## Abstract

We use a two-dimensional tsunami sediment transport model to study the source of the 2004 earthquake. To test the model behavior, numerical experiment on sediment deposition and erosion is performed using various hypothetical parameters of tsunami wavelength, topographic slope, and sediment supply. The numerical experiment results show that erosion and deposition are strongly influenced by the tsunami wavelength and the topographic slope. The model is used to compute the spatial distribution of tsunami deposit thickness produced by the 2004 Indian Ocean over an actual elevation datasets in the coastal area of Lhok Nga, Banda Aceh, Indonesia. The model produced simulated tsunami deposits that have similar thicknesses with the measured data along a surveyed transect. Then we estimate a simple fault model for the southern portion of the 2004 earthquake using tsunami sediment transport simulations. The simulated tsunami run-up from the fault model is very close to the measured run-up. This result indicates that a source process of a large earthquake that generates a large tsunami has a potential to be estimated using sediment deposit distribution data.

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## 1. Introduction

Large tsunamis can deposit sand layers up to several tens centimeters and distributed sand layers several kilometers inland. For example, in case of the 2004 Indian Ocean tsunami, the maximum tsunami height at Lhok Nga, west of Banda Aceh, Indonesia, was 35 m, and the inundation distance reached 6 km inland (Tsuji et al., 2005; Paris et al., 2007). The tsunami deposited sand up to 80 cm thick and left mud up to 5 km inland (Moore et al., 2006).

Recent tsunami sediment deposits have been studied by Nishimura and Miyaji (1995) in Hokkaido, Dawson (1994) in Java, Gelfenbaum and Jaffe (2003) in Papua New Guinea, Moore et al. (2006) and Paris et al. (2007) in Aceh, Sumatra, and MacInnes et al. (2009) in Kuril Islands. The interactions between the tsunami, the topography, and the sediment source affect the spatial distribution and characteristics of tsunami deposits (Dawson, 1994; Gelfenbaum and Jaffe, 2003). Jaffe and Gelfenbuam (2007) developed a simple model for tsunami sedimentation that can be applied to calculate tsunami flow speed from the thickness and grain size of a tsunami deposit. Two-dimensional simulation models that simulate topographic change induced by tsunami previously developed by Takahashi et al. (1999, 2000), Fujii et al. (2008), and Nishihata et al. (2005). A three-dimensional hydrostatic shallow water model, the C-HYDRO3D, was developed to study tsunami sediment transport at coastal areas near a harbor (Kihara and Matsuyama, 2010). Another three-dimensional numerical model, the Deflt3D, was used to simulate the inundation and sediment transport of tsunami over measured and idealized coastal morphologies at Fagafue Bay, American Samoa (Apotsos et al., 2011a) and Kuala Meurisi, Sumatra, Indonesia (Apotsos et al., 2011b).

On the other hand, when those tsunami deposits were buried and preserved, they became the geological records of past tsunamis (Paris et al., 2007). Paleotsunami deposits have been used to estimate the recurrence interval of great earthquakes (Atwater, 1987; Minoura et al., 2001; Jankaew et al., 2008; Monecke et al., 2008). Paleo-tsunami deposits have been used to study the type and size of a pre-historical earthquake (Minoura et al., 2001; Namegaya et al., 2010). Namegaya et al. (2010) compare the simulated inundation area with the distribution area of the 869 Jogan tsunami deposits to determine that the earthquake was an interplate earthquake with magnitude of *M*_{w} 8.4. Those previous studies assumed that the tsunami inundation area is the same as the tsunami deposits distribution area. However, the tsunami inundation area can be much larger than the deposited area (Goto et al., 2011).

In this paper, we try to show that the tsunami deposits data are useful to find the source processes of pre-historical large earthquake. Source processes of pre-historical large earthquakes are one of key data need to be found for earthquake prediction researches. Numerical simulation of tsunami sediment transport from the co-seismic deformation using the earthquake fault model is needed for explaining the tsunami deposit data. First, to understand the interaction between tsunami hydraulics and sediment transport with a given input parameters, we perform one-dimensional sediment transport numerical experiment. Then, we test our technique to explain the actual tsunami deposit data at Lhok Nga, Banda Aceh, Indonesia, due to the 2004 Sumatra-Andaman earthquake using a two-dimensional sediment transport simulation. Finally, we try to find the possibility to estimate the slip amount on an assumed simple fault model from sediment transport simulations.

