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1\documentclass[a4paper]{article}
2% to be added when submitted to ocean dynamics
3%\documentclass[smallcondensed,draft]{svjour3}
4%\usepackage{mathptmx}
5%\journalname{Ocean Dynamics}
6
7\usepackage{graphicx}
8\usepackage{hyperref}
9\usepackage{amsfonts}
10\usepackage{url}      % for URLs and DOIs
11\newcommand{\doi}[1]{\url{http://dx.doi.org/#1}}
12
13
14%----------title-------------%
15\title{Benchmarking Tsunami Models using the December 2004 Indian
16  Ocean Tsunami and its Impact at Patong Beach}
17
18%-------authors-----------
19\author{J.~D. Jakeman \and O. Nielsen \and K. VanPutten \and
20  D. Burbidge \and R. Mleczko \and N. Horspool}
21
22% to be added when submitted to ocean dynamics
23%\institute{J.~D. Jakeman \at
24%       The Australian National University, Canberra, \textsc{Australia}\
25%       \email{john.jakeman@anu.edu.au}
26%       \and
27%       O. Nielsen \and R. Mleczko \and D. Burbidge \and K. VanPutten \and N. Horspool \at
28%       Geoscience Australia, Canberra, \textsc{Australia}
29%}
30
31
32%================Start of Document================
33\begin{document}
34\maketitle
35%------Abstract--------------
36\begin{abstract}
37In this paper a new benchmark for tsunami model validation is
38proposed. The benchmark is based upon the 2004 Indian Ocean tsunami,
39which provides a uniquely large amount of observational data for model
40comparison. Unlike the small number of existing benchmarks, the
41proposed test validates all three stages of tsunami evolution -
42generation, propagation and inundation. Specifically we use geodetic
43measurements of the Sumatra--Andaman earthquake to validate the
44tsunami source, altimetry data from the \textsc{jason} satellite to
45test open ocean propagation, eye-witness accounts to assess near shore
46propagation and a detailed inundation survey of Patong Bay, Thailand
47to compare model and observed inundation. Furthermore we utilise this
48benchmark to further validate the hydrodynamic modelling tool
49\textsc{anuga} which is used to simulate the tsunami
50inundation. Important buildings and other structures were incorporated
51into the underlying computational mesh and shown to have a large
52influence of inundation extent. Sensitivity analysis also showed that
53the model predictions are comparatively insensitive to large changes
54in friction and small perturbations in wave weight at the 100 m depth
55contour.
56% to be added when submitted to ocean dynamics
57%\keywords{Tsunami \and modelling \and validation and verification \and benchmark}
58\end{abstract}
59
60\tableofcontents
61%================Section===========================
62
63\section{Introduction}
64Tsunami are a potential hazard to coastal communities all over the
65world. A number of recent large events have increased community and
66scientific awareness of the need for effective detection, forecasting,
67and emergency preparedness. Probabilistic, geological, hydrodynamic,
68and economic models are required to predict the location and
69likelihood of an event, the initial sea floor deformation and
70subsequent propagation and inundation of the tsunami, the
71effectiveness of hazard mitigation procedures and the economic impact
72of such measures and the event itself. Here we focus on modelling of
73the physical processes. For discussion on economic and decision based
74models refer to~\cite{} and the references therein.
75
76Various approaches are currently used to assess the potential impact
77of tsunami. These methods differ in both the formulation used to
78describe the evolution of the tsunami and the numerical methods used
79to solve the governing equations. However any legitimate model must
80address each of the three distinct stages of tsunami evolution---
81generation, propagation and inundation. Geological models must be used
82to provide estimates of initial sea floor and ocean surface
83deformation. The complexity of these models range from empirical to
84non-linear three-dimensional mechanical models. The shallow water wave
85equations, linearised shallow water wave equations, and
86Boussinesq-type equations are frequently used to simulate tsunami
87propagation. These models are typically used to predict quantities
88such as arrival times, wave speeds and heights, and inundation extents
89which are used to develop efficient hazard mitigation plans.
90
91Inaccuracies in model prediction can result in inappropriate
92evacuation plans and town zoning which may result in loss of life and
93large financial losses. Consequently tsunami models must undergo
94sufficient end-to-end testing to increase scientific and community
95confidence in the model predictions.
96
97Complete confidence in a model of a physical system cannot be
98established.  One can only hope to state under what conditions the
99model hypothesis holds true. Specifically the utility of a model can
100be assessed through a process of verification and
101validation. Verification assesses the accuracy of the numerical method
102used to solve the governing equations and validation is used to
103investigate whether the model adequately represents the physical
104system~\cite{bates01}. Together these processes can be used to
105establish the likelihood that a model represents a legitimate
106hypothesis.
107
108The sources of data used to validate and verify a model can be
109separated into three main categories; analytical solutions, scale
110experiments and field measurements. Analytical solutions of the
111governing equations of a model, if available, provide the best means
112of verifying any numerical model. However, analytical solutions are
113frequently limited to a small set of idealised examples that do not
114completely capture the more complex behaviour of `real' events. Scale
115experiments, typically in the form of wave-tank experiments, provide a
116much more realistic source of data that better captures the complex
117dynamics of flows such as those generated by tsunami, whilst allowing
118control of the event and much easier and accurate measurement of the
119tsunami properties. Comparison of numerical predictions with field
120data provides the most stringent test. The use of field data increases
121the generality and significance of conclusions made regarding model
122utility. On the other hand, it must be noted that the use of field
123data also significantly increases the uncertainty of the validation
124experiment that may constrain the ability to make unequivocal
125statements~\cite{bates01}.
126
127Currently, the extent of tsunami related field data is limited. The
128cost of tsunami monitoring programs, bathymetry and topography surveys
129prohibits the collection of data in many of the regions in which
130tsunamis pose greatest threat. The resulting lack of data has limited
131the number of field data sets available to validate tsunami
132models. Synolakis et. al~\cite{synolakis07} have developed a set of
133standards, criteria and procedures for evaluating numerical models of
134tsunami. They propose three analytical solutions to help identify the
135validity of a model and five scale comparisons (wave-tank benchmarks)
136and two field events to assess model veracity.
137
138The first field data benchmark introduced by Synolakis compares model
139results against observed data from the Hokkaido-Nansei-Oki tsunami
140that occurred around Okushiri Island, Japan on the 12th of July
1411993. This tsunami provides an example of extreme runup generated from
142reflections and constructive interference resulting from local
143topography and bathymetry. The benchmark consists of two tide gauge
144records and numerous spatially distributed point sites at which
145modelled maximum runup elevations can be compared. The second
146benchmark is based upon the Rat Islands Tsunami that occurred off the
147coast of Alaska on the 17th of November 2003. The Rat island tsunami
148provides a good test for real-time forecasting models since tsunami
149was recorded at three tsunameters. The test requires matching the
150propagation model data with the DART recording to constrain the
151tsunami source model and then using it to reproduce the tide gauge
152record at Hilo.
