\section{Modelling the Event}\label{sec:models}
Numerous models are currently used to model and predict tsunami
generation, propagation and inundation. These range in solving different
equations and employing different methodologies with some examples
being~\cite{titov97a,satake95,zhang08}. Here we introduce the
modelling methodology employed by Geoscience Australia to 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.

\subsection{Generation}\label{sec:modelGeneration}
There are various approaches to modelling the expected crustal
deformation from an earthquake. Most approaches model the
earthquake as a dislocation in a linear elastic medium. Here we use
the method of Wang et al~\cite{wang03}. 
%One of the main advantages
%of their method is that it allows the dislocation to be located in a
%stratified linear elastic half-space with an arbitrary number of
%layers. Other methods (such as those based on Okada's equations) can
%only model the dislocation in a homogeneous elastic half space, or can
%only include a limited number of layers, and thus cannot model the
%effect of the depth dependence of the elasticity of the
%Earth~\cite{wang03}. The original versions of the codes described here
%are available from \url{http://www.iamg.org/CGEditor/index.htm}. The
%first program, \textsc{edgrn}, calculates elastic Green's function for
%a set of point sources at a regular set of depths out to a specified
%distance. The equations controlling the deformation are solved by
%using a combination of Hankel's transform and Wang et al's
%implementation of the Thomson-Haskell propagator
%algorithm~\cite{wang03}. Once the Green's functions are calculated 
%a slightly modified version of \textsc{edcmp}\footnote{For this study, 
%we have made minor modifications
%to \textsc{edcmp} in order for it to provide output in a file format
%compatible with the propagation code in the following section. Otherwise it
%is similar to the original code.} is used to calculate the sea
%floor deformation for a specific subfault. This second code
%discretises the subfault into a set of unit sources and sums the
%elastic Green's functions calculated from \textsc{edgrn} for all the
%unit sources on the fault plane in order to calculate the final static
%deformation caused by a two dimensional dislocation along the
%subfault. This step is possible because of the linearity of the
%governing equations.
%In order to calculate the crustal deformation using these codes 
%a model that describes the variation in elastic
%properties with depth and a slip model of the earthquake to describe
%the dislocation is required. 
In order to calculate the crustal deformation a model that describes the 
variation in elastic properties with depth and a slip model of the 
earthquake to describe the dislocation is required. 
The elastic parameters used for this study are the
same as those in Table 2 of Burbidge et al~\cite{burbidge08}. For the slip
model, there are many possible models for the 2004 Andaman--Sumatran
earthquake to select from
~\cite{chlieh07,asavanant08,arcas06,grilli07,ioualalen07}. Some are
determined from various geological surveys of the site. Others solve
an inverse problem that calibrates the source based upon the tsunami
wave signal, the seismic signal and/or even the run-up. 
The source
parameters used here to simulate the 2004 Indian Ocean tsunami were
taken from the slip model G-M9.15 of Chlieh
et al~\cite{chlieh07}. This model was created by inversion of wide
range of geodetic and seismic data. The slip model consists 
of 686~20~km~x~20~km subsegments each with a different slip, strike and dip
angle. The dip subfaults range from $17.5^\circ$ in the north and $12^\circ$ in
the south. Refer to Chlieh et al~\cite{chlieh07} for a detailed
discussion of this model and its derivation. %Note that the geodetic
%data used in the validation was also included by~\cite{chlieh07} in
%the inversion used to find G-M9.15. Thus the validation is not
%completely independent. However, a reasonable validation would still
%show that the crustal deformation and elastic properties model used
%here is at least as valid as the one used by Chlieh
%et al~\cite{chlieh07} and can reproduce the observations just as
%accurately.

\subsection{Open water propagation}\label{sec:modelPropagation}
The \textsc{ursga} model described below was used to simulate the
propagation of the 2004 Indian Ocean tsunami across the open ocean, based on a
discrete representation of the initial deformation of the sea floor, as
described in Section~\ref{sec:modelGeneration}. For the models shown
here, the uplift is assumed to be instantaneous and creates an initial 
displacement of the ocean surface of the same size and amplitude as the 
co-seismic sea floor deformation. \textsc{ursga} is well suited to 
modelling propagation over large domains and is used to propagate the tsunami 
until it reaches shallow water, typically the $100$ m depth contour.
%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.

\subsubsection{URSGA}\label{sec:ursga}
\textsc{ursga} is a hydrodynamic code that models the propagation of
the tsunami in deep water using a finite difference method on a staggered grid.
It solves the depth integrated linear or nonlinear shallow water equations in
spherical co-ordinates with friction and Coriolis terms. The code is
based on Satake~\cite{satake95} with significant modifications made by
the \textsc{urs} corporation, Thio et al~\cite{thio08} and Geoscience
Australia, Burbidge et al~\cite{burbidge08}. 
The tsunami was propagated via the nested
grid system described in Section \ref{sec:propagation data} where 
the coarse grids were used in the open ocean and the finest
resolution grid was employed in the region closest to Patong City. 
\textsc{Ursga} is not publicly available.

\subsection{Shallow water propagation and inundation}
\label{sec:modelInundation}
The utility of the \textsc{ursga} model decreases with water depth
unless an intricate sequence of nested grids is employed. In
comparison \textsc{anuga}, described below, is designed to produce
robust and accurate predictions of inundation, but is less
suitable for earthquake source modelling and large study areas because
it is based on projected spatial coordinates. Consequently, the
Geoscience Australia tsunami modelling methodology is based on a
hybrid approach using models like \textsc{ursga} for tsunami
propagation up to an offshore depth contour, typically $100$ m.
%Specifically we use the \textsc{ursga} model to simulate the
%propagation of the 2004 Indian Ocean tsunami in the deep ocean, based
%on a discrete representation of the initial deformation of the sea
%floor, described in Section~\ref{sec:modelGeneration}.
The wave signal and the velocity field is then used as a 
time varying boundary condition for
the \textsc{anuga} inundation simulation.
% A description of \textsc{anuga} is the following section.

\subsubsection{ANUGA}\label{sec:anuga}
\textsc{Anuga} is a Free and Open Source hydrodynamic inundation tool that
solves the conserved form of the depth-integrated nonlinear shallow
water wave equations using a Finite-Volume scheme on an 
unstructured triangular mesh. 
The scheme, first
presented by Zoppou and Roberts~\cite{zoppou99}, is a high-resolution
Godunov-type method that uses the rotational invariance property of
the shallow water equations to transform the two-dimensional problem
into local one-dimensional problems. These local Riemann problems are
then solved using the semi-discrete central-upwind scheme of Kurganov
et al~\cite{kurganov01} for solving one-dimensional conservation
equations. The numerical scheme is presented in detail in 
Roberts and Zoppou~\cite{zoppou00,roberts00} and
Nielsen et al~\cite{nielsen05}. An important capability of the
finite-volume scheme is that discontinuities in all conserved quantities 
are allowed at every edge in the mesh. This means that the tool is 
well suited to adequately resolving hydraulic jumps, transcritical flows and 
the process of wetting and drying. Consequently, \textsc{anuga} 
is suitable for
simulating water flow onto a beach or dry land and around structures
such as buildings. \textsc{anuga} has been validated against 
%a number of analytical solutions !!!Analytical solutions have not been published. Ask Steve. 
the wave tank simulation of the 1993 Okushiri
Island tsunami~\cite{nielsen05,roberts06} and 
dam break experiments~\cite{baldock07}.
More information on \textsc{anuga} and how to obtain it are available from \url{https://datamining.anu.edu.au/anuga}.

