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Source Constraints and Model Simulation

of the December 26, 2004, Indian Ocean TsunamiStéphan T. Grilli, M.ASCE1; Mansour Ioualalen2; Jack Asavanant3; Fengyan Shi4;

James T. Kirby, M.ASCE4; and Philip Watts5

Abstract: The December 26, 2004 tsunami was perhaps the most devastating tsunami in recorded history, causing over 200,000 fatalitiesand widespread destruction in countries bordering the Indian Ocean. It was generated by the third largest earthquake on record�Mw=9.1–9.3� and was a truly global event, with significant wave action felt around the world. Many measurements of this event weremade with seismometers, tide gauges, global positioning system stations, and a few satellite overpasses. There were numerous eyewitnessobservations and video digital recordings of coastal tsunami impact, as well as subsequent coastal field surveys of runup and flooding. Afew ship-based expeditions also took place in the months following the event, to measure and map seafloor disturbances in the epicenterarea. Based on these various data sets, recent seismic analysis estimates of rupture propagation speed, and other seismological andgeological constraints, we develop a calibrated tsunami source, in terms of coseismic seafloor displacement and rupture timing along1,200 km of the Andaman–Sunda trench. This source is used to build a numerical model of tsunami generation, propagation, and coastalflooding for the December 26, 2004 event. Frequency dispersion effects having been identified in the deep water tsunami wavetrain, wesimulate tsunami propagation and coastal impact with a fully nonlinear and dispersive Boussinesq model �FUNWAVE�. The tsunamisource is specified in this model as a series of discrete, properly parameterized, dislocation source segments �Okada, 1985, Bull. Seismol.Soc. Am., 75�4�, 1135–1154�, triggered in a time sequence spanning about 1,200 s. ETOPO2’s bottom bathymetry and land topographyare specified in the modeled ocean basin, supplemented by more accurate and denser data in selected coastal areas �e.g., Thailand�. A1 min grid is used for tsunami simulations over the Indian Ocean basin, which is fine enough to model tsunami generation and propagationto nearshore areas. Surface elevations simulated in the model agree well, in both amplitude and timing, with measurements at tide gauges,one satellite transect, and ranges of runup values. These results validate our tsunami source and simulations of the December 26, 2004event and indicate these can be used to conduct more detailed case studies, for specific coastal areas. In fact, part of the development ofour proposed source already benefitted from such regional simulations performed on a finer grid �15 s�, as part of a Thailand case study,in which higher frequency waves could be modeled �Ioualalen et al. 2006, J. Geophys. Res., in press�. Finally, by running a non-dispersiveversion of FUNWAVE, we estimate dispersive effects on maximum deep water elevations to be more than 20% in some areas. We believethat work such as this, in which we achieve a better understanding through modeling of the catastrophic December 26, 2004 event, willhelp the scientific community better predict and mitigate any such future disaster. This will be achieved through a combination offorecasting models with adequate warning systems, and proper education of the local populations. Such work must be urgently done inlight of the certitude that large, potentially tsunamogenic, earthquakes occur along all similar megathrust faults, with a periodicity of a fewcenturies.

DOI: XXXX

CE Database subject headings: Tsunamis; Surface waters; Earthquakes; Hydrodynamics; Wave propagation; Wave runup; Numeri-cal models; Geophysical surveys; Simulation models.

Introduction

The December 26, 2004 Indian Ocean tsunami was likely themost devastating tsunami in recorded history, causing over200,000 fatalities in more than ten countries across the entireIndian Ocean basin, with tens of thousands reported missing and

over 1 million left homeless �Kawata et al. 2005; Yalciner et al.2005a�. The tsunami was a truly global event, with significantwave activity recorded around the world, for which the IndianOcean in fact only represented near-field tsunami wave propaga-tion �Titov et al. 2005�. The tsunami was generated in the Bay ofBengal by the third largest earthquakes ever recorded, with a

1Professor, Dept. of Ocean Engineering, Univ. of Rhode Island,Narragansett, RI 02882 �Corresponding author�. E-mail: [email protected]

2Geosciences Azur �CNRS-IRD�, Villefranche-sur-mer, France.3Dept. of Mathematics, Chulalongkorn Univ., Bangkok 10330,

Thailand.4Center for Applied Coastal Research, Univ. of Delaware, Newark,

DE 19761.5Applied Fluids Engineering, Inc., 5710 E. 7th St., Long Beach,

CA 90803.Note. Discussion open until April 1, 2008. Separate discussions must

be submitted for individual papers. To extend the closing date by onemonth, a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possiblepublication on February 3, 2006; approved on July 17, 2006. This paperis part of the Journal of Waterway, Port, Coastal, and Ocean Engineer-ing, Vol. 133, No. 6, November 1, 2007. ©ASCE, ISSN 0733-950X/2007/6-1–XXXX/$25.00.

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moment magnitude Mw=9.1–9.3 �Ammon et al. 2005; Lay et al.2005; Park et al. 2005; Stein and Okal 2005�. The tsunami sourcewas located along the Sunda and Andaman trenches �Fig. 1�,which mark the approximate boundary between the Indian–Australian and Eurasian/Andaman plates, the former plate sub-ducting under the latter at 5–6 cm/year with a largely East–Westdirection of convergence. The Bay of Bengal consists mostly ofthe Indian–Australian plate, with a sequence of islands runningnorth-south along the eastern edge of the bay, denoting the plateboundaries and the edge of the subduction zone. In the Bay ofBengal, sediments from rivers contribute to a massive sedimentfan that covers the entire downgoing plate from north to south,whose motion creates a large accretionary wedge east of the sub-duction zone �Davis et al. 1983�.

Characteristics of Rupture and Seabed Deformation

The December 26, 2004 event started with a main shock at 0 h58 min 53 s Greenwich mean time �GMT�, when the locked fault

between the plates ruptured at the megathrust earthquake’s hypo-center, located 3.32°N and 95.85°E, i.e., �160 km west ofSumatra, at a depth of 25–30 km, liberating strain accumulatedfrom subduction since the last large earthquakes occurred in thearea, in 1861 and 1881. The main shock epicenter is marked onFig. 1.

Seismic inversion models �e.g., Ammon et al. 2005� indicatethat the main shock, or rupture, propagated northward from theepicenter, parallel to the trenches, at a shear wave speed of2–3 km/s, thus covering the �1,200 km of the ruptured faultlength in about 500 s. �This value was confirmed by hydroacous-tic measurements �de Groot-Hedlin 2005�.� The same models pre-dict that the elastic rebound associated with the earthquake causedthe seabed to uplift by as much as 6 m or subside by up to thesame amount, slightly more in some areas, over a region100–150 km wide around the subduction zone �Ammon et al.2005; Lay et al. 2005�. Maximum uplift ��O�10 m; Bilham 2005��and subsidence �a so-called asperity� occurred west of Banda

Fig. 1. Tsunami simulation grid designed for Bay of Bengal using ETOPO2 bathymetry and topography �contours every 500 m�, with locationof five independent rupture segments, S1-S5 �Table 1�: �*� location of December 26, 2004 earthquake epicenter; �� locations of tide gauges; and�� location of yacht Mercator. �• • • • � JASON 1’s satellite transect.

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Aceh at the northern tip of Sumatra, around 5°N. Sea levelchanges in Andaman and Nicobar Islands, in the North, indicatethat the coseismic crustal deformation extended that far north�Kayanne et al. 2005; Satake 2005b�, further confirming thesource length. The seafloor motion displaced an estimated 30 km3

of water on the ocean surface, causing the killer tsunami in theprocess �Kawata et al. 2005�.

Seismic inversion models also predict that fault slip was sig-nificantly nonuniform along the rupture length: up to 15–25 mslip in the bottom two-thirds of the rupture zone and much less inthe north. Global positioning system �GPS� measurements con-firm these interpretations of rupture and deformation processesand also show that fault slip was deep and nonhomogeneous, verysmall in the south, very large off of Sumatra’s Northern tip andPhuket �about 200 km north of the epicenter�, and decreasing tosmaller values beyond 7°N �Chlieh et al. 2005; Vigny et al.2005�. Distributions of aftershocks and seafloor deformationsshow an arched rupture zone, with a minimum of 3–4 separatesubzones or segments, which also correspond to the time progres-sion of the rupture along the fault �Ammon et al. 2005; Lay et al.2005; de Groot-Hedlin 2005; Tanioka et al. 2005�.