153
154In this paper we develop a field data benchmark to be used in
155conjunction with the other tests proposed by Synolakis et
156al.~\cite{synolakis07} to validate and verify tsunami models. Unlike
157the aforementioned tests, the proposed benchmark allows evaluation of
158model structure during all three distinctive stages of the evolution
159of a tsunami. The benchmark consists of geodetic measurements of the
160Sumatra--Andaman earthquake which are used to validate the description
161of the tsunami source, altimetry data from the JASON satellite to test
162open ocean propagation, eye-witness accounts to assess near shore
163propagation and a detailed inundation survey of Patong Bay, Thailand
164to compare model and observed inundation. A description of the data
165required to construct the benchmark is given in
166Section~\ref{sec:data}.
167
168An associated aim of this paper is to illustrate the use of this new
169benchmark to validate an operational tsunami inundation model called
170\textsc{anuga} used by Geoscience Australia. A description of
171\textsc{anuga} is given in Secion~\ref{sec:models} and the validation
172results are given in Secion~\ref{sec:results}.
173
174The numerical models used to model tsunami are extremely
175computationally intensive. Full resolution models of the entire
176evolution process will often take a number of days to
177run. Consequently the uncertainty in model predictions is difficult to
178quantify. However model uncertainty should not be ignored. Section
179~\ref{sec:sensitivity} provides a simple sensitivity analysis that can
180be used to investigate the sensitivity of model predictions to model
181parameters.
182
183%================Section===========================
184\section{Data}\label{sec:data}
185The sheer magnitude of the 2004 Sumatra-Andaman earthquake and the
186devastation caused by the subsequent tsunami have generated much
187scientific interest. As a result an unusually large amount of post
188seismic data has been collected and documented. Data sets from
189seismometers, tide gauges, \textsc{gps} surveys, satellite overpasses,
190subsequent coastal field surveys of run-up and flooding, and
191measurements of coseismic displacements and bathymetry from ship-based
192expeditions, have now been made
193available.%~\cite{vigny05,amnon05,kawata05,liu05}. 
194In this section we
195present the data necessary to implement the proposed benchmark
196corresponding to each of the three stages of the tsunami's evolution.
197
198\subsection{Generation}\label{sec:gen_data}
199All tsunami are generated from an initial disturbance of the ocean
200which develops into a low frequency wave that propagates outwards from
201the source. The initial deformation of the water surface is most
202commonly caused by coseismic displacement of the sea floor, but
203submarine mass failures, landslides, volcanoes or asteroids can also
204cause tsunami. In this section we detail the information we used in
205this study to validate models of the sea floor deformation generated
206by the 2004 Sumatra--Andaman earthquake.
207
208The 2004 Sumatra--Andaman tsunami was generated by severe coseismic
209displacement of the sea floor as a result of one of the largest
210earthquakes on record. The mega-thrust earthquake started on the 26
211December 2004 at 0h58'53'' UTC (or just before 8 am local time)
212approximately 70 km offshore North Sumatra
213(\url{http://earthquake.usgs.gov/eqcenter/eqinthenews/2004/usslav}). The
214rupture propagated 1000-1300 km along the Sumatra-Andaman trench to
215the north at a rate of 2.5-3 km.s$^{-1}$ and lasted approximately 8-10
216minutes~\cite{ammon05}. Estimates of the moment magnitude of this
217event range from about 9.1 to 9.3~\cite{chlieh07,stein07}.
218
219The unusually large surface deformation caused by this earthquakes
220means that there were a range of different geodetic measurements of
221the surface deformation available. These include field measurements of
222uplifted or subsided coral heads, continuous or campaign \textsc{GPS}
223measurements and remote sensing measurements of uplift or subsidence
224(see~\cite{chlieh07} and references therein). Here we use the the near
225field estimates of vertical deformation in northwestern Sumatra and
226the Nicobar-Andaman islands collated by~\cite{chlieh07} to validate
227that our crustal deformation model of the 2004 Sumatra--Andaman
228earthquake is producing reasonable results. Note that the geodetic
229data used here is a combination of the vertical deformation that
230happened in the $\sim$10 minutes of the earthquake plus the
231deformation that followed in the days following the earthquake before
232each particular measurement was actually made (typically of order
233days). Therefore some of the observations may not contain the purely
234co-seismic deformation but could include some post-seismic deformation
235as well~\cite{chlieh07}.
236
237%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.
238
239%\begin{figure}[ht]
240%\begin{center}
241%\includegraphics[width=8.0cm,keepaspectratio=true]{geodeticMeasurements.jpg}
242%\caption{Near field geodetic measurements used to validate tsunami generation. FIXME: Insert appropriate figure here}
243%\label{fig:geodeticMeasurements}
244%\end{center}
245%\end{figure}
246
247\subsection{Propagation}
248Once generated a tsunami will propagate outwards from the source until
249it encounters the shallow water bordering coastal regions. This period
250of the tsunami evolution is referred to as the propagation stage. The
251height and velocity of the tsunami is dependent on the local
252bathymetry in the regions through which the wave travels and the size
253of the initial wave. This section details the bathymetry data needed
254to model the tsunami propagation and the satellite altimetry transects
255used here to validate open ocean tsunami models.
256
257\subsubsection{Bathymetry Data}
258A number of raw data sets were obtained, analysed and checked for
259quality and subsequently gridded for easier visualisation and input
260into the tsunami models. The resulting grid data is relatively coarse
261in the deeper water and becomes progressively finer as the distance to
262Patong Bay decreases.
263
264The nested bathymetry grid was generated from:
265\begin{itemize}
266\item A two arc minute grid data set covering the Bay of Bengal,
267  DBDB2, obtained from US Naval Research Labs;
268\item A 3 second arc grid covering the whole of the Andaman Sea based
269  on Thai Navy charts no 45 and no 362; and
270\item A one second grid created from the digitised Thai Navy
271  bathymetry chart, no 358. which covers Patong Bay and the
272  immediately adjacent regions.
273\end{itemize}
274
275The final bathymetry data set consists of four nested grids obtained
276via interpolation and resampling of the aforementioned data sets. The
277four grids are shown in Figure~\ref{fig:nested_grids}.  The coarsest
278bathymetry was obtained by interpolating the DBDB2 grid to a 27 second
279arc grid. A subsection of this region was then replaced by 9 second
280data which was generated by sub-sampling the 3 second of arc grid from
281NOAA. A subset of the 9 second grid was replaced by the 3 second
282data. Finally, the one second grid was used to approximate the
283bathymetry in Patong Bay and the immediately adjacent regions. Any
284points that deviated from the general trend near the boundary were
285deleted.
