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%----------title-------------%
%\title{Inundation Modelling of the December 2004 Indian Ocean Tsunami}
\title{Benchmarking Tsunami Models using the December 2004 Indian Ocean Tsunami and its Impact at Patong Beach}

%-------authors-----------
\author{J.~D. Jakeman\thanks{The Australian National University, Canberra, \textsc{Australia}.
\protect\url{mailto:john.jakeman@anu.edu.au}}
\and O.Nielsen\thanks{Geoscience Australia, Canberra, \textsc{Australia}}
\and R. Mleczko\footnotemark[2]
\and D. Burbidge\footnotemark[2]
\and K. VanPutten\footnotemark[2]
\and N. Horspool\footnotemark[2]
}

%================Start of Document================
\begin{document}
\maketitle
%------Abstract--------------
\begin{abstract}
In this paper a new benchmark for tsunami model validation is proposed. The benchmark is based upon the 2004 Indian Ocean tsunami, which provides a uniquely large amount of observational data for model comparison. Unlike the small number of existing benchmarks, the proposed test validates all three stages of tsunami evolution - generation, propagation and inundation. Specifically we use geodetic measurements of the Sumatra--Andaman earthquake to validate the tsunami source, altimetry data from the \textsc{jason} satellite to test open ocean propagation, eye-witness accounts to assess near shore propagation and a detailed inundation survey of Patong Bay, Thailand to compare model and observed inundation. Furthermore we utilise this benchmark to further validate the hydrodynamic modelling tool \textsc{anuga}  which is used to simulate the tsunami inundation and run rain-induced floods.
\end{abstract}

\tableofcontents
%================Section===========================

\section{Introduction}
Tsunami are a potential hazard to coastal communities all over the world. A number of recent large events have increased community and scientific awareness of the need for effective tsunami hazard mitigation. Tsunami modelling is major component of hazard mitigation, which involves detection, forecasting, and emergency preparedness~\cite{synolakis05}. Accurate models can be used to provide information that increases the effectiveness of action undertaken before the event to minimise damage (early warning systems, breakwalls etc.) and protocols put in place to be followed when the flood waters subside.

Several approaches are currently used to model tsunami propagation and inundation. These methods differ in both the formulation used to describe the evolution of the tsunami and the numerical methods used to solve the governing equations. The shallow water wave equations, linearised shallow water wave equations, and Boussinesq-type equations are frequently used to simulate tsunami propagation. The nonlinear nature of these equations, the highly variable nature of the phenomena that they describe and the complex reality of the geometry they operate in, necessitate the use of numerical models. These models are typically used to predict quantities such as arrival times, wave speeds and heights and inundation extents which are used to develop efficient hazard mitigation plans. Inaccuracies in model prediction can result in inappropriate evacuation plans and town zoning which may result in loss of life and large financial losses. Consequently tsunami models must undergo sufficient testing to increase scientific and community confidence in the model predictions.

Complete confidence in a model of a physical system frequently in general cannot be established.  One can only hope to state under what conditions the model hypothesis holds true. Specifically the utility of a model can be assessed through a process of verification and validation. Verification assesses the accuracy of the numerical method used to solve the governing equations and validation is used to investigate whether the model adequately represents the physical system~\cite{bates01}. Together these processes can be used to establish the likelihood that that a model is a legitimate hypothesis.

The sources of data used to validate and verify a model can be separated into three main categories; analytical solutions, scale experiments and field measurements. Analytical solutions of the governing equations of a model, if available, provide the best means of verifying any numerical model. However analytical solutions to the governing equations used here are frequently limited to a small set of idealised examples that do not completely capture the more complex behaviour of `real' events. For tsunami inundation, scale experiments, typically in the form of wave-tank experiments, provide a much more realistic source of data that better captures the complex dynamics of natural tsunami, whilst allowing control of the event and much easier and accurate measurement of the tsunami properties. However comparison of numerical predictions with field data provides the most stringent test of model veracity. The use of field data increases the generality and significance of conclusions made regarding model utility. However the use of field data also significantly increases the uncertainty of the validation experiment that may constrain the ability to make unequivocal statements~\cite{bates01}.

Currently the extent of tsunami related field data is limited. The cost of tsunami monitoring programs, bathymetry and topography surveys prohibits the collection of data in many of the regions in which tsunamis pose greatest threat. The resulting lack of data has limited the number of field data sets available to validate tsunami models. Synolakis et. al~\cite{synolakis07} have developed a set of standards, criteria and procedures for evaluating numerical models of tsunami. They propose three analytical solutions to help identify the validity of a model and  five scale comparisons (wave-tank benchmarks) and two field events to assess model veracity. 

The first field data benchmark introduced by Synolakis compares model results against observed data from the Hokkaido-Nansei-Oki tsunami that occurred around Okushiri Island, Japan on the 12th of July 1993. This tsunami provides an example of extreme runup generated from reflections and constructive interference resulting from local topography and bathymetry. The benchmark consists of two tide gauge records and numerous spatially distributed point sites at which modelled maximum runup elevations can be compared. The second benchmark is based upon the Rat Islands Tsunami that occurred off the coast of Alaska on the 17th of November 2003. The Rat island tsunami provides a good test for real-time forecasting models since tsunami was recorded at three tsunameters. The test requires matching the propagation model data with the DART recording to constrain the tsunami source model and using a propagation model to to reproduce the tide gauge record at Hilo.

