Rupture length and duration of the 2004 Aceh-Sumatra earthquake

from the phases of the Earth’s gravest free oscillations

S. Lambotte, L. Rivera, and J. HindererInstitut de Physique du Globe de Strasbourg, UMR CNRS-ULP 7516, Strasbourg, France

Received 25 July 2005; revised 10 October 2005; accepted 20 December 2005; published 7 February 2006.

[1] The Aceh-Sumatra 2004 earthquake strongly excitedthe Earth’s free oscillations. Well separated split multipletsprovide useful information on the earthquake source.Particularly, the phases of split singlets constrain theduration, rupture length and mean rupture velocity. Weanalyze the initial phases of some of the Earth’s gravest freeoscillations (0S2, 0S3, 0S0 and 1S0) in order to constrain thespace-time finiteness of the source. We use recordings ofvertical broadband seismometers and superconductinggravimeters from several worldwide geophysicalnetworks. We estimate a rupture length of about 1220 km,a source time duration of about 500 s, and a mean rupturevelocity of 2.4 km/s. Citation: Lambotte, S., L. Rivera, and

J. Hinderer (2006), Rupture length and duration of the 2004

Aceh-Sumatra earthquake from the phases of the Earth’s gravest

free oscillations, Geophys. Res. Lett., 33, L03307, doi:10.1029/

2005GL024090.

1. Introduction

[2] Soon after the great Chilean 1960 earthquake, Benioffet al. [1961], Ness et al. [1961] and Alsop et al. [1961]reported accurate measurements of spectral peaks that wereassociated with the Earth’s gravest free oscillations. Record-ings of this event also provided the first observationalevidence for splitting of the lowest frequency modes. Thissplitting was attributed to the Earth’s rotation by Backus andGilbert [1961] and Pekeris et al. [1961]. Splitting effect dueto ellipticity was introduced later on by Usami and Sato[1962] and Dahlen [1968].[3] The 2004 Aceh-Sumatra earthquake is the largest

earthquake ever recorded since the 1960 Chilean earthquakeand the 1964 Alaskan earthquake. Initial teleseismic body-wave studies suggested a rupture length of about 400–600km(e.g., J. Chen, Preliminary result of the 04/12/26 (Mw 9.0), offW coast of northern Sumatra earthquake, 2005, available athttp://www.gps.caltech.edu/�jichen/Earthquake/2004/aceh/aceh.html; Y. Yagi, Preliminary results of rupture processfor 2004 off coast of northern Sumatra giant earthquake,2005, available at http://iisee.kenken.go.jp/staff/ yagi/eq/Sumatra2004/Sumatra2004.html; Y. Yamanaka, 04/12/26 offW. coast of N. Sumatra, 2005, available at http://www.eri.u-tokyo.ac.jp/sanchu/Seismo_Note/2004/EIC161ea.html).Subsequent studies using long period data [Park et al.,2005b; Stein and Okal, 2005; Ammon et al., 2005], GPSdata [Vigny et al., 2005] and short period data [Ni et al.,2005; Lomax, 2005] suggest a longer rupture of over1200–1300 km. This is consistent with the location of

the aftershocks that extend more than 1000 km northwardfrom the epicenter.[4] Free oscillations are widely used to study the Earth’s

structure [e.g., Ritzwoller et al., 1986; Woodhouse et al.,1986; Smith and Masters, 1989; Widmer et al., 1992]. Freeoscillations with periods between 50 and 300 s are also usedto study source parameters of moderate and large earth-quakes [e.g., Dziewonski et al., 1981]. The gravest normalmodes can also provide information on the focal mecha-nism, seismic moment [Abe, 1970; Geller and Stein, 1977;Kedar et al., 1994; Stein and Okal, 2005], duration andlength of the earthquake rupture. Phases of free oscillationsare seldom explicitly studied, except for the radial modes,[see Park et al., 2005a]. We demonstrate in this study how itis possible to retrieve the spatio-temporal extension of therupture using the initial phases of some of the gravest modes(0S2, 0S3, 0S0 and 1S0) excited by the Aceh-Sumatra 2004earthquake.

