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1 Source Constraints and Model Simulation of the December 26, 2004, Indian Ocean Tsunami Stéphan T. Grilli, M.ASCE 1 ; Mansour Ioualalen 2 ; Jack Asavanant 3 ; Fengyan Shi 4 ; James T. Kirby, M.ASCE 4 ; and Philip Watts Abstract: The December 26, 2004 tsunami was perhaps the most devastating tsunami in recorded history, causing over 200,000 fatalities 7 and widespread destruction in countries bordering the Indian Ocean. It was generated by the third largest earthquake on record 8 M w = and was a truly global event, with significant wave action felt around the world. Many measurements of this event were 9 made with seismometers, tide gauges, global positioning system stations, and a few satellite overpasses. There were numerous eyewitness 10 observations and video digital recordings of coastal tsunami impact, as well as subsequent coastal field surveys of runup and flooding. A 11 few ship-based expeditions also took place in the months following the event, to measure and map seafloor disturbances in the epicenter 12 area. Based on these various data sets, recent seismic analysis estimates of rupture propagation speed, and other seismological and 13 geological constraints, we develop a calibrated tsunami source, in terms of coseismic seafloor displacement and rupture timing along 14 1,200 km of the Andaman Sunda trench. This source is used to build a numerical model of tsunami generation, propagation, and coastal 15 flooding for the December 26, 2004 event. Frequency dispersion effects having been identified in the deep water tsunami wavetrain, we 16 simulate tsunami propagation and coastal impact with a fully nonlinear and dispersive Boussinesq model FUNWAVE. The tsunami 17 source is specified in this model as a series of discrete, properly parameterized, dislocation source segments Okada, 1985, Bull. Seismol. 18 Soc. Am., 75 4, , triggered in a time sequence spanning about 1,200 s. ETOPO2 s bottom bathymetry and land topography 19 are specified in the modeled ocean basin, supplemented by more accurate and denser data in selected coastal areas e.g., Thailand. A 20 1 min grid is used for tsunami simulations over the Indian Ocean basin, which is fine enough to model tsunami generation and propagation 21 to nearshore areas. Surface elevations simulated in the model agree well, in both amplitude and timing, with measurements at tide gauges, 22 one satellite transect, and ranges of runup values. These results validate our tsunami source and simulations of the December 26, event and indicate these can be used to conduct more detailed case studies, for specific coastal areas. In fact, part of the development of 24 our proposed source already benefitted from such regional simulations performed on a finer grid 15 s, as part of a Thailand case study, 25 in which higher frequency waves could be modeled Ioualalen et al. 2006, J. Geophys. Res., in press. Finally, by running a non-dispersive 26 version of FUNWAVE, we estimate dispersive effects on maximum deep water elevations to be more than 20% in some areas. We believe 27 that work such as this, in which we achieve a better understanding through modeling of the catastrophic December 26, 2004 event, will 28 help the scientific community better predict and mitigate any such future disaster. This will be achieved through a combination of 29 forecasting models with adequate warning systems, and proper education of the local populations. Such work must be urgently done in 30 light of the certitude that large, potentially tsunamogenic, earthquakes occur along all similar megathrust faults, with a periodicity of a few 31 centuries. 32 DOI: XXXX 33 CE Database subject headings: Tsunamis; Surface waters; Earthquakes; Hydrodynamics; Wave propagation; Wave runup; Numerical models; Geophysical surveys; Simulation models Introduction 38 The December 26, 2004 Indian Ocean tsunami was likely the 39 most devastating tsunami in recorded history, causing over ,000 fatalities in more than ten countries across the entire 41 Indian Ocean basin, with tens of thousands reported missing and over 1 million left homeless Kawata et al. 2005; Yalciner et al a. The tsunami was a truly global event, with significant 43 wave activity recorded around the world, for which the Indian 44 Ocean in fact only represented near-field tsunami wave propagation Titov et al The tsunami was generated in the Bay of Bengal by the third largest earthquakes ever recorded, with a 47 1 Professor, Dept. of Ocean Engineering, Univ. of Rhode Island, Narragansett, RI Corresponding author. grilli@ oce.uri.edu 2 Geosciences Azur CNRS-IRD, Villefranche-sur-mer, France. 3 Dept. of Mathematics, Chulalongkorn Univ., Bangkok 10330, Thailand. 4 Center for Applied Coastal Research, Univ. of Delaware, Newark, DE Applied Fluids Engineering, Inc., 5710 E. 7th St., Long Beach, CA Note. Discussion open until April 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on February 3, 2006; approved on July 17, This paper is part of the Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 133, No. 6, November 1, ASCE, ISSN X/ 2007/6-1 XXXX/$ JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007 / 1

