Total Predictive Uncertainty Research Articles

Assessing the Effect of Uncertainties in Pollutant Wash-Off Dynamics in Stormwater Source-Control Systems Modeling: Consequences of Using an Inappropriate Error Model

This study investigates the effect of uncertainties associated with pollutant wash-off dynamics in the context of stormwater management practices modelling. A formal Bayesian approach is adopted for the calibration and the uncertainty analysis of a commonly used wash-off model, under (1) the unverified assumption of homoscedastic, independent and normally distributed residuals and (2) using a more correct heteroscedastic and autoregressive error model. The results obtained for each of these approaches are compared and the uncertainty associated with water quality modelling is later propagated through a conceptual Best Management Practices (BMP) model, for various stormwater management scenarios, so as to assess the effect of this uncertainty for BMP modelling and clarify the benefits of a robust description of error structure. This study indicates that the violation of the statistical asumptions about the residuals may result in unreliable estimation of model parameters and total predictive uncertainty. The effect of the uncertainty in the intra-event variability of concentations in runoff is however found to have only a limited effect on the outputs of the BMP model, regadless of the error model adopted for calibration.

Effect of formal and informal likelihood functions on uncertainty assessment in a single event rainfall-runoff model

In the present study, DREAM(ZS), Differential Evolution Adaptive Metropolis combined with both formal and informal likelihood functions, is used to investigate uncertainty of parameters of the HEC-HMS model in Tamar watershed, Golestan province, Iran.In order to assess the uncertainty of 24 parameters used in HMS, three flood events were used to calibrate and one flood event was used to validate the posterior distributions. Moreover, performance of seven different likelihood functions (L1–L7) was assessed by means of DREAM(ZS)approach. Four likelihood functions, L1–L4, Nash–Sutcliffe (NS) efficiency, Normalized absolute error (NAE), Index of agreement (IOA), and Chiew–McMahon efficiency (CM), is considered as informal, whereas remaining (L5–L7) is represented in formal category. L5 focuses on the relationship between the traditional least squares fitting and the Bayesian inference, and L6, is a hetereoscedastic maximum likelihood error (HMLE) estimator. Finally, in likelihood function L7, serial dependence of residual errors is accounted using a first-order autoregressive (AR) model of the residuals.According to the results, sensitivities of the parameters strongly depend on the likelihood function, and vary for different likelihood functions. Most of the parameters were better defined by formal likelihood functions L5 and L7 and showed a high sensitivity to model performance. Posterior cumulative distributions corresponding to the informal likelihood functions L1, L2, L3, L4 and the formal likelihood function L6 are approximately the same for most of the sub-basins, and these likelihood functions depict almost a similar effect on sensitivity of parameters. 95% total prediction uncertainty bounds bracketed most of the observed data. Considering all the statistical indicators and criteria of uncertainty assessment, including RMSE, KGE, NS, P-factor and R-factor, results showed that DREAM(ZS) algorithm performed better under formal likelihood functions L5 and L7, but likelihood function L5 may result in biased and unreliable estimation of parameters due to violation of the residualerror assumptions. Thus, likelihood function L7 provides posterior distribution of model parameters credibly and therefore can be employed for further applications.

Global Optimization Under Uncertainty and Uncertainty Quantification Applied to Tractor-Trailer Base Flaps

Using a global optimization evolutionary algorithm (EA), propagating aleatory and epistemic uncertainty within the optimization loop, and using computational fluid dynamics (CFD), this study determines a design for a 3D tractor-trailer base (back-end) drag reduction device that reduces the wind-averaged drag coefficient by 41% at 57 mph (92 km/h). Because it is optimized under uncertainty, this design is relatively insensitive to uncertain wind speed and direction and uncertain deflection angles due to mounting accuracy and static aeroelastic loading. The model includes five design variables with generous constraints, and this study additionally includes the uncertain effects on drag prediction due to truck speed and elevation, steady Reynolds-averaged Navier–Stokes (RANS) approximation, and numerical approximation. This study uses the Design Analysis Kit for Optimization and Terascale Applications (DAKOTA) optimization and uncertainty quantification (UQ) framework to interface the RANS flow solver, grid generator, and optimization algorithm. The computational model is a simplified full-scale tractor-trailer with flow at highway speed. For the optimized design, the estimate of total predictive uncertainty is +15/−42%; 8–10% of this uncertainty comes from model form (computation versus experiment); 3–7% from model input (wind speed and direction, flap angle, and truck speed); and +0.0/−28.5% from numerical approximation (due to the relatively coarse, 6 × 106 cell grid). Relative comparison of designs to the no-flaps baseline should have considerably less uncertainty because numerical error and input variation are nearly eliminated and model form differences are reduced. The total predictive uncertainty is also presented in the form of a probability box, which may be used to decide how to improve the model and reduce uncertainty.

