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\title{Application of SMF surface elevation function in inundation modelling}
\date{}

\begin{document}

\maketitle


Geoscience Australia (GA) is a federal government agency playing a
critical role in enabling government and the community to make
information decisions about exploration of resources, the management
of the environment, the safety of critical infrastructure and the
resultant wellbeing of all Australians. GA does this by producing
first-class geoscientific information and knowledge.

The Risk Research Group (RRG) within GA is researching natural and
human-caused hazards to enhance Australia's risk mitigation
capabilities through policy and decision-maker support. The group is
working with other agencies to develop and collect information on
natural disasters, and develop risk models for forecasting the
impact of future hazard events.

In a recent inundation study, we implemented the surface elevation
function as described in equation 14 of Watts et al 2005, [1], for a
slump tsunami scenario. Investigating the long term behaviour of the
system, it was found that water was being lost from the system when
the slump was added to the system. Further investigation showed that
the depressed volume was greater than the volume displaced above the
water surface with approximately 2-3 \% loss. Figure 2 of [1] shows
a series of the surface elevation functions for various parameters
which indicate that volume is not conserved.

{\bf Question:}   Is there a physical explanation to why the volume
of the surface elevation function should not be zero?

Integrating equation 14 and solving to zero for $\kappa'$ ensures
the system volume is conserved. As a result,

$$\kappa' = [{\rm erf} ( \frac{(x - x_0)}{\sqrt \lambda_0 }) /
{\rm erf} ( \frac{(x - \Delta x - x_0)}{\sqrt \lambda_0 })]_{x_{\rm
min}}^{x_{\rm max}} \ .$$

\noindent The relationship between $\kappa$ and $\Delta_x$ can be
seen in Figure \ref{fig:vol_cons} where $\kappa$ approaches $\inf$
quickly.Additionally, it is not possible for $\kappa' = 0.83$ as
shown in Figure 2 of [1] as {\rm erf(x)} = 1 for ${\rm abs} x >
5.93$. For the example described in Figures 2 and 3 of [1], whilst
$\kappa'$ is technically less than 1 for $\Delta x < 5$ it is
effectively equal to 1 for $0 \le \Delta x \approx 5$.


Figure 2 in [1]
could then be reproduced for appropriate values of $\kappa'$ and $\Delta x$ to
ensure conservation of mass within the system. Using the above
formulation, the values of interest shown in Figure 2 of [1] would
be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in
Figure \ref{fig:eta_vary}.



\begin{figure}[hbt]

  %\centerline{ \includegraphics[width=75mm, height=75mm]{volume_conservation.eps}}

  \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.}
  \label{fig:vol_cons}
\end{figure}

\begin{figure}[hbt]

  %\centerline{ \includegraphics[width=75mm, height=75mm]{redo_figure.eps}}

  \caption{Surface elevation functions for
($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.}
  \label{fig:eta_vary}
\end{figure}

The impact onshore is altered if the surface elevation function is altered
as described. This is of course expected as there is an increased volume
of water which can propagate ashore. In one investigation, we saw little 
change to the inundation extent, but some significant increases in
maximum inundation depth in some locations.

Watts et al [1] also provide additional information on the value of
$\Delta x$; $x_0 - \Delta x \approx x_g$, where $x_g$ is formulated
as $x_g = d/\tan \theta + T/ \sin \theta$ (described as a gauge
located above the SMF initial submergence location in [2]). Here $d$
represents the depth at where the SMF is situated, $T$ the thickness
and $\theta$ the slope of the bed. As a result, $\kappa'$ can be
recast as

$$\kappa'  \approx {\rm erf} ( \frac{(x - x_0)}{\sqrt\lambda_0} ) / 
{\rm erf} ( \frac{(x - 2 x_0
- x_g)}{\sqrt \lambda_0 )}$$

\noindent thereby eliminating $\Delta x$ from the surface elevation
function, $\eta(x,y)$. Implementing this formulation for values in
[2] (T = 0.052m, d = 0.259m) provides the following figure
describing the relationship between $x_0$ and $\kappa'$.

%{\caption Utilising $x_g$ in determining $\kappa'$ to ensure volume
%conservation}

{\bf Question:}   Is this a realistic substitution?

{\bf TO DO:} Need a discussion in here on "characteristic distance
of motion".

\section{References}

[1] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J., 2005,
Tsunami generation by submarine mass failure Part II: Predictive
equations and case studies, Journal of Waterway, Port, Coastal, and
Ocean Engineering, 131, 298 - 310.

[2] Grilli, S.T. and Watts, P., 2005, Tsunami generation by
submarine mass failure Part I: Modeling, experimental validation,
and sensitivity analyses, Journal of Waterway, Port, Coastal, and
Ocean Engineering, 131, 283 - 297.


\end{document}