2 Fig. 1. Tsunami simulation grid designed for Bay of Bengal using ETOPO2 bathymetry and topography contours every 500 m, with location of 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. 48 moment magnitude M w = Ammon et al. 2005; Lay et al ; Park et al. 2005; Stein and Okal The tsunami source 50 was located along the Sunda and Andaman trenches Fig. 1, 51 which mark the approximate boundary between the Indian 52 Australian and Eurasian/Andaman plates, the former plate subducting under the latter at 5 6 cm/year with a largely East West direction of convergence. The Bay of Bengal consists mostly of 55 the Indian Australian plate, with a sequence of islands running 56 north-south along the eastern edge of the bay, denoting the plate 57 boundaries and the edge of the subduction zone. In the Bay of 58 Bengal, sediments from rivers contribute to a massive sediment 59 fan that covers the entire downgoing plate from north to south, 60 whose motion creates a large accretionary wedge east of the subduction zone Davis et al Characteristics of Rupture and Seabed Deformation 63 The December 26, 2004 event started with a main shock at 0 h min 53 s Greenwich mean time GMT, when the locked fault between the plates ruptured at the megathrust earthquake s hypocenter, located 3.32 N and E, i.e., 160 km west of Sumatra, at a depth of km, liberating strain accumulated 67 from subduction since the last large earthquakes occurred in the 68 area, in 1861 and The main shock epicenter is marked on 69 Fig Seismic inversion models e.g., Ammon et al indicate 71 that the main shock, or rupture, propagated northward from the 72 epicenter, parallel to the trenches, at a shear wave speed of km/s, thus covering the 1,200 km of the ruptured fault 74 length in about 500 s. This value was confirmed by hydroacoustic measurements de Groot-Hedlin The same models pre dict that the elastic rebound associated with the earthquake caused 77 the seabed to uplift by as much as 6 m or subside by up to the 78 same amount, slightly more in some areas, over a region km wide around the subduction zone Ammon et al ; Lay et al Maximum uplift O 10 m; Bilham and subsidence a so-called asperity occurred west of Banda 82 2 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007

3 Aceh at the northern tip of Sumatra, around 5 N. Sea level changes in Andaman and Nicobar Islands, in the North, indicate that the coseismic crustal deformation extended that far north Kayanne et al. 2005; Satake 2005b, further confirming the source length. The seafloor motion displaced an estimated 30 km 3 of water on the ocean surface, causing the killer tsunami in the process Kawata et al Seismic inversion models also predict that fault slip was significantly nonuniform along the rupture length: up to m slip in the bottom two-thirds of the rupture zone and much less in the north. Global positioning system GPS measurements confirm these interpretations of rupture and deformation processes and also show that fault slip was deep and nonhomogeneous, very small in the south, very large off of Sumatra s Northern tip and Phuket about 200 km north of the epicenter, and decreasing to smaller values beyond 7 N Chlieh et al. 2005; Vigny et al Distributions of aftershocks and seafloor deformations show an arched rupture zone, with a minimum of 3 4 separate subzones or segments, which also correspond to the time progression of the rupture along the fault Ammon et al. 2005; Lay et al. 2005; de Groot-Hedlin 2005; Tanioka et al We point out here that, in such a large event, there can be many faults that experience rupture along the subduction zone, and especially along secondary structures running from the subduction zone up to the seabed, within the accretionary wedge. These secondary structures are evident, for instance, in the 3 km high face of stepped or echelon thrust faults rising above the Sunda subduction trench in the southern part, and in the rough tapestry or fabric of the seafloor on the overriding plate over the whole rupture zone McNeill et al. 2005; Henstock et al Seismic profiles using twin air guns and direct video recording using a remotely operated vehicle ROV were made across some of these structures during the Sumatra Earthquake and Tsunami Offshore Survey cruise in May 2005 SEATOS Moran et al. 2005; Mosher et al. 2005; report.html, and confirmed the existence of complex systems of faults. It is along these secondary faults that co-seismic displacement from the main shock is expressed, with many local variations about uplift/subsidence values calculated in seismic inversion models, in which simplifying assumptions are made regarding seabed and subduction zone geology. We will also see later that constraints on the tsunami source from surface elevation measurements will lead us to reduce the speed of co-seismic seabed deformation to much less than the deep shear wave speed predicted by these models. 128 Characteristics of Tsunami and Its Coastal Effects 129 Many direct measurements of the generated tsunami and its 130 coastal effects were made during the December 26, 2004 event, 131 including a few satellite overpasses e.g., JASON 1; Gower 2005; 132 Kulikov 2005 and tide gauge records see for instance The latter 134 provided approximate tsunami arrival times for many locations 135 in the Indian Ocean, of which we selected seven, for which accurate digital records were readily available. Such tide gauge records were used in the first few days following the event, when 138 little detailed seismic information was available, to quickly estimate the tsunami source area through inverse propagation of the tsunami leading wave, at the long wave speed. Thus, Satake a, for instance, found using only tsunami arrival times at 142 Vishakapatnam, India 156 min and Cocos Islands 140 min, 143 that the main area for tsunami generation was the bottom 500 km of the rupture zone outlined in Fig. 1. Similar analyses that further constrained the tsunami source area were performed later using arrival times at more gauges. Perhaps for the first time in the history of tsunami science, there were numerous detailed eyewitness observations of coastal tsunami impact in the form of video digital recordings. These provided visual estimates of wave height and, in some cases, rough arrival times of successive tsunami waves e.g., After completing field surveys, such video recordings were further processed by some of the international scientific teams to estimate tsunami flow velocity over land e.g., Vatvani et al. 2005, in Banda Aceh. In the weeks and months following the event, multiple international scientific teams surveyed coastal areas impacted by the tsunami, documenting damage, measuring runup and inundation, and assembling careful reconstructions of wave activity. Given