Effect of Finite Frequency Bandwidth Limitation on Evaluations of Seismic Radiation Energy of the 1999 Chi-Chi Earthquake

1 Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan, ROC 2 Institute of Geophysics, National Central University, Chung-Li, Taiwan, ROC 3 National Science and Technology Center for Disaster Reduction, Taipei, Taiwan, ROC * Corresponding author address: Dr. Jeen-Hwa Wang, Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan, ROC; E-mail: jhwang@earth.sinica.edu.tw doi: 10.3319/TAO.2007.18.3.567(T) Based on the ω and ω source models, we explore the effect on estimates of seismic radiation energy, Es , caused by finite frequency bandwidth limitation of source spectra. Let fc be the corner frequency of a source spectrum and fu and fc are, respectively, the upper and lower bounds of a frequency band in use. Results show that the effect depends on fu/fc and fl/fc , and Es is under-estimated when fl > 0 and fu < ∞. When fu/fc < 20, the effect is sensitive to both fl/fc and fu/fc for the ω −2 source model, but mainly to fl/fc for the ω model. When fu/fc > 20, the effect is insensitive only to fu/fc for the two models. Let Es’ be the seismic radiation energy estimated without removal of finite frequency bandwidth limitation. Results show: (1) Es’/Es first slightly increases and then decreases with increasing fc ; or (2) Es’/Es monotonously decrease with increasing fc. For the 1999 Ms 7.6 Chi-Chi earthquake, Taiwan, Es was under-estimated by Hwang et al. (2001), and the degree of under-estimates varies from station to station. (


INTRODUCTION
Seismic radiation energy, E s , is an important parameter quantifying an earthquake (cf.Wang 2006).However, estimates of E s can be influenced by the source spectrum, seismic radiation patterns, seismic-wave attenuation, surface amplification, site effect, instrumental response, and noise.A correct evaluation of E s will help seismologists to understand source behavior more exactly.Boore (1988), Di Bona and Rovelli (1988), and Singh and Ordaz (1994) stressed that E s is underestimated when high-frequency signals are not included.Thus, the E s measured from local seismograms is usually larger than that done from teleseismic data (Bolt 1986;Smith et al. 1991;Singh and Ordaz 1994;Hwang et al. 2001;Huang et al. 2002).In principle, E s is measured for f = 0 -∞ Hz, while in practice the measurement can be made only for f 1 f f u due to limitation in instrumental response and noise.This results in so-called finite frequency bandwidth limitation (denoted ffbl hereafter).Ide and Beroza (2001) theoretically studied such an effect in a high-frequency regime.Based on the ω −2 source model, Wang (2004) studied the effect in both low-and high-frequency regimes.Both studies show underestimation of E s due to the ffbl-effect.
For the 1999 M s 7.6 Chi-Chi earthquake, Wang (2004) made corrections only based on the ω −2 source model.However, Huang and Wang (2002) stressed that a ω −3 model must be taken into account for the northern fault plane of the earthquake.In this work, the ffbl-effects of source spectrum for both high-and low-frequency regimes on estimates of E s based on the ω −2 and ω −3 source models (Aki 1967; Brune 1970) will be discussed in detail.Theoretical results will be applied to correct estimates of the seismic radiation energy of the Chi-Chi earthquake.