Claims Run-Off Uncertainty: The Full Picture

The aim of this contribution is to revisit, clarify and complete the picture of uncertainty estimates in the chain-ladder (CL) claims reserving method. Therefore, we consider the conditional mean square error of prediction (MSEP) of the total prediction uncertainty (using Mack’s formula) and the one of the one-year prediction uncertainty (using the Merz-Wuthrich formula). We discuss and compare these two formulas and we derive the missing pieces that close the gap between the one-year and total run-off uncertainty view.

Considering rating curve uncertainty in water level predictions

Abstract. Streamflow cannot be measured directly and is typically derived with a rating curve model. Unfortunately, this causes uncertainties in the streamflow data and also influences the calibration of rainfall-runoff models if they are conditioned on such data. However, it is currently unknown to what extent these uncertainties propagate to rainfall-runoff predictions. This study therefore presents a quantitative approach to rigorously consider the impact of the rating curve on the prediction uncertainty of water levels. The uncertainty analysis is performed within a formal Bayesian framework and the contributions of rating curve versus rainfall-runoff model parameters to the total predictive uncertainty are addressed. A major benefit of the approach is its independence from the applied rainfall-runoff model and rating curve. In addition, it only requires already existing hydrometric data. The approach was successfully demonstrated on a small catchment in Poland, where a dedicated monitoring campaign was performed in 2011. The results of our case study indicate that the uncertainty in calibration data derived by the rating curve method may be of the same relevance as rainfall-runoff model parameters themselves. A conceptual limitation of the approach presented is that it is limited to water level predictions. Nevertheless, regarding flood level predictions, the Bayesian framework seems very promising because it (i) enables the modeler to incorporate informal knowledge from easily accessible information and (ii) better assesses the individual error contributions. Especially the latter is important to improve the predictive capability of hydrological models.

Predicting morphological changes DS New Naga-Hammadi Barrage for extreme Nile flood flows: A Monte Carlo analysis

While construction of the Aswan High Dam (AHD) has stopped concurrent flooding events, River Nile is still subject to low intensity flood waves resulting from controlled release of water from the dam reservoir. Analysis of flow released from New Naga-Hammadi Barrage, which is located at 3460km downstream AHD indicated an increase in magnitude of flood released from the barrage in the past 10years. A 2D numerical mobile bed model is utilized to investigate the possible morphological changes in the downstream of Naga-Hammadi Barrage from possible higher flood releases. Monte Carlo simulation analyses (MCS) is applied to the deterministic results of the 2D model to account for and assess the uncertainty of sediment parameters and formulations in addition to sacristy of field measurements. Results showed that the predicted volume of erosion yielded the highest uncertainty and variation from deterministic run, while navigation velocity yielded the least uncertainty. Furthermore, the error budget method is used to rank various sediment parameters for their contribution in the total prediction uncertainty. It is found that the suspended sediment contributed to output uncertainty more than other sediment parameters followed by bed load with 10% less order of magnitude.

Towards a comprehensive approach to parameter estimation in land surface parameterization schemes

AbstractIn climate models, the land–atmosphere interactions are described numerically by land surface parameterization (LSP) schemes. The continuing improvement in realism in these schemes comes at the expense of the need to specify a large number of parameters that are either directly measured or estimated. Also, an emerging problem is whether the relationships used in LSPs are universal and globally applicable. One plausible approach to evaluate this is to first minimize uncertainty in model parameters by calibration. In this paper, we conduct a comprehensive analysis of some model diagnostics using a slightly modified version of the Simple Biosphere 3 model for a variety of biomes located mainly in the Amazon. First, the degree of influence of each individual parameter in simulating surface fluxes is identified. Next, we estimate parameters using a multi‐operator genetic algorithm applied in a multi‐objective context and evaluate simulations of energy and carbon fluxes against observations. Compared with the default parameter sets, these parameter estimates improve the partitioning of energy fluxes in forest and cropland sites and provide better simulations of daytime increases in assimilation of net carbon during the dry season at forest sites. Finally, a detailed assessment of the parameter estimation problem was performed by accounting for the decomposition of the mean squared error to the total model uncertainty. Analysis of the total prediction uncertainty reveals that the parameter adjustments significantly improve reproduction of the mean and variability of the flux time series at all sites and generally remove seasonality of the errors but do not improve dynamical properties. Our results demonstrate that error decomposition provides a meaningful and intuitive way to understand differences in model performance. To make further advancements in the knowledge of these models, we encourage the LSP community to adopt similar approaches in the future. Copyright © 2012 John Wiley & Sons, Ltd.

