnitude distribution of the new data sets
and the Rathje et al. (1998) data set. Because of the recent large magnitude earthquakes
in Turkey and Taiwan, the current data sets consist of significantly more M w⬎7.0 motions than the 1998 data set. The number of motions in the Tm, Tavg , and To data sets
increased in the M w 6.5 to 6.9 range because of new information regarding the site classification of strong motion stations from the 1994 Northridge (M w⫽6.7) earthquake. At
magnitudes less than 6.5, the Tm data set did not change significantly from the 1998
study. However, eliminating the filter-frequency requirement for the To data set resulted
in significantly more motions at magnitudes less than 6.5. As a result, the Tm and Tavg ,
and To data sets have different magnitude distributions. It should be noted that a similar
filter-frequency requirement was incorporated by Abrahamson and Silva (1997) in developing their attenuation relationship for spectral acceleration. Their filter frequency requirements resulted in a data set that varied with spectral period. Because there are very
few M w⬍5.5 motions in the Tm data set, the Tm relationship should be used with caution
at magnitudes below 5.5.
For each strong motion station, the two orthogonal horizontal components of motion
were combined to represent a single data point. For Tm , the Fourier amplitude coefficients of the two orthogonal components were combined using the Euclidean norm
(冑(X1)2⫹(X2)2) and the resulting Fourier Amplitude Spectrum was used in the Tm calculation. The Euclidean norm was used because Fourier amplitude space is a vector
space. For Tavg and To , each acceleration response spectrum was first normalized by its
PGA and then combined using the geometric mean (冑X1X2) of the two components. The
geometric mean was used to be consistent with the methods used in attenuation relationship development. The resulting response spectrum was used to compute Tavg and To .
The Simplified Geotechnical Site (SGS) classification system described by
Rodriguez-Marek et al. (2001) was utilized to classify site conditions at the strong mo-
126
E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
Table 1. Data set used for regression analysis
Earthquake
Imperial Valley
Kern County
San Francisco
Parkfield
Borrego Mountain
Lytle Creek
San Fernando
Point Mugu
Hollister
Oroville
Oroville
Oroville
Fruili, Italy
Gazli, USSR
Friuli, Italy
Santa Barbara
Tabas, Iran
Coyote Lake
Imperial Valley
Imperial Valley
Imperial Valley
Livermore
Livermore
Anza
Mammoth Lakes
Victoria, Mexico
Mammoth Lakes
Irpinia, Italy
Irpinia, Italy
Taiwan (SMART#5)
Corinth, Greece
Westmorland
Coalinga
Coalinga
Coalinga
Coalinga
Coalinga
Coalinga
Borah Peak, ID
Borah Peak, ID
Morgan Hill
Lazio-Abruzzo, Italy
Bishop (Rnd Val)
Nahanni, Canada
Hollister
Taiwan (SMART#40)
Date
Mw
No. of records
Tm and Tavg
No. of records
To
1940 0519
1952 0721
1957 0322
1966 0628
1968 0409
1970 0912
1971 0209
1973 0221
1974 1128
1975 0801
1975 0802
1975 0808
1976 0506
1976 0517
1976 0915
1978 0813
1978 0916
1979 0806
1979 1015
1979 1015
1979 1016
1980 0124
1980 0127
1980 0225
1980 0527
1980 0609
1980 0611
1980 1123
1980 1123
1981 0129
1981 0224
1981 0426
1983 0502
1983 0509
1983 0611
1983 0709
1983 0722
1983 0722
1983 1028
1983 1029
1984 0424
1984 0507
1984 1123
1985 1223
1986 0126
1986 0520
7.0
7.4
5.3
6.1
6.8
5.4
6.6
5.8
5.2
6.0
5.0
4.7
6.5
6.8
6.1
6.0
7.4
5.7
6.5
5.2
5.5
5.8
5.4
4.9
4.9
6.4
5.0
6.9
6.2
5.7
6.7
5.8
6.4
5.0
5.3
5.2
5.8
4.9
6.9
5.1
6.2
5.9
5.8
6.8
5.4
6.4
1
2
—
5
4
—
29
1
—
—
—
—
4
—
2
2
4
9
31
—
—
6
2
—
2
4
—
5
5
6
1
3
45
—
—
—
8
—
2
2
23
—
—
3
—
8
1
5
1
5
5
10
44
1
3
1
2
9
5
1
4
2
4
10
31
15
1
7
8
5
13
4
9
9
9
7
1
6
46
20
5
13
11
2
2
2
26
3
1
3
3
8
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
