In Extremis

We review our studies of the statistics of return intervals and extreme events (block maxima) in long-term correlated data sets, characterized by a power-law decaying autocorrelation function with correlation exponent γ between 0 and 1, for different distributions (Gaussian, exponential, power-law, and log-normal). For the return intervals, the long-term memory leads (i) to a stretched exponential distribution (Weibull distribution), with an exponent equal to γ, (ii) to long-term correlations among the return intervals themselves, yielding clustering of both small and large return intervals, and (iii) to an anomalous behavior of the mean residual time to the next event that depends on the history and increases with the elapsed time in a counterintuitive way. We present an analytical scaling approach and demonstrate that all these features can be seen in long climate records. For the extreme events we studied how the long-term correlations in data sets with Gaussian and exponential distribution densities affect the extreme value statistics, i.e., the statistics of maxima values within time segments of fixed duration R. We found numerically that (i) the integrated distribution function of the maxima converges to a Gumbel distribution for large R similar to uncorrelated signals, (ii) the deviations for finite R depend on both the initial distribution of the records and on their correlation properties, (iii) the maxima series exhibit long-term correlations similar to those of the original data, and most notably (iv) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the previous maximum. Finally we evaluate the effect of long-term correlations on the estimation of centennial events, which is an important task in hydrological risk estimation. We show that most of the effects revealed in artificial data can also be found in real hydro- and climatological data series.

Climate risk is the probability of adverse effects from extreme values of variables in the climate system. Because climate changes, so can the various types of climate risk (floods, storms, etc.) change. This field is of strong socioeconomic relevance. Estimates of climate risk variations come from instrumental, proxy and documentary records of past climate extremes and projections of future extremes. Kernel estimation is a powerful statistical technique for quantifying trends in climate risk. It is not parametrically restricted and allows realistic, non-monotonic trends. The bootstrap is a computing-intensive statistical resampling method used here to provide a confidence band around the estimated risk curve. Confidence bands, like error bars, are essential for a reliable assessment whether changes and trends are significant or came by chance into the data. This methodology is presented using reconstructed flood records of the central European rivers Elbe, Oder and Werra over the past five centuries. Trends in flood risk differ among rivers and also between hydrological seasons. The scientific conclusion is that flood risk analysis has to take into account the high spatial variability from orographic rainfall, as well as different hydrological regimes in winter and summer. In an ideal co-operation between experts, quantitative knowledge with uncertainty ranges (like the estimated trends in climate risk) should form the deliverable from scientists to policy makers and decision takers.

Standard flood return level estimation is based on extreme value analysis assuming independent extremes, i.e. fitting a model to excesses over a threshold or to annual maximum discharge. The assumption of independence might not be justifiable in many practical applications. The dependence of the daily run-off observations might in some cases be carried forward to the annual maximum discharge. Unfortunately, using the autocorrelation function, this effect is hard to detect in a short maxima series. One consequence of dependent annual maxima is an increasing uncertainty of the return level estimates. This is illustrated using a simulation study. The confidence intervals obtained from the asymptotic distribution of the maximum likelihood estimator (MLE) for the generalized extreme value distribution (GEV) turned out to be too small to capture the resulting variability. In order to obtain more reliable confidence intervals, we compare four bootstrap strategies, out of which one yields promising results. The performance of this semi-parametric bootstrap strategy is studied in more detail. We exemplify this approach with a case study: a confidence limit for a 100-year return level estimate from a run-off series in southern Germany was calculated and compared to the result obtained using the asymptotic distribution of the MLE.

The reconstruction of historical extreme hydrometeorological events contributes to a validation of extreme value statistics. This can mitigate several uncertainties in the flood risk analysis, e.g. in calculating possible discharges with extreme value statistics which are based on short reference data series [4.5]. The presented case study of the extreme flood of 1824 in the Neckar catchment and their triggering precipitation patterns can take place in a recent flood risk management and can be used to validate the results in trend and extreme value analysis of hydrometeorological time series.

For planning, development, and sustainable management of water resources, applications of new and advanced methodologies are essential for design flood estimation. The L-moments are a recent development within statistics and offer significant advantages over ordinary product moments. Regional flood frequency relationships are developed based on the L-moment approach. The annual maximum peak floods data are screened using the discordancy measure (D _i }), and homogeneity of the region is tested employing the L-moment-based heterogeneity measure (H). For computing heterogeneity measure H, 500 simulations are performed using the κ-distribution . Twelve frequency distributions namely extreme value (EV1), generalized extreme value (GEV), logistic (LOS), generalized logistic (GLO), normal (NOR), generalized normal (GNO), uniform (UNF), Pearson type-III (PE3), exponential (EXP), generalized Pareto (GPA), κ- (KAP), and five-parameter Wakeby (WAK) are employed. Based on the L-moment ratio diagram and \(|Z_{i}^{\mathrm{dist}}|\)-statistic criteria, GNO is identified as the robust frequency distribution for the study area. For estimation of floods of various return periods for gauged catchments of the study area, the regional flood frequency relationship is developed using the L-moment-based GNO distribution. Also, for estimation of floods of various return periods for ungauged catchments, the regional flood frequency relationships developed for gauged catchments is coupled with the regional relationship between mean annual maximum peak flood and catchment area.