Bayesian uncertainty assessment of flood predictions in ungauged urban basins for conceptual rainfall-runoff models

Abstract. Urbanization and the resulting land-use change strongly affect the water cycle and runoff-processes in watersheds. Unfortunately, small urban watersheds, which are most affected by urban sprawl, are mostly ungauged. This makes it intrinsically difficult to assess the consequences of urbanization. Most of all, it is unclear how to reliably assess the predictive uncertainty given the structural deficits of the applied models. In this study, we therefore investigate the uncertainty of flood predictions in ungauged urban basins from structurally uncertain rainfall-runoff models. To this end, we suggest a procedure to explicitly account for input uncertainty and model structure deficits using Bayesian statistics with a continuous-time autoregressive error model. In addition, we propose a concise procedure to derive prior parameter distributions from base data and successfully apply the methodology to an urban catchment in Warsaw, Poland. Based on our results, we are able to demonstrate that the autoregressive error model greatly helps to meet the statistical assumptions and to compute reliable prediction intervals. In our study, we found that predicted peak flows were up to 7 times higher than observations. This was reduced to 5 times with Bayesian updating, using only few discharge measurements. In addition, our analysis suggests that imprecise rainfall information and model structure deficits contribute mostly to the total prediction uncertainty. In the future, flood predictions in ungauged basins will become more important due to ongoing urbanization as well as anthropogenic and climatic changes. Thus, providing reliable measures of uncertainty is crucial to support decision making.

Technical Note: The normal quantile transformation and its application in a flood forecasting system

Abstract. The Normal Quantile Transform (NQT) has been used in many hydrological and meteorological applications in order to make the Cumulated Distribution Function (CDF) of the observed, simulated and forecast river discharge, water level or precipitation data Gaussian. It is also the heart of the meta-Gaussian model for assessing the total predictive uncertainty of the Hydrological Uncertainty Processor (HUP) developed by Krzysztofowicz. In the field of geo-statistics this transformation is better known as the Normal-Score Transform. In this paper some possible problems caused by small sample sizes when applying the NQT in flood forecasting systems will be discussed and a novel way to solve the problem will be outlined by combining extreme value analysis and non-parametric regression methods. The method will be illustrated by examples of hydrological stream-flow forecasts.

A HOLISTIC APPROACH TO UNCERTAINTY QUANTIFICATION WITH APPLICATION TO SUPERSONIC NOZZLE THRUST

In modeling and simulation (MS i.e., a p-box. Implications of sampling over interval versus probabilistic uncertainties in the outer loop are discussed and examples are given showing the effects of the choice of uncertainty propagation and characterization methods. In addition to the uncertainties in model inputs, uncertainties also arise due to modeling deficiencies and numerical approximations. Modeling uncertainties can be reduced by performing additional experiments and numerical uncertainties can be reduced by using additional computational resources; thus, both sources of uncertainty can be modeled as epistemic and can be characterized as intervals and included in the total predictive uncertainty by appropriately broadening the prediction p-box. A simple example is given for the M&S predictions of supersonic nozzle thrust that incorporates and quantifies all three sources of uncertainty.

