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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 Dn 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 functions and Kendall correlation coefficient

CopulaCθ(u, v)θτ
G–H  [1, + ∞)  
Clayton  (0, + ∞)  
Frank  R\{0}  
CopulaCθ(u, v)θτ
G–H  [1, + ∞)  
Clayton  (0, + ∞)  
Frank  R\{0}  
Table 4

Estimated parameters of copulas and statistical tests' results

 ParameterStatistical test
CopulaθRMSES1n(pvalve)Dn (Dn,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) 
 ParameterStatistical test
CopulaθRMSES1n(pvalve)Dn (Dn,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

Comparison between the observed and simulated values for different copula functions.

Figure 5

Comparison between the observed and simulated values for different copula functions.

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