%Anuga validation publication % %Geoscience Australia and others 2007-2008 % Use the Elsevier LaTeX document class %\documentclass{elsart3p} % Two column %\documentclass{elsart1p} % One column %\documentclass[draft]{elsart} % Basic \documentclass{elsart} % Basic % Useful packages \usepackage{graphicx} % avoid epsfig or earlier such packages \usepackage{url} % for URLs and DOIs \usepackage{amsmath} % many want amsmath extensions \usepackage{amsfonts} \usepackage{underscore} \usepackage{natbib} % Suggested by the Elsevier style % Use \citep and \citet instead of \cite % Local LaTeX commands %\newcommand{\Python}{\textsc{Python}} %\newcommand{\VPython}{\textsc{VPython}} \newcommand{\pypar}{\textsc{mpi}} \newcommand{\Metis}{\textsc{Metis}} \newcommand{\mpi}{\textsc{mpi}} \newcommand{\UU}{\mathbf{U}} \newcommand{\VV}{\mathbf{V}} \newcommand{\EE}{\mathbf{E}} \newcommand{\GG}{\mathbf{G}} \newcommand{\FF}{\mathbf{F}} \newcommand{\HH}{\mathbf{H}} \newcommand{\SSS}{\mathbf{S}} \newcommand{\nn}{\mathbf{n}} \newcommand{\code}[1]{\texttt{#1}} \begin{document} \begin{frontmatter} \title{On The Validation of A Hydrodynamic Model} \author[GA]{D.~S.~Gray} \ead{Duncan.Gray@ga.gov.au} \author[GA]{O.~M.~Nielsen} \ead{Ole.Nielsen@ga.gov.au} \author[GA]{M.~J.~Sexton} \ead{Jane.Sexton@ga.gov.au} \author[GA]{L.~Fountain} \author[GA]{K.~VanPutten} \author[ANU]{S.~G.~Roberts} \ead{Stephen.Roberts@anu.edu.au} \author[UQ]{T.~Baldock} \ead{Tom.Baldock@uq.edu.au} \author[UQ]{M.~Barnes} \ead{Matthew.Barnes@uq.edu.au} \address[GA]{Georisk Project, Geospatial and Earh Monitoring Division, Geoscience Australia, Canberra, Australia} \address[ANU]{Department of Mathematics, Australian National University, Canberra, Australia} \address[UQ]{University of Queensland, Brisbane, Australia} % Use the \verb|abstract| environment. \begin{abstract} Modelling the effects on the built environment of natural hazards such as riverine flooding, storm surges and tsunami is critical for understanding their economic and social impact on our urban communities. Geoscience Australia and the Australian National University have developed a hydrodynamic inundation modelling tool called ANUGA to help simulate the impact of these hazards. The core of ANUGA is a Python implementation of a finite-volume method for solving the conservative form of the Shallow Water Wave equation. In this paper, a number of tests are performed to validate ANUGA. These tests range from benchmark problems to wave and flume tank examples. ANUGA is available as Open Source to enable free access to the software and allow the scientific community to use, validate and contribute to the software in the future. %This method allows the study area to be represented by an unstructured %mesh with variable resolution to suit the particular problem. The %conserved quantities are water level (stage) and horizontal momentum. %An important capability of ANUGA is that it can robustly model the %process of wetting and drying as water enters and leaves an area. This %means that it is suitable for simulating water flow onto a beach or %dry land and around structures such as buildings. \end{abstract} \begin{keyword} % keywords here, in the form: keyword \sep keyword % PACS codes here, in the form: \PACS code \sep code Hydrodynamic Modelling \sep Model validation \sep Finite-volumes \sep Shallow water wave equation \end{keyword} \date{\today()} \end{frontmatter} % Begin document in earnest \section{Introduction} \label{sec:intro} Hydrodynamic modelling allows impacts from flooding, storm-surge and tsunami to be better understood, their impacts to be anticipated and, with appropriate planning, their effects to be mitigated. A significant proportion of the Australian population reside in the coastal corridors, thus the potential of significant disruption and loss is real. The extent of inundation is critically linked to the event, tidal conditions, bathymetry and topography and it not feasible to make impact predictions using heuristics alone. Geoscience Australia in collaboration with the Mathematical Sciences Institute, Australian National University, is developing a software application called ANUGA to model the hydrodynamics of floods, storm surges and tsunami. These hazards are modelled using the conservative shallow water equations which are described in section~\ref{sec:model}. In ANUGA these equations are solved using a finite volume method as described in section~\ref{sec:model}. A more complete discussion of the method can