Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems

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. 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. 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. Theory of Copulas Copulas enable us to express the joint distribution of random variables in terms of their marginal distribution using the theorem of Sklar (1959): where: FX,Y(x,y) is the joint cdf of the random variables Fx(x), Fy(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 Outline –Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

5. Archimedian Copulas 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: jis the generator of the copula One-parameter Archimedian copula Gumbel-Hougaard Family: where: Parameter Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

6. 2-Parameter Copulas 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. 2-Parameter copula BB1 (Joe, 1997): where: Parameter Parameter models the upper tail dependence Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

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

8. Bivariate Frequency Analysis Non-exceedance probability: Exceedance probability exceeding x and y : Return period: Exceedance probability exceeding x or y : Return period: Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

9. Research Area Watershed of the river Unstrut: Catchment area: 6343 km² Highly vulnerable to floods Flood Retention System: Volume: ~ 100 Mio. m3 2 Reservoirs Polder system 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 Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

10. Methodology RIMAX joint research project: “Flood control management for the river Unstrut” Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

11. Generation of Flood Events for Risk Analysis 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 Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

12. Bivariate Analysis Flood Peak-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  Bivariate analysis of flood peak and volume Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

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. 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 Reservoir Straußfurt Reservoir Kelbra Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

15. Bivariate Analysis Flood Peak-Volume Parametric and nonparametric estimates of Archimedian copulas: 2-Parameter copula BB1: Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

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

17. Bivariate Analysis Flood Peak-Volume Joint return periods:  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 Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

18. Bivariate Analysis Flood Peak-Volume Critical Events at the reservoir Straußfurt Waterlevel > 150.3 m a.s.l.  Outflow > 200 m3s-1  Severe damages downstream TvX,Y>40 years: all selected events are critical events  Hydrol. risk is very high 25<TvX,Y<40 years: 3 of 5 selected events are critical events TvX,Y<25 years: 2 of 12 selected events are critical events  Hydrol. risk is low Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

19. Spatial Variability 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 Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

20. Bivariate Analysis of corresponding Flood Peaks Parametric and nonparametric estimates of KC(t) 1000000 simulated random samples from the copulas  Gumbel-Hougaard copula is used for further analysis Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

21. Bivariate Analysis of corresponding Flood Peaks Joint return periods:  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 Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis Research Area – Application -Conclusions

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. Acknowledgments • BMBF (Federal Ministry of Education and Research) / RIMAX • Unstrut-Project: TMLNU, MLU LSA, DWD Thank you very much for your attention! bastian.klein@rub.de www.ruhr-uni-bochum.de/hydrology