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Bivariate Flood Frequency Analysis of Upper Godavari River Flows Using Archimedean Copulas

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Abstract

In this paper, a copula based methodology is presented for flood frequency analysis of Upper Godavari River flows in India. By using the specific advantages of copula method in modeling the joint dependence structure of uncertain variables, this study applies Archimedean copulas for frequency analysis of flood characteristics annual peak flow, flood volume and flood duration. To determine the best fit marginal distributions for flood variables, few parametric and nonparametric probability distributions are examined and the best fit model is adopted for copula modeling. Four Archimedean family of copulas, namely Ali-Mikhail-Haq, Clayton, Gumbel-Hougaard and Frank copulas are evaluated for modeling the joint dependence of annual peak flow-volume, and flood volume-duration pairs. The performance of two parameter estimation methods, namely method-of-moments-like estimator based on inversion of Kendall’s tau and maximum pseudo-likelihood estimator for copulas are investigated. On performing Monte Carlo simulation to assess the performance of copula distributions in modeling the joint dependence structure of flood variables, it is found that the developed copula models are well representing the observed flood characteristics. From standard statistical tests, Frank copula has been identified as the best fitted copula for both bivariate models. The Frank copula function is used for obtaining joint and conditional return periods of flood characteristics, which can be useful for risk based design of water resources projects.

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Authors and Affiliations

  1. Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    M. Janga Reddy & Poulomi Ganguli

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  1. M. Janga Reddy
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  2. Poulomi Ganguli
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Correspondence to M. Janga Reddy.

Appendix

Appendix

1.1 A1 Simulation of Random Samples from Bivariate Archimedean Copulas

The generation of random variates (u, v) from Clayton and Frank copulas involves the following steps:

  1. 1.

    Generate two independent uniform U(0,1) random variates u and q.

  2. 2.

    Set \( v = C_u^{{ - 1}}\left( {q|u} \right) \), where \( C_u^{{ - 1}}\left( {q|u} \right) \) is the inverse of conditional distribution \( {C_u}(v) = {{{\partial C\left( {u,v} \right)}} \left/ {{\partial u}} \right.} \). The functional relationships are given in Table 7.

    • Clayton and Frank copulas have closed form relationships for \( C_u^{{ - 1}}\left( {q|u;\theta } \right) \), whereas Gumbel-Hougaard (G-H) copula does not has closed form relation, hence \( C_u^{{ - 1}}\left( {q|u} \right) \) is solved numerically.

  3. 3.

    The pair (u, v) forms the random vector drawn from the respective copula family C(u, v; θ).

  4. 4.

    The desired (simulated) sample is given by \( \left( {x,y} \right) = \left[ {F_X^{{ - 1}}(u),F_Y^{{ - 1}}(v)} \right] \), where \( F_X^{{ - 1}}\left( \bullet \right) \) and \( F_Y^{{ - 1}}\left( \bullet \right) \) are inverse of marginal distributions F X (x) and F Y (y) respectively.

Table 7 Functional relationships for conditional and inverse of conditional distributions for bivariate Archimedean copulas
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Reddy, M.J., Ganguli, P. Bivariate Flood Frequency Analysis of Upper Godavari River Flows Using Archimedean Copulas. Water Resour Manage 26, 3995–4018 (2012). https://doi.org/10.1007/s11269-012-0124-z

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