We point out here that, in such a large event, there can bemany faults that experience rupture along the subduction zone,and especially along secondary structures running from the sub-duction zone up to the seabed, within the accretionary wedge.These secondary structures are evident, for instance, in the 3 kmhigh face of stepped �or echelon� thrust faults rising above theSunda subduction trench in the southern part, and in the roughtapestry �or fabric� of the seafloor on the overriding plate over thewhole rupture zone �McNeill et al. 2005; Henstock et al. 2006�.Seismic profiles �using twin air guns� and direct video recording�using a remotely operated vehicle �ROV�� were made acrosssome of these structures during the Sumatra Earthquake andTsunami Offshore Survey cruise in May 2005 �SEATOS� �Moranet al. 2005; Mosher et al. 2005; �http://www.oce.uri.edu/seatos/report.html ��, and confirmed the existence of complex systemsof faults. It is along these secondary faults that co-seismic dis-placement from the main shock is expressed, with many localvariations about uplift/subsidence values calculated in seismic in-version models, in which simplifying assumptions are made re-garding seabed and subduction zone geology. We will also seelater that constraints on the tsunami source from surface elevationmeasurements will lead us to reduce the speed of co-seismic sea-bed deformation to much less than the deep shear wave speedpredicted by these models.

Characteristics of Tsunami and Its Coastal Effects

Many direct measurements of the generated tsunami and itscoastal effects were made during the December 26, 2004 event,including a few satellite overpasses �e.g., JASON 1; Gower 2005;Kulikov 2005� and tide gauge records �see for instance �http://www.pmel.noaa.gov/tsunami/sumatra20041226.html��. The latterprovided approximate tsunami arrival times for many locationsin the Indian Ocean, of which we selected seven, for which accu-rate digital records were readily available. Such tide gaugerecords were used in the first few days following the event, whenlittle detailed seismic information was available, to quickly esti-mate the tsunami source area through inverse propagation of thetsunami leading wave, at the long wave speed. Thus, Satake�2005a�, for instance, found using only tsunami arrival times atVishakapatnam, India �156 min� and Cocos Islands �140 min�,that the main area for tsunami generation was the bottom 500 km

of the rupture zone outlined in Fig. 1. Similar analyses that furtherconstrained the tsunami source area were performed later usingarrival times at more gauges.

Perhaps for the first time in the history of tsunami science,there were numerous detailed eyewitness observations of coastaltsunami impact in the form of video digital recordings. Theseprovided visual estimates of wave height and, in some cases,rough arrival times of successive tsunami waves �e.g., �http://www.waveofdestruction.org/tsunami-videos/ ��. After completingfield surveys, such video recordings were further processed bysome of the international scientific teams to estimate tsunami flowvelocity over land �e.g., Vatvani et al. 2005, in Banda Aceh�.

In the weeks and months following the event, multiple inter-national scientific teams surveyed coastal areas impacted by thetsunami, documenting damage, measuring runup and inundation,and assembling careful reconstructions of wave activity. Giventhe length of damaged coastline and number of countries in-volved, each team restricted their survey to a limited geographicalarea �Fritz and Synolakis 2005; Gusiakov 2005; Kawata et al.2005; Liu et al. 2005; Satake et al. 2005, 2006; Sannasiraj andSundar, 2005; Synolakis et al. 2005; Yalciner et al. 2005a,b;Yamada et al. 2005�. A few ship-based expeditions took place, inthe months following the event, to measure seafloor disturbancesin the epicenter area, notably the HMS Scott’s, a British Navyship that conducted a high resolution multibeam survey inJanuary–February 2005 of 40,000 km2 of the seafloor in the maintsunami generation area, north of the epicenter �McNeill et al.2005�, and SEATOS �Moran et al. 2005� already mentioned.

It appears from the various data sets available that, upon gen-eration and following the distribution of seafloor uplift and sub-sidence caused by the earthquake, the westward propagating tsu-nami had a leading elevation wave, subsequently hitting SriLanka, India, the Maldives and Somalia, whereas the eastwardpropagating tsunami had a leading depression wave, eventuallyimpacting Indonesia, Thailand, Malaysia, and Myanmar. Perform-ing global analyses of tide gauge data as well as numerical mod-eling �albeit linearized and on a coarse grid�, Titov et al. �2005�showed that the December 26, 2004 tsunami was very directionalin the cross-source �east-west� direction, due to a combination ofsource focusing—because of the long and narrow earthquakesource region—and bathymetric waveguides. This explains whythe tsunami caused serious damage and deaths as far as the eastcoast of Africa and why substantial wave energy propagated todistant coasts, including different oceans. In some cases, waveheights measured at far distant tide gauges were larger than thoseat some near-field gauges located in the long-source �south-north�direction. Bangladesh, for instance, which lies at the northern endof the Bay of Bengal, did not experience much tsunami effect andhad very few fatalities, despite being a low-lying country rela-tively near the epicenter.

A few, usually three, large tsunami waves were reported toarrive in most impacted coastal areas in the Indian Ocean, withone of the latter waves usually being the largest. Tsunami coastaleffects were the most severe in Banda Aceh, which is nearest thearea of maximum fault slip and seafloor uplift, causing the ma-jority of fatalities. The tsunami arrived in Banda Aceh within34 min of the start of the event, and runup and inundation reacheda 5–30 m height in most locations, advancing up to 4 km inlandwith current velocities up to 8 m/s �Kawata et al. 2005; Vatvaniet al. 2005; Yalciner et al. 2005a�. One of team even reportedmeasuring a 49 m inundation height in Rhiting, 5° 25�N about15 km southwest of Banda Aceh �Shibayama et al. 2005�. Mostimpacted next was Thailand, where the tsunami arrived at the

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southern tip of Phuket Island within 1 h 40 min of the start of theevent. Runup reached a 3–11 m height in most places, with thelargest runup measured on the west coast in Khao Lak �about 8°40� N�, 16–19.5 mdepending on the source, with 6–8 m/s cur-rent �Kawata et al. 2005; Satake et al. 2005, 2006; Yalciner et al.2005a�. By contrast, in Myanmar, just north of Thailand, runuponly reached 1–3 m, with the tsunami first arriving within 2 h30 min of the start of the event. These smaller values are likelydue to the smaller fault slip in the northern part of the rupture areaand to the protection offered the coast of Taninthayi Division byoffshore islands �Satake et al. 2005, 2006�. In Malaysia, Yalcineret al. �2005b� also report similar 1.5–3 m runup at most places,with one extreme 3.7 m value. On the western side of the IndianOcean, tsunami waves arrived in Sri Lanka and southeast India,2 h–2 h 30 min after the start of the event, causing 2.5–5 mrunup for the most part with a few extreme values reaching8–12 m in southern Sri Lanka �Liu et al. 2005; Synolakis et al.2005; Yalciner et al. 2005a�. In the Maldives, which are made ofa series of atolls, runup varied greatly depending on exposition,but is generally reported to have reached 2–3.4 m with thetsunami first arriving within 3 h 25 min of the start of theevent �Fritz and Synolakis 2005; Kawata et al. 2005�. Finally, inSomalia, where the tsunami arrived about 7–8 h after the startof the event, unexpectedly large runup values of 4.5–9 m weremeasured, which can in part be explained by the high tsunamidirectionality briefly discussed above �http://www.usc.edu/dept/tsunamis/2005/tsunamis/041226indianOcean/somalia/ �.