The evolution of earthquake-generated tsunamis has three distinctive stages: generation, propagation and run-up~\cite{titov97a}. To accurately model the evolution of a tsunami all three stages must be dealt with. In this section we present the data necessary to implement the proposed benchmark corresponding to each of the three stages of the tsunami's evolution.  In this paper we develop a field data benchmark to be used in conjunction with the other tests proposed by Synolakis et al.~\cite{synolakis07} to validate and verify tsunami models. Unlike the aforementioned tests, the proposed benchmark allows one to estimate the error in a model prediction for all three distinctive stages of the evolution of a tsunami: generation, propagation and run-up. The benchmark comprises of geodetic measurements of the Sumatra--Andaman earthquake to validate the tsunami source, altimetry data from the JASON satellite to test open ocean propagation, eye-witness accounts to assess near shore propagation and a detailed inundation survey of Patong Bay, Thailand to compare model and observed inundation. A description of this data is give in Section~\ref{sec:data}.

An associated aim of this paper is to illustrate the use of this new benchmark to validate an operational tsunami model called \textsc{anuga}. A description of \textsc{anuga} is given in Secion~\ref{sec:models} and the validation results are given in Secion~\ref{sec:results}. 

%================Section===========================
\section{Data}\label{sec:data}
The sheer magnitude of the 2004 Sumatra-Andaman earthquake and the devastation caused by the subsequent tsunami generated much scientific interest. As a result an unusually large amount of post seismic data has been collected and documented. Data sets from seismometers, tide gauges, \textsc{gps} surveys, satellite overpasses, subsequent coastal field surveys of run-up and flooding, and measurements of coseismic displacements and bathymetry from ship-based expeditions, have now been made available.%~\cite{vigny05,amnon05,kawata05,liu05}.

In this section we present the data necessary to implement the proposed benchmark corresponding to each of the three stages of the tsunami's evolution.  The final datasets are available on Sourceforge under the ANUGA project (\url{http://sourceforge.net/projects/anuga}). At the time
of writing the direct link is \url{http://tinyurl.com/patong2004-data}.
%
%\url{http://sourceforge.net/project/showfiles.php?group_id=172848&package_id=319323&release_id=677531}.

\subsection{Generation}\label{sec:gen_data}
All tsunami are generated from an initial disturbance of the ocean which develops into a low frequency wave that propagates outwards from the source. The initial deformation of the water surface is most commonly caused by coseismic displacement of the sea floor, but submarine mass failures, landslides, volcanoes or asteroids can also cause tsunami. In this section we detail the information we used in this study to validate models of the sea floor deformation generated by the 2004 Sumatra--Andaman earthquake.

The 2004 Sumatra--Andaman tsunami was generated by severe coseismic displacement of the sea floor as a result of one of the largest earthquakes on record. The mega-thrust earthquake started on the 26 December 2004 at 0h58'53'' UTC (or just befor 8 am local time) approximately 70 km offshore North Sumatra (\url{http://earthquake.usgs.gov/eqcenter/eqinthenews/2004/usslav}). The rupture propagated 1000-1300 km along the Sumatra-Andaman trench time to the north at a rate of 2.5-3 km.s$^{-1}$ and lasted approximately 8-10 minutes~\cite{ammon05}. Estimates of the moment magnitude of this event range from about 9.1 to 9.3~\cite{chlieh07, stein07}.

The unusually large surface deformation caused by this earthquakes means that there were a range of different geodetic measurements of the surface deformation available. These include field measurements of uplifted or subsided coral heads, continuous or campaign \textsc{GPS} measurments and remote sensing measurements of uplift or subsidence (see ~\cite{chlieh07} and references therein). Here we use the the near field estimates of vertical deformation in northwestern Sumatra and the Nicobar-Andaman islands collated by~\cite{chlieh07} to validate that our crustal deformation model of the 2004 Sumatra--Andaman earthquake is producing reasonable results. Note that the geodetic data used here is a combination of the vertical deformation that happened in the ~10 minutes of the earthquake plus the deformation that followed in the days following the earthquake before each particular measurement was actually made (typically of order days). Therefore some of the observations may not contain the purely co-seismic deformation but could include some post-seismic deformation as well~\cite{chlieh07}.

%DAVID: I commented out the figure since we can combine it with the model result without obscuring it. That will keep the number of figures down.

%\begin{figure}[ht]
%\begin{center}
%\includegraphics[width=8.0cm,keepaspectratio=true]{geodeticMeasurements.jpg}
%\caption{Near field geodetic measurements used to validate tsunami generation. FIXME: Insert appropriate figure here}
%\label{fig:geodeticMeasurements}
%\end{center}
%\end{figure}

\subsection{Propagation}
Once generated a tsunami will propagate outwards from the source until it finally encounters the shallow water bordering coastal regions. This period of the tsunami evolution is referred to as the propagation stage. The height and velocity of the tsunami is dependent on the local bathymetry in the regions through which the wave travels and the size of the initial wave. This section details the bathymetry data needed to model the tsunami propagation and the satellite altimetry transects used here to validate open ocean tsunami models.