2. Basics

[5] Deviations from spherical symmetry such as rotation,ellipticity and lateral heterogeneities remove the degeneracyand hence force free oscillations multiplets to split. Ingeneral, each singlet within a multiplet will have a slightlydifferent central frequency. Generally, to model the splitspectrum of a multiplet, it is necessary to account for theEarth’s rotation, ellipticity and heterogeneity. For the grav-est modes considered here, the main effect on splitting isdue to rotation and ellipticity. It can be computed byperturbation of the SNREI (spherically symmetric, non-rotating, elastic and isotropic) solution. Frequencies andquality factors are only dependent on the Earth’s structure.For the modes considered here, amplitudes depend mainlyon the seismic moment, the source mechanism, and thesource and station location.[6] For a point source, the Fourier transform of the

acceleration of an isolated multiplet of harmonic degreel can be written as a sum of Lorentzians [Dahlen andTromp, 1998],

aps x;wð Þ ¼Xm

amps x;wð Þ ¼Xm

M : E?m xps� �� �

sm xð Þ2 gm þ i w� wmð Þð Þ ; ð1Þ

where the subscript ps stands for point source, m is theazimuthal order, wm the singlet central frequency, M theseismic moment tensor, Em(xps) the strain of the singlet mevaluated at the source location xps, sm(x) the eigenfunctionof the singlet m evaluated at the station location x, and gmits attenuation rate, related to the quality factor Qm (gm =wm/(2Qm)). The symbol 8 is used to denote complex

GEOPHYSICAL RESEARCH LETTERS, VOL. 33, L03307, doi:10.1029/2005GL024090, 2006

Copyright 2006 by the American Geophysical Union.0094-8276/06/2005GL024090$05.00

L03307 1 of 4

conjugate, and : stands for tensor contraction. Thecontribution at wm of the Lorentzian centered at �wm

depends on Qm. For the high quality spheroidal modesconsidered here, it represents less than 0.1% of the totalamplitude. For the sake of brevity, in writing equation (1),we have omitted the contribution of negative frequencies.[7] For large earthquakes with fault lengths exceeding

several hundred kilometers, it is important to take intoaccount the finiteness of the source in the amplitude andphase computation. The result can be written as

amfs x;wð Þ ¼ amps x;wð ÞFm wð Þ ð2Þ

with Fm wð Þ ¼

Z L

0

M : E?m� �

xð Þe�iwt xð Þ dx

L M : E?m xps� �� � ; ð3Þ

where afs(x, w) is the spectrum of the acceleration for afinite source, L is the length of the source and t(x)represents the rupture delay of the point located at x on thefault. (e.g., t(x) = x/Vr in the case of a unidirectional andconstant velocity rupture). Fm(w) bears all the informationabout the kinematics of the rupture and is independent ofthe receiver location. Equation (2) allows us to predict theamplitude and phase at any station for any givenunidimensional rupture model. In the present work, weare specifically interested in the phase of the complexnumber Fm(w = wm), which we shall call the initial phase,Xm, of the singlet m.[8] Ben-Menahem and Singh [1980] (pp. 397–398)

derived, under certain assumptions, a simple analyticalrelation,

Fm ¼ sinXm

Xm

e�iXm ; Xm ¼ L

2Vr

wm þ mVr sin fr0 sin q0

!; ð4Þ

where r0 the Earth’s radius, q0 the epicentral co-latitude, fthe azimuth of the fault, and Vr the rupture velocity. Itassumes unidirectional rupture propagation, uniform dis-location and constant rupture velocity. Equation (4),although inaccurate, is appealing for its simplicity. The firstterm on the right-hand side of the phase expression in (4)depends on the time duration of the rupture and is nearlyindependent of m as wm varies slightly with m in a multiplet.