the length of damaged coastline and number of countries involved, each team restricted their survey to a limited geographical area 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 A few ship-based expeditions took place, in the months following the event, to measure seafloor disturbances in the epicenter area, notably the HMS Scott s, a British Navy ship that conducted a high resolution multibeam survey in January February 2005 of 40,000 km 2 of the seafloor in the main tsunami generation area, north of the epicenter McNeill et al. 2005, and SEATOS Moran et al already mentioned. It appears from the various data sets available that, upon generation and following the distribution of seafloor uplift and subsidence caused by the earthquake, the westward propagating tsunami had a leading elevation wave, subsequently hitting Sri Lanka, India, the Maldives and Somalia, whereas the eastward propagating tsunami had a leading depression wave, eventually impacting Indonesia, Thailand, Malaysia, and Myanmar. Performing global analyses of tide gauge data as well as numerical modeling albeit linearized and on a coarse grid, Titov et al showed that the December 26, 2004 tsunami was very directional in the cross-source east-west direction, due to a combination of source focusing because of the long and narrow earthquake source region and bathymetric waveguides. This explains why the tsunami caused serious damage and deaths as far as the east coast of Africa and why substantial wave energy propagated to distant coasts, including different oceans. In some cases, wave heights measured at far distant tide gauges were larger than those at some near-field gauges located in the long-source south-north direction. Bangladesh, for instance, which lies at the northern end of the Bay of Bengal, did not experience much tsunami effect and had very few fatalities, despite being a low-lying country relatively near the epicenter. A few, usually three, large tsunami waves were reported to arrive in most impacted coastal areas in the Indian Ocean, with one of the latter waves usually being the largest. Tsunami coastal effects were the most severe in Banda Aceh, which is nearest the area of maximum fault slip and seafloor uplift, causing the majority of fatalities. The tsunami arrived in Banda Aceh within 34 min of the start of the event, and runup and inundation reached a 5 30 m height in most locations, advancing up to 4 km inland with current velocities up to 8 m/s Kawata et al. 2005; Vatvani et al. 2005; Yalciner et al. 2005a. One of team even reported measuring a 49 m inundation height in Rhiting, 5 25 N about 15 km southwest of Banda Aceh Shibayama et al Most impacted next was Thailand, where the tsunami arrived at the JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007 / 3

4 southern tip of Phuket Island within 1 h 40 min of the start of the event. Runup reached a 3 11 m height in most places, with the largest runup measured on the west coast in Khao Lak about 8 40 N, mdepending on the source, with 6 8 m/s current Kawata et al. 2005; Satake et al. 2005, 2006; Yalciner et al. 2005a. By contrast, in Myanmar, just north of Thailand, runup only reached 1 3 m, with the tsunami first arriving within 2 h 30 min of the start of the event. These smaller values are likely due to the smaller fault slip in the northern part of the rupture area and to the protection offered the coast of Taninthayi Division by offshore islands Satake et al. 2005, In Malaysia, Yalciner et al. 2005b also report similar m runup at most places, with one extreme 3.7 m value. On the western side of the Indian Ocean, tsunami waves arrived in Sri Lanka and southeast India, 2 h 2 h 30 min after the start of the event, causing m runup for the most part with a few extreme values reaching 8 12 m in southern Sri Lanka Liu et al. 2005; Synolakis et al. 2005; Yalciner et al. 2005a. In the Maldives, which are made of a series of atolls, runup varied greatly depending on exposition, but is generally reported to have reached m with the tsunami first arriving within 3 h 25 min of the start of the event Fritz and Synolakis 2005; Kawata et al Finally, in Somalia, where the tsunami arrived about 7 8 h after the start of the event, unexpectedly large runup values of m were measured, which can in part be explained by the high tsunami directionality briefly discussed above tsunamis/2005/tsunamis/041226indianocean/somalia/. 234 Purpose of This Work 235 In this work, in light of the characteristics of the December 26, event briefly summarized above, we focus on constructing 237 and constraining a reasonable tsunami source based on available 238 geological, seismological, and tsunami elevation and timing data. 239 We use this source to perform tsunami simulations with a numerical model of long wave propagation, coastal inundation, and runup. Here, however, we only aim at explaining the large scale 242 tsunami propagation features measured during the event, as well 243 as overall coastal tsunami impact runup surveyed following the 244 event. In other work, reported elsewhere, we use our present 245 analyses to conduct more detailed case studies of coastal tsunami 246 impact on finer regional model grids, for selected areas such as 247 Thailand Ioualalen et al Results obtained in the latter 248 case not shown, particularly for higher frequency waves, were 249 already used to constrain the present source, in order to ensure 250 full consistency of the various simulations. 251 Since our goal is to later perform regional case studies for 252 western and southern Thailand, and northern Sumatra, where 253 maximum runup was observed, it should be pointed out that, in 254 our iterative development of tsunami sources, we gave priority to 255 data reflecting east-west tsunami propagation rather than northsouth 256 propagation. 