Bivariate Frequency Analyses of flood and Sediment Discharge Using Copulas

Extended Abstract
Background and Objectives
Flood is a natural disaster that causes a lot of damage in different parts of the world every year, because it has caused a lot of financial and human losses in many countries. One of the main problems of floods is soil erosion and the production of sediments that are transported along the river during the flood and destroy the buildings on the banks of the river, reduce the capacity of canals and reservoirs of dams, and damage agricultural lands. Since the amount of sediment in a river is a random variable that is a function of several correlated random characteristics such as flow rate, suspended load and bed. Therefore, estimation of sediment load based on univariate probabilistic analysis is not a reliable criterion. For this reason, multivariate analyzes are of special importance. The bivariate behavior of the flood discharge and the resulting sediment load depends on their joint cumulative distribution function, which can be implemented with the help of copula functions. Since hydrological variables are multidimensional and copula functions allow multidimensional analysis of variables and give us more information about hydrological processes. Therefore, the use of copula functions can be an important step in promoting hydrological research. Based on this, the aim of the current research is to investigate the two-variable behavior of flood discharge and sediment load using copula functions and to analyze the frequency of both variables in the Minab watershed.

Methodology
In this study, Archimedean and elliptical copula functions were used as a tool for bivariate analysis of flood (Qw) and sediment discharge (Qs). First, marginal distribution functions were fitted on flood (Qw) and sediment discharge (Qs) variables. Then, using conventional correlation methods such as spearman, Pearson and Kendal tau, the correlation between the variables was checked. In the next step, elliptic copula (T-Student and Gaussian) and Archimedean (Frank, Joy, Clayton, Gamble) functions were fitted to the variables based on the maximum likelihood method, and the best fitted function was the two variables of flood discharge and sediment. The copula was normal. After the coupling of the variables, the single and double return period of 2 to 1000 years in "AND" and "OR" mode and in conditional modes with scenarios T(Qf/Qs<=qs) and T(Qf/Qs>=qs) It was calculated and finally two-variable return period analysis was done based on return period tables and graphs.

Findings
The results of the correlation test showed that there is a positive and significant relationship between the investigated variables in the study area. Based on the results of Chi-plot and Kendall-plot, there is an acceptable correlation between the two investigated variables. The results of the copula fit showed that the normal copula has an acceptable performance on the investigated variables. Examining the return period of flood discharge and sediment discharge showed that the changes in sediment discharge are far greater than the changes in flood discharge in higher return periods, so that with the increase in the return period, the amount of flood discharge increases exponentially. The AND scenario gives much higher values than the return period of OR and is a suitable tool for risk analysis of hydrological events. Also, among the conditional scenarios, the scenario T(Qf/Qs>=qs) showed greater values of the return period than T(Qf/Qs<=qs). Finally, the risk analysis in the studied area showed that the risk of flooding and sedimentation in the two-year return period with the AND and OR scenarios is without risk, but for the two conditional cases, it has a low risk. Or in the 20-year return period, the risk of the 50-year project for all scenarios except the OR scenario and the rest of the scenarios have predicted critical conditions.

Conclusion
The results of the return period analysis showed that in the case of "AND" return period, the return period values are much larger than the "OR" values. In the same way, it has a higher risk. Also, considering the conditional scenarios were also significantly different. Therefore, ignoring the correlation between flood discharge and sediment may significantly overestimate or underestimate the actual amount of sediment, as a result of which the probability of the corresponding occurrence increases or decreases. Therefore, depending on the purpose and needs in the region, the following scenarios can be used.