The second term is a function of the fault length anddepends linearly on m. Consequently, for m = 0, the phasecontains information only on the time duration of thesource. For m 6¼ 0, the phase constrains both the rupturelength and the source duration.[9] However, equation (4) is only approximate since the

latitudinal variations in (M : Em8) have been neglected. It is

exact only for the radial modes or for purely E-W equatorialfaults. The integral in equation (3) can be calculated eithernumerically in the geographical reference frame, or analyt-ically in a reference frame in which the fault lies along theequator. In the latter approach, use of the zeroth ordereigenfunctions is implicit, and transformation back to thegeographical reference frame is achieved using the Wigner’srotation matrices [Feynman et al., 1965, pp. 18-9–18-13].

3. Data

[10] We use recordings from vertical very broad bandseismometers belonging to several worldwide networks(Geoscope, Iris-IDA, Berkeley, Iris-USGS and Iris-China)and from superconducting gravimeters of the GGP (GlobalGeodynamics Project) network. Figure 1 shows a map of allthe stations used in this study. The coverage is not homo-geneous since most of the stations are located at mid-latitudes in both hemispheres. We use atmospheric pressurerecordings from GGP stations to correct data with anadmittance of �3 nm/s2/hPa [Crossley et al., 1995]. Timeseries are highpass filtered with a cut-off frequency of4 hours to remove most of the Earth’s tide. They are alsocorrected for instrument response and converted to accel-erations. We use Hanning-tapered time series with a lengthof 20 days for calculation of spectra.

4. Method

[11] Using equation (1), for each multiplet, we simulta-neously search 2l + 1 complex amplitudes, 2l + 1 frequen-cies and 2l + 1 quality factors, in total 8l + 4 realparameters. To retrieve these parameters, we perform anonlinear fitting for each multiplet. In addition, we iteratethe fitting process to take advantage of the independence ofthe central frequencies and quality factors on the receiverlocation, so reducing the total number of independentparameters. The taper window is taken into considerationin the fitting process.[12] As an example, Figure 2 shows the results of fitting

of two multiplets for two different stations: BFO (BlackForest Observatory, Germany) and UNM (Mexico City).Depending on the mechanism and the epicentral location,all the spectral peaks are not equally excited, and there canbe large differences in the signal to noise ratio. As we cansee in Figure 2, singlets with jmj = 0, 1, 2 for 0S3 at UNMstation are only slightly above the background noise.Consequently in the following analysis, we only considerspectral peaks with S/N greater than 3.[13] In order to isolate the finiteness effect we proceed as

follows. We first compute theoretical amplitudes and phases[Dahlen and Tromp, 1998; Stein and Geller, 1977] for aninstantaneous point source mechanism using Harvard CMTfocal mechanism (aps

m in (1) and (2)). We then identify theresult of the fit to afs

m and use (2) to calculate Fm and in

Figure 1. Map of the stations used in this study. Colorscorrespond to different networks. The Aceh-Sumatra 2004earthquake is indicated by a triangle.

L03307 LAMBOTTE ET AL.: 2004 ACEH-SUMATRA EARTHQUAKE L03307

2 of 4

particular the initial phases Xm. Finally, these measurementsare compared with the predictions of expression (3) in orderto determine the length and the duration of the rupture. Theisotropic PREM model [Dziewonski and Anderson, 1981] isused for this computation. The rotation and ellipticityeffects are included, but the Earth’s lateral heterogeneityis ignored.[14] Both seismometers and gravimeters are simulta-

neously sensitive to the ground acceleration and to thevariations of Earth’s gravity field. The latter is significantfor the gravest modes [see Gilbert, 1980]. We take intoaccount these effects by including free-air changes in gravitydue to the radial displacement of the seismometer, andperturbation in the gravitational potential due to the redistri-bution of the Earth’s masses [Gilbert, 1980; Wahr, 1981].

