Empirical Relationships for FrequencyContent Parameters of EarthquakeGround Motions

Ellen M. Rathje,a) M.EERI, Fadi Faraj,b) Stephanie Russell,c) and JonathanD. Bray,d) M.EERI

The frequency content of an earthquake ground motion is important be-cause it affects the dynamic response of earth and structural systems. Fourscalar parameters that characterize the frequency content of strong groundmotions are (1) the mean period (Tm), (2) the average spectral period (Tavg),(3) the smoothed spectral predominant period (To), and (4) the predominantspectral period (Tp). Tm and Tavg distinguish the low frequency content ofground motions, while To is affected most by the high frequency content. Tpdoes not adequately describe the frequency content of a strong ground mo-tion and is not recommended. Empirical relationships are developed that pre-dict three parameters (Tm , Tavg , and To) as a function of earthquake magni-tude, site-to-source distance, site conditions, and rupture directivity. Therelationships are developed from a large strong-motion database that includesrecorded motions from the recent earthquakes in Turkey and Taiwan. The newrelationships 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 con-tent parameters and smaller standard error terms than conventional rockand soil site classes. Forward directivity significantly increases the fre-quency content parameters, particularly Tm and To , at distances less than 20km. Each of the frequency content parameters can be predicted with reason-able accuracy, but Tm is the preferred because it best distinguishes the fre-quency content of strong ground motions. [DOI: 10.1193/1.1643356]

INTRODUCTION

The dynamic response of geotechnical and structural systems subjected to earth-quake ground shaking is significantly affected by the frequency content of the inputearthquake ground motion. When the frequency content of an earthquake ground motionclosely matches the natural period of a geotechnical system (e.g., soil deposit, earthdam) 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 78712b) Graduate Research Assistant, Dept. of Civil Engineering, University of Texas, Austin, TX 78712c) Undergraduate Research Assistant, Dept. of Civil Engineering, University of Texas, Austin, TX 78712d) Professor, Dept. of Civil and Environmental Engineering, University of California, Berkeley, CA 94720119Earthquake Spectra, Volume 20, No. 1, pages 119144, February 2004; 2004, Earthquake Engineering Research Institute

120 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY1996, Chopra 2001). As a result, it is important to evaluate the frequency content of anearthquake ground motion and consider its effect on the dynamic response of a struc-ture.

Acceleration response spectra and Fourier Amplitude Spectra provide the most com-plete characterization of the frequency content of strong ground motions. These spectraprovide information regarding the distribution of motion across a range of frequencies.Several attenuation relationships are available that predict the full acceleration responsespectrum 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 tocharacterize the frequency content of a strong ground motion with a single, scalar pa-rameter. A scalar representation of frequency content allows the frequency content ofdifferent strong ground motions to be compared quickly and easily (Stewart et al.2001a). Additionally, a scalar frequency content parameter can be compared with thenatural period of a dynamic system to evaluate the possibility of resonance conditions oran enhanced dynamic response (e.g., Nadim and Whitman 1983, Bray et al. 1998, Rathjeet al. 1998). Recently, seismic design procedures for landslides and earth fills have in-corporated scalar frequency content parameters (e.g., Blake et al. 2002, Stewart et al.2003). These examples illustrate the usefulness of scalar frequency content parametersin engineering design and analysis.

Various scalar parameters have been used to characterize the frequency content ofstrong ground motions. Rathje et al. (1998) described several frequency content param-eters and developed predictive relationships for three parameters. In this paper the threefrequency content parameters identified by Rathje et al. (1998) are re-examined and afourth 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 Am-plitude Spectrum, averaging periods (over a specified frequency range) weighted by theFourier amplitudes. The average spectral period (Tavg) utilizes the 5% damped accelera-tion response spectrum and averages periods (over a specified frequency range) weightedby the spectral accelerations. The smoothed spectral predominant period (To) also uti-lizes the 5% damped acceleration response spectrum, but averages periods only over therange of spectral amplification (i.e., spectral acceleration greater than 1.2 times the peakground acceleration). Finally, the predominant spectral period (Tp) utilizes the accelera-tion response spectrum and is simply defined as the period of the maximum spectral ac-celeration.

This paper updates the predictive relationships developed by Rathje et al. (1998) forTm 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 relation-ships developed in this study improve upon the previous study in several ways. The dataset used in this study is significantly larger, with 835 recordings used for Tm and Tavg and1208 recordings used for To (previously, 306 records were used). The new model usesthree site classes that distinguish between rock, shallow soil, and deep soil, while theprevious study combined rock and shallow soil into a single site category. A recent de-velopment in earthquake engineering and engineering seismology is the recognition of

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 121fault 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 ac-counts 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 pro-cess used to establish the functional form and regression coefficients for the predictiverelationships. The newly developed relationships are compared with previous relation-ships, and pertinent observations are discussed.

