## 1. Introduction

[2] The threat of nonstationary hydrology has motivated significant research efforts investigating the potential impacts of climate change on regional hydrology and implications for local water resource systems. Despite these efforts, uncertainty in both future climate conditions and regional hydrologic response confounds the interpretation of results and diminishes their utility in water resources planning [*Lopez et al.*, 2009]. A systematic approach is required to account for the uncertainty in hydrologic impact assessments so that decision makers can consider adaptation strategies contextualized by the uncertainty in design statistics critical to the decision-making process. In this paper we propose a statistical framework that quantifies several sources of uncertainty in long-range projections of hydrologic alteration, including uncertainties in future climate, hydrologic model predictive skill, model parameterization, and sampling error of estimated hydrologic statistics. These uncertainties are integrated to develop a probabilistic description of potential alterations to regional hydrology useful for water resources planning.

[3] In the vast majority of studies, hydrologic alteration under climate change is assessed using future climate scenarios, as simulated by global circulation models (GCMs), that are downscaled to a location of interest and used to force a regional hydrologic model. The simulated hydrologic response is then compared to a baseline response based on historic climate data, and measures of hydrologic alteration are computed [*Gleick*, 1986]. There are multiple sources of uncertainty that degrade this process, including those associated with the GCMs (i.e., inaccuracy at subcontinental scales, inconsistencies across models, parameterization, uncertain boundary conditions, difficulty in assessing predictive skill), the ambiguity between different downscaling techniques, and the hydrologic model (i.e., model structure, input and output data used for calibration, parameterization) [*Wood et al.*, 1997]. GCM accuracy and consistency, along with the choice of downscaling methodology, are considered to be the primary sources of uncertainty and have garnered significant research attention [*Räisänen and Palmer*, 2001; *Palmer and Räisänen*, 2002; *Piani et al.*, 2005; *Stainforth et al.*, 2005; *Fowler et al.*, 2007; *Stainforth et al.*, 2007a; *Lopez et al.*, 2009]. Errors associated with the hydrologic model, however, have received less emphasis in studies considering hydrologic alteration under climate change. In the majority of climate change impact assessments, hydrologic simulations of future climate are treated largely as deterministic output that can be used to directly identify hydrologic alterations [*Chao*, 1999; *Hamlet and Lettenmaier*, 1999; *Lettenmaier et al.*, 1999; *Nijssen et al.*, 2001]. Some studies have explored the impacts of hydrologic model uncertainty on climate impact assessment results, but they often only investigate uncertainties in parameterization [*Arnell*, 1999; *Cameron et al.*, 2001; *Wilby*, 2005], model structure [*Boorman and Sefton*, 1997; *Jiang et al.*, 2007], or a combination of both [*Wilby and Harris*, 2006; *Kay et al.*, 2009; *Prudhomme and Davies*, 2009a, 2009b], and almost never formally account for prediction error, which can often dominate total model uncertainty [*Stedinger et al.*, 2008].

[4] While parameter and structural errors are important components of the total uncertainty in hydrologic model results, accounting for these uncertainties alone may not guarantee reliable predictive bounds for streamflow estimates. For a watershed exhibiting significant heterogeneity or unexplainable behavior, many types of hydrologic response may be challenging to simulate even with an ensemble of model structures or parameterizations. The assumption that a set of hydrologic models with multiple parameterizations is complete enough to reliably bound true hydrologic response is difficult to verify [*Renard et al.*, 2010]. This is especially true if the models struggle to reproduce certain aspects of the observed streamflow and exhibit errors that vary across the magnitude and timing of hydrologic responses. To generate reliable predictive bounds, a formal quantification of residual error is needed. If predictive uncertainty associated with the hydrologic model is not formally addressed and propagated through climate change impact analyses, claims of hydrologic alteration from such studies can be overstated and misguide water resources decision makers.

