Uncertainty analysis in systems evaluation and management has been considered as a logical aspect in engineering estimates in recent decades. Uncertainty generally implies that there is no complete knowledge of the behavior of a system and the specific values ​​of its variables. Currently, the uncertainty problem has been one of the topics of interest in research, decision-making and design in the field of water science and engineering. Furthermore, field observational evaluations have shown that hydrological phenomena are inherently random and can follow probabilistic functions. In the past, univariate probability models have been used to predict hydrological events. But today, it has been shown that hydrological phenomena such as precipitation, runoff, flood, and drought are random and multivariate and should be expressed by the characteristics of intensity, duration and amount. The efficiency of copula functions has been proven as an effective technique in multivariate analysis of hydrological events. This study was aimed to develop the probabilistic-fuzzy framework according to the existing gap in the previous studies. Therefore, long-term rainfall and runoff information were analyzed for generating the decision-making system in Barz Plain, Khuzestan province, Iran. Four parameters of maximum rainfall rate and depth, peak flow and volume of runoff have been analyzed to generate the time series of information.
The developed conceptual structure could be addressed in four steps. In the first step, an attempt was made to analyze the relationship between the flood hydrographs in the period from 1973 to 2018 in Barz Plain. Barz plain is located in the eastern north of Khuzestan province, Iran within 31° 18' to 31° 28' latitude and 50° 18' to 51° 26' longitude. Next, a probabilistic model was developed by the MATLAB program. In this model, marginal distribution functions, bivariate frequency analysis and calculation of return periods for rainfall characteristics were evaluated. Moreover, a fuzzy set analysis sub-program was prepared to transfer the effects of rainfall to the runoff. This process was developed within the framework of a multi-objective optimization program that was solved using the NSGAII method. The hydrological information used in this study includes the amount of rainfall and runoff recorded in daily time steps. Analysis of available data revealed about 37 rainfall events that led to the flood. Rainfall characteristics (maximum daily rate (mm/day) and cumulative rainfall depth in one event (mm)) and two characteristics of runoff hydrograph (hydrograph peak flow (m³/s) and runoff volume (million cubic meters (MCM)) were calculated from daily rainfall and runoff information. Therefore, a probabilistic decision model based on copula multivariate functions was developed to predict the variables at different return periods. The relationship between rainfall rate and depth with peak hydrograph flow and runoff volume for flood events over a 37-year period was formulated through fuzzy set theory. The feasible domain of the fuzzy problem was searched using a multi-objective optimization genetic algorithm based on the non-dominated sorting to find the extreme points. The obtained solutions were used as a fuzzy response to calculate the runoff of the Barz plain in Khuzestan province in southwestern Iran.
The relationship between rainfall and runoff characteristics showed that the maximum rainfall rate and the peak runoff discharge can be inspired by fuzzy theory and create a logical relationship. The results obtained by Chi-Squared, Anderson Darling and Kolmogorov Smirnov tests for the marginal frequency functions indicated that the Generalized Gamma, Log Normal, Generalized Extreme Values, and Log Pearson functions were the best options for estimating the maximum rate and depth of rainfall and peak flow and volume of runoff, respectively. Correlation coefficient was calculated to evaluate the bivariate model performance for the variables. The results of the maximum likelihood estimator to determine the superior joint function showed that Archimedean Clayton copula fited better than others for rainfall characteristics. Based on the developed concepts, the predicted runoff for Barz plain was estimated between 650 to 850 MCM for a 100-year return period. This return period is recommended for reservoir dam design. Consequently, in the 25-year return period, which is an appropriate time scale for flood diversion systems of water storage reservoirs, flood volume of 140 to 173 MCM with a maximum flow of 850 m³/s has been obtained. This flood can be caused by a rainfall of 78 mm/day or a depth of 137 mm.