## 1. Introduction and Scope

[2] Hydrologic models, no matter how sophisticated and spatially explicit, aggregate at some level of detail complex, spatially distributed vegetation and subsurface properties into much simpler homogeneous storages with transfer functions that describe the flow of water within and between these different compartments. These conceptual storages correspond to physically identifiable control volumes in real space, even though the boundaries of these control volumes are generally not known. A consequence of this aggregation process is that most of the parameters in these models cannot be inferred through direct observation in the field, but can only be meaningfully derived by calibration against an input-output record of the catchment response. In this process the parameters are adjusted in such a way that the model approximates as closely and consistently as possible the response of the catchment over some historical period of time. The parameters estimated in this manner represent effective conceptual representations of spatially and temporally heterogeneous watershed properties.

[3] The traditional approach to watershed model calibration assumes that the uncertainty in the input-output representation of the model is attributable primarily to uncertainty associated with the parameter values. This approach effectively neglects errors in forcing data, and assumes that model structural inadequacies can be described with relatively simple additive error structures. This is not realistic for real world applications, and it is therefore highly desirable to develop an inference methodology that treats all sources of error separately and appropriately. Such a method would help to better understand what is and what is not well understood about the catchments under study, and help provide meaningful uncertainty estimates on model predictions, state variables and parameters. Such an approach should also enhance the prospects of finding useful regionalization relationships between catchment properties and optimized model parameters, something that is desirable, especially within the context of the Predictions in Ungauged Basins (PUB) initiative [*Sivapalan*, 2003].

[4] In recent years, significant progress has been made toward the development of a systematic framework for uncertainty treatment. While initial methodologies have focused on methods to quantify parameter uncertainty only [*Beven and Binley*, 1992; *Freer et al.*, 1996; *Gupta et al.*, 1998; *Vrugt et al.*, 2003], recent emerging approaches include state space filtering [*Vrugt et al.*, 2005; *Moradkhani et al.*, 2005a, 2005b; *Slater and Clark*, 2006; *Vrugt et al.*, 2006a], multimodel averaging [*Butts et al.*, 2004; *Georgakakos et al.*, 2004; *Vrugt et al.*, 2006b; *Marshall et al.*, 2006; *Ajami et al.*, 2007; *Vrugt and Robinson*, 2007b] and Bayesian approaches [*Kavetski et al.*, 2006a, 2006b; *Kuczera et al.*, 2006; P. Reichert and J. Mieleitner, Analyzing input and structural uncertainty of a hydrological model with stochastic, time-dependent parameters, unpublished manuscript, 2008] to explicitly treat individual error sources, and assess predictive uncertainty distributions. Much progress has also been made in the description of forcing data error [*Clark and Slater*, 2006], development of a formal hierarchical framework to formulate, build and test conceptual watershed models [*Clark et al.*, 2008], and algorithms for efficient sampling of complex distributions [*Vrugt et al.*, 2003; *Vrugt and Robinson*, 2007a; *Vrugt et al.*, 2008a] to derive uncertainty estimates of state variables, parameters and model output predictions.

[5] This paper has two main contributions. First, a novel adaptive Markov chain Monte Carlo (MCMC) algorithm is introduced for efficiently estimating the posterior probability density function of parameters within a Bayesian framework. This method, entitled differential evolution adaptive Metropolis (DREAM), runs multiple chains simultaneously for global exploration, and automatically tunes the scale and orientation of the proposal distribution during the evolution to the posterior distribution. The DREAM scheme is an adaptation of the shuffled complex evolution Metropolis (SCEM-UA) [*Vrugt et al.*, 2003] global optimization algorithm that has the advantage of maintaining detailed balance and ergodicity while showing good efficiency on complex, highly nonlinear, and multimodal target distributions [*Vrugt et al.*, 2008a]. Second, the applicability of DREAM is demonstrated for analyzing forcing error during watershed model calibration. *Vrugt et al.* [2008b] extended the work presented in this paper to include model structural error as well through the use of a first-order autoregressive scheme of the error residuals.

[6] The framework presented herein has various elements in common with the Bayesian total error analysis (BATEA) approach of *Kavetski et al.* [2006a, 2006b], but uses a different inference methodology to estimate the model parameters and rainfall multipliers that characterize and describe forcing data error. In addition, this method generalizes the “do hydrology backward” approach introduced by *Kirchner* [2008] to second- and higher-order nonlinear dynamical catchment systems, and simultaneously provides uncertainty estimates of rainfall, model parameters and streamflow predictions. This approach is key to understanding how much information can be extracted from the observed discharge data, and quantifying the uncertainty associated with the inferred records of whole-catchment precipitation.

[7] The paper is organized as follows. Section 2 briefly discusses the general model calibration problem, and highlights the need for explicit treatment of forcing data error. Section 3 describes a parsimonious framework for describing forcing data error that is very similar to the methodology described by *Kavetski et al.* [2002]. Successful implementation of this method requires the availability of an efficient and robust parameter estimation method. Section 4 introduces the differential evolution adaptive Metropolis (DREAM) algorithm, which satisfies this requirement. Then section 5 demonstrates how DREAM can help to provide fundamental insights into rainfall uncertainty, and its effect on streamflow prediction uncertainty and the optimized values of the hydrologic model parameters. A summary with conclusions is presented in section 6.