\subsubsection{Bathymetry Data}
A number of raw data sets were obtained, analysed and checked for quality and subsequently gridded for easier visualisation and input into the tsunami models. The resulting grid data is relatively coarse in the deeper water and becomes progressively finer as the distance to Patong Bay decreases.

The nested bathymetry grid was generated from: 
\begin{itemize}
\item A two arc minute grid data set covering the Bay of Bengal, DBDB2, obtained from US Naval Research Labs; 
\item A 3 second arc grid covering the whole of the Andaman Sea which is based on Thai charts 45 and 362; and 
\item A one second grid created from the digitised Thai Navy bathymetry chart, no 358. which covers Patong Bay and the immediately adjacent regions.
\end{itemize}

The final bathymetry data set consits of four nested grids obtained via interpolation and resampling of the aforementioned data sets. The coarsest bathymetry was obtained by interpolating the DBDB2 grid to a 27 second arc grid. A subsection of this region was then replaced by 9 second data which was generated by sub-sampling the 3 second of arc grid from NOAA. A subset of the 9 second grid was replaced by the 3 second data. Finally a one second grid was used to approximate the bathymetry in Patong Bay and the immediately adjacent regions. This elevation data was created from the digitised Thai Navy bathymetry chart, no 358. Any points that deviated from the general trend near the boundary were deleted. See Figure~\ref{fig:nested_grids}.

The sub-sampling of larger grids was performed by using {\bf resample} a Generic Mapping Tools (\textsc{GMT}) program (\cite{wessel98}). The gridding of data was performed using {\bf Intrepid} a commercial geophysical processing package developed by Intrepid Geophysics. The gridding scheme employed the nearest neighbour algorithm followed by an application of minimum curvature akima spline smoothing. 

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm,keepaspectratio=true]{nested_grids}
\caption{Nested grids of elevation data.}
\label{fig:nested_grids}
\end{center}
\end{figure}

\subsubsection{JASON Satellite Altimetry}\label{sec:data_jason}
During the 26 December 2004 event, the Jason satellite tracked from north to south and over the equator at 02:55 UTC nearly two hours after the  earthquake \cite{gower05}. The satellite recorded the sea level anomaly compared to the average sea level from its previous five passes over the same region in the 20-30 days prior. 
%DB I suggest we combine with model data to reduce the number of figures. The satellite track is shown in Figure~\ref{fig:satelliteTrack}.

%\begin{figure}[ht]
%\begin{center}
%\includegraphics[width=8.0cm,keepaspectratio=true]{sateliteTrack.jpg}
%\caption{URS wave heights 120 minutes after the initial earthquake with the JASON satellite track and its observed sea level anomalies overlaid. Note the URS data has not been corrected for the flight path time. FIXME: should we just have track and not URS heights.}
%\label{fig:satelliteTrack}
%\end{center}
%\end{figure}

%\begin{figure}[ht]
%\begin{center}
%\includegraphics[width=8.0cm,keepaspectratio=true]{jasonAltimetry.jpg}
%\caption{JASON satellite altimetry seal level anomaly. FIXME: should we include figure here with just JASON altimetry.}
%\label{fig:jasonAltimetry}
%\end{center}
%\end{figure}

%FIXME: Can we compare the urs model against the TOPEX-poseidon satellite as well? DB No (we don't have the data currently).

\subsection{Inundation}
Inundation refers to the final stages of the evolution a tsunami. Specifically the propagation of the tsunami in shallow coastal water and the subsequent run-up on to the shoreline. This process is typically the most difficult of the three stages to model. Aside from requiring robust solvers which can simulate flow over dry land, this part of the modelling process requires high resolution and quality bathymetry data and high quality field measurements, which are often not available. For the proposed benchmark the authors have obtained a high resolution bathymetry and topography data set and a high quality inundation survey map which can be used to validate model inundation. These data sets are described here. In this section we also present eye-witness accounts which can be used to qualitatively validate tsunami inundation.

\subsubsection{Topography Data}
A one second grid was used to approximate the topography in Patong Bay. This elevation data was again created from the digitised Thai Navy bathymetry chart, no 358. A visualisation of the elevation data set used in Patong bay is shown in Figure~\ref{fig:patong_bathymetry}. The continuous topography is an interpolation of known elevation measured at the coloured dots.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm,keepaspectratio=true]{patong_bay_data.jpg}
\caption{Visualisation of the elevation data set used in Patong Bay showing data points, contours, rivers and roads draped over the final model.}
\label{fig:patong_bathymetry}
\end{center}
\end{figure}

\subsubsection{Buildings and Other Structures}
Human made build and structures can significantly effect tsunamni inundation. The location and size and number of floors of the buildings in Patong Bay were extracted from a GIS data set~\cite{}. The heights of these buildings was estimated assuming that each floor was 3m high.

\subsubsection{Eyewitness Accounts}
Eyewitness accounts detailed in~\cite{papadopoulos06} report that most people at Patong Beach observed an initial retreat of the shoreline of more than 100m followed a few minutes later by a strong wave (crest). Another less powerful wave arrived another five or ten minutes later. Eyewitness statements place the arrival time of the strong wave between 2 hours and 55 minutes to 3 hours and 5 minutes after the source rupture (09:55am to 10:05am local time). 

