This study aims to analyze the frequency of bivariate precipitation characteristics using Copula functions. for this purpose, daily rainfall data of Aloni station located in Khanmirza plain during the statistical period of 1986-2012 were used. After evaluating the rainfall events recorded at Aloni station in the study period (763 events), rainfall duration, rainfall depth, and then rainfall intensity of the events were calculated. Studies show that in the study area, usually rainfall events with an intensity of 5 mm/hr and more lead to floods, so in this study, the events that led to floods were selected to continue the calculations. Then, the common distributions in hydrology were fitted to each of the rainfall characteristics (duration, intensity, depth of rainfall) and the distributions that had the best fit to each of the rainfall characteristics were selected. Then, ten Copula functions were used to create a multivariate distribution of rainfall characteristics.
In this study, at first rainfall characteristics such as intensity, duration and depth were extracted for rainfall data leading to floods. Then the common margin distribution functions in hydrology were fitted to the characteristics. Then, after selecting the best margin distribution to create the cumulative distribution function (CDF) to create the multivariate distribution of rainfall characteristics, fitting the Copula functions of Clyton, Ali-Mikhaiel-Haq, Farli-Gumble-Morgan Stern, Frank, Galambos, Gamble-Hauggard, Placket, Filip-Gumble, Joe, and Gumble-Barnett on the mentioned variables were studied in pairs and for each pair of precipitation characteristics, the best Copula function was determined by comparing with the corresponding values of the empirical Copula. Then, using good criteria, the fit of the best Copula function for rainfall characteristics was determined. Since the condition for using Copula functions is the existence of a correlation between the studied features, so using Spearman, Pearson, and Kendall correlation coefficients, the correlation between the features was investigated also the cases of joint and conditional return periods, both probability and conditional and Kendall return period, which is basic concepts for analysis based on Copula functions, were evaluated.
The results of the analysis showed that the general extreme value distribution function (GEV) on rainfall characteristics (intensity, duration, depth) was known as the best distribution function and the results of the goodness of fit test showed that the Joe Copula function as The superior Copula function is based on the characteristics (intensity and duration) and (intensity and depth) and the Farli Gumble Morgan Stern Copula function was known as the superior Copula function on the depth and duration characteristics of rainfall. The results of both probability and conditional probability showed that when the flooding rainfall is 8 hours, the probability level will be 45 mm for the probability level of 0.2 and the probability of precipitation for the same level for the duration of is not necessary. It can be omitted15 hours. It will be 51 mm. The results of the Joint return period for “and” state showed that for the depth of rainfall of 60 mm and the intensity of rainfall of 60 mm/hr., the return period in the "and" state is less than 20 years. Based on the "or" mode for the same amount of intensity and depth of rainfall, the return period is less than 10 years (about 6 years). For a 25-year return period, provided the duration of the rainfall is 12.5 hours or more, the rainfall depth will be 75 mm.
Based on the results of comparing the values of theoretical Copulas with the corresponding values of empirical probability, the Joe Copula function was recognized as the superior Copula function to create a bivariate distribution of rainfall intensity and depth characteristics, as well as a pair of rainfall intensity and duration characteristics. Farli- Gumble - Morgan Stern Copula had a better fit for rainfall duration and depth data. Then, using superior fitted Copula functions, useful information such as probabilistic and conditional probability as well as joint and conditional return periods were extracted. The maximum rainfall depth recorded at Aloni station was 114.7 mm and its duration was 14.40 hours. The seasonal "or" is 60 years old. The results of the joint and conditional return periods in this study have been widely used in hydrological and water resources studies, including flood risk analysis, drought, watershed management, and rangeland management.
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