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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 (FIXME: Richar is chasing
402this up) which include footage of the tsunami in Patong Bay on the day
403of the Indian Ocean Tsunami. Both videos show an already inundated
404group of buildings, they then show what is to be assumed as the second
405and third waves approaching and further flooding the buildings and
406street.  The first video is in the very north filmed from what is
407believed to be the roof of the Novotel Hotel marked 'North' in Figure
408\ref{fig:gauge_locations}. The second video is in the very south
409filmed from the second story of a building next door to the Comfort
410Resort near the corner of Ruam Chai St and Thaweewong Road.  This
411location is marked 'south' in Figure \ref{fig:gauge_locations} and
412Figure~\ref{fig:video_flow} shows stills from this video. Both videos
413were used to estimate flow speeds and inundation dephts over time.
414
415\begin{figure}[ht]
416\begin{center}
417\includegraphics[width=6.0cm,keepaspectratio=true]{flow_rate_south_0_00sec.jpg}
418\includegraphics[width=6.0cm,keepaspectratio=true]{flow_rate_south_5_04sec.jpg}
419\includegraphics[width=6.0cm,keepaspectratio=true]{flow_rate_south_7_12sec.jpg}
420\includegraphics[width=6.0cm,keepaspectratio=true]{flow_rate_south_7_60sec.jpg}
421\caption{Four frames from a video where flow rate could be estimated,
422  circle indicates tracked debris, from top left: 0.0 sec, 5.0 s, 7.1
423  s, 7.6 s.}
424\label{fig:video_flow}
425\end{center}
426\end{figure}
427
428Flow rates were estimated using landmarks found in both videos and
429were found to be in the range of 5 to 7 metres per second (+/- 2 m/s)
430in the north and 0.5 to 2 metres per second (+/- 1 m/s) in the south.
431Water depths could also
432be estimated from the videos by the level at which water rose up the
433sides of buildings such as shops. Our estimates are in the order of
4341.5 to 2.0 metres (+/- 0.5 m).
435Fritz ~\cite{fritz06} performed a detailed
436analysis of video frames taken around Banda Aceh and arrived at flow
437speeds in the range of 2 to 5 m/s.
438
439\begin{figure}[ht]
440\begin{center}
441\includegraphics[width=8.0cm,keepaspectratio=true]{patongescapemap.jpg}
442\caption{Tsunami survey mapping the maximum observed inundation at
443  Patong beach courtesy of the Thai Department of Mineral Resources
444  \protect \cite{szczucinski06}.}
445\label{fig:patongescapemap}
446\end{center}
447\end{figure}
448
449\subsection{Validation Check-List}
450\label{sec:checkList}
451The data described in this section can be used to construct a
452benchmark to validate all three stages of the evolution of a
453tsunami. In particular we propose that a legitimate tsunami model
454should reproduce the following behaviour:
455\begin{itemize}
456 \item Reproduce the vertical deformation observed in north-western
457   Sumatra and along the Nicobar--Andaman islands, see
458   Section~\ref{sec:gen_data}.
459 \item Reproduce the \textsc{jason} satellite altimetry sea surface
460   anomalies, see Section~\ref{sec:data_jason}.
461 \item Reproduce the inundation survey map in Patong bay
462   (Figure~\ref{fig:patongescapemap}).
463 \item Simulate a leading depression followed by two distinct crests
464   of decreasing magnitude.
465 \item Predict the water depths and flow speeds, at the locations of
466   the eye-witness videos, that fall within the bounds obtained from
467   the videos.
468\end{itemize}
469
470Ideally, the model should also be compared to measured timeseries of
471waveheights and velocities but the authors are not aware of the
472availability of such data.
473
474
475%================Section===========================
476\section{Modelling the Event}\label{sec:models}
477Numerous models are currently used to model and predict tsunami
478generation, propagation and run-up~\cite{titov97a,satake95}. Here we
479introduce the modelling methodology employed by Geoscience Australia
480to illustrate the utility of the proposed benchmark. Geoscience
481Australia's tsunami modelling methodology comprises the three parts;
482generation, propagation and inundation
483(Sections~\ref{sec:modelGeneration},\ref{sec:modelPropagation} and
484\ref{sec:modelInundation} respectively).
