1. Bivariate Flood Frequency Analysis using Copula with Parametric and Nonparametric Marginals Subhankar Karmakar Slobodan P. Simonovic The University of Western Ontario The Institute for Catastrophic Loss Reduction

2. 4th ISFD Toronto 2Simonovic May 6-8 2008 Presentation outline Introduction Objectives of the study Study area Data and flood characteristics Marginal distributions of flood characteristics Parametric and nonparametric estimation Joint and conditional distributions using copula Conclusions

3. 4th ISFD Toronto 3Simonovic May 6-8 2008 Introduction Flood management (design, planning, operations) requires knowledge of flood event characteristics Flood characteristics Peak flow DurationVolume

4. 4th ISFD Toronto 4Simonovic May 6-8 2008 Introduction

5. 4th ISFD Toronto 5Simonovic May 6-8 2008 Introduction

6. 4th ISFD Toronto 6Simonovic May 6-8 2008 Study objectives To determine appropriate marginal distributions for peak flow, volume and duration using parametric and nonparametric approaches To define marginal distribution using orthonormal series method To apply the concept of copulas by selecting marginals from different families of probability density functions To establish joint and conditional distributions of different combinations of flood characteristics and corresponding return periods

7. 4th ISFD Toronto 7Simonovic May 6-8 2008 Study area Red River Basin 116,500 km 2 ( 89% in USA 11% in CDN) Flooding in the basin is natural phenomena Historical floods: 1826; 1950; 1997 Size of the basin and flow direction No single solution to the flood mitigation challenge

8. 4th ISFD Toronto 8Simonovic May 6-8 2008 Study area - data Daily streamflow data for 70 years (1936- 2005) Gauging station (05082500) - Grand Forks, North Dakota, US Location - latitude 47°55'37"N and longitude 97°01'44"W Drainage area - 30,100 square miles Contributing area - 26,300 square miles http://waterdata.usgs.gov

9. 4th ISFD Toronto 9Simonovic May 6-8 2008 Study area – flood characteristics Dependence between P, V and D P and V highly correlated All the correlations positive

10. 4th ISFD Toronto 10Simonovic May 6-8 2008 Marginals - parametric

11. 4th ISFD Toronto 11Simonovic May 6-8 2008 Marginals - nonparametric Nonparametric kernel estimation of flood frequency Orthonormal series method

12. 4th ISFD Toronto 12Simonovic May 6-8 2008 Results

13. 4th ISFD Toronto 13Simonovic May 6-8 2008 Results P follows gamma distribution (parametric) and V and D follow distribution function obtained from orthonormal series method (nonparametric) Mixed marginals

14. 4th ISFD Toronto 14Simonovic May 6-8 2008 Results Second test

15. 4th ISFD Toronto 15Simonovic May 6-8 2008 Copula An alternative way of modeling the correlation structure between random variables. They dissociate the correlation structure from the marginal distributions of the individual variables. n - dimensional distribution function can be written:

16. 4th ISFD Toronto 16Simonovic May 6-8 2008 Copula

17. 4th ISFD Toronto 17Simonovic May 6-8 2008 Results – joint distributions Peak flow - Volume

18. 4th ISFD Toronto 18Simonovic May 6-8 2008 Results – joint distributions Volume - Duration

19. 4th ISFD Toronto 19Simonovic May 6-8 2008 Results – joint distributions Peak flow - Duration

20. 4th ISFD Toronto 20Simonovic May 6-8 2008 Results – conditional distributions

21. 4th ISFD Toronto 21Simonovic May 6-8 2008 Results – conditional distributions

22. 4th ISFD Toronto 22Simonovic May 6-8 2008 Results – conditional distributions

23. 4th ISFD Toronto 23Simonovic May 6-8 2008 Results – return period

24. 4th ISFD Toronto 24Simonovic May 6-8 2008 Conclusions Concept of copula is used for evaluating joint distribution function with mixed marginal distributions - eliminates the restriction of selecting marginals for flood variables from the same family of probability density functions. Nonparametric methods (kernel density estimation and orthonormal series) are used to determine the distribution functions for peak flow, volume and duration. Nonparametric method based on orthonormal series is more appropriate than kernel estimation.