Short Note

The Relation between Ground Acceleration and Earthquake

Source Parameters: Theory and Observations

by Itzhak Lior and Alon Ziv

Abstract A simple relation between the root mean square (rms) of the ground ac-celeration and earthquake spectral (or source) parameters is introduced:

EQ-TARGET;temp:intralink-;;55;590 Arms ! "2π#2Ω0

f20!!!!!!!!!πκT

p"1$ πκf0

1:50:25#2;

in which Ω0 is the low-frequency displacement spectral plateau, f0 is the corner fre-quency, κ is an attenuation parameter, and T is the data interval. This result uses theomega-square model for far-field radiation and accounts for site-specific attenuation.The main advantage of the new relation with respect to that of Hanks (1979) is that itrelaxes the simplifying assumption that the spectral corner frequency is much smallerthan the maximum corner frequency resulting from attenuation, and that the spectrummay be approximated as being perfectly flat between the two frequencies. The newlyproposed relation is tested using a composite dataset of earthquake records from Ja-pan, California, Mexico, and Taiwan. Excellent agreement is found between observedand predicted ground acceleration for any combination of corner frequencies. Thus,use of the above relation enables the extrapolation of ground-motion prediction equa-tion inferred from the frequent small-magnitude earthquakes to the rare large magni-tudes. This capacity is extremely useful near slow-slip plate boundaries, where theseismic moment release rates are low.

Introduction

Understanding how ground motion depends on earth-quake source parameters is of interest to both earthquakeseismologists and earthquake engineers. The former canuse this understanding to constrain source parameters(e.g., Baltay et al., 2013; Lior et al., 2016), and the lattermay incorporate it into ground-motion prediction equa-tions that are key ingredients in any seismic-hazard assess-ment (e.g., Boore and Atkinson, 2008; Campbell andBozorgnia, 2008; Abrahamson et al., 2014). A commonlyutilized relation between ground acceleration and sourceparameters is

EQ-TARGET;temp:intralink-;df1;55;179AMHrms !

"2π#2Δτ106ρR

!!!!!!!!!!fmax

f0

s

"1#

(Hanks, 1979; McGuire and Hanks, 1980), in whichArms is the ground acceleration root mean square (rms), withthe superscript referring to McGuire and Hanks (1980), Δτ isthe stress drop, f0 is the corner frequency, and fmax is thefrequency above which the acceleration spectrum dropssteeply (Hanks, 1982). As several approximations were made

throughout the derivation of this expression, it is instructive torevisit this result and assess the extent to which relaxing thesimplifying assumptions underlying its derivation improvesground acceleration prediction.

In this study, an exact solution for Arms is obtained.Owing to its complex form, an analytical approximation tothe exact solution is provided. Finally, the performance of thenewly derived Arms prediction is compared with previous re-sults (Hanks, 1979; McGuire and Hanks, 1980; Hanks andMcGuire, 1981), using a composite dataset of earthquakerecords from Japan, California, Mexico, and Taiwan. Next,for the sake of completeness, the approach leading to equa-tion (1) is described, and the various approximations andsimplifying assumptions that enter its derivation are stated.

The Simplifying Assumptions Underlying AMHrms

The point of departure in the derivation of equation (1) isBrune’s earthquake source model (Brune, 1970), accordingto which the far-field spectral amplitude of the groundacceleration reads as

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Bulletin of the Seismological Society of America, Vol. 107, No. 2, pp. –, April 2017, doi: 10.1785/0120160251

EQ-TARGET;temp:intralink-;df2;55;733

!Ω"f # ! "2πf #2Ω0

1$ " ff0#2; "2#

in which f is the frequency, f0 is the corner frequency, andΩ0 is the spectral plateau of the low-frequency displacementspectrum. The two spectral parameters Ω0 and f0 holdfundamental information regarding the physical attributesof the earthquake source. The former is a function of the seis-mic moment and the hypocentral distance according to

EQ-TARGET;temp:intralink-;df3;55;627Ω0 !M0UφθFs

4πρC3SR

; "3#

with Uφθ being the radiation pattern, Fs being a free-surfacecorrection factor, CS being the S-wave velocity, R being thehypocentral distance, and ρ being the density. The cornerfrequency is related to the stress drop and the seismicmoment as