257 Numerical Model 258 One specificity of our modeling approach is the use, perhaps 259 for the first time for such a large scale event, of a fully nonlinear and dispersive Boussinesq long wave propagation model FUNWAVE, which was initially developed for modeling 262 ocean wave transformation from deep water to the coast, including breaking and runup Wei and Kirby 1995; Wei et al FUNWAVE retains information to O kh 2 in frequency dispersion and to all orders in nonlinearity a/h where k denotes a wave number, a denotes a wave amplitude, and h denotes a water depth scale. FUNWAVE also has a physical parametrization of dissipation processes including breaking, as well as an accurate moving inundation boundary algorithm, both of which are necessary to correctly estimate coastal tsunami effects and runup over land Chen et al. 2000; Kennedy et al Wei et al showed that the retention in FUNWAVE of nonlinear effects beyond the usual order in standard weakly nonlinear Boussinesq models is crucial to the correct modeling of shoaling solitary waves or undular bores on slopes, up to near breaking, and thus in the present case is important for modeling shoreline inundation. The presence of frequency dispersion in the model is important for the case of short or higher frequency wave propagation into relatively deeper water such as directly west of the December 26, 2004 event ruptured area, and allows for the mechanism of wave crest splitting during wave propagation over shallow bathymetry. FUNWAVE has been thoroughly validated and used to study small scale motions such as the propagation of waves in the nearshore Chen et al. 2000; Kennedy et al and the generation of wave-induced currents Chen et al. 2003, as well as regional scale tsunami propagation Day et al. 2005; Ioualalen et al. 2005; Watts et al. 2003; Waythomas and Watts Preliminary results for the modeling of the December 26, 2004 tsunami using the model have been reported by Watts et al A review of the theory behind FUNWAVE and other examples of its application are given by Kirby The Appendix gives a brief summary of equations implemented in the version of FUNWAVE used in this work. Considering the fairly small longitudinal and latitudinal extensions of the Bay of Bengal, which is our main area of interest, the present simulations were performed with a version of FUNWAVE implemented on a Cartesian grid. A spherical version of FUNWAVE, also including Coriolis corrections, has recently been derived and could be used in future work Kirby et al Sphericity corrections might play a role in simulating tsunami signals at far distant tide gauges i.e., the simulated tsunami would arrive too early in the Cartesian grid, while deviations due to Coriolis force might affect the tsunami propagation. As discussed above, FUNWAVE features more complete physics than standard models used for tsunami modeling, which are typically based on nondispersive linear or nonlinear shallow water wave equations NSWE. Specifically, for the December 26, 2004 tsunami, Kulikov 2005 performed a wavelet frequency analysis based on satellite altimetry data recorded in the Bay of Bengal in deep water, and showed the importance of dispersive effects on wave evolution. His results indicate that the leading edge of the wave components with order 10 km wavelength were significantly delayed in comparison with the much longer waves in the main wave front. He concluded that a long wave model including dispersion such as FUNWAVE should be used for this event. This result is not surprising in light of Ward s 1980 simple scaling analyses, which showed, in a constant 4,000 m deep ocean, that any wave of length less than a couple of hundred km should be dispersive. Okal 1982 had also similarly stressed the importance of modeling dispersion. Hence, in the present case, FUNWAVE can potentially yield more accurate results, given the same data and tsunami source parameters, than more standard models, which typically neglect dispersion / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007

5 325 Tsunami Source 326 Based on rupture parameters estimated by seismic inversion models i.e., slip and speed of rupture, and other seismological and geological constraints, some of these discussed above, we estimate a reasonable earthquake tsunami source for the December , 2004 event, in terms of magnitude and timing of the coseismic seafloor displacement along 1,200 km of the Andaman Sunda trench. We then iteratively refine this estimate by further 333 constraining the source and simulated tsunami to match salient 334 features of tide gauge and satellite track records. 335 Our earthquake tsunami source is based on the standard halfplane solution for an elastic dislocation with maximum slip Okada Thus, we define an oblique planar fault of horizontal length L and width W, with centroid located at latitude longitude x 0,y 0, and depth d of the earthquake at the centroid, 340 and discretize it into many small trapezoids. The vertical coseismic displacement on the ocean floor surrounding the fault is calculated by summing up contributions of point source elastic 343 solutions, based on the actual depth of each trapezoid. The shear 344 modulus can be specified as a function of depth and other 345 seismic and geological descriptors, although it will be assumed 346 to be constant in this work. Okada s solution is implemented 347 in Tsunami open and progressive initial conditions system 348 TOPICS, a software tool that provides the vertical co-seismic 349 displacements as outputs, as well as a characteristic tsunami 350 wavelength 0 smaller of the fault dimensions L or W and a 351 characteristic tsunami period T 0. A characteristic initial tsunami 352 amplitude 0 is defined as the minimum or maximum elevation found from the bottom coseismic displacement. The seismic moment M 0 is proportional to, but slightly less than, LW, 355 because a Gaussian slip distribution is assumed about the centroid. TOPICS allows for the superposition of multiple fault planes, which can be assembled into complex fault structures or 358 slip distributions. 