Purpose of This Work

In this work, in light of the characteristics of the December 26,2004 event briefly summarized above, we focus on constructingand constraining a reasonable tsunami source based on availablegeological, seismological, and tsunami elevation and timing data.We use this source to perform tsunami simulations with a numeri-cal model of long wave propagation, coastal inundation, andrunup. Here, however, we only aim at explaining the large scaletsunami propagation features measured during the event, as wellas overall coastal tsunami impact �runup� surveyed following theevent. In other work, reported elsewhere, we use our presentanalyses to conduct more detailed case studies of coastal tsunamiimpact on finer regional model grids, for selected areas such asThailand �Ioualalen et al. 2007�. Results obtained in the lattercase �not shown�, particularly for higher frequency waves, werealready used to constrain the present source, in order to ensurefull consistency of the various simulations.

Since our goal is to later perform regional case studies forwestern and southern Thailand, and northern Sumatra, wheremaximum runup was observed, it should be pointed out that, inour iterative development of tsunami sources, we gave priority todata reflecting east-west tsunami propagation rather than north-south propagation.

Numerical Model

One specificity of our modeling approach is the use, perhapsfor the first time for such a large scale event, of a fully non-linear and dispersive Boussinesq long wave propagation model�FUNWAVE�, which was initially developed for modelingocean wave transformation from deep water to the coast, includ-ing breaking and runup �Wei and Kirby 1995; Wei et al. 1995�.FUNWAVE retains information to O��kh�2� in frequency disper-

sion and to all orders in nonlinearity a /h �where k denotes a wavenumber, a denotes a wave amplitude, and h denotes a water depthscale�. FUNWAVE also has a physical parametrization of dis-sipation processes �including breaking�, as well as an accuratemoving inundation boundary algorithm, both of which are neces-sary to correctly estimate coastal tsunami effects and runup overland �Chen et al. 2000; Kennedy et al. 2000�.

Wei et al. �1995� showed that the retention in FUNWAVE ofnonlinear effects beyond the usual order in standard weakly non-linear Boussinesq models is crucial to the correct modeling ofshoaling solitary waves or undular bores on slopes, up to nearbreaking, and thus in the present case is important for modelingshoreline inundation. The presence of frequency dispersion in themodel is important for the case of short �or higher frequency�wave propagation into relatively deeper water �such as directlywest of the December 26, 2004 event ruptured area�, and allowsfor the mechanism of wave crest splitting during wave propaga-tion over shallow bathymetry.

FUNWAVE has been thoroughly validated and used to studysmall scale motions such as the propagation of waves in the near-shore �Chen et al. 2000; Kennedy et al. 2000� and the generationof wave-induced currents �Chen et al. 2003�, as well as regionalscale tsunami propagation �Day et al. 2005; Ioualalen et al. 2005;Watts et al. 2003; Waythomas and Watts 2003�. Preliminary re-sults for the modeling of the December 26, 2004 tsunami usingthe model have been reported by Watts et al. �2005�. A review ofthe theory behind FUNWAVE and other examples of its applica-tion are given by Kirby �2003�. The Appendix gives a brief sum-mary of equations implemented in the version of FUNWAVEused in this work.

Considering the fairly small longitudinal and latitudinalextensions of the Bay of Bengal, which is our main area of inter-est, the present simulations were performed with a version ofFUNWAVE implemented on a Cartesian grid. A spherical versionof FUNWAVE, also including Coriolis corrections, has recentlybeen derived and could be used in future work �Kirby et al. 2004�.Sphericity corrections might play a role in simulating tsunamisignals at far distant tide gauges �i.e., the simulated tsunamiwould arrive too early in the Cartesian grid�, while deviations dueto Coriolis force might affect the tsunami propagation.

As discussed above, FUNWAVE features more complete phys-ics than standard models used for tsunami modeling, which aretypically based on nondispersive linear or nonlinear shallowwater wave equations �NSWE�. Specifically, for the December26, 2004 tsunami, Kulikov �2005� performed a wavelet frequencyanalysis based on satellite altimetry data recorded in the Bay ofBengal in deep water, and showed the importance of dispersiveeffects on wave evolution. His results indicate that the leadingedge of the wave components with order 10 km wavelength weresignificantly delayed in comparison with the much longer wavesin the main wave front. He concluded that a long wave modelincluding dispersion �such as FUNWAVE� should be used for thisevent. This result is not surprising in light of Ward’s �1980�simple scaling analyses, which showed, in a constant 4,000 mdeep ocean, that any wave of length less than a couple of hundredkm should be dispersive. Okal �1982� had also similarly stressedthe importance of modeling dispersion. Hence, in the presentcase, FUNWAVE can potentially yield more accurate results,given the same data and tsunami source parameters, than morestandard models, which typically neglect dispersion.

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Tsunami Source

Based on rupture parameters estimated by seismic inversion mod-els �i.e., slip and speed of rupture�, and other seismological andgeological constraints, some of these discussed above, we esti-mate a reasonable earthquake tsunami source for the December26, 2004 event, in terms of magnitude and timing of the co-seismic seafloor displacement along 1,200 km of the Andaman-Sunda trench. We then iteratively refine this estimate by furtherconstraining the source and simulated tsunami to match salientfeatures of tide gauge and satellite track records.

Our earthquake tsunami source is based on the standard half-plane solution for an elastic dislocation with maximum slip ��Okada 1985�. Thus, we define an oblique planar fault of horizon-tal length L and width W, with centroid located at latitude-longitude �x0 ,y0�, and depth d of the earthquake at the centroid,and discretize it into many small trapezoids. The vertical co-seismic displacement on the ocean floor surrounding the fault iscalculated by summing up contributions of point source elasticsolutions, based on the actual depth of each trapezoid. The shearmodulus � can be specified as a function of depth and otherseismic and geological descriptors, although it will be assumedto be constant in this work. Okada’s solution is implementedin “Tsunami open and progressive initial conditions system”�TOPICS�, a software tool that provides the vertical co-seismicdisplacements as outputs, as well as a characteristic tsunamiwavelength �0 �smaller of the fault dimensions L or W� and acharacteristic tsunami period T0. A characteristic initial tsunamiamplitude �0 is defined as the minimum or maximum eleva-tion found from the bottom coseismic displacement. The seismicmoment M0 is proportional to, but slightly less than, �LW�,because a Gaussian slip distribution is assumed about the cen-troid. TOPICS allows for the superposition of multiple faultplanes, which can be assembled into complex fault structures orslip distributions.

To perform tsunami simulations with the propagation modelFUNWAVE, we first define a model grid and specify the bottombathymetry in the modeled ocean basin. We then trigger a seriesof discrete, properly parameterized, Okada’s sources, in a timesequence spanning the selected rupture duration. In doing so, fol-lowing the standard procedure, we assume that each source,which represents the final co-seismic bottom deformation inducedby the earthquake over a given area, or fault segment, is instan-taneously reproduced as an ocean surface elevation, with thewater having no initial velocity. To facilitate such simulations, wecombine TOPICS and FUNWAVE into a single integrated model,referred to as GEOWAVE, in which the tsunami sources calcu-lated by TOPICS for a tsunami event are transferred and linearlysuperimposed into FUNWAVE, as an initial free surface condi-tion. �The application of this methodology to landslide tsunamisources is detailed in Watts et al. 2003�.

Geological and Seismological Constraints

The geologic structures responsible for the December 26, 2004event are approximately identified in the offshore bathymetry bythe Andaman-Sunda trench, unless they are buried under loosesediment. As mentioned in the “Introduction,” these structures aregenerally described as the Indian-Australian �or downgoing� platesubducting beneath the Eurasian/Andaman �or overriding� plate,with a largely east-west direction of convergence. In the Bay of

Bengal the morphology of the seafloor is thus an expression of thethree-dimensional tectonic structures that exist, as well as the tec-tonic processes that are taking place at depth.

In our initial modeling of the December 26, 2004 event �Wattset al. 2005�, given the bathymetry of the Bay of Bengal, thegeometry of the subduction zone, and distributions of rupture andaftershocks provided by initial seismic inversion models �TaniokaPersonal communication 2005�, we first identified four fault seg-ments with different morphologies and earthquake parameters.These four segments were L=220, 410, 300, and 350 km long,making up the 1,200 km of ruptured subduction zone, and wereidentified by their unique shape and orientation. Four Okadasources, corresponding to each of these segments, were specifiedin FUNWAVE to simulate the event. These were triggered attime t0=0, 105, 223, and 331 s, corresponding to a rupture speedinitially estimated at 3 km/s. The simulated tsunami agreed rea-sonably well with arrival times at seven tide gauges, reproducedsalient features of JASON 1’s satellite transect, and predictedgeneral ranges of variations of measured coastal runups �Wattset al. 2005�. Details, however, were not well simulated, suchas amplitudes and periods of successive tsunami waves arrivingat tide gauges and the front and back of the satellite transectelevations.

