## 1. Introduction

[2] Hydrologic modeling has benefited from significant developments over the past two decades, including dramatic growths in computational power, ever increasing availability of distributed hydrologic observations, and improved understanding of the physics and dynamics of the hydrologic system. This has led to the building of higher levels of complexity into hydrologic models, and an advance from lumped, conceptual models toward semidistributed and distributed physics-based models. Paradoxically, while these advances reflect our growing understanding, they have also increased the need for concrete methods to deal with the increasing uncertainty associated with the models themselves, and with the observations required for driving and evaluating the models. It is now being broadly recognized that proper consideration of uncertainty in hydrologic predictions is essential for purposes of both research and operational modeling [*Wagener and Gupta*, 2005]. The value of a hydrologic prediction to water resources and other relevant decision-making processes is limited if reasonable estimates of the corresponding predictive uncertainty are not provided [e.g., *Georgakakos et al.*, 2004].

[3] To adequately address uncertainty in hydrologic modeling, there are three distinct yet related aspects to be considered: understanding, quantification, and reduction of uncertainty. Arguably, understanding uncertainty is an integral part of any application of uncertainty quantification and/or reduction. Many uncertainty analysis frameworks have been introduced in the hydrologic literature, including the generalized likelihood uncertainty estimation (GLUE) methodology [*Beven and Binley*, 1992], the Bayesian recursive estimation technique (BaRE) [*Thiemann et al.*, 2001], the Shuffled Complex Evolution Metropolis algorithm (SCEM) [*Vrugt et al.*, 2003a], the multiobjective extension of SCEM [*Vrugt et al.*, 2003b], the dynamic identifiability analysis framework (DYNIA) [*Wagener et al.*, 2003], the maximum likelihood Bayesian averaging method (MLBMA) [*Neuman*, 2003], the dual state-parameter estimation methods [*Moradkhani et al.*, 2005a, 2005b], and the simultaneous optimization and data assimilation algorithm (SODA) [*Vrugt et al.*, 2005]. However, few of these methods completely address all the above three critical aspects of uncertainty analysis in an explicit and cohesive way.

[4] Methods of probabilistic prediction and data assimilation (DA) for quantification and reduction of state uncertainty have been extensively explored in the atmospheric and oceanic sciences [e.g., *Daley*, 1991; *Courtier et al.*, 1993; *Anderson and Anderson*, 1999]. Their application in the hydrological sciences is relatively new, although deterministic hydrological prediction and parameter estimation have become reasonably mature. Nevertheless, the hydrologic literature has seen various applications of data assimilation and/or uncertainty analysis in hydrology ranging from characterization of soil moisture and/or surface energy balance [e.g., *Entekhabi et al.*, 1994; *Houser et al.*, 1998; *Entekhabi et al.*, 1999; *Galantowicz et al.*, 1999; *Boni et al.*, 2001; *Walker et al.*, 2001; *Reichle et al.*, 2001a, 2001b, 2002a, 2002b; *Margulis et al.*, 2002; *Dunne and Entekhabi*, 2005], to rainfall-runoff modeling [e.g., *Restrepo*, 1985; *Moradkhani et al.*, 2005a, 2005b; *Vrugt et al.*, 2005], to flood foresting [e.g., *Kitanidis and Bras*, 1980; *Young*, 2002], to estimation of hydraulic conductivity [e.g., *Katul et al.*, 1993; *Lee et al.*, 1993], to groundwater flow and transport problems [e.g., *Eigbe et al.*, 1998; *Graham and McLaughlin*, 1991; *McLaughlin et al.*, 1993], to estimation of water table elevations [e.g., *Van Geer et al.*, 1991; *Yangxiao et al.*, 1991], and to water quality modeling [e.g., *Beck*, 1987].

[5] One critical issue for hydrologic modeling is how the DA methods used in atmospheric and related sciences can best be adapted and combined with hydrologic methods to cope with the uncertainties arising from hydrologic modeling in a cohesive, systematic way to maximally reduce and adequately quantify the predictive hydrologic uncertainty [*Krzysztofowicz*, 1999; *Mantovan and Todini*, 2006]. Although general principles and techniques on addressing hydrologic uncertainty are emerging in the literature, there exist no well-accepted guidelines about how to actually implement these principles and techniques in various hydrologic settings. In this paper we discuss the sources of uncertainty in hydrological modeling from a systems perspective, illustrate in detail some of the common DA methods that have been used to quantify and reduce hydrological uncertainty, and propose a (preliminary) hierarchical data assimilation framework for systematically addressing the various types of uncertainties as a way to move forward. It is worth noting that this paper does not attempt to provide a comprehensive review of the literature regarding all the methods, applications, and issues related to data assimilation in hydrology; instead, we aim to present to the readers an illustrative and integrated (rather than fragmented) picture of the state of the art of hydrological data assimilation from a systems perspective.

[6] The paper is organized as follows: Section 2 discusses the three important aspects in addressing hydrologic uncertainty, i.e., understanding, quantifying, and reducing uncertainty; in section 3 we present an integrated view of uncertainty in hydrologic modeling from a systems perspective; Bayes' theorem and its application to data assimilation are discussed in section 4; sections 5, 6, 7, and 8 are devoted to reviews of the common methods that have been used to approach problems of system identification, parameter estimation, state estimation, and simultaneous state and parameter estimation, respectively; an integrated Bayesian hierarchical framework for handling all hydrologic uncertainty in a cohesive, systematic manner is proposed in section 9; and the paper closes with some general discussions and recommendations for future research in section 10.