359 To perform tsunami simulations with the propagation model 360 FUNWAVE, we first define a model grid and specify the bottom 361 bathymetry in the modeled ocean basin. We then trigger a series 362 of discrete, properly parameterized, Okada s sources, in a time 363 sequence spanning the selected rupture duration. In doing so, following the standard procedure, we assume that each source, which represents the final co-seismic bottom deformation induced 366 by the earthquake over a given area, or fault segment, is instantaneously reproduced as an ocean surface elevation, with the water having no initial velocity. To facilitate such simulations, we 369 combine TOPICS and FUNWAVE into a single integrated model, 370 referred to as GEOWAVE, in which the tsunami sources calculated by TOPICS for a tsunami event are transferred and linearly superimposed into FUNWAVE, as an initial free surface condition. The application of this methodology to landslide tsunami sources is detailed in Watts et al Geological and Seismological Constraints 376 The geologic structures responsible for the December 26, event are approximately identified in the offshore bathymetry by 378 the Andaman-Sunda trench, unless they are buried under loose 379 sediment. As mentioned in the Introduction, these structures are 380 generally described as the Indian-Australian or downgoing plate 381 subducting beneath the Eurasian/Andaman or overriding plate, 382 with a largely east-west direction of convergence. In the Bay of Bengal the morphology of the seafloor is thus an expression of the three-dimensional tectonic structures that exist, as well as the tectonic processes that are taking place at depth. In our initial modeling of the December 26, 2004 event Watts et al. 2005, given the bathymetry of the Bay of Bengal, the geometry of the subduction zone, and distributions of rupture and aftershocks provided by initial seismic inversion models Tanioka Personal communication 2005, we first identified four fault segments 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 were identified by their unique shape and orientation. Four Okada sources, corresponding to each of these segments, were specified in FUNWAVE to simulate the event. These were triggered at time t 0 =0, 105, 223, and 331 s, corresponding to a rupture speed initially estimated at 3 km/ s. The simulated tsunami agreed reasonably well with arrival times at seven tide gauges, reproduced salient features of JASON 1 s satellite transect, and predicted general ranges of variations of measured coastal runups Watts et al Details, however, were not well simulated, such as amplitudes and periods of successive tsunami waves arriving at tide gauges and the front and back of the satellite transect elevations. In this work, we gradually refined our initial tsunami sources by integrating further constraints from seismic inversion models Ammon et al. 2005; Lay et al ; GPS data Chlieh et al. 2005; Vigny et al. 2005, and other detailed seismological and geological analyses performed for the event. As discussed in the Introduction, we then iteratively adjusted the source parameters for the generated tsunami to better match observations. In doing so, however, we tried to have as small a number of sources/ segments as possible, in order to both reduce the number of free parameters to adjust and limit the generation of spurious tsunami waves at discrete segment junctions. This led us to replace our middle two segments by three segments and thus use a total of five segments that both better match the shape of the ruptured area and known rupture parameters. Let us consider each segment in turn Table 1; Fig. 1 : 1. Segment 1 L=220 km covers the southern arc of the ruptured subduction zone, facing in a general SW direction of tsunami propagation, perpendicular to rupture, and roughly extends NW of the epicenter. The faulting trends north along two relatively sharp bends, one to the north and one to the south of the segment. Here, the overriding plate is at its steepest, and the water depth is largest along the ruptured subduction zone, at around h=5, 100 m in the deepest part of the Java trench. 2. Segments 2 and 3 cover a long L=150 and 390 km and relatively straight section of the subduction zone in a NNW direction along the trench. The most notable feature is the nearly uniform profile of the overriding plate in the northern Segment 3, with a steep rise from the subduction trench to a shallow ridge, followed by a descent into a deeper basin farther east. The southern, shorter and wider Segment 2, covers the slip asperity, predicted off Banda Aceh in seismic inversion models, corresponding to a larger maximum slip responsible for the largest coastal runups measured in and around Banda Aceh. Direct effects of this large slip in the form of seafloor uplift may have been observed during the SEATOS cruise in the so-called ditch feature Moran et al Segments 4 and 5 L=150 and 350 km feature a marked change in orientation and shape, notably a widening of JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007 / 5

6 Table 1. Tsunami Source Parameters Used in TOPICS for Okada s 1985 Source Segments S1 S5 Shown in Fig. 1. Total Surface Elevation Computed Using 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 x 0 longitude y 0 latitude d km degs degs degs m L km W km t 0 s Pa M 0 J km T 0 min m 3.27; ; ; ; ; Note: A 60 s rising time is included in time delay of segment rupture from earthquake time in t 0 and maximum slip is Gaussian distributed and drops by 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 five segments is M = or M = the distance between the subduction zone and the basin to 447 the east. The basin is narrower here, more in the form of a 448 trench. The ridge is shallow enough to form a number 449 of small islands. Segment 4 is facing Northern Thailand, 450 where very large runup was measured, e.g., in Khao Lak. 451 In Segment 5, a significant number of larger islands the 452 Andaman Islands are formed on the overriding plate these 453 