11. Confidence Intervals for Flood Return Level Estimates assuming ...

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234 11 Flood Level Confidence Intervalsvisible in a short sample of a process, it might still be present. Accounting forthis dependence improves the estimation of confidence limits.In the next step, we model the correlation structure of the maxima serieswith a Farima process. Again, we do not expect this series being more adequatelymodelled by a more complex process than the daily data. We use Hicto choose a model among orders p max ≤ p = 3, q max ≤ q = 1. The smallest valuesfor the Hic is attained for a Farima[0, d, 0] model with d = 0.398 ±0.010.The goodness-of-fit yields a p-value of p = 0.319 indicating a suitable model.Combining Distribution and ACF. Having the model for the Acf of themaxima series, we are now able to generate a bootstrap ensemble of artificialdata sets according to the semi-parametric strategy bootstrap sp as described inSect. 11.4.5. We use Iaaft to combine the results of the Farima simulationwith the Box-Cox transformed resampled empirical maxima series. In the laststep the ensemble is restored to the original scale of measurement by invertingthe Box-Cox transform we started with.Calculating the Confidence Limit. Subsequently, the 100-year return levelˆr ∗ (or 0.99 quantile) is estimated for each ensemble member yielding the desireddistribution for the 100-year return level estimator shown in Fig. 11.11.From this distribution we obtain an estimate for a (1 − α)% one-sided upperconfidence limit r α using order statistics. r α is calculated from order statisticsas r α = ˆr 100 − (ˆr(N+1)α ∗ − ˆr 100) [11.17], where the ˆr i ∗ are sorted in ascendingorder. With an ensemble size N ensemble = 9 999 we ensure (N + 1)α being aninteger for common choices of α.To facilitate the comparison of the 95% confidence limits obtained fromthe bootstrap ensemble (rboot 0.95 ≈ 148m3 /s) and the asymptotic distribution(rasymp 0.95 ≈ 110m3 /s) they are marked as vertical lines in Fig. 11.11. The bootstrap95% confidence level rboot 0.95 clearly exceeds the quantile expected fromthe asymptotic distribution confirming the substantial increase in uncertaintydue to dependence. Furthermore, the tails of the bootstrap ensemble decayslower than the tails of the asymptotic distribution. The interpretation of sucha confidence level is the following: In 95% of 100-year return level estimatesthe expected (“true”) 100-year return level will not exceed the 95% confidencelimit.11.7 DiscussionThe approach is presented in the framework of Gev modelling of annual maximausing maximum likelihood. The concept can also be applied in the contextof other models for the maxima distribution (e.g., log-normal) or also differentparameter estimation strategies (e.g., probability weighted moments). Further-
Henning W. Rust et al. 235VilsbiburgDensity0.00 0.01 0.02 0.03 0.04 0.05Asympt.Bootstrap EnsembleAsymptotic DistributionEmpirical Return Level95% QuantilesBootstrap40 60 80 100 120 140 160100−year return levelFig. 11.11. Frequency distribution of the 100-year return level estimates from the bootstrap spensemble with 9 999 members for Vilsbiburg as histogram (grey) and density estimate (solid line)compared to the asymptotic distribution of the Ml estimator derived from the Fisher informationmatrix (dashed). The 100-year return level estimate from the empirical maxima series is marked asdotted vertical line. The 95% quantiles of the asymptotic and bootstrap distributions are indicatedas dashed and solid vertical lines, respectively.more, it is conceivable to extend the class of models describing the dependenceto Farima models with dependent driving noise (Farima-Garch [11.20]) orseasonal models [11.40,11.44].The modelling approach using Farima[p,d,q] models and a subsequent adjustmentof the values has been investigated in more detail by [11.61]. Usingsimulation studies, it was demonstrated that the combination of Farima modelsand the Iaaft is able to reproduce also other characteristics of time seriesthen the distribution and power spectrum. Also the increment distribution andstructure functions for river run-off are reasonably well recovered.In the approach described, we obtain a model for the Acf of the maximaseries only with the help of a model of the daily series. The longer this dailyseries is the more reliable the model will be. Regarding the uncertainty of
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