[5] In a related line of research, predictive uncertainty in hydrologic modeling has been extensively explored and mature methods for quantifying error have been developed. Early efforts focused on pseudo-Bayesian methods [*Beven and Binley*, 1992; *Beven and Freer*, 2001], and later more formal Bayesian techniques emerged to properly account for both residual and parameter uncertainties [*Bates and Campbell*, 2001; *Marshall et al.*, 2004; *Stedinger et al.*, 2008; *Schoups and Vrugt*, 2010]. Further studies have dissected model error into its component parts, investigating the impacts of uncertain input and response data on model predictions [*Kavetski et al.*, 2006a, 2006b; *Thyer et al.*, 2009; *Renard et al.*, 2010]. Other innovative approaches for assessing hydrologic model uncertainty include Bayesian recursive estimation [*Thiemann et al.*, 2001], Bayesian hierarchical mixture of experts [*Marshall et al.*, 2007], and simultaneous parameter optimization and data assimilation [*Vrugt et al.*, 2005; *Clark and Vrugt*, 2006]. These techniques can be extended to climate impact studies to quantify the total uncertainty in hydrologic models and demonstrate the extent to which it obscures the differences between future and baseline hydrologic conditions.

[6] To the authors' knowledge, only one study has attempted to simultaneously quantify hydrologic model prediction and parameterization error and then propagate that uncertainty through climate impact assessments of hydrologic alteration [*Khan and Coulibaly*, 2010]. This study employed a Bayesian neural network rainfall-runoff model to explore climate-impacted hydrology. In this study, the posterior distribution of model parameters and the final distribution of model predictions were assumed Gaussian to improve the tractability of Bayesian integrals, despite the availability of Markov chain Monte Carlo (MCMC) sampling procedures that allow for more complex and accurate distributional assumptions. More importantly, uncertainty bounds were only generated for the streamflow trace generated using the mean of ensemble climate members, rather than for each climate member individually. This approach artificially deflates the true uncertainty in future hydrologic model projections because hydrologic model error should be integrated with the range of uncertainties stemming from GCMs and downscaling techniques.

[7] The study presented here will contribute to the science of hydrologic uncertainty analysis under climate change by developing a framework in which hydrologic model error is formally characterized and appropriately integrated with other sources of future climate uncertainty to better quantify the total uncertainty of hydrologic alterations under future climates. This allows a comparison of the range of projected changes in streamflow due to climate change to be compared with the uncertainty due to hydrologic model error. Hydrologic model prediction error is formally characterized with an appropriate likelihood function and combined with prior distributions of model parameters using Bayes' Theorem. MCMC sampling is used to evaluate the posterior distributions of hydrologic and error model parameters. Reliable uncertainty bounds for streamflow estimates are constructed from the integration of parameter and residual uncertainties and evaluated over the historic record. The Bayesian hydrologic modeling framework is then extended to a climate change impact assessment. Ensembles of baseline and future climate data are downscaled from GCMs and used to drive simulations of streamflow over parameter samples drawn from the posterior space. While GCM projections do not fully capture climate change uncertainty, the range of climate projections can be described as an estimate of the irreducible range of climate uncertainty, a minimum bound [*Stainforth et al.*, 2007a; *Wilby and Dessai*, 2010]. Time series of streamflow statistics are generated from baseline and future ensembles of simulated flows. Appropriate probability distributions are then fit to these statistics, enabling the estimation of streamflow quantiles and their sampling error for the ensemble of baseline and future conditions. Quantile estimates are directly compared between baseline and future scenarios in the context of their cumulative uncertainties. The framework can be used to highlight the complex interactions between different sources of uncertainty and their effects on future estimates of design flow statistics used in decision making. An application of this framework is presented for the White River Basin in Vermont using a version of the monthly ABCD hydrology model [*Thomas*, 1981] with a snow component.

[8] The paper will proceed as follows. Section 2provides background on Bayesian inference techniques in rainfall-runoff modeling and their potential use for error propagation in future hydrologic simulations.Section 3 delineates the methodology used to quantify the total uncertainty of hydrologic alteration under future climate change scenarios. The methodology is applied and results presented in section 4, and the study concludes in section 5 with a discussion of future research needs.