FREQUENCY CONTENT CHARACTERIZATION

Only a few previous studies have addressed predicting the frequency content ofstrong ground motions, and most of these previous studies have chosen predominantspectral period (Tp) as the parameter to characterize frequency content. Gutenberg andRichter (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 as-sumed to correspond to the predominant spectral period. Seed et al. (1969) suggested arelationship between Tp and distance for different earthquake magnitudes based on alimited data set. Idriss (1991) modified the Seed et al. (1969) relationships based on re-corded strong motions from the 1989 Loma Prieta (Mw56.9) earthquake. Rathje et al.(1998) performed a systematic, statistical empirical study of scalar frequency contentparameters for a relatively large database of recorded motions. In the 1998 study, 306strong ground motions from 20 earthquake events in active plate-margin regions wereused to develop predictive relationships for Tp , as well as mean period (Tm) andsmoothed spectral predominant period (To). The Rathje et al. (1998) study concludedthat Tm and To better characterize the frequency content of strong ground motions thanTp , and that these two parameters can be more accurately estimated. This paper re-examines the scalar frequency content parameters described in Rathje et al. (1998) andintroduces another parameter, the average spectral period (Tavg). These parameters aredefined and discussed below.

Tm is computed from the Fourier Amplitude Spectrum and is defined as

Tm5(iCi

2~1/f i!

(iCi

2 for 0.25 Hz < f i < 20 Hz, with Df < 0.05 Hz (1)

where Ci are the Fourier amplitude coefficients, f i are the discrete fast Fourier transform(FFT) frequencies between 0.25 and 20 Hz, and Df is the frequency interval used in theFFT computation. The discrete FFT frequencies are equally spaced in the frequency do-main, 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 un-usually large frequency intervals are used. Using a theoretical model for the Fourier Am-plitude Spectrum of an earthquake ground motion (Brune 1970, 1971), it was found thata stable value of Tm (i.e., the same value of Tm is computed) was obtained for frequencyintervals smaller than 0.05 Hz. The frequency interval (Df ) is related to the time step(Dt) and number of points (N) in a time series by Df51/(NDt). For common strong-motion 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. BRAYsponding to Df50.05 Hz is 1000, 2000, and 4000, respectively. To ensure a stable valueof Tm is calculated for recorded strong ground motions, motions should contain at leastthe minimum number of points indicated above or should be augmented with zeroes toattain these minimum values. A benefit to using the FAS to define a scalar frequencycontent parameter is that the Fourier amplitude coefficients at each frequency are mutu-ally independent. This is not the case for spectral acceleration.

The frequency content parameters Tp , To , and Tavg are based on the 5% damped ac-celeration response spectrum. Tp is simply defined as the period of the maximum spec-tral acceleration. To is computed as

To5(iTilnSSa~Ti!PGA D(i

lnSSa~Ti!PGA

D for Ti withSa

PGA> 1.2, D log Ti < 0.02 (2)

where Ti are the discrete periods in the acceleration response spectrum equally spacedon a log axis, Sa(Ti) are the spectral accelerations at periods Ti , and PGA is the peakground acceleration. Only periods where the spectral acceleration is greater than 1.2PGA are used in the computation in Equation 2. In essence, To attempts to define thepeak in the response spectrum by smoothing the spectral accelerations over the rangewhere Sa is greater than 1.2PGA. Consequently, To is similar to Tp . However, To rep-resents an improvement over Tp because it smoothes the response spectrum to find itspeak. In the previous study (Rathje et al. 1998), Tp displayed the largest variability be-cause it represents only one point in the response spectrum. To can provide similar in-formation 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 ofstrong motions and may be best suited for structures that are sensitive to motions in thisfrequency range (e.g., nuclear reactors). Tavg is computed as

Tavg5(iTiSSa~Ti!PGA D

2

(iSSa~Ti!PGA

D2 for 0.05 s < Ti < 4 s, DTi < 0.05 s (3)where Ti are the discrete periods in the acceleration response spectrum equally spacedon an arithmetic axis, Sa(Ti) are the spectral accelerations at periods Ti , and PGA is thepeak ground acceleration. As defined, Tavg is similar to Tm , except that the periods areequally spaced on an arithmetic axis and spectral acceleration is used in lieu of Fourieramplitudes.

It is important to note how the period spacing in an acceleration response spectrumaffects scalar frequency content parameters. When periods are spaced equally on anarithmetic axis, the spectral accelerations at long periods are not independent of one an-other. These spectral accelerations are not independent because a response spectrum rep-resents the response of single-degree-of-freedom (SDOF) oscillators and the frequencybandwidth 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 islarger, 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 123function of almost the same frequency bandwidth of the motion. Consequently, whenaveraging spectral accelerations at periods equally spaced on an arithmetic axis, the lowfrequency content of the motion is weighted more heavily than the high frequency con-tent. This consequence is advantageous if one wishes to emphasize the low frequencycontent of a strong motion.

Figure 1 shows the normalized acceleration response spectra (Sa /PGA), Fourier Am-plitude Spectra, and associated frequency content parameters for three strong motionsrecorded during the 1999 Chi-Chi, Taiwan, (Mw57.6) earthquake. These motions are theTCU048 (shallow soil, rupture distance 14.4 km), the TCU047 (shallow soil, rupturedistance 33 km), and the TCU129 (deep soil, rupture distance 1.2 km) recordings. The

Figure 1. (a) Normalized acceleration response spectra (Sa /PGA) and (b) Fourier AmplitudeSpectra for the TCU048, TCU047, and TCU129 motions recorded during the 1999 Chi-Chiearthquake.