In this work, we gradually refined our initial tsunami sourcesby integrating further constraints from seismic inversion models�Ammon et al. 2005; Lay et al. 2005�; GPS data �Chlieh et al.2005; Vigny et al. 2005�, and other detailed seismological andgeological analyses performed for the event. As discussed in the“Introduction,” we then iteratively adjusted the source parametersfor the generated tsunami to better match observations. In doingso, however, we tried to have as small a number of sources/segments as possible, in order to both reduce the number of freeparameters to adjust and limit the generation of spurious tsunamiwaves at discrete segment junctions. This led us to replace ourmiddle two segments by three segments and thus use a total offive segments that both better match the shape of the rupturedarea and known rupture parameters. Let us consider each segmentin turn �Table 1; Fig. 1�:1. Segment 1 �L=220 km� covers the southern arc of the rup-

tured subduction zone, facing in a general SW direction �oftsunami propagation�, perpendicular to rupture, and roughlyextends NW of the epicenter. The faulting trends north alongtwo relatively sharp bends, one to the north and one to thesouth of the segment. Here, the overriding plate is at itssteepest, and the water depth is largest along the rupturedsubduction zone, at around h=5, 100 m in the deepest part ofthe Java trench.

2. Segments 2 and 3 cover a long �L=150 and 390 km� andrelatively straight section of the subduction zone in a NNWdirection along the trench. The most notable feature is thenearly uniform profile of the overriding plate in the northernSegment 3, with a steep rise from the subduction trench to ashallow ridge, followed by a descent into a deeper basinfarther east. The southern, shorter and wider Segment 2,covers the slip asperity, predicted off Banda Aceh in seismicinversion models, corresponding to a larger maximum slipresponsible for the largest coastal runups measured inand around Banda Aceh. Direct effects of this large slip inthe form of seafloor uplift may have been observed duringthe SEATOS cruise in the so-called “ditch” feature �Moranet al. 2005�.

3. Segments 4 and 5 �L=150 and 350 km� feature a markedchange in orientation and shape, notably a widening of

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the distance between the subduction zone and the basin tothe east. The basin is narrower here, more in the form of atrench. The ridge is shallow enough to form a numberof small islands. Segment 4 is facing Northern Thailand,where very large runup was measured, e.g., in Khao Lak.In Segment 5, a significant number of larger islands �theAndaman Islands� are formed on the overriding plate �theseare better visible in Fig. 2�.

Finally, and this is one of our important findings, in order tomatch the arrival times of successive tsunami waves measured atfar distant tide gauges, and at the same time reproduce the tail ofJASON 1’s satellite transect in the simulation, the tsunamisources corresponding to the five selected segments must be trig-gered over about 1,200 s. This is a much longer time than usedearlier, corresponding to a speed of co-seismic bottom deforma-tion only averaging 0.8 km/s, i.e., much smaller than rupturespeed. Some of this reduction in speed can be explained by so-called rising time effects �Heaton 1990�: the combination of rup-ture �i.e., slip� propagation along the fault plane at �1 m/s and oflateral rupture propagation in the segment width direction, at2–3 km/s, yields in our case an average rising time of 60 s.However much of this reduction in speed remains unexplained.The ocean bottom in the subduction zone and accretionary prismis made of a number of layers of different materials, with quitedifferent geological properties and shear moduli �. In the softersediment of the accretionary prism, the shear wave speed is typi-cally smaller, 0.6–0.8 km/s �e.g., Kramer 1996�. Hence, while itis well established that the earthquake recorded at seismographsdid proceed at a deep shear wave speed of 2–3 km/s, one mightconjecture that, because of the significant accretionary prism infront of the subducting plate, it would appear that the surfacerupture, responsible for the co-seismic bottom deformation thatgenerated the tsunami, occurred solely in softer sediment andhence at a much lower speed of propagation. Finally, it is impor-tant to point out that two other modeling groups independentlyreached a similar conclusion that the apparent rupture speed mustbe reduced in the tsunami propagation simulations, as comparedto predictions of seismic inversion models or hydroacoustic mea-surements. Satake et al. �2005, 2006� and Fujii and Satake �2006�,

using over 20 independent Okada tsunami sources with param-eters optimized using a linear inversion algorithm, found that itwas necessary to reduce the rupture speed to about 1 km/s for thegenerated tsunami to match JASON 1’s and two other satellitetransects and the many tide gauge data. Tanioka et al. �2005�similarly significantly reduced the rupture speed.

Tsunami Source Parameters

We define five separate Okada tsunami sources for the fivesegments S1–S5 shown in Fig. 1 and detailed above. We triggereach source at increasing time t0 in FUNWAVE, according to the�reduced� speed of propagation of co-seismic bottom deformationfound necessary to match observations, i.e., over about 1,200 sfrom south to north.

The earthquake parameters for each tsunami source arelisted in Table 1. The total seismic moment released isM0=0.76�1023 J, equivalent to Mw= �log M0−9� / log 32=9.22,with �=4.0�1010 Pa. To reduce the number of free parameters inOkada’s dislocation sources, in the absence of accurate geologicalinformation, we initially assumed all segments to have a rakeangle �=90° and a dip angle �=12°, such as to reproduce thecorrect distances between seafloor features. We adjusted the strikeangle � to closely follow the bottom bathymetry �Fig. 1�. Thewidth W of each segment, which also represents the characteristictsunami wavelength �0 and hence is proportional to the charac-teristic tsunami period T0��0 /�gh �where h is the local depthand g is gravity�, was initially selected based on the distributionof seafloor deformation obtained in seismic inversion models dis-cussed above. The width was then iteratively adjusted for thesimulations to better reproduce the main tsunami periods mea-sured at tide gauges; thus W generally reduced from 130 km inthe south to 95 km in the north. Based on slip distributions pre-dicted in seismic inversion models and GPS data �Ammon et al.2005; Vigny et al. 2005�, the earthquake depth d was fixed at25 km, and maximum slip � was set to 12–18 m, except inSegment 2 where it was increased to 23 m to model the asperityoff Banda Aceh.

Table 1. Tsunami Source Parameters Used in TOPICS for Okada’s �1985� Source Segments S1–S5 Shown in Fig. 1. Total Surface Elevation ComputedUsing These Sources is Shown in Fig. 2. Time Delay of Segment Rupture from Earthquake Time.

Parameters Segment 1 Segment 2 Segment 3 Segment 4 Segment 5

x0 �longitude� 94.57 93.90 93.21 92.60 92.87

y0 �latitude� 3.83 5.22 7.41 9.70 11.70

d �km� 25 25 25 25 25

� �degs� 323° 348° 338° 356° 10°

� �degs� 90° 90° 90° 90° 90°

� �degs� 12° 12° 12° 12° 12°

� �m� 18 23 12 12 12

L �km� 220 150 390 150 350

W �km� 130 130 120 95 95

t0 �s� 60 272 588 913 1273

� �Pa� 4.0�1010 4.0�1010 4.0�1010 4.0�1010 4.0�1010

M0 �J� 1.85�1022 1.58�1022 2.05�1022 0.61�1022 1.46�1022

�0 �km� 130 130 120 95 95

T0 �min� 24.77 17.46 23.30 18.72 18.72

�0 �m� −3.27; +7.02 −3.84; +8.59 −2.33; +4.72 −2.08; +4.49 −2.31; +4.60

Note: A 60 s rising time is included in time delay of segment rupture from earthquake time in t0 and maximum slip � is Gaussian distributed and dropsby 50% from each segment’s centroid to L km from it. Initial time t=0 corresponds to 0 h 58 min 53 s GMT. The total seismic moment of all fivesegments is M =7.55�1022 or M =9.25.