be found in \citet{Nielsen2005} where the model and solution technique is validated on a standard tsunami benchmark data set or in \citet{Roberts2007} where the numerical method and parallelisation of ANUGA is discussed. This modelling capability is part of Geoscience Australia's ongoing research effort to model and understand the potential impact from natural hazards in order to reduce their impact on Australian communities \citep{Nielsen2006}. ANUGA is currently being trialled for flood modelling \citep{Rigby2008}. The validity of other hydrodynamic models have been reported elsewhere, with \citet{Hubbard02} providing an excellent review of 1D and 2D models and associated validation tests. They described the evolution of these models from fixed, nested to adaptive grids and the ability of the solvers to cope with the moving shoreline. They highlighted the difficulty in verifying the nonlinear shallow water equations themselves as the only standard analytical solution is that of \citet{Carrier58} that is strictly for non-breaking waves. Further, whilst there is a 2D analytic solution from \citet{Thacker81}, it appears that the circular island wave tank example of Briggs et al will become the standard data set to verify the equations. This paper will describe the validation outputs in a similar way to \citet{Hubbard02} to present an exhaustive validation of the numerical model. Further to these tests, we will incorporate a test to verify friction values. The tests reported in this paper are: \begin{itemize} \item Verification against the 1D analytical solution of Carrier and Greenspan (p~\pageref{sec:carrier}) \item Testing against 1D (flume) data sets to verify wave height and velocity (p~\pageref{sec:stage and velocity}) \item Determining friction values from 1D flume data sets (p~\pageref{sec:friction}) \item Validation against a genuinely 2D analytical solution of the model equations (p~\ref{sec:XXX}) \item Testing against the 2D Okushiri benchmark problem (p~\pageref{sec:okushiri}) \item Testing against the 2D data sets modelling wave run-up around a circular island by Briggs et al. (p~\pageref{sec:circular island}) \end{itemize} Throughout the paper, qualitative comparisons will be drawn against other models. Moreover, all source code necessary to reproduce the results reported in this paper is available as part of the ANUGA distribution in the form of a test suite. It is thus possible for anyone to readily verify that the implementation meets the requirements set out by these benchmarks. %Hubbard and Dodd's model, OTT-2D, has some similarities to ANUGA, and %whilst the mesh can be refined, it is based on rectangular mesh. %The ANUGA model and numerical scheme is briefly described in %section~\ref{sec:model}. A more detailed description of the numerical %scheme and software implementation can be found in \citet{Nielsen2005} and %\citet{Roberts2007}. The six case studies to validation and verify ANUGA will be presented in section~\ref{sec:validation}, with the conclusions outlined in section~\ref{sec:conclusions}. NOTE: This is just a brain dump at the moment and needs to be incorporated properly in the text somewhere. Need some discussion on Bousssinesq type models - Boussinesq equations get the nonlinearity and dispersive effects to a high degree of accuracy moving wet-dry boundary algorithms - applicability to coastal engineering Fuhrman and Madesn 2008 \cite{Fuhrman2008}do validation - they have a Boussinesq type model, finite difference (therefore needing a supercomputer), 4th order, four stage RK time stepping scheme. their tests are (1) nonlinear run-up on periodic and transient waves on a sloping beach with excellent comparison to analytic solutions (2) 2d parabolic basin (3) solitary wave evolution through 2d triangular channel (4) solitary wave evolution on conical island (we need to compare to their computation time and note they use a vertical exaggeration for their images) excellent accuracy mentioned - but what is it - what does it mean? of interest is that they mention mass conservation and calculate it throughout the simulations Kim et al \cite{DaiHong2007} use Riemann solver - talk about improved accuracy by using 2nd order upwind scheme. Use finite volume on a structured mesh. Do parabolic basic and circular island. Needed? Delis et all 2008 \cite{Delis2008}- finite volume, Godunov-type explicit scheme coupled with Roe's approximate Riemann solver. It accurately describes breaking waves as bores or