An application of the statistical framework described above is presented for the White River Basin, located in central Vermont. Records of monthly precipitation, temperature, and potential evapotranspiration are used to drive a Bayesian calibration of a conceptual rainfall-runoff model of the basin. An adaptation of the ABCD conceptual hydrologic model that incorporates a new snow modeling scheme is chosen for this purpose. After calibration, posterior distributions of both hydrologic and error model parameters are examined for convergence, and a probabilistic evaluation of the error model is presented to ensure the distribution of model residuals is well characterized. After the model is evaluated, the framework for climate impact assessments is applied to an ensemble of transient GCM climate scenarios.
4.2. ABCD Hydrologic Model
 An altered version of the ABCD hydrologic model is considered to model monthly streamflow in the White River Basin. The original ABCD model is a four parameter (a,b,c,d), conceptual rainfall-runoff model designed through a control volume analysis on upper soil moisture zone storage [Thomas, 1981]. The model converts monthly averaged precipitation and potential evapotranspiration into estimates of monthly streamflow by diverting water between two soil storage zones, losses to evapotranspiration, and the stream. The model has been recommended as an effective parsimonious model with physically meaningful parameters capable of efficiently reproducing monthly water balance dynamics in both theory [Vogel and Sankarasubramanian, 2003] and practice [Alley, 1984; Vandewiele et al., 1992]. A detailed review of the original ABCD model formulation can be found in [Fernandez et al., 2000].
 A snow component similar to that of Martinez and Gupta  was added to the ABCD model to simulate the snow accumulation/melt processes that dominate much of the hydrologic cycle in northern latitude watersheds. A snow storage zone is added that stores all incoming precipitation as snow water equivalent during times of year when the temperature falls below a threshold TSnow. A second threshold, Train, delimits the temperature above which all precipitation falls as rain. When temperatures rise above Train, all water held in the snow storage zone melts and is added to incoming precipitation for that month. This threshold melt process is highly representative of springtime hydrology seen in northern New England rivers. When monthly temperatures fall between Train and Tsnow, a fraction of the incoming precipitation for that month enters the snow storage component, and the remainder falls as rain. In addition, a fraction of the water held as snow is available for melt and is added to the effective rainfall for that month. The rate of melt is given by the parameter e. The total snow melt in time t is given by
where St-1 is the water stored as snow in the previous month, Ptot,t is the total precipitation, Tt is the mean monthly temperature, and fract is the fraction of precipitation that falls as snow, equal to . The water stored as snow in month t is given by
 The effective precipitation input to the model (precipitation available for runoff, soil zone storage, ET, etc.) is then given by
 In total, three parameters are used to represent snow accumulation/melt processes, bringing the total number of model parameters to seven (a, b, c, d, e, Train, Tsnow). During calibration, the parameter Tsnow is not directly calibrated because its prior distribution would have to be conditioned on the value of Train to ensure it took a smaller value. To circumvent this issue, a nonnegative parameter dif = Train − Tsnow is used, from which Tsnow can be directly computed.
 Martinez and Gupta performed a thorough analysis on the suitability of a similar snow-augmented ABCD model structure for catchments throughout the United States, testing the model using several diagnostic statistics including Nash-Sutcliffe efficiency, bias, and variance error. That study found that the snow-augmented ABCD model structure significantly improves results for snow-dominated watersheds in New England and is a suitable structure for many catchments in the region, supporting its use in this study.
4.3. Bayesian Calibration and Evaluation
 Historic, monthly averages of precipitation and maximum, minimum, and mean daily temperatures were gathered for the basin over the period of January 1980 to December 2005 from the gridded observed meteorological data set produced by Maurer et al. . Average monthly streamflows were collected from the U.S. Geological Survey (USGS) West Hartford gauge (ID 01144000) located at the mouth of the White River. Monthly averages of maximum, minimum, and mean daily temperatures were combined with estimates of monthly extraterrestrial solar radiation to produce a time series of potential evapotranspiration using the Hargreaves method [Hargreaves and Samani, 1982]. Solar radiation was calculated using the method presented by Allen et al. .
 Based on past hydrologic modeling experience for monthly flows in the New England region, a normal distribution with mean zero and standard deviation σ was initially chosen to characterize the sampling distribution of the residuals of the natural logarithms of observations and model predictions (hereafter referred to simply as model residuals)
where εln = ln(Q) − ln( ). The likelihood function for the observed streamflow values, Q, is then given by
 The prior for the unknown parameter σ was set to a gamma distribution with known shape λ = 1 and scale ζ = 2.5 parameters. The posterior of this parameter characterizes the level of uncertainty in hydrologic model estimates. A verification of the chosen sampling distribution for model residuals is described below.