\subsubsection{Video Evidence}
Two videos were sourced from the internet which include footage of the tsunami in Patong Bay on the day of the Indian Ocean Tsunami. Both videos show an
already inundated town, they then show what is to be assumed as the second and third waves approaching and further flooding the town. The first video
is in the very north of the town filmed from what is believed to be the roof of the Novotel Hotel. The second video is in the very south of the town
filmed from a building next door to the Comort Resort.The figure shows
a video from from which we made our estimates. Crude flow rates can be
estimated with landmarks and the use of a GIS and were found to be in
the range of 5 to 7 metres per second (+/- 2 m/s) in the north and 0.5
to 2 metres per second (+/- 1 m/s) in the south. This is in agreement
with results from our simulations. Our modelled flow rates show
maximum values in the order of 0.2 to 2.6 m/s in the south and 0.1 to
3.3 m/s for the north as shown in the figures. Water depths could also be estimated from the videos by the level at which water rose up the sides of buildings such as shops. Our estimates are in the order of 1.5 to 2.0 metres (+/- 0.5 m). This is in the same range as our modelled maximum depths of 1.4 m in the north and 1.5 m in the south as seen in the figure. Fritz [20] did a detailed analysis of video frames taken around Banda Aceh and arrived at flow speeds in the range of 2 to 5 m/s.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=10.0cm,keepaspectratio=true]{gauges_hotels_speed.jpg}
\includegraphics[width=10.0cm,keepaspectratio=true]{gauges_hotels_depths.jpg}
\caption{modelled gauges near the hotels showing flow speed and depths.}
\label{fig:hotel_guages}
\end{center}
\end{figure}

\subsubsection{Inundation Survey}
Tsunami run-up is often the cause of the largest financial and human losses yet run-up data that can be used to validate model runup predictions is scarce. Of the two field benchmarks proposed by Synolakis only the Okishiri benchmark facilitates comparison between modelled and observed run-up. One of the major strengths of the benchmark proposed here is that modelled runup can be compared to an inundation survey which maps the maximum run-up along an entire coast line rather than at a series of discrete sites. The survey map is shown in Figure~\ref{fig:patongescapemap} and plots the maximum run-up of the 2004 tsunami in Patong bay. Refer to Szczucinski et al~\cite{szczucinski06} for further details.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm,keepaspectratio=true]{patongescapemap.jpg}
\caption{Tsunami survey mapping the maximum observed inundation at Patong beach courtesy of the Thai Department of Mineral Resources \protect \cite{szczucinski06}.}
\label{fig:patongescapemap}
\end{center}
\end{figure}

\subsection{Validation Check-List}
\label{sec:checkList}
The data described in this section can be used to construct a benchmark to validate all three stages of the evolution of a tsunami. In particular we propose that a legitimate tsunami model should reproduce the following behaviour:
\begin{itemize}
 \item Reproduce the vertical deformation observed in north-western Sumatra and along the Nicobar--Andaman islands, see Section~\ref{sec:gen_data}.
 \item Reproduce the \textsc{jason} satellite altimetry sea surface anomalies, see Section~\ref{sec:data_jason}.
 \item Simulate a leading depression followed by two distinct crests of decreasing magnitude.
 \item Predict the arrival time of the first crest should arrive at Patong beach between 2 hours and 55 minutes to 3 hours and 5 minutes after the initial rupture of the source. The subsequent crest arrive five to ten minutes later.
 \item Reproduce the inundation survey map in Patong bay (Figure~\ref{fig:patongescapemap}).
\end{itemize}

%================Section===========================
\section{Modelling the Event}\label{sec:models}
Numerous models are currently used to model and predict tsunami generation, propagation and run-up\cite{titov97a,satake95}. Here we introduce the modelling methodology employed by Geoscience Australia to illustrate the utility of the proposed benchmark. Geoscience Australia's tsunami model can again be decomposed into three parts which simulate generation, propagation and inundation (Sections~\ref{sec:modelGeneration},\ref{sec:modelPropagation} and \ref{sec:modelInundation} respectively).

\subsection{Generation}\label{sec:modelGeneration}

There are various approaches to modelling the expected crustal deformation from an earthquake at depth. Most approaches model the earthquake as a dislocation in linear, elastic medium. Here we use the method of~\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 arbitary 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 we use a slightly modified version of \textsc{EDCMP} 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. For this study, we have made minor modifications to \textsc{EDCMP} in order for it to output in a file format compatible with the propagation code in the following section but it is otherwise the similar to the original code.

In order to calculate the crustal deformation using these codes we thus need to have a model describing the variation in elastic properties with depth and a slip model of the earthquake to describe the dislocation. The elastic parameters used for this study are the same as those in Table 2 of \cite{burbidge08}. For the slip model, there are many possible models for the 2004 Andaman--Sumatran earthquake to choose from ~\cite{chlieh07,asavanant08,arcas06,grilli07,ioualalen07}. Some are determined from various geological surveys of the site, others solve an inverse problem which calibrates the source based upon the tsunami wave signal, the seismic signal and/or the runup. The source parameters used here to simulate the 2004 Indian Ocean Tsunami were taken from model G-M9.15 from~\cite{chlieh07}. This model was created by inversion of wide range of geodetic and seismic data. The slip model consists of 686 20km x 20km subsegments each with a different slip, strike and dip angle. The dip subfaults go from $17.5^0$ in the north and $12^0$ in the south. Refer to~\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 successful validation would still show that the crustal deformation and elastic properties model used here is at least as valid as the one used in~\cite{chlieh07} and can reproduce the observations just as accurately.

