Research On The Parameter Estimation Methods Of Frequency Distribution For Univariate Hydrological Sequence

Hydrological frequency calculation aims to reveal the statistic laws of hydrological event from existing hydrological data based on Hydrologic statistics and some mathematical methods,and to calculate the hydrological design value for a given return period,providing scientific basis for the planning,design and management of hydraulic engineering.During long-term research and practice,hydrologists have accumulated a lot of experience in hydrological frequency calculation.However,due to the complexity of hydrological events,researchers still encounter many difficulties.Based on the extensive literature analysis,this thesis reviewed the research progress around several issues of the current hydrological frequency calculation and summarized some problems existing in the present studies.For these problems,this paper carried out some researches and proposed some new theoretical methods.The hydro-meteorological data in Yellow River Basin were selected as case studies to validate the usefulness of the proposed methods.The main contents are summarized as follows:1.To conduct the frequency calculation of hydrological series containing zero values,six distributions were constructed using the unit impulse response functions of a Linearized Kinematic Diffusion model(KD)and a Muskingum Flood Routing model(MK)。The Probability Weighted Moments(PWM)of the distributions(except the three-parameter distribution of KD,KD3)were derived accordingly.The distribution parameters for the monthly precipitation series which contains zero values were estimated using PWM method,Method of Moments(MOM)、Maximum Likelihood Method(MLM)and Simulated Annealing algorithm(SA)method.The existed methods(including the distribution of total probability theory,the partial L-moments(PLM)of censored samples,the frequency proportional method and the multiplication type II distribution)were employed for frequency calculation as well.Finally,the fitting effect of each method was evaluatied and compared.Results show that the two-parameter Gamma distribution(GA2)with parameters estimated by Principle of Maximum Entropy(POME)and PWM methods gives better fits when the total probability theory was used.The Partial L-moments(PLM)of three-parameter Generalized Pareto distribution(GP3)and two-parameter Weibull distribution(WB2)perform better than other distributions considered in the study when using censoring sample theory.The Pearson type III distribution(P III),WB2 and GP3 distributions show satisfactory fitting results when using frequency proportional method.The multiplication type II distribution generally performs well.Both PWM and SA methods of KD2 fit the series better than the remaining methods,and the SA method of KD3 is better than MOM and MLM methods.The PWM and SA methods of the MK-based distributions fit the series better than MOM and MLM methods,and the distribution with two-parameter Generalized Pareto distribution(DGP2)as conditional distribution fits the series better than other distributions.Overall,the KD-and MK-based distributions fit the series better than the traditional methods.2.For the calculation of short-term hydrological sequence,a Copula-based composite likelihood function(Copula-based Composite Likelihood Approach,CBCLA)incorporating both the concurrent and non-concurrent periods of data series in an integrated framework was provided.In this method,the bivariate model of concurrent part was constructed by means of Copula.The variance-covariance matrix of the CBCLA parameter estimates was derived to calculate the standard errors and confidence intervals of design values.Monte Carlo simulation was performed to analyze the statistical properties of CBCLA method,and the annual precipitation series were used as case study to testify its applicability.Compared with the results of univariate method,CBCLA method achieved more precise parameter estimates and a reduction in uncertainty in design precipitation values of a relative shorter series.In addition,the fitting effect of a longer series can also be improved by this method.In general,the CBCLA method can fully utilize the hydrological information and enhance the accuracy of the quantile estimates for a relative shorter series.3.To quantify the uncertainty of hydrological design values caused by the parameter estimators’ error,the confidence interval calculation of design values on the basis of POME parameter estimates were studied.The calculation formulas of variances and covariance estimated by POME method for parameters of the P III、GA2 and Extreme Value type I(EVI)distributions were derived,and the feasibility of the formulas was testified by Monte Carlo simulation.The standard errors and confidence intervals of design precipitation values were then calculated and compared with the results of MOM and MLM methods.Results show that the POME method is superior to MOM and MLM methods in fitting precipitation series,and gives the smallest standard errors and confidence interval widths of design values.In addition,the results of POME and CBCLA methods are comparable,and the POME method is more convenient.4.In order to improve the fitting effect of the two ends of frequency curves,the theory of truncated distribution was studied.The relationships between the moments of the left-truncated P III distribution and the moments of P III distribution,the relationships between the moments of the left-and right-truncated GA2 distribution and the moments of GA2 distribution were derived for parameter estimation.The left-truncated WB2 distribution and its MOM and MLM methods were also provided.The distribution parameters of annual average flow series and annual maximum flow series were calculated using truncated samples when giving different truncating levels,and the fitting effects of the smaller and the larger observations in the sequence were evaluated respectively.Results show that left-and right-truncated distributions both fit the data well.The right-truncated GA2 distribution and the higher-censored PLM of GA2 distribution can both improve the fitting effect of the smaller average annual flow values.The fitting effects of large maximum flows were improved when left-truncated P III,GA2 and WB2 distributions,and the lower-censored PLM of GA2 distribution were employed.The left-truncated GA2 distribution and lower-censored PLM of GA2 are superior to other methods.5.Research on the parameter estimation methods of Generalized Gamma distribution(GG)and Generalized Beta distribution of the second type(GB2).The PWM of GG distribution and Probability Weighted Mixed Moments(PWMIXM)of both GG and GB2 distributions were derived.The PWM and PWMIXM methods were then employed to estimated distribution parameters for annual maximum flow series.The performance of PWM and PWMIXM methods of GG distribution and the performance of PWM of GB2 distribution were both compared with some traditional methods.Then a further comparison were made among GG,GB2,P III and Generalized Extreme Value distribution(GEV).Results show that the GG distribution fits the whole series well when PWM and PWMIXM methods were employed for parameter estimation,and PWMIXM method gives a better fit for larger values than the remaining methods.GB2 distribution fits the whole series well with parameters estimated by MLM and POME methods,and POME method fits the larger values better than other candidate methods.In general,the GB2 distribution and the left-truncated GA2 distribution fit the larger maximum flow values better than other distributions considered in the study.