 Past studies were used to inform prior distributions for the hydrologic model parameters a, b, c, and e [Alley, 1984; Vandewiele et al., 1992; Fernandez et al., 2000; Martinez and Gupta, 2010], and the remaining model parameters (d, Train, dif) were given vague priors in the form of uniform distributions or normal distributions with large variances. Initial states were also calibrated in the model to avoid any parameter biases from incorrect initial conditions. The slice sampler was chosen for the MCMC sampling and was implemented in the JAGS programming language (M. Plummer, rjags: Bayesian graphical models using MCMC, R package version 2.2.0-4, http://CRAN.R-project.org/package=rjags). Three chains were used in the sampling, and the Gelman and Rubin factor was used to test for convergence [Gelman and Rubin, 1992]. Calibration was implemented over the period between January 1980 and December 1999, leaving 6 years of data for evaluation. Table 1 summarizes the prior and posterior distributions for all parameters inferred in the MCMC sampling, as well as allowable ranges for each parameter. Figure 3a shows the history plots of parameter a for the three chains, and Figure 3b presents histograms of the prior and posterior distributions of parameter a. For all model parameters, the Gelman and Rubin convergence factor was within 0.005 of 1, suggesting that convergence was reached for all calibrated parameters.
Table 1. Summary of Prior and Posterior Distributions for All Model Parametersa
|Parameter||Allowable Range||Prior Distributions||Posterior Distribution|
|First Quartile||Median||Mean||Third Quartile|
|a||(0, 1)||Beta (a = 1.2, b = 0.6)||0.982||0.984||0.984||0.986|
|b (mm)||(0, ∞)||Normal (μ = 300, ϕ = 100)||303||310||310||316|
|c||(0, 1)||Beta (a = 0.6, b = 1.2)||0.14||0.18||0.18||0.22|
|d||(0, 1)||Uniform (a = 0, b = 1)||0.45||0.66||0.74||0.90|
|e||(0, 1)||Beta (a = 0.8, b = 1.8)||0.141||0.205||0.206||0.268|
|Train (°C)||(−∞, ∞)||Normal (μ = 0, ϕ = 4)||−1.65||−1.48||−1.47||−1.31|
|dif (°C)||(0, ∞)||Uniform (a = 0.01, b = 20)||12.9||13.9||14.0||15.0|
|σ(ln(mm))||(0, ∞)||Gamma (λ =1, ζ = 2.5)||0.13||0.14||0.14||0.15|
Figure 3. MCMC and model error diagnostics, including (a) the history plot for parameter a shown for the three MCMC chains, (b) a histogram of the prior (red) and posterior (black) distribution for parameter a, (c) a Q-Q plot showing sample quantiles of model errorεln against theoretical quantiles of a standard normal distribution, and (d) the autocorrelation function of model errors εln.
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 Figure 3c presents a normal probability plot of the model errors εln generated from the hydrologic simulation under the median posterior parameter set over the evaluation period (January 2000 to December 2005), and Figure 3dshows their autocorrelation coefficients. Results from the Q-Q plot suggest that model residuals follow a normal distribution relatively well. Most autocorrelation coefficients inFigure 3d are insignificant, including that at lag 1. There are some coefficients that exhibit small but significant values, particular at seasonal lag times. An autocorrelation component could be added to the error model, but this would require additional parameters to be estimated in the calibration, creating a tradeoff between problem dimensionality and error model accuracy. The seasonal autocorrelation seen in Figure 3d is rather low and not considered worth the increased dimensionality needed to model its behavior. Therefore, the original choice of a normal error model with no autocorrelation component for εln was considered adequate for this modeling exercise.
 Figure 4shows the observed monthly streamflow for the last 5 years of calibration and the entire evaluation period, as well as model estimates generated by the median values of the posteriors for hydrologic model parameters. The Nash-Sutcliffe efficiency (NSE), mean flow bias, and variance error for simulated streamflow using the median parameter set equals 0.82, −1.4%, and +6.6% for the calibration period and 0.67, −5.2%, and −15.1% for the evaluation period. The bias and variance errors are expressed as a percentage of observed values. These performance statistics are considered either “good” or “acceptable” in other hydrologic modeling studies [Martinez and Gupta, 2010]. Also shown in Figure 4 are error bounds consistent with the 2.5th and 97.5th percentiles of streamflow estimates, calculated according to equation 3. Observed data from the calibration and evaluation periods fell outside the 95% predictive interval 3.3% and 6.7% of the time, respectively, again suggesting that the error model adopted is appropriate for this application.