127
Table 1. (cont.). Data set used for regression analysis
Earthquake
N. Palm Springs
Chalfant Valley
Chalfant Valley
Chalfant Valley
Chalfant Valley
Whittier Narrows
Whittier Narrows
Superstition Hills
Superstition Hills
Spitak, Armenia
Loma Prieta
Griva, Greece
Erzican, Turkey
Roermond, Netherlands
Cape Mendocino
Landers
Northridge
Kobe, Japan
Kozani, Greece
Kozani, Greece
Kozani, Greece
Dinar, Turkey
Kocaeli, Turkey
Chi-Chi, Taiwan
Duzce, Turkey
Date
Mw
No. of records
Tm and Tavg
No. of records
To
1986 0708
1986 0720
1986 0721
1986 0721
1986 0731
1987 1001
1987 1004
1987 1124
1987 1124
1988 1207
1989 1018
1990 1221
1992 0313
1992 0413
1992 0425
1992 0628
1994 0117
1995 0116
1995 0515
1995 0517
1995 0519
1995 1001
1999 0817
1999 0920
1999 1112
6.0
5.9
6.2
5.6
5.8
6.0
5.3
6.7
6.3
6.8
6.9
5.9
6.9
5.3
7.1
7.3
6.7
6.9
6.6
5.3
5.1
6.4
7.4
7.6
7.1
10
—
9
—
—
16
—
4
1
—
56
—
1
—
5
56
106
11
—
—
1
1
20
306
13
32
5
11
3
2
114
11
6
1
1
56
1
1
1
6
68
140
11
2
1
2
3
20
306
21
tion stations. The SGS classification system differentiates between rock (site class B, soil
depth <6 m), weathered/soft rock and shallow stiff soil (site class C, soil depth <60 m),
and deep stiff soil (site class D, soil depth >60 m). These depth criteria are loosely based
on site period (T⬍0.2 s for Class B, T⬍0.8 s for Class C, and T⬍2.0 s for Class D) and
were established by considering the statistical difference between motions recorded during the Northridge and Loma Prieta earthquakes for each site class. Stations classified as
soft soil (site class E, soft soil thickness >3 m) were not used in this study. Rathje et al.
(1998) combined site classes B and C into a single rock/shallow soil category. However,
recent research (e.g., Rodriguez-Marek et al. 2001) indicates that there is a significant
difference between strong motions recorded at rock versus shallow soil sites, and therefore, these site categories are separated.
Rodriguez-Marek et al. (2001) provided the site classifications for the Loma Prieta
(1989) and Northridge (1994) earthquakes, Rathje et al. (2003) classified sites from the
recent Kocaeli (1999) and Duzce (1999) earthquakes, and Stewart et al. (2001b) provided classifications for the remaining stations in the Los Angeles Basin and San
Francisco Bay Area. For the data recorded during the Chi-Chi (1999) earthquake, the
128
E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
Figure 2. Comparison of Tm and Tavg , To , and Rathje et al. (1998) data sets.
Lee et al. (2001) site classification was employed with the following qualitative descriptions: (1) class B includes Miocene and older strata, limestone, igneous, and metamorphic rock, (2) class C includes Pliocene and Pleistocene strata, conglomerates, pyroclastic rocks, and geomorphologic lateritic terraces, and (3) class D includes late Pleistocene
and Holocene strata, geomorphologic fluvial terraces, stiff clays, and sandy soils with
average SPT⭓15 in the upper 30 m. The distribution of data across site classes for the
Tm , Tavg and To data sets were almost identical, with 15% B sites, 27% C sites, and 58%
D sites. Figure 3 shows the distribution of data with respect to earthquake magnitude,
distance, and site class for the Tm, Tavg , and To data sets.
DEVELOPMENT OF FUNCTIONAL FORM OF EMPIRICAL MODEL
THEORETICAL MODEL
To develop an appropriate functional form for the empirical relationships for the frequency content parameters, a theoretical earthquake point source model that predicts a
Fourier Amplitude Spectrum was used. Although the functional form developed from the
point source model is only theoretically applicable to Tm , it has been found that this
functional form works well for Tavg and To also (Rathje et al. 1998).