## 2. Tsunami Sediment Transport Model

To simulate tsunami propagation and inundation, the TUNAMI-N2 (IUGG/IOC TIME Project) model is used. The model solves the nonlinear shallow water equations that are based on the conservation of mass and momentum using a finite difference scheme in the Cartesian coordinate system (Goto et al., 1997). Those equations are the following:

where *t* is time, *h* is water level, *M* and *N* are discharge fluxes along x and *y* axes, *g* is the gravitational acceleration, *n* is Manning’s roughness, and *D* is the total water depth given by *h* + *d*. Boundary conditions for tsunami run-up computation are determined by the judgment of a cell for being submerged or dry. Discharge across the boundary between two cells is calculated if the ground height in the dry cell is lower than the water level in the submerged cell; otherwise, the discharge is considered to be zero (Imamura, 1996).

Among the sediment transport models mentioned in the previous section, we choose to build and use a two-dimensional model based on the method by Takahashi et al. (2000) because it is rather simple compare to the three-dimensional models. Another reason is that a two-dimensional model requires less computer time per run compare to a three-dimensional model. The magnitude of the 2004 tsunami required us to simulate sediment transport over a large area, so a two-dimensional model is preferred. Takahashi et al. (2000) assumed two distinct layers of suspended load layer and bed load layer in their model and used the concept of exchange rate that connect the two layers. The exchange rate is a balance between sediment settling down from suspended load layer and the sediment rising up to the suspension. This method was applied to study the sediment transport of the 1960 Chile tsunami in Kesennuma Bay, Japan (Takahashi et al., 2000).

To simulate sediment transported by tsunami, the continuity equations for bed load transport and suspended transport are solved numerically. The model can be used to evaluate tsunami induced erosion and deposition for a single grain diameter of sediment. The followings are the continuity equations in the bed load layer and the suspended layer.

In the above equations, *Z*_{B} is bed level, *λ* is porosity, *Q*_{Bx} and *Q*_{By} are bed load rate in −*x* and −*y* directions, *W*_{ex} is exchange load, and *C*̄ is the mean concentration of sediment in the suspended load layer. The mean concentration of the suspended load is limited up to 1%.

Takahashi et al. (2000) proposed the following equations for the bed load and exchange rate due to a tsunami.

In the above equations, *q*_{B} is bed load rate, is the submerged specific gravity of the sand particle, *ρ*_{s} and *ρ* are the densities of sediment and the fluid, respectively, *d* is the mean grain diameter, the is Shields number, *u*_{*} is shear velocity, and *ω*_{0} is the settling velocity.

Watanabe et al. (1984) showed that the bed load transport is affected by the topography and is a function of the diffusion coefficient of sediment (*ε*_{z}), which is assumed to be 2.0 (Tanaka et al., 1989). The equations of bed load rate in −*x* and −*y* directions due to the effect of topography (Watanabe et al., 1984) are the following:

The settling velocity of a particle is related to the particle shape, particle size, specific gravity, and the kinematic viscosity of the sediment. The settling velocity of a sphere in a fluid at rest can be estimated by solving the balance between the gravitational force and the drag resistance.

A simple formula to estimate the settling velocity of natural sediment particles has been obtained from the previous work of Dietrich (1982), which can be used for a given sediment diameter, shape factor, and roundness. In case of no information on shape and roundness factor, the shape factor of 0.7 and roundness value of 3.5 can be used for natural sediment particles (Jiménez and Madsen, 2003). The submerged specific gravity is assumed to be 1.65. The kinematic viscosity corresponds to fresh water at specified temperature.

## 3. Numerical Experiment

### 3.1 Method

Tsunami waveforms are simulated at a point with a distance of 2 km from the coastline of a hypothetical bathymetry by using different fault models. The fault models have different fault widths of 25 km, 50 km, and 100 km, respectively. Each of the fault models has a slip amount that is set to generate tsunami waveforms with amplitudes of about 6 m. The simulated tsunami waveforms of TW1, TW2 and TW3 have negative wave front and wave periods of about 36, 42, and 65 min, respectively (Fig. 1). These tsunami waveforms are used as inputs of sediment transport simulations in this numerical experiment.

We design three geometries of hypothetical topography of Topo1, Topo2, and Topo3 that have topographic slopes of 8:800, 5:800, and 3:800, respectively (Fig. 2). Each of the topography has three types of landforms, which are beach, coastal plain and hill. Steeper topographic slope (hill) on each of the hypothetical topography is designated to stop a tsunami for inundating farther inland. The slope of the bathymetry used for all designs is the same (slope = 3:200) with maximum depth of 30 m (Fig. 2). The cross-shore length of the elevation profiles is 3 km with modeling grid size of 20 m.

