Calculation of Hydrological Random Variables and Probability Distribution Based on Copula Function

Songbai Song, Xiaojun Wang

Abstract


Hydrological random variables and distribution calculation are important contents for deducing design flood area composition, design flood downstream of cascade reservoirs, etc. They are of great importance for planning, design and management of water conservancy and hydropower projects downstream of reservoirs, and urban flood control risk assessment, etc. The traditional hydrological random variables and distribution are derived from the function distribution of two-dimensional variables. The marginal distribution must be of the same type, and its application is limited. According to the definition of two-dimensional random variables and probability distribution, this paper uses the Copula function and product variation to change the original principle, and strictly deduces the calculation formula of two-dimensional phase distribution according to machine variation and probability.  The distribution probability calculation formula of the sum of variables under the two commonly used marginal distributions of Gamma distribution and P-Ⅲ distribution is only one-dimensional integration of conditional Copula function, thus avoiding information distortion of data conversion by probability combination discrete summation method and overcoming the requirement of traditional multivariate distribution that the marginal distribution is of the same type. Taking the 3h flood volume composition from Shuibuya Reservoir to Geheyan Reservoir in Qingjiang River Basin as an example, the calculation method of hydrological random variables and distribution is given. The model and calculation method in this paper are expected to provide theoretical support for the composition of design flood areas and the calculation of design flood downstream of cascade reservoirs in China.

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References


Ministry of Water Resources, Yangtze River Water Conservancy Commission, Hydrographic Bureau, Ministry of Water Resources, Nanjing Institute of Hydrology and Water Resources. Design Flood Calculation Manual for Water Conservancy and Hydropower Projects [M]. Beijing: China Water Conservancy and Hydropower Publishing House, "2001.

Zhang Yuan Jubilee. Independent Quantity GAMMA Joint Distribution Function of Sum and Difference of Distribution Random Variables [J]. Journal of Hefei University of Technology, "1983(3):53-59 .

Huang Nong. A Combined Frequency Numerical Method for Calculating the Sum of Two Independent γ Distribution Random Variables [J]. Journal of Hefei University of Technology, "1987,9(5):110-115 .

Xie Xiaoping, Huang Lingzhi, Xi Qiuyi, et al. Research on the Formation of Flood Control Zones Based on JC Method [J]. Journal of Hydrodynamics, 2006,25(6):125-129 .

Huang Lingzhi, Xie Xiaoping, Huang Qiang, et al. JC Method [J]] Journal of Natural Disasters, 2006,15(4): "in Research on Design Flood Area Composition of Cascade Reservoirs 163-167 .

Yan Baowei, Lei Shizhong, Hu Si-yi, et al. Study on Design Flood Area Composition Based on Copula Function [J]. Journal of Hydroelectric Power Generation, 2010,29(6):60-65 .

Li Tianyuan, Guo Shenglian, Liu Zhangjun, et al. Research on Method of Calculating Flood Water under Cascade Water Reservoirs [J]. Journal of Hydrology, 2014,45(6):641-648 .

Liu Zhangjun, Guo Shenglian, Li Tianyuan, et al. Research on the Establishment of Flood Control Zones for Cascade Water Reservoirs Based on the Copula Function [J]. Research on Water Resources.2014,3(2):124-135 .

Liu Zhangjun, Guo Shenglian, Li Tianyuan, et al. Study on the Method of Inter-district Estimation Based on Setting up Flood Control Zones [J]. Journal of Hydrology, 2015,46(5):543-549 .

Liu Zhangjun, Guo Shenglian, Li Tianyuan, et al. General formula for calculating the most probable area composition of design flood of cascade reservoirs [J]. advances in water science, 2014,25(4):549-558 .

Li Fei, Guo Shenglian, Li Tianyuan, et al. Study on Composition Method of Discontinuous Series Design Flood Areas [J]. Hydroelectric Energy Science, 2011,29(5):47-49 .

NADARAJAH S . Products and ratios for a bivariate gamma distribution[J]. Journal of Applied Mathematics and Computing,2005,24:581-595 .

NADARAJAH S . A bivariate distribution with gamma and beta marginals with application to drought data[J].Journal of Applied Statistics,2009,36:277-301 .

NADARAJAH S . A bivariate gamma model for drough[t J]. Water Resources Research,2007,43:749-759 .

NADARAJAH S . A bivariate Pareto model for drough[t ment,2008,23:811-822 .J]. Stochastic Environmental Research and Risk Assess⁃

NADARAJAH S,GUPTA A K . Cherian’s bivariate gamma distributioins as a model for drought data[J]. Agro⁃ ciencia-Mexico,2006,40:483-490 .

NADARAJAH S,GUPTA A K . Friday and Patil’s bivariate exponential distributioin with application to drought data[J]. Water Resources Management,2006,20:749-759 .

NADARAJAH S,GUPTA A K . Intensity-duration models based on bivariate gama distributions[J]. Hiroshima Mathematical Journal,2006,36:387-395 .

NADARAJAH S,KOTZ S . Sums,products and ratios for Downton’s bivariate exponential distribution[J]. Sto⁃ chastic Environmental Research and Risk Assessment,2006,20:164-170 .

Huang Zhenping, Chen Yuanfang. Hydrostatistics [M]. Beijing: China Water Resources and Hydropower Press, 2011.

Song Songbai, Cai Huanjie, Jin Juliang, et al. Copulas Function Theory and Its Application in Hydrology [M]. Beijing: Science Press, 2012.

NELSEN R B . An Introduction to Copulas[M]. Springer,New York,1999 .

Xu Shiliang. FORTRAN Common Algorithm Assembly [M].2 Edition. Beijing: Tsinghua University Publishing House, 1997.

Zou Hai. New Calculation Method in Optimal Design [M]. Beijing: New Times Publishing House, 1982.


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