<%namespace name="tools" file="tools.tex"/>
<%! import os %>
\chapter{Model Performance Statistics}
\label{app:ModelPerformanceStatistics}
\section{Introduction}

In this Appendix the theory behind the Model Performance Statistics (MPS) used in the XBeach skillbed is explained. The MPS are used to quantify the performance of model results based on a comparison with measurement data. Different MPS parameters are used as each parameter has its own characteristics. 

First an overview is given of the MPS parameters used in the XBeach skillbed, summarized in table form including some basic characteristics. Consequently, each MPS parameters listed in the overview table is further explained in separate sections.

\section{MPS parameters}

An overview of the MPS parameters used in the XBeach skillbed is given in \autoref{tab:MPS}.

\begin{table}[H]
  \centering
  \begin{table}[!tbp]
    \caption{MPS parameters\label{table}}
    \label{tab:MPS}
    \begin{center}
      \begin{tabular}{llll}\hline\hline
        Parameter & Description & Ranges\\\
        \hline
        ME \& STD & Mean Error \& Standard Deviation & 0: perfect prediction\\\
        Rel. bias & Systematic error relative to the mean & low value: good performance\\\
        Sci & Scatter Index & low values: performance\\\
        \hline
      \end{tabular}
    \end{center}
  \end{table}
\end{table}

Each parameter listed in the table is further explained in the following paragraphs.

\section{Mean Error \& Standard Deviation}

The Mean Error (ME) and the Standard Deviation (STD) of the error of a timeseries are a useful measure to quantify model performance for parameters such as wave heights or water levels. The SD is in general not so useful when applied to morphological parameters such as the bed leve evolution.

\begin{equation}ME=\frac{1}{N}\sum_{i=1}^N(f_{comp.,i}-f_{meas.,i})\end{equation}
\begin{equation}STD=\sqrt{\frac{1}{N-1}\sum_{i=2}^N(f_{comp.,i}-f_{meas.,i}-ME)^{2}}\end{equation}

\section{Relative Bias}

The Relative Bias (Rel. Bias) is the systematic error relative to the mean. Relative low values of the mean can cause high vales of the Rel. Bias.

\begin{equation}Rel. Bias=\frac{\sum_{i=1}^N(f_{comp.,i}-f_{meas.,i})}{\sum_{i=1}^N\bar f_{meas.}}\end{equation}

\section{Scatter Index}

The Scatter index (SCI) is the standard deviation relative to the mean value of the measured signal. Relative low values of the mean can cause high vales of the SCI. 

\begin{equation}Sci=\frac{\sqrt{\frac{1}{N-1}\sum_{i=2}^N(f_{comp.,i}-f_{meas.,i}-ME)^{2}}}{\bar f_{meas.}}\end{equation}


\chapter{Morphology indicators}
\label{app:MorphologyIndicators}



\section{Erosion volume}
The dune erosion volume is defined as the volume per running meter between the initial bed level and the bed level at a given time. For both the initial bed level and the bed level at a given time, the maximum water level is applied as lower limit (See \autoref{fig:overview_BOI_erosion}).

${tools.image(os.path.join(data_path,'indicators/'), 'overview_BOI_erosion.png','Definition of erosion volume',label='overview_BOI_erosion',width=150)}

\section{Dune front retreat}
The dune front retreat is defined as the horizontal displacement at a given reference height in the considered time period. The reference height is defined as the 1.5 times the grid resolution below the maximum initial bed level for the lab experiments(See \autoref{fig:overview_BOI_ds}). This height is representative for the dune front and low enough to capture all the dune front of all the observed profiles. The dune height can reduce during an experiment due to erosion. For each of the field cases a different characteristic dune height is applied since the profiles are very different. This definition of the dune retreat for the field cases is given in each section. 

${tools.image(os.path.join(data_path,'indicators/'), 'overview_BOI_ds.png','Dune retreat indicator.',label='overview_BOI_ds',width=150)}

\section{Berm slope}
The berm slope is defined as the mean slope in the deposition zone. The upper limit is defined as the maximum water level and the lower limit is equal to the most seaward point of the deposition zone, where the deposition is equal to 50\% of the maximum vertical deposition (See \autoref{fig:overview_BOI_slope}). The mean of this slope is computed after interpolating the bed level to a uniform grid. This interpolation is required to prevent that the mean berm slope is affected by the spatial variation in the grid.

${tools.image(os.path.join(data_path,'indicators/'), 'overview_BOI_slope.png','Berm slope indicator',label = 'overview_BOI_slope',width=150)}