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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

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Whats it all about? Changing risk. PresentFuture

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Whats it all about? Changing risk. PresentFuture PastPresent

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

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

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

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

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

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

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

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

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

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

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

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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)

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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

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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

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

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

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

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3 Systematic uncertainties PresentFuture 3.3Instationarity

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

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

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

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

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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

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THANKS!

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

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

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

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

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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)

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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)

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

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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.

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

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

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