124 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAYacceleration response spectra (Figure 1a) indicate that TCU048 and TCU047 have simi-lar spectral shapes in the period range of 0.01 to 1.0 s, but TCU129 drops off quickly atperiods greater than 0.35 s. Additionally, TCU048 demonstrates the largest spectral ac-celerations at long periods. The Fourier Amplitude Spectra reveal similar trends (Figure1b); TCU129 contains the most high frequency energy, while TCU048 contains the mostlong period energy. However, the enhanced high frequency energy in TCU129 is moreeasily identified in the Fourier Amplitude Spectrum.

Considering the scalar frequency content parameters, the values of Tp are very simi-lar 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 moderatefrequencies. 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 TCU047because To considers the spectral accelerations throughout the range of spectral accel-eration amplification (i.e., where Sa.1.2PGA). However, To cannot distinguish betweenTCU048 and TCU047, which differ significantly at long periods, because the normalizedspectral accelerations for TCU048 are below 1.2 at periods beyond 1.0 s. The values ofTavg and Tm better differentiate between all three motions, with TCU129 displaying thesmallest 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 scalarfrequency content parameter rather than using an acceleration response spectrum. TheFourier amplitudes represent the amplitudes of harmonic waves that make up anacceleration-time history and each Fourier amplitude coefficient is mutually exclusive.An acceleration response spectrum is a collection of maximum responses of dampedSDOF oscillators. As a result, the acceleration response spectrum is not a direct repre-sentation of an acceleration-time history. Each spectral acceleration is affected by abandwidth of frequencies in the ground motion, which makes spectral accelerations atlong periods more similar to one another. Additionally, the shape of the Fourier Ampli-tude Spectrum for strong ground motions provides more information regarding the longperiod content of a motion because the Fourier amplitude coefficients do not decline asquickly as the spectral acceleration values at long periods (i.e., T.1.0 s). As a result, Tmbest characterizes the frequency content of a strong ground motion over moderate tolong periods. However, engineers who feel more comfortable with response spectra ver-sus Fourier Amplitude Spectra may find Tavg to be most useful to characterize the mod-erate to long period content of a strong ground motion. To is most affected by the low tomoderate period content of ground motions, and may be most useful if these periods areof engineering interest. Finally, Tp does not adequately describe the frequency content ofan 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 ofPacific Engineering and Analysis and are available from the Pacific Earthquake Engi-neering Research Center strong motion database (http://peer.berkeley.edu/smcat). Thedata 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 125amount of new data, including over 300 motions from the 1999 Chi-Chi (Mw57.6)earthquake. Further, additional motions from previous earthquakes were included in thisstudy due to new information regarding site classification. As a result, 1208 motionsfrom 71 events ranging in magnitude from 4.7 to 7.6 were available for this study. Anadditional constraint was imposed on motions for the Tm and Tavg data sets based on theusable frequency band of the record, as indicated by the low-pass and high-pass filterfrequencies applied during processing. Motions were discarded if the high-pass filterwas greater than 0.3 Hz or the low-pass filter was less than 10 Hz, because filters atthese frequencies significantly affect the wave amplitudes within the frequency rangeused in the Tm and Tavg calculation (i.e., 0.25 to 20 Hz). This constraint mainly affectedmotions from smaller magnitude earthquakes because these motions have less long pe-riod energy and often require high-pass filters greater than 0.3 Hz to eliminate noise. Asa result, 835 motions from 44 events ranging in magnitude from 4.9 to 7.6 were used todevelop the predictive relationship for Tm and Tavg . Table 1 lists the events and numberof recordings per event included in the data sets for this study. It should be noted thatdifferent tectonic settings (e.g., stable versus active tectonic regions) were not consid-ered because all of the motions came from reverse and strike-slip events in active tec-tonic regions.

Figure 2 provides a comparison of the magnitude distribution of the new data setsand the Rathje et al. (1998) data set. Because of the recent large magnitude earthquakesin Turkey and Taiwan, the current data sets consist of significantly more Mw.7.0 mo-tions than the 1998 data set. The number of motions in the Tm, Tavg , and To data setsincreased in the Mw 6.5 to 6.9 range because of new information regarding the site clas-sification of strong motion stations from the 1994 Northridge (Mw56.7) earthquake. Atmagnitudes less than 6.5, the Tm data set did not change significantly from the 1998study. However, eliminating the filter-frequency requirement for the To data set resultedin 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 similarfilter-frequency requirement was incorporated by Abrahamson and Silva (1997) in de-veloping their attenuation relationship for spectral acceleration. Their filter frequency re-quirements resulted in a data set that varied with spectral period. Because there are veryfew Mw,5.5 motions in the Tm data set, the Tm relationship should be used with cautionat magnitudes below 5.5.

For each strong motion station, the two orthogonal horizontal components of motionwere combined to represent a single data point. For Tm , the Fourier amplitude coeffi-cients of the two orthogonal components were combined using the Euclidean norm(A(X1)21(X2)2) and the resulting Fourier Amplitude Spectrum was used in the Tm cal-culation. The Euclidean norm was used because Fourier amplitude space is a vectorspace. For Tavg and To , each acceleration response spectrum was first normalized by itsPGA and then combined using the geometric mean (AX1X2) of the two components. Thegeometric mean was used to be consistent with the methods used in attenuation relation-ship development. The resulting response spectrum was used to compute Tavg and To .