hydraulic jumps and conserves volume across flow discontinuties - is this just a result of finite volume? They also show mass conservation for most of the simulations similar range of validation tests that compare well - our job to compare to these as well \section{Mathematical model, numerical scheme and implementation} \label{sec:model} The ANUGA model is based on the shallow water wave equations which are widely regarded as suitable for modelling 2D flows subject to the assumptions that horizontal scales (e.g. wave lengths) greatly exceed the depth, vertical velocities are negligible and the fluid is treated as inviscid and incompressible. See e.g. the classical texts \citet{Stoker57} and \citet{Peregrine67} for the background or \citet{Roberts1999} for more details on the mathematical model used by ANUGA. The conservation form of the shallow water wave equations used in ANUGA are: \[ \frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial x}+\frac{\partial \GG}{\partial y}=\SSS \] where $\UU=\left[ {{\begin{array}{*{20}c} h & {uh} & {vh} \\ \end{array} }} \right]^T$ is the vector of conserved quantities; water depth $h$, $x$-momentum $uh$ and $y$-momentum $vh$. Other quantities entering the system are bed elevation $z$ and stage (absolute water level above a reference datum such as Mean Sea Level) $w$, where the relation $w = z + h$ holds true at all times. The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given by \[ \EE=\left[ {{\begin{array}{*{20}c} {uh} \hfill \\ {u^2h+gh^2/2} \hfill \\ {uvh} \hfill \\ \end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c} {vh} \hfill \\ {vuh} \hfill \\ {v^2h+gh^2/2} \hfill \\ \end{array} }} \right] \] and the source term (which includes gravity and friction) is given by \[ \SSS=\left[ {{\begin{array}{*{20}c} 0 \hfill \\ -{gh(z_{x} + S_{fx} )} \hfill \\ -{gh(z_{y} + S_{fy} )} \hfill \\ \end{array} }} \right] \] where $S_f$ is the bed friction. The friction term is modelled using Manning's resistance law \[ S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy} =\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}} \] in which $\eta$ is the Manning resistance coefficient. %%As demonstrated in our papers, \cite{modsim2005,Roberts1999} these %%equations provide an excellent model of flows associated with %%inundation such as dam breaks and tsunamis. Question - how do we %%know it is excellent? ANUGA uses a finite-volume method as described in \citet{Roberts2007} where the study area is represented by an unstructured triangular mesh in which the vector of conserved quantities $\UU$ is maintained and updated over time. The flexibility afforded by allowing unstructed meshes rather than fixed resolution grids is the ability for the user to refine the mesh in areas of interest while leaving other areas coarse and thereby conserving computational resources. The approach used in ANUGA are distinguished from many other implementations (e.g. \citet{Hubbard02} or \citet{Zhang07}) by the following features: \begin{itemize} \item The fluxes across each edge are computed using the semi-discrete central-upwind scheme for approximating the Riemann problem proposed by \citet{KurNP2001}. This scheme deals with different flow regimes such as shocks, rarefactions and sub to super critical flow transitions using one general approach. We have found this scheme to be pleasingly simple, robust and efficient. \item ANUGA does not employ a shoreline detection algorithm as the central-upwind scheme is capable of resolving fluxes arising between wet and dry cells. ANUGA does optionally bypass unnecessary computations for dry-dry cell boundaries purely to improve performance. \item ANUGA employs a second order spatial reconstruction of triangles to produce a piece-wise linear function construction of the conserved quantities. This function is allowed to be discontinuous across the edges of the cells, but the slope of this function is limited to avoid artificially introduced oscillations. This approach provides good approximation of steep gradients in the solution. However, where the depths are very small compared to the bed-slope a linear combination between second order and first order reconstructions is employed to guarantee numerical stability that may arise form very small depths. \end{itemize} In the computations presented in this paper we use an explicit Euler time stepping method with variable timestepping subject to the CFL condition: \[ \delta t = \min_k \frac{r_k}{v_k} \] where $r_k$ refers