Introduction: A flood is a multivariate natural event with a two-way correlation among its variables, and the damage caused by it becomes more severe when a flood with a high peak flow, a large volume, and for a more extended period occurs. For this reason, the duration of floods is essential for the design of hydraulic structures related to floods (Salarpour, M., et al. 2013, 2015). However, the damages caused by floods are not only affected by these three variables; thus, it is necessary to investigate other variables affecting floods, such as sediment. The sediment changes the cross-section and reduces the volume of the river bed. Using copula functions for more than two variables causes extensiveness and complexity in calculations; therefore, Bad Ford and Cook recommended vine copulas (C-vine and D-vine) in 2001. Vine copulas have a high flexibility and can convert multivariate distributions into bivariate distributions, thus reducing the amount of calculations (Bedford and Cooke 2001, 2002). Shafaei et al. (2016) used the C-vine structure for four-dimensional modeling variables of peak discharge, volume, base time, and flood peak time. Latif and Mustafa (2020) investigated the flood in the Kelantan River basin using Vine copulas (C-Vine and D-Vine) and three variables: peak discharge, flood volume, and duration. Amini et al. (2021) used Vine structures (C-Vine and D-Vine) and variables of peak discharge, volume, base time, and time to reach the flood peak; they investigated the flood in Landi station in three-dimensional and four-dimensional ways.

This research uses Vine copulas and four variables of peak flow, volume, duration, and flood sediment to analyze the four variables of the flood phenomenon in the Qarasu River catchment.

Methodology: The catchment area of the Qarasu River is located northwest of the Karkhe basin, Fig. 1. To analyze the flooding process in this catchment, we extracted four essential characteristics of the flood, i.e., peak flow, volume, duration, and flood sediment first, by drawing the flood hydrograph and, with the help of equations 1 and 2. Then, we determined the most appropriate marginal distribution for these variables, made their pair models based on the Tau-Kendall correlation coefficient (Eq. 3) and copula functions, applied the Cramer von Mises test and the criteria of the logarithm of the likelihood function, root mean square error, and Akaike and Bayesian information (Eqs 12-17) to obtain their coefficient relationship and the most suitable copula function. The parameter of the copula function is also obtained from the maximum pseudo-likelihood method. All codes are written in R software. Then, Vine copula (C-Vine and D-Vine) were used for the four-variable modeling of these variables. Finally, the univariate and combined return periods are calculated and compared using the qualified copula functions. To compute the joint return period, among the existing methods, the joint return period methods in the {or} mode and the joint return period in the {and} mode are selected (Eq. 24-31).

Results and Discussion: Table 3 shows the statistical characteristics of the flood data for the variables of peak flow, volume, duration, and sediment in the Qarasu River catchment. Table 4 indicates that the best marginal distribution is the log-normal distribution for the peak flow variable and the Weibull distribution for the volume, duration, and sediment variables. Table 5 illustrates that the correlation between all pairs of variables was solid and positive. For all pairs of variables, the amount of the Kramer von Mises test and the criteria of the logarithm of the likelihood function, Akaike and Bayesian information, and the root mean square error were computed to determine the most appropriate copula function for each pair of variables, for example, Table 6 shows the results for the pair of peak discharge and flood volume. Then, a comparison was conducted between four dimensions of C-vine and D-vine structures, indicating that the C-vine structure performed better than the D-vine structure. Finally, calculating and comparing the single-variable and four-variables return periods determined that as the number of variables increases, the return period of {or} state decreases, but the return period of {and} state increases, Table 10. Using the return period in the case of {and} will increase the project's safety because the return period means more peak discharge and, hence, the construction of hydraulic structures with higher dimensions and higher strength.

The key of the analysis is that the sediment variable plays a role in the occurrence of floods in this basin, and the study of four variables for flood management in this region is more comprehensive than the analysis of two and three variables.

Conclusion: The results show that the four-variable flood analysis is more accurate than the analysis of two and three variables, and it effectively describes the relationship between the sediment variable and the variables of peak discharge, volume, and duration. Moreover, the sediment has an essential effect on floods in this area. Furthermore, the joint return period {or} state is smaller than the joint return period of {and} state and the univariate return periods. Therefore, the joint return period of {and} is preferred in large projects requiring high safety. In general, using four or more variables to simulate the correlation between flood variables with high accuracy is suggested. Also, using the Vine copula is very efficient for multivariate flood analysis.

Keywords: Qarasu River, Flood characteristics, Sediment, Family of elliptic and Archimedean copulas, return period.