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The maximum vertical seafloor uplift subsidence predicted byTOPICS for each source is listed in Table 1 and varies in therange �0=−3.8–8.6 m, which is consistent with the range of val-ues estimated by the seismic inversion models. The total co-seismic seafloor vertical displacement obtained for the five com-bined tsunami sources is depicted in Fig. 2, with uplift andsubsidence contours plotted at a ±1 m spacing. We note rightaway the similarity of our source with the uplift-subsidence con-tours inferred from seismic inversion models �e.g., Ammon et al.2005�. The maximum uplift �9 m is predicted west of the north-ern tip of Sumatra, in Segment 2. In the northern Andaman Is-lands the uplift is about 1–2 m, and slightly more for the middleand southern islands. Such uplifts of the Andaman Island wereconfirmed by field surveys �Kayanne et al. 2005�. The five tsu-nami sources do not merge perfectly at all locations with oneanother, as a result of the division of the source in discrete seg-ments, although this fact disappears from the wave front in modelsimulations, within a few minutes of tsunami propagation. Wealso note that the source for each segment has a slightly differentshape of co-seismic displacement. These differences arise largelyout of the variations in width and slip between each segment, andare intended to mimic seafloor bathymetry and the known features

of the rupture. Although the source in Fig. 2 does not perfectlymatch all seafloor features, it captures all major characteristicsof the seafloor morphology and, as we will see, the generatedtsunami agrees well with observed data.

The first tsunami source, for Segment 1, is triggered at the startof the numerical simulation, t0=60 s, the chosen rising time. Wethen calculate the delay between the triggering of subsequenttsunami sources from the distance between epicentral locationsalong the rupture path to the segment center. In doing so, asdiscussed before, we assume a velocity of bottom deformationcaused by the rupture down to about a third of what was predictedby seismic inversion models for the shear wave speed, i.e.,0.87 km/s in the south and 0.70 km/s in the north, with an aver-age shear wave speed of 0.8 km/s. This yields the triggeringtimes listed in Table 1.

Tsunami Simulations

We simulate the December 26, 2004 tsunami propagation in theBay of Bengal using FUNWAVE, with the main purpose of bothconstraining and validating our tsunami source. In the simula-tions, we specify the five earthquake tsunami sources in a timesequence, with parameters such as listed in Table 1, correspond-ing to the five rupture segments S1–S5 shown in Fig. 1. Perform-ing more than 15 such simulations, we compared simulatedtsunami elevations with data measured at tide gauges, one satel-lite transect, and coastal runup. Source parameters were itera-tively adjusted in light of these comparisons, which eventuallyyielded parameters listed in Table 1.

Construction of Model Grids

In order to include all relevant tide gauges in the Bay of Bengal,but minimize grid size while achieving maximum resolution, themodel grid used for the ocean scale basin covers the area depictedin Fig. 1, from 72° to 102°E in longitude and from 13.0°S to23.5°N in latitude. Simulations are performed on a 1��1� grid,or about 1.85�1.85 km, which yields a grid with 1,793 by 2,191points. At this resolution, the time step was selected at 1.2 s.Open boundary conditions were specified in FUNWAVE on allocean boundaries. We constructed the numerical grid in the Bayof Bengal by interpolating the ETOPO2 bathymetry and topogra-phy data at grid points. Additionally, because of work performedfor our case studies, we digitized and merged with this data set,denser and more accurate bathymetry and topography data pro-vided by the Royal Thai Navy for coastal Thailand �Fig. 3 showswhere such points were used�. In Fig 1, the resulting bathymetriccontours are plotted every 1,000 m.

Note that the mean water level specified in the model didnot include effects of tides. Tides would slightly affect tsunamipropagation, mostly in very shallow water, through small changesin depth and, hence, propagation speed. Most of the tide gaugerecords used in the following �those from the University ofHawaii Sea Level Center �UHSLC�� were already provided with anet tsunami signal, obtained after tide removal, and we similarlyprocessed the Royal Thai Navy �RTN� tide gauge at Taphao Noi.Runup observations used in the following were tide corrected aswell by the various post-tsunami survey teams. Consequently thecomparison between observations and the simulation results isconsistent.

Fig. 2. Total tsunami source elevation computed for combination offive Okada sources, with parameters listed in Table 1. Thick �—�lines indicate uplift and �- - - -� subsidence, contoured every 1 m; thin�—-� lines show bathymetric contours every 500 m.

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Tsunami Simulation Results

The maximum simulated tsunami elevations above sea level aredepicted in Fig. 4 with details given in Fig. 5. As expected fromother work �e.g., Titov et al. 2005; Watts et al. 2005�, the tsunamiradiation patterns in Fig. 4 show high directionality, both becauseof the source length and in relation with various features ofthe seafloor. To the west, tsunami propagation depends on thesediment fan that covers most of the Bay of Bengal. To theeast, a much more complex pattern emerges due to interferenceand interactions of multiple wave fronts propagating to andamong various shorelines. Shallower water near the Andamanand Nicobar Islands also strongly affects east-west tsunamipropagation.

Satellite TransectIn Fig. 6, we compare model results with estimates of surfaceelevation measured along JASON 1’s satellite transect, shown inFig. 1. This estimate was obtained along satellite track No. 129,by calculating the difference between the anomaly of the sea sur-face elevation for transect 109, measured during the tsunamievent, and the one of transect 108 measured about 10 days before�Gower 2005; Kulikov 2005�. The satellite traveled on thistransect from 12°S and 20°N, between 2 h 51 min and 3 h02 min UTC, or about 2 h after the start of the event. Each dotin Fig. 1 represents a numerical gauge whose time series wascalculated in FUNWAVE during tsunami simulations. The actualmotion of the satellite over time, given in Gower �2005� �it took

about 8 min for the satellite to travel from 5°S to 20°N, duringtsunami propagation� is then used to select the relevant numericaldata for each gauge along the transect. Measured satellite eleva-tions appear quite noisy between 0° and 8°N, which suggest ahigh variability �intraseasonal� of the geostrophic current fieldstructure in the area.

Except for a small spatial shift at some locations, the overallagreement between measurements and simulations is quite good

Fig. 3. Locations of bathymetric data �adapted from Royal ThaiNavy map data� and depth contours �every 50 m� used for coastalThailand to construct numerical grid

Fig. 4. Maximum tsunami elevations in Bay of Bengal simulatedwith FUNWAVE, using source of Fig. 1 and Table 1 �scale is inmeters�

Fig. 5. Details of Fig. 4, maximum tsunami elevations simulatedwith FUNWAVE in Northern Sumatra, the Andaman Islands, andThailand �scale is in meters�

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in Fig. 6, which is encouraging considering the uncertainties inthe location of the MWL and the noise in the satellite data. Thetwo leading tsunami crests are well resolved in the model�3–5° S�, although they are slightly smaller than measured. Theoverall crest to trough difference �i.e., wave height� of the leadingwaves is well predicted, at about 1.1 m as compared to 1.2 m inthe data. The next main trough to crest height is also well pre-dicted, to about 1 m, in between 2°S and 2°N, and some of thesmaller oscillations are well resolved as well. Finally, the agree-ment with the tail of the satellite data, north of 5°N is quiteremarkable, particularly north of 10°N, where the tsunami is dueto generation in the northernmost Segments 4 and 5 and is some-what affected by the Andaman Islands. This in itself justifies theslower timing we adopted for the triggering of our sources, thanpredicted by seismic inversion models alone. The main discrep-ancies between simulations and observations are observed in be-tween 2° and 5°N, an area for which, due to its directionality�Fig. 4�, the tsunami is generated by Segment 3 in the model. Thissegment is the longest in our tsunami source, and the time lagbetween actual start of uplift at its southern end and the modeledstart at t0, considering each segment is treated as a single source,may explain some of the spatial lag seen in the simulated satellitetrack. This could be improved by using a larger number of seg-ments in the source. Also, in this region, simulations are morestrongly affected by propagation through the Nicobar Islands,where errors in the ETOPO2 shallow water bathymetry near theislands may affect the accuracy of the simulated tsunami. Finally,the use of discrete sources, whose edges tend to produce spurioussecondary waves bouncing off the islands and disturbing the mainlower frequency wavetrain, can also be responsible in part forthese discrepancies.

