Application Of Four-Parameter Exponential Gamma Distribution In Hydrology

Four-parameter Exponential Gamma Distribution is a probability distribution model proposed by Sun Jiliang and Qin Dayong of China Institute of Water Resources and Hydropower Research.When its parameters are given some specific values,it can be transformed into more than 10 common probability distributions,including Log-normal distribution,Weibull distribution,Gamma distribution and Rayleigh distribution.Fourparameter Exponential Gamma distribution has flexible statistical characteristics and strong versatility.It can be used to describe the events of complex climatic conditions and river basin characteristics.At the same time,it can well fit the sequence which hydrological enginer pay more attention to serious tailing phenomena.Although Four-parameter Exponential Gamma Distribution has vivid statistical characteristics and strong universality,there are many difficulties in the parameter estimation process.The parameter estimation methods commonly used for the Four-parameter Exponential Gamma Distribution include the moment method(MOM),the maximum likelihood method(ML),the probability weight moment method(PWM),and principle of maximum entropy(POME).Because the difficulties in solving the four unknown parameters of the distribution,it is limited in practical use.There are still problems such as large calculation error of the moment method,approximate calculation of the probability weight moment method,and lack of comparative research.Based on the above problems,this paper focused on the parameter estimation methods of the Four-parameter Exponential Gamma Distribution,and applied them to flood frequency analysis and precipitation frequency analysis,compared the fitting effects of each parameter estimation method with current P-III distribution synthetically.The research mainly includes three aspects:(a)Based on the Exponential Gamma distribution,8 parameter estimation methods,calculation formula and sample calculation formula with historical flood were derived.(b)The statistical performance and design accuracy of 8 parameter estimation methods were evaluated by Monte Carlo simulation.(c)Taking precipitation data of 90 meteorological stations in Weihe River Basin and flood data of 6 hydrological stations in Ankang,Linjiang,Donglan,Shaping,Lanzhou and Jiangjie River as examples,AIC,R-square and RMSE were used as the evaluation criteria to analyze the of the Four-parameter Exponential Gamma Distribution.The fitting results of 8 methods based on the Fourparameter Exponential Gamma Distribution were compared with the fitting results of the PIII distribution.The main results and conclutions are as follows:(1)Based on eight commonly used parameter estimation methods,the formulas for calculating sample data and design values of the Four-parameter Exponential Gamma Distribution with historical floods were derived,and the explicit integral expression of S functions of the Four-parameter Exponential Gamma Distribution is presented for calculating the third-order probability weight moment S functions.(2)The statistical performance and design accuracy of 8 parameter estimation methods were evaluated by Monte Carlo simulation.The results show that under the same scenario mode,the precision of optimal alignment method,MOM and L-M is the best method,followed by PWM and ENT,while PWMM,MOMM and ML are the worst.The statistical performance and design accuracy of the eight parameter estimation methods are most affected by the sample size.The larger the sample size is,the better the statistical performance and the higher the design accuracy have.In the case of only increasing the number of historical floods,the statistical performance and design accuracy of the eight parameter estimation methods can not be improved.(3)Based on 8 methods of the Four-parameter Exponential Gamma Distribution,the fitting condition of precipitation series and flood series were analyzed.The fitting results show that in the precipitation frequency analysis and flood frequency analysis,the fitting effect of frequency curve determined by optimum fitting line method,PWM and L-M is better than that of P-III distribution,while the fitting effect of MOM and MOMM is worse.In the precipitation frequency analysis,ML is better than that of flood frequency analysis.In the flood frequency analysis,ENT has better fitting effect.(4)Based on the 8 methods of the Four-parameter Exponential Gamma Distribution,the fitting condition of precipitation series and flood series were quantitatively evaluated by three error criteria.The results of error analysis show that the optimal fitting line method and PWM are better than P-III distribution in frequency analysis of precipitation,and the error of design value is smaller than that of P-III distribution.In the flood frequency analysis,the fitting effect of PWM and L-M is better,and the error of design value is smaller.(5)The results of fitting analysis and error analysis show that it is feasible to use the Exponential Gamma distribution as a substitute curve for P-III distribution,and it has better universality and fitting effect.