LETTER TO THE EDITOR Response to comment by Keith Beven on ‘‘Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?’’

This is our reply to the comment by Beven (2008) on our paper ‘‘Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?’’ recently published in Stochastic Environmental Research and Risk Assessment. There is strong disagreement in the hydrologic literature whether an appropriate framework for uncertainty estimation should have its roots within a proper statistical (Bayesian) context or such a framework should be based on a different philosophy and use non-statistical methodologies for assessing model predictive uncertainty. The stated goal of our paper (Vrugt et al. 2008) was to establish some common ground between these different viewpoints. In so doing, we highlighted that, under a variety of conditions, both Bayesian and informal Bayesian methods, such as the generalized likelihood uncertainty estimation (GLUE) method, can retrieve very similar estimates of total predictive uncertainty. Indeed, the GLUE results were based only on weighted simulations from the set of behavioral models, whereas the formal Bayesian results using Markov chain Monte Carlo (MCMC) simulations with DREAM used a first-order autocorrelated error model with explicit information from the streamflow observations. Beven (2008) argues that this comparison is simply inappropriate and invalid. We disagree with his assessment. First, our paper (Vrugt et al. 2008) was not intended to explore different implementations of 1-day-ahead forecasting as Beven (2008) seems to believe, but just compares two different approaches for streamflow uncertainty estimation. The formal Bayesian approach requires an explicit expression for input, parameter, model structural and calibration data error. In our paper, we simply took one possible vanilla implementation of a formal Bayesian approach that used explicit information from the streamflow observations to quantify model structural error. Thus, our implementations of GLUE and formal Bayes were indeed different. Nevertheless, they both obtained fairly similar estimates of total predictive streamflow uncertainty. This is a rather unexpected result, which, we believe, speaks strongly in favor of the continued use of GLUE by practitioners, as discussed and highlighted on page 13, paragraph ‘‘If the interest is in estimating ...’’ of our paper. Second, the goal of our paper was not to advocate which uncertainty estimation method is more appropriate and should be used, but rather (page 2, right column, lines 7–12 from top of our paper) to establish common ground between statistical and non-statistical methods for uncertainty estimation. At various places throughout the manuscript, we discussed the advantages and limitations of both approaches, starting from paragraph 2 to 3 of the J. A. Vrugt (&) Center for NonLinear Studies (CNLS), Los Alamos National Laboratory (LANL), Mail Stop B258, Los Alamos, NM 87545, USA e-mail: [email protected]