The Brune (1970, 1971) ␻2 point source model was employed to consider the theoretical magnitude and distance dependence for Tm . The Fourier amplitudes (cm/s) predicted by the Brune (1970, 1971) model are given by
冋
册
␲•f•R
f2
Mo
•exp共⫺␲•␬•f 兲•exp ⫺
C共 f 兲⫽0.78•
2•
f
R
␤oQ共 f 兲
1⫹
fc
冉冊
(4)
with f c⫽4.9⫻106␤o共⌬␴/M o兲1/3
(5)
log M o⫽1.5M w⫹16.05
(6)
Q共 f 兲⫽Q0 f n
(7)
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
129
Figure 3. Distribution of data for (a) Tm and Tavg data set and (b) To data set.
where f represents frequency (Hz); f c is the corner frequency (Hz); M o is the seismic
moment (dyne-cm); M w is moment magnitude; R is distance from the point source (km);
␬ is a factor that represents the damping of seismic waves as they propagate through the
crust (s); ␤o is the shear wave velocity of the crust at the source depth (km/s); ⌬␴ represents the stress drop of the source (bars); and Q(f ) is a frequency dependent quality
factor, representing inelastic attenuation in the crust.
Using source parameters appropriate for the western United States (⌬␴⫽80 bars,
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E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
Figure 4. Theoretical (a) distance and (b) magnitude dependencies for Tm .
␤o⫽3.2 km/s, ␬⫽0.035 s, Qo⫽300, n⫽0.6; Boore and Joyner 1997; Abrahamson, personal communication 1997), Equations 4 through 7 were used to generate Fourier Amplitude Spectra for earthquake magnitudes between 5 and 8 and distances between 1 and
100 km. The resulting Fourier Amplitude Spectra were used to compute Tm for each M w
and R pair. The theoretical distance and magnitude dependencies developed from this
procedure are shown in Figure 4. The data in Figure 4 are plotted on a semi-log axis and
indicate that the distance dependence for ln(Tm) is essentially linear. The magnitude dependence for ln(Tm) is nonlinear (Figure 4b), with a linear dependence at smaller magnitudes and almost no magnitude dependence at magnitudes greater than 7.5. There is no
magnitude dependence at larger magnitudes because at these magnitudes the corner frequency (f c), which controls the low frequency components of motion, is outside the frequency range used for the Tm computation. Consequently, the additional low frequency
energy generated by earthquakes greater than 7.5 is outside the frequency range of the
Tm calculation and does not affect Tm . The nonlinear magnitude dependence can be approximated by a linear relationship at magnitudes less than 7.25 and a constant relationship at magnitudes greater than 7.25. Rathje et al. (1998) inferred linear distance and
magnitude dependencies for Tm , rather than ln(Tm), from the theoretical source model.
Either of these models could be used, but the ln(Tm) linear distance and magnitude dependencies were used in this study because they facilitate the regression analyses. Additionally, ground motion parameters have been shown to be log-normally distributed
(e.g., Abrahamson 1988), which makes the ln(Tm) regression more desirable.
EXTENSIONS OF THEORETICAL MODEL
The Brune (1970, 1971) point source model provides guidance regarding the distance and magnitude dependencies to be incorporated in the empirical model. However,
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
131
the point source model does not offer information regarding the effect of site conditions
or fault rupture directivity. Previous studies and observations from recorded strong
ground motions were used to develop the methodology for treatment of site conditions
and fault rupture directivity.
Site conditions significantly affect the frequency content of strong ground motions
because the dynamic response of soil sites enhances the long period components of
ground shaking. Rathje et al. (1998) used only two site categories (rock/shallow soil and
deep soil) and found that strong ground motions at deep soil sites have larger values of
Tm and To than strong motions at ‘‘rock’’ sites. These larger values of Tm and To are a
direct result of the dynamic response of deep soil deposits. Observations from the 1999
Chi-Chi earthquake (Faraj 2002) show a significant difference between the frequency
contents of motions recorded at rock sites (Site Class B) and motions recorded at shallow soil sites (Site Class C). Similar observations have been made for spectral acceleration (Rodriguez-Marek et al. 2001). As a result, this study used three categories to distinguish site conditions (B—rock, C—weathered/soft rock and shallow stiff soil, and
D—deep stiff soil). The function incorporated in the empirical model that accounts for
site conditions is not distance or magnitude dependent.