to the radius of the inscribed circle of triangle $k$, $v_k$ refers to the maximal velocity calculated from fluxes passing in or out of triangle $k$ and $\delta t$ is the resulting 'safe' timestep to be used for the next iteration. ANUGA utilises a general velocity limiter described in the manual which guarantees a gradual compression of computed velocities in the presence of very shallow depths: \begin{equation} \hat{u} = \frac{\mu}{h + h_0/h}, \bigskip \hat{v} = \frac{\nu}{h + h_0/h}, \end{equation} where $h_0$ is a regularisation parameter that controls the minimal magnitude of the denominator. The default value is $h_0 = 10^{-6}$. ANUGA is mostly written in the object-oriented programming language Python with computationally intensive parts implemented as highly optimised shared objects written in C. Python is known for its clarity, elegance, efficiency and reliability. Complex software can be built in Python without undue distractions arising from idiosyncrasies of the underlying software language syntax. In addition, Python's automatic memory management, dynamic typing, object model and vast number of libraries means that ANUGA scripts can be produced quickly and can be adapted fairly easily to changing requirements. \section{Validation} \label{sec:validation} Validation is an ongoing process and the purpose of this paper is to describe a range of tests that validate ANUGA as a hydrodynamic model. This section will describe the six tests outlined in section~\ref{sec:intro}. Run times where specified measure the model time only and exclude model setup, data conversions etc. All examples were timed on a a 2GHz 64-bit Dual-Core AMD Opteron(tm) series 2212 Linux server. %This is a tornado compute node (cat /proc/cpuinfo). \subsection{1D analytical validation} Tom Baldock has done something here for that NSW report \subsection{Stage and Velocity Validation in a Flume} \label{sec:stage and velocity} This section will describe tilting flume tank experiments that were conducted at the Gordon McKay Hydraulics Laboratory at the University of Queensland that confirm ANUGA's ability to estimate wave height and velocity. The same flume tank simulations were also used to explore Manning's friction and this will be described in the next section. The flume was set up for dam-break experiments, having a water reservior at one end. The flume was glass-sided, 3m long, 0.4m in wide, and 0.4m deep, with a PVC bottom. The reservoir in the flume was 0.75m long. For this experiment the reservoir water was 0.2m deep. At time zero the reservoir gate is manually opened and the water flows into the other side of the flume. The water ran up a flume slope of 0.03 m/m. To accurately model the bed surface a Manning's friction value of 0.01, representing PVC was used. % Neale, L.C. and R.E. Price. Flow characteristics of PVC sewer pipe. % Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE. % pp. 109-129. 1964. Acoustic displacement sensors that produced a voltage that changed with the water depth was positioned 0.4m from the reservoir gate. The water velocity was measured with an Acoustic Doppler Velocimeter 0.45m from the reservoir gate. This sensor only produced reliable results 4 seconds after the reservoir gate opened, due to limitations of the sensor. % Validation UQ flume % at X:\anuga_validation\uq_sloped_flume_2008 % run run_dam.py to create sww file and .csv files % run plot.py to create graphs heere automatically % The Coasts and Ports '2007 paper is in TRIM d2007-17186 \begin{figure}[htbp] \centerline{\includegraphics[width=4in]{uq-flume-depth}} \caption{Comparison of wave tank and ANUGA water height at .4 m from the gate}\label{fig:uq-flume-depth} \end{figure} \begin{figure}[htbp] \centerline{\includegraphics[width=4in]{uq-flume-velocity}} \caption{Comparison of wave tank and ANUGA water velocity at .45 m from the gate}\label{fig:uq-flume-velocity} \end{figure} Figure~\ref{fig:uq-flume-depth} shows that ANUGA predicts the actual water depth very well, although there is an initial drop in water depth within the first second that is not simulated by ANUGA. Water depth and velocity are coupled as described by the nonlinear shallow water equations, thus if one of these quantities accurately estimates the measured values, we would expect the same for the other quantity. This is demonstrated in Figure~\ref{fig:uq-flume-velocity} where the water velocity is also predicted accurately. Sediment transport studies rely on water velocity estimates in