Figure 4. Time series of streamflow during calibration (left of vertical dashed line) and evaluation phases (right of vertical dashed line). Only a portion of the calibration time period is shown for clarity.
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 An additional evaluation procedure was conducted to further evaluate the adequacy of the error model. The details of the procedure are given by Laio and Tamea . In brief, the procedure tests whether probabilistic predictions for a set of streamflow observations are adequate in a statistical sense. To conduct the test, the cumulative distribution function of predicted streamflow at time t is evaluated with respect to the observation qt at t via a probability integral transform, vt = Pt(qt). If the probabilistic predictions of streamflow are suitable then the vt values will be mutually independent and distributed uniformly between 0 and 1. To test uniformity, a probability plot can be employed to graphically examine how well the distribution of vt values matches a U(0,1) distribution. The condition of mutual independence can be tested using the Kendall's tau test of independence.
 The probability plot of vt values versus a theoretical uniform distribution are shown in Figure 5, along with Kolmogorov confidence bands at the 95% confidence level. The distribution of vt values match that of a U(0,1) distribution very well, satisfying the first condition of the test. In addition, the condition of mutual independence was met under the Kendall's tau test of independence (p value of 0.81), satisfying the second condition of the test. These results provide further support for the error model chosen in this application.
Figure 5. Q-Q plot of the sample quantiles of the vt values versus those of a U(0,1) distribution. Kolmogorov confidence bands (dashed lines) at the 95% confidence level are also shown.
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4.5. Projections of Hydrologic Response With Uncertainty
 The Z = 73 baseline (1950–1999) and future (2050–2099) climate scenarios taken from the CMIP3 data set were each used to drive an ensemble of K = 5000 hydrologic model simulations, each with different parameter sets drawn from the posterior distributions developed in section 4.3. Four different annual streamflow statistics (Y) were considered in the analysis, including average January, March, April, and October streamflows. These monthly statistics were chosen because they exhibit a wide range of changes under future climate and highlight the importance of including hydrologic model error in climate impact assessments. These statistics were assumed to follow a lognormal distribution, similar to the observed historic streamflow data. This assumption was validated for each of these statistics under a large sample of climate scenarios and parameter sets using probability plots. Sampling error in the quantiles of these statistics was estimated using D = 1000 different estimates for the mean and standard deviation of the fitted lognormal distributions drawn from their posterior distributions. Results are presented as follows. The isolated effects of hydrologic model residual error on the estimation of these statistics are considered first. The integration of uncertainties from the range of climate projections, model residual error, model parameterization, and sampling uncertainty are then addressed. An analysis of alteration in different monthly statistics is then presented in the context of their integrated uncertainty estimates.
 Figure 7 presents the pdf of a fitted lognormal distribution to January monthly streamflows developed from one GCM scenario over the baseline period forced with one sample of hydrologic and error model parameters. Two pdf's are shown, one developed from the original streamflow trace, and a second developed from the same trace after being perturbed with noise generated from the error model. The variability in both future climate and parameter estimates is omitted by considering only one climate trace and parameter set, therefore isolating the effects of residual error on the distribution of the January flow statistic. As expected, the addition of residual error to the simulated streamflow trace causes the spread in January flows to increase. Addition of residual uncertainty to the model output appropriately adjusts the data so that it better represents the actual precision with which we can estimate characteristics of the streamflow statistic. Since the error model is logarithmic, the spread increases more at higher streamflow values than it does at lower values, suggesting different levels of precision for different magnitudes of flow. Interestingly, this highlights one of the difficulties in the choice of error model. While a transformation might make the data more tractable for a given error model, the application of that error model may lead to asymmetric uncertainty estimates after the transformation is reversed.
Figure 7. Probability density functions of baseline January monthly streamflow with (red dashed line) and without (black solid line) a perturbation with noise generated from the error model. Only one GCM scenario (z = 1) and parameter set (k = 1) were used to generate the streamflow trace.