5. Results and Discussion

[15] Using stations shown on Figure 1, we have analyzedthe spectral properties of some of the Earth’s gravest normalmodes (0S2, 0S3, 0S0 and 1S0) excited by the 2004 Aceh-Sumatra earthquake. By comparing the complex amplitudesof the individual singlets with the theoretical predictions for

an instantaneous point source, we can isolate the initialphase Xm of each singlet. Figures 3, 4 and 5 show the resultsfor 0S0, 0S2 m = ±2, and 0S3 m = ±3.[16] The average initial phase for 0S0 is �67.6�, which

corresponds to a rupture duration of 460 s. More precisely,we measure a difference of 230 s between the nucleationtime and the 0S0 centroid time. This value is very similar tothe one obtained by Park et al. [2005b]. By using the initialphases of each couple of singlets (l, +m) and (l, �m) (form 6¼ 0), it is possible to calculate both the length and durationof the source if estimates of the epicentral co-latitude q0 andthe fault strike f are available. We use a latitude of 3�N and afault strike of 330�N [Lay et al., 2005]. In principle, only onestation is sufficient to find the rupture length and the timeduration of the source.[17] Table 1 summarizes the results of our calculations.

The average values are a fault length of 1220 ± 50 km and arupture duration of 500 ± 30 s. These results are consistentwith various studies using short period data [Ni et al., 2005;Lomax, 2005], GPS data [Vigny et al., 2005], and longperiod data [Park et al., 2005b; Ammon et al., 2005]. Thesereferences present a detailed reconstruction of the sourceprocess with a centroid clearly offset to the south. If such isthe case, it is somewhat surprising that, with ultra-longperiod data, we obtain results similar to the short periodestimates.[18] Since we are using only ultra-long period data, the

model obtained here is of course simplified. Possibly, this iswhy we obtain slightly shorter source durations with 0S0 and

1S0. On the other hand, 0S3 and 0S2, which have periodsmuch longer than the obtained rupture duration, yieldcomparable results both in rupture duration and fault length(see Table 1). In conclusion, with our free oscillations data

Figure 2. Results of the fitting process for two multipletsat two different stations (0S2 for BFO station (Black ForestObservatory, Germany) on left panels and 0S3 for UNMstation (Mexico City) on right ones). Red curves correspondto the result of the fitting process, and the blue onescorrespond to the data spectrum. From top to bottom, panelsrepresent the amplitude, the real part and the imaginary partof the spectrum. The small line in the top right corner of toppanels represents the frequency resolution corresponding tothe original non-padded time series.

Figure 3. Initial phases for 0S0 (m = 0) with respect to apoint source. The right diagram shows the distribution ofinitial phases.

Figure 4. Initial phases for 0S2 m = ±2 with respect to apoint source. Red and green colors correspond, respectively,to m = +2 and m = �2. The right diagram shows thedistribution of initial phases.

Figure 5. Initial phases for 0S3 m = ±3 with respect to apoint source. Yellow and blue colors correspond, respec-tively, to m = +3 and m = �3. The right diagram shows thedistribution of initial phases.

L03307 LAMBOTTE ET AL.: 2004 ACEH-SUMATRA EARTHQUAKE L03307

3 of 4

we do not observe slip after 500 s. If there was any, it musthave been much smaller or too slow to efficiently excite theEarth’s free oscillations. To obtain more precise details onthe source geometry and rupture history, it seems natural toattempt a similar analysis at higher frequencies. However,this is a difficult task. First, lateral heterogeneity is expectedto be significant for higher harmonic degrees, leading tocomplex splitting patterns. Second, the broadening of spec-tral peaks due to attenuation is of the order of the splitfrequency separation, limiting our ability to resolve indi-vidual peaks.[19] With estimates of rupture length and duration, we

can evaluate the mean rupture velocity. The mean rupturevelocity is about 2.4 km/s for the Aceh-Sumatra 2004earthquake. This value agrees with the model proposed byVigny et al. [2005] on the basis of modeling fast GPSobservations.

