Bivariate Rainfall and Runoff Analysis Using Entropy and Copula Theories

Entropy - Tập 14 Số 9 - Trang 1784-1812
Lan Zhang1, Vijay P. Singh2,3
1Department of Civil Engineering, University of Akron, Akron, OH 44325, USA
2Department of Biological and Agricultural Engineering, Texas A&M University, College Station, TX 77843, USA
3Department of Civil and Environmental Engineering, Texas A&M University, College Station, TX, 77843, USA

Tóm tắt

Multivariate hydrologic frequency analysis has been widely studied using: (1) commonly known joint distributions or copula functions with the assumption of univariate variables being independently identically distributed (I.I.D.) random variables; or (2) directly applying the entropy theory-based framework. However, for the I.I.D. univariate random variable assumption, the univariate variable may be considered as independently distributed, but it may not be identically distributed; and secondly, the commonly applied Pearson’s coefficient of correlation (g) is not able to capture the nonlinear dependence structure that usually exists. Thus, this study attempts to combine the copula theory with the entropy theory for bivariate rainfall and runoff analysis. The entropy theory is applied to derive the univariate rainfall and runoff distributions. It permits the incorporation of given or known information, codified in the form of constraints and results in a universal solution of univariate probability distributions. The copula theory is applied to determine the joint rainfall-runoff distribution. Application of the copula theory results in: (i) the detection of the nonlinear dependence between the correlated random variables-rainfall and runoff, and (ii) capturing the tail dependence for risk analysis through joint return period and conditional return period of rainfall and runoff. The methodology is validated using annual daily maximum rainfall and the corresponding daily runoff (discharge) data collected from watersheds near Riesel, Texas (small agricultural experimental watersheds) and Cuyahoga River watershed, Ohio.

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Tài liệu tham khảo

Haan, C.T., and Wilson, B.N. (, January May). Another look at the joint probability of rainfall and runoff. Hydrologic Frequency Modeling, Proceedings of the International Symposium on Flood Frequency and Risk Analyses, Baton Rouge, LA, USA.

Singh, 1991, Derivation of bivariate probability density functions with exponential marginals, Stoch. Hydrol. Hydraul., 5, 55, 10.1007/BF01544178

Yue, 1999, The Gumbel mixed model for flood frequency analysis, J. Hydrol., 226, 88, 10.1016/S0022-1694(99)00168-7

Yue, 2001, A bivariate extreme value distribution applied to flood frequency analysis, Nord. Hydrol., 32, 49, 10.2166/nh.2001.0004

Yue, 2001, The bivariate lognormal distribution to model a multivariate flood episode, Hydrol. Processes, 14, 2575, 10.1002/1099-1085(20001015)14:14<2575::AID-HYP115>3.0.CO;2-L

Pegram, 2009, Copula based multisite model for daily precipitation simulations, Hydrol. Earth Syst. Sci., 13, 2299, 10.5194/hess-13-2299-2009

Evin, 2008, A new rainfall model based on the Neyman-Scott process using cubic copulas, Water Resour. Res., 44, W03433, 10.1029/2007WR006054

Salvadori, 2005, Bivariate statsitical approach to check adequacy of dam spillway, J. Hydro. Eng., 10, 1084

Favre, 2004, Multivariate hydrological frequency analysis using copulas, Water Resour. Res., 40, W01101, 10.1029/2003WR002456

Genest, 2007, Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data, Water Resour. Res., 43, W09401, 10.1029/2006WR005275

Genest, 2007, Everything you always wanted to know about copula modeling but were afraid to ask, J. Hydro. Eng., 12, 347, 10.1061/(ASCE)1084-0699(2007)12:4(347)

Grimaldi, 2006, Asymmetric copula in multivariate flood frequency analysis, Adv. Water resour., 29, 1155, 10.1016/j.advwatres.2005.09.005

Kao, 2007, A Bivariate frequency analysis of extreme rainfall with implications for design, J. Geophys. Res., 112, D13119, 10.1029/2007JD008522

Kao, 2007, Probabilistic structure of storm surface runoff considering the dependence between average intensity and storm duration of rainfall events, Water Resour. Res., 43, W06410, 10.1029/2006WR005564

Serinaldi, 2009, Probabilistic characterization of drought properties through copulas, J. Phys. Chem. Earth, 34, 596, 10.1016/j.pce.2008.09.004

Song, 2010, Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data, Stoch. Environ. Res. Risk Assess, 24, 425, 10.1007/s00477-009-0331-1

Song, 2010, Frequency analysis of droughts using the Plackett copula and parameter estimation by genetic algorithm, Stoch. Environ. Res. Risk Assess, 24, 783, 10.1007/s00477-010-0364-5

Vandenberghe, 2010, Fitting bivariate copulas to the dependence structure between storm characteristics: A detailed analysis based on 105 year 10 min rainfall, Water Resour. Res., 46, W01512, 10.1029/2009WR007857

Vandenberghe, 2011, A comparative copula-based bivariate frequency analysis of observed and simulated storm events: A case study on Bartlett-Lewis modeled rainfall, Water Resour. Res., 47, W07529, 10.1029/2009WR008388

Wang, 2009, Copula-based flood frequency (COEF) analysis at the confluences of river systems, Hydrol. Process., 23, 1471, 10.1002/hyp.7273

Zhang, 2006, Bivariate flood frequency analysis using the copula method, J. Hydrol. Eng., 11, 150, 10.1061/(ASCE)1084-0699(2006)11:2(150)

