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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
4equations and employing different methodologies with some examples
5being~\cite{titov97a,satake95,zhang08}. Here we introduce the
6modelling methodology employed by Geoscience Australia to illustrate
7the utility of the proposed benchmark. The methodology used by
8Geoscience Australia has three distinct components. Firstly an
9appropriate model is used to approximate the initial sea surface
10deformation. This model is chosen according to the cause of the intial
11disturbance. The resulting wave is propagated using the \textsc{ursga}
12model (see Section~\ref{sec:ursga}) in the deep ocean until the wave
13reaches shallow water, typically the $100$ m depth contour. The ocean
14surface profile along this contour is used as a time varying boundary
15condition for the \textsc{anuga} model (see Section~\ref{sec:anuga})
16which simulates the propagation of the tsunami within the shallow
17water and the subsequent inundation of the land. This three part
18methodology roughly follows the three stages of tsunami evolution. The
19components used to model each stage of evolution are described in more
20detail below.
23There are various approaches to modelling the expected crustal
24deformation from an earthquake. Most approaches model the
25earthquake as a dislocation in a linear elastic medium. Here we use
26the method of Wang et al~\cite{wang03}.
27%One of the main advantages
28%of their method is that it allows the dislocation to be located in a
29%stratified linear elastic half-space with an arbitrary number of
30%layers. Other methods (such as those based on Okada's equations) can
31%only model the dislocation in a homogeneous elastic half space, or can
32%only include a limited number of layers, and thus cannot model the
33%effect of the depth dependence of the elasticity of the
34%Earth~\cite{wang03}. The original versions of the codes described here
35%are available from \url{}. The
36%first program, \textsc{edgrn}, calculates elastic Green's function for
37%a set of point sources at a regular set of depths out to a specified
38%distance. The equations controlling the deformation are solved by
39%using a combination of Hankel's transform and Wang et al's
40%implementation of the Thomson-Haskell propagator
41%algorithm~\cite{wang03}. Once the Green's functions are calculated
42%a slightly modified version of \textsc{edcmp}\footnote{For this study,
43%we have made minor modifications
44%to \textsc{edcmp} in order for it to provide output in a file format
45%compatible with the propagation code in the following section. Otherwise it
46%is similar to the original code.} is used to calculate the sea
47%floor deformation for a specific subfault. This second code
48%discretises the subfault into a set of unit sources and sums the
49%elastic Green's functions calculated from \textsc{edgrn} for all the
50%unit sources on the fault plane in order to calculate the final static
51%deformation caused by a two dimensional dislocation along the
52%subfault. This step is possible because of the linearity of the
53%governing equations.
54%In order to calculate the crustal deformation using these codes
55%a model that describes the variation in elastic
56%properties with depth and a slip model of the earthquake to describe
57%the dislocation is required.
58In order to calculate the crustal deformation a model that describes the
59variation in elastic properties with depth and a slip model of the
60earthquake to describe the dislocation is required.
61The elastic parameters used for this study are the
62same as those in Table 2 of Burbidge et al~\cite{burbidge08}. For the slip
63model, there are many possible models for the 2004 Andaman--Sumatran
64earthquake to select from
65~\cite{chlieh07,asavanant08,arcas06,grilli07,ioualalen07}. Some are
66determined from various geological surveys of the site. Others solve
67an inverse problem which calibrates the source based upon the tsunami
68wave signal, the seismic signal and/or even the run-up.
