Bayesian Probabilistic Hydrological Forecast Based On Copula Function

Hydrological forecasts widely used at present are typically deterministic,which usually provide to users in the form of point estimation that ignores the uncertainty and cannot meet the demand of risk information for decision makers.Quantitative description and estimation of the inherent uncertainty of hydrological forecast in terms of probability distribution and the corresponding probabilistic hydrological forecast is not only more scientific and reasonable in theory,but also can increase economic and social benefits.Hydrological forecasting services are now trending toward providing users with probabilistic forecasts instead of traditional deterministic forecasts.Professor Roman Krzysztofowicz developed a series of Bayesian probabilistic hydrological forecast methods,in which the meta-Guassian model relying on the normal quantile transform(NQT)and linear-normal hypothesis were commonly applied to model the prior density and likelihood function.The main objective and innovation of this thesis is to replace the meta-Guassian model with Copula function to obtain the analytical expressions of prior density,likelihood function and posterior density,thereby a set of Bayesian probabilistic hydrological forecast methods based on Copula function were proposed.Three Gorges reservoir(TGR)basin was selected as case study throughout the thesis.The main research works and findings were summarized as follows:(1)The research developments and trends of Bayesian probabilistic forecasting methods were reviewed and discussed.Then,application progress of Copula function in hydrology was also summarized.It is pointed out that Copula function can be a powerful mathematical tool to describe and model the prior density and likelihood function in Bayesian probabilistic forecasting methods.(2)The distribution types,parameter estimation and goodness-of-fit test methods of marginal distribution for observed and forecasted discharges were presented.Subsequently,the definition of Copula function,several commonly used Copula function in hydrology and their parameter estimation and goodness-of-fit test methods were introduced.(3)A Copula-based Bayesian Forecast(CBF)processor was proposed using Copula function to derive the analytical expression of likelihood function and adopted to obtain the probabilistic inflow forecasts of the TGR.It is shown that the CBF processor is very flexible and widely applicable by relaxing linear-normal hypothesis.The accuracy of posterior median forecasts obtained by the CBF is better than those of the deterministic forecasts,and especially the relative error(RE)of the total runoff has been reduced considerably.Probabilistic forecasts by the CBF processor are effective and reliable,the continuous rank probability score(CRPS)values for 1d,2d and 3d lead-times are decreased by 9.12%,14.35%and 15.65%,respectively.(4)On the basis of the CBF processor,a Copula-based Hydrologic Uncertainty(CHU)processor was developed by constructing the prior density and likelihood function conditioned on a state variable,i.e.the current observed discharge.The CHU processor was adopted to produce the probabilistic forecasts of TGR inflow and compared with that of the CBF processor.The results demonstate that the improvement on the structure of the prior density and likelihood function has enriched information,thus the relevant result is more scientific and reasonable.The accuracy of posterior median forecasting associated with the CHU processor is slightly superior to that of the CBF processor.Compared with the CBF processor,the CRPS values for 1d,2d and 3d lead-times are decreased by 2.46%,3.03%and 4.14%,respectively.(5)The Copula-based Bayesian Transition Forecast(CBTF)method was proposed on the strength of one-step Markov transition density function,thus a Copula-based Multivariate Hydrologic Uncertainty(CMHU)processor was developed.The results show that CBTF method can quantitatively describe transition probability density and distribution functions between inflows in adjacent lead-times.The CMHU processor can provide the posterior joint probability density function of actual inflow process considering the inherent dependence structure,conditioned on the observed discharge at the forecast time and deterministic forecasting process.The proposed CBTF method and CMHU processor not only offer useful tools for analyzing the time evolution characteristics of hydrological forecasting uncertainty,but also effectively support reservoir operation decisions.(6)The Copula-based Bayesian Extremum Forecast(CBEF)method was proposed by utilizing outputs of CBTF method.This was implemented by theoretical derivation of exceed probability distribution function of maximum discharges within lead time interval,which was solved by numerical integration.The results indicate that three kinds of forecast products provided by the CBEF method,including exceed probability distribution functions,iso-probability quantile series and distribution of time to flooding can display and convey the uncertainty to decision makers adequately,which are useful for flood warning system.