Tide GaugesTsunami elevations were measured at various coastal tide gaugesin the Indian Ocean �Merrifield et al. 2005�, of which we useseven locations marked in Fig. 1, for which accurate digital datawere available. Data shown in Fig. 7 are for three tide gauges inthe Maldives �Hannimaadhoo, Male, Gan; the northern two beingin direct line of sight along the main direction of tsunami propa-gation from the source�; Diego Garcia, south of the Maldives;Columbo, on the sheltered west side of Sri Lanka; Cocos Island,directly south of the tsunami source; and Taphao-Noi on the eastcoast of Thailand but on the sheltered east side of Phuket. Inaddition, the tsunami was recorded with a depth echo sounder bythe Belgian yacht “Mercator” which was anchored 1 mile off NaiHarn Bay �SW of Phuket�, in approximately 12 m of water atthe time of the event. Table 2 lists the tide gauge and yacht namesand their approximate locations. Fig. 7 shows both measured and

simulated time series at the tide gauges and the yacht. The actualdata points are marked by circles and we see that the time reso-lution varies between tide gauges, from 1 to 6 min, Taphao Noibeing manually digitized. In the latter case, this introduces asignificant filtering of the tsunami signal. Note in Fig. 7�e� thatthe tide gauge failed in Columbo right after the arrival of the firsttsunami crest. Also note, for the first six tide gauges, measuredelevations were filtered by applying a moving average over a 120,240, 240, 360, 120, and 60 s time window. For sake of compari-son, a similar filter was applied to the simulated tsunami eleva-tions shown in Fig. 7.

Table 2 lists computed and observed arrival times of the tsu-nami at the gauges, which we define as the time of the extremumof the first depression or elevation wave, whichever comes first.Estimated depth h1 at the gauges is also given. Simulated andmeasured arrival times agree well in most cases. The simulatedtsunami usually arrives slightly too early, by up to 3 min, exceptas expected from the above discussions on sphericity and Corioliseffects, at the two southernmost locations, Diego Garcia andCocos, where the simulated tsunami arrives 16 and 11 min tooearly, respectively. In addition, due to the coarse 1.85 km gridsize used in the model, with respect to coastal waters, the depth h0

of the boundary grid cell where the tide gauge is located does nottypically match the actual tide gauge �or yacht� depth h1 but isusually larger. This means that part of the slowing down of thetsunami, roughly proportional to �gh in shallow water, is notcorrectly modeled and having this present would produce a slighttsunami delay in the simulations.

More specifically, in Figs. 7�a and b�, we see that, except for agauge resolution effect, the agreement is good between simula-tions and observations at the two northern tide gauges in theMaldives, Hannimaadhoo and Male, for the elevation and periodof the first three waves. A good match is expected at these gauges,as they lie on a fairly direct path of tsunami propagation, orthogo-nal to the source axis �Figs. 1 and 4�. At Gan, farther south,Fig. 7�c� shows that the agreement is reasonable for the first crestbut not so good for later waves. However, this gauge is locatedwithin a somewhat protected area, which yields a weaker signalquite affected by local coastal topography not resolved in themodel. Except for a time shift, the agreement is reasonable atDiego Garcia, for the first two waves in Fig. 7�d�. In Columbo, inFig. 7�e�, the agreement for the first crest before the tide gaugefailed is quite good, particularly considering the tsunami had topropagate around the southern tip of Sri Lanka to reach the tidegauge, very much like an edge wave �e.g., Liu et al. 1998�. InCocos, in Fig. 7�f�, despite the southern location off the maindirection of tsunami propagation, the agreement is quite good in

Fig. 6. Comparison of tsunami elevation measured with satellite altimetry by JASON 1 �- - - -� and results of model simulation with: FUNWAVE�——�; and NSWE �– - –�

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amplitude and period for the first three waves, except for a timeshift. The tide gauge in Cocos is located inside a lagoon in shal-low water, and part of the time shift can be explained by the poorrepresentation of slowing down effects of waves in very shallowwater in the model. In Taphao Noi, east of Phuket in southernThailand, a depression wave first arrives, as expected, and the

agreement is quite good in amplitude and period for the first twowaves. The comparison of the arrival times is not relevant be-cause Tsuji et al. �2006� found significant time lags between therecord and the predicted tide.

Finally, in Fig. 7�h�, we see that the yacht Mercator also ex-perienced an initial depression wave, followed by three waves of

Fig. 7. Comparison of tsunami elevation measured �–�–� and simulated with: FUNWAVE �—-�; NSWE �– - –�, at tide gauges and yacht, markedin Fig. 1: �a� Hannimaadhoo; �b� Male; �c� Gan; �d� Diego Garcia; �e� Columbo; �f� Cocos Island; �g� Taphao-Noi; and �h� Mercator yacht

Table 2. Comparison of Observed Tsunami Arrival Times and Simulated with FUNWAVE, at Tide Gauges and Yacht �Fig. 1�. Tide Gauge Depth IsAssumed 5 m When Unknown. Data for First Five Gauges Are from University of Hawaii Sea Level Center; Cocos’ Data Are from National Tidal Center,Australia; and Taphao-Noi’s Data Are from Hydrographic Department, Royal Thai Navy.

LocationsCoordinates�Lat., Long.�

Model arrivaltime

Data arrivaltime

Depth h1

�m�

Hannimaadhoo, Maldives �6.767, 73.167� 3 h 39 min 3 h 40 min 5

Male, Maldives �4.233, 73.540� 3 h 26 min 3 h 25 min 4

Gan, Maldives �−0.667,73.172� 3 h 25 min 3 h 28 min 5

Diego Garcia �−7.233,72.435� 3 h 39 min 3 h 54 min 5

Columbo, Sri Lanka �7.000, 79.835� 2 h 56 min 2 h 59 min 5

Cocos Island �−12.133,96.877� 2 h 16 min 2 h 27 min 5

Taphao-Noi, Thailand �7.833, 98.417� 2 h 15 min 2 h 18 min 5

Mercator, Phuket �7.733, 98.283� 1 h 49 min 1 h 48 min 12

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elevation, with a maximum trough-to-peak height of 5.6 m. Theyacht was anchored near areas of Thailand that experienced verylarge waves and runup. The depression wave in the model arrives1 h 49 min after the earthquake, only 1 min after that measuredon the yacht, and its amplitude is almost correct. The first crestelevation is also well predicted, but with a slight time shift in theperiod, and so is the third crest, with a larger time shift. Thesecond crest, however, is almost entirely missing from thesesimulations. At this stage it is fair to state that the Mercator signalis not fully explained yet in these simulations, beyond the firstwave. More work is being performed as part of the Thailand casestudy to address these discrepancies, on a finer regional grid�Ioualalen et al. 2007�. As a final remark, as expected from seis-mological considerations, all of the tide gauge data, both modeledand observed, show leading elevation waves on the western sideof the Bay of Bengal, i.e., on the side of the uplift in the tsunamisource. By contrast, the record for the yacht Mercator and for thetide gauge at Taphao Noi, both on the eastern side near Phuket,close to the source area where subsidence occurred, show a lead-ing depression wave. This agrees with all of the eyewitnessobservations, pictures, and movies that showed that the oceanretreated on the eastern side, in Thailand, before the tsunami ar-rived, but did not do so on the western side.