Another important feature in sedimentation process is sediment supply. Coastal areas usually formed by sandy beaches, dunes, cliffs, and rocky outcrops, it is also likely to have estuaries, lagoons, and river deltas (Bird, 2008). Sandy beaches are easy to be eroded by tsunamis while soil-formed lands are reluctant to erosion. Here we design four different geometries of bed conditions of Sp1, Sp2, Sp3, and Sp4 (Fig. 3). The geometries are applied to the topography of the Topo2. Each of the geometries contains movable bed layer and unmovable bed layer. The movable bed layer described by two types of limits, the first is the vertical limit as deep as 5 m from the surface that is applied to all geometries, and the second is the horizontal limit. The Sp1 does not have horizontal limit within the modeling area and therefore have homogenous thickness of movable bed of 5 m. In Sp2, the initial topography is not movable from 140 m to the farther inland. In Sp3, the initial topography is not movable from the shoreline to the farther inland. The Sp4 has the same inland limit as Sp2 but the offshore limit is set to be as far as 740 m from the shoreline.

### 3.2 Results

#### 3.2.1 Effect of tsunami wavelength

The results of sediment transport simulations on the three land slopes show that the tsunami with shorter wavelength generates more erosion than the tsunami with longer wavelengths (Fig. 4). This is due to the energy by the tsunami with shorter wavelength that induces larger flow acceleration than the tsunami with longer wavelength. The tsunami with longer wavelength distributes sand layer more smoothly along the coastal plain than the tsunami with shorter wavelength (Fig. 5). The tsunami with shorter wavelength accumulates more sand deposits near the hill at the back of the beach than the tsunami with longer wavelength (Fig. 5).

#### 3.2.2 Effect of topographic slope

Simulation results on the three geometries topography show that the erosions are located near the shoreline and depositions are on the coastal plain (Fig. 5). Erosion usually occurs around the slope break (Fig. 5) where spatial flow acceleration can be large. The eroded land is located near the beach where large bed load transport and large amount of sand rising to suspension immediately after the tsunami hit the shore and during backwash. The calculated volume of erosion is larger when using topography with sharper slope break, which in this case is the topography with milder slope (Fig. 4).

The simulated sand deposits thickness increase near the steeper slope topography (hill) that stops the tsunami for inundating farther inland. This is because more sand particles fall on the bed due to accumulation and saturation of suspended concentration in the water columns near the hill. Comparison of bed changes using the three topography shows that the sand layer is distributed more smoothly when using milder topographic slope. The comparisons using tsunami waveforms with different wavelengths also show the same result.

#### 3.2.3 Effect of sediment supply

The calculated volumes of deposition decrease from Sp1 to Sp2 and Sp2 to Sp3 in which the amount of movable sand on the bed decreases. The deposited sand when using the geometry Sp3 where there is no sand supply in the onshore direction is significantly smaller compare to that using the geometries of Sp1, Sp2, and Sp4. Comparison of the calculated volumes of deposition using the geometries of Sp2 and Sp4 shows that the limitation of sediment supply offshore does not affect much the deposition on land (Fig. 6). These results show that the deposition on land seems to be influenced by supply of sediment near-shore and onshore, but not significantly by supply of sediment in the offshore area.

## 4. Case Study of the 2004 Indian Ocean Tsunami

### 4.1 Data

Tsunami sand deposits were collected by Moore et al. (2006) along a 400 m transect in Lhok Nga, Banda Aceh, Indonesia. The measured transect originates at the shoreline at 5°26′32.5″N and 95°14′22.8″E. Along the transect they measured topographic profile and thickness of sediment deposited by the 2004 tsunami. The sand layer is distributed along the profile from 100 to 400 m from shoreline with maximum sediment thickness of 20 cm. The mean grain diameters of sediment samples range from 0.3 to 0.9 mm. In this study, these data are used for comparison to verify the simulation result. The tsunami run-up height of about 14 m at 400 m from shoreline is estimated from the limit of sediment deposit.

Elevation datasets that include both bathymetry and topography are required for tsunami inundation and sediment transport simulations. For bathymetry data, the General Bathymetric Chart of the Oceans (GEBCO) dataset with 30 arc-sec resolution and nautical charts around west coast of Banda Aceh are used. To reproduce the topography dataset around the surveyed transect, the BAKOSURTANAL topographic contour map and the measured ground elevations by Moore et al. (2006) are combined. Then the elevation datasets for tsunami inundation and sediment transport simulation is obtained by combining the bathymetry and the topography datasets.