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 To develop comprehensive uncertainty bounds around future hydrologic statistics, the residual error of the hydrologic model needs to be integrated with uncertainties in model parameterization, future climate projections, and sampling error. Figure 8 shows 95% predictive intervals for quantile estimates of baseline period January streamflow plotted against nonexceedance probabilities for different considerations of uncertainty. Figures 8a–8c show the isolated contributions of climate uncertainty, hydrologic model parameter and residual error, and sampling error to the uncertainty of quantile estimates, respectively. The range of quantile estimates in Figure 8a stems from the ensemble of baseline climate scenarios run over the median hydrologic model parameter set without the addition of residual noise. The range in Figure 8b was developed for only one ensemble member of baseline climate, but both parameter and residual uncertainties from the hydrologic model were considered. The influence of hydrologic model parameter and residual errors are aggregated and presented together in Figure 8b in order to represent the total added uncertainty from the hydrologic model. In Figure 8c, one baseline climate scenario was used to drive the hydrologic model with the median parameter set and no additional noise, but sampling uncertainty was calculated for each quantile. We note that the ranges of uncertainty in Figures 8a–8c are dependent on the climate ensemble member or parameter set that was held constant during their development and are thus only used to illustrate the range of isolated uncertainty bounds. Figures 8d–8f show the predictive bounds for quantile estimates when climate, hydrologic model, and sampling uncertainties are integrated together. Figure 8d is the same as Figure 8a, but Figure 8e shows the uncertainty bounds for quantile estimates when climate uncertainty, parameter uncertainty, and residual uncertainty are considered simultaneously. Figure 8f shows the total integrated uncertainty with sampling error considered as well.
Figure 8. (a–c) Isolated and (d–f) integrated 95% predictive intervals for quantiles of January streamflow over the baseline period. Uncertainty originating from a range of climate scenarios, parameter and residual errors in the hydrologic model, and sampling error are shown in isolation in Figures 8a, 8b, and 8c, respectively. Climate uncertainty in Figure 8a is repeated in Figure 8d, the integration of climate, parameter, and residual uncertainties is presented in Figure 8e, and Figure 8f shows the cumulative uncertainty after sampling error is also considered.
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 When comparing isolated and integrated uncertainties, it immediately becomes clear that uncertainties from climate projections, hydrologic model parameter and residual error, and sampling error cannot be independently added to generate reliable predictive bounds for estimates of hydrologic statistics and their properties. This is seen in Figure 8e and 8f, in which the range of uncertainty for many quantiles, particularly the larger ones, is greater than the sum of the uncertainties of their component parts (Figures 8a–8c). This property highlights the dependence of uncertainty bounds on the interactions between the different sources of uncertainty.
 This is a particularly important point, so we present a simplified example to emphasize it here. Consider a normalized streamflow quantile, Yp, with zero mean and a variance conditional on either isolated climate uncertainty ( ) or hydrologic modeling uncertainty ( ). Assuming Yp is normally distributed, a (1 − α) predictive interval under isolated climate uncertainty and isolated hydrologic modeling uncertainty could be respectively written as [ , ] and [ , ], where is the ( ) percentile of the standard normal distribution. Now assume that Yp can be expressed under the simple additive model Yp = εc + εh, where εc ∼ N(0, ) and εh ∼ N(0, ). Assuming that variations in Yp stemming from climate and hydrologic modeling uncertainty are independent, we would expect that the total variance of Yp would equal the sum of the isolated variances, . However, the predictive interval for Yp under integrated climate and hydrologic modeling uncertainty would be given as , , which does not correspond to the sum of the two isolated intervals above because . Therefore, even under the simplifying assumption that variations in Yp can be described by the simple additive model above, we would not expect uncertainty intervals to be additive. Thus, there is no reason to believe that uncertainty intervals would be additive given a more complex situation in which variations in Yp can be influenced by the interactions of different sources of uncertainty within a hydrologic modeling framework.
 The dependence of variations in Yp on interactions between different sources of uncertainty can be traced to several contributing factors. First, the hydrologic model being considered is nonlinear, so different parameterizations of that model will result in nonlinear responses to a given climate. When those various parameterizations are used to simulate hydrologic response over a range of climates, there is the potential that the combination of an extreme climate ensemble member and parameter set will lead to significantly different streamflow responses than that seen under just climate or parameter uncertainty alone. Another source of dependency arises from the interaction between the error model and the ensemble of climate members. Because the error model used in this application is based on a logarithmic transformation, the uncertainty of large quantile values becomes highly skewed to the right after residual uncertainty is accounted. If an ensemble climate member leads to slightly larger quantile values for the streamflow statistic being considered, the residual error estimated for those larger quantiles could lead to the significant expansion of their predictive bounds. Finally, there are significant interactions between sampling error estimation and both hydrologic and climate model uncertainties. Sampling error uncertainty bounds will grow with the uncertainty in the parameters of the distribution used to model the streamflow statistic. The sampling distributions of these parameters will likely change when climate and hydrologic model uncertainties are considered, causing the magnitude of sampling error to change with respect to its range when considered in isolation.