The Simplified Geotechnical Site (SGS) classification system described byRodriguez-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. BRAYTable 1. Data set used for regression analysis

Earthquake Date MwNo. of records

Tm and Tavg

No. of recordsTo

Imperial Valley 1940 0519 7.0 1 1Kern County 1952 0721 7.4 2 5San Francisco 1957 0322 5.3 1Parkfield 1966 0628 6.1 5 5Borrego Mountain 1968 0409 6.8 4 5Lytle Creek 1970 0912 5.4 10San Fernando 1971 0209 6.6 29 44Point Mugu 1973 0221 5.8 1 1Hollister 1974 1128 5.2 3Oroville 1975 0801 6.0 1Oroville 1975 0802 5.0 2Oroville 1975 0808 4.7 9Fruili, Italy 1976 0506 6.5 4 5Gazli, USSR 1976 0517 6.8 1Friuli, Italy 1976 0915 6.1 2 4Santa Barbara 1978 0813 6.0 2 2Tabas, Iran 1978 0916 7.4 4 4Coyote Lake 1979 0806 5.7 9 10Imperial Valley 1979 1015 6.5 31 31Imperial Valley 1979 1015 5.2 15Imperial Valley 1979 1016 5.5 1Livermore 1980 0124 5.8 6 7Livermore 1980 0127 5.4 2 8Anza 1980 0225 4.9 5Mammoth Lakes 1980 0527 4.9 2 13Victoria, Mexico 1980 0609 6.4 4 4Mammoth Lakes 1980 0611 5.0 9Irpinia, Italy 1980 1123 6.9 5 9Irpinia, Italy 1980 1123 6.2 5 9Taiwan (SMART#5) 1981 0129 5.7 6 7Corinth, Greece 1981 0224 6.7 1 1Westmorland 1981 0426 5.8 3 6Coalinga 1983 0502 6.4 45 46Coalinga 1983 0509 5.0 20Coalinga 1983 0611 5.3 5Coalinga 1983 0709 5.2 13Coalinga 1983 0722 5.8 8 11Coalinga 1983 0722 4.9 2Borah Peak, ID 1983 1028 6.9 2 2Borah Peak, ID 1983 1029 5.1 2 2Morgan Hill 1984 0424 6.2 23 26Lazio-Abruzzo, Italy 1984 0507 5.9 3Bishop (Rnd Val) 1984 1123 5.8 1Nahanni, Canada 1985 1223 6.8 3 3Hollister 1986 0126 5.4 3Taiwan (SMART#40) 1986 0520 6.4 8 8

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 127tion stations. The SGS classification system differentiates between rock (site class B, soildepth 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 significantdifference between strong motions recorded at rock versus shallow soil sites, and there-fore, 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 therecent Kocaeli (1999) and Duzce (1999) earthquakes, and Stewart et al. (2001b) pro-vided classifications for the remaining stations in the Los Angeles Basin and SanFrancisco Bay Area. For the data recorded during the Chi-Chi (1999) earthquake, the

Table 1. (cont.). Data set used for regression analysis

Earthquake Date MwNo. of records

Tm and Tavg

No. of recordsTo

N. Palm Springs 1986 0708 6.0 10 32Chalfant Valley 1986 0720 5.9 5Chalfant Valley 1986 0721 6.2 9 11Chalfant Valley 1986 0721 5.6 3Chalfant Valley 1986 0731 5.8 2Whittier Narrows 1987 1001 6.0 16 114Whittier Narrows 1987 1004 5.3 11Superstition Hills 1987 1124 6.7 4 6Superstition Hills 1987 1124 6.3 1 1Spitak, Armenia 1988 1207 6.8 1Loma Prieta 1989 1018 6.9 56 56Griva, Greece 1990 1221 5.9 1Erzican, Turkey 1992 0313 6.9 1 1Roermond, Netherlands 1992 0413 5.3 1Cape Mendocino 1992 0425 7.1 5 6Landers 1992 0628 7.3 56 68Northridge 1994 0117 6.7 106 140Kobe, Japan 1995 0116 6.9 11 11Kozani, Greece 1995 0515 6.6 2Kozani, Greece 1995 0517 5.3 1Kozani, Greece 1995 0519 5.1 1 2Dinar, Turkey 1995 1001 6.4 1 3Kocaeli, Turkey 1999 0817 7.4 20 20Chi-Chi, Taiwan 1999 0920 7.6 306 306Duzce, Turkey 1999 1112 7.1 13 21

128 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAYLee et al. (2001) site classification was employed with the following qualitative descrip-tions: (1) class B includes Miocene and older strata, limestone, igneous, and metamor-phic rock, (2) class C includes Pliocene and Pleistocene strata, conglomerates, pyroclas-tic rocks, and geomorphologic lateritic terraces, and (3) class D includes late Pleistoceneand Holocene strata, geomorphologic fluvial terraces, stiff clays, and sandy soils withaverage SPT>15 in the upper 30 m. The distribution of data across site classes for theTm , 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 fre-quency content parameters, a theoretical earthquake point source model that predicts aFourier Amplitude Spectrum was used. Although the functional form developed from thepoint source model is only theoretically applicable to Tm , it has been found that thisfunctional form works well for Tavg and To also (Rathje et al. 1998).

The Brune (1970, 1971) v2 point source model was employed to consider the theo-retical magnitude and distance dependence for Tm . The Fourier amplitudes (cm/s) pre-dicted by the Brune (1970, 1971) model are given by

C~f !50.78f2

11S ffcD2

MoR

exp~2pkf !expF2 pfRboQ~f !G (4)with fc54.9310

6bo~Ds/Mo!1/3 (5)

log Mo51.5Mw116.05 (6)

Q~f !5Q0fn (7)

Figure 2. Comparison of Tm and Tavg , To , and Rathje et al. (1998) data sets.