\subsection{Propagation}\label{sec:modelPropagation}
We use the \textsc{ursga} model to simulate the propagation of the 2004 tsunami in the deep ocean ocean, based on a discrete representation of the initial deformation of the sea floor, described in Section~\ref{sec:modelGeneration}. For the models shown here, we assume that the uplift is instantaneous and creates a wave of the same size and amplitude as the co-seismic sea floor deformation.

\subsubsection{URSGA}
\textsc{ursga} is a hydrodynamic code that models the propagation of the tsunami in deep water using the finite difference method to solve 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~\cite{thio08} and Geoscience Australia~\cite{burbidge08}. The tsunami is propagated via a staggered grid system. Coarse grids are used in the open ocean and the finest resolution grid is employed in the region of most interest. \textsc{Ursga} is not publicly available. 

\subsection{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} is designed to produce robust and accurate predictions of on-shore inundation in mind, but is less suitable for earthquake source modelling and large study areas. Consequently, the Geoscience Australia tsunami modelling methodology is based on a hybrid approach using models like \textsc{ursga} for tsunami propagation up to a 100m depth contour. 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 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}
\textsc{Anuga} is an Open Source hydrodynamic inundation tool that solves the depth integrated nonlinear shallow water wave equations. The scheme used by \textsc{anuga}, 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 Zoppou and Roberts~\cite{zoppou99}, Zoppou and Roberts~\cite{zoppou00}, and Roberts and Zoppou~\cite{roberts00}, Nielsen et al.~\cite{nielsen05}. An important capability of the software is that it can model the process of wetting and drying as water enters and leaves an area. This means that it is suitable for simulating water flow onto a beach or dry land and around structures such as buildings. It is also capable of adequately resolving hydraulic jumps due to the ability of the finite-volume method to handle discontinuities. \textsc{Anuga} has been validated against a number of analytical solutions and the wave tank simulation of the 1993 Okushiri Island tsunami~\cite{roberts06,nielsen05}.

%================Section===========================
\section{Results}\label{sec:results}
This section presents validates the modelling practice of Geoscience Australia against the new proposed benchmarks. The criteria outlined in Section\ref{sec:checkList} are addressed for each three stages of tsunami evolution.

\subsection{Generation}\label{modelGeneration}
The location and magnitude of the sea floor displacement associated with the 2004 Sumatra--Andaman tsunami calculated from the G-M9.15 model  of~\cite{chlieh07} is shown in Figure~\ref{fig:surface_deformation}. The magnitude of the sea floor displacement ranges from about $-3.0$ to $5.0$ metres. The region near the fault is predicted to uplift, while that further away from the fault subsides. Also shown in Figure~\ref{fig:surface_deformation} are the areas that were observed to uplift (arrows pointing up) or subside (arrows point down) during and immediately after the earthquake. Most of this data comes uplifted or subsided coral heads. The length of vector increases with the magniutude of the displacement, the length corresponding to 1m of observed motion is shown in the top right corner of the figure. As can be seen, the source model detailed in Section~\ref{sec:modelGeneration} produces a crustal deformation that matches the vertical displacements in the Nicobar-Andaman islands and Sumatra very well. Uplifted regions are close to the fault and subsided regions are further away. The crosses on Figure~\ref{fig:surface_deformation} show estimates of the pivot line from the remote sensing data~\cite{chlieh07} and they follow the predicted pivgot line quite accurately. The average difference between the observed motion and the predicted motion (including the pivot line points) is only 0.06m, well below the typical error of the observations of between 0.25 and 1.0m. However, the occasional point has quite a large error (over 1m), for example a couple uplifted/subsided points appear to be on a wrong side of the predicted pivot line~\ref{fig:surface_deformation}. These points are rare, most fit the predicted vertical motion very well. The excellence of the fit is not surprising, since the original slip model was chosen by~\cite{chlieh07} to fit this (and the seismic data) well. However, this does demonstrate that \textsc{edgrn} and our modified version of \textsc{edstat} can reproduce the correct pattern of vertical deformation very well when the slip distribution is well constrained and when reasonable values for the elastic properties are used.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm,keepaspectratio=true]{surface_deformation.jpg}
%\includegraphics[totalheight=0.3\textheight,width=0.8\textwidth]{surface_deformation.jpg}
\caption{Location and magnitude of the vertical component of the sea floor displacement associated with the 2004 Indian Ocean tsunami based on the slip model, G-M9.15. The black arrows which point up show areas observed to uplift during and immediately after the earthquake, those point down are locations which subsided. The length of increases with the magnitude of the deformation. The arrow length corresponding to 1m of deformation is shown in the top right hand corner of the figure. The crosses marks show the location of the pivot line (the region between the uplift and subsided region where the uplift is zero) derived from remote sensing. All the observational data come from the dataset collated by~\cite{chlieh07}.}
\label{fig:surface_deformation}
\end{center}
\end{figure}

\subsection{Propagation}
The deformation results described in Section~\ref{sec:modelGeneration} was then used to provide an profile of the initial ocean surface displacement. This wave was used as an initial condition for \textsc{ursga} and was propagated the tsunami throughtout the Bay of Bengal. The rectangular computational domain of the largest grid extended from 90$^0$ to 100$^0$East and 0 to 15$^0$North and contained 1335$\times$1996 finite difference points. Inside this grid, a nested sequence of grids was used. The grid resolution of the nested grids went from 27 arc seconds in the coarsest grid, down to 9 arc seconds in the second grid, 3 arc seconds in the thrid grid and finally 1 arc second in the finest grid near Patong. The computational domain is shown in Figure\ref{fig:computational_domain}.