are better visible in Fig Finally, and this is one of our important findings, in order to 455 match the arrival times of successive tsunami waves measured at 456 far distant tide gauges, and at the same time reproduce the tail of 457 JASON 1 s satellite transect in the simulation, the tsunami 458 sources corresponding to the five selected segments must be triggered over about 1,200 s. This is a much longer time than used earlier, corresponding to a speed of co-seismic bottom deformation only averaging 0.8 km/ s, i.e., much smaller than rupture speed. Some of this reduction in speed can be explained by socalled rising time effects Heaton 1990 : the combination of rup ture i.e., slip propagation along the fault plane at 1 m/s and of 465 lateral rupture propagation in the segment width direction, at km/s, yields in our case an average rising time of 60 s. 467 However much of this reduction in speed remains unexplained. 468 The ocean bottom in the subduction zone and accretionary prism 469 is made of a number of layers of different materials, with quite 470 different geological properties and shear moduli. In the softer 471 sediment of the accretionary prism, the shear wave speed is typically smaller, km/s e.g., Kramer Hence, while it is well established that the earthquake recorded at seismographs 474 did proceed at a deep shear wave speed of 2 3 km/s, one might 475 conjecture that, because of the significant accretionary prism in 476 front of the subducting plate, it would appear that the surface 477 rupture, responsible for the co-seismic bottom deformation that 478 generated the tsunami, occurred solely in softer sediment and 479 hence at a much lower speed of propagation. Finally, it is important to point out that two other modeling groups independently reached a similar conclusion that the apparent rupture speed must 482 be reduced in the tsunami propagation simulations, as compared 483 to predictions of seismic inversion models or hydroacoustic measurements. Satake et al. 2005, 2006 and Fujii and Satake , using over 20 independent Okada tsunami sources with parameters optimized using a linear inversion algorithm, found that it was necessary to reduce the rupture speed to about 1 km/s for the 487 generated tsunami to match JASON 1 s and two other satellite 488 transects and the many tide gauge data. Tanioka et al similarly significantly reduced the rupture speed. 490 Tsunami Source Parameters We define five separate Okada tsunami sources for the five 492 segments S1 S5 shown in Fig. 1 and detailed above. We trigger 493 each source at increasing time t 0 in FUNWAVE, according to the 494 reduced speed of propagation of co-seismic bottom deformation 495 found necessary to match observations, i.e., over about 1,200 s 496 from south to north. 497 The earthquake parameters for each tsunami source are 498 listed in Table 1. The total seismic moment released is 499 M 0 = J, equivalent to M w = log M 0 9 /log 32=9.22, 500 with = Pa. To reduce the number of free parameters in 501 Okada s dislocation sources, in the absence of accurate geological 502 information, we initially assumed all segments to have a rake 503 angle =90 and a dip angle =12, such as to reproduce the 504 correct distances between seafloor features. We adjusted the strike 505 angle to closely follow the bottom bathymetry Fig. 1. The 506 width W of each segment, which also represents the characteristic 507 tsunami wavelength 0 and hence is proportional to the characteristic tsunami period T 0 0 / gh where h is the local depth and g is gravity, was initially selected based on the distribution 510 of seafloor deformation obtained in seismic inversion models discussed above. The width was then iteratively adjusted for the simulations to better reproduce the main tsunami periods measured at tide gauges; thus W generally reduced from 130 km in the south to 95 km in the north. Based on slip distributions predicted in seismic inversion models and GPS data Ammon et al ; Vigny et al. 2005, the earthquake depth d was fixed at km, and maximum slip was set to m, except in 518 Segment 2 where it was increased to 23 m to model the asperity 519 off Banda Aceh / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007

7 of the rupture. Although the source in Fig. 2 does not perfectly match all seafloor features, it captures all major characteristics of the seafloor morphology and, as we will see, the generated tsunami agrees well with observed data. The first tsunami source, for Segment 1, is triggered at the start of the numerical simulation, t 0 =60 s, the chosen rising time. We then calculate the delay between the triggering of subsequent tsunami sources from the distance between epicentral locations along the rupture path to the segment center. In doing so, as discussed before, we assume a velocity of bottom deformation caused by the rupture down to about a third of what was predicted by 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 average shear wave speed of 0.8 km/ s. This yields the triggering times listed in Table Tsunami Simulations 558 We simulate the December 26, 2004 tsunami propagation in the 559 Bay of Bengal using FUNWAVE, with the main purpose of both 560 constraining and validating our tsunami source. In the simulations, we specify the five earthquake tsunami sources in a time sequence, with parameters such as listed in Table 1, corresponding to the five rupture segments S1 S5 shown in Fig. 1. Perform ing more than 15 such simulations, we compared simulated 565 tsunami elevations with data measured at tide gauges, one satellite transect, and coastal runup. Source parameters were itera tively adjusted in light of these comparisons, which eventually 568 yielded parameters listed in Table Construction of Model Grids 570 Fig. 2. Total tsunami source elevation computed for combination of five 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. 