Assessment of disgn-flows in water management, Classical methods, nonstationary and multidimensional extensions of Extreme Value Modeling (EVM)

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1 Assessment of disgn-flows in water management, Classical methods, nonstationary and multidimensional extensions of Extreme Value Modeling (EVM) Dr. Winfried Willems, IAWG

2 Outline Classical Approach, short intro Part A Classical (stationary) vs. instationary EVM Limitations of the stationary approach Principles the instationary approach Assessment of design-flows Examples Summary Part B Classical (univariate) vs. multivariate EVM Limitations of the univariate approach Principles of the multivariate approch Assessment of design-flows Examples Summary

3 Classical Approach Design-Flows: prob.-based discharge Low Flows e.g. NM7Q(50) Floods e.g. HQ(100) statistical deterministic DVWK 120, 121 DVWK 251

4 PIK Potsdam 12/.2005 Classical Approach Pegel Andernach/Rhein Pegel Dresden/Elbe Variable Zeitraum Variable NM7Q(j,4,3), HQ(j,11,10), Zeitraum /s] HQ [m[m 3 /s] NMxQ Zeit Zeit (a) (a) HyStat HyStat IAWG IAWG

5 Classical Approach P [%] Pegel Pegel DUESSELDORF/Rhein RUHRORT/Rhein Variable HQ(j,11,10), Wahrscheinlichkeit Zeitraum Log-Normal 3 (LM) Log-Normal 3 (ML) Pearson 3 (M) Pearson 3 (LM) Log-Pearson 3 (M) Weibull 3 (M) Weibull 3 (ML) Gumbel (M) 90 Gumbel (LM) Gumbel (ML) 80 Verall.Extremwert (M) Verall.Extremwert (LM) Verall.Extremwert (ML) Rossi (ML) 2 40 Plot.Pos.(Hosking) Log-Normal 3 (M) Log-Normal 3 (LM) 30 Log-Normal 3 (ML) Pearson 3 (M) Pearson 3 (LM) 10 Log-Pearson 3 (M) 90 Weibull 3 (M) 10 5 Weibull 3 (ML) 2 Gumbel (M) 80 Gumbel (LM) 5 1 Gumbel (ML) 0.5 Verall.Extremwert (M) Verall.Extremwert (LM) Verall.Extremwert (ML) 0.05 Rossi (ML) 2 Plot.Pos.(Hosking) Q [m 3 /s] HyStat/HQ-NQ 1.06 IAWG 2005 Jährlichkeit Jährlichkeit Tn [a] Tn [a]

6 Limitations of classical approach Limitation Uncertainty Nonstationarity Multidimensionality Solution Regional EVM Instationary EVM Multivariate EVM

7 Part A: stationary - instationary

8 Limitations of classical approach

9 Limitations of classical approach PIK Potsdam 12/.2005 Q two nonlinear No linear instationarities trends Jump trend and one jump Urbanisation? flood detention basins? combined effects? In anthropogenic affected catchments, instationarities may be the rule, not the exception!!! t

10 Principles of instationary approach or: Q ~ GEV ( µ, σ, γ ) Q ~ GEV( µ ( t), σ, γ) µ ( t) µ ( p, t) = a+ bt Q ~ GEV ( µ ( p, t), σ, γ ) Location Scale Shape = b0 + b1 sin(2πp /12) + b2 cos(2πp /12) + b3 Q ~ GEV( µ ( p, t), σ( t), γ ) Q ~ GEV( µ ( p, t, NAO), σ( t), γ ) t Periodicities, jumps and trends are parts of the probabilty model

11 Is the instationarity significant? PIK Potsdam 12/.2005! D(M1,M2)= 2(L2(M2)-L1(M1)) Model Parameter Log- Likelihood µ, σ, γ M1: GEVML-S µ ( p), σ, γ D Df 2 (Df,1-) M2: GEVML-P L µ ( p, t), σ, γ M3: GEVML P L T L