Toward a reliable decomposition of predictive uncertainty in hydrological modeling: Characterizing rainfall errors using conditional simulation

This study explores the decomposition of predictive uncertainty in hydrological modeling into its contributing sources. This is pursued by developing data‐based probability models describing uncertainties in rainfall and runoff data and incorporating them into the Bayesian total error analysis methodology (BATEA). A case study based on the Yzeron catchment (France) and the conceptual rainfall‐runoff model GR4J is presented. It exploits a calibration period where dense rain gauge data are available to characterize the uncertainty in the catchment average rainfall using geostatistical conditional simulation. The inclusion of information about rainfall and runoff data uncertainties overcomes ill‐posedness problems and enables simultaneous estimation of forcing and structural errors as part of the Bayesian inference. This yields more reliable predictions than approaches that ignore or lump different sources of uncertainty in a simplistic way (e.g., standard least squares). It is shown that independently derived data quality estimates are needed to decompose the total uncertainty in the runoff predictions into the individual contributions of rainfall, runoff, and structural errors. In this case study, the total predictive uncertainty appears dominated by structural errors. Although further research is needed to interpret and verify this decomposition, it can provide strategic guidance for investments in environmental data collection and/or modeling improvement. More generally, this study demonstrates the power of the Bayesian paradigm to improve the reliability of environmental modeling using independent estimates of sampling and instrumental data uncertainties.

Predictive uncertainty of chaotic daily streamflow using ensemble wavelet networks approach

Perfect or even mediocre weather predictions over a long period are almost impossible because of the ultimate growth of a small initial error into a significant one. Even though the sensitivity of initial conditions limits the predictability in chaotic systems, an ensemble of prediction from different possible initial conditions and also a prediction algorithm capable of resolving the fine structure of the chaotic attractor can reduce the prediction uncertainty to some extent. All of the traditional chaotic prediction methods in hydrology are based on single optimum initial condition local models which can model the sudden divergence of the trajectories with different local functions. Conceptually, global models are ineffective in modeling the highly unstable structure of the chaotic attractor. This paper focuses on an ensemble prediction approach by reconstructing the phase space using different combinations of chaotic parameters, i.e., embedding dimension and delay time to quantify the uncertainty in initial conditions. The ensemble approach is implemented through a local learning wavelet network model with a global feed‐forward neural network structure for the phase space prediction of chaotic streamflow series. Quantification of uncertainties in future predictions are done by creating an ensemble of predictions with wavelet network using a range of plausible embedding dimensions and delay times. The ensemble approach is proved to be 50% more efficient than the single prediction for both local approximation and wavelet network approaches. The wavelet network approach has proved to be 30%–50% more superior to the local approximation approach. Compared to the traditional local approximation approach with single initial condition, the total predictive uncertainty in the streamflow is reduced when modeled with ensemble wavelet networks for different lead times. Localization property of wavelets, utilizing different dilation and translation parameters, helps in capturing most of the statistical properties of the observed data. The need for taking into account all plausible initial conditions and also bringing together the characteristics of both local and global approaches to model the unstable yet ordered chaotic attractor of a hydrologic series is clearly demonstrated.

Confronting uncertainty in model-based geostatistics using Markov Chain Monte Carlo simulation

This paper demonstrates for the first time the use of Markov Chain Monte Carlo (MCMC) simulation for parameter inference in model-based soil geostatistics. We implemented the recently developed DiffeRential Evolution Adaptive Metropolis (DREAM) algorithm to jointly summarize the posterior distribution of variogram parameters and the coefficients of a linear spatial model, and derive estimates of predictive uncertainty. The DREAM method runs multiple different Markov chains in parallel and jumps in each chain are generated from a discrete proposal distribution containing a fixed multiple of the difference of the states of randomly chosen pairs of other chains. This approach automatically scales the orientation and scale of the proposal distribution, and is especially designed to maintain detailed balance and ergodicity, thereby generating an exact approximation of the posterior probability density function (pdf) of the parameters of the linear model and variogram. This approach is tested using three different data sets from Australia involving variogram estimation of soil thickness, kriging of soil pH, and spatial prediction of soil organic carbon content. The results showed some advantages of MCMC over the conventional method of moments and residual maximum likelihood (REML) estimation. The posterior pdf derived with MCMC conveys important information about parameter uncertainty, multi-dimensional parameter correlation, and thus how many significant parameters are warranted by the calibration data. Parameter uncertainty constitutes only a small part of total prediction uncertainty for the case studies considered here. The prediction accuracies using MCMC and REML are similar. The variogram estimated using conventional approaches (method of moments, and without simulation) lies within the 95% prediction uncertainty interval of the posterior distribution derived with DREAM. Altogether our results show that conventional kriging and regression-kriging still remain a viable option for production mapping.