DEFINITION AND METHODOLOGY FOR MEASURING E S
The source spectra of earthquakes are mainly controlled by the low-frequency spectral level ( Ω o ) and corner frequency (f c ) (Aki 1967).Theory and observations show that when f > f c , the spectral amplitude decays in a power-law function like f −α .Commonly accepted power- law functions have either f -2 or f -3 , which are, respectively, referred to as the ω −2 and ω −3 source models, where ω π = 2 f (Aki 1967;Brune 1970).Let d(t) and v(t) be the source displacement and velocity, respectively.Their Fourier transforms are, respectively, D(f) and V(f).D(f) can be approximated by D + for the ω −3 one (cf.Beresnev and Atkinson  1997).Hence, the approximations of V(f) are, respectively: (1) ~ (2) can be approximated individually by a piece-wise linear function (Fig. 1).E s is calculated by the following expression: where ρ and β are, respectively, the density and the S-wave velocity.In principle, the first integral is performed from −∞ to +∞ in the time domain and the second one from 0 to +∞ in the frequency domain.Define: Thus, E I s V = 4πρβ .Wang (2004) derived the formulas to show the ffbl-effect based on the ω −2 model.For the purpose of comparison, his formulas are shown again below.In the following, we add a subscript 'o' to denote a quantity obtained through integration from −∞ and +∞ sec in the time domain or from 0 to ∞ Hz in the frequency domain.Inserting Eqs. ( 1) and (2) into Eq.( 4), respectively, leads to: where the subscript is 2 for the ω −2 model and 3 for the ω −3 model.Clearly, I V2o = 4I V3o .When integration is made only in a finite frequency band from f l to f u , with f l < f c < f u , which is in between two dashed-dotted lines as shown in Fig. 1, the ffbl-effect exists.When f c /f l = f u /f c , for the ω −2 model the high-frequency cut-off part with f > f u is almost equal to that from the low- frequency one with f < f l ; while for the ω −3 model the former is smaller than the latter.Inserting Eqs. ( 1) and (2) into Eq.( 4), with f l < f c < f u , respectively, gives:

THE EFFECT DUE TO FINITE FREQUENCY BANDWIDTH LIMITATION
where the integral range is of from f 1 to f u .After integration, Eqs. ( 7) and ( 8), respectively, becomes: where When f l = 0 and f u → ∞, F V2 = 1 and F V3 = 1, and, thus, I V2 = I V2o and I V2 = I V3o .
Hereafter, let E s and E s ' be the values of seismic radiation energy estimated, respectively, with and without removal of the ffbl-effect.From Eqs. ( 9) -( 12), the energy ratio of E s ' to E s is: for the ω −2 model and: for the ω −3 model.The variations of E s2 '/E s and E s3 '/E s with f l /f c are made only for f l /f c < 1 and f l /f c > 1 under the request of f l < f c < f u .In other words, the calculations are made when f l / f c = 0.05 -0.95 and f u /f c = 2 to 20, with a difference of 2. The plots for ten values of f u /f c are shown, respectively, in Fig. 2 (for E s2 '/E s ) and Fig. 3 (for E s3 '/E s ), where the dotted line displays the energy ratio of 1, without ffbl.In Figs. 2 and 3, all curves are below the dotted line with E s '/E s = 1, and, thus, E s2 '/E s and E s3 '/E s are both smaller than 1, with a maximum of about 0.937 for E s2 '/E s and 0.999 for E s3 '/E s .Obviously, the ffbl-effect yields an under-estimation of seismic radiation energy.E s2 '/E s and  E s3 '/E s both decrease with increasing f l /f c , and the amount of the decreasing rate increases with f l /f c .For fixed f c , decreases in E s2 '/E s and E s3 '/E s with increasing f l /f c lead to increases in E s2 '/E s and E s3 '/E s with decreasing f l .This indicates that an increase in the width of the low-frequency regime improves estimation of E s .When f l /f c < 0.4 for E s2 '/E s and f l /f c < 0.2 for E s3 '/E s , the curves are almost flat for all f u /f c .This means that f l = 0.4f c for E s2 '/E s and f l = 0.2f c for E s3 '/E s are the individual optimum lower bounds to lead to a stable value of E s .E s2 '/E s and E s3 '/E s both increase with f u /f c .The curves are close to one another for E s2 '/E s when f u /f c 10 and for E s3 '/E s when f u /f c 4, thus indicating that f u = 10f c for E s2 '/E s and f u = 4f c for E s3 '/E s are both large enough to lead to a stable estimate of E s .For fixed f c , increases in E s2 '/ E s and E s3 '/E s with f u /f c yield increases in E s2 '/E s and E s3 '/E s with f u , thus indicating that an increase in the width of high-frequency regime improves estimates of E s .This is consistent with others' (Boore 1988;Di Bona and Rovelli 1988;Singh and Ordaz 1994;Ide and Beroza 2001).
Figures. 2 and 3 show that for fixed f l , decreases in E s2 '/E s and E s3 '/E s with increasing f l /f c lead to increases in E s2 '/E s and E s3 '/E s with f c , thus implying that the ffbl-effect in the lowfrequency regime gives a greater underestimate of E s for events with lower f c than for those with higher f c .This effect is stronger for the ω −3 model than the ω −2 model.For fixed f u , increases in E s2 '/E s and E s3 '/E s with f u /f c result in increases in E s2 '/E s and E s3 '/E s with decreasing f c , thus showing that the ffbl-effect in the high-frequency regime yields a bigger underestimate of E s for events with higher f c than for those with lower f c .When both f l and f u are finite and fixed, an increase in f c will lead to a decrease in both f l /f c and f u /f c .Hence, the variation of E s2 '/E s and E s3 '/E s with f c can be either of the following two types: (1) the ratio first slightly increases and then decreases with increasing f c ; and (2) the ratio monotonously decrease with increasing f c .