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 129where f represents frequency (Hz); fc is the corner frequency (Hz); Mo is the seismicmoment (dyne-cm); Mw is moment magnitude; R is distance from the point source (km);k is a factor that represents the damping of seismic waves as they propagate through thecrust (s); bo is the shear wave velocity of the crust at the source depth (km/s); Ds rep-resents the stress drop of the source (bars); and Q(f ) is a frequency dependent qualityfactor, representing inelastic attenuation in the crust.

Using source parameters appropriate for the western United States (Ds580 bars,

Figure 3. Distribution of data for (a) Tm and Tavg data set and (b) To data set.

130 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAYbo53.2 km/s, k50.035 s, Qo5300, n50.6; Boore and Joyner 1997; Abrahamson, per-sonal communication 1997), Equations 4 through 7 were used to generate Fourier Am-plitude Spectra for earthquake magnitudes between 5 and 8 and distances between 1 and100 km. The resulting Fourier Amplitude Spectra were used to compute Tm for each Mwand R pair. The theoretical distance and magnitude dependencies developed from thisprocedure are shown in Figure 4. The data in Figure 4 are plotted on a semi-log axis andindicate that the distance dependence for ln(Tm) is essentially linear. The magnitude de-pendence for ln(Tm) is nonlinear (Figure 4b), with a linear dependence at smaller mag-nitudes and almost no magnitude dependence at magnitudes greater than 7.5. There is nomagnitude dependence at larger magnitudes because at these magnitudes the corner fre-quency (fc), which controls the low frequency components of motion, is outside the fre-quency range used for the Tm computation. Consequently, the additional low frequencyenergy generated by earthquakes greater than 7.5 is outside the frequency range of theTm calculation and does not affect Tm . The nonlinear magnitude dependence can be ap-proximated by a linear relationship at magnitudes less than 7.25 and a constant relation-ship at magnitudes greater than 7.25. Rathje et al. (1998) inferred linear distance andmagnitude 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 de-pendencies were used in this study because they facilitate the regression analyses. Ad-ditionally, 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 dis-tance and magnitude dependencies to be incorporated in the empirical model. However,

Figure 4. Theoretical (a) distance and (b) magnitude dependencies for Tm .

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 131the point source model does not offer information regarding the effect of site conditionsor fault rupture directivity. Previous studies and observations from recorded strongground motions were used to develop the methodology for treatment of site conditionsand fault rupture directivity.

Site conditions significantly affect the frequency content of strong ground motionsbecause the dynamic response of soil sites enhances the long period components ofground shaking. Rathje et al. (1998) used only two site categories (rock/shallow soil anddeep soil) and found that strong ground motions at deep soil sites have larger values ofTm and To than strong motions at rock sites. These larger values of Tm and To are adirect result of the dynamic response of deep soil deposits. Observations from the 1999Chi-Chi earthquake (Faraj 2002) show a significant difference between the frequencycontents of motions recorded at rock sites (Site Class B) and motions recorded at shal-low soil sites (Site Class C). Similar observations have been made for spectral accelera-tion (Rodriguez-Marek et al. 2001). As a result, this study used three categories to dis-tinguish site conditions (Brock, Cweathered/soft rock and shallow stiff soil, andDdeep stiff soil). The function incorporated in the empirical model that accounts forsite conditions is not distance or magnitude dependent.

Previous research (e.g., Somerville et al. 1997) has shown that fault rupture direc-tivity affects the amplitudes and durations of strong ground motions. When the fault rup-ture propagates towards a site and the slip direction is aligned with the rupture direction(called forward directivity), recorded motions generally exhibit three distinct character-istics due to constructive interference of shear waves. These three characteristics are (1)enhanced long period motion, (2) fault normal components of motion greater than faultparallel components of motion at long periods, and (3) shorter duration. The change instrong motion amplitudes at long periods affects the frequency content of strong mo-tions, and thus should affect the scalar frequency content parameters. This study willconsider only the effect of the enhanced long period motion on the frequency contentparameters, and this effect is primarily a result of the fault rupture propagating towardsa 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 azimuthangle and length ratio definitions for dip-slip and strike-slip faulting are illustrated inFigure 5. For the purposes of this study, forward directivity motions will be identifiedsolely by their geometric location with respect to the fault rupture. Specifically, record-ings will be designated as forward directivity motions if more than one-half of the faultruptured towards the site (length ratio >0.5) and the azimuth is less than or equal to 30degrees. This strict geometric definition of forward directivity will result in some mo-tions being classified as forward directivity although they do not display classic forwarddirectivity characteristics. However, using a consistent spatial definition of forward di-rectivity will provide an unbiased estimate of the frequency content parameters in thezone of expected forward directivity.

