Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov chain Monte Carlo simulation

There is increasing consensus in the hydrologic literature that an appropriate framework for streamflow forecasting and simulation should include explicit recognition of forcing and parameter and model structural error. This paper presents a novel Markov chain Monte Carlo (MCMC) sampler, entitled differential evolution adaptive Metropolis (DREAM), that is especially designed to efficiently estimate the posterior probability density function of hydrologic model parameters in complex, high-dimensional sampling problems. This MCMC scheme adaptively updates the scale and orientation of the proposal distribution during sampling and maintains detailed balance and ergodicity. It is then demonstrated how DREAM can be used to analyze forcing data error during watershed model calibration using a five-parameter rainfall-runoff model with streamflow data from two different catchments. Explicit treatment of precipitation error during hydrologic model calibration not only results in prediction uncertainty bounds that are more appropriate but also significantly alters the posterior distribution of the watershed model parameters. This has significant implications for regionalization studies. The approach also provides important new ways to estimate areal average watershed precipitation, information that is of utmost importance for testing hydrologic theory, diagnosing structural errors in models, and appropriately benchmarking rainfall measurement devices.