the region where the sensors cannot provide this data. With water velocity being accurately predicted, studies such as sediment transport can now use reliable estimates. \subsection{Okushiri Wavetank Validation} \label{sec:okushiri} As part of the Third International Workshop on Long-wave Runup Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four benchmark problems were specified to allow the comparison of numerical, analytical and physical models with laboratory and field data. One of these problems describes a wave tank simulation of the 1993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A significant feature of this tsunami was a maximum run-up of 32~m observed at the head of the Monai Valley. This run-up was not uniform along the coast and is thought to have resulted from a particular topographic effect. Among other features, simulations of the Hokkaido tsunami should capture this run-up phenomenon. This dataset has been used by to validate tsunami models by a number of tsunami scientists. Examples include Titov ... lit review here on who has used this example for verification (Leharne?) \begin{figure}[htbp] %\centerline{\includegraphics[width=4in]{okushiri-gauge-5.eps}} \centerline{\includegraphics[width=4in]{ch5.png}} \centerline{\includegraphics[width=4in]{ch7.png}} \centerline{\includegraphics[width=4in]{ch9.png}} \caption{Comparison of wave tank and ANUGA water stages at gauge 5,7 and 9.}\label{fig:val} \end{figure} \begin{figure}[htbp] \centerline{\includegraphics[width=4in]{okushiri-model.jpg}} \caption{Complex reflection patterns and run-up into Monai Valley simulated by ANUGA and visualised using our netcdf OSG viewer.}\label{fig:run} \end{figure} The wave tank simulation of the Hokkaido tsunami was used as the first scenario for validating ANUGA. The dataset provided bathymetry and topography along with initial water depth and the wave specifications. The dataset also contained water depth time series from three wave gauges situated offshore from the simulated inundation area. The ANUGA model comprised $41404$ triangles and took about $1330$ s to run on the test platform described in Section~\ref{sec:validation}. The script to run this example is available in the ANUGA distribution in the subdirectory \code{anuga_validation/automated_validation_tests/okushiri_tank_validation}. Figure~\ref{fig:val} compares the observed wave tank and modelled ANUGA water depth (stage height) at one of the gauges. The plots show good agreement between the two time series, with ANUGA closely modelling the initial draw down, the wave shoulder and the subsequent reflections. The discrepancy between modelled and simulated data in the first 10 seconds is due to the initial condition in the physical tank not being uniformly zero. Similarly good comparisons are evident with data from the other two gauges. Additionally, ANUGA replicates exceptionally well the 32~m Monai Valley run-up, and demonstrates its occurrence to be due to the interaction of the tsunami wave with two juxtaposed valleys above the coastline. The run-up is depicted in Figure~\ref{fig:run}. This successful replication of the tsunami wave tank simulation on a complex 3D beach is a positive first step in validating the ANUGA modelling capability. \subsection{Runup of solitary wave on circular island wavetank validation} \label{sec:circular island} This section will describe the ANUGA results for the experiments conducted by Briggs et al (1995). Here, a 30x25m basin with a conical island is situated near the centre and a directional wavemaker is used to produce planar solitary waves of specified crest lenghts and heights. A series of gauges were distributed within the experimental setup. As described by Hubbard and Dodd \cite{Hubbard02}, a number of researchers have used this benchmark problem to test their numerical models. {\bf Jane: check whether these results are now avilable as they were not in 2002}. Hubbard and Dodd \cite{Hubbard02} note that a particular 3D model appears to obtain slightly better results than the 2D ones reported but that 3D models are unlikely to be competitive in terms of computing power for applications in coastal engineering at least. Choi et al \cite{Choi07} use a 3D RANS model (based on the Navier-Stokes equations) for the same problem and find a very good comparison with laboratory and 2D numerical results. An obvious advantage of the 3D model is its ability to investigate the velocity field and Choi et al also report on the limitation