Trends and Extreme Values of River Discharge Time Series
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TRENDS AND EXTREME VALUES OF RIVER DISCHARGE TIME SERIES Dipl.-Syst. Wiss. Malaak Kallache Potsdam, April 2007 Dissertation zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) Eingereicht im Fachbereich Geowissenschaften der Fakultat¨ II – Biologie, Chemie und Geowissenschaften der Universitat¨ Bayreuth, Deutschland Dissertation submitted for obtaining the degree of Doctor rerum naturalium in the department of Geosciences at the faculty of Biology, Chemistry and Geosciences of the University of Bayreuth, Germany Vollstandiger¨ Abdruck der von der Fakultat¨ Biologie, Chemie und Geowissenschaften der Universitat¨ Bayreuth genehmigten Dissertation zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.). Die vorliegende Arbeit wurde in der Zeit vom November 2002 bis November 2007 am Lehrstuhl fur¨ Okologische¨ Modellbildung der Universitat¨ Bayreuth angefertigt. Die Arbeit wurde eingereicht am 16. April 2007 Das Rigorosum fand statt am 02. November 2007 Der Prufungsausschuss¨ bestand aus: Prof. Dr. C. Beierkuhnlein (Vorsitzender) Prof. Dr. H. Lange (Erstgutachter) Prof. Dr. H.J. Schellnhuber (Zweitgutachter) Prof. Dr. M. Hauhs PD Dr. G. Lischeid .....you cannot step twice into the same river Heraclitus CONTENTS Abstract xi Zusammenfassung xiii 1 Motivation 1 1.1 Climatechange .................................. 1 1.2 Influence of climate change on the hydrological cycle . ....... 2 1.3 Approachestoassesstrends . 3 1.4 Aimsandoutlineofthethesis . 5 2 Basic Methods and Data 9 2.1 Stochasticprocesses.............................. 9 2.1.1 Basicdefinitions ............................. 9 2.1.2 Autoregressivemovingaverageprocesses . 10 2.1.3 FractionalARIMAprocesses . 11 2.2 Waveletanalysis.................................. 13 2.2.1 Discretewavelettransform . 13 2.2.2 Daubechiesleastasymmetricwaveletfilters . 17 2.2.3 Waveletvariance ............................. 19 2.3 Extremevaluestatistics. 20 2.3.1 Modelsforblockmaxima . 20 2.3.2 Thresholdexcessmodels . 23 2.3.3 Comparison of block maxima and threshold excesses approach . 29 2.3.4 Non-stationaryextremevaluemodels . 29 2.4 Parameterestimation.............................. 34 2.4.1 FARIMAprocessparameterestimation . 34 2.4.2 Pointprocessparameterestimation . 35 2.4.3 Consistencyandefficiency . 36 2.5 Modelselection .................................. 38 2.5.1 Akaikeinformationcriterion . 38 2.5.2 Deviancestatistic ............................. 38 2.6 Modelvalidation–goodness-of-fit . 39 2.6.1 FARIMAgoodness-of-fittest . 39 2.6.2 Poissonprocessgoodness-of-fittest . 40 2.6.3 Kolmogorov-Smirnovtest . 40 2.6.4 Probabilityplotandquantileplot. 41 2.7 Data......................................... 42 2.8 Preprocessingthedata ............................. 44 2.8.1 Seasonaleffects .............................. 44 2.8.2 Declustering................................ 46 3 Trendsinmeanvaluesconsideringauto-correlation 49 3.1 Trenddetectionapproaches . 49 vii viii CONTENTS 3.2 Trenddefinition.................................. 50 3.2.1 Stochasticversusdeterministictrends . 50 3.2.2 Trendsandauto-correlation . 51 3.2.3 Trend shape and trend detection methods . 53 3.3 Trendestimate................................... 55 3.3.1 Separatingscale.............................. 55 3.3.2 Boundaryconditions........................... 56 3.3.3 Stochasticcomponent .......................... 56 3.3.4 Varianceofthetrendestimate. 58 3.3.5 Consistencyandefficiency . 59 3.3.6 Pointwiseconfidenceintervals . 60 3.4 Testforsignificanceofthetrend. 60 3.4.1 Powerofthetrendtest.......................... 62 3.5 Application,resultsanddiscussion . ..... 63 3.5.1 RiverNeckarbasin ............................ 63 3.5.2 Comparison with Mann-Kendall trend test (Danube River basin) . 68 4 Trends in extremes 73 4.1 Trend detection approaches for extremes . ...... 73 4.2 Trenddefinition.................................. 74 4.2.1 Trend shape and trend detection methods . 74 4.3 Choiceofextremes ................................ 76 4.3.1 Choiceofseason ............................. 76 4.3.2 Thresholdselection............................ 76 4.3.3 Declustering................................ 76 4.4 Trendestimate................................... 77 4.4.1 Trendextrapolationinthefuture . 78 4.5 Pointprocessapproach. 78 4.6 Test for significance of the trend in extremes . ....... 80 4.6.1 Powerofthetrendtest.......................... 80 4.6.2 Simulating the distribution of the deviance statistic . ........ 83 4.7 Application,resultsanddiscussion . ..... 86 5 Conclusion and perspectives 93 5.1 Trendsinmeanvalues .............................. 93 5.2 Trendsinextremevalues. 95 5.3 Finalremarksandoutlook . 