Previous research (e.g., Somerville et al. 1997) has shown that fault rupture directivity affects the amplitudes and durations of strong ground motions. When the fault rupture propagates towards a site and the slip direction is aligned with the rupture direction
(called forward directivity), recorded motions generally exhibit three distinct characteristics due to constructive interference of shear waves. These three characteristics are (1)
enhanced long period motion, (2) fault normal components of motion greater than fault
parallel components of motion at long periods, and (3) shorter duration. The change in
strong motion amplitudes at long periods affects the frequency content of strong motions, and thus should affect the scalar frequency content parameters. This study will
consider only the effect of the enhanced long period motion on the frequency content
parameters, and this effect is primarily a result of the fault rupture propagating towards
a site.
The important rupture directivity parameters, as defined by Somerville et al. (1997),
are the azimuth angle (angle between the fault rupture plane and the ray path to the site)
and the length ratio (the fraction of the fault that ruptures towards the site). The azimuth
angle and length ratio definitions for dip-slip and strike-slip faulting are illustrated in
Figure 5. For the purposes of this study, forward directivity motions will be identified
solely by their geometric location with respect to the fault rupture. Specifically, recordings will be designated as forward directivity motions if more than one-half of the fault
ruptured towards the site (length ratio ⭓0.5) and the azimuth is less than or equal to 30
degrees. This strict geometric definition of forward directivity will result in some motions being classified as forward directivity although they do not display classic forward
directivity characteristics. However, using a consistent spatial definition of forward directivity will provide an unbiased estimate of the frequency content parameters in the
zone of expected forward directivity.
To evaluate whether forward rupture directivity significantly affects scalar frequency
content parameters, recorded strong ground motions from the 1979 Imperial Valley
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E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
Figure 5. Fault rupture directivity parameters (azimuth and length ratio) for dip-slip and strikeslip faults (adapted from Somerville et al. 1997).
(M w⫽6.5) earthquake, the 1989 Loma Prieta (M w⫽6.9) earthquake, and the 1994
Northridge (M w⫽6.7) earthquake were considered. For these earthquakes, recordings
were designated forward directivity motions based on their geometric locations, as discussed previously. The azimuth angles and length ratios for the recordings were obtained
from Stewart (personal communication 2002). For each motion, the residual with respect
to the Rathje et al. (1998) relationship [ln(recorded Tm) — ln(1998 model)] was computed and plotted versus distance. These data are shown in Figure 6, along with mean
values computed within overlapping distance bins. The data indicate a significant under
prediction of Tm (i.e., residual greater than zero) by the Rathje et al. (1998) model within
about 15 to 20 km of the fault. In this distance range, the variation of the residuals with
distance is approximately linear. At distances greater than 20 km, the mean of the re-
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
133
Figure 6. Tm residuals versus distance for Imperial Valley, Loma Prieta, and Northridge earthquakes.
siduals is close to zero, indicating no significant directivity effects beyond 20 km. Similar trends were observed for Tavg and To . In addition to the results shown in Figure 6, the
entire Tm data set was evaluated to assess whether the forward directivity motions (as
defined above) were significantly different from the rest of the data set. The results indicated these data were significantly and statistically different (Faraj 2002). Based on
these observations, a forward directivity function that decays linearly with distance (up
to 20 km) was incorporated in the empirical model. Based on these criteria, 98 recordings within the data set were identified as forward directivity motions. These recordings
encompass earthquake magnitudes between 6.0 and 7.6, distances between 0.1 and 20.0
km, and all site classes (8% B sites, 30% C sites, and 62% D sites).