286
287The sub-sampling of larger grids was performed by using {\bf resample}
288a Generic Mapping Tools (\textsc{GMT}) program (\cite{wessel98}). The
289gridding of data was performed using {\bf Intrepid} a commercial
290geophysical processing package developed by Intrepid Geophysics. The
291gridding scheme employed the nearest neighbour algorithm followed by
292an application of minimum curvature akima spline smoothing FIXME(Ole):
293Need \cite{} here.
294
295\begin{figure}[ht]
296\begin{center}
297\includegraphics[width=0.75\textwidth,keepaspectratio=true]{nested_grids}
298\caption{Nested grids of elevation data.}
299\label{fig:nested_grids}
300\end{center}
301\end{figure}
302
303\subsubsection{JASON Satellite Altimetry}\label{sec:data_jason}
304During the 26 December 2004 event, the Jason satellite tracked from
305north to south and over the equator at 02:55 UTC nearly two hours
306after the earthquake \cite{gower05}. The satellite recorded the sea
307level anomaly compared to the average sea level from its previous five
308passes over the same region in the 20-30 days prior.  This data was
309used to validate the propagation stage in Section
310\ref{sec:resultsPropagation}.
311%DB I suggest we combine with model data to reduce the number of figures. The satellite track is shown in Figure~\ref{fig:satelliteTrack}.
312
313%\begin{figure}[ht]
314%\begin{center}
315%\includegraphics[width=8.0cm,keepaspectratio=true]{sateliteTrack.jpg}
316%\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.}
317%\label{fig:satelliteTrack}
318%\end{center}
319%\end{figure}
320
321%\begin{figure}[ht]
322%\begin{center}
323%\includegraphics[width=8.0cm,keepaspectratio=true]{jasonAltimetry.jpg}
324%\caption{JASON satellite altimetry seal level anomaly. FIXME: should we include figure here with just JASON altimetry.}
325%\label{fig:jasonAltimetry}
326%\end{center}
327%\end{figure}
328
329%FIXME: Can we compare the urs model against the TOPEX-poseidon satellite as well? DB No (we don't have the data currently).
330
331\subsection{Inundation}
332Inundation refers to the final stages of the evolution a tsunami and
333covers the propagation of the tsunami in shallow coastal water and the
334subsequent run-up on to land. This process is typically the most
335difficult of the three stages to model due to thin layers of water
336flowing rapidly over dry land.  Aside from requiring robust solvers
337which can simulate such complex flow patterns, this part of the
338modelling process also requires high resolution and quality elevation
339data which is often not available. In the case of model validation
340high quality field measurements are also required. For the proposed
341benchmark the authors have obtained a high resolution bathymetry and
342topography data set and a high quality inundation survey map from the
343CCOP in Thailand (\cite{szczucinski06}) which can be used to validate
344model inundation. 
345
346The datasets necessary for reproducing the results
347of the inundation stage are available on Sourceforge under the ANUGA
348project (\url{http://sourceforge.net/projects/anuga}). At the time of
349writing the direct link is \url{http://tinyurl.com/patong2004-data}.
350%
351%\url{http://sourceforge.net/project/showfiles.php?group_id=172848&package_id=319323&release_id=677531}.
352
353In this section we also present eye-witness accounts which can be used to qualitatively validate tsunami inundation.
354
355\subsubsection{Topography Data}
356A one second grid was used to approximate the topography in Patong
357Bay. This elevation data was again created from the digitised Thai
358Navy bathymetry chart, no 358. A visualisation of the elevation data
359set used in Patong bay is shown in
360Figure~\ref{fig:patong_bathymetry}. The continuous topography is an
361interpolation of known elevation measured at the coloured dots.
362
363\begin{figure}[ht]
364\begin{center}
365\includegraphics[width=8.0cm,keepaspectratio=true]{patong_bay_data.jpg}
366\caption{Visualisation of the elevation data set used in Patong Bay showing data points, contours, rivers and roads draped over the final model.}
367\label{fig:patong_bathymetry}
368\end{center}
369\end{figure}
370
371\subsubsection{Buildings and Other Structures}
372Human made build and structures can significantly effect tsunami
373inundation. The location and size and number of floors of the
374buildings in Patong Bay were extracted from a GIS data set provided by
375the CCOP in Thailand (see acknowledgements at the end of this paper).
376The heights of these
377buildings were estimated assuming that each floor has a height of 3 m.
378
379\subsubsection{Inundation Survey}
380Tsunami run-up is often the cause of the largest financial and human
381losses yet run-up data that can be used to validate model runup
382predictions is scarce. Of the two field benchmarks proposed by
383Synolakis only the Okushiri benchmark facilitates comparison between
384modelled and observed run-up. One of the major strengths of the
385benchmark proposed here is that modelled runup can be compared to an
386inundation survey which maps the maximum run-up along an entire coast
387line rather than at a series of discrete sites. The survey map is
388shown in Figure~\ref{fig:patongescapemap} and plots the maximum run-up
389of the 2004 tsunami in Patong bay. Refer to Szczucinski et
390al~\cite{szczucinski06} for further details.
391
392\subsubsection{Eyewitness Accounts}\label{sec:eyewitness data}
393Eyewitness accounts detailed in~\cite{papadopoulos06}
394report that most people at Patong Beach observed an initial retreat of
395the shoreline of more than 100 m followed a few minutes later by a
396strong wave (crest). Another less powerful wave arrived another five
397or ten minutes later. Eyewitness statements place the arrival time of
398the strong wave between 2 hours and 55 minutes to 3 hours and 5
399minutes after the source rupture (09:55am to 10:05am local time).
400
401Two videos were sourced from the internet\footnote{The footage is
402widely available and can for example be obtained from
403\url{http://www.archive.org/download/patong_bavarian/patong_bavaria.wmv}
404(Comfort Hotel) and
405\url{http://www.archive.org/download/tsunami_patong_beach/tsunami_patong_beach.wmv}
406(Novotel)}
407%http://wizbangblog.com/content/2005/01/01/wizbang-tsunami.php
408which include footage of the tsunami in Patong Bay on the day
409of the Indian Ocean Tsunami. Both videos show an already inundated
410group of buildings, they then show what is to be assumed as the second
411and third waves approaching and further flooding the buildings and
412street.  The first video is in the very north filmed from what is
413believed to be the roof of the Novotel Hotel marked 'North' in Figure
414\ref{fig:gauge_locations}. The second video is in the very south
415filmed from the second story of a building next door to the Comfort
416Resort near the corner of Ruam Chai St and Thaweewong Road.  This
417location is marked 'south' in Figure \ref{fig:gauge_locations} and
418Figure~\ref{fig:video_flow} shows stills from this video. Both videos
419were used to estimate flow speeds and inundation dephts over time.
