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30 | %\title{Application of SMF surface elevation function in inundation modelling} |
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31 | \date{} |
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32 | |
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33 | \begin{document} |
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34 | |
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35 | %\maketitle |
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36 | |
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37 | May 2006 |
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38 | |
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39 | Dr Phil Watts |
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40 | |
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41 | Applied Fluids Engineering |
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42 | |
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43 | Long Beach California |
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44 | |
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45 | USA |
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46 | |
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47 | phil.watts@appliedfluids.com |
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48 | |
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49 | Dear Phil, |
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50 | \parindent 15pt |
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51 | |
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52 | {\bf Ref: Application of sediment mass failure surface elevation function |
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53 | in inundation modelling} |
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54 | |
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55 | We work at Geoscience Australia (GA) in the Risk Research Group |
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56 | researching risks posed by a range of natural hazards |
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57 | (http://www.ga.gov.au/urban/projects/risk/index.jsp). |
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58 | Due to recent |
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59 | events and Australia's apparent vulnerabiliy to tsunami hazards, |
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60 | we are investigating the tsunami risk to Australia. To understand |
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61 | impact ashore, we have developed in conjunction |
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62 | with the Australian National University, a hydrodynamic model called |
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63 | ANUGA which uses the finite volume technique, [1]. |
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64 | |
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65 | A recent tsunami inundation study called for the tsunami source to |
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66 | be a slump and as such, we implemented the surface elevation |
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67 | function as described in Watts et al 2005, [2]. We found this a very useful |
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68 | way to incorporate another tsunami-genic event to our understanding |
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69 | of tsunami risk. In trying |
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70 | to implement this function however, we had some questions; |
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71 | |
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72 | \begin{itemize} |
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73 | \item |
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74 | Is there a physical explanation to why the total volume |
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75 | of the surface elevation function should not be zero? |
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76 | \item |
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77 | Should $\eta_{\rm min}$ used in the surface elevation function |
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78 | be | ${\eta_{\rm min}}$ | instead? |
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79 | \item |
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80 | Is the substitution of $x_g$ into the elevation |
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81 | function realistic? |
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82 | \end{itemize} |
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83 | |
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84 | Investigating the long term behaviour of the |
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85 | system, we found that water was being lost from the system when |
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86 | the slump was added to the system. Further investigation showed that |
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87 | the depressed volume was greater than the volume displaced above the |
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88 | water surface with approximately 2-3 \% loss. You can see from |
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89 | Figure 2 of [2] that the |
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90 | surface elevation function $\eta(x,y)$ indicates that |
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91 | the total volume is not conserved. |
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92 | |
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93 | However, we can alleviate this issue by finding the appropriate set of |
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94 | parameters which |
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95 | will conserve volume. Setting the integral of the elevation function to zero |
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96 | and solving for $\kappa'$ yields the result, |
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97 | $$\kappa' = [ |
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98 | {\rm erf} ( \frac{x - x_0 } {\sqrt \lambda_0 } ) / |
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99 | {\rm erf} ( \frac{x - \Delta x - x_0}{\sqrt \lambda_0 }) |
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100 | ]_{x_{\rm min}}^{x_{\rm max}} \ .$$ |
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101 | |
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102 | \noindent The relationship between $\kappa'$ and $\Delta x$ is shown in |
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103 | Figure \ref{fig:vol_cons}. It must be noted, that whilst |
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104 | $\kappa'$ is technically less than 1 for $\Delta x < 5.93$ it is |
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105 | effectively equal to 1 for those values. From this calculation, it would |
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106 | seem then that there is no appropriate $\Delta x$ for $\kappa'$ = 0.83 |
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107 | (a parameter used in [2]) satisfying conservation of volume. |
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108 | |
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109 | We've reproduced Figure 2 in [2] |
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110 | for appropriate values of $\kappa'$ and $\Delta x$ to |
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111 | ensure volume conservation within the system. Using the above |
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112 | formulation, the values of interest shown in Figure 2 in [2] would |
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113 | be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in |
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114 | Figure \ref{fig:eta_vary}. Note, this has not been scaled by $\eta_{\rm min}$. |
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115 | |
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116 | |
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117 | \begin{figure} |
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118 | |
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119 | \centerline{ \includegraphics[width=75mm, height=50mm]{volume_conservation.png}} |
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120 | |
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121 | \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.} |
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122 | \label{fig:vol_cons} |
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123 | \end{figure} |
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124 | |