Figure \ref{fig:jasonComparison} provides a comparison of the \textsc{ursga} predicted surface elevation with the JASON satellite altimetry data. The \textsc{ursga} model replicates the amplitude and timing of the first peak and trough well. However the model does not resolve the double peak of the first wave. Also note that the \textsc{ursga} model prediction of the ocean surface elevation becomes out of phase with the JASON data at 3 to 7 degrees latitude. Chlieh et al~\cite{chlieh07} also observe these misfits and suggest it is caused by a reflected wave from the Aceh Peninsula that is not resolved in the model due to insufficient resolution of the computational mesh and bathymetry data. This is also a limitation of the model presented here, but probably could be improved by nesting grids near Aceh. 

\begin{figure}[ht]
\begin{center}
\includegraphics[width=12.0cm,keepaspectratio=true]{jasonComparison.jpg}
\caption{Comparison of the \textsc{ursga} predicted surface elevation with the JASON satellite altimetry data. The \textsc{ursga} wave heights have been corrected for the time the satellite passed overhead compared to JASON sea level anomaly. 
}
\label{fig:jasonComparison}
\end{center}
\end{figure}

\subsection{Inundation}
After propagating the tsunami in the open ocean using \textsc{ursga} the approximated ocean and surface elevation and tsunami velocities were extracted and used to construct a boundary condition for the \textsc{anuga} model. The interface betwen the \textsc{ursga} and \textsc{anuga} models was chosen to roughly follow the 100m depth contour along the west coast of Phuket Island. The computational domain is shown in Figure \ref{fig:computational_domain}
\begin{figure}[ht]
\begin{center}
%\includegraphics[width=5.0cm,keepaspectratio=true]{extent_of_ursga_model.jpg}
%\includegraphics[width=5.0cm,keepaspectratio=true]{ursgaDomain.jpg}
\includegraphics[width=5.0cm,keepaspectratio=true]{extent_of_ANUGA_model.jpg}
\caption{Computational domain of the URSGA simulation (inset: white and black squares and main: black square) and the \textsc{anuga} simulation (main and inset: red polygon)}
\label{fig:computational_domain}
\end{center}
\end{figure}

The domain was discretised into approximately 350,000 triangles. The resolution of the grid was increased in certain regions to efficiently increase the accuracy of the simulation. The grid resolution ranged between a maximum triangle area of $1\times 10^5$ m$^2$ near the Western ocean boundary to $3$ m$^2$ in the small regions surrounding the inundation region in Patong Bay. Due to a lack of available data, friction was set to a constant throughout the computational domain. For the reference simulation a Manning's coefficient of 0.01 was chosen based upon previous numerical experiments conducted by the authors (FIXME: Citation Tom Baldock?? Or Duncan??).

The boundary condition at each side of the domain towards the south and the north where no data was available was chosen as a transmissive boundary condition effectively replicating the time dependent wave height present just inside the computational domain. Momentum was set to zero. Other choices include applying the mean tide value as a Dirichlet type boundary condition but experiments as well as the result of the verification reported here showed that this approach tends to under estimate the tsunami impact due to the tempering of the wave near the side boundaries. 

During the \textsc{anuga} simulation the tide was kept constant at $0.80$m. This value was chosen to correspond to the tidal height specified by the Thai Navy tide charts (\url{http://www.navy.mi.th/hydro/}) at the time the tsunami arrived at Patong Bay. Although the tsunami propagated for approximately 3 hours before it reach Patong Bay, the period of time during which the wave propagated through the \textsc{anuga} domain is much smaller. Consequently the assumption of constant tide height is reasonable

FIXME(John): Need a commentary on the dynamics of what is being observed and whether it aligns with eye witness observations.
Both the URS model and the \textsc{anuga} inundation model shows that the event comprises a train of waves some with preceding drawdown effects Add details of waveform with a graph from URL and a gauge from \textsc{anuga} and discuss. This will come from the work of Kristy (plots of water depth and speed at two locations on beach and one of shore) and Richard (approximating water depth and wave speeds from videos). 

Maximum onshore inundation elevation was simulated throughout the entire Patong Bay region. Figure~\ref{fig:inundationcomparison1cm} shows very good agreement between the measured and simulated inundation. The \textsc{anuga} simulation determines a region to be inundated if at some point in time it was covered by at least 1cm of water. This precision in field measurements is impossible to obtain. The inundation boundary is determined by observing water marks and other signs left by the receding waters. The precision of the observed inundation map is, most likely, at least an order of magnitude worse than the \textsc{anuga} simulation. The simulated inundation based upon a 10cm threshold is shown in Figure~\ref{fig:inundationcomparison1cm}. An inundation threshold of 10cm was selected for all future simulations to reflect the likely accuracy of the survey and subsequently facilitate a more appropriate comparison between the modelled and observed inundation area. 