EQ-TARGET;temp:intralink-;df4;55;532f0 ! kCs

"16

7

ΔτM0

#1=3

; "4#

with k being a constant. From equation (2), the accelerationspectral amplitude increases as f2 at frequencies below thecorner frequency and approaches asymptotically to"2πf0#2Ω0 above it (Fig. 1). In practice, however, the accel-eration spectral plateau is attenuated due to anelastic andnear-site effects (dotted line in Fig. 1), and is truncated bythe Nyquist frequency. While for local earthquakes, the effectof the distance-dependent anelastic attenuation is usually mi-nor (Wu et al., 2005; Wu and Zhao, 2006; Lior et al., 2016),that of the near-site attenuation is not. The spectral amplitudesubject to site-specific attenuation may be expressed as

EQ-TARGET;temp:intralink-;df5;55;369

!Ω"f # ! "2πf #2Ω0

1$ " ff0#2exp"−πκf #; "5#

in which κ is a site-specific attenuation parameter (Andersonand Hough, 1984). This attenuation introduces an additionalcorner frequency that is hereafter referred to as fκ:

EQ-TARGET;temp:intralink-;df6;55;299fκ !1

πκ"6#

(Hanks, 1979). The rms of the ground-motion accelerationmay be obtained by inserting the ground acceleration spec-trum !Ω"f # into Parseval’s theorem:

EQ-TARGET;temp:intralink-;df7;55;228Arms !

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!R∞−∞ j !Ω"f#j2df

T

s

; "7#

with T being the data interval. From this point onward,Hanks (1979) and McGuire and Hanks (1980) disregardedthe possibility that f0 could be larger than fκ. By settingfκ to be equal to fmax, and approximating the accelerationspectrum as being equal to "2πf0#2Ω0 between f0 andfmax (thick black line in Fig. 1), they obtained

EQ-TARGET;temp:intralink-;df8;55;111AMH1rms ! "2π#2Ω0

f2:50!!!!T

p

!!!!!!!!!!!!!!!!!!fmax

f0− 1

s

; "8#

in which MH1 stands for McGuire and Hanks’ first set ofapproximations. For most moderate-to-large earthquakes,it is expected that fmax will be much larger than f0, in whichcase the above expression may be further simplified as

EQ-TARGET;temp:intralink-;df9;313;391AMH2rms ! "2π#2Ω0

f2:50!!!!T

p

!!!!!!!!!!fmax

f0

s

; "9#

in which MH2 signifies McGuire and Hanks’ additionalapproximation, namely fmax ≫ f0. Further setting the datainterval to be equal to the rupture duration, 1=f0, and sub-stituting equations (3) and (4) into equation (9) leads toequation (1).

In summary, equation (1) is model based (Brune, 1970),and rests on two simplifying assumptions: (1) fmax ≫ f0,and (2) the spectrum may be approximated as being equalto "2πf0#2Ω0 between f0 and fmax. Hanks and McGuire(1981) further assumed that the peak ground acceleration(PGA) occurs within 1=f0 from the time of the first S-wavearrival, and related Arms to PGA.

Exact Solution and Analytical Approximationfor Arms

The exact solution for Arms is obtained via insertion of!Ω"f # (equation 5) into equation (7) as follows:

EQ-TARGET;temp:intralink-;df10;313;133Aexactrms !"2π#2Ω0

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2

T

Z ∞

0

f4

"1$" ff0#2#2

exp"−2πκf#dfs

: "10#

The solution of the above integral is

Frequency

Hanks’ approximation

Figure 1. Schematic diagram illustrating the effect of near-siteattenuation on the acceleration spectra. The omega-square modeland the near-site attenuation are indicated by solid and dashed graylines, respectively, and the acceleration spectrum subject to near-siteattenuation is indicated by a dotted line. The horizontal thick black linebetween f0 and fκ represents Hanks’ approximation (Hanks, 1979).

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EQ-TARGET;temp:intralink-;df11;55;727Aexactrms !"2π#2Ω0

1!!!!!!2T

p"πκ#2:5

α20!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2−2α0Ci"2α0#%2α0cos"2α0#$3sin"2α0#&$%πα0−2α0Si"2α0#&%2α0sin"2α0#−3cos"2α0#&

p

"11#

(corrected equation A3 of Luco, 1985), with α0 ! πκf0 !f0=fκ, and Ci and Si being the cosine integral function andthe sine integral function, respectively. For α0 → 0, the sit-uation considered in Hanks (1979), the rms of the ground ac-celeration is asymptotic to "2π#2Ω0α20="

!!!!T

p"πκ#2:5#. Whereas

for α0 → ∞, the ground acceleration intensity is independentof f0 and is asymptotic to "2π#2Ω0