485
486\subsection{Generation}\label{sec:modelGeneration}
487
488There are various approaches to modelling the expected crustal
489deformation from an earthquake at depth. Most approaches model the
490earthquake as a dislocation in a linear, elastic medium. Here we use
491the method of Wang et. al.~\cite{wang03}. One of the main advantages
492of their method is that it allows the dislocation to be located in a
493stratified linear elastic half-space with an arbitrary number of
494layers. Other methods (such as those based on Okada's equations) can
495only model the dislocation in a homogeneous elastic half space, or can
496only include a limited number of layers, and thus cannot model the
497effect of the depth dependence of the elasticity of the
498Earth~\cite{wang03}. The original versions of the codes described here
499are available from \url{http://www.iamg.org/CGEditor/index.htm}. The
500first program, \textsc{edgrn}, calculates elastic Green's function for
501a set of point sources at a regular set of depths out to a specified
502distance. The equations controlling the deformation are solved by
503using a combination of Hankel's transform and Wang et al's
504implementation of the Thomson-Haskell propagator
505algorithm~\cite{wang03}. Once the Green's functions are calculated we
506use a slightly modified version of \textsc{edcmp} to calculate the sea
507floor deformation for a specific subfault. This second code
508discretises the subfault into a set of unit sources and sums the
509elastic Green's functions calculated from \textsc{edgrn} for all the
510unit sources on the fault plane in order to calculate the final static
511deformation caused by a two dimensional dislocation along the
512subfault. This step is possible because of the linearity of the
513governing equations. For this study, we have made minor modifications
514to \textsc{edcmp} in order for it to output in a file format
515compatible with the propagation code in the following section but it
516is otherwise the similar to the original code.
517
518In order to calculate the crustal deformation using these codes we
519thus need to have a model describing the variation in elastic
520properties with depth and a slip model of the earthquake to describe
521the dislocation. The elastic parameters used for this study are the
522same as those in Table 2 of Burbidge~\cite{burbidge08}. For the slip
523model, there are many possible models for the 2004 Andaman--Sumatran
524earthquake to choose from
525~\cite{chlieh07,asavanant08,arcas06,grilli07,ioualalen07}. Some are
526determined from various geological surveys of the site, others solve
527an inverse problem which calibrates the source based upon the tsunami
528wave signal, the seismic signal and/or the runup. The source
529parameters used here to simulate the 2004 Indian Ocean tsunami were
530taken from the slip model G-M9.15 from Chlieh
531et. al.~\cite{chlieh07}. This model was created by inversion of wide
532range of geodetic and seismic data. The slip model consists of 686
53320km x 20km subsegments each with a different slip, strike and dip
534angle. The dip subfaults go from $17.5^0$ in the north and $12^0$ in
535the south. Refer to Chlieh et. al.~\cite{chlieh07} for a detailed
536discussion of this model and its derivation. Note that the geodetic
537data used in the validation was also included by~\cite{chlieh07} in
538the inversion used to find G-M9.15, thus the validation is not
539completely independent. However, a successful validation would still
540show that the crustal deformation and elastic properties model used
541here is at least as valid as the one used by Chlieh
542et. al.~\cite{chlieh07} and can reproduce the observations just as
543accurately.
544
545\subsection{Propagation}\label{sec:modelPropagation}
546We use the \textsc{ursga} model described below to simulate the
547propagation of the 2004 tsunami in the deep ocean ocean, based on a
548discrete representation of the initial deformation of the sea floor,
549described in Section~\ref{sec:modelGeneration}. For the models shown
550here, we assume that the uplift is instantaneous and creates a wave of
551the same size and amplitude as the co-seismic sea floor deformation.
552
553\subsubsection{URSGA}
554\textsc{ursga} is a hydrodynamic code that models the propagation of
555the tsunami in deep water using a finite difference method on a staggered grid
556to solve
557the depth integrated linear or nonlinear shallow water equations in
558spherical co-ordinates with friction and Coriolis terms. The code is
559based on Satake~\cite{satake95} with significant modifications made by
560the \textsc{urs} corporation~\cite{thio08} and Geoscience
561Australia~\cite{burbidge08}. The tsunami is propagated via the nested
562grid system. Coarse grids are used in the open ocean and the finest
563resolution grid is employed in the region of most
564interest. \textsc{Ursga} is not publicly available.