EMPIRICAL RELATIONSHIPS
REGRESSION RESULTS
A random-effects model (Brillinger and Preisler 1984) was used to determine the
regression coefficients for the empirical models for the three frequency content parameters. Random-effects modeling has been used previously to develop empirical attenuation relationships for spectral acceleration (e.g., Abrahamson and Youngs 1992, Abrahamson and Silva 1997). In random-effects modeling, the error is divided into intra-event
and inter-event terms. The intra-event residual (␧) represents the difference between any
single data point and the median prediction for that event, while the inter-event residual
(␩) represents the deviation of the median prediction for a single event from the median
prediction based on the entire data set. The intra-event and inter-event error terms are
assumed normally distributed with mean of zero and standard errors of ␴ and ␶, respectively. The total error for the model is computed as ␴total⫽冑␴2⫹␶2. The statistical data
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E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
Table 2. Regression coefficients and standard errors for the regression coefficients
Value for Tm
Regression
Value for Tavg
Regression
Value of To
Regression
c1
−1.00
(0.055)
−0.89
(0.046)
−1.78
(0.041)
c2
0.18
(0.062)
0.29
(0.047)
0.30
(0.043)
c3
0.0038
(0.00036)
0.0030
(0.0003)
0.0045
(0.00031)
c4
0.078
(0.040)
0.07
(0.036)
0.15
(0.033)
c5
0.27
(0.037)
0.25
(0.033)
0.33
(0.030)
c6
0.40
(0.069)
0.37
(0.062)
0.24
(0.065)
Coefficient
analysis software ‘‘R’’ (R 1.4.0—A Programming Environment for Data Analysis and
Graphics, 2001) was used for all random-effects regression analyses. This program contains a built in subroutine for random-effects modeling.
The general functional forms incorporated in this study are
ln共T兲⫽c1⫹c2•共 M w⫺6兲⫹c3•R⫹c4•SC⫹c5•SD⫹c6•共1⫺R/20兲•FD
(8)
for 5.0⭐M w⭐7.25 for Tm
for 4.7⭐M w⭐7.6 for Tavg and To
ln共T兲⫽c1⫹c2•共7.25⫺6兲⫹c3•R⫹c4•SC⫹c5•SD⫹c6•共1⫺R/20兲•FD
(9)
for M w⬎7.25 for Tm
where T is the appropriate frequency content parameter (Tm , Tavg , or To , in units of s);
M w is moment magnitude; R is the closest distance to the fault rupture plane (km); SC
and SD are indicator variables that designate site class (SC⫽SD⫽0 for Site Class B, SC
⫽1 and SD⫽0 for Site Class C, SC⫽0 and SD⫽1 for Site Class D); and FD is an indicator variable that designates forward directivity conditions (FD⫽1 for sites with M w
⭓6.0, R⭐20.0 km, azimuth angle ⭐30°, and rupture length ratio ⭓0.5, FD=0 otherwise). The error term for Equations 8 and 9 has a mean of zero and standard deviation
of ␴total . Note that a magnitude cutoff is employed for Tm by limiting the M w value to
7.25 for all M w⬎7.25 (Equation 9). There is no theoretical magnitude cutoff for Tavg or
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
135
Figure 7. Intra-event Tm residuals versus magnitude and distance for site classes B, C, and D.
To , but no data from magnitudes greater than 7.6 were used to develop the relationships.
The authors recommend that for magnitudes greater than 7.6, the values calculated at a
magnitude of 7.6 be used.
The regression coefficients for the empirical models for Tm , Tavg , and To are listed in
Table 2, along with the standard errors for each coefficient. Each regression coefficient
is statistically significant at a level smaller than 0.0001 (p⬍0.0001 that the coefficient is
equal to 0, using hypothesis testing and the statistical t distribution, Devore 1991), except for coefficient c4 for Tm and Tavg . These parameters are not as statistically significant (p⬃0.06), but the authors believe that c4 is sufficiently significant to remain in the
predictive equations. The coefficient c4 controls the site effect for shallow soil sites (Site
Class C). The smaller statistical significance of c4 for Tm and Tavg indicates that shallow
soil sites do not change Tm and Tavg as significantly as shallow soil sites change To . This
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E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
Figure 8. Intra-event Tavg and To residuals versus magnitude and distance for site classes B, C,
and D.
result is expected because Tm and Tavg are affected most by the long period energy in a
strong motion and shallow soil sites tend to amplify short periods more than long periods.
The intra-event residuals for Tm are plotted versus magnitude and distance for the
three site classes in Figures 7a–f. As a whole, the intra-event residuals have a mean of
zero and standard deviation of ␴. However, the data in Figure 7 indicate that the intraevent standard deviation varies with site class, with more scatter observed for Site Class
B than for site classes C or D. Computing the standard deviation of the intra-event Tm
residuals for each site class, Site Class B displays the largest error (␴B⫽0.42), followed
by Site Class C (␴C⫽0.38) and Site Class D (␴D⫽0.31). The statistical F test was used
to evaluate whether the standard errors for each site class were statistically different (DeTable 3. Intra-event and inter-event standard error terms for Tm , Tavg , and
To
␴B
␴C
␴D
␶ (All site classes)
Error Terms
for Tm
Error Terms
for Tavg
Error Terms
for To
0.42
0.38
0.31
0.17
0.38
0.36
0.29
0.13
0.38
0.33
0.31
0.22
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
137
Figure 9. Inter-event residuals versus magnitude for Tm , Tavg , and To .