!!!!!!!1:5

p="

!!!!T

p"πκ#2:5#. Given

the complex form of the above result, it is sensible to seek asimpler expression. An analytical approximation that satisfiesthe two asymptotic solutions is

EQ-TARGET;temp:intralink-;df12a;55;550A∼exactrms ! "2π#2Ω0

α20!!!!T

p"πκ#2:5"1$ α0

1:50:25#2; "12a#

or equivalently

EQ-TARGET;temp:intralink-;df12b;55;492A∼exactrms ! "2π#2Ω0

f20!!!!!!!!!πκT

p"1$ πκf0

1:50:25#2; "12b#

with the ∼exact signifying analytically approximated exactsolution. Inspection of normalized Arms as a function of α0(Fig. 2) reveals an excellent agreement between the exactand the analytically approximated solution when f0 is muchlarger than or much smaller than fκ, and a maximum misfit of6.5% when f0 ∼ fκ. The agreement between the exact solu-tion and the two approximate solutions, MH1 (equation 8) andMH2 (equation 9), is excellent when f0 ≪ fκ (i.e., α0 → 0),and that with MH1 is still quite good when f0 ∼ fκ. Beyondtheir range of validity, in which f0 ≥ fκ (i.e., α0 > 1), MH2diverges rapidly from the exact solution, and MH1 becomesimaginary.

Finally, substituting equations (3) and (4) into (12b)yields an equivalent expression for A∼exact

rms in terms of the seis-mic moment, the stress drop, and the hypocentral distance as

EQ-TARGET;temp:intralink-;df13;55;262A∼exactrms ! β

M1=30 Δτ2=3!!!!!!κT

pR

="1$ 1:5−0:25πκkCS

"16Δτ7M0

#1=3

#2;

"13#

in which β ! 4πUϕθFs"16=7#2=3"kCS#2="!!!π

p4ρC3

S#. In thenext section, the goodness of the A∼exact

rms prediction is com-pared with that of AMH1

rms and AMH2rms using a composite dataset

of earthquake records.

Data and Analysis

The Composite Catalog

The dataset used in this study (Fig. 3) consists of 6311three-component seismograms, with signal-to-noise ratiothat is greater than 20 and whose hypocentral distances

are less than 60 km. It includes 3042 velocity seismogramsand 286 accelerograms recorded by the California IntegratedSeismic Network, 2917 accelerograms recorded by Japan’sK-NET and KiK-net surface accelerometers, and 66 accel-erograms from the Pacific Earthquake Engineering Re-search–Next Generation Attenuation of Ground Motions(PEER NGA) database (Ancheta et al., 2014). These dataare associated with 542 southern California earthquakeswhose magnitude is between 3 and 6, 196 Japan earthquakeswhose magnitude is between 4 and 7.3, the 1999 Mw 7.6Chi-Chi earthquake, and the 2010 Mw 7.2 El Mayor–Cucapah earthquake. P-phase arrivals were picked manually,zero-offset corrections were applied, and accelerationtime series were high-pass filtered at 0.02 Hz. The rmsof the ground acceleration Arms is calculated accordingto

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Pni!1%AZ2

i $ AE2i $ AN2

i &=np

, in which AZ, AE, andAN are the acceleration amplitudes along the vertical, east,and north directions, respectively, and n is the number ofsamples within the data interval. The start and end timesof the data intervals were determined with the objective ofincluding the direct S waves that are emitted from the source,while excluding, as much as possible, the weaker coda wavesthat follow. With this in mind, the start time of the data in-terval is set to be equal to the first S-wave arrival Tstart !TP $ R=8, with TP being the time of the first P-phase arrivaland R being the hypocentral distance (in kilometers, derivedfrom the earthquake catalog); the duration of the data intervalis set to be equal to the sum of the S-wave travel time and therupture duration Tend − Tstart ! R=CS $ 1=f0, with CSbeing an average S-wave velocity of 3:2 km=s, and the cornerfrequency is estimated from equation (4) using a stress dropof 1 MPa (later, the corner frequency will be obtainedthrough spectral inversion). Thus, the data interval of smallearthquakes (i.e., short rupture durations) ends approxi-mately when the elapsed time since the earthquake origintime is twice the S-wave travel time, which according tocoda-wave studies marks the onset of coda waves (Rautianand Khalturin, 1978; Herraiz and Espinosa, 1987). The ad-dition of the distance-dependent term R=CS to the data in-terval accounts for the increase of the wave-packet spreadwith distance from the earthquake source (Boore andThompson, 2014).