521 The maximum vertical seafloor uplift subsidence predicted by 522 TOPICS for each source is listed in Table 1 and varies in the 523 range 0 = m, which is consistent with the range of values estimated by the seismic inversion models. The total co seismic seafloor vertical displacement obtained for the five combined tsunami sources is depicted in Fig. 2, with uplift and subsidence contours plotted at a ±1 m spacing. We note right 528 away the similarity of our source with the uplift-subsidence contours inferred from seismic inversion models e.g., Ammon et al The maximum uplift 9 m is predicted west of the northern tip of Sumatra, in Segment 2. In the northern Andaman Is lands the uplift is about 1 2 m, and slightly more for the middle 533 and southern islands. Such uplifts of the Andaman Island were 534 confirmed by field surveys Kayanne et al The five tsunami sources do not merge perfectly at all locations with one another, as a result of the division of the source in discrete segments, although this fact disappears from the wave front in model simulations, within a few minutes of tsunami propagation. We 539 also note that the source for each segment has a slightly different 540 shape of co-seismic displacement. These differences arise largely 541 out of the variations in width and slip between each segment, and 542 are intended to mimic seafloor bathymetry and the known features In order to include all relevant tide gauges in the Bay of Bengal, 571 but minimize grid size while achieving maximum resolution, the 572 model grid used for the ocean scale basin covers the area depicted 573 in Fig. 1, from 72 to 102 E in longitude and from 13.0 S to N in latitude. Simulations are performed on a 1 1 grid, 575 or about km, which yields a grid with 1,793 by 2, points. At this resolution, the time step was selected at 1.2 s. 577 Open boundary conditions were specified in FUNWAVE on all 578 ocean boundaries. We constructed the numerical grid in the Bay 579 of Bengal by interpolating the ETOPO2 bathymetry and topography data at grid points. Additionally, because of work performed for our case studies, we digitized and merged with this data set, 582 denser and more accurate bathymetry and topography data provided by the Royal Thai Navy for coastal Thailand Fig. 3 shows where such points were used. In Fig 1, the resulting bathymetric 585 contours are plotted every 1,000 m. 586 Note that the mean water level specified in the model did 587 not include effects of tides. Tides would slightly affect tsunami 588 propagation, mostly in very shallow water, through small changes 589 in depth and, hence, propagation speed. Most of the tide gauge 590 records used in the following those from the University of 591 Hawaii Sea Level Center UHSLC were already provided with a 592 net tsunami signal, obtained after tide removal, and we similarly 593 processed the Royal Thai Navy RTN tide gauge at Taphao Noi. 594 Runup observations used in the following were tide corrected as 595 well by the various post-tsunami survey teams. Consequently the 596 comparison between observations and the simulation results is 597 consistent. 598 JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007 / 7

8 Fig. 4. Maximum tsunami elevations in Bay of Bengal simulated with FUNWAVE, using source of Fig. 1 and Table 1 scale is in meters Fig. 3. Locations of bathymetric data adapted from Royal Thai Navy map data and depth contours every 50 m used for coastal Thailand to construct numerical grid 599 Tsunami Simulation Results 600 The maximum simulated tsunami elevations above sea level are 601 depicted in Fig. 4 with details given in Fig. 5. As expected from 602 other work e.g., Titov et al. 2005; Watts et al. 2005, the tsunami 603 radiation patterns in Fig. 4 show high directionality, both because 604 of the source length and in relation with various features of 605 the seafloor. To the west, tsunami propagation depends on the 606 sediment fan that covers most of the Bay of Bengal. To the 607 east, a much more complex pattern emerges due to interference 608 and interactions of multiple wave fronts propagating to and 609 among various shorelines. Shallower water near the Andaman 610 and Nicobar Islands also strongly affects east-west tsunami 611 propagation. about 8 min for the satellite to travel from 5 S to 20 N, during 625 tsunami propagation is then used to select the relevant numerical 626 data for each gauge along the transect. Measured satellite elevations appear quite noisy between 0 and 8 N, which suggest a high variability intraseasonal of the geostrophic current field 629 structure in the area. 630 Except for a small spatial shift at some locations, the overall 631 agreement between measurements and simulations is quite good Satellite Transect 613 In Fig. 6, we compare model results with estimates of surface 614 elevation measured along JASON 1 s satellite transect, shown in 615 Fig. 1. This estimate was obtained along satellite track No. 129, 616 by calculating the difference between the anomaly of the sea surface elevation for transect 109, measured during the tsunami event, and the one of transect 108 measured about 10 days before 619 Gower 2005; Kulikov The satellite traveled on this 620 transect from 12 S and 20 N, between 2 h 51 min and 3 h min UTC, or about 2 h after the start of the event. Each dot 622 in Fig. 1 represents a numerical gauge whose time series was 623 calculated in FUNWAVE during tsunami simulations. The actual 624 motion of the satellite over time, given in Gower 2005 it took Fig. 5. Details of Fig. 4, maximum tsunami elevations simulated with FUNWAVE in Northern Sumatra, the Andaman Islands, and Thailand scale is in meters 8 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007

