T RENDS IN EXTREME VALUES

In the second part of this thesis, we developed a methodological framework to assess extremes of hydro-meteorological data which accounts for non-stationarity and

auto-correlation. An unjustified assumption of stationarity may lead to considerable under-estimation of the probability of a disastrous extreme event (see, e.g., Coles et al. 2002). In our approach we take the skewed distribution of the extremes into account and apply the Fisher-Tippett theorem. By doing so, we are able to examine independent observations with almost arbitrary distribution: The convergence of their extreme values towards the generalized Pareto distribution (GPD) is theoretically well founded. The extremes are de-rived as excesses over a threshold and are modelled using a point process (cf. Coles 2001).

This allows for a sound modelling of the frequency of occurrence and the magnitude of the extreme values. The auto-correlation structure in the extremes is eliminated by thin-ning out the data, and non-stationarity is incorporated by allowing for non-stationary model parameters. The most suitable model is then selected out of a class of models which comprises the stationary model and models with a variety of polynomial and ex-ponential trend assumptions. Thus, trend detection is transfered to a model selection problem.

Regarding methodological aspects of our extreme value assessment framework, we evaluate its adequacy and its power to detect trends by simulation studies. The point process representing the extreme events is a combination of two components: An in-homogeneous Poisson process with possibly non-stationary rate of occurrence to model the frequency of occurrence of the extreme events, and the GPD with a possibly non-stationary scale parameter to derive the magnitude of these extreme events. From our empirical data we obtain sets of extreme values with a size ranging from 30 to 120. We find that this is the minimum size to get reliable results. Very good results are obtained for estimating and simulating the frequency of occurrence of extreme events and testing for a trend in the rate of occurrence. We also reliably detect the strong trends in the mag-nitude of the extremes. However, to be able to detect even weak trends in the magmag-nitude as well, there should be rather 200 extrema at hand. We therefore assume that we did not find every weak trend being present in the Danube River basin. To cross-check our procedure, we compared it to results obtained by using the generalized extreme value (GEV) distribution. Here the extremes are derived by drawing maxima from intervals of equal length and we chose the standard interval length of 365 days. In those cases where both approaches suggest a stationary model as best suiting one, we find good agreement between both approaches. However, they differ regarding the significance of trends: Of-ten the GEV approach suggests a non-stationary model where the point process approach does not and vice versa. We consider the point process framework to be more reliable, because we found that the set of threshold excesses contains the more extreme events of our empirical data and therefore better represents the extremes.

We analysed extremes of daily discharge measurements of about 50 stations within the Danube River basin. Thereby we focused on the winter season and analysed the jointly covered time period between 1941 and 2000. Basic assumptions of standard ex-treme value theory such as independent and identically distributed exex-tremes are often not fulfilled by hydrological data. Discharge is influenced by complex dynamical pro-cesses (precipitation, snow accumulation and snowmelt, evapotranspiration, et cetera), therefore a univariate distribution with two or three parameters is not necessarily a good approximation of its multifaced nature (Engeland et al. 2004). Indeed, we had to remove auto-correlations in all discharge records of the Danube River basin and one third of the stations exhibit also non-stationary extremes. Importantly, the estimates of the extreme value distribution parameters may be biased in case a stationary model is used to repre-sent non-stationary extremes. The magnitude of this bias depends on the trend strength.

By using the framework proposed in this thesis, we are able to augment the number of empirical records which can be adequately analysed.

Concerning the characteristics of the extreme values of the Danube River basin, we give particular attention to the estimated shape parameter of the GPD. The shape pa-rameter determines the “thickness” of the upper tail of the flood frequency distribution and thus helps to characterise the properties of extreme floods. Dependent on the mag-nitude of this shape parameter, the extreme value distribution can be distinguished into three types. We found all three of them being present in the Danube River catchment.

About 25% of the gauges possess a Weibull distribution, i.e., a finite endpoint can be de-termined. About 50% of the records are compatible with being Gumbel distributed, and the region exhibits 25% discharge records with heavy-tailed extrema. This Frechet distri-bution deserves special interest, because arbitrarily high and more frequent outliers than in the Gumble case have to be expected here. We found a heterogeneous spatial distri-bution of the different extreme value distridistri-butions. The estimate of the shape parameter was always lower using the most suitable, non-stationary point process model than us-ing the stationary model. This supports the theoretical findus-ing that a mixture of Gumbel distributions (that is the shape parameter is compatible with being zero) is capable to mimic a heavy tailed distribution (here the shape parameter is larger than zero). We con-clude that in case a stationary model is solely fitted to the data, heavy-tailed distributions may falsely be detected. This may result in the estimation of too high return levels and therefore too high construction costs of flood protection buildings.

Regarding the trend tendencies, the frequency of occurrence of extreme events always increases when it is detected to be non-stationary. In case a change in the magnitude of the extreme events is found, we observe decreasing and increasing tendencies. Al-together about one third of the 47 analysed records of the Danube River basin exhibit non-stationarity. To demonstrate the relevance of the changes which arise when allow-ing for non-stationary extreme value models, we exemplarily chose return levels. They are an important assessment measure for water management authorities. We identified changes up to 100% for the probability of exceedance of the 100-year return level. This implies a potential doubling of the damage costs, because the 100-year return level is then actually expected to be exceeded twice within 100 years. The spatial pattern of the trend in extremes is not immediately interpretable. Nevertheless, most of the detected trend tendencies are increasing, which might justify the activity of the Bavarian water management authorities, who anticipate an increase in flood magnitude due to climate change and therefore introduced a climate change factor of 1.15 in 2006, that is, every design value is increased by 15% (BlfW 2006) .

Hydrological design which takes non-stationary conditions into account is a direct consequence of accepting the idea of environmental change. As the approach we provide is fully parametric, in principle it provides the possibility to extrapolate the trends into the future. However, in some cases the estimated trends were found to deviate substan-tially at the beginning and the end of the time series. Therefore, we decided to proceed cautiously. We considered trends within the measurement period, but we did not extrap-olate them. This is in line with other authors (cf. B´ardossy and Filiz 2005).

Our methodology provides several ways to supply uncertainty intervals along with each parameter estimate and assessment measure, namely the delta method, the profile likelihood or bootstrapping. The resulting confidence intervals show that the

examina-tion of uncertainty is a crucial prerequisite for the evaluaexamina-tion and interpretaexamina-tion of the results of extreme value analysis.

We conclude that the application of an extended extreme value analysis framework, as presented in this thesis, is necessary to adequately assess non-stationary extreme val-ues. We demonstrated the usability, the potential and the limits of our framework. Trends in frequency and magnitude of extremes of river discharge are anticipated because of cli-mate change and we already find a noteworthy fraction of the empirical records analysed exhibiting non-stationary extremes.