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125 | \begin{figure}[hbt] |
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126 | |
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127 | \centerline{ \includegraphics[width=75mm, height=50mm]{redo_figure.png}} |
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128 | |
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129 | \caption{Surface elevation functions for |
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130 | ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.} |
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131 | \label{fig:eta_vary} |
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132 | \end{figure} |
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133 | |
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134 | For our particular test case, changing the surface elevation function |
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135 | in this way increases the inundation depth ashore by a factor greater than |
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136 | the initial water loss of 2-3 \%. |
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137 | |
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138 | Turning to our question regarding the scaling of the surface elevation |
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139 | function formulation, we see that $\eta_{\rm min}$ is always negative |
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140 | and hence |
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141 | $- \eta_{O,3D} / \eta_{\rm min}$ would be always positive. This |
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142 | would change the form of $\eta(x,y)$ and place the depressed volume behind |
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143 | the submarine mass failure. Should then $\eta_{\rm min}$ be replaced |
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144 | by |$\eta_{\rm min}$|? |
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145 | |
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146 | Our final question is whether it is appropriate to substitute |
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147 | the formulation for $x_g$ into the surface elevation function using |
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148 | $x_0 - \Delta x \approx x_g$. |
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149 | ($x_g$ is formulated |
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150 | as $x_g = d/\tan \theta + T/ \sin \theta$ which is described as a gauge |
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151 | located above the submarine mass failure |
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152 | initial submergence location in [3].) In this |
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153 | way, $\kappa'$ as described above would not |
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154 | be dependent on $\Delta x$, nor the subsequent surface elevation function. |
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155 | |
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156 | |
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157 | We are continuing to seek out validation data sets to improve the |
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158 | accuracy of our model. We recently had success in validating |
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159 | the model against the Benchmark Problem $\#$2 Tsunami Run-up |
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160 | onto a complex 3-dimensional beach, as provided to the 3rd |
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161 | International Workshop on Long Wave Run-up in 2004, see [1]. |
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162 | We note in [4] your proposal for others to employ the benchmark |
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163 | cases described there for experimental or numerical work. |
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164 | Your model has been compared with the laboratory experiments in 2003 [5] and |
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165 | again in 2005 [3] with fairly good agreement. Given |
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166 | the numerical model you implemented was the boundary element method, we would |
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167 | be very interested in comparing our finite volume model using the |
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168 | approximated surface elevation function with your |
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169 | experimental results. Would it therefore be possible for you to provide the |
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170 | experimental time series for comparison with ANUGA? |
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171 | |
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172 | \parindent 0pt |
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173 | |
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174 | Thanks for your time and we look forward to your response. |
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175 | |
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176 | Yours sincerely, |
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177 | |
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178 | Jane Sexton, Ole Nielsen, Adrian Hitchman and Trevor Dhu. |
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179 | |
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180 | Risk Research Group, Geoscience Australia. |
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181 | |
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182 | \newpage |
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183 | {\bf References} |
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184 | |
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185 | [1] Nielsen, O., S. Robers, D. Gray, A. McPherson, and A. Hitchman (2005) |
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186 | Hydrodynamic modelling of coastal inundation, MODSIM 2005 International |
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187 | Congress on Modelling and Simulation. Modelling and Simulation Society |
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188 | of Australian and New Zealand, 518-523, \newline URL: |
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189 | http://www.mssanz.org.au/modsim05/papers/nielsen.pdf |
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190 | |
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191 | [2] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J. (2005), |
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192 | Tsunami generation by submarine mass failure Part II: Predictive |
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193 | equations and case studies, Journal of Waterway, Port, Coastal, and |
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194 | Ocean Engineering, 131, 298 - 310. |
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195 | |
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196 | [3] Grilli, S.T. and Watts, P. (2005), Tsunami generation by |
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197 | submarine mass failure Part I: Modeling, experimental validation, |
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198 | and sensitivity analyses, Journal of Waterway, Port, Coastal, and |
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199 | Ocean Engineering, 131, 283 - 297. |
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200 | |
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201 | [4] Watts, P., Imamura, F. and Grilli, S. (2000) |
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202 | Comparing Model Simulations of Three Benchmark Tsunami Generation, |
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203 | Science of Tsunami Hazards, 18, 2, 107-123. |
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204 | |
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205 | [5] Enet, F., Grilli, S.T. and Watts, P. (2003), Laboratory Experiments for |
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206 | Tsunamis Generated by Underwater Landslides: |
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207 | Comparison with Numerical Modeling, |
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208 | Proceedings of the Thirteenth (2003) International Offshore and |
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209 | Polar Engineering Conference. The International Society of Offshore and |
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210 | Polar Engineers. |
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211 | \end{document} |
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