1. Hannover, 28 November 2006 Fehlergrenzen von Extremwerten des Wetters Errors bounds in extreme weather analyses Manfred Mudelsee University of Leipzig, Germany Climate Risk Analysis, Halle (S), Germany

2. Whats it all about? Changing risk. PresentFuture

3. Whats it all about? Changing risk. PresentFuture PastPresent

4. Message 1 Climate science:no certainty, no proofs. Rather: hypothesis tests, parameter estimates.

5. Message 2 Parameter estimates (e.g., of flood risk) without realistic error bars are useless.

6. Basics Theoretical example: o daily runoff values o one year, n = 365 What is the maximum value in a year?

7. Basics Theoretical example: 5% > 3500 m 3 s –1 return period = 20 years

8. Objective Return period estimation using data risk estimation temporal changes expected damages

9. Structure of talk 1Return period estimation 2Statistical uncertainties 3 Systematic uncertainties Example: Elbe

10. 1 Return period estimation f(x)f(x) x

11. f(x)f(x) x Johnson et al. (1995) Continuous Univariate Distributions, Vol. 2, Wiley.

12. 1 Return period estimation f(x)f(x) x

13. f(x)f(x) x maximize L GEV maximize likelihood that GEV model produced data

14. 1 Return period estimation: Example Elbe, Dresden, 1852–2002, summer, annual maxima (n = 151) HQ 100 = 3921 m 3 s –1

15. 2 Statistical uncertainties nfinite GEV parameter estimation errors > 0 return period estimation error > 0 How large is error? 1. Theoretical derivation 2. Simulation Johnson et al. (1995)

16. 2 Statistical uncertainties: Simulation Jackknife simulation: Step 1:Remove randomly one point Step 2:Fit GEV, estimate return period Step 3:Go to Step 1 until 400 simulated return periods exist Step 4:Take STD over simulations

17. 2 Statistical uncertainties: Example Elbe, Dresden, 1852–2002, summer, annual maxima (n = 151) Jackknife simulations of HQ 100 : 38863962389539033960 39023920391139573961 39533959388639613959 39363892383839573871 HQ 100 = 3921 m 3 s –1 Mean = 3923 m 3 s –1 STD = 38 m 3 s –1

18. 3 Systematic uncertainties 3.1 Model suitability fitted GEV empirical (kernel density)

19. 3 Systematic uncertainties 3.2Data errors: WQ relation Mudelsee et al. (2006) Hydrol. Sci. J. 51:818–833.Werra

20. 3 Systematic uncertainties 3.2Data errors: Simulation Step 1: Q sim (i) = Q(i) + δQ WQ (i) Step 2: CombineQ sim (i) with jackknife

21. 3 Systematic uncertainties PresentFuture 3.3Instationarity

22. 3 Systematic uncertainties Mudelsee et al. (2003) Nature 425:166–169. 3.3Instationarity

23. 3 Systematic uncertainties 3.3Instationarity = the real challenge! Time-dependent GEV parameters Work in progress...

24. Message 1 Climate science:no certainty, no proofs. Rather: hypothesis tests, parameter estimates.

25. Message 2 Parameter estimates (e.g., of flood risk) without realistic error bars are useless.

26. Message 2 Parameter estimates (e.g., of flood risk) without realistic error bars are useless. Case 1Q 100 = 3921 m 3 s –1 ± ??? Case 2Q 100 = 3921 m 3 s –1 ± 38 m 3 s –1 Case 3Q 100 = 3921 m 3 s –1 ± 300 m 3 s –1

27. THANKS!

28. Example 2: Extremes, X out (T) Elbe, winter, class 2–3 h CV = 35 yr

29. Bootstrap resample (with replacement, same size) Elbe, winter, class 2–3 h CV = 35 yr Example 2: Extremes, X out (T)

30. Bootstrap resample (with replacement, same size) Elbe, winter, class 2–3 h CV = 35 yr Example 2: Extremes, X out (T)

31. Bootstrap resample (with replacement, same size) 2nd Bootstrap resample Elbe, winter, class 2–3 h CV = 35 yr Example 2: Extremes, X out (T)

32. Bootstrap resample (with replacement, same size) 2nd Bootstrap resample 2000 Bootstrap resamples Elbe, winter, class 2–3 h CV = 35 yr Example 2: Extremes, X out (T)

33. Elbe, winter, class 2–3 h CV = 35 yr Bootstrap resample (with replacement, same size) 2nd Bootstrap resample 2000 Bootstrap resamples Example 2: Extremes, X out (T)

34. Elbe, winter, class 2–3 h CV = 35 yr Example 2: Extremes, X out (T) 90% bootstrap confidence band

35. Elbe, winter, class 2–3 h CV = 35 yr Example 2: Extremes, X out (T) 90% bootstrap confidence band Cowling et al. (1996) J. Am. Statist. Assoc. 91:1516. Mudelsee et al. (2004) J. Geophys. Res. 109:D23101.

36. Example 2: Extremes, X out (T) Mudelsee et al. (2003) Nature 425:166.

37. References http://www.uni-leipzig.de/~meteo/MUDELSEE/ http://www.climate-risk-analysis.com