To evaluate whether forward rupture directivity significantly affects scalar frequencycontent parameters, recorded strong ground motions from the 1979 Imperial Valley

132 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAY(Mw56.5) earthquake, the 1989 Loma Prieta (Mw56.9) earthquake, and the 1994Northridge (Mw56.7) earthquake were considered. For these earthquakes, recordingswere designated forward directivity motions based on their geometric locations, as dis-cussed previously. The azimuth angles and length ratios for the recordings were obtainedfrom Stewart (personal communication 2002). For each motion, the residual with respectto the Rathje et al. (1998) relationship [ln(recorded Tm) ln(1998 model)] was com-puted and plotted versus distance. These data are shown in Figure 6, along with meanvalues computed within overlapping distance bins. The data indicate a significant underprediction of Tm (i.e., residual greater than zero) by the Rathje et al. (1998) model withinabout 15 to 20 km of the fault. In this distance range, the variation of the residuals withdistance is approximately linear. At distances greater than 20 km, the mean of the re-

Figure 5. Fault rupture directivity parameters (azimuth and length ratio) for dip-slip and strike-slip faults (adapted from Somerville et al. 1997).

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 133siduals is close to zero, indicating no significant directivity effects beyond 20 km. Simi-lar trends were observed for Tavg and To . In addition to the results shown in Figure 6, theentire Tm data set was evaluated to assess whether the forward directivity motions (asdefined above) were significantly different from the rest of the data set. The results in-dicated these data were significantly and statistically different (Faraj 2002). Based onthese observations, a forward directivity function that decays linearly with distance (upto 20 km) was incorporated in the empirical model. Based on these criteria, 98 record-ings within the data set were identified as forward directivity motions. These recordingsencompass earthquake magnitudes between 6.0 and 7.6, distances between 0.1 and 20.0km, 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 theregression coefficients for the empirical models for the three frequency content param-eters. Random-effects modeling has been used previously to develop empirical attenua-tion relationships for spectral acceleration (e.g., Abrahamson and Youngs 1992, Abraha-mson and Silva 1997). In random-effects modeling, the error is divided into intra-eventand inter-event terms. The intra-event residual () represents the difference between anysingle data point and the median prediction for that event, while the inter-event residual(h) represents the deviation of the median prediction for a single event from the medianprediction based on the entire data set. The intra-event and inter-event error terms areassumed normally distributed with mean of zero and standard errors of s and t, respec-tively. The total error for the model is computed as stotal5As21t2. The statistical data

Figure 6. Tm residuals versus distance for Imperial Valley, Loma Prieta, and Northridge earth-quakes.

134 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAYanalysis software R (R 1.4.0A Programming Environment for Data Analysis andGraphics, 2001) was used for all random-effects regression analyses. This program con-tains a built in subroutine for random-effects modeling.

The general functional forms incorporated in this study are

ln~T!5c11c2~Mw26!1c3R1c4SC1c5SD1c6~12R/20!FD (8)

for 5.0

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 135To , 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 amagnitude of 7.6 be used.

The regression coefficients for the empirical models for Tm , Tavg , and To are listed inTable 2, along with the standard errors for each coefficient. Each regression coefficientis statistically significant at a level smaller than 0.0001 (p,0.0001 that the coefficient isequal to 0, using hypothesis testing and the statistical t distribution, Devore 1991), ex-cept for coefficient c4 for Tm and Tavg . These parameters are not as statistically signifi-cant (p;0.06), but the authors believe that c4 is sufficiently significant to remain in thepredictive equations. The coefficient c4 controls the site effect for shallow soil sites (SiteClass C). The smaller statistical significance of c4 for Tm and Tavg indicates that shallowsoil sites do not change Tm and Tavg as significantly as shallow soil sites change To . This

Figure 7. Intra-event Tm residuals versus magnitude and distance for site classes B, C, and D.

136 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAYresult is expected because Tm and Tavg are affected most by the long period energy in astrong motion and shallow soil sites tend to amplify short periods more than long peri-ods.

The intra-event residuals for Tm are plotted versus magnitude and distance for thethree site classes in Figures 7af. As a whole, the intra-event residuals have a mean ofzero and standard deviation of s. However, the data in Figure 7 indicate that the intra-event standard deviation varies with site class, with more scatter observed for Site ClassB than for site classes C or D. Computing the standard deviation of the intra-event Tmresiduals for each site class, Site Class B displays the largest error (sB50.42), followedby Site Class C (sC50.38) and Site Class D (sD50.31). The statistical F test was usedto evaluate whether the standard errors for each site class were statistically different (De-

Figure 8. Intra-event Tavg and To residuals versus magnitude and distance for site classes B, C,and D.

Table 3. Intra-event and inter-event standard error terms for Tm , Tavg , andTo

Error Termsfor Tm

Error Termsfor Tavg

Error Termsfor To

sB 0.42 0.38 0.38sC 0.38 0.36 0.33sD 0.31 0.29 0.31

t (All site classes) 0.17 0.13 0.22

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 137vore 1991). A probability level of 0.05 was chosen to define statistical significance. Thedifference between sB and sC is not statistically significant (p50.12 that sB

2 5sC2 ), but

the differences with Site Class D are statistically significant (p,1024 that sB2 5sD

2 , sC2

5sD2 ). 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 sitecategory. In contrast, the deep stiff soil sites of Site Class D cover a relatively smallerrange of shear wave velocities and deep soil deposits provide a consistent filter to in-coming ground motions, resulting in a less variable response and smaller s. The intra-event residuals for Tavg and To are plotted versus magnitude and distance for the threesite classes in Figure 8, and the intra-event standard error terms for each site class arelisted in Table 3. The intra-event error terms for Tavg are slightly smaller than for Tm .However, the trends regarding the relative differences of sB , sC , and sD for Tavg aresimilar as for Tm , with no statistical significance between sB and sC (p50.23 that sB

2

5sC2 ) and statistically significant differences with sD (p,10

24 that sB2 5sD

2 , sC2 5sD

2 ).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, sB is statistically significant fromthe others (p50.02 that sB

2 5sC2 , p;1024 that sB

2 5sD2 ), but sC and sD are not statisti-

cally significant from one another (p50.08 that sC2 5sD

2 ).