According to physical laws, the water-holding capacity of the atmosphere and hence the potential for intense precipitation increases with warming. Since a robust warming signal and a number of large rain-caused floods have been observed recently, it is of paramount importance to examine whether there has been an increasing trend in intense precipitation and flood flow. However, even if widespread increases in observed intense precipitation have been reported in many areas, the analysis of annual maximum river flow records does not detect an ubiquitous and coherent increasing trend. This is in disagreement with some projections for the future, where increasing intense precipitation and flood hazard are expected. One can conclude that flood process is complex, influenced by several non-climatic factors, and can be caused by several generating mechanisms, which are affected in different ways by climate change. Hence, issuing a flat-rate statement on change in flood hazard is not justified.

This chapter reviews some methods for studying changes in the occurrence of extreme river flows. Some general background information on change detection is given. Changes in various characteristics of high flows are investigated using a variety of approaches. Although many cases of statistically significant changes are found, they go in different directions and are of different character in different regions. There does not seem to be a clear, uniform signal of change, except for, possibly, the occurrence of most extreme flows that seems to be increasing in the recent decades.

This chapter proposes and applies an extreme value assessment framework, which allows for auto-correlation and non-stationarity in the extremes. This is, e.g., useful to assess the anticipated intensification of the hydrological cycle due to climate change. The costs related to more frequent or more severe floods are enormous. Therefore, an adequate estimation of these hazards and the related uncertainties is of major concern. Exceedances over a threshold are assumed to be distributed according to a generalised Pareto distribution and we use a point process to approximate the data. In order to eliminate auto-correlation, the data are thinned out. Contrary to ordinary extreme value statistics, potential non-stationarity is included by allowing the model parameters to vary with time. By this, changes in frequency and magnitude of the extremes can be tracked. The model which best suits the data is selected out of a set of models which comprises the stationary model and models with a variety of polynomial and exponential trend assumptions. Analysing winter discharge data of about 50 gauges within the Danube River basin, we find trends in the extremes in about one-third of the gauges examined. The spatial pattern of the trends is not immediately interpretable. We observe neighbouring gauges often to display distinct behaviour, possibly due to non-climatic factors such as changes in land use or soil conditions. Importantly, assuming stationary models for non-stationary extremes results in biased assessment measures. The magnitude of the bias depends on the trend strength and we find up to 100% increase for the 100-year return level. The results obtained are a basis for process-oriented, physical interpretation of the trends. Moreover, common practice of water management authorities can be improved by applying the proposed methods, and costs for flood protection buildings can be calculated with higher accuracy.

Application of a generalized time series decomposition technique shows that observed German monthly precipitation time series can be interpreted as a realization of a Gumbel-distributed random variable with time-dependent location parameter and time-dependent scale parameter. The achieved complete analytical description of the series, that is, the probability density function (PDF) for every time step of the observation period, allows probability assessments of extreme values for any threshold at any time. So, we found in the western part of Germany that climate is getting more extreme in winter. Both the probability for exceeding the 95th percentile and the probability for falling under the 5th percentile are increasing. Contrary results are found in summer. The spread of the distribution is shrinking. But in the south, relatively high precipitation sums become more likely and relatively low precipitation sums become more unlikely in turn of the twentieth century.

Applying the Pettitt test we study long river runoff records from gauges in southern Germany and find significant change points . Theoretically, a change point represents a sudden change in the statistics of a record. Using detrended fluctuation analysis , we also find – in agreement with previous studies – pronounced long-term temporal autocorrelations in the considered records. The results of both approaches indicate a relation between the occurrence of change points and the strength of long-term correlations. In order to clarify a possible connection, we further analyse with both methods artificial long-term correlated records and find for weak long-term correlations already highly significant change points . The significance dramatically increases with the strength of the long-term correlations.

We studied and compared the autocorrelation behaviour and the temporal multifractal properties of long daily river discharge and precipitation records from 42 hydrological stations and 99 meteorological stations around the globe. To determine the scaling behaviour in the presence of trends , we applied detrended fluctuation analysis (DFA) and multifractal DFA . We found that the daily runoffs are characterised by a power-law decay of the autocorrelation function above some crossover time that usually is several weeks.

Hydrological discharge time series are known to depict low-frequency oscillations, long-range statistical dependencies, and pronounced nonlinearities. A better understanding of this runoff behaviour on regional scales is crucial for a variety of water management purposes and flood risk assessments. We aimed at extracting long-term components which influence simultaneously a set of southern German runoff records.

Climatic time series, in general, and hydrological time series, in particular, exhibit pronounced annual periodicity. This periodicity and its corresponding harmonics affect the nonlinear properties of the relevant time series (i.e. the long-term volatility correlations and the width of the multifractal spectrum) and thus have to be filtered out before studying fractal and volatility properties. We compare several filtering techniques and find that in order to eliminate the periodicity effects on the nonlinear properties of the hydrological time series, it is necessary to filter out the seasonal standard deviation in addition to the filtering of the seasonal mean, with conservation of linear two-point correlations . We name the proposed filtering technique “phase substitution”, because it employs the Fourier phases of the series. The obtained results still indicate nonlinearity of the river data, its strength being weaker than under previously used techniques.

Hydrological processes are characterised by spatio-temporal patterns with certain correlations in both time and space. Thus, time series of related quantities recorded at different locations show relevant correlations if these locations are influenced by the same patterns. The actual strength and temporal as well as spatial extension of these correlations depend crucially on the considered observable (temperature, runoff, precipitation, etc.). We analyse whether the corresponding interrelationships change significantly in the presence of changing environmental conditions. For this purpose, we systematically study a variety of measures which quantify the statistical dependence between the components of bi- and multivariate hydrological records. As a particular example, we consider runoff time series from an ensemble of gauges in the Upper Main catchment area in southern Germany. The qualitative behaviour of spatial correlations and their changes during extreme weather events are intensively discussed.