
\section{Discussion of Results}
%


\subsection{ Oslo}
Oslo is surge-dominant and has the largest amount of data of the locations presented in this chapter. Both these factors mean that all the methods applied should perform quite well here. Surge-dominance means that the non-stochastic tidal component of the measured sea levels does not interfere to a large degree, and a large amount of data is naturally desirable. As for possible sources of error, the removal of the post-glacial rebound trend is certainly worth mentioning, and the same procedure as in the report by \cite{Hansen2000} was followed.

Figure~\ref{disc_retlevels_oslo} shows that return levels are very similar for all methods for 5, 10 and 20 year periods, while larger differences arise for the 100 and 200 year periods. Note that the lines between the indicated points are only drawn for visual purposes.

Figure~\ref{disc_200levels_oslo} shows the 200 year return level estimates, together with 95\% confidence intervals. As seen, the Gumbel model is producing the largest estimate, at 266 cm, while the GEV model gives the smallest, at 249 cm. This difference is not large, and all methods contain the results from the other methods within their confidence interval. Also note that the ACER method has the shortest confidence interval, indicating perhaps a higher estimation accuracy. 
\begin{figure}[ht]
\centering
\subfigure[Return levels for 5, 10, 20, 100 and 200 year periods.]{
\includegraphics[scale=0.32]{EWL/Figures/oslo_TM_100_2.pdf}%
\label{disc_retlevels_oslo}
}
\subfigure[200 year return levels and confidence bounds.]{
\includegraphics[scale=0.32]{EWL/Figures/oslo_TM_100_3.pdf}%
\label{disc_200levels_oslo}
}
\caption{Comparison between return levels in Oslo.}%
\label{disc_retplot_oslo}%
\end{figure}
%



\subsection{ Heimsjø}

For Heimsjø, the non-stochastic tidal component dominates, and estimating the extreme value distribution becomes more difficult. This is seen in the larger differences between methods. The RJP method, in particular, disagrees markedly for the three first periods of 5, 10 and 20 years. This is shown in Figure~\ref{disc_retlevels_heimsjo}. Since the ACER method provides unbiased estimates of the exact in sample values, it can be concluded that the RJP method estimates are inaccurate at these return periods. However, the RJP method performs much better for the 100 and 200 year levels, where it basically is in line with all models but the Gumbel one. 

Figure~\ref{disc_200levels_heimsjo} shows better the difference between the 200 year estimates, and highlights the extreme upper bound of the estimate obtained by the RJP method. The other confidence bounds look minuscule in comparison, but this is because of the scale of the bounds of the RJP method. The Gumbel model produces a significantly higher return level compared to the other methods. However, the difference between the highest estimate (Gumbel) and the lowest (POT) for the longest return period is only 18~cm. It is seen that the ACER method again provides the smallest confidence interval.
\begin{figure}[ht]
\centering
\subfigure[Return levels for 5, 10, 20, 100 and 200 year periods.]{
\includegraphics[scale=0.32]{EWL/Figures/heimsjo_TM_230_2.pdf}%
\label{disc_retlevels_heimsjo}
}
\subfigure[200 year return levels and confidence bounds.]{
\includegraphics[scale=0.32]{EWL/Figures/heimsjo_TM_230_3.pdf}%
\label{disc_200levels_heimsjo}
}
\caption{Comparison between return levels in Heimsjø. }%
\label{disc_retplot_heimsjo}%
\end{figure}



\subsection{ Honningsvåg}

Mostly the same pattern as in Heimsjø is shown for Honningsvåg, but with a greater scatter of the results. All methods except the RJP method agree at the 5 year return level, but diverge at the higher levels. The RJP method starts out estimating lower return levels, but ends up in the same region as the ACER estimate for the 200 year period. The Gumbel and POT methods give by far the largest estimates, while the GEV method produces by far the lowest.