vore 1991). A probability level of 0.05 was chosen to define statistical significance. The
difference between ␴B and ␴C is not statistically significant (p⫽0.12 that ␴B2⫽␴C2), but
2
the differences with Site Class D are statistically significant (p⬍10⫺4 that ␴B2⫽␴D
, ␴C2
2
⫽␴D). Site Class B exhibits the largest error, most likely because of the relatively large
range of dynamic stiffnesses (i.e., shear wave velocities) encompassed in the rock site
category. In contrast, the deep stiff soil sites of Site Class D cover a relatively smaller
range of shear wave velocities and deep soil deposits provide a consistent filter to incoming ground motions, resulting in a less variable response and smaller ␴. The intraevent residuals for Tavg and To are plotted versus magnitude and distance for the three
site classes in Figure 8, and the intra-event standard error terms for each site class are
listed in Table 3. The intra-event error terms for Tavg are slightly smaller than for Tm .
However, the trends regarding the relative differences of ␴B , ␴C , and ␴D for Tavg are
similar as for Tm , with no statistical significance between ␴B and ␴C (p⫽0.23 that ␴B2
⫽␴C2) and statistically significant differences with ␴D (p⬍10⫺4 that ␴B2⫽␴D2, ␴C2⫽␴D2).
For To , the error terms are similar in magnitude to the Tm and Tavg error terms. However,
when comparing the error terms for each site class, ␴B is statistically significant from
2
the others (p⫽0.02 that ␴B2⫽␴C2, p⬃10⫺4 that ␴B2⫽␴D
), but ␴C and ␴D are not statisti2
2
cally significant from one another (p⫽0.08 that ␴C⫽␴D
).
The total standard error for a random-effects model also incorporates the inter-event
error, ␶. The inter-event residuals for Tm , Tavg , and To are plotted versus magnitude in
Figure 9 and the inter-event error terms (␶) for each parameter are listed in Table 3. The
inter-event residuals in Figure 9 display significantly less variability than the intra-event
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E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
Figure 10. Median predictions of Tm , Tavg , and To for Site Class B and different magnitude
events.
residuals in figures 7 and 8. As a result, the values of ␶ are much smaller than the values
of ␴ (Table 3), which is consistent with results from attenuation relationships for spectral
acceleration (e.g., Abrahamson and Silva 1997). Tavg displays the smallest inter-event
error (␶⫽0.13), while Tm and To exhibit larger values of inter-event error (␶⫽0.17 and
0.22, respectively). A reduction in ␴ and ␶ with magnitude was also investigated based
on the observations of Youngs et al. (1995), but no significant trends were observed.
Given the intra-event error term for the appropriate site class (␴siteclass) and the interevent error term (␶) from Table 3, the total standard error can be computed as
2
␴total⫽冑␴siteclass
⫹␶2
(10)
where siteclass⫽B, C, or D
DISCUSSION
The predicted median relationships for Tm , Tavg , and To versus distance are shown in
Figure 10 for Site Class B and magnitudes 5.5, 6.5, and 7.5. Tm and Tavg generally take
on the same values at smaller magnitudes, but Tavg displays a larger magnitude dependence. To is much smaller than Tm and Tavg at each distance and magnitude, which is
expected because To samples a different frequency range than Tm and Tavg (Figure 1).
The magnitude dependence for To is similar to that for Tavg and significantly larger than
that for Tm . These differences are indicated by the spacing between the curves for each
magnitude in Figure 10 and the regression coefficient c2 in Table 2. The larger magnitude dependence for To and Tavg versus Tm may indicate that the response spectrum is
more affected by earthquake magnitude than the Fourier Amplitude Spectrum.
The empirical relationships for Tm shown in Figure 10 are different than the theoretical relationships shown in Figure 4. At M w⫽5.5 the relationships are similar, but the
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
139
Figure 11. Effect of forward directivity on Tm values.
theoretical relationships predict larger values of Tm at larger magnitudes. This difference
can be attributed to the Brune source spectrum used for the theoretical relationship. The
Brune spectrum assumes a point source, which overpredicts the Fourier amplitude coefficients at long periods (Boore 1983, Atkinson and Silva 1997). The overpredicted Fourier amplitudes produce larger theoretical Tm values that are not realistic. The empirical
data confirm this assertion.