RE-EVALUATION OF E S OF THE 1999 CHI-CHI EARTHQUAKE
The M s 7.6 Chi-Chi earthquake, which ruptured the Chelungpu fault, struck central Taiwan on 20 September 1999.The epicenter and the fault trace are displayed in Fig. 4. The values of f c and Ω o at four near-fault stations evaluated by Hwang et al. (2001) are f c = 0.064 -0.193 Hz and Ω o = 89.4 -2350.0cm.The values of f c and Ω o are shown in columns 2 and 3 of Table 1.They also estimated the values of E s , which is equivalent to E s2 ' for the ω −2 model and E s3 ' for the ω −3 source model in this study and denoted by E s ' in column 8 of Table 1, at four near-fault seismic stations (see Fig. 4) based on two sets of f l and f u : (1) f l = 0.03 and f u = 1.0 Hz at TCU102 and TCU052; and (2) f l = 0.03 and f u = 3.0 Hz at TCU076 and TCU129.The values of f l and f u used are shown in columns 4 and 5 of Table 1.In order to obtain a reliable value of E s , they eliminated the effects caused by seismic radiation patterns, seismic-wave attenuation, surface amplification, site effect, and instrumental response.Wang (2004) re-evaluated the values of E s estimated by Hwang et al. (2001) through the removal of the ffbl-effect based on the ω −2 model.His values of E s2 '/E s and Es are shown in parentheses of columns 9 and 10 in Table 1.
From the values of f c , f l , and f u at the four stations, the ratios of f l /f c and f u /f c are calculated and given in column 6 and 7 of Table 1: f l /f c of from 0.155 to 0.469 and f u /f c of from 8.197 to Table 1.The values of several parameters at four near-fault seismic stations.In columns 9 and 10, E s '/E s and E s , respectively, includes E s2 '/E s and E s for the ω −2 model and E s3 '/E s and E s for the ω −3 model.The values of E s '/E s and E s not inside the parenthesis are, respectively, E s2 '/E s and E s taken from Wang (2004).The values of E s '/E s and E s inside the parenthesis are, respectively, E s3 '/E s and E s of this study.
18.750.The values of E s2 '/E s and E s re-evaluated by Wang (2004) based on the ω −3 model are shown in the parentheses of columns 9 and 10 in Table 1.Clearly, the ffbl-effect results in an underestimate of E s , and the underestimate is higher at two northern stations than at the southern ones.We calculate the values of E s '/E s and E s at two northern stations using Eqs.( 12) and ( 13) based on the ω −3 model.Results are shown in the parentheses of columns 9 and 10 of Table 1.
Obviously, the results are opposite to those evaluated based on the ω −2 model.The difference is bigger at TCU102 and smaller at TCU052.Based on the ω −3 model, the value of E s at TCU102 estimated by Hwang et al. (2001) is good enough.
To examine the problem in advance, we plot the variations of energy ratio with f c in the range 0.05 -0.20 Hz in Fig. 5 for two sets of f l and f u : (1) f l = 0.03 and f u = 1.0 Hz for northern seismic stations; and (2) f l = 0.03 and f u = 3.0 Hz for southern ones.The dashed and solid lines represent, respectively, E s2 '/E s and E s3 '/E s for the northern stations; and the dashed-dotted line shows E s2 '/E s for the southern stations.The estimated results are also plotted by an open circle or a cross attached with a station code in Fig. 5.
In Fig. 5, E s2 '/E s first increases and then decreases with increasing f c .Whereas, E s3 '/E s first increases with f c and then becomes flat when f c > 0.14 Hz.The variations are as expected as mentioned above.The three variations are all below the dotted line with E s '/E s = 1, thus showing underestimate of E s at the four seismic stations.The solid line intersects the dashed and dashed-dotted ones at f c = 0.065 and f c = 0.090 Hz, respectively.Hence, at the northern stations the underestimate of E s is smaller from the ω −2 model than from the ω −3 model when f c < 0.065 Hz, and opposite when f c > 0.065 Hz.The difference between the effects from the two models is small at TCU052 and large at TCU102.Underestimation of E s is smaller at the southern stations than at the northern ones when f c < 0.09 Hz, and opposite when f c > 0.09 Hz.