An animation of this simulation is available on the ANUGA website at \url{https://datamining.anu.edu.au/anuga} or directly from \url{http://tinyurl.com/patong2004}.

%\url{https://datamining.anu.edu.au/anuga/attachment/wiki/AnugaPublications/patong_2004_indian_ocean_tsunami_ANUGA_animation.mov}.




\begin{figure}[ht]
\begin{center}
\includegraphics[width=5.0cm,keepaspectratio=true]{Depth_small_transmissive_d0.jpg}
\includegraphics[width=5.0cm,keepaspectratio=true]{sensitivity_reference.jpg}
\caption{Simulated inundation versus observed inundation using an inundation threshold of 1cm (left) and 10cm (right). FIXME: NEED Graph for 10cm}
\label{fig:inundationcomparison1cm}
\end{center}
\end{figure}

Here we introduce the measure
\begin{equation}
A(I_{in})=\frac{A(I_m\cap I_o)}{A(I_o)}
\end{equation}
to quantify the fraction of the area $A(I_{in})$ of observed inundation region $I_o$ captured by the model $I_m$. Another useful measure is the fraction of the modelled inundation area that falls outside the observed inundation area given by the formula
\begin{equation}
A(I_{out})=\frac{A(I_m\setminus (I_m\cap I_o))}{A(I_o)}
\end{equation}
These values for the two aforementioned simulations are given in Table~\ref{table:inundationAreas}

Additional causes of the discrepancies between the survey data and the modelled inundated include: unknown distribution of surface roughness, inappropriate parameterisation of the source model, effect of humans structures on flow, as well as uncertainties in the elevation data, effects of erosion and deposition by the tsunami event, measurement errors, and missing data in the field survey data itself. The impact of some of these sources of uncertainties are is investigated in Section~\ref{sec:sensitivity}


\section{Dynamic behaviour of the tsunami}

Figure \ref{fig:gauge_locations} shows four locations where time series have been extracted from the model. The two offshore timeseries are shown in Figure \ref{fig:offshore_timeseries} and
the two onshore timeseries are shown in Figure \ref{fig:onshore_timeseres}. The latter coincide with locations where video footage from the event is available.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.0cm,keepaspectratio=true]{tide_gauge_locations.jpg}
\caption{Location of timeseries extracted from the model output}
\label{fig:gauge_locations}
\end{center}
\end{figure}


\begin{figure}[ht]
\begin{center}
%\includegraphics[width=7.0cm,keepaspectratio=true]{ }
\caption{Timeseries obtained from the two offshore locations shown in Figure \protect \ref{fig_gauge_locations}}
\label{fig:offshore_timeseries}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
%\includegraphics[width=7.0cm,keepaspectratio=true]{ }
\caption{Timeseries obtained from the two onshore locations shown in Figure \protect \ref{fig_gauge_locations}}
\label{fig:onshore_timeseries}
\end{center}
\end{figure}


%================Section===========================
\section{Sensitivity Analysis}
\label{sec:sensitivity}
This section shows how model maximum inundation varies with: different values of Manning's friction coefficient; changing waveheight at the ANUGA boundary (where it was coupled with the URSGA model); and finally the presence and absence of buildings in the elevation dataset.

%========================Friction==========================%
\subsection{Friction}
The first study investigated the impact of surface roughness on the predicted run-up. According to Schoellte~\cite{schoettle2007} appropriate values of Manning's coefficient range from 0.007 to 0.030 for tsunami propagation over a sandy sea floor.  Consequently we simulated the maximum onshore inundation using the a Manning's coefficient of 0.0003 and 0.03. The resulting run-up is shown in Figures
\ref{fig:sensitivity_friction} and  the maximum flow speeds\ref{fig:sensitivity_friction_speed}. These figurers show that the on-shoer inundation extent decreases with increasing friction and that small perturbations in the friction cause bounded changes in the output. This is consistent with the conclusions of Synolakis~\cite{synolakis05} who states that the long wavelength of tsunami tends to mean that the friction is less important in comparison to the motion of the wave.

%========================Wave-Height==========================%
\subsection{Input Wave Height}\label{sec:waveheightSA}
The effect of the wave-height used as input to the inundation model \textsc{anuga} was also investigated.  Figure\ref{fig:sensitivity_boundary} indicates that the inundation severity is directly proportional to the boundary waveheight but small perturbations in the input wave-height of 10cm seem to have little effect on the final on-shore run-up. Obviously larger perturbations will have greater impact, however unlike the uncertainty in the friction, the range of uncertainty in the propagation wave-height is hard to estimate.



%========================Buildings==========================%
\subsection{Buildings and Other Structures}
The presence of buildings has the greatest influence on the maximum on-shore inundation extent. Figure \ref{fig:sensitivity_nobuildings} and shows the maximum run-up in the presence and absence of buildings. It is apparent that the inundation is much more severe when the presence of man made structures and buildings are ignored. Maximal flow speeds for these two model parameterisations are shown in Figure~\ref{fig:sensitivity_nobuildings_speed}.

\begin{table}
\begin{center}
\label{table:inundationAreas}
\caption{$A(I_{in})$ and $A(I_{out})$ of the reference simulation and all sensitivity studies}
\begin{tabular}{|c|c|c|}
\hline
 & $A(I_{in})$ & $A(I_{out})$ \\ 
\hline\hline
Reference & 0.76 & 0.22\\ 
Min. Friction & × & \\ 
Max. Friction & × & \\ 
Min. Wave-Height× & × & \\
Max. Wave-Height× & × & \\
No Buildings × & × & \\
\hline 
\end{tabular}
\end{center}
\end{table}

%================Section===========================

\section{Conclusion}
This paper proposes an additional field data benchmark for the verification of tsunami inundation models. Currently, there is a scarcity of appropriate validation datasets due to a lack of well documented historical tsunami impacts. FIXME(John): Complete...