### 4.2 Method

The computational area of the tsunami numerical simulation ranges from 88.5°E to 100°E and from 0° to 17°N. Different grid sizes of 1.8 km (domain A), 600 m (domain B), 200 m (domain C), 67 m (domain D), and 22 m (domain E) are respectively used to compute the 2004 tsunami (Fig. 7). Nested grid system is used to connect between grid systems with different grid sizes. The finest grid system is for coastal area around Lhok Nga, Banda Aceh. The sediment transport computation is done only in the finest grid system (domain E).

The sediment transport simulation of the 2004 tsunami is done using different sediment grain diameters of 0.3 mm, 0.5 mm, and 0.8 mm that are within the range of mean grain diameters in the study area measured by Moore et al. (2006). Each of sediment grain diameters is simulated separately. Computation time interval (Δ*t*) of 0.5 sec is selected for the sediment transport simulation in the finest grid system. Detailed topography data before the tsunami is required in order to properly compute the bed change due to the tsunami. Unfortunately, such data in this study area does not exist. Because detail topography data and the supply of sediment along the surveyed transect before the tsunami are unknown, therefore we assume the basal limit as the initial topography that is not movable by the tsunami. Movable bed layer is set to be from 30 m inland to offshore until the limit of modeling domain.

To verify the sediment transport model, we run numerical simulation of sediment transported by the 2004 Indian Ocean tsunami in Lhok Nga, Banda Aceh, Indonesia. Slip distribution for the 2004 Sumatra-Andaman earthquake was estimated using tsunami waveforms data and sea surface heights captured by satellite by previous studies (Fine et al., 2005; Hirata et al., 2006; Tanioka et al., 2006; Tanioka and Iwasaki, 2006; Fujii and Satake, 2007). In this study we use the source model estimated by Tanioka and Iwasaki (2006) because it can well explain the tsunami waveform data and satellite altimetry data.

The bottom deformation of the 2004 earthquake is calculated by Okada (1985) formula using slip distribution from Tanioka and Iwasaki (2006). The rupture area of the earthquake extends all the way to the trench axis, so the generated bottom deformation has sharp uplift near the trench axis. In this case, it is not appropriate to assume that the sea-surface deformation is the same as the sea-bottom deformation. Instead of using the assumption, the sea surface deformation is calculated by Kajiura (1963) formula using the bottom deformation.

### 4.3 Results

#### 4.3.1 The 2004 tsunami inundation and sediment deposits simulations

The simulated tsunami run-up of 12 m is close to the measured run-up from the deposition limit at elevation of 14 m. The observed tsunami inundation at the study area is explained very well by the simulated tsunami inundation from the slip distribution (Fig. 8). We compare and analyze the simulated sediment deposits and measured sediment deposits thicknesses along the surveyed transect by Moore et al. (2006). Simulated sediment deposits distribution using sediment grain diameter of 0.8 mm is closer to the measured deposits distribution compare to that using the other grain diameters.

The simulated sediment deposits with grain diameter of 0.8 mm are distributed from 100 to 400 m along the topographic profile (Fig. 9). The simulation shows that there is no sediment deposited on the beach closer than about 80 m from the shoreline. The simulated sand deposits thicken in front and on the back of a dune formation at 160 m from shoreline, which is consistent with the observation. The simulated sediment thickness from 300 to 350 m inland is very close to the observation. However, there are some disagreements between simulated sediment thicknesses and observations. The simulated thickness on the back of the dune from 200 to 250 m inland is overestimated the observation. Farther back before the hill from 350 to 400 m inland, the simulated sediment thickness is underestimated the observation.

The sedimentation process at a modeling grid that is located 220 m inland is analyzed using time series plots of flow depth, concentration of suspended sediment, bed load rate, bed change, and cumulative bed change (Fig. 10). The tsunami flow depth rapidly increases from 0 to 5 m within less than one minute (from about 46 to 47 min). The tsunami inundates with flow depth around 5 m from 47 to 53 min, and then the tsunami flow depth gradually decreases. When tsunami flow depths are relatively steady, the sediment settles to the bottom, consequently the concentration of suspended sediment decreases and the sediment on bed gradually thickens. During tsunami backwash, the deposited sediment is eroded by the tsunami and as a result, the concentration of suspended sediment increases. Large bed load rate occur when the tsunami flow depth is rapidly change. The simulation result shows that the deposited sand is transported by the tsunami in suspension.

