%================Section===========================
\section{Sensitivity Analysis}
\label{sec:sensitivity}
The numerical models used to simulate tsunami impact 
are computationally intensive and high resolution models of the entire
evolution process will often take a number of days to
compute. Consequently, the uncertainty in model predictions is difficult to
quantify as it would require a very large number of runs. 
However, model uncertainty should not be ignored. The aim of this section is 
not to provide a detailed investigation of sensitivity but to rather 
illustrate that changes in important parameters produce behaviour consistent with the known physics and that 
small changes in these parameters produce bounded variations in the output.

This section investigates the effect of different values of Manning's
friction coefficient, changing waveheight at the 100 m depth contour,
and the presence and absence of buildings in the elevation dataset on
model maximum inundation as computed by \textsc{anuga}. 

The reference model is the one reported in 
Figure~\ref{fig:inundationcomparison1cm} (right) with a friction coefficient of 0.01, buildings included and the boundary condition produced by the 
\textsc{ursga} model.

%========================Friction==========================%
\subsection{Friction}
\label{sec:friction sensitivity}
The first sensitivity study investigated the impact of surface roughness on the
predicted run-up. According to Schoettle~\cite{schoettle2007}
appropriate values of Manning's coefficient range from 0.007 to 0.03
for tsunami propagation over a sandy sea floor and the reference model
uses a value of 0.01. To investigate sensitivity to this parameter,
we simulated the maximum onshore inundation using a Manning's
coefficient of 0.0003 and 0.03. The resulting inundation maps are
shown in Figure~\ref{fig:sensitivity_friction} 
and the maximum flow speeds in Figure~\ref{fig:sensitivity_friction_speed}.
The figure, along with Table~\ref{table:inundationAreas},
shows, as expected, that the on-shore inundation extent decreases with increasing
friction and that small perturbations in the friction cause bounded
changes in the output. This is consistent with the conclusions of
Synolakis~\cite{synolakis05} et al, who state that the long wavelength of
tsunami tends to mean that friction is less important in
comparison to the motion of the wave.

%========================Wave-Height==========================%
\subsection{Input Wave Height}\label{sec:waveheightSA}
Wave heights in the open ocean are 
generally well predicted by the generation and propagation models such as
\textsc{ursga} as demonstrated in Section \ref{sec:resultsPropagation} 
and also in \cite{thomas2009} assuming that the source parameters chosen appropriately. 
Nevertheless, the effect of errors in 
the wave height used as input to the inundation model \textsc{anuga}
was investigated by perturbing the 
amplitude of the input wave by $\pm$10 cm. This value was chosen to be consistent 
with the expected error in the amplitude predicted by the propagation model 
and amounts to about $\pm$5\% of the maximal waveheight at the boundary.

Figure~\ref{fig:sensitivity_boundary}, Figure~\ref{fig:sensitivity_boundary_speed}
and Table~\ref{table:inundationAreas} 
indicate that the inundation severity is directly proportional to the 
boundary waveheight but small
perturbations in the input wave height of 10 cm appear to have little
effect on the final inundated area. Obviously larger perturbations
would have greater impact.




%========================Buildings==========================%
\subsection{Buildings and Other Structures}
The presence or absence of physical buildings in the elevation model was also 
investigated.
Figure~\ref{fig:sensitivity_buildings} shows the inundated area 
in the presence and absence of buildings.
From 
Table~\ref{table:inundationAreas} it is apparent that densely built-up 
areas act as dissipators greatly reducing the inundated area. 
Figure~\ref{fig:sensitivity_buildings_speed} show the associated flow speeds in the presence and absence of buildings. It is evident that flow speeds 
tend to increase in passages between buildings but slow down in areas behind them as compared to the bare earth scenario.
These results suggest that, when possible, the presence of 
human-made structures should be included into the model topography.
Furthermore, these results also 
indicate that simply matching point sites with much lower resolution meshes or, indeed, 
areas of artificially high friction
than used here is an over simplification. Such simulations cannot capture the
fine detail that so clearly affects inundation depth and flow speeds.



\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
 & $\rho_{in}$ & $\rho_{out}$ \\ 
\hline\hline
Reference model & 0.79 & 0.20\\ 
Friction = 0.0003 & 0.83 & 0.26 \\ 
Friction = 0.03 & 0.67 & 0.09\\ 
Boundary wave hight minus 10 cm & 0.77 & 0.17 \\
Boundary wave hight plus 10 cm & 0.82 & 0.22 \\
No Buildings & 0.94 & 0.44 \\
\hline 
\end{tabular}
\end{center}
\caption{$\rho_{in}$ and $\rho_{out}$ of the reference simulation and all sensitivity studies.}

\label{table:inundationAreas}
\end{table}
