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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~\cite{titov97a,satake95}. Here we
4introduce the modelling methodology employed by Geoscience Australia
5to 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 distrubance. The resulting wave is propagated using \textsc{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 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.
8There are various approaches to modelling the expected crustal
9deformation from an earthquake. Most approaches model the
10earthquake as a dislocation in a linear elastic medium. Here we use
11the method of Wang et al~\cite{wang03}.
12%One of the main advantages
13%of their method is that it allows the dislocation to be located in a
14%stratified linear elastic half-space with an arbitrary number of
15%layers. Other methods (such as those based on Okada's equations) can
16%only model the dislocation in a homogeneous elastic half space, or can
17%only include a limited number of layers, and thus cannot model the
18%effect of the depth dependence of the elasticity of the
19%Earth~\cite{wang03}. The original versions of the codes described here
20%are available from \url{}. The
21%first program, \textsc{edgrn}, calculates elastic Green's function for
22%a set of point sources at a regular set of depths out to a specified
23%distance. The equations controlling the deformation are solved by
24%using a combination of Hankel's transform and Wang et al's
25%implementation of the Thomson-Haskell propagator
26%algorithm~\cite{wang03}. Once the Green's functions are calculated
27%a slightly modified version of \textsc{edcmp}\footnote{For this study,
28%we have made minor modifications
29%to \textsc{edcmp} in order for it to provide output in a file format
30%compatible with the propagation code in the following section. Otherwise it
31%is similar to the original code.} is used to calculate the sea
32%floor deformation for a specific subfault. This second code
33%discretises the subfault into a set of unit sources and sums the
34%elastic Green's functions calculated from \textsc{edgrn} for all the
35%unit sources on the fault plane in order to calculate the final static
36%deformation caused by a two dimensional dislocation along the
37%subfault. This step is possible because of the linearity of the
38%governing equations.
39%In order to calculate the crustal deformation using these codes
40%a model that describes the variation in elastic
41%properties with depth and a slip model of the earthquake to describe
42%the dislocation is required.
43In order to calculate the crustal deformation a model that describes the
44variation in elastic properties with depth and a slip model of the
45earthquake to describe the dislocation is required.
46The elastic parameters used for this study are the
47same as those in Table 2 of Burbidge et al~\cite{burbidge08}. For the slip
48model, there are many possible models for the 2004 Andaman--Sumatran
49earthquake to select from
50~\cite{chlieh07,asavanant08,arcas06,grilli07,ioualalen07}. Some are
51determined from various geological surveys of the site. Others solve
52an inverse problem which calibrates the source based upon the tsunami
53wave signal, the seismic signal and/or even the run-up.
54The source
55parameters used here to simulate the 2004 Indian Ocean tsunami were
56taken from the slip model G-M9.15 of Chlieh
57et al~\cite{chlieh07}. This model was created by inversion of wide
58range of geodetic and seismic data. The slip model consists of 686
5920km x 20km subsegments each with a different slip, strike and dip
60angle. The dip subfaults go from $17.5^0$ in the north and $12^0$ in
61the south. Refer to Chlieh et al~\cite{chlieh07} for a detailed
62discussion of this model and its derivation. %Note that the geodetic
63%data used in the validation was also included by~\cite{chlieh07} in
64%the inversion used to find G-M9.15. Thus the validation is not
65%completely independent. However, a reasonable validation would still
66%show that the crustal deformation and elastic properties model used
67%here is at least as valid as the one used by Chlieh
68%et al~\cite{chlieh07} and can reproduce the observations just as
71\subsection{Deep water propagation}\label{sec:modelPropagation}
72The \textsc{ursga} model described below was used to simulate the
73propagation of the 2004 Indian Ocean tsunami across the open ocean, based on a
74discrete representation of the initial deformation of the sea floor, as
75described in Section~\ref{sec:modelGeneration}. For the models shown
76here, the uplift is assumed to be instantaneous and creates an initial
77displacement of the ocean surface of the same size and amplitude as the
78co-seismic sea floor deformation. \textsc{ursga} is well suited to
79modelling propagation over large domains and is used to propagate the tsunami
80until it reaches shallow water, typically the $100$m depth contour.
81%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.
84\textsc{ursga} is a hydrodynamic code that models the propagation of
85the tsunami in deep water using a finite difference method on a staggered grid.
86It solves the depth integrated linear or nonlinear shallow water equations in
87spherical co-ordinates with friction and Coriolis terms. The code is
88based on Satake~\cite{satake95} with significant modifications made by
89the \textsc{urs} corporation, Thio et al~\cite{thio08} and Geoscience
90Australia, Burbidge et al~\cite{burbidge08}.
91The tsunami was propagated via the nested
92grid system described in Section \ref{sec:propagation data} where
93the coarse grids were used in the open ocean and the finest
94resolution grid was employed in the region closest to Patong bay.
95\textsc{Ursga} is not publicly available.
97\subsection{Shallow water propagation and inundation}
99The utility of the \textsc{ursga} model decreases with water depth
100unless an intricate sequence of nested grids is employed. In
101comparison \textsc{anuga}, described below, is designed to produce
102robust and accurate predictions of on-shore inundation, but is less
103suitable for earthquake source modelling and large study areas because
104it is based on projected spatial coordinates. Consequently, the
105Geoscience Australia tsunami modelling methodology is based on a
106hybrid approach using models like \textsc{ursga} for tsunami
107propagation up to an offshore depth contour, typically 100 m.
108%Specifically we use the \textsc{ursga} model to simulate the
109%propagation of the 2004 Indian Ocean tsunami in the deep ocean, based
110%on a discrete representation of the initial deformation of the sea
111%floor, described in Section~\ref{sec:modelGeneration}.
112The wave signal and the velocity field is then used as a
113time varying boundary condition for
114the \textsc{anuga} inundation simulation.
115% A description of \textsc{anuga} is the following section.
118\textsc{Anuga} is a Free and Open Source hydrodynamic inundation tool that
119solves the conserved form of the depth-integrated nonlinear shallow
120water wave equations using a Finite-Volume scheme on an
121unstructured triangular mesh.
122The scheme, first
123presented by Zoppou and Roberts~\cite{zoppou99}, is a high-resolution
124Godunov-type method that uses the rotational invariance property of
125the shallow water equations to transform the two-dimensional problem
126into local one-dimensional problems. These local Riemann problems are
127then solved using the semi-discrete central-upwind scheme of Kurganov
128et al~\cite{kurganov01} for solving one-dimensional conservation
129equations. The numerical scheme is presented in detail in
130Roberts and Zoppou~\cite{zoppou00,roberts00} and
131Nielsen et al~\cite{nielsen05}. An important capability of the
132finite-volume scheme is that discontinuities in all conserved quantities
133are allowed at every edge in the mesh. This means that the tool is
134well suited to adequately resolving hydraulic jumps, transcritical flows and
135the process of wetting and drying. This means that \textsc{Anuga} 
136is suitable for
137simulating water flow onto a beach or dry land and around structures
138such as buildings. \textsc{Anuga} has been validated against
139%a number of analytical solutions and 
140%FIXME (Ole): Analytical solutions have not been published. Ask Steve.
141the wave tank simulation of the 1993 Okushiri
142Island tsunami~\cite{nielsen05,roberts06}.
143FIXME (Ole): Add reference to Tom Baldock's Dam Break valiadation of ANUGA.
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