6. Conclusion

[20] Initial phases of the split free oscillations can providerobust information on the spatio-temporal finiteness of largeearthquakes. They allow us to estimate fault length, sourceduration, and mean rupture velocity. Usually, for the gravestmodes, only the amplitudes are studied to constrain thesource mechanism and the seismic moment. Phases arerarely considered explicitly. For the 2004 Aceh-Sumatraearthquake, initial phases of singlets of the gravest modesallow us to estimate a fault length of 1220 ± 50 km, a sourcetime duration of about 500 ± 30 s and a mean rupturevelocity of about 2.4 ± 0.2 km/s. These results are inagreement with other studies based on both long or shortperiod data.

[21] Acknowledgments. We are grateful to GEOSCOPE, IRIS, GGPand BERKELEY for supplying high quality data. Bob Geller and RuediWidmer-Schnidrig provided us with very in-depth reviews. We alsoacknowledge G. Masters for making available minos for the calculationof eigenfrequencies and eigenfunctions.

ReferencesAbe, K. (1970), Determination of seismic moment and energy from theEarth’s free oscillation, Phys. Earth Planet. Inter., 4, 49–61.

Alsop, L. E., G. H. Sutton, and M. Ewing (1961), Free oscillations of theEarth observed on strain and pendulum seismographs, J. Geophys. Res.,66, 631–641.

Ammon, C. J., et al. (2005), Rupture process of the 2004 Sumatra-Anda-man earthquake, Science, 308, 1133–1139.

Backus, G., and F. Gilbert (1961), The rotational splitting of the free oscil-lations of the Earth, in Proc. Natl. Acad. Sci. U. S. A., 47, 362–371.

Benioff, H., F. Press, and S. W. Smith (1961), Excitation of the free oscilla-tions of the Earth by earthquakes, J. Geophys. Res., 66, 605–620.

Ben-Menahem, A., and S. J. Singh (1980), Seismic Waves and Sources, 2nded., Dover, Mineola, N. Y.

Crossley, D., O. Jensen, and J. Hinderer (1995), Effective barometricadmittance and gravity residuals, Phys. Earth Planet. Inter., 90, 221–241.

Dahlen, F. A. (1968), The normal modes of a rotating, elliptical Earth,Geosphys. J. R. Astron. Soc., 16, 329–367.

Dahlen, F. A., and J. Tromp (1998), Theoretical Global Seismology,Princeton Univ. Press, Princeton, N. J.

Dziewonski, A., and D. L. Anderson (1981), Preliminary reference Earthmodel, Phys. Earth Planet. Inter., 25, 297–356.

Dziewonski, A., T. A. Chou, and J. H. Woodhouse (1981), Determinationof earthquake source parameters from waveform data for studies of globaland regional seismicity, J. Geophys. Res., 86, 2825–2852.

Feynman, R., R. B. Leighton, and M. Sands (1965), The Feynman Lectureson Physics, vol. III, Quantum Mechanics, Addison-Wesley, Boston,Mass.

Geller, R., and S. Stein (1977), Split free oscillation amplitudes for the 1960Chilean and 1964 Alaskan earthquakes, Bull. Seismol. Soc. Am., 67,651–660.

Gilbert, F. (1980), An introduction to low-frequency seismology, in Fisicadell’interno della Terra, Rendiconti della Scuola Internazionale di Fisica‘‘Enrico Fermi’’, edited by A. M. Dziewonski and E. Boschi, pp. 41–81,Elsevier, New York.

Kedar, S., S. Watada, and T. Tanimoto (1994), The 1989 Macquarie Ridgeearthquake: Seismic moment estimation from long-period free oscilla-tions, J. Geophys. Res., 99, 17,893–17,907.

Lay, T., et al. (2005), The great Sumatra-Andaman earthquake of26 December 2004, Science, 308, 1127–1133.