69The source
70parameters used here to simulate the 2004 Indian Ocean tsunami were
71taken from the slip model G-M9.15 of Chlieh
72et al~\cite{chlieh07}. This model was created by inversion of wide
73range of geodetic and seismic data. The slip model consists of 686
7420 km x 20 km subsegments each with a different slip, strike and dip
75angle. The dip subfaults go from $17.5^0$ in the north and $12^0$ in
76the south. Refer to Chlieh et al~\cite{chlieh07} for a detailed
77discussion of this model and its derivation. %Note that the geodetic
78%data used in the validation was also included by~\cite{chlieh07} in
79%the inversion used to find G-M9.15. Thus the validation is not
80%completely independent. However, a reasonable validation would still
81%show that the crustal deformation and elastic properties model used
82%here is at least as valid as the one used by Chlieh
83%et al~\cite{chlieh07} and can reproduce the observations just as
86\subsection{Deep water propagation}\label{sec:modelPropagation}
87The \textsc{ursga} model described below was used to simulate the
88propagation of the 2004 Indian Ocean tsunami across the open ocean, based on a
89discrete representation of the initial deformation of the sea floor, as
90described in Section~\ref{sec:modelGeneration}. For the models shown
91here, the uplift is assumed to be instantaneous and creates an initial
92displacement of the ocean surface of the same size and amplitude as the
93co-seismic sea floor deformation. \textsc{ursga} is well suited to
94modelling propagation over large domains and is used to propagate the tsunami
95until it reaches shallow water, typically the $100$ m depth contour.
96%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.
99\textsc{ursga} is a hydrodynamic code that models the propagation of
100the tsunami in deep water using a finite difference method on a staggered grid.
101It solves the depth integrated linear or nonlinear shallow water equations in
102spherical co-ordinates with friction and Coriolis terms. The code is
103based on Satake~\cite{satake95} with significant modifications made by
104the \textsc{urs} corporation, Thio et al~\cite{thio08} and Geoscience
105Australia, Burbidge et al~\cite{burbidge08}.
106The tsunami was propagated via the nested
107grid system described in Section \ref{sec:propagation data} where
108the coarse grids were used in the open ocean and the finest
109resolution grid was employed in the region closest to Patong bay.
110\textsc{Ursga} is not publicly available.
112\subsection{Shallow water propagation and inundation}
114The utility of the \textsc{ursga} model decreases with water depth
115unless an intricate sequence of nested grids is employed. In
116comparison \textsc{anuga}, described below, is designed to produce
117robust and accurate predictions of on-shore inundation, but is less
118suitable for earthquake source modelling and large study areas because
119it is based on projected spatial coordinates. Consequently, the
120Geoscience Australia tsunami modelling methodology is based on a
121hybrid approach using models like \textsc{ursga} for tsunami
122propagation up to an offshore depth contour, typically $100$ m.
123%Specifically we use the \textsc{ursga} model to simulate the
124%propagation of the 2004 Indian Ocean tsunami in the deep ocean, based
125%on a discrete representation of the initial deformation of the sea
126%floor, described in Section~\ref{sec:modelGeneration}.
127The wave signal and the velocity field is then used as a
128time varying boundary condition for
129the \textsc{anuga} inundation simulation.
130% A description of \textsc{anuga} is the following section.
133\textsc{Anuga} is a Free and Open Source hydrodynamic inundation tool that
134solves the conserved form of the depth-integrated nonlinear shallow
135water wave equations using a Finite-Volume scheme on an
136unstructured triangular mesh.
137The scheme, first
138presented by Zoppou and Roberts~\cite{zoppou99}, is a high-resolution
139Godunov-type method that uses the rotational invariance property of
140the shallow water equations to transform the two-dimensional problem
141into local one-dimensional problems. These local Riemann problems are
142then solved using the semi-discrete central-upwind scheme of Kurganov
143et al~\cite{kurganov01} for solving one-dimensional conservation
144equations. The numerical scheme is presented in detail in
145Roberts and Zoppou~\cite{zoppou00,roberts00} and
146Nielsen et al~\cite{nielsen05}. An important capability of the
147finite-volume scheme is that discontinuities in all conserved quantities
148are allowed at every edge in the mesh. This means that the tool is
149well suited to adequately resolving hydraulic jumps, transcritical flows and
150the process of wetting and drying. Consequently, \textsc{anuga} 
151is suitable for
152simulating water flow onto a beach or dry land and around structures
153such as buildings. \textsc{anuga} has been validated against
154%a number of analytical solutions !!!Analytical solutions have not been published. Ask Steve.
155the wave tank simulation of the 1993 Okushiri
156Island tsunami~\cite{nielsen05,roberts06} and
157dam break experiments~\cite{baldock07}.
158More information on \textsc{anuga} and how to obtain it are available from \url{}.
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