Zhang, 2007, Bivariate rainfall frequency distributions using Archimedean copulas, J. Hydrol., 332, 93, 10.1016/j.jhydrol.2006.06.033

Zhang, 2007, Gumbel-Hougaard copula for trivariate rainfall frequency analysis, J. Hydrol. Eng., 12, 409, 10.1061/(ASCE)1084-0699(2007)12:4(409)

Zhang, 2007, Trivariate flood frequency analysis using the Gumbel-Hougaard copula, J. Hydrol. Eng., 12, 431, 10.1061/(ASCE)1084-0699(2007)12:4(431)

Agrawal, 2005, Maximum Entropy-based Conditional Probability Distribution Runoff Model, Biosystem Eng., 90, 103, 10.1016/j.biosystemseng.2004.08.003

Hao, 2011, Single-site monthly streamflow simulation using entropy theory, Water Resour. Res., 47, W09528, 10.1029/2010WR010208

Krstanovic, 1993, A Real-Time Flood Forecasting Model Based on Maximum-Entropy Spectral Analysis: I. Development, Water Resour. Mgmt., 7, 109, 10.1007/BF00872477

Krstanovic, 1993, A Real-Time Flood Forecasting Model Based on Maximum-Entropy Spectral Analysis: II. Application, Water Resour. Mgmt., 7, 131, 10.1007/BF00872478

Singh, 1987, A stochastic model for sediment yield using the principle of maximum entropy, Water Resour. Res., 23, 78l, 10.1029/WR023i005p00781

Chang, 2010, Wind speed and power density analysis based on mixture weibull and maximum entropy distributions, Int. J. Appl. Sci. Eng., 8, 39

Papalexiou, 2012, Entropy based derivation of probability distributions: A case study to daily rainfall, Adv. Water Resour., 45, 51, 10.1016/j.advwatres.2011.11.007

Singh, V.P. (1998). Entropy-Based Parameter Estimation in Hdyrology, Kluwer Academic Publishers.

Singh, 2010, Entropy theory for derivation of infiltration equations, Water Resour. Res., 46, W03527, 10.1029/2009WR008193

Singh, 2010, Entropy theory for movement of moisture in soils, Water Resour. Res., 46, W03516, 10.1029/2009WR008288

Singh, 2010, Derivation of rating curves using entropy theory, Trans. ASABE, 53, 1811, 10.13031/2013.35807

Singh, 2011, Hydrologic synthesis using entropy theory: Review, J. Hydrol. Eng., 16, 421, 10.1061/(ASCE)HE.1943-5584.0000332

Shannon, 1948, The mathematical theory of communications, Bell Syst. Tech. J., 27, 379, 10.1002/j.1538-7305.1948.tb01338.x

Jaynes, 1957, Information theory and statistical mechanics, I, Phys. Rev., 106, 620, 10.1103/PhysRev.106.620

Jaynes, 1957, Information theory and statistical mechanics, II, Phys. Rev., 108, 171, 10.1103/PhysRev.108.171

Kapur, J.N. (1989). Maximum Entropy Models in Science and Engineering, John Wiley & Sons INC.. [1st ed.].

Kapur, J.N., and Kesavan, H.K. (1992). Entropy Optimization Principles with Applications, Academic Press. [1st ed.].

Zellner, 1988, Calculation of maximum entropy distributions and approximation of marginal posterior distributions, J. Econometrics, 37, 95, 10.1016/0304-4076(88)90002-4

Sklar, 1959, Fonctions de repartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229

Nelsen, R.B. (2006). An Introduction to Copulas, Springer Science+Business Media, Inc. [2nd ed.].

Poulin, 2007, Important of tail dependence in bivariate frequency analysis, J. Hydrol. Eng., 12, 394, 10.1061/(ASCE)1084-0699(2007)12:4(394)

Abberger, 2005, A simple graphical method to explore tail dependence in stock-return pairs, Appl. Financial Economics, 15, 43, 10.1080/0960310042000280429

Frahm, 2005, Estimating the tail dependence coefficient: properties and pitfalls, Insur. Math. Econ., 37, 80, 10.1016/j.insmatheco.2005.05.008

Cole, 1999, Dependence measures for extreme value analysis, Extremes, 2, 339, 10.1023/A:1009963131610

Joe, H. (1997). Multivariate Models and Dependence Concepts, Chapman & Hall/CRC. [1st ed.].

Genest, 1997, Bivariate distributions with given extreme value attractor, J. Multivariate Anal., 72, 567

Salvadori, 2011, On the return period and design in a multivariate framework, Hydrol. Earth Syst. Sci., 15, 3293, 10.5194/hess-15-3293-2011

Miller, 1956, Table of percentage points of Kolmogorov statistics, J. Am. Stat. Assoc., 51, 111, 10.1080/01621459.1956.10501314

NIST Engineering statistics handbook, Available online: http://www.itl.nist.gov/div898/handbook/.

Genest, 2006, Goodness-of-fit procedures for copula models based on the integral probability transformation, Scand. J. Stat., 33, 337, 10.1111/j.1467-9469.2006.00470.x

Genest, 2009, Goodness-of-fit tests for copulas: A review and a power study, Insur. Math. Econ., 44, 199, 10.1016/j.insmatheco.2007.10.005

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Entropy

Tập 14 Số 9

1784-1812

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