98 Bibliography 99 Symbol Index 109 Appendix 111 A-1 Spectral density and auto-covariance function of a FARIMA process . 111 A-2 Simulation study to evaluate the likelihood estimation of point processes . 113 A-3 Generationofnon-stationarypointprocesses . ........ 119 A-4 DanubeRiverbasin: Thresholdandclustersize . 123 A-5 Software ...................................... 125 Danksagung 127 LIST OF FIGURES 2.1 Example of a FD(δ) and AR(1) series and auto-correlation function . 13 2.2 ChirpsignalandWPS .............................. 15 2.3 MRAofchirpsignal ............................... 17 2.4 Daubechiesleastasymmetricwavelets . ..... 18 2.5 GEVdistribution ................................. 21 2.6 Countingprocessexample. 25 2.7 Thresholdselection. MeanResidualLifePlot. ....... 27 2.8 Threshold selection. GPD fit over range of thresholds. ........ 28 2.9 Comparison of block maxima and threshold excesses . ...... 30 2.10 Returnlevelcalculation . 32 2.11 Lengthoftimeseries .............................. 43 2.12 SizeofNeckarRiversub-catchments . ..... 43 2.13 Size of Danube River sub-catchments . 43 2.14 Gauges in Bavaria and Baden-Wurttemberg..................¨ 45 2.15 Referenceyear................................... 46 2.16 Declusteringalgorithm. 47 3.1 Comparison of short- and long-range correlation. ........ 52 3.2 TrendshapeoftheDreisamRiveratEbnet. 53 3.3 Parameter estimation under trend or filtering . ....... 57 3.4 Wavelet trend estimate and CI’s for different models . ........ 59 3.5 Wavelet trend estimate for the Fox River at Wayland . ....... 62 3.6 Powerofthetrendtest .............................. 64 3.7 Influenceoftrendelimination . 65 3.8 Auto-correlationintheDanubebasin. 69 3.9 Comparison of Mann-Kendall and CGP trend test . 70 4.1 Validation of non-stationary point process . ........ 79 4.2 PowerofthetrendtestforthePoissonprocess . ...... 81 4.3 PowerofthetrendtestfortheGPDmodel. 82 4.4 Powerofthetrendtestwithsimulateddistribution . ........ 85 4.5 Distribution of the shape parameter ξ ..................... 86 4.6 Trend in frequency of occurrence of extreme events . ....... 87 4.7 Trend in mean and variance of extreme events . 88 4.8 Estimates of the mean of the extremes of the Naab River at Heitzenhofen . 89 4.9 100-year return level for the Naab River at Heitzenhofen . ......... 90 4.10 pp-plot and qq-plot for the Naab River at Heitzenhofen . ......... 90 4.11 Change of probability of excessing a 100-year return level.......... 91 4.12 Floodchangesandfloodprotectionstrategies . ....... 92 A-2.1 Comparison of non-stationary and stationary PP fits in small windows . 114 A-2.2 Goodnessofpointprocessfit . 115 A-2.3 Estimated slope of non-stationary σ ..................... 116 ix x LIST OF FIGURES A-3.1 Test of simulation of non-stationary point processes (µ).......... 120 A-3.2 Test of simulation of non-stationary point processes (σ and ξ) ...... 120 A-3.3 Test of simulation of non-stationary point processes (µ).......... 121 A-3.4 Test of simulation of non-stationary point processes (σ and ξ) ...... 121 ABSTRACT The central goal of this thesis is the development and application of sound data analysis frameworks to assess trends in mean and extreme river run-off. This includes the con- sideration of the auto-correlation structure (memory) of the data. Such an approach is very useful to, e.g., assess the anticipated intensification of the hydrological cycle due to anthropogenic climate change. The costs related to more frequent or more severe floods and droughts are enormous. Therefore an adequate estimation of these hazards and the related uncertainties is of major concern. We analyse discharge of basins of the Danube River and Neckar River in Southern Germany. Thereby we compare our frameworks with methods common in hydrology with respect to the calculation of assessment mea- sures used by water management authorities. Furthermore, we look for spatial patterns of trends. In the first part of this thesis, we evaluate trends of average discharge data. We use a trend test, where the auto-correlation structure of the data is explicitly modelled using stochastic FARIMA processes. This is a crucial task, because auto-correlations are capable of producing spurious trends. We assume that the data consists of an additive combina- tion of natural variability, which is represented by the stochastic process, and potentially a deterministic trend component. The trend is estimated using wavelets and represents the fluctuation of the data on large time scales. In case the trend adds more variability to the data than the natural variability is likely to generate we consider the trend as significant. Hydrological data possesses short- and long-term memory, due to, e.g., weather regimes or consequences of large scale atmospheric patterns. This is confirmed by our results when analysing about 90 discharge records