565
566\subsection{Inundation}\label{sec:modelInundation}
567The utility of the \textsc{ursga} model decreases with water depth
568unless an intricate sequence of nested grids is employed. In
569comparison \textsc{anuga}, described below, is designed to produce
570robust and accurate predictions of on-shore inundation, but is less
571suitable for earthquake source modelling and large study areas because
572it is based on projected spatial coordinates. Consequently, the
573Geoscience Australia tsunami modelling methodology is based on a
574hybrid approach using models like \textsc{ursga} for tsunami
575propagation up to a 100 m depth contour.
576%Specifically we use the \textsc{ursga} model to simulate the
577%propagation of the 2004 Indian Ocean tsunami in the deep ocean, based
578%on a discrete representation of the initial deformation of the sea
579%floor, described in Section~\ref{sec:modelGeneration}.
580The wave signal is then used as a time varying boundary condition for
581the \textsc{anuga} inundation simulation.
582% A description of \textsc{anuga} is the following section.
583
584\subsubsection{ANUGA}
585\textsc{Anuga} is an Open Source hydrodynamic inundation tool that
586solves the conserved form of the depth integrated nonlinear shallow
587water wave equations. The scheme used by \textsc{anuga}, first
588presented by Zoppou and Roberts~\cite{zoppou99}, is a high-resolution
589Godunov-type method that uses the rotational invariance property of
590the shallow water equations to transform the two-dimensional problem
591into local one-dimensional problems. These local Riemann problems are
592then solved using the semi-discrete central-upwind scheme of Kurganov
593et al.~\cite{kurganov01} for solving one-dimensional conservation
594equations. The numerical scheme is presented in detail in Zoppou and
595Roberts~\cite{zoppou99}, Roberts and Zoppou~\cite{roberts00}, and
596Nielsen et al.~\cite{nielsen05}. An important capability of the
597software is that it can model the process of wetting and drying as
598water enters and leaves an area. This means that it is suitable for
599simulating water flow onto a beach or dry land and around structures
600such as buildings. It is also capable of adequately resolving
601hydraulic jumps due to the ability of the finite-volume method to
602handle discontinuities. The numerical scheme can also handle
603transitions between sub-critical and super-critical flow regimes
604seamlessly. \textsc{Anuga} has been validated against a number of
605analytical solutions and the wave tank simulation of the 1993 Okushiri
606Island tsunami~\cite{nielsen05,roberts06}.
607
608%================Section===========================
609\section{Results}\label{sec:results}
610This 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.
611
612\subsection{Generation}\label{modelGeneration}
613The location and magnitude of the sea floor displacement associated
614with the 2004 Sumatra--Andaman tsunami calculated from the G-M9.15
615model of~\cite{chlieh07} is shown in
616Figure~\ref{fig:surface_deformation}. The magnitude of the sea floor
617displacement ranges from about $-3.0$ to $5.0$ metres. The region near
618the fault is predicted to uplift, while that further away from the
619fault subsides. Also shown in Figure~\ref{fig:surface_deformation} are
620the areas that were observed to uplift (arrows pointing up) or subside
621(arrows point down) during and immediately after the earthquake. Most
622of this data comes uplifted or subsided coral heads. The length of
623vector increases with the magnitude of the displacement, the length
624corresponding to 1m of observed motion is shown in the top right
625corner of the figure. As can be seen, the source model detailed in
626Section~\ref{sec:modelGeneration} produces a crustal deformation that
627matches the vertical displacements in the Nicobar-Andaman islands and
628Sumatra very well. Uplifted regions are close to the fault and
629subsided regions are further away. The crosses on
630Figure~\ref{fig:surface_deformation} show estimates of the pivot line
631from the remote sensing data~\cite{chlieh07} and they follow the
632predicted pivot line quite accurately. The average difference between
633the observed motion and the predicted motion (including the pivot line
634points) is only 0.06 m, well below the typical error of the
635observations of between 0.25 and 1.0 m. However, the occasional point
636has quite a large error (over 1 m), for example a couple
637uplifted/subsided points appear to be on a wrong side of the predicted
638pivot line~\ref{fig:surface_deformation}. The excellence of the fit is
639not surprising, since the original slip model was chosen
640by~\cite{chlieh07} to fit this (and the seismic data) well. However,
641this does demonstrate that \textsc{edgrn} and our modified version of
642\textsc{edstat} can reproduce the correct pattern of vertical
643deformation very well when the slip distribution is well constrained
644and when reasonable values for the elastic properties are used.
