Some statistical inferences of parameter in MCMC approach and the application in uncertainty analysis of hydrological simulation

Abstract

Markov Chain Monte Carlo (MCMC) method has been increasingly popular in uncertainty analysis of hydrological simulation. In MCMC approach, deviations between model outputs and observations are commonly assumed to follow Gaussian distribution with zero medium and constant standard deviation σ2. However, the estimation of σ2 is a difficulty in terms of that it was assigned subjectively in previous studies, hindering the improvement of performance for uncertainty assessment. This work systemically investigates the statistical meaning of parameter σ2. σ could be expressed as the product of data length and two standard deviations, one of which is for observations (i.e. σObs) and the other for Nash-Sutcliffe Coefficient of Efficiency (NSCE) (i.e.σs). A new label called Confidence Level of Model (CLM) is developed to interpretσs. The natural logarithm of the posterior probability distribution for NSCE is a first-order linear equation associated with CLM. The CLM could be employed to guide the construction of σs and then the estimation of σ2. Uncertainty analysis of a flow duration curve (FDC) model is conducted using the MCMC method based on CLM, and the generalized likelihood uncertainty estimation (GLUE) method is employed for comparison. Results show that the CLM affects the MCMC results by three kinds of trade-offs, and the MCMC method based on CLM performs well in generating regular posterior distributions of model parameters and discharges. The MCMC method also yields narrow and symmetrical confidence intervals. Findings of this paper could interpret typical uncertainty behaviors commonly existing in hydrological modeling, and provide beneficial insights for the uncertainty analysis of other environmental modeling.