1 | \section{Modelling the Event}\label{sec:models} |
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2 | Numerous models are currently used to model and predict tsunami |
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3 | generation, propagation and run-up. These range in solving different |
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4 | equations and employing different methodologies with some examples |
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5 | being~\cite{titov97a,satake95,zhang08}. Here we introduce the |
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6 | modelling methodology employed by Geoscience Australia to illustrate |
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7 | the utility of the proposed benchmark. The methodology used by |
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8 | Geoscience Australia has three distinct components. Firstly an |
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9 | appropriate model is used to approximate the initial sea surface |
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10 | deformation. This model is chosen according to the cause of the intial |
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11 | disturbance. The resulting wave is propagated using the \textsc{ursga} |
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12 | model (see Section~\ref{sec:ursga}) in the deep ocean until the wave |
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13 | reaches shallow water, typically the $100$ m depth contour. The ocean |
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14 | surface profile along this contour is used as a time varying boundary |
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15 | condition for the \textsc{anuga} model (see Section~\ref{sec:anuga}) |
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16 | which simulates the propagation of the tsunami within the shallow |
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17 | water and the subsequent inundation of the land. This three part |
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18 | methodology roughly follows the three stages of tsunami evolution. The |
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19 | components used to model each stage of evolution are described in more |
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20 | detail below. |
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21 | |
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22 | \subsection{Generation}\label{sec:modelGeneration} |
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23 | There are various approaches to modelling the expected crustal |
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24 | deformation from an earthquake. Most approaches model the |
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25 | earthquake as a dislocation in a linear elastic medium. Here we use |
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26 | the method of Wang et al~\cite{wang03}. |
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27 | %One of the main advantages |
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28 | %of their method is that it allows the dislocation to be located in a |
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29 | %stratified linear elastic half-space with an arbitrary number of |
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30 | %layers. Other methods (such as those based on Okada's equations) can |
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31 | %only model the dislocation in a homogeneous elastic half space, or can |
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32 | %only include a limited number of layers, and thus cannot model the |
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33 | %effect of the depth dependence of the elasticity of the |
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34 | %Earth~\cite{wang03}. The original versions of the codes described here |
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35 | %are available from \url{http://www.iamg.org/CGEditor/index.htm}. The |
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36 | %first program, \textsc{edgrn}, calculates elastic Green's function for |
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37 | %a set of point sources at a regular set of depths out to a specified |
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38 | %distance. The equations controlling the deformation are solved by |
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39 | %using a combination of Hankel's transform and Wang et al's |
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40 | %implementation of the Thomson-Haskell propagator |
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41 | %algorithm~\cite{wang03}. Once the Green's functions are calculated |
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42 | %a slightly modified version of \textsc{edcmp}\footnote{For this study, |
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43 | %we have made minor modifications |
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44 | %to \textsc{edcmp} in order for it to provide output in a file format |
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45 | %compatible with the propagation code in the following section. Otherwise it |
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46 | %is similar to the original code.} is used to calculate the sea |
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47 | %floor deformation for a specific subfault. This second code |
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48 | %discretises the subfault into a set of unit sources and sums the |
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49 | %elastic Green's functions calculated from \textsc{edgrn} for all the |
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50 | %unit sources on the fault plane in order to calculate the final static |
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51 | %deformation caused by a two dimensional dislocation along the |
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52 | %subfault. This step is possible because of the linearity of the |
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53 | %governing equations. |
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54 | %In order to calculate the crustal deformation using these codes |
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55 | %a model that describes the variation in elastic |
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56 | %properties with depth and a slip model of the earthquake to describe |
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57 | %the dislocation is required. |
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58 | In order to calculate the crustal deformation a model that describes the |
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59 | variation in elastic properties with depth and a slip model of the |
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60 | earthquake to describe the dislocation is required. |
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61 | The elastic parameters used for this study are the |
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62 | same as those in Table 2 of Burbidge et al~\cite{burbidge08}. For the slip |
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63 | model, there are many possible models for the 2004 Andaman--Sumatran |
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64 | earthquake to select from |
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65 | ~\cite{chlieh07,asavanant08,arcas06,grilli07,ioualalen07}. Some are |
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66 | determined from various geological surveys of the site. Others solve |
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67 | an inverse problem which calibrates the source based upon the tsunami |
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68 | wave signal, the seismic signal and/or even the run-up. |
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69 | The source |
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70 | parameters used here to simulate the 2004 Indian Ocean tsunami were |
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71 | taken from the slip model G-M9.15 of Chlieh |
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72 | et al~\cite{chlieh07}. This model was created by inversion of wide |
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73 | range of geodetic and seismic data. The slip model consists |
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74 | of 686~20~km~x~20~km subsegments each with a different slip, strike and dip |
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75 | angle. The dip subfaults go from $17.5^\circ$ in the north and $12^\circ$ in |
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76 | the south. Refer to Chlieh et al~\cite{chlieh07} for a detailed |
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77 | discussion of this model and its derivation. %Note that the geodetic |
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78 | %data used in the validation was also included by~\cite{chlieh07} in |
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79 | %the inversion used to find G-M9.15. Thus the validation is not |
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80 | %completely independent. However, a reasonable validation would still |
