1. Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems B. Klein, M. Pahlow, Y. Hundecha, C. Gattke and A. Schumann Institute of Hydrology, Water Resources Management and Environmental Engineering Ruhr-University Bochum, Germany

2. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 2 Outline Introduction Theory of Copulas Bivariate Frequency Analysis Research Area Application Conclusions Outline – Introduction – Theory of Copulae – Bivariate Frequency Analysis Research Area – Application -Conclusions

3. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 3 Introduction To analyze flood control systems via risk analysis a lot of different hydrological scenarios have to be considered Probabilities have to be assigned to these events Univariate probability analysis in terms of flood peaks can lead to an over- or underestimation of the risk associated with a given flood.  Multivariate analysis of flood properties such as flood peak, volume, shape and duration Considerably more data is required for the multivariate case  In practice the application is mainly reduced to the bivariate case. Traditional bivariate probability distributions have a large drawback: Marginal distributions have to be from the same family  Analysis via copulas Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

4. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 4 Theory of Copulas Outline –Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Copulas enable us to express the joint distribution of random variables in terms of their marginal distribution using the theorem of Sklar (1959): where:F X,Y (x,y) is the joint cdf of the random variables F x (x), F y (y) are the marginal cdf‘s of the random variables C is a copula function such that: C: [0,1]²  [0,1] C(u,v) = 0 if at least one of the arguments is 0 C(u,1)=u and C(1,v)=v

5. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 5 Archimedian Copulas Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions A large variety of Copulas are available to model the dependence structure of the random variables (Nelson, 2006; Joe, 1997), such as Archimedian copulas: where:  is the generator of the copula One-parameter Archimedian copula Gumbel-Hougaard Family: where: Parameter

6. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 6 2-Parameter Copulas Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions 2-Parameter copula BB1 (Joe, 1997): 2-Parameter copulas might be used to capture more than one type of dependence, one parameter models the upper tail dependence and the other the lower tail dependence. where: Parameter Parameter models the upper tail dependence

7. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 7 Parameter Estimation & Evaluation Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Other estimation methods: Spearman‘s Rho, Kendalls Tau, IFM- (Inference from margins) method Rank-based Maximum Pseudolikelihood: Evaluation of the appropriate family of copulas, comparison of parametric and nonparametric estimate of: ( Genest and Rivest, 1993) Archimedian copulas:

8. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 8 Bivariate Frequency Analysis Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Non-exceedance probability: Exceedance probability exceeding x and y : Return period: Exceedance probability exceeding x or y :

9. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 9 Research Area Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Flood Retention System: Volume: ~ 100 Mio. m 3 Watershed of the river Unstrut: 2 Reservoirs Polder system Highly vulnerable to floods RIMAX joint research project: “Flood control management for the river Unstrut”  Analysis, optimization and extension of the flood control system through an integrated flood risk assessment instrument Catchment area: 6343 km²

10. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 10 Methodology Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions RIMAX joint research project: “Flood control management for the river Unstrut”

11. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 11 Generation of Flood Events for Risk Analysis Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Stochastic generation of 10x1000 years daily precipitation Daily water balance simulation with a semi-distributed model (following the HBV concept) Selection of representative events with return periods between 25 to 1000 years Disaggregation of the daily precipitation to hourly values for the selected events Simulation of hourly flood hydrographs via an event-based rainfall-runoff model

12. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 12 Bivariate Analysis Flood Peak-Volume Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions  Bivariate analysis of flood peak and volume Univariate probability analysis in terms of flood peaks can lead to an over- or underestimation of the risk associated with a given flood: Peak Return Period T = 100 a

13. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 13 Bivariate Analysis Flood Peak-Volume Marginal distributions of the flood peaks: Generalized Extreme Value (GEV) distribution Parameter estimation method: Reservoir Straußfurt: L-Moments Reservoir Kelbra: Product moments Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

14. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 14 Bivariate Analysis Flood Peak-Volume Marginal distributions of the flood volumes: Generalized Extreme Value (GEV) distribution using the method of product moments as parameter estimation method Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Reservoir StraußfurtReservoir Kelbra

15. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 15 Bivariate Analysis Flood Peak-Volume Parametric and nonparametric estimates of Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Archimedian copulas:2-Parameter copula BB1:

16. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 16 Bivariate Analysis Flood Peak-Volume 1000000 simulated random pairs (X,Y) from the copulas Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions  Only the Gumbel-Hougaard copula and the BB1 copula can model the dependence structure of the data  BB1 copula provides a better fit to the data

17. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 17 Bivariate Analysis Flood Peak-Volume Joint return periods: Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions  A large variety of different hydrological scenarios is considered in design E.g. return period of flood peak of about 100 years at reservoir Straußfurt, the corresponding return periods of the flood volumes ranges between 25 and 2000 years

18. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 18 Bivariate Analysis Flood Peak-Volume Critical Events at the reservoir Straußfurt Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Waterlevel > 150.3 m a.s.l.  Outflow > 200 m 3 s -1  Severe damages downstream T v X,Y >40 years: all selected events are critical events  Hydrol. risk is very high 25<T v X,Y <40 years: 3 of 5 selected events are critical events T v X,Y <25 years: 2 of 12 selected events are critical events  Hydrol. risk is low

19. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 19 Spatial Variability Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Catchment area with two main tributaries: What overall probability should be assigned to events for risk analysis? Two reservoirs are situated within the two main tributaries  Reservoir operation alters extreme value statistics downstream  Gages downstream can’t be used for categorization of the events  Bivariate Analysis of the corresponding inflow peaks to the two reservoirs to consider the spatial variability of the events

20. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 20 Bivariate Analysis of corresponding Flood Peaks 1000000 simulated random samples from the copulas Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions Parametric and nonparametric estimates of K C (t)  Gumbel-Hougaard copula is used for further analysis

21. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 21 Bivariate Analysis of corresponding Flood Peaks Joint return periods: Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions  A large variety of different hydrological scenarios is considered in design E.g. Return period of about 100 years at reservoir Straußfurt, the return periods of the corresponding flood peaks at the reservoir Kelbra ranges between 10 and 500 years

22. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 22 Conclusions A methodology to categorize hydrological events based on copulas is presented The joint probability of corresponding flood peak and volume is analyzed to consider flood properties in risk analysis Critical events for flood protection structures such as reservoirs can be identified via copulas The spatial variability of the events is described via the joint probability of the corresponding peaks at the two reservoirs Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

23. ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems 23  BMBF (Federal Ministry of Education and Research) / RIMAX  Unstrut-Project: TMLNU, MLU LSA, DWD Acknowledgments Thank you very much for your attention! bastian.klein@rub.de www.ruhr-uni-bochum.de/hydrology