RunupAs shown in Figs. 4 and 5, the largest runups are predicted in thesimulations near Banda Aceh in northern Sumatra, in westernThailand around the Nicobar–Andaman Islands, and on the east-ern side of India and Sri Lanka. This is also where the majordestruction caused by the tsunami was observed. Table 3 providesmaximum runup values calculated and observed, for a few spe-cific locations in these coastal areas. The simulated runup valuesare in good agreement with observations made during a numberof field surveys �e.g., Fritz and Synolakis 2005; Gusiakov 2005;Kawata et al. 2005; Liu et al. 2005; Satake et al. 2005, 2006;Sannasiraj and Sundar 2005; Synolakis et al. 2005; Yalciner et al.2005a,b; Yamada et al. 2005�. Regarding extreme runup valueswe note, in particular, the good agreement for the northern coastof Aceh �Sumatra�, and Khao Lak �Thailand�. The latter is a lo-cation for which detailed coastal topography was specified in themodel grid �Fig. 3�. The largest runup values, measured on thewest coast of Banda Aceh, are underpredicted by 50% in themodel. These runups however occurred through a combination of

deep inland flooding and tsunami interaction with complex topo-graphic features, which focused on waves and enhanced runup�Yalciner et al. 2005b�. Such features are not accurately repre-sented in the ETOPO2 data set �Satake et al. 2006� and are notresolved well enough in the 1� model grid. To simulate theseextreme runups, much finer grids should be used in the area ofBanda Aceh, together with accurate nearshore bathymetry andtopography.

Nevertheless, in view of the overall agreement of simulatedand observed runup values, we believe that our basin scale tsu-nami simulation captures well the key features of the actual event.

DispersionTo estimate dispersive effects, the same simulation was run usinga version of FUNWAVE that solves NSWE. Both grid dataand numerical methods were identical but some terms were can-celed in Boussinesq equations �see the Appendix�. Fig. 8 showsrelative differences calculated in percent between wave elevationscomputed with NSWE and FUNWAVE models, normalized byFUNWAVE results. To limit noise, these results were only com-puted for elevations greater than 1 m �the 1 m limit is marked onthe figure�. In general, in the deeper water region surrounding theWSW direction of main tsunami propagation west of the source,NSWE results are larger than FUNWAVE results by up to 23%.Less than 1% of the NSWE results in this region are smaller thanFUNWAVE’s and these are not shown in the figure. These find-ings illustrate the known property of NSWE waves to over-steepen, whereas in FUNWAVE smaller oscillations are shed be-hind the leading waves due to dispersion, somewhat reducingtheir amplitude. East of the source, near Sumatra and Thailand,dispersive effects are less important, likely due to shorter dis-tances of propagation in deep water not letting dispersive effectssignificantly express themselves. These observations are furtherconfirmed by satellite transect and tide gauge results.

Surface elevations were calculated along the satellite transectwith the NSWE model and plotted in Fig. 6. Differences withFUNWAVE results are mainly visible south of the equator. InNSWE results, as expected, the first peak of the leading wave isslightly taller and steeper, while the second peak is much smaller�almost half trough to crest height� than in FUNWAVE’s results.The same is true for the third and fourth peak in the wavetrain,with the latter almost disappearing in NSWE results. Surfaceelevations at tide gauges were calculated as well using the NSWE

Table 3. Simulation Results at Shore and Runup Ranges Measured in Field Surveys at Few Key Locations �Kawata et al. 2005; Yalciner et al. 2005a,b;Yamada et al. 2005�

Locations �Long. E, Lat. N�Boussinesq model

�m�NSWE model

�m�

Fieldsurveys

�m�

Aceh �N coast�, Indonesia �95.323,5.570� 9.38 9.33 10–11

Aceh �N coast�, Indonesia �95.284,5.556� 14.44 14.40 10–16

Aceh �W coast�, Indonesia �95.247,5.458� 16.92 16.94 24–35

Galle, Sri Lanka �80.475,5.974� 2.97 3.23 2–3

SE coast, Sri Lanka �81.816,7.427� 6.71 8.13 5–10

Chennai, India �80.279,13.021� 2.45 2.43 2–3

Nagappaattinam, India �79.740,10.865� 4.98 4.67 2–3.5

Pulikat, India �80.333,13.383� 2.63 2.62 3.45

Kamala Bch., Phuket, Thailand �98.275,7.973� 3.46 3.47 4.5–5.3

Patong Bch., Phuket, Thailand �98.276,7.900� 2.46 2.48 4.8–5.5

Kho Phi Phi, Thailand �98.777,7.739� 3.67 3.68 4.6–5.5

Khao Lak, Thailand �98.268,8.857� 13.82 13.88 15.77

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model and plotted in Fig. 7. In the far distant Maldives, inFigs. 7�a, b, and c�, NSWE results overpredict both the first crestand trough in the tsunami wavetrain, with some of the secondaryoscillations even disappearing �Gan�. Similar observations can bemade for Diego Garcia and Cocos. Surprisingly, there is almostno difference for Columbo, maybe because, for a large part ofits propagation, the tsunami propagates around the southern tipof Sri Lanka, in the manner of an edge wave, in shallow water�i.e., without dispersion; this can be clearly seen in animationsof model results�. For Taphao Noi and the Mercator yacht inThailand, NSWE and FUNWAVE results are almost identicallikely because, as mentioned above, the distance of tsunamipropagation in deep water between the source and the continentalshelf is quite short east of the source, and the tsunami is essen-tially nondispersive in shallow water.

Conclusions

Using a variety of seismological, geological, seafloor morphol-ogy, and tsunami elevation constraints, we developed a tsunamisource made of five properly parameterized dislocations sources�Okada 1985�, for the December 26, 2004 tsunami. These sourcessimulate the co-seismic bottom deformation, caused by the earth-quake, that propagated along a 1,200 km long rupture zone of theAndaman-Sunda trench. Our seafloor deformation agrees wellwith predictions of seismic inversion models as well as GPS data.The only significant discrepancy with these results, which is alsoa significant finding �corroborated by other investigators�, is thatour apparent rupture speed must be reduced to only �0.8 km/s inorder for the simulations to better match tsunami observations.We give a tentative explanation for this speed reduction, in rela-

tion with the much reduced shear wave speed in the softer sedi-ment making the accretionary prism, in front of the subductionzone. Thus, it would appear as if the near surface rupture, respon-sible for the co-seismic bottom deformation that generated thetsunami, occurred solely in softer sediment and hence at a lowerspeed of propagation than the much deeper earthquake rupture.

We simulate the tsunami by specifying the five dislocationsources as a time sequence of free surface elevations �lasting�1,200 s� in a higher-order Boussinesq model �FUNWAVE�. Themodel grid has a 1��1� regular mesh, mostly constructed withthe ETOPO2 bathymetry and topography �except in coastal Thai-land where more accurate data are used�. We find reasonableagreement between model simulations and measured elevations ata few shallow water tide gauges, a deep water satellite transect,and runup values observed at many shoreline locations. Consid-ering the data available at the present stage, to both construct themodel grid and constrain the tsunami source, and the mostly smalldifferences found between observed and simulated tsunami eleva-tions and timing, we believe that we have developed a reasonablefad ad hoc source for the December 26, 2004 event. This sourcecould certainly be further improved, for instance by using a largernumber of segments �as in Satake et al. 2006�, but it is not clearthat the larger number of free parameters this would involvewould be realistically constrained and that simulation resultswould yield better predictions of coastal inundation and runup,which are the ultimate important outcome of such simulations.

Dispersive effects were quantified in the simulations by run-ning a version of FUNWAVE solving NSWE. Differences of upto �20% in surface elevations, between Boussinesq and NSWEsimulations, mostly occur west of the source, in deeper water.Almost no differences occur east of the source, at least for thegrid size we used, likely because the deep water propagation dis-tance is very short, before the tsunami reaches the shallow conti-nental shelf over which dispersive effects essentially disappear.Frequency dispersion effects are always present but can be quitesubtle in the case of a seismically generated tsunami, where theyare expressed in the details of, for example, a sequence of wavecrests in the tsunami wave packet �this can be seen, for instance,in the satellite transect results�. In view of the apparently smalldispersive effects found here, it could be argued that the use of afully nonlinear Boussinesq equation model is overkill in the con-text of a general basin-scale tsunami model. However, it is ourfeeling that the generality of the modeling framework provided bythe model is advantageous in that it automatically covers most ofthe range of effects of interest, from propagation out of the gen-eration region, through propagation at ocean basin scale, to runupand inundation at affected shorelines. Rather than switching fromone model regime to another as we move through this range, weuse a single comprehensive model that adapts automatically tocover each range. Besides, computationally, such simulations canbe run in a matter of a few hours on a small computer cluster and,hence, are no longer prohibitive.

