Abstract: The multi-dimensionality of the flood phenomenon causes us to not be able to get an accurate estimation of flood characteristics with one-variable analysis. Moreover, multi-dimensional analysis usually includes the variables of peak flow, volume, and duration of flood. Meanwhile, other variables, such as sediment, can affect the occurrence of floods. This research examined the effect of sediment on the occurrence of floods using copula functions of two variables of volume and flood sediment in the Qarasu River catchment in a statistical period of 41 years. The results revealed that the Weibull distribution is the most suitable marginal distribution, and the Frank copula function can well create the bivariate distribution of volume and flood sediment. Furthermore, the estimated return period of the bivariate showed that the sediment parameter has a significant effect on the return period. However, it is decreased in the case of {or} and increased in the case of {and}.

Introduction: Flood frequency analysis is necessary for properly designing hydraulic structures related to flooding and reducing damages caused by it (Nashwan et al., 2018). In the past, only a single variable analysis of flood peak discharge was used to investigate the probable flood hazard. While the hydrological processes, including the flood phenomenon, are usually multi-dimensional and by studying them individually, due to the dependence of these variables together, correct results will not be obtained (Cunnane,1988; Laio et al., 2011; Dawdy et al., 2012). For this multivariate analysis, it is recommended to use the copula functions presented by Sklar in 1959 (Afsharypour et al., 2018; Favre et al., 2004; Papaioannou et al., 2016; Salvadori, and De Michele, 2004; Sraj et al., 2015). These functions were first used in 2003 by De-Michele and Salvadori (2003) to analyze rainfall (with two variables, intensity, and duration of rainfall) in hydrology. After that, they were also used to analyze drought and flood phenomena (Shiau, 2006; Omidi et al., 2010; da Rocha et al., 2020; Poonia et al., 2021).
Zhang and Singh (2006) used copula functions and three variables of peak discharge, flood volume, and duration in order to analyze the frequency of two flood variables. Chen and Lin (2016), Using Archimedean copula functions and the data of two variables, peak flow and flood volume, investigated the frequency of bivariate discontinuous floods in Dadu River, China. In order to investigate the risk of flooding in Arizona, Zhong et al. (2020) used the data of peak discharge, duration, rainfall indicators, the family of Archimedean and elliptic joint functions, and the Apriori Algorithm.
An overview of the conducted research indicates that 1) the use of copula functions to analyze the frequency of flood phenomenon is very efficient, and 2) so far, the frequency analysis of two flood variables has not been done by using two variables of flood, volume and sediment, and copula functions. Sediment reduces the river transfer capacity and the efficiency of flood control structures (Hooke, 2019; Yuntian et al., 2019).
In this research, elliptic and Archimedean copula functions are used to analyze the flood frequency caused by the two variables of flood volume and sediment in the Qarasu River catchment.

Methodology: The catchment area of the Qarasu River is in the west of Iran (‎Fig. 2). For the bivariate analysis of floods in this basin, we first drew a schematic hydrograph like ‎Fig. 2 using Excel software. Then, we determined the most appropriate marginal distribution for both variables with the help of Easyfit software. After that, we obtained the correlation between the variables through Kendall tau (Eq. ‏(2)) and Spearman (Eq. ‏(3)) correlation coefficients. Then, we fitted all elliptic and Archimedean copula functions on this pair of variables, and by using the Cramer von Mises test and the criteria of the logarithm of the likelihood function, the Root Mean Square Error and Akaike and Bayesian information (Eqs. ‏(14)-‏(19)) we determined the most suitable copula function. Finally, using the selected copula function, we obtained the bivariate return period in two states {or} and {and} (Eqs. ‏(24) and ‏(25)) and compared it with the single variable return period (Eqs. ‏(20) and ‏(21)). All these steps are done using coding in R software.
Results and Discussion: To determine the values of flood volume and sediment variables, the daily flow data from 1974 to 2004 belonging to Doab-Merg station were used, and the statistical characteristics of these variables are summarized in ‎Table 3. ‎Table 4 illustates that the Weibull distribution the best marginal distributions for both flood volume and sediment variables. The positive and strong correlation between the two variables of volume and flood sediment (‎Table 5) showed that the method of copula functions can be used to model these two variables. The magnitudes of Cramer von Mises test and the logarithm of the likelihood function, the Root Mean Square Error, and Akaike and Bayesian information, show the Frank copula function is a more appropriate function in displaying the dependence structure between the two variables of volume and flood sediment (‎Table 6). Then, using the Frank copula function, the common return period was calculated in {or} and {and} modes. It was found that the common return period in the {and} state is greater than the {or} state and the single variable return period and vice versa (‏جدول 7). This is a sign of the variable impact of sediment on the flood return period.
Finally, by analyzing the results obtained from the return period in univariate and bivariate mode, it can be said that the use of bivariate analysis of the flood phenomenon is much more efficient than the univariate mode and reduces possible flood risks. Therefore, it is suggested that more variables be used in future research to analyze the flood phenomenon.

Conclusion: Overall, the results show that 1) Flood bivariate analysis is very efficient because it well describes the dependence between the volume and sediment variables of the flood, and the more the number of these variables increases, the more detailed the analysis. 2) In addition, the sediment variable has an effect on the occurrence of floods in the Qarasu catchment area. Because it has a positive and strong correlation with the flood volume variable, by considering it, the common return period in the {and} state became larger than the common return period in the {or} state and the single variable return period. 3) it is also more suitable for designing aspects to use the results of the return period in {and} mode.