12 Design flows In the case of significant instationarities, design flows are no longer constant but a function of the variable t that describes the instationarity HQ (T) -> HQ(T,t)

13 Examples

14 Instationary EVM, summary stationary EVM is a special case of instationary EVM Model identification an significance test is a part of instationary EVM, instationary EVM should be standard approach In the case of significant instationarities, design flows are no longer constant HQ(T) -> HQ(T,t)

15 Part B: univariate - multivariate

16 Limitations of class. (univariate) EVM Definition of Low-Flow-Variables NM7Q Looking at discharge Keeping duration constant

17 Applications of Multivariate EVM 1. Correlated variables at one gage 2. One Variable at many gages

18 Applications of Multivariate EVM 1. Correlated Variables, examples Low flow duration and mean deficit Duration, Discharge and Deficit-Volume Flood peak and flood volume Flood peak, Flood volume and flood duration discharge and temperature discharge, temperature und nitrate

19 Applications of multivariate EVM 2. One Variable at many gages, examples Low flow at Danube and Regnitz ( RMD-Überleitung ) Floods at several gages in a larger area ( accumulation loss )

20 When to use Multivariate EVM? Risk unidimensional moredimensional Univariate EVM Multivariate EVM

21 Principles, models Selection of multivariate Models: Multivariate Normal Distribution Multivariate GEV Copulas

22 Multivariate Normal

23 Transformation to Normal Prellberg y = log(( x + c) min( x)) log(max( x) ( x + c)) Box-Cox-Transformation y = λ x 1, λ log x, λ λ = 0 0

24 Copulas Variables may be derived from any continous distributions correlation structures may be complex (e.g. nonsymmetric) Copulas are multidimensional distribution functions that link the univariate marginal distributions

25 Complex correlations structures PIK Potsdam 12/.2005 r = 0.5 r = 0.5 r = 0.5 symmetric nonsymmetric

26 Copulas 1. Fit appropiate distribution F j (X j ) for each X 1,... X n 2. Estimate empirical Copula C(): P ( X, x,..., X x ) = 1 1 n n 3. Fit appropriate theoretical Copula 4. Application, e.g. Monte-Carlo-Simulation Recurrence - Period - Isolines C( F ( X ),..., 1 1 F n ( X )) n

27 Theoretical Copulas Elliptic (sym.) e.g. Normal-Copula t-copula Farly-Gumbel-Morgenstern-C. Archimed. (asym.) e.g. Gumbel-Copula Nelsen-Copula

28 Design-flows Univariate: PÜ = P(X > x); T = 1 / PÜ Multivariate (here bivariate) : PÜ or = P(X 1 > x 1 X 2 > x 2 ); T or = 1 / PÜ or PÜ and = P(X 1 > x 1 X 2 > x 2 ) ; T and T-Isolines!! = 1 / PÜ and

29 Example 1: Duration and Deficit PIK Potsdam 12/.2005 Gage: Ingolstadt / Donau Measurements from: Duration [d] mean DEFICIT [m 3 /s] Correlation: 0.76 Jul Jul

30 Histograms of Low-Flow-Variables

31 Transformation to Normal

32 Boxplot MNV, DURATION

33 Boxplot MNV, DEFICIT

34 Copula-model, marginals

35 Copula-model, variable DURATION

36 Copula-model, variable DEFICIT

37 T-Isolines

38 Example 2: one variable, many gages

39 Example 2: one variable, many gages Korrelationen 75%-Quantile Mittelwerte

40 Multivariate EVM, summary Use if risk is multidimensional Multivariate Normal, if all vars can be transformed to Normal Multivariate GEV, if all vars are GEV-distributed Copula-Models almost always applicable Design-Flows are defined by isolines in the bivariate case

41 Outlook It is possible to combine - Regional EVM and - Instationary EVM and - Multivariate EVM!!

42 Thanks for attention!!

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