Figure~\ref{disc_retlevels_honning} compares the 5, 10, 20, 100 and 200 year estimates for all methods, while Figure~\ref{disc_200levels_honning} shows the comparatively large difference between the 200 year estimates. The POT estimate for that period is ~30~cm larger than the corresponding GEV estimate. Again the ACER method provides the smallest confidence interval.
\begin{figure}[ht]
\centering
\subfigure[Return levels for 5, 10, 20, 100 and 200 year periods.]{
\includegraphics[scale=0.32]{EWL/Figures/honningsvag_TM_260_2.pdf}%
\label{disc_retlevels_honning}
}
\subfigure[200 year return levels and confidence bounds.]{
\includegraphics[scale=0.32]{EWL/Figures/honningsvag_TM_260_3.pdf}%
\label{disc_200levels_honning}
}
\caption{Comparison between return levels in Honningsvåg.}%
\label{disc_retplot_honning}%
\end{figure}
%

\subsection{ Comments}

The Annual Maxima method and the Peaks-over-Threshold method are both widely known and much applied methods. However, they both have some possible defects. As mentioned, the AM method throws away much of the data, and may end up fitting a poor model since all locations have less than 100 years of observed maxima to draw from. The POT method allows for the use of a great deal more data, but is subjected to two individual choices: the extreme threshold and the empirical clustering rule. The former of the choices has available supportive literature \citep{cole01}, but the selection of the clustering rule has little general theory, and some experience is a valuable asset in dealing with this problem. 

The Revised Joint Probabilities (RJP) method is less widely applied than the first two methods, but has seen application to a number of British locations by the creators and proponents of the method, cf. \cite{tawn92,Dixon1994,Haigh2010}. But because the literature is more sparse, and from fewer sources, it is more difficult to fully assess the reliability of this method. 

As for the estimated return period levels by the RJP method, a notable feature is what appears to be underestimated short period return levels in the tide-dominant locations of Heimsjø and Honningsvåg. This is not an artifact from the tide-surge interaction modeling, since corresponding return levels were equally low when the estimation was performed without such modeling. The POT method is used to estimate surge parameters, which was not done by \cite{Dixon1994} in earlier work. This could perhaps affect results. 

However, their methods are adopted in declustering the data using $r=30$, which is quite different from the choice made in the POT analysis. It was done to emulate the method as used by \cite{Dixon1994}, but could possibly have benefited from more careful consideration. Other choices that need to be made are the functions with which tide-surge interaction is corrected for and the quantiles that are deemed extreme. The correction functions were made to the specifications in \cite{Dixon1994}, while the quantiles used were taken from the preceding POT section which provides a generally thorough explanation of the threshold selection. 

In total, it is difficult to recommend a method that has the likely defect of underestimating short period return levels. Although it agrees more with the other methods for higher return levels, this is an area where there is a large margin of error anyway. The method is also somewhat difficult to implement, since Eq.~\eqref{eq:rjpmdistrev} needs to be solved numerically. Tide measurements also need to be available. Using such data is, however, a strong point of the model, if applied correctly. More data are then used in the estimation. This attempt to incorporate sea level specific data and methodology into the return level estimation is arguably the most desirable quality of the method. 

The ACER method is the most recently developed method, and it is therefore interesting to see that it produces results very similar to those by more conventional methods. In the return level plots in this section it is seen that the ACER method produces 200 year estimates in the middle of the range of the methods used here.

Two choices need to be made when estimating return levels with the ACER method; the tail marker $\eta_0$ and the ACER function $\varepsilon_k$ used for the parameter estimation. It was found that as long as data around or above the 90-95\% quantile were used, quite consistent results were obtained. 

The ACER method seems to be a viable alternative for extreme sea level estimation, and with the developed methodology for parameter estimation and construction of confidence intervals it is quite easy to implement. One of the attractive features of the ACER method is its diagnostic power. By plotting the ACER functions of relevant order, it can be decided which order is necessary to capture the effect of dependence in the data on the extreme value statistics. 

In summary, the ACER method seems to be an attractive method, with its quite easily implemented methodology and attractive statistical properties. When looking at 200 year return levels, it usually agrees with the other data-intensive methods, that is the RJP method and the POT method. While the empirical ACER functions provide a completely general nonparametric representation of the extreme value distribution given by the data, for the purpose of extrapolation to long return period levels a specific class of parametric distributions were introduced. This class of distributions specifically targets the cases where the Gumbel distribution is the appropriate asymptotic extreme value distribution. In the context of this chapter, this assumption is justified based on the underlying water level statistics, which invariably belongs to the domain of attraction of an asymptotic Gumbel distribution for the extremes. 