The total standard error for a random-effects model also incorporates the inter-eventerror, t. The inter-event residuals for Tm , Tavg , and To are plotted versus magnitude inFigure 9 and the inter-event error terms (t) for each parameter are listed in Table 3. Theinter-event residuals in Figure 9 display significantly less variability than the intra-event

Figure 9. Inter-event residuals versus magnitude for Tm , Tavg , and To .

138 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAYresiduals in figures 7 and 8. As a result, the values of t are much smaller than the valuesof s (Table 3), which is consistent with results from attenuation relationships for spectralacceleration (e.g., Abrahamson and Silva 1997). Tavg displays the smallest inter-eventerror (t50.13), while Tm and To exhibit larger values of inter-event error (t50.17 and0.22, respectively). A reduction in s and t with magnitude was also investigated basedon the observations of Youngs et al. (1995), but no significant trends were observed.

Given the intra-event error term for the appropriate site class (ssiteclass) and the inter-event error term (t) from Table 3, the total standard error can be computed as

stotal5Assiteclass2 1t2 (10)

where siteclass5B, C, or D

DISCUSSION

The predicted median relationships for Tm , Tavg , and To versus distance are shown inFigure 10 for Site Class B and magnitudes 5.5, 6.5, and 7.5. Tm and Tavg generally takeon the same values at smaller magnitudes, but Tavg displays a larger magnitude depen-dence. To is much smaller than Tm and Tavg at each distance and magnitude, which isexpected 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 thanthat for Tm . These differences are indicated by the spacing between the curves for eachmagnitude in Figure 10 and the regression coefficient c2 in Table 2. The larger magni-tude dependence for To and Tavg versus Tm may indicate that the response spectrum ismore affected by earthquake magnitude than the Fourier Amplitude Spectrum.

The empirical relationships for Tm shown in Figure 10 are different than the theoret-ical relationships shown in Figure 4. At Mw55.5 the relationships are similar, but the

Figure 10. Median predictions of Tm , Tavg , and To for Site Class B and different magnitudeevents.

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 139theoretical relationships predict larger values of Tm at larger magnitudes. This differencecan be attributed to the Brune source spectrum used for the theoretical relationship. TheBrune spectrum assumes a point source, which overpredicts the Fourier amplitude coef-ficients at long periods (Boore 1983, Atkinson and Silva 1997). The overpredicted Fou-rier amplitudes produce larger theoretical Tm values that are not realistic. The empiricaldata confirm this assertion.

The effect of forward directivity on the predicted Tm relationship is illustrated in Fig-ure 11. At distances less than 20 km, the predicted Tm for a forward directivity motion isas much as 50% larger than for a non-directivity motion. This difference is largest closeto the fault rupture plane. Because Tm and Tavg better incorporate the long period com-ponents of motion, they are more affected by forward directivity than To . This result isapparent from the larger c6 regression coefficients for Tm and Tavg than for To (c6 con-trols the directivity effect, equations 8 and 9, Table 2). It is commonly assumed that for-ward 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 thestatistical significance of regression coefficient c6 (Table 2). This trend is most likely at-tributed to the fact that spectral acceleration at a given period is affected by the motionat a range of periods centered about the given period. If forward directivity affects somecomponents of motion within that range of periods, To will be affected.

Figure 12 provides a comparison of the relationships developed in this study withthose developed by Rathje et al. (1998). The effect of site conditions on predicted valuesof Tm for a magnitude 7 event is shown in Figure 12a. For the current study, Site ClassC displays a slightly larger value of Tm than Site Class B. However, the difference ismore significant for Site Class D, where the enhanced long period energy generated by

Figure 11. Effect of forward directivity on Tm values.

140 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAYthe dynamic response of deep soil sites results in a 30% increase in Tm compared withSite Class B. The site class effect for Tavg is similar to that of Tm , but the site class effectis larger for To . Site effects for each frequency content parameter can be comparedthrough the regression coefficients c4 and c5 in Table 2. Based on these coefficients, Tois 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 systemthat combined rock (Site Class B) and shallow soil/weathered rock (Site Class C) into asingle rock category. As a result, the rock results from Rathje et al. (1998) agree bestwith the Site Class C results from this study. Site Class B falls about 10% below the

Figure 12. Comparison of relationships from this study and Rathje et al. (1998).

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 141previous rock/shallow soil relationship. Because the 1998 deep soil category is consis-tent with the current Site Class D category, these relationships compare favorably. Com-parisons between the current and 1998 studies for To are similar to those for the Tm re-lationships.