645
646\begin{figure}[ht]
647\begin{center}
648\includegraphics[width=5cm,keepaspectratio=true]{surface_deformation.jpg}
649%\includegraphics[totalheight=0.3\textheight,width=0.8\textwidth]{surface_deformation.jpg}
650\caption{Location and magnitude of the vertical component of the sea
651  floor displacement associated with the 2004 Indian Ocean tsunami
652  based on the slip model, G-M9.15. The black arrows which point up
653  show areas observed to uplift during and immediately after the
654  earthquake, those point down are locations which subsided. The
655  length of increases with the magnitude of the deformation. The arrow
656  length corresponding to 1 m of deformation is shown in the top right
657  hand corner of the figure. The crosses marks show the location of
658  the pivot line (the region between the uplift and subsided region
659  where the uplift is zero) derived from remote sensing. All the
660  observational data come from the dataset collated
661  by~\cite{chlieh07}.}
662\label{fig:surface_deformation}
663\end{center}
664\end{figure}
665
666\subsection{Propagation}\label{sec:resultsPropagation}
667The deformation results described in Section~\ref{sec:modelGeneration}
668were used to provide a profile of the initial ocean surface
669displacement. This wave was used as an initial condition for
670\textsc{ursga} and was propagated throughout the Bay of Bengal. The
671rectangular computational domain of the largest grid extended from
67290$^0$ to 100$^0$ East and 0 to 15$^0$ North and contained
6731335$\times$1996 finite difference points. Inside this grid, a nested
674sequence of grids was used. The grid resolution of the nested grids
675went from 27 arc seconds in the coarsest grid, down to 9 arc seconds
676in the second grid, 3 arc seconds in the third grid and finally 1 arc
677second in the finest grid near Patong. The computational domain is
678shown in Figure~\ref{fig:computational_domain}.
679
680Figure \ref{fig:jasonComparison} provides a comparison of the
681\textsc{ursga} predicted sea surface elevation with the JASON
682satellite altimetry data. The \textsc{ursga} model replicates the
683amplitude and timing of the the wave observed at 2.5 degrees South,
684but underestimates the amplitude of the wave further to the south at
6854 degrees South. In the model, the southern most of these two waves
686appears only as a small bump in the cross section of the model shown
687in Figure~\ref{fig:jasonComparison} instead of being a distinct peak
688as can be seen in the satellite data. Also note
689that the \textsc{ursga} model prediction of the ocean surface
690elevation becomes out of phase with the JASON data at 3 to 7 degrees
691latitude. Chlieh et al~\cite{chlieh07} also observe these misfits and
692suggest it is caused by a reflected wave from the Aceh Peninsula that
693is not resolved in the model due to insufficient resolution of the
694computational mesh and bathymetry data. This is also a limitation of
695the model presented here, but probably could be improved by nesting
696grids near Aceh.
697
698\begin{figure}[ht]
699\begin{center}
700\includegraphics[width=12.0cm,keepaspectratio=true]{jasonComparison.jpg}
701\caption{Comparison of the \textsc{ursga} predicted surface elevation
702  with the JASON satellite altimetry data. The \textsc{ursga} wave
703  heights have been corrected for the time the satellite passed
704  overhead compared to JASON sea level anomaly.  }
705\label{fig:jasonComparison}
706\end{center}
707\end{figure}
708
709\subsection{Inundation}
710After propagating the tsunami in the open ocean using \textsc{ursga}
711the approximated ocean and surface elevation and horisontal flow
712velocities were extracted and used to construct a boundary condition
713for the \textsc{anuga} model. The interface betwen the \textsc{ursga}
714and \textsc{anuga} models was chosen to roughly follow the 100 m depth
715contour along the west coast of Phuket Island. The computational
716domain is shown in Figure \ref{fig:computational_domain}
717\begin{figure}[ht]
718\begin{center}
719%\includegraphics[width=5.0cm,keepaspectratio=true]{extent_of_ursga_model.jpg}
720%\includegraphics[width=5.0cm,keepaspectratio=true]{ursgaDomain.jpg}
721\includegraphics[width=5.0cm,keepaspectratio=true]{extent_of_ANUGA_model.jpg}
722\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)}
723\label{fig:computational_domain}
724\end{center}
725\end{figure}
726
727The domain was discretised into 386,338 triangles. The resolution of
728the grid was increased in certain regions to efficiently increase the
729accuracy of the simulation. The grid resolution ranged between a
730maximum triangle area of $1\times 10^5$ m$^2$ near the Western ocean
731boundary to $20$ m$^2$ in the small regions surrounding the inundation
732region in Patong Bay. Due to a lack of available data, friction was
733set to a constant throughout the computational domain. For the
734reference simulation a Manning's coefficient of 0.01 was chosen to
735represent a small resistance to the water flow. See Section
736\ref{sec:friction sensitivity} for details on model sensitivity to
737this parameter.