9 Fig. 6. Comparison of tsunami elevation measured with satellite altimetry by JASON and results of model simulation with: FUNWAVE ; and NSWE in Fig. 6, which is encouraging considering the uncertainties in 634 the location of the MWL and the noise in the satellite data. The 635 two leading tsunami crests are well resolved in the model S, although they are slightly smaller than measured. The 637 overall crest to trough difference i.e., wave height of the leading 638 waves is well predicted, at about 1.1 m as compared to 1.2 m in 639 the data. The next main trough to crest height is also well predicted, to about 1 m, in between 2 S and 2 N, and some of the smaller oscillations are well resolved as well. Finally, the agreement with the tail of the satellite data, north of 5 N is quite remarkable, particularly north of 10 N, where the tsunami is due 644 to generation in the northernmost Segments 4 and 5 and is somewhat affected by the Andaman Islands. This in itself justifies the slower timing we adopted for the triggering of our sources, than 647 predicted by seismic inversion models alone. The main discrepancies between simulations and observations are observed in be tween 2 and 5 N, an area for which, due to its directionality 650 Fig. 4, the tsunami is generated by Segment 3 in the model. This 651 segment is the longest in our tsunami source, and the time lag 652 between actual start of uplift at its southern end and the modeled 653 start at t 0, considering each segment is treated as a single source, 654 may explain some of the spatial lag seen in the simulated satellite 655 track. This could be improved by using a larger number of segments in the source. Also, in this region, simulations are more strongly affected by propagation through the Nicobar Islands, 658 where errors in the ETOPO2 shallow water bathymetry near the 659 islands may affect the accuracy of the simulated tsunami. Finally, 660 the use of discrete sources, whose edges tend to produce spurious 661 secondary waves bouncing off the islands and disturbing the main 662 lower frequency wavetrain, can also be responsible in part for 663 these discrepancies. 664 Tide Gauges 665 Tsunami elevations were measured at various coastal tide gauges 666 in the Indian Ocean Merrifield et al. 2005, of which we use 667 seven locations marked in Fig. 1, for which accurate digital data 668 were available. Data shown in Fig. 7 are for three tide gauges in 669 the Maldives Hannimaadhoo, Male, Gan; the northern two being 670 in direct line of sight along the main direction of tsunami propagation from the source ; Diego Garcia, south of the Maldives; Columbo, on the sheltered west side of Sri Lanka; Cocos Island, 673 directly south of the tsunami source; and Taphao-Noi on the east 674 coast of Thailand but on the sheltered east side of Phuket. In 675 addition, the tsunami was recorded with a depth echo sounder by 676 the Belgian yacht Mercator which was anchored 1 mile off Nai 677 Harn Bay SW of Phuket, in approximately 12 m of water at 678 the time of the event. Table 2 lists the tide gauge and yacht names 679 and their approximate locations. Fig. 7 shows both measured and simulated time series at the tide gauges and the yacht. The actual 680 data points are marked by circles and we see that the time resolution varies between tide gauges, from 1 to 6 min, Taphao Noi being manually digitized. In the latter case, this introduces a 683 significant filtering of the tsunami signal. Note in Fig. 7 e that 684 the tide gauge failed in Columbo right after the arrival of the first 685 tsunami crest. Also note, for the first six tide gauges, measured 686 elevations were filtered by applying a moving average over a 120, , 240, 360, 120, and 60 s time window. For sake of comparison, a similar filter was applied to the simulated tsunami eleva tions shown in Fig Table 2 lists computed and observed arrival times of the tsunami at the gauges, which we define as the time of the extremum of the first depression or elevation wave, whichever comes first. 693 Estimated depth h 1 at the gauges is also given. Simulated and 694 measured arrival times agree well in most cases. The simulated 695 tsunami usually arrives slightly too early, by up to 3 min, except 696 as expected from the above discussions on sphericity and Coriolis 697 effects, at the two southernmost locations, Diego Garcia and 698 Cocos, where the simulated tsunami arrives 16 and 11 min too 699 early, respectively. In addition, due to the coarse 1.85 km grid 700 size used in the model, with respect to coastal waters, the depth h of the boundary grid cell where the tide gauge is located does not 702 typically match the actual tide gauge or yacht depth h 1 but is 703 usually larger. This means that part of the slowing down of the 704 tsunami, roughly proportional to gh in shallow water, is not 705 correctly modeled and having this present would produce a slight 706 tsunami delay in the simulations. 707 More specifically, in Figs. 7 a and b, we see that, except for a 708 gauge resolution effect, the agreement is good between simulations and observations at the two northern tide gauges in the Maldives, Hannimaadhoo and Male, for the elevation and period 711 of the first three waves. A good match is expected at these gauges, 712 as they lie on a fairly direct path of tsunami propagation, orthogonal to the source axis Figs. 1 and 4. At Gan, farther south, Fig. 7 c shows that the agreement is reasonable for the first crest 715 but not so good for later waves. However, this gauge is located 716 within a somewhat protected area, which yields a weaker signal 717 quite affected by local coastal topography not resolved in the 718 model. Except for a time shift, the agreement is reasonable at 719 Diego Garcia, for the first two waves in Fig. 7 d. In Columbo, in 720 Fig. 7 e, the agreement for the first crest before the tide gauge 721 failed is quite good, particularly considering the tsunami had to 722 propagate around the southern tip of Sri Lanka to reach the tide 723 gauge, very much like an edge wave e.g., Liu et al In724 Cocos, in Fig. 7 f, despite the southern location off the main 725 direction of tsunami propagation, the agreement is quite good in 726 JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING ASCE / NOVEMBER/DECEMBER 2007 / 9