The magnitude dependencies for Tm , Tavg , and To are shown in Figure 12b for SiteClass D and R520 km. Site Class D was used for this comparison because this site classis most consistent between the current study and the 1998 study. The new Tm relation-ship is very similar to the 1998 relationship, despite the addition of over 500 new mo-tions. The change in the To relationship is more dramatic. The To relationship shifteddown at small magnitudes, without significant shifting at larger magnitudes. As a result,the current relationship indicates that To varies with magnitude considerably more thanTm . This observation is in contrast with the Rathje et al. (1998) study, which found thatTm increased with magnitude at a faster rate than To . The discrepancy is most likely theresult of using different data sets for Tm versus To in this study, rather than using thesame 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 Mw,6.5 than the To data set and there is almost no Tm and Tavg data for Mw,5.5 (Figure 2).The difference in the data sets is a direct result of the filter frequency requirements forTm and Tavg , particularly the high-pass filter frequency constraint that requires a high-pass filter less than 0.3 Hz. Most motions from small magnitude events do not meet thisconstraint because they do not contain significant low frequency (i.e., long period) en-ergy, and therefore require high-pass filters greater than 0.3 Hz to eliminate low fre-quency noise. The few small magnitude motions that do meet the Tm and Tavg high-passfilter requirement tend to have significant long period energy for that magnitude, other-wise they would have required a high-pass filter greater than 0.3 Hz. Consequently, theTm and Tavg data may be somewhat biased towards larger Tm and Tavg values at smallmagnitudes. When the filter-frequency requirements were dropped for To , the number ofsmall magnitude motions available for the regression increased considerably (Figure 2).The addition of these motions from small magnitude events resulted in smaller predictedvalues of To at small magnitudes for this study compared with the 1998 study (Figure12b). These differences indicate that the filter-frequency requirement somewhat biasesthe Tm and Tavg relationship at small magnitudes. Again, it should be noted that a similarfilter-frequency requirement has been incorporated by others (e.g., Abrahamson andSilva 1997) in developing attenuation relationships for spectral acceleration and theserelationships at long periods are similarly biased.

CONCLUSIONS

The frequency content of an earthquake ground motion is an important ground mo-tion parameter because it affects the dynamic response of geotechnical and structuralsystems. A scalar representation of frequency content is useful in earthquake engineer-ing practice to evaluate possible conditions for dynamic resonance or enhanced dynamicresponse. Several recently developed seismic design procedures have incorporated scalarfrequency content parameters (e.g., Bray et al. 1998, Stewart et al. 2003).

142 E. M. RATHJE, F. FARAJ, S. RUSSELL, AND J. D. BRAYThis paper described the development of empirical relationships that predict threefrequency 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 theFourier Amplitude Spectrum, while Tavg and To are based on the acceleration responsespectrum. The predominant spectral period (Tp) was also considered, but this parameterdid not adequately describe the frequency content of a strong ground motion. The func-tional form of the empirical relationship for Tm was developed from a theoretical pointsource model that predicts the Fourier Amplitude Spectrum as a function of magnitudeand distance. This functional form was further extended to account for the effects of siteconditions and fault rupture directivity. The recorded data supported the use of three siteclasses: rock (Site Class B), weathered/soft rock and shallow soil (Site Class C), anddeep soil (Site Class D). Additionally, near-fault strong ground motion recordings indi-cated a significant increase in Tm for sites experiencing forward directivity. Similar ob-servations regarding site conditions and directivity were made for Tavg and To . The samefunctional form developed for Tm was used for Tavg and To , except for a magnitude cut-off.

The developed predictive relationships expand and enhance the relationships devel-oped 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 therecent large magnitude earthquakes in Turkey and Taiwan. Three site classes were incor-porated rather than two, and the effects of fault rupture directivity were included. Addi-tionally, the random-effects model was used in the regression, which allows the variabil-ity within events (intra-event variability) and between events (inter-event variability) tobe 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 forwarddirectivity. Based on the magnitude distribution of the various data sets, the Tm and Tavgrelationships are most reliable for Mw55.57.6 and the To relationship is most reliablefor Mw54.77.6. The intra-event error terms (s) vary as a function of site class, withSite Class D displaying the least intra-event variability. The inter-event error terms (t)indicate that Tavg has the smallest inter-event variability. Neither s nor t were observedto vary with earthquake magnitude. Each scalar frequency content parameter in thisstudy 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 wherethis frequency range is of interest (e.g., nuclear reactors). Tm and Tavg best account forthe long period content of strong motions, and they best differentiate the long periodcontent between different motions. Although these parameters are based on differentrepresentations of a strong ground motion (Fourier Amplitude Spectrum vs. accelerationresponse spectrum), they take on similar values and are similarly affected by site classand directivity. However, the Fourier Amplitude Spectrum is a more robust representa-tion of a strong motion, and therefore, Tm is considered the preferred frequency contentparameter.

ACKNOWLEDGMENTS

Financial support was provided through the Pacific Earthquake Engineering Re-search Center Lifelines Program under projects 1C01. Additional support was provided

EMPIRICAL RELATIONSHIPS FOR FREQUENCY CONTENT PARAMETERS OF EARTHQUAKE MOTIONS 143by the National Science Foundation through an REU (Research Experience for Under-graduates) supplement under grant CMS-9875430 and by the David and Lucile PackardFoundation. The authors also wish to thank Dr. Walt Silva of Pacific Engineering andAnalysis 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 classifica-tion information, and Dr. Jonathan Stewart of UCLA for providing directivity param-eters for the strong motion database. Particularly insightful comments were provided byseveral anonymous reviewers.

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(Received 6 September 2002; accepted 20 June 2003)