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81 | %show that the crustal deformation and elastic properties model used |
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82 | %here is at least as valid as the one used by Chlieh |
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83 | %et al~\cite{chlieh07} and can reproduce the observations just as |
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84 | %accurately. |
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85 | |
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86 | \subsection{Deep water propagation}\label{sec:modelPropagation} |
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87 | The \textsc{ursga} model described below was used to simulate the |
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88 | propagation of the 2004 Indian Ocean tsunami across the open ocean, based on a |
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89 | discrete representation of the initial deformation of the sea floor, as |
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90 | described in Section~\ref{sec:modelGeneration}. For the models shown |
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91 | here, the uplift is assumed to be instantaneous and creates an initial |
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92 | displacement of the ocean surface of the same size and amplitude as the |
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93 | co-seismic sea floor deformation. \textsc{ursga} is well suited to |
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94 | modelling propagation over large domains and is used to propagate the tsunami |
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95 | until it reaches shallow water, typically the $100$ m depth contour. |
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96 | %The propagation of the tsunami in shallow water ($<100$m) and inundation are modelled using a hydrodynamic package called \textsc{ursga}. This package is ideally suited to shallow water propagation and inundation as it accurately simulates flow over dry land and is based upon an irregular triangular grid which can be refined in areas of interest. |
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97 | |
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98 | \subsubsection{URSGA}\label{sec:ursga} |
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99 | \textsc{ursga} is a hydrodynamic code that models the propagation of |
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100 | the tsunami in deep water using a finite difference method on a staggered grid. |
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101 | It solves the depth integrated linear or nonlinear shallow water equations in |
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102 | spherical co-ordinates with friction and Coriolis terms. The code is |
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103 | based on Satake~\cite{satake95} with significant modifications made by |
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104 | the \textsc{urs} corporation, Thio et al~\cite{thio08} and Geoscience |
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105 | Australia, Burbidge et al~\cite{burbidge08}. |
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106 | The tsunami was propagated via the nested |
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107 | grid system described in Section \ref{sec:propagation data} where |
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108 | the coarse grids were used in the open ocean and the finest |
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109 | resolution grid was employed in the region closest to Patong bay. |
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110 | \textsc{Ursga} is not publicly available. |
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111 | |
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112 | \subsection{Shallow water propagation and inundation} |
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113 | \label{sec:modelInundation} |
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114 | The utility of the \textsc{ursga} model decreases with water depth |
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115 | unless an intricate sequence of nested grids is employed. In |
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116 | comparison \textsc{anuga}, described below, is designed to produce |
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117 | robust and accurate predictions of on-shore inundation, but is less |
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118 | suitable for earthquake source modelling and large study areas because |
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119 | it is based on projected spatial coordinates. Consequently, the |
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120 | Geoscience Australia tsunami modelling methodology is based on a |
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121 | hybrid approach using models like \textsc{ursga} for tsunami |
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122 | propagation up to an offshore depth contour, typically $100$ m. |
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123 | %Specifically we use the \textsc{ursga} model to simulate the |
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124 | %propagation of the 2004 Indian Ocean tsunami in the deep ocean, based |
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125 | %on a discrete representation of the initial deformation of the sea |
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126 | %floor, described in Section~\ref{sec:modelGeneration}. |
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127 | The wave signal and the velocity field is then used as a |
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128 | time varying boundary condition for |
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129 | the \textsc{anuga} inundation simulation. |
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130 | % A description of \textsc{anuga} is the following section. |
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131 | |
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132 | \subsubsection{ANUGA}\label{sec:anuga} |
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133 | \textsc{Anuga} is a Free and Open Source hydrodynamic inundation tool that |
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134 | solves the conserved form of the depth-integrated nonlinear shallow |
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135 | water wave equations using a Finite-Volume scheme on an |
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136 | unstructured triangular mesh. |
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137 | The scheme, first |
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138 | presented by Zoppou and Roberts~\cite{zoppou99}, is a high-resolution |
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139 | Godunov-type method that uses the rotational invariance property of |
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140 | the shallow water equations to transform the two-dimensional problem |
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141 | into local one-dimensional problems. These local Riemann problems are |
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142 | then solved using the semi-discrete central-upwind scheme of Kurganov |
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143 | et al~\cite{kurganov01} for solving one-dimensional conservation |
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144 | equations. The numerical scheme is presented in detail in |
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145 | Roberts and Zoppou~\cite{zoppou00,roberts00} and |
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146 | Nielsen et al~\cite{nielsen05}. An important capability of the |
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147 | finite-volume scheme is that discontinuities in all conserved quantities |
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148 | are allowed at every edge in the mesh. This means that the tool is |
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149 | well suited to adequately resolving hydraulic jumps, transcritical flows and |
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150 | the process of wetting and drying. Consequently, \textsc{anuga} |
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151 | is suitable for |
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152 | simulating water flow onto a beach or dry land and around structures |
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153 | such as buildings. \textsc{anuga} has been validated against |
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154 | %a number of analytical solutions !!!Analytical solutions have not been published. Ask Steve. |
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155 | the wave tank simulation of the 1993 Okushiri |
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156 | Island tsunami~\cite{nielsen05,roberts06} and |
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157 | dam break experiments~\cite{baldock07}. |
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158 | More information on \textsc{anuga} and how to obtain it are available from \url{https://datamining.anu.edu.au/anuga}. |
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159 | |
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