738
739
740The boundary condition at each side of the domain towards the south
741and the north where no data was available was chosen as a transmissive
742boundary condition effectively replicating the time dependent wave
743height present just inside the computational domain. Momentum was set
744to zero. Other choices include applying the mean tide value as a
745Dirichlet type boundary condition but experiments as well as the
746result of the verification reported here showed that this approach
747tends to under estimate the tsunami impact due to the tempering of the
748wave near the side boundaries whereas the transmissive boundary
749condition robustly preserves the wave.
750
751During the \textsc{anuga} simulation the tide was kept constant at
752$0.80$ m. This value was chosen to correspond to the tidal height
753specified by the Thai Navy tide charts
754(\url{http://www.navy.mi.th/hydro/}) at the time the tsunami arrived
755at Patong Bay. Although the tsunami propagated for approximately 3
756hours before it reach Patong Bay, the period of time during which the
757wave propagated through the \textsc{anuga} domain is much
758smaller. Consequently the assumption of constant tide height is
759reasonable
760
761%FIXME (Ole): Perhaps rephrase a bit as the 1cm vs 10cm is hard to
762%understand.  Remove figure using 1cm inundation
763Maximum onshore inundation elevation was computed from the model throughout the entire Patong Bay
764region. Figure~\ref{fig:inundationcomparison1cm} shows very good
765agreement between the measured and simulated inundation.  However these results are dependent on
766the classification used to determine whether a region in in the numerical simulation was inundated.
767In Figure~\ref{fig:inundationcomparison1cm} a point in the computational domain was deemed
768inundated if at some point in time it was covered by at least 1cm of water.
769However the precision of the field measurements is most likely different to the 1cm used
770to determine the simulated inundation. The inundation boundary generated by the on-site survey
771was determined by observing water marks and other signs left by the receding waters. Consequently
772the measurement error along the inundation boundary of the survey varies significantly. However it is
773impossible to quantify this error. Figure~\ref{fig:inundationcomparison1cm} shows the simulated
774inundation using a threshold of 10cm. An inundation threshold of 10cm was selected for
775the current and all future simulations to reflect the likely
776accuracy of the survey and subsequently facilitate a more appropriate
777comparison between the modelled and observed inundation area.
778
779An 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}.
780
781%\url{https://datamining.anu.edu.au/anuga/attachment/wiki/AnugaPublications/patong_2004_indian_ocean_tsunami_ANUGA_animation.mov}.
782
783\begin{figure}[ht]
784\begin{center}
785\includegraphics[width=5.0cm,keepaspectratio=true]{Depth_small_transmissive_d0.jpg}
786\includegraphics[width=5.0cm,keepaspectratio=true]{sensitivity_reference.jpg}
787\caption{Simulated inundation versus observed inundation using an inundation threshold of 1cm (left) and 10cm (right). FIXME: NEED Graph for 10cm}
788\label{fig:inundationcomparison1cm}
789\end{center}
790\end{figure}
791
792To quantify the agreement between observed and simulated inundation we
793introduce the measure
794\begin{equation}
795\rho_{in}=\frac{A(I_m\cap I_o)}{A(I_o)}
796\end{equation}
797representing the ratio $\rho_{in}$ of observed
798inundation region $I_o$ captured by the model $I_m$. Another useful
799measure is the fraction of the modelled inundation area that falls
800outside the observed inundation area given by the formula
801\begin{equation}
802\rho_{out}=\frac{A(I_m\setminus (I_m\cap I_o))}{A(I_o)}
803\end{equation}
804These values for the two aforementioned simulations are given in
805Table~\ref{table:inundationAreas} %FIXME (Ole): The left hand side of
806%these equations are not areas - consider another symbol.
