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1\section{Modelling the Event}\label{sec:models}
2Numerous models are currently used to model and predict tsunami
3generation, propagation and run-up. These range in solving different equations and employing
4different methodologies with some examples being~\cite{titov97a,satake95,zhang08}. Here we
5introduce the modelling methodology employed by Geoscience Australia
6to illustrate the utility of the proposed benchmark. The methodology used by Geoscience Australia has three distinct components. Firstly an appropriate model is used to approximate the initial sea surface deformation. This model is chosen according to the cause of the intial disturbance. The resulting wave is propagated using the \textsc{ursga} model (see Section~\ref{sec:ursga}) in the deep ocean until the wave reaches shallow water, typically the $100$ m depth contour. The ocean surface profile along this contour is used as a time varying boundary condition for the \textsc{anuga} model (see Section~\ref{sec:anuga}) which simulates the propagation of the tsunami within the shallow water and the subsequent inundation of the land. This three part methodology roughly follows the three stages of tsunami evolution. The components used to model each stage of evolution are described in more detail below.
9There are various approaches to modelling the expected crustal
10deformation from an earthquake. Most approaches model the
11earthquake as a dislocation in a linear elastic medium. Here we use
12the method of Wang et al~\cite{wang03}.
13%One of the main advantages
14%of their method is that it allows the dislocation to be located in a
15%stratified linear elastic half-space with an arbitrary number of
16%layers. Other methods (such as those based on Okada's equations) can
17%only model the dislocation in a homogeneous elastic half space, or can
18%only include a limited number of layers, and thus cannot model the
19%effect of the depth dependence of the elasticity of the
20%Earth~\cite{wang03}. The original versions of the codes described here
21%are available from \url{}. The
22%first program, \textsc{edgrn}, calculates elastic Green's function for
23%a set of point sources at a regular set of depths out to a specified
24%distance. The equations controlling the deformation are solved by
25%using a combination of Hankel's transform and Wang et al's
26%implementation of the Thomson-Haskell propagator
27%algorithm~\cite{wang03}. Once the Green's functions are calculated
28%a slightly modified version of \textsc{edcmp}\footnote{For this study,
29%we have made minor modifications
30%to \textsc{edcmp} in order for it to provide output in a file format
31%compatible with the propagation code in the following section. Otherwise it
32%is similar to the original code.} is used to calculate the sea
33%floor deformation for a specific subfault. This second code
34%discretises the subfault into a set of unit sources and sums the
35%elastic Green's functions calculated from \textsc{edgrn} for all the
36%unit sources on the fault plane in order to calculate the final static
37%deformation caused by a two dimensional dislocation along the
38%subfault. This step is possible because of the linearity of the
39%governing equations.
40%In order to calculate the crustal deformation using these codes
41%a model that describes the variation in elastic
42%properties with depth and a slip model of the earthquake to describe
43%the dislocation is required.
44In order to calculate the crustal deformation a model that describes the
45variation in elastic properties with depth and a slip model of the
46earthquake to describe the dislocation is required.
47The elastic parameters used for this study are the
48same as those in Table 2 of Burbidge et al~\cite{burbidge08}. For the slip
49model, there are many possible models for the 2004 Andaman--Sumatran
50earthquake to select from
51~\cite{chlieh07,asavanant08,arcas06,grilli07,ioualalen07}. Some are
52determined from various geological surveys of the site. Others solve
53an inverse problem which calibrates the source based upon the tsunami
54wave signal, the seismic signal and/or even the run-up.
55The source
56parameters used here to simulate the 2004 Indian Ocean tsunami were
57taken from the slip model G-M9.15 of Chlieh
58et al~\cite{chlieh07}. This model was created by inversion of wide
59range of geodetic and seismic data. The slip model consists of 686
6020 km x 20 km subsegments each with a different slip, strike and dip
61angle. The dip subfaults go from $17.5^0$ in the north and $12^0$ in
62the south. Refer to Chlieh et al~\cite{chlieh07} for a detailed
63discussion of this model and its derivation. %Note that the geodetic
64%data used in the validation was also included by~\cite{chlieh07} in
65%the inversion used to find G-M9.15. Thus the validation is not
66%completely independent. However, a reasonable validation would still
67%show that the crustal deformation and elastic properties model used
68%here is at least as valid as the one used by Chlieh
69%et al~\cite{chlieh07} and can reproduce the observations just as
72\subsection{Deep water propagation}\label{sec:modelPropagation}
73The \textsc{ursga} model described below was used to simulate the
74propagation of the 2004 Indian Ocean tsunami across the open ocean, based on a
75discrete representation of the initial deformation of the sea floor, as
76described in Section~\ref{sec:modelGeneration}. For the models shown
77here, the uplift is assumed to be instantaneous and creates an initial
78displacement of the ocean surface of the same size and amplitude as the
79co-seismic sea floor deformation. \textsc{ursga} is well suited to
80modelling propagation over large domains and is used to propagate the tsunami
81until it reaches shallow water, typically the $100$m depth contour.