Our 1� model grid is quite fine for a basin scale simulation butis still not refined enough near certain coastlines �e.g., northernSumatra� to capture the runup process in detail. Likewise, theshallow water bathymetry and coastline details were not availableor used in this work �except for coastal Thailand�. We expect torefine the model grid and the bathymetry/topography further inthe near future, to conduct regional case studies, and further ex-ploit the significant capabilities of our Boussinesq wave propaga-tion and inundation model �e.g., Ioualalen et al. 2007�. In theselocal studies, we will use the tsunami source that has been con-strained and validated in the present work. The main modeling

Fig. 8. Maximum tsunami amplitudes simulated in Bay of Bengalusing FUNWAVE �gray shading in background; scale is in meters�.Contoured percentages indicate relative differences between NSWEand FUNWAVE results in areas where amplitude is �1 m: �—� 10%;thick �– –� 23%.

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challenge is to move across a sequence of spatial resolutionsneeded to resolve wave crests as they move from the deep oceaninto complex coastal environments. This hierarchical sequencingof cascading model scales has not yet been implemented in apractical model and should be an important focus of future work.Refining the model grid towards the coast thus could be done byusing nested grids, where forcing of the simulations in the finerregional grid is obtained from simulation results in the coarser,basin scale grid �e.g., Titov and Synolakis 1998 for the NSWE�.Another approach is to use unstructured or boundary fitted grids.The latter have already been implemented in a newer version ofFUNWAVE �Shi et al. 2001�.

Another area for improvement of simulations is the inclusionof sphericity and Coriolis effects in FUNWAVE. Kirby et al.�2004� have presented initial results of such an implementation.Although, for the December 26, 2004 event, this would likely notaffect near field results, particularly in Indonesia and Thailand,this could clearly be important to simulate far field observations.Reduction of the arrival time shift at far distant tide gauges, inparticular, would be expected in such simulations. In any case,these additions would be required to apply FUNWAVE in largerbasins such as the Pacific Ocean.

Finally, regarding the source itself, there is little doubt that afully dynamic source, rather than a series of static sources speci-fied in a time sequence in the model, would yield much bettersimulation results, even given the same basic parameters. Thereason has to do with both the long duration of the December 26,2004 rupture processes and, as mentioned above, the initial per-turbation introduced in the wavefield by using discrete, discon-tinuous source segments. A dynamic source could actually bespecified as a bottom boundary condition, directly in the Bouss-inesq model. A last area of tsunami source improvement is therefinement of Okada’s �1985� homogeneous half-plane elasticdislocation solution to include an inhomogeneous medium morerepresentative of a subduction zone. Some direct observations ofseafloor disturbance off of Sumatra, made during the SEATOScruise �Moran et al. 2005; Mosher et al. 2005�, in fact indicatethe possibility that much larger uplift ��12 m� may have oc-curred than predicted here. Masterlak �2005� performed numeri-cal simulations for a dislocation source, in which the medium ismade of three different materials representing deep and shallowlayers, and the softer accretionary prism material. For the sameparameters as an Okada source, he obtained up to twice the sea-floor deformation of a purely homogeneous medium. Such asource could greatly affect tsunami generation in some areas andhelp explain some of the extreme runup values observed for theDecember 26, 2004 event �e.g., 49 m in Sumatra�.

We believe that work such as this, in which we achievea better understanding through modeling of the catastrophicDecember 26, 2004 event, will help the scientific community bet-ter predict and mitigate any such future disaster. This will beachieved through a combination of forecasting models withadequate warning systems, and proper education of the localpopulations. Such work must be urgently done in light of thecertitude that large, potentially tsunamogenic, earthquakes occuralong all similar megathrust faults, with a periodicity of a fewcenturies.

Acknowledgments

The writers would like to gratefully acknowledge the followingorganizations in Thailand: the NECTEC center �Bangkok� for

the use of their computer cluster and the Chulalongkorn Univer-sity Tsunami Center, through Dr. P. Charusiri, for providing uswith useful field survey data, and the Marine Dept., throughDr. A. Sanitawong, for providing them with digitalized topogra-phy and bathymetry data sets. M. Vallee from GSA is acknowl-edged for discussions on seismology and M. Merrifield fromUHSLC for kindly providing them with the tide gauge records.S. T. Grilli, J. T. Kirby, and F. Shi acknowledge continuingsupport from the Office of Naval Research, Coastal GeosciencesProgram. M. Ioualalen gratefully acknowledges IRD for grantinghim a 4-month visit to Chulalongkorn University, Math Dept.,and AVIC colleagues for having hosted him. He also gratefullyacknowledges the Agence Nationale pour la Recherche �ANR� forsupporting this work through TSUMOD Grant No. ANR-05-CATT-016-02.

Appendix: Boussinesq Model Equations

The Boussinesq model �BM� equations implemented in FUN-WAVE are based on the work of Wei et al. �1995�, with exten-sions to cover bottom friction, breaking, and shoreline runup ef-fects developed by Chen et al. �2000� and Kennedy et al. �2000�.The equation for volume conservation is given by �subscript tindicates a partial time derivative�

�t + �h · M = 0 �1�

where ��x ,y , t�surface displacement away from local meandepth h�x ,y�; �hgradient in horizontal coordinates �x ,y�; andMdepth-integrated horizontal volume flux. M is given by

M = ��u� + z�2

2−

1

6�h2 − h� + �2��hA + z� +

1

2�h − ��hB�

�2�

Here, u�horizontal velocity at an elevation z� defined with zoriented upwards from the free surface, taken to be z�=−0.531h,following Wei et al. �1995�. A and B are functions of velocitygiven by

A = �h · u� �3�

B = �h · �hu�� 4�

The factors and � were introduced by Kennedy et al. �2000�and Chen et al. �2000� to implement a porous �i.e., absorbing�beach method, used to keep the subaerial portion of the modelgrid computationally active and to simplify the calculation ofrunup on dry shorelines. These factors are given by

= � 1, � z*

� + �1 − ��e��−z*�/h0, � � z* �5�

and

� = �� − z*� + ��z* + h0� +�1 − ��h0

��1 − e−��1+z*/h0�� , � z*

�� + h0� +�1 − ��h0

�e��−z*�/h0�1 − e−��1+�/h0�� , � � z*

�6�

The porous layer depth h0 must be deeper than depth of maxi-mum wave rundown during a calculation. The choice of z* isdiscussed by Kennedy et al. �2000�. Here we use parameter val-

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ues �=0.08 and �=25, based on studies of a number of tsunamirunup events �Watts et al. 2003; Day et al. 2005�.

The momentum equations are given by

u�t + �u� · �h�u� + g�h� + V1 + V2 + R f − Rb = 0 �7�

where V1 and V2 represent dispersive effects and are given by

V1 =z�

2

2�hAt + z��hBt − �h��2

2At + �Bt� �8�

V2 = �h��z� − ��u� · �h�B +1

2�z�

2 − �2��u� · �h�A�+

1

2�h��B + �A�2� �9�

Rb and R fforces arising from wave breaking and bottom fric-tion, respectively, and are explained in Kennedy et al. �2000�.

NSWEs follow from the above by neglecting terms represent-ing frequency dispersion, leading to the results

M = �u� �10�

V1 = V2 = 0 �11�

This formulation is included as a regular option in the FUNWAVEcode.

FUNWAVE as used here does not include a moving bottom,and tsunami signals are introduced as static surface elevation dis-placements. In the event of multiple sources with staggered initialtimes, each source is introduced by linearly superimposing it onthe already evolving wave field. This effectively eliminates anynonlinear effects arising during the initial upthrust of the watercolumn above each source.

The version of FUNWAVE used here does not utilize the re-vision of dispersive terms discussed by Chen et al. �2003�, whichincorporates an improved representation of vorticity. The problemstudied here, with propagation based on an initial static source, isessentially irrotational, and differences between the models wouldnot be apparent until after interaction between the tsunami wavefront and inundated shorelines.

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