807
808Discrepancies between the survey data and the modelled inundated
809include: unknown distribution of surface roughness, inappropriate
810parameterisation of the source model, effect of humans structures on
811flow, as well as uncertainties in the elevation data, effects of
812erosion and deposition by the tsunami event, measurement errors, and
813missing data in the field survey data itself. The impact of some of
814these sources of uncertainties are is investigated in
815Section~\ref{sec:sensitivity}
816
817\subsection{Eye-witness accounts}
818Figure \ref{fig:gauge_locations} shows four locations where time
819series have been extracted from the model. The two offshore timeseries
820are shown in Figure \ref{fig:offshore_timeseries} and the two onshore
821timeseries are shown in Figure \ref{fig:onshore_timeseries}. The
822latter coincide with locations where video footage from the event is
823available as described in Section \ref{sec:eyewitness data}.
824
825\begin{figure}[ht]
826\begin{center}
827\includegraphics[width=7.0cm,keepaspectratio=true]{tide_gauge_locations.jpg}
828\caption{Location of timeseries extracted from the model output}
829\label{fig:gauge_locations}
830\end{center}
831\end{figure}
832
833
834\begin{figure}[ht]
835\begin{center}
836\includegraphics[width=10.0cm,keepaspectratio=true]{gauge_bay_depth.jpg}
837\includegraphics[width=10.0cm,keepaspectratio=true]{gauge_bay_speed.jpg}
838\caption{Timeseries obtained from the two offshore locations shown in Figure \protect \ref{fig:gauge_locations}}
839\end{center}
840\label{fig:offshore_timeseries}
841\end{figure}
842
843\begin{figure}[ht]
844\begin{center}
845\includegraphics[width=10.0cm,keepaspectratio=true]{gauges_hotels_depths.jpg}
846\includegraphics[width=10.0cm,keepaspectratio=true]{gauges_hotels_speed.jpg}
847\caption{Timeseries obtained from the two onshore locations shown in Figure \protect \ref{fig:gauge_locations}}
848\end{center}
849\label{fig:onshore_timeseries}
850\end{figure}
851
852
853The estimated depths and flow rates given in Section \ref{sec:eyewitness data} are shown
854together with the modelled depths and flow rates obtained from the model in
855Table \ref{tab:depth and flow comparisons}.
856
857\begin{table}
858\[
859  \begin{array}{|l|cc|cc|}
860  \hline
861                 & \multicolumn{2}{|c|}{\mbox{Depth [m]}}
862                 & \multicolumn{2}{c|}{\mbox{Flow [m/s]}} \\ 
863                 & \mbox{Observed} & \mbox{Modelled}
864                 & \mbox{Observed} & \mbox{Modelled} \\ \cline{2-5}                 
865    \mbox{North} & 1.5-2 & 1.4 & 5-7 & 0.1 - 3.3 \\
866    \mbox{South} & 1.5-2 & 1.5 & 0.5-2 & 0.2 - 2.6 \\ \hline
867  \end{array}
868\]
869\label{tab:depth and flow comparisons}
870\end{table} 
871
872%can be estimated with landmarks found in
873%satellite imagery and the use of a GIS and were found to be in the
874%range of 5 to 7 metres per second (+/- 2 m/s) in the north and 0.5 to
875%2 metres per second (+/- 1 m/s) in the south.
876
877Given the uncertainties in both model and observations, there is agreement
878between the values obtained from the videos and the simulations.
879
880% Our modelled flow rates show
881%maximum values in the order of 0.2 to 2.6 m/s in the south and 0.1 to
882%3.3 m/s for the north as shown in the figures. Water depths could also
883%be estimated from the videos by the level at which water rose up the
884%sides of buildings such as shops. Our estimates are in the order of
885%1.5 to 2.0 metres (+/- 0.5 m). This is in the same range as our
886%modelled maximum depths of 1.4 m in the north and 1.5 m in the south
887%as seen in the figure.
888
889
890
891
892
893%================Section===========================
894\section{Sensitivity Analysis}
895\label{sec:sensitivity}
896This section investigates the effect of different values of Manning's
897friction coefficient, changing waveheight at the 100 m depth contour,
898and the presence and absence of buildings in the elevation dataset on
899model maximum inundation.