420
421\begin{figure}[ht]
422\begin{center}
423\includegraphics[width=6.0cm,keepaspectratio=true]{flow_rate_south_0_00sec.jpg}
424\includegraphics[width=6.0cm,keepaspectratio=true]{flow_rate_south_5_04sec.jpg}
425\includegraphics[width=6.0cm,keepaspectratio=true]{flow_rate_south_7_12sec.jpg}
426\includegraphics[width=6.0cm,keepaspectratio=true]{flow_rate_south_7_60sec.jpg}
427\caption{Four frames from a video where flow rate could be estimated,
428  circle indicates tracked debris, from top left: 0.0 sec, 5.0 s, 7.1
429  s, 7.6 s.}
430\label{fig:video_flow}
431\end{center}
432\end{figure}
433
434Flow rates were estimated using landmarks found in both videos and
435were found to be in the range of 5 to 7 metres per second (+/- 2 m/s)
436in the north and 0.5 to 2 metres per second (+/- 1 m/s) in the south.
437Water depths could also
438be estimated from the videos by the level at which water rose up the
439sides of buildings such as shops. Our estimates are in the order of
4401.5 to 2.0 metres (+/- 0.5 m).
441Fritz ~\cite{fritz06} performed a detailed
442analysis of video frames taken around Banda Aceh and arrived at flow
443speeds in the range of 2 to 5 m/s.
444
445\begin{figure}[ht]
446\begin{center}
447\includegraphics[width=8.0cm,keepaspectratio=true]{patongescapemap.jpg}
448\caption{Tsunami survey mapping the maximum observed inundation at
449  Patong beach courtesy of the Thai Department of Mineral Resources
450  \protect \cite{szczucinski06}.}
451\label{fig:patongescapemap}
452\end{center}
453\end{figure}
454
455\subsection{Validation Check-List}
456\label{sec:checkList}
457The data described in this section can be used to construct a
458benchmark to validate all three stages of the evolution of a
459tsunami. In particular we propose that a legitimate tsunami model
460should reproduce the following behaviour:
461\begin{itemize}
462 \item Reproduce the vertical deformation observed in north-western
463   Sumatra and along the Nicobar--Andaman islands, see
464   Section~\ref{sec:gen_data}.
465 \item Reproduce the \textsc{jason} satellite altimetry sea surface
466   anomalies, see Section~\ref{sec:data_jason}.
467 \item Reproduce the inundation survey map in Patong bay
468   (Figure~\ref{fig:patongescapemap}).
469 \item Simulate a leading depression followed by two distinct crests
470   of decreasing magnitude.
471 \item Predict the water depths and flow speeds, at the locations of
472   the eye-witness videos, that fall within the bounds obtained from
473   the videos.
474\end{itemize}
475
476Ideally, the model should also be compared to measured timeseries of
477waveheights and velocities but the authors are not aware of the
478availability of such data.
479
480
481%================Section===========================
482\section{Modelling the Event}\label{sec:models}
483Numerous models are currently used to model and predict tsunami
484generation, propagation and run-up~\cite{titov97a,satake95}. Here we
485introduce the modelling methodology employed by Geoscience Australia
486to illustrate the utility of the proposed benchmark. Geoscience
487Australia's tsunami modelling methodology comprises the three parts;
488generation, propagation and inundation
489(Sections~\ref{sec:modelGeneration},\ref{sec:modelPropagation} and
490\ref{sec:modelInundation} respectively).
491
492\subsection{Generation}\label{sec:modelGeneration}
493
494There are various approaches to modelling the expected crustal
495deformation from an earthquake at depth. Most approaches model the
496earthquake as a dislocation in a linear, elastic medium. Here we use
497the method of Wang et. al.~\cite{wang03}. One of the main advantages
498of their method is that it allows the dislocation to be located in a
499stratified linear elastic half-space with an arbitrary number of
500layers. Other methods (such as those based on Okada's equations) can
501only model the dislocation in a homogeneous elastic half space, or can
502only include a limited number of layers, and thus cannot model the
503effect of the depth dependence of the elasticity of the
504Earth~\cite{wang03}. The original versions of the codes described here
505are available from \url{http://www.iamg.org/CGEditor/index.htm}. The
506first program, \textsc{edgrn}, calculates elastic Green's function for
507a set of point sources at a regular set of depths out to a specified
508distance. The equations controlling the deformation are solved by
509using a combination of Hankel's transform and Wang et al's
510implementation of the Thomson-Haskell propagator
511algorithm~\cite{wang03}. Once the Green's functions are calculated we
512use a slightly modified version of \textsc{edcmp} to calculate the sea
513floor deformation for a specific subfault. This second code
514discretises the subfault into a set of unit sources and sums the
515elastic Green's functions calculated from \textsc{edgrn} for all the
516unit sources on the fault plane in order to calculate the final static
517deformation caused by a two dimensional dislocation along the
518subfault. This step is possible because of the linearity of the
519governing equations. For this study, we have made minor modifications
520to \textsc{edcmp} in order for it to output in a file format
521compatible with the propagation code in the following section but it
522is otherwise the similar to the original code.
523
524In order to calculate the crustal deformation using these codes we
525thus need to have a model describing the variation in elastic
526properties with depth and a slip model of the earthquake to describe
527the dislocation. The elastic parameters used for this study are the
528same as those in Table 2 of Burbidge~\cite{burbidge08}. For the slip
529model, there are many possible models for the 2004 Andaman--Sumatran
530earthquake to choose from
531~\cite{chlieh07,asavanant08,arcas06,grilli07,ioualalen07}. Some are
532determined from various geological surveys of the site, others solve
533an inverse problem which calibrates the source based upon the tsunami
534wave signal, the seismic signal and/or the runup. The source
535parameters used here to simulate the 2004 Indian Ocean tsunami were
536taken from the slip model G-M9.15 from Chlieh
537et. al.~\cite{chlieh07}. This model was created by inversion of wide
538range of geodetic and seismic data. The slip model consists of 686
53920km x 20km subsegments each with a different slip, strike and dip
540angle. The dip subfaults go from $17.5^0$ in the north and $12^0$ in
541the south. Refer to Chlieh et. al.~\cite{chlieh07} for a detailed
542discussion of this model and its derivation. Note that the geodetic
543data used in the validation was also included by~\cite{chlieh07} in
544the inversion used to find G-M9.15, thus the validation is not
545completely independent. However, a successful validation would still
546show that the crustal deformation and elastic properties model used
547here is at least as valid as the one used by Chlieh
548et. al.~\cite{chlieh07} and can reproduce the observations just as
549accurately.
