The G–H, Clayton, and Frank copulas of the Archimedean family (Table 3) were applied to construct the joint distribution of flood peak and volume. The parameters of bivariate copulas were estimated using the Kendall correlation coefficient (*τ*), and the expressions which describe the relationship between the estimated parameters and Kendall correlation coefficient are also shown in Table 3. The result of the Kendall tau is 0.66 in Geheyan reservoir. The results of the Cramer–von Mises test and K–S test *D*_{n} for the pair are presented in Table 4, and *P*-values were calculated based on the parametric bootstrap or multipliers procedure with 10,000 runs (Kojadinovic *et al.* 2011; Sraj *et al.* 2015). The root mean square error (*RMSE*) was also calculated and listed in Table 4. The graphical goodness-of-fit test between simulated (sample size 10,000) and observed values is plotted in Figure 5. On the basis of the statistical tests and graphical fitting (Figure 5), it is found that the G–H copula is much more suitable for constructing the joint distribution of peak discharge and flood volume. This finding is in accordance with other authors (e.g., Genest & Mackay 1986; Poulin *et al.* 2007; Sraj *et al.* 2015; Zhang *et al.* 2015b).

Table 3

Copula . | C(_{θ}u, v)
. | θ ∈
. | τ
. |
---|---|---|---|

G–H | [1, + ∞) | ||

Clayton | (0, + ∞) | ||

Frank | R\{0} |

Copula . | C(_{θ}u, v)
. | θ ∈
. | τ
. |
---|---|---|---|

G–H | [1, + ∞) | ||

Clayton | (0, + ∞) | ||

Frank | R\{0} |

Table 4

. | Parameter . | Statistical test . | . | |
---|---|---|---|---|

Copula . | θ
. | RMSE . | S^{1}_{n}(p − valve)
. | D_{n} (D_{n,0.95})
. |

G–H | 2.98 | 0.068 | 0.011 (0.44) | 0.035 (0.215) |

Clayton | 3.95 | 0.074 | 0.021 (0.17) | 0.048 (0.215) |

Frank | 9.93 | 0.071 | 0.015 (0.28) | 0.043 (0.215) |

. | Parameter . | Statistical test . | . | |
---|---|---|---|---|

Copula . | θ
. | RMSE . | S^{1}_{n}(p − valve)
. | D_{n} (D_{n,0.95})
. |

G–H | 2.98 | 0.068 | 0.011 (0.44) | 0.035 (0.215) |

Clayton | 3.95 | 0.074 | 0.021 (0.17) | 0.048 (0.215) |

Frank | 9.93 | 0.071 | 0.015 (0.28) | 0.043 (0.215) |

Figure 5

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