900
901%========================Friction==========================%
902\subsection{Friction}
903\label{sec:friction sensitivity}
904The first study investigated the impact of surface roughness on the
905predicted run-up. According to Schoettle~\cite{schoettle2007}
906appropriate values of Manning's coefficient range from 0.007 to 0.030
907for tsunami propagation over a sandy sea floor and the reference model
908uses a value of 0.01.  To investigate sensitivity to this parameter,
909we simulated the maximum onshore inundation using the a Manning's
910coefficient of 0.0003 and 0.03. The resulting inundation maps are
911shown in Figure~\ref{fig:sensitivity_friction} and the maximum flow
912speeds in Figure~\ref{fig:sensitivity_friction_speed}. These figures
913show that the on-shore inundation extent decreases with increasing
914friction and that small perturbations in the friction cause bounded
915changes in the output. This is consistent with the conclusions of
916Synolakis~\cite{synolakis05} who states that the long wavelength of
917tsunami tends to mean that the friction is less important in
918comparison to the motion of the wave.
919
920%========================Wave-Height==========================%
921\subsection{Input Wave Height}\label{sec:waveheightSA}
922The effect of the wave-height used as input to the inundation model
923\textsc{anuga} was also investigated.
924Figure~\ref{fig:sensitivity_boundary} indicates that the inundation
925severity is directly proportional to the boundary waveheight but small
926perturbations in the input wave-height of 10 cm appear to have little
927effect on the final on-shore run-up. Obviously larger perturbations
928will have greater impact. However, this value is generally well
929predicted by the generation and propagation models such as
930\textsc{ursga}. See e.g. \cite{FIXME} Toshi Baba's validation study at
931Kuril islands.
932
933
934
935%========================Buildings==========================%
936\subsection{Buildings and Other Structures}
937The presence of buildings has the greatest influence on the maximum
938on-shore inundation extent. Figure~\ref{fig:sensitivity_nobuildings}
939shows the maximum run-up in the presence and absence of buildings. It
940is apparent that the inundation is much more severe when the presence
941of man made structures and buildings are ignored. Maximal flow speeds
942for these two model parameterisations are shown in
943Figure~\ref{fig:sensitivity_nobuildings_speed}.
944
945\begin{table}
946\begin{center}
947\label{table:inundationAreas}
948\caption{$\rho_{in}$ and $\rho_{out}$ of the reference simulation and all sensitivity studies}
949\begin{tabular}{|c|c|c|}
950\hline
951 & $\rho_{in}$ & $\rho_{out}$ \\ 
952\hline\hline
953Reference & 0.76 & 0.22\\ 
954Min. Friction & Ã— & \\ 
955Max. Friction & Ã— & \\ 
956Min. Wave-Height× & Ã— & \\
957Max. Wave-Height× & Ã— & \\
958No Buildings × & Ã— & \\
959\hline 
960\end{tabular}
961\end{center}
962\end{table}
963
964%================Section===========================
965
966\section{Conclusion}
967This paper proposes an additional field data benchmark for the
968verification of tsunami inundation models. Currently, there is a
969scarcity of appropriate validation datasets due to a lack of well
970documented historical tsunami impacts. The benchmark proposed here
971utilises the uniquely large amount of observational data for model
972comparison obtained during, and immediately following, the
973Sumatra--Andaman tsunami of 26th December 2004. Unlike the small
974number of existing benchmarks, the proposed test validates all three
975stages of tsunami evolution - generation, propagation and
976inundation. In an attempt to provide higher visability and easier
977accessibility for tsunami benchmark problems the data used to
978construct the proposed benchmark is documented and freely available at
979\url{http://tinyurl.com/patong2004-data}.
980
981This study also shows that the tsunami impact modelling methodology
982adopted is sane and able to predict inundation extents with reasonable
983accuracy.  An associated aim of this paper was to further validate the
984hydrodynamic modelling tool \textsc{anuga} which is used to simulate
985the tsunami inundation and run rain-induced floods. Model predictions
986matched well geodetic measurements of the Sumatra--Andaman earthquake,
987altimetry data from the \textsc{jason}, eye-witness accounts of wave
988front arrival times and flow speeds and a detailed inundation survey
989of Patong Bay, Thailand.