550
551\subsection{Propagation}\label{sec:modelPropagation}
552We use the \textsc{ursga} model described below to simulate the
553propagation of the 2004 tsunami in the deep ocean ocean, based on a
554discrete representation of the initial deformation of the sea floor,
555described in Section~\ref{sec:modelGeneration}. For the models shown
556here, we assume that the uplift is instantaneous and creates a wave of
557the same size and amplitude as the co-seismic sea floor deformation.
558
559\subsubsection{URSGA}
560\textsc{ursga} is a hydrodynamic code that models the propagation of
561the tsunami in deep water using a finite difference method on a staggered grid
562to solve
563the depth integrated linear or nonlinear shallow water equations in
564spherical co-ordinates with friction and Coriolis terms. The code is
565based on Satake~\cite{satake95} with significant modifications made by
566the \textsc{urs} corporation~\cite{thio08} and Geoscience
567Australia~\cite{burbidge08}. The tsunami is propagated via the nested
568grid system. Coarse grids are used in the open ocean and the finest
569resolution grid is employed in the region of most
570interest. \textsc{Ursga} is not publicly available.
571
572\subsection{Inundation}\label{sec:modelInundation}
573The utility of the \textsc{ursga} model decreases with water depth
574unless an intricate sequence of nested grids is employed. In
575comparison \textsc{anuga}, described below, is designed to produce
576robust and accurate predictions of on-shore inundation, but is less
577suitable for earthquake source modelling and large study areas because
578it is based on projected spatial coordinates. Consequently, the
579Geoscience Australia tsunami modelling methodology is based on a
580hybrid approach using models like \textsc{ursga} for tsunami
581propagation up to a 100 m depth contour.
582%Specifically we use the \textsc{ursga} model to simulate the
583%propagation of the 2004 Indian Ocean tsunami in the deep ocean, based
584%on a discrete representation of the initial deformation of the sea
585%floor, described in Section~\ref{sec:modelGeneration}.
586The wave signal is then used as a time varying boundary condition for
587the \textsc{anuga} inundation simulation.
588% A description of \textsc{anuga} is the following section.
589
590\subsubsection{ANUGA}
591\textsc{Anuga} is an Open Source hydrodynamic inundation tool that
592solves the conserved form of the depth integrated nonlinear shallow
593water wave equations. The scheme used by \textsc{anuga}, first
594presented by Zoppou and Roberts~\cite{zoppou99}, is a high-resolution
595Godunov-type method that uses the rotational invariance property of
596the shallow water equations to transform the two-dimensional problem
597into local one-dimensional problems. These local Riemann problems are
598then solved using the semi-discrete central-upwind scheme of Kurganov
599et al.~\cite{kurganov01} for solving one-dimensional conservation
600equations. The numerical scheme is presented in detail in Zoppou and
601Roberts~\cite{zoppou99}, Roberts and Zoppou~\cite{roberts00}, and
602Nielsen et al.~\cite{nielsen05}. An important capability of the
603software is that it can model the process of wetting and drying as
604water enters and leaves an area. This means that it is suitable for
605simulating water flow onto a beach or dry land and around structures
606such as buildings. It is also capable of adequately resolving
607hydraulic jumps due to the ability of the finite-volume method to
608handle discontinuities. The numerical scheme can also handle
609transitions between sub-critical and super-critical flow regimes
610seamlessly. \textsc{Anuga} has been validated against a number of
611analytical solutions and the wave tank simulation of the 1993 Okushiri
612Island tsunami~\cite{nielsen05,roberts06}.
613
614%================Section===========================
615\section{Results}\label{sec:results}
616This section presents a validation of 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.
617
618\subsection{Generation}\label{modelGeneration}
619The location and magnitude of the sea floor displacement associated
620with the 2004 Sumatra--Andaman tsunami calculated from the G-M9.15
621model of~\cite{chlieh07} is shown in
622Figure~\ref{fig:surface_deformation}. The magnitude of the sea floor
623displacement ranges from about $-3.0$ to $5.0$ metres. The region near
624the fault is predicted to uplift, while that further away from the
625fault subsides. Also shown in Figure~\ref{fig:surface_deformation} are
626the areas that were observed to uplift (arrows pointing up) or subside
627(arrows point down) during and immediately after the earthquake. Most
628of this data comes uplifted or subsided coral heads. The length of
629vector increases with the magnitude of the displacement, the length
630corresponding to 1 m of observed motion is shown in the top right
631corner of the figure. As can be seen, the source model detailed in
632Section~\ref{sec:modelGeneration} produces a crustal deformation that
633matches the vertical displacements in the Nicobar-Andaman islands and
634Sumatra very well. Uplifted regions are close to the fault and
635subsided regions are further away. The crosses on
636Figure~\ref{fig:surface_deformation} show estimates of the pivot line
637from the remote sensing data~\cite{chlieh07} and they follow the
638predicted pivot line quite accurately. The average difference between
639the observed motion and the predicted motion (including the pivot line
640points) is only 0.06 m, well below the typical error of the
641observations of between 0.25 and 1.0 m. However, the occasional point
642has quite a large error (over 1 m), for example a couple
643uplifted/subsided points appear to be on a wrong side of the predicted
644pivot line~\ref{fig:surface_deformation}. The excellence of the fit is
645not surprising, since the original slip model was chosen
646by~\cite{chlieh07} to fit this (and the seismic data) well. However,
647this does demonstrate that \textsc{edgrn} and our modified version of
648\textsc{edstat} can reproduce the correct pattern of vertical
649deformation very well when the slip distribution is well constrained
650and when reasonable values for the elastic properties are used.
651
652\begin{figure}[ht]
653\begin{center}
654\includegraphics[width=5cm,keepaspectratio=true]{surface_deformation.jpg}
655%\includegraphics[totalheight=0.3\textheight,width=0.8\textwidth]{surface_deformation.jpg}
656\caption{Location and magnitude of the vertical component of the sea
657  floor displacement associated with the 2004 Indian Ocean tsunami
658  based on the slip model, G-M9.15. The black arrows which point up
659  show areas observed to uplift during and immediately after the
660  earthquake, those point down are locations which subsided. The
661  length of increases with the magnitude of the deformation. The arrow
662  length corresponding to 1 m of deformation is shown in the top right
663  hand corner of the figure. The crosses marks show the location of
664  the pivot line (the region between the uplift and subsided region
665  where the uplift is zero) derived from remote sensing. All the
666  observational data come from the dataset collated
667  by~\cite{chlieh07}.}
668\label{fig:surface_deformation}
669\end{center}
670\end{figure}
671
672\subsection{Propagation}\label{sec:resultsPropagation}
673The deformation results described in Section~\ref{sec:modelGeneration}
674were used to provide a profile of the initial ocean surface
675displacement. This wave was used as an initial condition for
676\textsc{ursga} and was propagated throughout the Bay of Bengal. The
677rectangular computational domain of the largest grid extended from
67890$^0$ to 100$^0$ East and 0 to 15$^0$ North and contained
6791335$\times$1996 finite difference points. Inside this grid, a nested
680sequence of grids was used. The grid resolution of the nested grids
681went from 27 arc seconds in the coarsest grid, down to 9 arc seconds
682in the second grid, 3 arc seconds in the third grid and finally 1 arc
683second in the finest grid near Patong. The computational domain is
684shown in Figure~\ref{fig:computational_domain}.