of depth-averaged 2D models for run-up simulations of this type. Once results are availble, need to compare to Hubbard and Dodd and draw any conclusions from nested rectangular grid vs unstructured gird. Figure \ref{fig:circular screenshots} shows a sequence of screenshots depicting the evolution of the solitary wave as it hits the circular island. \begin{figure}[htbp] \centerline{ \includegraphics[width=5cm]{circular1.png} \includegraphics[width=5cm]{circular2.png}} \centerline{ \includegraphics[width=5cm]{circular3.png} \includegraphics[width=5cm]{circular4.png}} \centerline{ \includegraphics[width=5cm]{circular5.png} \includegraphics[width=5cm]{circular6.png}} \centerline{ \includegraphics[width=5cm]{circular7.png} \includegraphics[width=5cm]{circular8.png}} \centerline{ \includegraphics[width=5cm]{circular9.png} \includegraphics[width=5cm]{circular10.png}} \caption{Screenshots of the evolution of solitary wave around circular island.} \label{fig:circular screenshots} \end{figure} \clearpage \subsection{Flume tank validation before and after breaking waves} % The Hinwood report is in TRIM: D2008-97610 and in georisk_model/inundation/data/flumes/Hinwood2008 % Photo material is photos_movies under that directory To explicitly determine if ANUGA can model waves after breaking several experiments were conducted at the Monash University Institute for Sustainable Water Resources using a wave flume. The experiments were designed to produce a variety of breaking waves. The experiments were conducted on a 2.5$^\circ$ and a 1.5$^\circ$ plane beach slope set-up in a glass-sided wave flume of 40m in length, 1.0m wide and 1.6m deep. The wave generator can generate waves up to 0.6m in height, with a period range of 0.3 - 7.0 seconds. Four scenarios with different combinations of wave height and wave period were used, with each test being repeated. A variety of measurements were taken during each test. Mid-depth water velocity and wave height were measured on the approach section. The water height at several points along the flume were measured using pressure transducers. The wave profile was video recorded, this determined the location of breaking waves. All the tests produced 4 to 7 waves. Generally the first wave did not break, with subsequent waves breaking; accept for scenario 2, for which the first 3 waves did not break. Scenario 1 produced plunging breakers. Scenario 3 produced collapsing breakers. All other scenarios produced spilling breakers. Details of the tests performed are given in Table \ref{tab:hinwoodSummary}. \begin{table} \caption{Details of the Monash University experiments.} % Can't get right \begin{center} \begin{tabular}{ c p{3cm} p{3cm} p{3cm} } \hline Test Name & Beach slope nominal, \emph{degrees} & Water depth offshore, \emph{mm } & Wave frequency nominal, \emph{Hz} \\ \hline S1R1 & 3.5 & 400 & 0.200 \\ \hline S1R2 & 3.5 & 400 & 0.200 \\ \hline S2R1 & 3.5 & 400 & 0.125 \\ \hline S2R2 & 3.5 & 400 & 0.125 \\ \hline S3R1 & 1.5 & 336 & 0.200 \\ \hline S3R2 & 1.5 & 336 & 0.200 \\ \hline S4R1 & 1.5 & 336 & 0.125 \\ \hline S4R2 & 1.5 & 336 & 0.125 \\ \hline % Mapping of new names to old names % S1R2 T1R3 % S1R1 T1R5 % S2R1 T2R7 % S2R2 T2R8 % S3R2 T3R28 % S3R1 T3R29 % S4R2 T4R31 % S4R1 T4R32 \end{tabular} \label{tab:hinwoodSummary} \end{center} \end{table} All of these tests were simulated using ANUGA. The Mid-depth water velocity and wave height measured on the approach section were used as boundary conditions for the ANUGA simulations. The origin of the z coordinate was the still water line, positive upwards. The origin of the x coordinate was the toe of the beach, x measured positive shorewards A Manning's friction coefficient of zero was used. To quantify the difference between the simulated stage and the experimental stage the Root Mean Square Deviation (RMSD) (\cite{Kobayshi2000}) was used \[ RMSD =\sqrt {\frac{1 }{n} \displaystyle\sum_{i=1}^{n}{(x_i - y_i)}^2} \] Figures \ref{fig:S1-rmsd} to \ref{fig:S4-rmsd} show the RMSD of each sensor for all tests and the location where each wave broke. The RMSD is calculated over the time of the experiment. % To create these figures goto \anuga_work\development\Hinwood_2008 % do python validation_graphs.py \begin{figure}[htbp] \centerline{\includegraphics[width=4in]{S1-rmsd}} \caption{RMSD of stage between the wave tank and ANUGA for S1R1 and S1R2. Horizontal lines represent the x location of breaking waves.