Entropy Based Hydro-statistical Approaches Considering Sample Conditions

Entropy theory based hydro-statistical approach has been increasingly applied in hydrology and water resources.From the perspective of statistics,entropy based applications are generally of two main types:one is entropy estimation in "descriptive study",where one uses entropy to quantify and analyze the uncertainty of a hydrological system;the other is the principle of maximum entropy(POME)in "inferential study,"where one uses the POME in hydrological frequency analysis and stochastic simulation,etc.In recent years,entropy-copula coupled research has been more popular in hydro-statistics.Entropy based hydro-statistical approach essentially copes with data-based issues.Therefore the validity of method and accuracy of results are constrained with sample conditions.There are at least two issues concerning sample conditions:one is the insufficiency of observed data for limited hydrological stations and data length,which is insufficiency of sample and it may reduce the accuracy of entropy estimators;the other is complexity of hydrological distributed patterns for increasing research targets and non-uniformity of hydrological variables,which is complexity of sample distribution and it makes accurate POME inference a big challenge.For the issue of insufficiency of sample,this research considers it in entropy estimation and fxrther entropy based hydrological uncertainty analysis.That is,from the perspective of method,this research introduces two innovative entropy estimator,Chao-Shen estimator,the shrinkage estimator,and to compare the statistical performance of them with the conventional maximum likelihood estimator,for hydrological simulation scenarios under the small-sample condition.Results indicate the more accuracy of the shrinkage estimator.From the perspective of application,this research presents the multi-scale moving entropy-based hydrological analyses(MM-EHA)to reveal the uncertainty of hydrological systems.By the MM-EHA,uncertainty of streamflow systems is analyzed on the Yichang and Hankou of Yangtze River,and Huayuankou and Lijin of the Yellow River,China.In addition,correlation analysis of entropy and relevant uncertainty statistics is also employed in order to explore their intrinsic connections.The insufficiency of sample is very common in hydrological practice,and the powerful shrinkage estimator is worthy to be popularized in more entropy based hydrological researches.For the issue of complexity of sample distribution,this research considers the determination of optimal moment constraints in POME modeling framework.That is,from the perspective of method,this research simulates the convergence of different moments based on Monte Carlo simulation and proposes a flexible TEA(Theoretical-empirical analysis)approach to determine the optimal order of moments,and finally proposes an OM-POME(Optimal-moment principle of maximum entropy)framework.Then coupled with copula function,a further OMME-C(Optimal-Moment constrained Maximum Entropy-Copula)multivariate modelling framework has been constructed.From the perspective of application,the first is to verify good performance in streamflow stochastic simulation of the OM-POME technique with hydrological case of(1)extreme value of water level from Yichang in the Yangtze River and(2)streamflow from Huayuankou in the Yellow River.The second is to model and analyze dependence patterns of three types of hydro-meteorological events(single station water level-streamflow,multi-station streamflow and multi-station precipitation)from representative stations in the Yangtze River and the Yellow River based on the constructed OM-POME copulas.Results indicate that the POME based distribution with optimal moment constraints is more accurate and thus suitable in probabilistic inference with complex sample distribution,and coupled with copula,the OM-POME copula which can capture more patterns which traditional correlation coefficients cannot reflect,provides an efficient way in other scenarios concerning multivariate modelling.