The effect of forward directivity on the predicted Tm relationship is illustrated in Figure 11. At distances less than 20 km, the predicted Tm for a forward directivity motion is
as much as 50% larger than for a non-directivity motion. This difference is largest close
to the fault rupture plane. Because Tm and Tavg better incorporate the long period components of motion, they are more affected by forward directivity than To . This result is
apparent from the larger c6 regression coefficients for Tm and Tavg than for To (c6 controls the directivity effect, equations 8 and 9, Table 2). It is commonly assumed that forward directivity only affects spectral periods greater than about 0.6 s (Somerville et al.
1997), which would suggest that forward directivity should not significantly affect To .
Nevertheless, the recorded data support a forward-directivity effect on To based on the
statistical significance of regression coefficient c6 (Table 2). This trend is most likely attributed to the fact that spectral acceleration at a given period is affected by the motion
at a range of periods centered about the given period. If forward directivity affects some
components of motion within that range of periods, To will be affected.
Figure 12 provides a comparison of the relationships developed in this study with
those developed by Rathje et al. (1998). The effect of site conditions on predicted values
of Tm for a magnitude 7 event is shown in Figure 12a. For the current study, Site Class
C displays a slightly larger value of Tm than Site Class B. However, the difference is
more significant for Site Class D, where the enhanced long period energy generated by
140
E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
Figure 12. Comparison of relationships from this study and Rathje et al. (1998).
the dynamic response of deep soil sites results in a 30% increase in Tm compared with
Site Class B. The site class effect for Tavg is similar to that of Tm , but the site class effect
is larger for To . Site effects for each frequency content parameter can be compared
through the regression coefficients c4 and c5 in Table 2. Based on these coefficients, To
is most affected by site class.
The predicted values of Tm for rock/shallow soil and deep soil sites from Rathje et al.
(1998) also are shown in Figure 12a. The 1998 study used a site classification system
that combined rock (Site Class B) and shallow soil/weathered rock (Site Class C) into a
single ‘‘rock’’ category. As a result, the rock results from Rathje et al. (1998) agree best
with the Site Class C results from this study. Site Class B falls about 10% below the
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
141
previous rock/shallow soil relationship. Because the 1998 deep soil category is consistent with the current Site Class D category, these relationships compare favorably. Comparisons between the current and 1998 studies for To are similar to those for the Tm relationships.
The magnitude dependencies for Tm , Tavg , and To are shown in Figure 12b for Site
Class D and R⫽20 km. Site Class D was used for this comparison because this site class
is most consistent between the current study and the 1998 study. The new Tm relationship is very similar to the 1998 relationship, despite the addition of over 500 new motions. The change in the To relationship is more dramatic. The To relationship shifted
down at small magnitudes, without significant shifting at larger magnitudes. As a result,
the current relationship indicates that To varies with magnitude considerably more than
Tm . This observation is in contrast with the Rathje et al. (1998) study, which found that
Tm increased with magnitude at a faster rate than To . The discrepancy is most likely the
result of using different data sets for Tm versus To in this study, rather than using the
same data sets for all parameters, as was done in the 1998 study (Figure 2).
The Tm and Tavg data set in this study contains considerably fewer motions at M w
⬍6.5 than the To data set and there is almost no Tm and Tavg data for M w⬍5.5 (Figure 2).
The difference in the data sets is a direct result of the filter frequency requirements for
Tm and Tavg , particularly the high-pass filter frequency constraint that requires a highpass filter less than 0.3 Hz. Most motions from small magnitude events do not meet this
constraint because they do not contain significant low frequency (i.e., long period) energy, and therefore require high-pass filters greater than 0.3 Hz to eliminate low frequency noise. The few small magnitude motions that do meet the Tm and Tavg high-pass
filter requirement tend to have significant long period energy for that magnitude, otherwise they would have required a high-pass filter greater than 0.3 Hz. Consequently, the
Tm and Tavg data may be somewhat biased towards larger Tm and Tavg values at small
magnitudes. When the filter-frequency requirements were dropped for To , the number of
small magnitude motions available for the regression increased considerably (Figure 2).