685
686Figure \ref{fig:jasonComparison} provides a comparison of the
687\textsc{ursga} predicted sea surface elevation with the JASON
688satellite altimetry data. The \textsc{ursga} model replicates the
689amplitude and timing of the the wave observed at 2.5 degrees South,
690but underestimates the amplitude of the wave further to the south at
6914 degrees South. In the model, the southern most of these two waves
692appears only as a small bump in the cross section of the model shown
693in Figure~\ref{fig:jasonComparison} instead of being a distinct peak
694as can be seen in the satellite data. Also note
695that the \textsc{ursga} model prediction of the ocean surface
696elevation becomes out of phase with the JASON data at 3 to 7 degrees
697latitude. Chlieh et al~\cite{chlieh07} also observe these misfits and
698suggest it is caused by a reflected wave from the Aceh Peninsula that
699is not resolved in the model due to insufficient resolution of the
700computational mesh and bathymetry data. This is also a limitation of
701the model presented here, but probably could be improved by nesting
702grids near Aceh.
703
704\begin{figure}[ht]
705\begin{center}
706\includegraphics[width=12.0cm,keepaspectratio=true]{jasonComparison.jpg}
707\caption{Comparison of the \textsc{ursga} predicted surface elevation
708  with the JASON satellite altimetry data. The \textsc{ursga} wave
709  heights have been corrected for the time the satellite passed
710  overhead compared to JASON sea level anomaly.  }
711\label{fig:jasonComparison}
712\end{center}
713\end{figure}
714
715\subsection{Inundation}
716After propagating the tsunami in the open ocean using \textsc{ursga}
717the approximated ocean and surface elevation and horisontal flow
718velocities were extracted and used to construct a boundary condition
719for the \textsc{anuga} model. The interface betwen the \textsc{ursga}
720and \textsc{anuga} models was chosen to roughly follow the 100 m depth
721contour along the west coast of Phuket Island. The computational
722domain is shown in Figure \ref{fig:computational_domain}
723\begin{figure}[ht]
724\begin{center}
725%\includegraphics[width=5.0cm,keepaspectratio=true]{extent_of_ursga_model.jpg}
726%\includegraphics[width=5.0cm,keepaspectratio=true]{ursgaDomain.jpg}
727\includegraphics[width=5.0cm,keepaspectratio=true]{extent_of_ANUGA_model.jpg}
728\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)}
729\label{fig:computational_domain}
730\end{center}
731\end{figure}
732
733The domain was discretised into 386,338 triangles. The resolution of
734the grid was increased in certain regions to efficiently increase the
735accuracy of the simulation. The grid resolution ranged between a
736maximum triangle area of $1\times 10^5$ m$^2$ near the Western ocean
737boundary to $20$ m$^2$ in the small regions surrounding the inundation
738region in Patong Bay. Due to a lack of available data, friction was
739set to a constant throughout the computational domain. For the
740reference simulation a Manning's coefficient of 0.01 was chosen to
741represent a small resistance to the water flow. See Section
742\ref{sec:friction sensitivity} for details on model sensitivity to
743this parameter.
744
745
746The boundary condition at each side of the domain towards the south
747and the north where no data was available was chosen as a transmissive
748boundary condition effectively replicating the time dependent wave
749height present just inside the computational domain. Momentum was set
750to zero. Other choices include applying the mean tide value as a
751Dirichlet type boundary condition but experiments as well as the
752result of the verification reported here showed that this approach
753tends to under estimate the tsunami impact due to the tempering of the
754wave near the side boundaries whereas the transmissive boundary
755condition robustly preserves the wave.
756
757During the \textsc{anuga} simulation the tide was kept constant at
758$0.80$ m. This value was chosen to correspond to the tidal height
759specified by the Thai Navy tide charts
760(\url{http://www.navy.mi.th/hydro/}) at the time the tsunami arrived
761at Patong Bay. Although the tsunami propagated for approximately 3
762hours before it reach Patong Bay, the period of time during which the
763wave propagated through the \textsc{anuga} domain is much
764smaller. Consequently the assumption of constant tide height is
765reasonable
766
767%FIXME (Ole): Perhaps rephrase a bit as the 1cm vs 10cm is hard to
768%understand.  Remove figure using 1cm inundation
769Maximum onshore inundation elevation was computed from the model
770throughout the entire Patong Bay region.
771Figure~\ref{fig:inundationcomparison1cm} (left) shows very good
772agreement between the measured and simulated inundation. However
773these results are dependent on the classification used to determine
774whether a region in the numerical simulation was inundated. In
775Figure~\ref{fig:inundationcomparison1cm} (left) a point in the computational
776domain was deemed inundated if at some point in time it was covered by
777at least 1 cm of water. However, the precision of the inundation boundary
778generated by the on-site survey is most likely less than that as it
779was determined by observing water marks and other signs
780left by the receding waters. Consequently the measurement error along
781the inundation boundary of the survey is likely to vary significantly
782and somewhat unpredictably.
783Consequently, an inundation threshold of 10 cm was selected for inundation
784extents reported in this paper to reflect
785the more likely accuracy of the survey and subsequently facilitate a more
786appropriate comparison between the modelled and observed inundation
787area.
788Figure~\ref{fig:inundationcomparison1cm} (right) shows the simulated
789inundation using a larger threshold of 10 cm.
790
791
792An 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}.
793
794%\url{https://datamining.anu.edu.au/anuga/attachment/wiki/AnugaPublications/patong_2004_indian_ocean_tsunami_ANUGA_animation.mov}.