From the sensitivity studies it appears that the presence or absence of buildings is the most important parameter followed by the right choice of friction whereas a small perturbation in the waveheight at the ANUGA boundary has comparatively little effect on the model results.

%================Acknowledgement===================
\section*{Acknowledgements}
This project was undertaken at Geoscience Australia and the Department of Mathematics, The Australian National University. The authors would like to thank Niran Chaimanee from the CCOP, Thailand for providing the post 2004 tsunami survey data and the elevation data for Patong beach, Prapasri Asawakun from the Suranaree University of Technology and Parida Kuneepong for supporting this work; and Drew Whitehouse from the Australian National University for preparing the animation.

\section{Appendix}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_minus10}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_plus10}
\caption{Model results with wav height at ANUGA boundary artificially modified 
to asses sensitivities. The first image is the reference inundation extent as reported in Section \protect \ref{sec:results} while the second and third show the inundation results if the wave at the ANUGA boundary is reduced or increased by 10cm respectively. The inundation severity varies in proportion to the boundary waveheight, but the model results are only slightly sensitive to this parameter for the range of values tested.}
\label{fig:sensitivity_boundary}
\end{center}
\end{figure}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_minus10_speed}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_plus10_speed}
\caption{The maximal flow speeds for the same model parameterisations found in  Figure \protect \ref{fig:sensitivity_boundary}.}
\label{fig:sensitivity_boundary_speed}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_nobuildings}
\caption{This figure shows the effect of having buildings as part of the 
elevation data set.
The first image is the reference inundation extent as reported in Section \protect \ref{sec:results} where buildings were included. The second shows the inundation results for a model entirely without buildings.
As expected, the absence of buildings will increase the inundation extent 
beyond what was surveyed.}
\label{fig:sensitivity_nobuildings}
\end{center}
\end{figure}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_nobuildings_speed}
\caption{The maximal flow speeds for the same model parameterisations found in Figure \protect \ref{fig:sensitivity_nobuildings}.}
\label{fig:sensitivity_nobuildings_speed}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f0003}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f03}
\caption{Model results for different values of Manning's friction coefficient.
The first image is the reference inundation extent as reported in Section \protect \ref{sec:results} where the friction value $0.01$ was used across the 
entire domain while the second and third show the inundation results for friction values of 0.0003 and 0.03 respectively. The inundation extent increases for the lower friction value while the higher slows the flow and decreases the inundation extent. Ideally, friction should vary across the entire domain depending on terrain and vegetation, but this is beyond the scope of this study.}
\label{fig:sensitivity_friction}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f0003_speed}
\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f03_speed}
\caption{The maximal flow speeds for the same model parameterisations found in Figure \protect \ref{fig:sensitivity_friction}.}
\label{fig:sensitivity_friction_speed}
\end{center}
\end{figure}

\clearpage

%====================Bibliography==================
\bibliographystyle{plain}
\bibliography{tsunami07}
\end{document}


===================
NOTES TO BE REMOVED

Main source of uncertainty arises from inaccuracies in initial condition (source), inaccurate bathymetry data, to a lesser extent friction

scale comparisons (laboratory benchmarking):
Scale differences are not believed to be important. scale experiments generally do not have same bootom friction characteristics as real scenario but has not proven to be a problem. The long wavelength of tsunami tends to mean that the friction is less important in comparison to the motion of the wave
Single wave on a simple beach
Solitary wave on composite beach
Conical island
Monai Valley
Landslide

Field benchmarking:
Most important verification process
Hydrodynamic inversion to predict the source is an ill posed problem
12 July 1993 Hokkaido-Nansei-Oki tsunami around Okushiri Island Japan extreme runup height of 31.7m was found at the tip of a narrow gully with the small cove at Monai
17 November 2003 Rat Islands Tsunami

Construction of more than one model can reveal biases in a single model. Two types of comparisons 1 between those that are conceptually similar and those that re different. In former case interested in how choice of numerical solver and discretisation effects results and the later can help determine the level of physical process representation necessary to represent an observed data set.

Moving to field data increases the generality and significance of scientific evidence obtained. However we also significantly increase the uncertainty of the validation experiment that may constrain the ability to make unequivacol statements. E.g. in bathymetry source condition friction.

The two field data benchmarks are very useful but only capture a small subset of possible tsunami behaviours and do not assess all three stages of tsunami evolution (generation,propagation and inundation) together. The type and size of a tsunami source, propagation extent, and local bathymetry and topography all affect the energy, waveform and subsequent inundation of a tsunami. Consequently additional field data benchmarks, such as the one proposed here, that further capture the variability and sensitivity of the real world system would be useful to allow model developers verify their models and subsequently use their models with greater confidence.
