# Changeset 2687 for documentation/experimentation/smf.tex

Ignore:
Timestamp:
Apr 10, 2006, 5:20:59 PM (17 years ago)
Message:

updating working documents

File:
1 edited

### Legend:

Unmodified
 r2681 \usepackage{lscape} %landcape pages support %\input{definitions} \title{Application of SMF surface elevation function in inundation modelling} the system volume is conserved. As a result, $$\kappa' = {\rm erf} ( (x - x0)/ \sqrt \lambda_0 ) / {\rm erf} ( (x - \Delta x - x0)/ \sqrt \lambda_0 )]_{x_min}^{x_max}$$ $$\kappa' = [{\rm erf} ( (x - x_0)/ \sqrt \lambda_0 ) / {\rm erf} ( (x - \Delta x - x_0)/ \sqrt \lambda_0 )]_{x_{\rm min}}^{x_{\rm max}} \ .$$ \noindent with $\kappa' \ge 1$ for $\Delta x \ge 0$. Figure 2 in [1] could then be reproduced for appropriate values of $\kappa'$ to \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 ($\kappa', \Delta x) = (1,0) and (1.2, 0.0167)$. This function becomes unbounded for small $\Delta x$ thereby limiting both parameters. be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in Figure \ref{fig:eta_vary}. %\caption Relationship between \kappa' and \delta x to ensure volume %conservation \begin{figure}[hbt] %\centerline{ \includegraphics[width=75mm, height=75mm]{volume_conservation.ps}} \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.ps}} \caption{Surface elevation functions for ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.} \label{fig:eta_vary} \end{figure} {\bf TO DO:} Need a discussion in here on whether the recast as $$\kappa' \approx {\rm erf} ( (x - x0)/ \sqrt\lambda_0 ) / {\rm erf} ( (x - 2 x0$$\kappa'  \approx {\rm erf} ( (x - x_0)/ \sqrt\lambda_0 ) / {\rm erf} ( (x - 2 x_0 - x_g)/ \sqrt \lambda_0 )