} \label{fig:S1-rmsd} \end{figure} \begin{figure}[htbp] \centerline{\includegraphics[width=4in]{S2-rmsd}} \caption{RMSD of stage between the wave tank and ANUGA for S2R1 and S2R2. Horizontal lines represent the x location of breaking waves.} \label{fig:S2-rmsd} \end{figure} \begin{figure}[htbp] \centerline{\includegraphics[width=4in]{S3-rmsd}} \caption{RMSD of stage between the wave tank and ANUGA for S3R1 and S3R2. Horizontal lines represent the x location of breaking waves. The circles represent gauges shown in \ref{fig:S3-stage-compares}} % More, circles represent gauges shown in %\protect{\ref{fig:S3-stage-compares}} Again, circles represent gauges %shown in \ref{fig:S3-stage-compares}} \label{fig:S3-rmsd} \end{figure} \begin{figure}[htbp] \centerline{\includegraphics[width=4in]{S4-rmsd}} \caption{RMSD of stage between the wave tank and ANUGA for S4R1 and S4R2. Horizontal lines represent the x location of breaking waves.} \label{fig:S4-rmsd} \end{figure} For a more direct comparision between the simulation and the experiment the water stages at three gauges, generally the initial, final and worst fit, were compared in Figures \ref{fig:S1-stage-compare} to \ref{fig:S4-stage-compare}. \begin{figure}[htbp] \centerline{\includegraphics[width=5in]{S1-stage-compare}} \caption{Comparison of wave tank (solid line) and ANUGA (broken line) water stages at three gauges for S1R1.} \label{fig:S1-stage-compare} \end{figure} \begin{figure}[htbp] \centerline{\includegraphics[width=5in]{S2-stage-compare}} \caption{Comparison of wave tank (solid line) and ANUGA (broken line) water stages at three gauges for S2R1.} \label{fig:S2-stage-compare} \end{figure} \begin{figure}[htbp] \centerline{\includegraphics[width=5in]{S3-stage-compare}} \caption{Comparison of wave tank (solid line) and ANUGA (broken line) water stages at three gauges for S3R1.} \label{fig:S3-stage-compare} \end{figure} \begin{figure}[htbp] \centerline{\includegraphics[width=5in]{S4-stage-compare}} \caption{Comparison of wave tank (solid line) and ANUGA (broken line) water stages at three gauges for S4R1.} \label{fig:S4-stage-compare} \end{figure} Overall these results show an excellent level of agreement between predicted and measured stage. The RMSD figures generally show a decrease in accuracy, the further the gauge is from the initial condition, untill wave breaking. Generally after wave breaking the RMSD value decreases. This is a clear indication of ANUGA accurately predicting the stage after the wave has broken. There are several points worth emphasising here. Overall all of the RMSD values are good. There is not much difference between the worst and best gauges (-0.7 m and 5.6m) for S1R1, for example. A decrease in RMSD does not necesarily mean the accuracy of ANUGA is improving. For example, in S4R1 the drop in RMSD between gauges 7.6 and 11.6 is partially due to vertical water motion effecting gauge 7.6 (vertical water motion creates an artificial pressure spike which is not representative of the physical wave (Michael Hughes)) and a decrease in the time period where waves are being measured, as opposed to still water, for gauge 11.6 (Comment: This means that due the late arrival of the wave most of the comparison will have very low RMSD error). Additionally, sensors near the wave run-up have a lower amplitude than the wave at breaking, which can result in a low RMSD, which may not be the case if the results were relative, see gauge 5.6 and 7.6 \ref{fig:S1-stage-compare}. \label{sec:Hinwood} \clearpage \section{Conclusions} \label{sec:conclusions} ANUGA is a flexible and robust modelling system that simulates hydrodynamics by solving the shallow water wave equation in a triangular mesh. It can model the process of wetting and drying as water enters and leaves an area and is capable of capturing hydraulic shocks due to the ability of the finite-volume method to accommodate discontinuities in the solution. ANUGA can take as input bathymetric and topographic datasets and simulate the behaviour of riverine flooding, storm surge, tsunami or even dam breaks. Initial validation using wave tank data supports ANUGA's ability to model complex scenarios. Further validation will be pursued as additional datasets become available. The ANUGA source code and validation case studies reported here are available at \url{http://sourceforge.net/projects/anuga}. something about use on flood modelling community and their validation initiatives %\bibliographystyle{plainnat} \bibliographystyle{elsart-harv} \bibliography{anuga-bibliography} \end{document}