Modeling κ, f0, and Ω0

The spectral parameters κ, f0, and Ω0 are obtained intwo steps. First, κ is modeled by fitting the high-frequencyportion of the acceleration spectra to

EQ-TARGET;temp:intralink-;df14;313;99 ln"A"f ## ! a − πκf; 10 Hz < f < 25 Hz; "14#

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with A"f # being the spectral amplitudes, and a and κ beingthe fitting parameters (Anderson and Hough, 1984). The fitto the above function is done for frequencies between 10 and25 Hz (Oth et al., 2011). The distribution of individual re-cords and station-average κ are shown in Figure 4a. Averageκ for California and Japan are similar and equal to 0.04 s.This κ distribution is similar to previously reported distribu-tions from California, Japan, and Taiwan (Oth et al., 2011;Van Houtte et al., 2011).

After having determined κ, the remaining spectralparameters f0 and Ω0 are determined via grid-search ap-proach, with the objective function being the absolute dif-ference between the logarithm of the observed spectra andthe logarithm of the attenuated Brune’s predicted spectra(equation 5), with κ set to be equal to the individual recordvalue obtained in the previous step. This approach is equiv-alent to solving f0 and Ω0 using equation (2) and theκ-corrected spectra. Furthermore, because here κ is deter-mined per seismogram, it embodies both site and path ef-fects (Anderson and Hough, 1984). The above analysisindicates that the f0 to fκ ratio of most of the Mw <5 andabout 50% of the 5 ≤ Mw < 6:5 are inconsistent with theMH2 approximation (Fig. 4b).

Comparison between Observed and Predicted Arms

The performances of A∼exactrms and AMH2

rms predictions arecompared in Figure 5. Similar comparison with Aexact

rms is notshown, because it is not visually distinguishable from A∼exact

rms .A comparison with AMH1

rms is not shown either, because thisexpression is mathematically restricted to f0 < fκ, and thuscannot be directly compared with the others. The standard de-viations of the discrepancies between observed and predictedArms (reported at the top panels of Fig. 5) indicate that A∼exact

rms

exhibits smaller discrepancies than AMH2rms . Inspection of these

discrepancies as a function of α0 reveals similarly good agree-ment between observed and predicted Arms for the two predic-tions when f0 ≪ fκ. Not surprisingly, the discrepancybetween the observed and the MH2-predicted Arms is α0 de-pendent, and the prediction ofAMH2

rms becomes less reliablewithincreasing α0 beyond ∼0:1. In contrast, the discrepancy be-tween the observed and the ∼exact-predicted Arms remainsfairly small when the small corner-frequency condition is notmet. Thus, the robustness of A∼exact

rms for ground-motion predic-tion for all α0 values is established. It must be emphasized thatthe MH solutions were never intended to apply for f0 > fκ,and the observed behavior is as expected.

PGA–Arms Relation

The PGA is a key parameter in earthquake-hazard analy-ses, design of building codes, algorithms for earthquake early

R [ ]

Count

Mag

nitu

de

Figure 3. Waveform distribution according to magnitude andhypocentral distances derived from the earthquake catalogs. Redtriangles indicate seismograms recorded by the California Inte-grated Seismic Network, green circles indicate accelerograms re-corded by the K-NET and KiK-net surface accelerometers, andblue squares indicate accelerograms from the Pacific EarthquakeEngineering Research–Next Generation Attenuation of GroundMotions (PEER NGA) database associated with two earthquakesfrom Taiwan and Mexico.

log

ofD

iscr

epan

cy(a)

(b)

Figure 2. Comparison between Aexactrms and the three approximate

solutions A∼exactrms , AMH1

rms , and AMH2rms (Arms is the ground acceleration

root mean square [rms], MH1 is the McGuire and Hanks (1980) firstset of approximations, and MH2 signifies McGuire and Hanks’ ad-ditional approximation). (a) The logarithm of normalized Arms,hArmsi ! Arms

!!!!T

p"πκ#2:5=""2π#2Ω0#, as a function of the logarithm

of α0. (b) Discrepancy between the logarithm of Aexactrms and A∼exact

rms ,AMH1rms , and AMH2

rms , as a function of the logarithm of α0.