Lomax, A. (2005), Rapid estimation of rupture extent for large earthquakes:Application to the 2004, M9 Sumatra-Adaman mega-thurst, Geophys.Res. Lett., 32, L10314, doi:10.1029/2005GL022437.

Ness, N. F., J. C. Harrison, and L. B. Slichter (1961), Observations of thefree oscillations of the Earth, J. Geophys. Res., 66, 621–629.

Ni, S., H. Kanamori, and D. Helmberger (2005), Energy radiation from theSumatra earthquake, Nature, 434, 582.

Park, J., K. Anderson, R. Aster, R. Butler, T. Lay, and D. Simpson (2005),Global seismographic network records the great Sumatra-Adaman earth-quake, Eos Trans. AGU, 86(6), 57, doi:10.1029/2005EO060001.

Park, J., et al. (2005b), Earth’s free oscillations excited by the 26 December2004 Sumatra-Adaman earthquake, Science, 308, 1139–1144.

Pekeris, C. L., Z. Alterman, and H. Jarosch (1961), Rotational multiplets inthe spectrum of the Earth, Phys. Rev., 122, 1692–1700.

Ritzwoller, M., G. Masters, and F. Gilbert (1986), Observations of anom-alous splitting and their interpretation in terms of aspherical structure,J. Geophys. Res., 91, 10,203–10,228.

Smith, M. F., and G. Masters (1989), Aspherical structure constraints fromfree oscillation frequency and attenuation measurements, J. Geophys.Res., 94, 1953–1976.

Stein, S., and R. Geller (1977), Amplitudes of the Earth’s split normalmodes, J. Phys. Earth, 25, 117–142.

Stein, S., and E. Okal (2005), Speed and size of the Sumatra earthquake,Nature, 434, 581.

Usami, T., and Y. Sato (1962), Torsional oscillation of a homogeneouselastic spheroid, Bull. Seismol. Soc. Am., 52, 469–484.

Vigny, C., et al. (2005), Insight into the 2004 Sumatra-Andaman earthquakefrom GPS measurements in southeast Asia, Nature, 436, 201–206.

Wahr, J. M. (1981), Body tides on an elliptical, rotating, elastic and ocean-less Earth, Geophys. J. R. Astron. Soc., 64, 677–703.

Widmer, R., G. Master, and F. Gilbert (1992), Observably splitmultiplets - data analysis and interpretation in terms of large-scaleaspherical structure, Geophys. J. Int., 111, 559–576.

Woodhouse, J. H., D. Giardini, and X.-D. Li (1986), Evidence for innercore anisotropy from splitting in free oscillations data, Geophys. Res.Lett., 13, 1549–1552.

�����������������������J. Hinderer, S. Lambotte, and L. Rivera, Institut de Physique du Globe

de Strasbourg, UMR CNRS-ULP 7516, 5 rue Rene Descartes, F-67084Strasbourg Cedex, France. (sophie.lambotte@eost.u-strasbg.fr)

Table 1. Measured Initial Phase F, Source Duration Tr, and

Rupture Length L for Each Mode (or Pair of Modes)a

Mode m Xm Tr, s L, km

1S0 0 �120.7� 411 . . .

0S0 0 �67.6� 460 . . .

0S3 �2 �27.4� 511 1190

0S3 +2 �63.8� 511 1190

0S3 �3 �48.9� 500 1210

0S3 +3 �35.0� 500 1210

0S2 0 �28.2� 512 . . .

0S2 �2 �34.6� 532 1254

0S2 +2 �24.4� 532 1254aOnly modes with jmj 6¼ 0 allow us to estimate L. 0S2 m = ±1 and 0S3 m =

0 are systematically too small to allow for the use of the phase (see, forexample, Figure 2).

L03307 LAMBOTTE ET AL.: 2004 ACEH-SUMATRA EARTHQUAKE L03307

4 of 4