CONCLUSION
The ffbl-effect of source spectrum on estimation of seismic radiation energy, E s , is analyzed theoretically on the basis of the ω −2 and ω −3 source models.Such an effect depends on f l /f c and f u /f c .Numerical results obviously show that E s are underestimated for all f l /f c and f u /f c .An increase in the frequency bandwidth including either the high-or low-frequency regime will increase reliability of estimating E s .When f u /f c < 20, the effect is sensitive to both f l /f c and f u /f c for the ω −2 model, but mainly to f l /f c for the ω −3 model.When f u /f c > 20, the effect is insensitive to f u /f c for the two models.For the two source models, E s '/E s depends on f c in either: (1) E s '/E s first slightly increases and then decreases with increasing f c ; or (2) E s '/E s monotonously decreases with increasing f c .Numerical results also suggest that Fig. 2 or 3 together with Fig. 5 can help us to select an appropriate frequency band for estimating a reliable value of seismic radiation energy.The values of f l and f u leading to an optimum estimate of E s are f l = 0.4f c and f u = 10f c for the ω −2 model and f l = 0.2f c and f u =4f c the ω −3 model.
For the 1999 M s 7.6 Chi-Chi, Taiwan, earthquake, the revised values of E s show that E s was underestimated by Hwang et al. (2001).However, the degree of underestimates varies from station to station.At the northern stations underestimation of E s is smaller for the ω −2 model than for the ω −3 model when f c < 0.065 Hz, and opposite when f c > 0.065 Hz.The differ- ence between the effects from the two models is small at TCU052 and large at TCU102.Underestimation of E s is smaller at the southern stations than at the northern ones when f c < 0.09 Hz, and opposite when f c > 0.09 Hz.The values of E s at the four near-field are 9.9 × 10 22 erg at TCU102, 2.9 × 10 23 erg at TCU052, 1.1 × 10 22 erg at TCU076, and 7.2 × 10 21 erg at TCU129.

Fig. 1 .
Fig. 1.The log-log plots of the normalized, simplified velocity spectra, V(f) versus frequency, f: the dashed and dotted lines, respectively, for the f -1 and f -2 source velocity models.The two vertical dashed-dotted lines display the frequency band in use.

Fig. 2 .
Fig. 2. The variations of E s2 '/E s with f l /f c (from 0.05 to 0.95) for ten values of f u /f c (from 2 to 20).The dotted line represents E s2 '/E s = 1.

Fig. 3 .
Fig. 3.The variations of E s3 '/E s with f l /f c (from 0.05 to 0.95) for ten values of f u /f c (from 2 to 20).The dotted line represents E s3 '/E s = 1.

Fig. 4 .
Fig. 4. A map showing the epicenter (in a solid star) of the 1999 Chi-Chi earthquake, Chelungpu fault (in a solid line), and four nearfault seismic stations (in solid triangles).

Fig. 5 .
Fig. 5.The variations of energy ratio with f c for various values of f l and f u as mentioned in the text: the dashed and solid lines, respectively, for E s2 '/E s and E s3 '/E s at the northern stations, and the dashed-dotted line for E s2 '/E s at the southern ones.The related values at four near-fault seismic stations for the Chi-Chi earthquake are displayed by an open circle or a cross attached with a station code.The dotted line represents E s '/E s = 1.