## 5. Discussion

In this study we demonstrate a way to estimate slip amount on an assumed fault model of the 2004 earthquake using tsunami deposit data. A simple fault model of the 2004 earthquake is assumed to be a shallow dipping thrust fault (strike = 340°, dip = 10°, rake = 90°) with the shallowest part of the fault in contact with the ocean bottom. The fault length and width are 300 km and 150 km, respectively (Fig. 7). The fault is located at the southern portion of the 2004 earthquake source area. The location and size of the fault is based on previous studies on source model of the 2004 earthquake (e.g. Lay et al., 2005; Hirata et al., 2006; Tanioka and Iwasaki, 2006; Chlieh et al., 2007; Fujii and Satake, 2007). The sediment transport simulations in the study area using different fault lengths produce similar results, using sediment deposits at only one location is not enough to estimate a fault length. There is trade-off between fault width and slip amount, but such analysis is beyond the scope of this study.

To estimate the slip amount on the simple fault model by sediment transport simulations, we run simulations of sediment transport using five slip amounts of 5 m, 10 m, 15 m, 20 m, and 30 m. Then each of the simulated sediment deposits is compared with the measured sediment deposits thicknesses along the surveyed transect by Moore et al. (2006). A range of slip amount on the single fault model of the 2004 earthquake can be estimated by comparing the measured sediment deposits thickness with the simulated ones. The sediment transport simulations are done using sediment grain diameter of 0.8 mm as suggested in the previous section.

The comparisons of the measured and simulated sediment deposits for the single fault models using different slip amounts along the surveyed transect by Moore et al. (2006) are shown in Fig. 11. The sand layers deposited by the tsunami generated from the fault with slip amounts of 20 m and 30 m are much thicker than the measured deposits thickness. There is no sand deposited on the profile from the fault with 5 m slip because the tsunami is too small. The slip amounts on the single fault model of 10 m and 15 m generated tsunamis that deposited sediment layers with maximum thickness that are comparable with the maximum thickness of the measured sediment layer. Maximum run-up heights generated by the slip amounts of 5 m, 10 m, 15 m, 20 m, and 30 m are 5 m, 12 m, 15 m, 20 m, and 25 m respectively. The slip amounts of 10 m and 15 m generate maximum tsunami run-ups that are close to the observation of 14 m.

The above result suggests that comparisons between the measured and the simulated sand layer thickness distribution can be used to estimate the slip amount on the single fault of the 2004 earthquake, although some disagreements are still exist between the simulated and the measured sand deposit distribution. The disagreements can be caused by at least three factors, which are unknown detailed topography data before the tsunami, unknown sediment supply geometry before the tsunami, and the limitation in the sediment transport model. The numerical experiment shows that topography data influence strongly on the location of erosion and deposition. It also shown that sand supply near-shore and onshore strongly influences the volume of deposition inland. These suggest that, to simulate sediment transport properly, it is important to have detailed topography data and geomorphology data of coastal areas before and after tsunami.

## 6. Conclusions

The effect of tsunami wavelength, topographic slope, and sediment supply are examined using the numerical simulation. Tsunami with longer wavelength distributes sediment layer more smoothly along the coastal plain, while tsunami with shorter wavelength causes more erosion at the beach and deposit more sediment near a hill. Comparison of bed changes at the three different hypothetical topographies with different slopes shows that the volume of erosion around the beach is higher when using topography with sharper slope break. The erosion usually occurs near the slope break where the spatial flow acceleration is large. The depositions on land seem to be influenced significantly by supply of sediment near-shore and onshore, but not significantly by supply of sediment offshore.

The observed tsunami inundation in Lhok Nga, Banda Aceh, due to the 2004 great Sumatra earthquake is explained well by the simulated tsunami inundation from the slip distribution estimated by Tanioka and Iwasaki (2006). The simulated sediment layer distributed along the profile from 80 to 400 m inland with maximum sediment deposits thickness of 35 cm, which is close to the measured sediment deposits thickness. The simulation result shows that the deposited sediment deposits were mostly transported by the tsunami in suspension.

Slip amount on the assumed fault model of the 2004 earthquake is estimated to be between 10 m and 15 m from the sand deposits along the transect in Lhok Nga. The fault models with slip amount of 10 m and 15 m can generate tsunami run-ups of 12 m and 15 m, respectively, which are close to the measured run-up of 14 m. This indicates that tsunami run-up heights can be estimated from sand deposit distribution data. Furthermore, if tsunami sand deposit distributions for many transects are available, the source process of a large earthquake that generates a large tsunami can be estimated.

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## Acknowledgments

We thank Kazuhisa Goto and two anonymous reviewers for their helpful comments and suggestions.

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Gusman, A.R., Tanioka, Y. & Takahashi, T. Numerical experiment and a case study of sediment transport simulation of the 2004 Indian Ocean tsunami in Lhok Nga, Banda Aceh, Indonesia.
*Earth Planet Sp* **64**, 3 (2012). https://doi.org/10.5047/eps.2011.10.009

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DOI: https://doi.org/10.5047/eps.2011.10.009