Bivariate Flood Frequency Analysis Using Copula Function

Transcription

1 Bivariate Flood Frequency Analysis Using Copula Function Presented by : Dilip K. Bishwkarma (student,msw,ioe Pulchok Campus) ( Er, Department of Irrigation, GoN) 17 th Nov Outlines Importance of Multivariate analysis Introduction to Copula Case study Result Conclusion References 2 1

2 Importance of Multivariate analysis What will be the impact of the flow in structure? We should Consider impact of both Peak and Volume. Which is only can be done by Bivariate FFA. 3 Google Map 4 2

3 Importance of Multivariate analysis Some flood events with a peak of return period of 100-years could be less damaging than floods with both peak and volume of return period of 10- years.( Veronika et.al ). simple univariate approach could lead to severe underestimation of the risk associated to a given event [Raynal-Villasenor and Salas, 1987; Bruneau et al., 1994]. single-variable hydrological frequency analysis can only provide limited assessment of these events [Yue et al., 2001]. Importance of Multivariate analysis Univariate Not suficient to know peak only Kite, 1978; Cunnane, 1987; Rao and Hamed, 2000 Multivariate flood frequency analysis( Consider volume and duration ) Ashkar and Rousselle, 1982; Krstanovic and Singh, 1987; Sackl and Bergmann, 1987; Singh and Singh, 1991; Yue et al., 1999, 2001) but with many restriction? MultiVariate by Concept of Copula is best solution used recently (Favre et al., 2004; Zhang and Singh, 2006; Grimaldi and Serinaldi, 2006; Zhang and Singh, 2007) 6 3

4 Introduction what is Copula? Sklar s theorem (1959) is the foundation of the concept of the copula. Sklar showed that every n-dimensionaldistribution function H(x1,,xd) could be written as: H(x1,...,xd ) =C(F1(x1),...,Fd (xd )).(1) H(x, y) = C(F(x), F( y)) =C (u,v) For two dimensional (bivariate) distribution function If F(x) and F(y) are continuous, then the copula function C is unique and has the following representation: C(u, v) = H(F 1 (u), F 1 (v)) (2) 0 u, v 1, where the F 1 (u) and F 1 (v) are inverse distribution functions of the marginal.(nelsen, 2006). 7 Introduction Family of Copula : 1) Elliptical Copulas 2) The Farlie-Gumbel-Morgenstern family of copulas. 3) Normal copulas 4) Shuffles of M Family 5) Archimedean Copulas Among these family Archimedean one is the very popular class used in hydrological application (De Michele et al., 2005; Favre et al., 2004; Genest and Favre, 2007; Zhang and Singh, 2006) 8 4

5 Diagrammatic representation Data with Peak discharge, Volume and Duration Best fit Distribution & Finding PDF & CDF( F(x) & F(y) of Peak & Volume Best Selection of Copula &Find Joint CDF H(x, Y) Gumbel, Normal, Log Normal, Gringorten plotting position formula (Gringorten, 1963),RMSE, AIC (Akaike Information Criaterian ) Chi square test For Nonparametric kernel density estimation used for determination of PDF for hydrologic variables Lall et al. (1996), Find Return Period 9 PDF can be calculated by Kernel Function 10 5

6 11 Most frequently used Archimedean copulas in hydrology 12 6

7 Kendall s tau is also Given as Where C = no of Concordant And D= no of Discordant pair Schweizer and Wolff (1981) 13 Joint return period for two variables defined more authors (Salvadori, 2004; Salvadori and De Michele, 2006; Shiau, 2003) and it can be written in the form of: 14 7

8 Case study To illustrate the methodology developed in the present study for flood frequency analysis, 70 years ( ) of daily streamflow data for Red River at Grand Forks in North Dakota, US, is used. The data is collected from U. S. Geological Survey (USGS) gauging station ( ) (Karmakar, S., Simonovic, S.P., 2007.) 15 MATLAB Used for Analysis Case study Probability Density For peak & Volume 16 8

9 Case study Joint probability distribution volume. function for peakflow and 17 Case study Joint return period for peak flow and volume. 18 9

10 Conclusion In the present work an extensive selection of marginal distribution functions for flood variables is performed by parametric and nonparametric methods, and concept of copula is used for evaluating bivariate food frequency analysis with mixed marginal distributions. 19 References Anne-Catherine Favre, Salaheddine El Adlouni, Luc Perreault, Nathalie Thie monge, and Bernard Bobe e,multivariate hydrological frequency analysis using copulas Received 4 July 2003; revised 3 October 2003; accepted 21 G. Salvadori, Frequency analysis via copulas: Theoretical aspects and applications to hydrological events, WATER RESOURCES Jenq-Tzong Shiau & Reza Modarres & Saralees Nadarajah, Assessing Multi-site Drought Connections in Iran Using Empirical Copula, Environ Model Assess (2012) 17: DOI /s Jun Yan, Enjoy the Joy of Copulas: With a Package copula, Journal of Statistical SoftwareOctober 2007, Volume 21, Issue 4. Nelsen, R. B. (1999), An Introduction to Copulas, Springer, New York. Karmakar, S., Simonovic, S.P., Flood frequency analysis using copula with mixed distributions. Project report No The University of Western Ontario, Department of Civil and Environmental Engineering. London, Ontario, Canada 58 p. Karmakar, S., Simonovic, S.P., Bivariate flood frequency analysis. Part 1: Determination of marginals by parametric and nonparametric techniques. J. Flood Risk Manag., 1, RESEARCH, VOL. 40, W12511, doi: /2004wr003133, 2004 Salvatore Grimaldi, Francesco Serinaldi, Asymmetric copula in multivariate flood frequency analysis, Advances in Water Resources 29 (2006) Salvatore grimaldi & Francesco serinaldi, Design hyetograph analysis with 3-copula function, Hydrological Sciences Journal des Sciences Hydrologiques, 51(2) April Veronika Bačová Mitková, Dana Halmová,Joint modeling of flood peak discharges, volume and duration: a case study of the Danube River in Bratislava J. Hydrol. Hydromech., 62, 2014, 3, DOI: /johh

11 Thank You 21 11