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Cou

ntIndividual record: 0.042 s

Station average: 0.042 s

Cou

nt

(a) (b)

Figure 4. Histograms showing the distribution of κ and α0. (a) The distributions of record-specific and station-average κ values areindicated in light and dark gray, respectively. (b) The distribution of α0 according to magnitude bins. The area left of the vertical dashedline indicates the region where the f0 ≪ fκ approximation is valid.

log(log(

= 0.357

Dis

crep

ancy

= 0.175

Dis

crep

ancy

(a) (b)

(c) (d)

Figure 5. Observed versus predicted Arms. (a) The logarithm of observed Arms as a function of the logarithms of A∼exactrms . (b) The logarithm

of observed Arms as a function of the logarithms of AMH2rms . (c) log"Arms# − log"A∼exact

rms # as functions of α0. (d) log"Arms# − log"AMH2rms # as

functions of α0.

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warning, and more. Thus, it is instructive to establish a re-lation between PGA and Arms. Hanks and McGuire (1981)obtained the following equation:

EQ-TARGET;temp:intralink-;df15;55;400

PGAArms

!

!!!!!!!!!!!!!!!!!!!!!!!!!!

2 ln"2fmax

f0

#s

: "15#

The above relation has been validated against large datasets(e.g., Baltay et al., 2013), including the data used in thisstudy (not shown). Yet, because it is mathematically re-stricted to f0 < 2fmax, it is useful to establish a PGA–Armsrelation that is assumption free. The log–log diagram of PGAas a function of Arms is shown in Figure 6, with the formercalculated according to max"

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!AZ2

i $ AE2i $ AN2

i

p#. Re-

gression analysis yields the following scaling:

EQ-TARGET;temp:intralink-;df16;55;248PGA ! 4:7"'1:4#Arms: "16#

The above empirical relation may be combined with equa-tion (13) to get a ground-motion prediction equation for PGA.

Summary

A simple relation between Arms and source (or spectral)parameters is introduced that is based on Brune’s omega-square model for far-field radiation (Brune, 1970). The mainadvantage of the new relation with respect to the commonlyutilized AMH

rms relation (Hanks, 1979; McGuire and Hanks,1980) is that it relaxes the simplifying assumption fmax ≫ f0and the assumption that the spectrum may be approximated asbeing perfectly flat between f0 and fmax.

The prediction of the newly derived relation is comparedwith that of AMH

rms using a composite dataset of earthquakerecords from Japan, California, Mexico, and Taiwan. Thespectral parameters κ, f0, and Ω0 are obtained in two stepsvia spectral inversion. On average, the values of κ for Cal-ifornia and Japan are similar and are close to 0.04 s (Fig. 4a).The ratio between f0 and fκ of most Mw <5 and half of the5 ≤ Mw < 6:5 are inconsistent with the MH2 approximation(Fig. 4b). Thus, use of A∼exact

rms , whose prediction is nearlyindependent of α0, is advantageous.

In summary, the availability of a single formulation thatis valid for any value of α0 provides a powerful tool forground-motion prediction, as it enables the extrapolation ofGMPE inferred from the frequent small-magnitude earth-quakes to the rare large magnitudes. This capacity is extremelyimportant near slow-slip plate boundaries, where the seismicmoment release rates are low.

Data and Resources

The data used in this study were obtained from theSouthern California Earthquake Data Center (Caltech. Data-set, http://scedc.caltech.edu/, last accessed January 2016),the K-NET and KiK-net strong-motion networks (http://www.kyoshin.bosai.go.jp/, last accessed January 2016), andfrom the Pacific Earthquake Engineering Research (PEER)Ground Motion Database (http://ngawest2.berkeley.edu/site/,last accessed January 2016).

Acknowledgments

We thank Associate Editor Hiroshi Kawase for his work and insightfulcomments. We also thank two anonymous reviewers for their very construc-tive remarks. This research was supported by the Ministry of Energy andWater Resources Grant Number 214-17-019.

References

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log( [ ])

log(

[])

a = 0.67= 0.131

Figure 6. The logarithm of peak ground acceleration (PGA) as afunction of the logarithm of acceleration root mean square. The grayline indicates a linear regression to log"PGA# ! a$ log"Arms#, thefitting coefficient a, and standard deviation are reported at the topleft corner.

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Campbell, K. W., and Y. Bozorgnia (2008). NGA ground motion model forthe geometric mean horizontal component of PGA, PGV, PGD and 5%damped linear elastic response spectra for periods ranging from 0.01 to10 s, Earthq. Spectra 24, 139–172.

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Department of GeosciencesTel-Aviv UniversityP.O. Box 39040Tel-Aviv 6997801, Israelitzhaklior22@gmail.com

Manuscript received 2 August 2016;Published Online 3 January 2017

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