The addition of these motions from small magnitude events resulted in smaller predicted
values of To at small magnitudes for this study compared with the 1998 study (Figure
12b). These differences indicate that the filter-frequency requirement somewhat biases
the Tm and Tavg relationship at small magnitudes. Again, it should be noted that a similar
filter-frequency requirement has been incorporated by others (e.g., Abrahamson and
Silva 1997) in developing attenuation relationships for spectral acceleration and these
relationships at long periods are similarly biased.
CONCLUSIONS
The frequency content of an earthquake ground motion is an important ground motion parameter because it affects the dynamic response of geotechnical and structural
systems. A scalar representation of frequency content is useful in earthquake engineering practice to evaluate possible conditions for dynamic resonance or enhanced dynamic
response. Several recently developed seismic design procedures have incorporated scalar
frequency content parameters (e.g., Bray et al. 1998, Stewart et al. 2003).
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E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY
This paper described the development of empirical relationships that predict three
frequency content parameters (Tm , Tavg , and To) as a function of earthquake magnitude,
site-to-source distance, site conditions, and fault rupture directivity. Tm is based on the
Fourier Amplitude Spectrum, while Tavg and To are based on the acceleration response
spectrum. The predominant spectral period (Tp) was also considered, but this parameter
did not adequately describe the frequency content of a strong ground motion. The functional form of the empirical relationship for Tm was developed from a theoretical point
source model that predicts the Fourier Amplitude Spectrum as a function of magnitude
and distance. This functional form was further extended to account for the effects of site
conditions and fault rupture directivity. The recorded data supported the use of three site
classes: rock (Site Class B), weathered/soft rock and shallow soil (Site Class C), and
deep soil (Site Class D). Additionally, near-fault strong ground motion recordings indicated a significant increase in Tm for sites experiencing forward directivity. Similar observations regarding site conditions and directivity were made for Tavg and To . The same
functional form developed for Tm was used for Tavg and To , except for a magnitude cutoff.
The developed predictive relationships expand and enhance the relationships developed by Rathje et al. (1998). A new frequency content parameter, Tavg , was also defined.
A significantly larger strong motion data set was utilized that included motions from the
recent large magnitude earthquakes in Turkey and Taiwan. Three site classes were incorporated rather than two, and the effects of fault rupture directivity were included. Additionally, the random-effects model was used in the regression, which allows the variability within events (intra-event variability) and between events (inter-event variability) to
be treated separately. The final regression results provide relationships that predict Tm ,
Tavg , and To as a function of magnitude, site-to-source distance, site class, and forward
directivity. Based on the magnitude distribution of the various data sets, the Tm and Tavg
relationships are most reliable for M w⫽5.5–7.6 and the To relationship is most reliable
for M w⫽4.7–7.6. The intra-event error terms (␴) vary as a function of site class, with
Site Class D displaying the least intra-event variability. The inter-event error terms (␶)
indicate that Tavg has the smallest inter-event variability. Neither ␴ nor ␶ were observed
to vary with earthquake magnitude. Each scalar frequency content parameter in this
study is appropriate for different conditions. To is most sensitive to the high frequency
(low period) content of strong ground motions and may be best suited for projects where
this frequency range is of interest (e.g., nuclear reactors). Tm and Tavg best account for
the long period content of strong motions, and they best differentiate the long period
content between different motions. Although these parameters are based on different
representations of a strong ground motion (Fourier Amplitude Spectrum vs. acceleration
response spectrum), they take on similar values and are similarly affected by site class
and directivity. However, the Fourier Amplitude Spectrum is a more robust representation of a strong motion, and therefore, Tm is considered the preferred frequency content
parameter.
ACKNOWLEDGMENTS
Financial support was provided through the Pacific Earthquake Engineering Research Center Lifelines Program under projects 1C01. Additional support was provided
EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS
143
by the National Science Foundation through an REU (Research Experience for Undergraduates) supplement under grant CMS-9875430 and by the David and Lucile Packard
Foundation. The authors also wish to thank Dr. Walt Silva of Pacific Engineering and
Analysis for providing the processed Chi-Chi motions, Ms. Thaleia Travasarou of U.C.
Berkeley for her assistance in using the ‘‘R’’ program and for providing site classification information, and Dr. Jonathan Stewart of UCLA for providing directivity parameters for the strong motion database. Particularly insightful comments were provided by
several anonymous reviewers.
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(Received 6 September 2002; accepted 20 June 2003)