82%The propagation of the tsunami in shallow water ($<100$m) and inundation are modelled using a hydrodynamic package called \textsc{ursga}. This package is ideally suited to shallow water propagation and inundation as it accurately simulates flow over dry land and is based upon an irregular triangular grid which can be refined in areas of interest.
85\textsc{ursga} is a hydrodynamic code that models the propagation of
86the tsunami in deep water using a finite difference method on a staggered grid.
87It solves the depth integrated linear or nonlinear shallow water equations in
88spherical co-ordinates with friction and Coriolis terms. The code is
89based on Satake~\cite{satake95} with significant modifications made by
90the \textsc{urs} corporation, Thio et al~\cite{thio08} and Geoscience
91Australia, Burbidge et al~\cite{burbidge08}.
92The tsunami was propagated via the nested
93grid system described in Section \ref{sec:propagation data} where
94the coarse grids were used in the open ocean and the finest
95resolution grid was employed in the region closest to Patong bay.
96\textsc{Ursga} is not publicly available.
98\subsection{Shallow water propagation and inundation}
100The utility of the \textsc{ursga} model decreases with water depth
101unless an intricate sequence of nested grids is employed. In
102comparison \textsc{anuga}, described below, is designed to produce
103robust and accurate predictions of on-shore inundation, but is less
104suitable for earthquake source modelling and large study areas because
105it is based on projected spatial coordinates. Consequently, the
106Geoscience Australia tsunami modelling methodology is based on a
107hybrid approach using models like \textsc{ursga} for tsunami
108propagation up to an offshore depth contour, typically 100 m.
109%Specifically we use the \textsc{ursga} model to simulate the
110%propagation of the 2004 Indian Ocean tsunami in the deep ocean, based
111%on a discrete representation of the initial deformation of the sea
112%floor, described in Section~\ref{sec:modelGeneration}.
113The wave signal and the velocity field is then used as a
114time varying boundary condition for
115the \textsc{anuga} inundation simulation.
116% A description of \textsc{anuga} is the following section.
119\textsc{Anuga} is a Free and Open Source hydrodynamic inundation tool that
120solves the conserved form of the depth-integrated nonlinear shallow
121water wave equations using a Finite-Volume scheme on an
122unstructured triangular mesh.
123The scheme, first
124presented by Zoppou and Roberts~\cite{zoppou99}, is a high-resolution
125Godunov-type method that uses the rotational invariance property of
126the shallow water equations to transform the two-dimensional problem
127into local one-dimensional problems. These local Riemann problems are
128then solved using the semi-discrete central-upwind scheme of Kurganov
129et al~\cite{kurganov01} for solving one-dimensional conservation
130equations. The numerical scheme is presented in detail in
131Roberts and Zoppou~\cite{zoppou00,roberts00} and
132Nielsen et al~\cite{nielsen05}. An important capability of the
133finite-volume scheme is that discontinuities in all conserved quantities
134are allowed at every edge in the mesh. This means that the tool is
135well suited to adequately resolving hydraulic jumps, transcritical flows and
136the process of wetting and drying. Consequently, \textsc{anuga} 
137is suitable for
138simulating water flow onto a beach or dry land and around structures
139such as buildings. \textsc{anuga} has been validated against
140%a number of analytical solutions !!!Analytical solutions have not been published. Ask Steve.
141the wave tank simulation of the 1993 Okushiri
142Island tsunami~\cite{nielsen05,roberts06} and
143dam break experiments~\cite{baldock07}.
144More information on \textsc{anuga} and how to obtain it are available from \url{}.
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