795
796\begin{figure}[ht]
797\begin{center}
798\includegraphics[width=5.0cm,keepaspectratio=true]{Depth_small_transmissive_d0.jpg}
799\includegraphics[width=5.0cm,keepaspectratio=true]{sensitivity_reference.jpg}
800\caption{Simulated inundation versus observed inundation using an inundation threshold of 1cm (left) and 10cm (right). FIXME: NEED Graph for 10cm}
801\label{fig:inundationcomparison1cm}
802\end{center}
803\end{figure}
804
805To quantify the agreement between observed and simulated inundation we
806introduce the measure
807\begin{equation}
808\rho_{in}=\frac{A(I_m\cap I_o)}{A(I_o)}
809\end{equation}
810representing the ratio $\rho_{in}$ of observed
811inundation region $I_o$ captured by the model $I_m$. Another useful
812measure is the fraction of the modelled inundation area that falls
813outside the observed inundation area given by the formula
814\begin{equation}
815\rho_{out}=\frac{A(I_m\setminus (I_m\cap I_o))}{A(I_o)}
816\end{equation}
817These values for the two aforementioned simulations are given in
818Table~\ref{table:inundationAreas} %FIXME (Ole): The left hand side of
819%these equations are not areas - consider another symbol.
820
821Discrepancies between the survey data and the modelled inundated
822include: unknown distribution of surface roughness, inappropriate
823parameterisation of the source model, effect of humans structures on
824flow, as well as uncertainties in the elevation data, effects of
825erosion and deposition by the tsunami event, measurement errors, and
826missing data in the field survey data itself. The impact of some of
827these sources of uncertainties are is investigated in
828Section~\ref{sec:sensitivity}
829
830\subsection{Eye-witness accounts}
831Figure \ref{fig:gauge_locations} shows four locations where time
832series have been extracted from the model. The two offshore timeseries
833are shown in Figure \ref{fig:offshore_timeseries} and the two onshore
834timeseries are shown in Figure \ref{fig:onshore_timeseries}. The
835latter coincide with locations where video footage from the event is
836available as described in Section \ref{sec:eyewitness data}.
837
838\begin{figure}[ht]
839\begin{center}
840\includegraphics[width=7.0cm,keepaspectratio=true]{tide_gauge_locations.jpg}
841\caption{Location of timeseries extracted from the model output}
842\label{fig:gauge_locations}
843\end{center}
844\end{figure}
845
846
847\begin{figure}[ht]
848\begin{center}
849\includegraphics[width=10.0cm,keepaspectratio=true]{gauge_bay_depth.jpg}
850\includegraphics[width=10.0cm,keepaspectratio=true]{gauge_bay_speed.jpg}
851\caption{Timeseries obtained from the two offshore locations shown in Figure \protect \ref{fig:gauge_locations}}
852\end{center}
853\label{fig:offshore_timeseries}
854\end{figure}
855
856\begin{figure}[ht]
857\begin{center}
858\includegraphics[width=10.0cm,keepaspectratio=true]{gauges_hotels_depths.jpg}
859\includegraphics[width=10.0cm,keepaspectratio=true]{gauges_hotels_speed.jpg}
860\caption{Timeseries obtained from the two onshore locations shown in Figure \protect \ref{fig:gauge_locations}}
861\end{center}
862\label{fig:onshore_timeseries}
863\end{figure}
864
865
866The estimated depths and flow rates given in Section \ref{sec:eyewitness data} are shown
867together with the modelled depths and flow rates obtained from the model in
868Table \ref{tab:depth and flow comparisons}.
869
870\begin{table}
871\[
872  \begin{array}{|l|cc|cc|}
873  \hline
874                 & \multicolumn{2}{|c|}{\mbox{Depth [m]}}
875                 & \multicolumn{2}{c|}{\mbox{Flow [m/s]}} \\ 
876                 & \mbox{Observed} & \mbox{Modelled}
877                 & \mbox{Observed} & \mbox{Modelled} \\ \cline{2-5}                 
878    \mbox{North} & 1.5-2 & 1.4 & 5-7 & 0.1 - 3.3 \\
879    \mbox{South} & 1.5-2 & 1.5 & 0.5-2 & 0.2 - 2.6 \\ \hline
880  \end{array}
881\]
882\label{tab:depth and flow comparisons}
883\end{table} 
884
885%can be estimated with landmarks found in
886%satellite imagery and the use of a GIS and were found to be in the
887%range of 5 to 7 metres per second (+/- 2 m/s) in the north and 0.5 to
888%2 metres per second (+/- 1 m/s) in the south.
889
890Given the uncertainties in both model and observations, there is agreement
891between the values obtained from the videos and the simulations.
892
893% Our modelled flow rates show
894%maximum values in the order of 0.2 to 2.6 m/s in the south and 0.1 to
895%3.3 m/s for the north as shown in the figures. Water depths could also
896%be estimated from the videos by the level at which water rose up the
897%sides of buildings such as shops. Our estimates are in the order of
898%1.5 to 2.0 metres (+/- 0.5 m). This is in the same range as our
899%modelled maximum depths of 1.4 m in the north and 1.5 m in the south
900%as seen in the figure.
901
902
903
904
905
906%================Section===========================
907\section{Sensitivity Analysis}
908\label{sec:sensitivity}
909This section investigates the effect of different values of Manning's
910friction coefficient, changing waveheight at the 100 m depth contour,
911and the presence and absence of buildings in the elevation dataset on
912model maximum inundation.
913
914%========================Friction==========================%
915\subsection{Friction}
916\label{sec:friction sensitivity}
917The first study investigated the impact of surface roughness on the
918predicted run-up. According to Schoettle~\cite{schoettle2007}
919appropriate values of Manning's coefficient range from 0.007 to 0.030
920for tsunami propagation over a sandy sea floor and the reference model
921uses a value of 0.01.  To investigate sensitivity to this parameter,
922we simulated the maximum onshore inundation using the a Manning's
923coefficient of 0.0003 and 0.03. The resulting inundation maps are
924shown in Figure~\ref{fig:sensitivity_friction} and the maximum flow
925speeds in Figure~\ref{fig:sensitivity_friction_speed}. These figures
926show that the on-shore inundation extent decreases with increasing
927friction and that small perturbations in the friction cause bounded
928changes in the output. This is consistent with the conclusions of
929Synolakis~\cite{synolakis05} who states that the long wavelength of
930tsunami tends to mean that the friction is less important in
931comparison to the motion of the wave.
932
933%========================Wave-Height==========================%
934\subsection{Input Wave Height}\label{sec:waveheightSA}
935The effect of the wave-height used as input to the inundation model
936\textsc{anuga} was also investigated.
937Figure~\ref{fig:sensitivity_boundary} indicates that the inundation
938severity is directly proportional to the boundary waveheight but small
939perturbations in the input wave-height of 10 cm appear to have little
940effect on the final on-shore run-up. Obviously larger perturbations
941will have greater impact. However, this value is generally well
942predicted by the generation and propagation models such as
943\textsc{ursga}. See e.g. \cite{} FIXME Toshi Baba's validation study at
944Kuril islands.