 After aggregating the uncertainties from climate scenarios, the hydrologic model, and sampling error, it becomes evident that some quantile values for certain streamflow statistics can only be estimated with limited precision. This is shown for the cumulative error under baseline climate conditions in Figure 8f. In the case of future climate conditions, the range of climate projections becomes far more significant. Figure 9 compares the cumulative uncertainty of January monthly flows evaluated over the historic and future climate conditions. Figure 9a is the same as in Figure 8f, but Figure 9b shows the uncertainty in future climate projections.
Figure 9. Integrated 95% predictive bounds for January flow quantiles under the (a) baseline and (b) future periods.
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 Two primary differences arise between the baseline and future cumulative uncertainties for January flow quantiles. First, the underlying climate uncertainty is far greater under the future scenarios than those of the baseline. This is expected because the baseline climate projections are all directly mapped to the historical trace of temperature and precipitation via downscaling. Thus, the range of historical projections does not model climate uncertainty or even climate model uncertainty but rather is an artifact of the bias correction method. Consequently, the range of future projections also does not model the uncertainty of future climate or even the model uncertainty of future climate projections. Nonetheless, the range of climate projections is commonly used to provide some sense of the uncertainty in the projections that arise due to model error and internal variability and are used for that purpose. That range, albeit a minimum range of climate uncertainty, significantly increases the uncertainty in the quantile estimates relative to hydrologic modeling uncertainty as shown in the comparison between Figures 9a and 9b. Second, the sampling error for larger quantiles is vastly greater for the future scenarios than for the baseline. This is due to the greater spread of January flows under future conditions and its influence on sampling error estimates. Overall, it is clear that the cumulative uncertainty for quantile estimates of this statistic is much greater for the future than it is under baseline conditions.
 Quantile estimates can be directly compared between baseline and future scenarios in the context of their cumulative uncertainties to help determine the level of confidence that can be associated with their possible alteration under climate change. Figure 10 presents the cumulative uncertainty of quantile estimates of monthly streamflow statistics in the White River for future and baseline conditions. Here, no distinction is made between the different sources of uncertainty (e.g., climate, hydrologic, or sampling errors). Rather, the cumulative 95% predictive intervals for flow quantiles under baseline and future conditions are overlaid on each other to provide a representation of whether changes in streamflow under climate change exceed the range of uncertainty that arises during the modeling process. Less overlap between predictive intervals of flow quantiles under baseline and future conditions provides greater confidence that the flow quantile will actual differ under future climate conditions. Figure 10a shows that there are significant differences between the distributions of January flows in the baseline and future periods even after accounting for cumulative modeling uncertainties. Results suggest that climate projections of January flows are significantly higher in the future than in the present, likely due to a shift in the snowfall to precipitation ratio driven by increased wintertime temperatures. Over most January quantiles, approximately half of the bounded region for future conditions lies completely outside the range of baseline uncertainty. This suggests that this range of climate changes rises to a level that is well above the baseline uncertainty.
Figure 10. Integrated 95% predictive bounds in flow quantiles for baseline and future periods for the months of (a) January, (b) March, (c) April, and (d) October.
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 Figures 10b and 10c show results for March and April average streamflows, respectively. The range of climate projections show March flows increasing in the future while April flows decrease. These changes are consistent with earlier snowmelt occurrences and decreases in snowpack storage that historically have persisted into the later spring. Interestingly, the highest quantiles of March flows for the future period show minor departures from those of the baseline; differences become more noticeable for flows below the 95th percentile. This is not the case for April flows, which show more significant departures between baseline and future flows at the highest quantiles. This suggests that more confidence can be associated with shifts in the highest flows during April than in March. This is likely because snowpack, a driving factor of the largest spring flows, is consistently reduced in April under all future hydroclimatic projections, but is more variable across the projections in the month of March.
 Figure 10d shows results for the month of October. The range of climate projections exceeds only minutely the baseline uncertainty bounds for October quantiles. The spread in the future period for most quantiles extends both below and above that of the baseline period, although the changes are extremely small except for the higher quantiles. These results suggest that no real change in most October flow quantiles are projected in this set of CMIP3 climate changes.