Abstract: The catchment area of Qarasu river is always exposed to floods. To prevent possible damages, factors affecting the occurrence of floods in this area should be investigated. In this research, the dependence between the two main flood variables of peak flow and volume was determined during a statistical period of 41 years using copula functions. The results showed that the most suitable marginal distributions for both variables are Log-normal and Weibull, respectively. moreover, normal copula function is the most capable function to depict the structure of dependence between them, according to goodness of fit tests. Finally, applying the selected copula illustrated that the joint return periods in the state {or} is less than the joint return periods in the {and} and single variable state.

Keywords: Flood hydrograph, Bivariate analysis, Goodness of fit, Univariate return period, Joint return periods.

Introduction: Flooding is a natural event that if not managed properly, it will pose a serious risk (Stedinger, 1993). Hydrological processes including the flood phenomenon are often multidimensional. The classic multidimensional models where the variables are dependent do not lead to precise results, the use of copula functions is recommended (Afsharypour, z., et al., 2018; Chebana, F., & Ouarda, T. B., 2011; Favre, A., et al., 2004; Papaioannou, G., et al., 2016; Salvadori, G. and C. De Michele, 2004; Sklar, 1959; Sraj, M et al., 2015). Although, copula functions introduced by Sklar in 1959, Dirmichele and Salvadori (2003) used copula functions in hydrology to model the dependence between the two variables of rainfall intensity and duration. Salari et al. (2015) investigated the bivariate flood in Karon river using copula functions. For this purpose, three flood variables, i.e., peak flow, volume and continuity of the flood were considered two by two (peak flow-volume, continuity-volume and peak flow-continuation). The bivariate copula functions applied for suding the flood behavior at Danba station in China using Archimedesian copula functions and bivariate peak discharge and flood volume (Chen, Y. and P. Lin, 2016), the effect of flood on the dam overflow (Afsharypour, z., et al., 2018).
The catchment area of Qarasu river is always exposed to floods, this research applies a bivariate copula function considering the peak discharge and flood volume to investigate the flood behavior and return period of flood in Qarasu catchment.

Methodology: The catchment area of Qarasu river eperincing floods, located in the North-West of Karkheh basin, Iran (Fig. 1); has experienced flood. We applied a bivariate Capula function, governing by Eq. 1, of peak flow and volume of flood measured at Doab-Merg to study the flood phenomenon in the cachment. We began with determining the flood characteristics (peak discharge and volume) by drawing the flood hydrograph in Excel software, a schematic hydrograph illustrated in Fig. 2. Then, we qualified the most appropriate marginal distribution for the both variables. It was followed by choosing the most suitable copula function according the best correlation between the two variables of peak discharge and flood volume. Correlation between the variables was obtained by Tau Kendall's correlation coefficient relationship and the most suitable copula function was ditermined based on the Kramer von Mises test and the criteria of the logarithm of the likelihood function, root mean square error, and Akaike and Bayesian information, used Eqs 8-13. The detailed function parameter is also obtained from the maximum pseudo-likelihood method. All required codes were provided in R software. Finally, applying the qualified copula functions, the single variable and bivariate returns are calculated and compared with each other. To calculate the joint return period, among the existing methods, the methods of joint return period in the {or} state and joint return period in the {and} state are selected.
Results and Discussion: Table 2 shows statistical characteristics of aquired flood data for the main characteristics of floods, peak flow and volume, in the catchment area of Qarasu river. The best marginal distributions of peak flow and volume variables, are log-normal and Weibel distributions, respectively, Table 3. Furthermore, the fitness analysis of Clayton, Gamble, Normal (Gaussi), Joe, Frank and T copula functions using Kramer von Mises test and the criteria of logarithm of likelihood function, root mean square error, Akaike information and Bayesian (Table 4), lead to found that the normal copula function is the best copula function to show the dependence structure between peak discharge and flood volume variables. Applying the bivariate-normal copula function resulted in the joint return period in the {or} state is smaller than the joint return period in the {and} state and the single variable return period, Table 5. On the contrary, the joint return period in the {and} state is lower than the {or} state and the single variable return period, Table 5.
In general, if in a project safety takes precedence over cost, it is better to design based on the joint return period in the {or} state, and if the cost takes precedence over safety, it is better to design based on the joint return period in the {and} state. Because in {or} state, when one of the flood variables exceeds its threshold, it considers that flood as dangerous and is ready to dream with more possibilities. But in the case of a flood, it is dangerous if both variables exceed their thresholds. These results have been confirmed in the past by other researchers such as (Afsharypour, z., et al., 2018) and (Rahimi, L., et al. 2014).
Finally, by comparing the results obtained from the return period in univariate and bivariate mode, we can see that the univariate analysis of flood is not an accurate and comprehensive analysis, and for proper analysis, the use of bivariate and more will give us better results, because flood is a multivariate phenomenon. It is next and its variables affect each other. Therefore, it is suggested to use more variables for flood analysis in future researches.

Conclusion: Comparing the results obtained from univariate and bivariate modes shows that the normal-bivariate flood analysis is crucially precise and is efficiently describe the dependence between the flood variables of peak flow and volume. Thus, it is suggested to use bivariate or more to simulate the correlation between flood variables more accurately. Moreover, the joint return period in the {or} state is smaller than the joint return period in the {and} state and the single variable return period, which is convenient to design where safety is prioritized over cost in a project.