945
946
947
948%========================Buildings==========================%
949\subsection{Buildings and Other Structures}
950The presence of buildings has the greatest influence on the maximum
951on-shore inundation extent. Figure~\ref{fig:sensitivity_nobuildings}
952shows the maximum run-up in the presence and absence of buildings. It
953is apparent that the inundation is much more severe when the presence
954of man made structures and buildings are ignored. Maximal flow speeds
955for these two model parameterisations are shown in
956Figure~\ref{fig:sensitivity_nobuildings_speed}.
957
958\begin{table}
959\begin{center}
960\label{table:inundationAreas}
961\caption{$\rho_{in}$ and $\rho_{out}$ of the reference simulation and all sensitivity studies}
962\begin{tabular}{|c|c|c|}
963\hline
964 & $\rho_{in}$ & $\rho_{out}$ \\ 
965\hline\hline
966Reference & 0.76 & 0.22\\ 
967Min. Friction & Ã— & \\ 
968Max. Friction & Ã— & \\ 
969Min. Wave-Height× & Ã— & \\
970Max. Wave-Height× & Ã— & \\
971No Buildings × & Ã— & \\
972\hline 
973\end{tabular}
974\end{center}
975\end{table}
976
977%================Section===========================
978
979\section{Conclusion}
980This paper proposes an additional field data benchmark for the
981verification of tsunami inundation models. Currently, there is a
982scarcity of appropriate validation datasets due to a lack of well
983documented historical tsunami impacts. The benchmark proposed here
984utilises the uniquely large amount of observational data for model
985comparison obtained during, and immediately following, the
986Sumatra--Andaman tsunami of 26th December 2004. Unlike the small
987number of existing benchmarks, the proposed test validates all three
988stages of tsunami evolution - generation, propagation and
989inundation. In an attempt to provide higher visability and easier
990accessibility for tsunami benchmark problems the data used to
991construct the proposed benchmark is documented and freely available at
992\url{http://tinyurl.com/patong2004-data}.
993
994This study also shows that the tsunami impact modelling methodology
995adopted is sane and able to predict inundation extents with reasonable
996accuracy.  An associated aim of this paper was to further validate the
997hydrodynamic modelling tool \textsc{anuga} which is used to simulate
998the tsunami inundation and run rain-induced floods. Model predictions
999matched well geodetic measurements of the Sumatra--Andaman earthquake,
1000altimetry data from the \textsc{jason}, eye-witness accounts of wave
1001front arrival times and flow speeds and a detailed inundation survey
1002of Patong Bay, Thailand.
1003
1004A simple sensitivity analysis was performed to assess the influence of
1005small changes in friction, wave-height at the 100 m depth contour and
1006the presence of buildings and other structures on the model
1007predictions. The presence of buildings has the greatest influence on
1008the simulated inundation extent. The value of friction and small
1009perturbations in the waveheight at the ANUGA boundary have
1010comparatively little effect on the model results.
1011
1012%================Acknowledgement===================
1013\section*{Acknowledgements}
1014This project was undertaken at Geoscience Australia and the Department
1015of Mathematics, The Australian National University. The authors would
1016like to thank Niran Chaimanee from the CCOP, Thailand for providing
1017the post 2004 tsunami survey data, building footprints, aereal
1018photography and the elevation data for Patong beach, Prapasri Asawakun
1019from the Suranaree University of Technology and Parida Kuneepong for
1020supporting this work; and Drew Whitehouse from the Australian National
1021University for preparing the animation.
1022
1023\section{Appendix}
1024\begin{figure}[ht]
1025\begin{center}
1026\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
1027\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_minus10}
1028\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_plus10}
1029\caption{Model results with wave height at ANUGA boundary artificially
1030  modified to asses sensitivities. The first image is the reference
1031  inundation extent as reported in Section \protect \ref{sec:results}
1032  while the second and third show the inundation results if the wave
1033  at the ANUGA boundary is reduced or increased by 10cm
1034  respectively. The inundation severity varies in proportion to the
1035  boundary waveheight, but the model results are only slightly
1036  sensitive to this parameter for the range of values tested.}
1037\label{fig:sensitivity_boundary}
1038\end{center}
1039\end{figure}
1040
1041
1042\begin{figure}[ht]
1043\begin{center}
1044\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
1045\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_minus10_speed}
1046\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_plus10_speed}
1047\caption{The maximal flow speeds for the same model parameterisations found in  Figure \protect \ref{fig:sensitivity_boundary}.}
1048\label{fig:sensitivity_boundary_speed}
1049\end{center}
1050\end{figure}
1051
1052\begin{figure}[ht]
1053\begin{center}
1054\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
1055\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_nobuildings}
1056\caption{This figure shows the effect of having buildings as part of
1057  the elevation data set.  The first image is the reference inundation
1058  extent as reported in Section \protect \ref{sec:results} where
1059  buildings were included. The second shows the inundation results for
1060  a model entirely without buildings.  As expected, the absence of
1061  buildings will increase the inundation extent beyond what was
1062  surveyed.}
1063\label{fig:sensitivity_nobuildings}
1064\end{center}
1065\end{figure}
1066
1067
1068\begin{figure}[ht]
1069\begin{center}
1070\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
1071\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_nobuildings_speed}
1072\caption{The maximal flow speeds for the same model parameterisations
1073  found in Figure \protect \ref{fig:sensitivity_nobuildings}.}
1074\label{fig:sensitivity_nobuildings_speed}
1075\end{center}
1076\end{figure}
1077
1078\begin{figure}[ht]
1079\begin{center}
1080\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
1081\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f0003}
1082\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f03}
1083\caption{Model results for different values of Manning's friction
1084  coefficient. The first image is the reference inundation extent as
1085  reported in Section \protect \ref{sec:results} where the friction
1086  value $0.01$ was used across the entire domain while the second and
1087  third show the inundation results for friction values of 0.0003 and
1088  0.03 respectively. The inundation extent increases for the lower
1089  friction value while the higher slows the flow and decreases the
1090  inundation extent. Ideally, friction should vary across the entire
1091  domain depending on terrain and vegetation, but this is beyond the
1092  scope of this study.}
1093\label{fig:sensitivity_friction}
1094\end{center}
1095\end{figure}
1096
1097\begin{figure}[ht]
1098\begin{center}
1099\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
1100\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f0003_speed}
1101\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f03_speed}
1102\caption{The maximal flow speeds for the same model parameterisations found in Figure \protect \ref{fig:sensitivity_friction}.}
1103\label{fig:sensitivity_friction_speed}
1104\end{center}
1105\end{figure}
1106
1107\clearpage
1108
1109%====================Bibliography==================
1110\bibliographystyle{spmpsci}
1111\bibliography{tsunami07}
1112\end{document}
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