990
991A simple sensitivity analysis was performed to assess the influence of
992small changes in friction, wave-height at the 100 m depth contour and
993the presence of buildings and other structures on the model
994predictions. The presence of buildings has the greatest influence on
995the simulated inundation extent. The value of friction and small
996perturbations in the waveheight at the ANUGA boundary have
997comparatively little effect on the model results.
998
999%================Acknowledgement===================
1000\section*{Acknowledgements}
1001This project was undertaken at Geoscience Australia and the Department
1002of Mathematics, The Australian National University. The authors would
1003like to thank Niran Chaimanee from the CCOP, Thailand for providing
1004the post 2004 tsunami survey data, building footprints, aereal
1005photography and the elevation data for Patong beach, Prapasri Asawakun
1006from the Suranaree University of Technology and Parida Kuneepong for
1007supporting this work; and Drew Whitehouse from the Australian National
1008University for preparing the animation.
1009
1010\section{Appendix}
1011\begin{figure}[ht]
1012\begin{center}
1013\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
1014\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_minus10}
1015\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_plus10}
1016\caption{Model results with wave height at ANUGA boundary artificially
1017  modified to asses sensitivities. The first image is the reference
1018  inundation extent as reported in Section \protect \ref{sec:results}
1019  while the second and third show the inundation results if the wave
1020  at the ANUGA boundary is reduced or increased by 10cm
1021  respectively. The inundation severity varies in proportion to the
1022  boundary waveheight, but the model results are only slightly
1023  sensitive to this parameter for the range of values tested.}
1024\label{fig:sensitivity_boundary}
1025\end{center}
1026\end{figure}
1027
1028
1029\begin{figure}[ht]
1030\begin{center}
1031\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
1032\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_minus10_speed}
1033\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_plus10_speed}
1034\caption{The maximal flow speeds for the same model parameterisations found in  Figure \protect \ref{fig:sensitivity_boundary}.}
1035\label{fig:sensitivity_boundary_speed}
1036\end{center}
1037\end{figure}
1038
1039\begin{figure}[ht]
1040\begin{center}
1041\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
1042\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_nobuildings}
1043\caption{This figure shows the effect of having buildings as part of
1044  the elevation data set.  The first image is the reference inundation
1045  extent as reported in Section \protect \ref{sec:results} where
1046  buildings were included. The second shows the inundation results for
1047  a model entirely without buildings.  As expected, the absence of
1048  buildings will increase the inundation extent beyond what was
1049  surveyed.}
1050\label{fig:sensitivity_nobuildings}
1051\end{center}
1052\end{figure}
1053
1054
1055\begin{figure}[ht]
1056\begin{center}
1057\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
1058\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_nobuildings_speed}
1059\caption{The maximal flow speeds for the same model parameterisations
1060  found in Figure \protect \ref{fig:sensitivity_nobuildings}.}
1061\label{fig:sensitivity_nobuildings_speed}
1062\end{center}
1063\end{figure}
1064
1065\begin{figure}[ht]
1066\begin{center}
1067\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference}
1068\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f0003}
1069\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f03}
1070\caption{Model results for different values of Manning's friction
1071  coefficient. The first image is the reference inundation extent as
1072  reported in Section \protect \ref{sec:results} where the friction
1073  value $0.01$ was used across the entire domain while the second and
1074  third show the inundation results for friction values of 0.0003 and
1075  0.03 respectively. The inundation extent increases for the lower
1076  friction value while the higher slows the flow and decreases the
1077  inundation extent. Ideally, friction should vary across the entire
1078  domain depending on terrain and vegetation, but this is beyond the
1079  scope of this study.}
1080\label{fig:sensitivity_friction}
1081\end{center}
1082\end{figure}
1083
1084\begin{figure}[ht]
1085\begin{center}
1086\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_reference_speed}
1087\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f0003_speed}
1088\includegraphics[width=3.5cm,keepaspectratio=true]{sensitivity_f03_speed}
1089\caption{The maximal flow speeds for the same model parameterisations found in Figure \protect \ref{fig:sensitivity_friction}.}
1090\label{fig:sensitivity_friction_speed}
1091\end{center}
1092\end{figure}
1093
1094\clearpage
1095
1096%====================Bibliography==================
1097\bibliographystyle{spmpsci}
1098\bibliography{tsunami07}
1099\end{document}
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