A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing

An overview of a comprehensive framework is given for estimating the predictive uncertainty of scientific computing applications. The framework is comprehensive in the sense that it treats both types of uncertainty (aleatory and epistemic), incorporates uncertainty due to the mathematical form of the model, and it provides a procedure for including estimates of numerical error in the predictive uncertainty. Aleatory (random) uncertainties in model inputs are treated as random variables, while epistemic (lack of knowledge) uncertainties are treated as intervals with no assumed probability distributions. Approaches for propagating both types of uncertainties through the model to the system response quantities of interest are briefly discussed. Numerical approximation errors (due to discretization, iteration, and computer round off) are estimated using verification techniques, and the conversion of these errors into epistemic uncertainties is discussed. Model form uncertainty is quantified using (a) model validation procedures, i.e., statistical comparisons of model predictions to available experimental data, and (b) extrapolation of this uncertainty structure to points in the application domain where experimental data do not exist. Finally, methods for conveying the total predictive uncertainty to decision makers are presented. The different steps in the predictive uncertainty framework are illustrated using a simple example in computational fluid dynamics applied to a hypersonic wind tunnel.

Prediction intervals to account for uncertainties in neural network predictions: Methodology and application in bus travel time prediction

Neural networks have been employed in a multitude of transportation engineering applications because of their powerful capabilities to replicate patterns in field data. Predictions are always subject to uncertainty arising from two sources: model structure and training data. For each prediction point, the former can be quantified by a confidence interval, whereas total prediction uncertainty can be represented by constructing a prediction interval. While confidence intervals are well known in the transportation engineering context, very little attention has been paid to construction of prediction intervals for neural networks. The proposed methodology in this paper provides a foundation for constructing prediction intervals for neural networks and quantifying the extent that each source of uncertainty contributes to total prediction uncertainty. The application of the proposed methodology to predict bus travel time over four bus route sections in Melbourne, Australia, leads to quantitative decomposition of total prediction uncertainty into the component sources. Overall, the results demonstrate the capability of the proposed method to provide robust prediction intervals.

A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non‐Gaussian errors

Estimation of parameter and predictive uncertainty of hydrologic models has traditionally relied on several simplifying assumptions. Residual errors are often assumed to be independent and to be adequately described by a Gaussian probability distribution with a mean of zero and a constant variance. Here we investigate to what extent estimates of parameter and predictive uncertainty are affected when these assumptions are relaxed. A formal generalized likelihood function is presented, which extends the applicability of previously used likelihood functions to situations where residual errors are correlated, heteroscedastic, and non‐Gaussian with varying degrees of kurtosis and skewness. The approach focuses on a correct statistical description of the data and the total model residuals, without separating out various error sources. Application to Bayesian uncertainty analysis of a conceptual rainfall‐runoff model simultaneously identifies the hydrologic model parameters and the appropriate statistical distribution of the residual errors. When applied to daily rainfall‐runoff data from a humid basin we find that (1) residual errors are much better described by a heteroscedastic, first‐order, auto‐correlated error model with a Laplacian distribution function characterized by heavier tails than a Gaussian distribution; and (2) compared to a standard least‐squares approach, proper representation of the statistical distribution of residual errors yields tighter predictive uncertainty bands and different parameter uncertainty estimates that are less sensitive to the particular time period used for inference. Application to daily rainfall‐runoff data from a semiarid basin with more significant residual errors and systematic underprediction of peak flows shows that (1) multiplicative bias factors can be used to compensate for some of the largest errors and (2) a skewed error distribution yields improved estimates of predictive uncertainty in this semiarid basin with near‐zero flows. We conclude that the presented methodology provides improved estimates of parameter and total prediction uncertainty and should be useful for handling complex residual errors in other hydrologic regression models as well.

Propagation of uncertainties in coupled hydro-meteorological forecasting systems: A stochastic approach for the assessment of the total predictive uncertainty

The pressure on the scientific community to provide medium term flood forecasts with associated meaningful predictive uncertainty estimations has increased in recent years. A technique for assessing this uncertainty in hydro-meteorological forecasting systems is presented. In those, the uncertainties generally propagate from an atmospheric model through a rainfall-runoff model. Consequently, it appears to be difficult to isolate the errors that stem from the individual model components. In this study, the integrated flood forecasting system uses the 7-day rainfall and temperature forecast of the American atmospheric GFS model (deterministic run) as forcing data in a conceptual hydrologic model (deterministic run) coupled with a linear error model in order to predict river discharge. The linear error model is added to the hydrologic model run, in order to take advantage of the correlation in time between forecasting errors, thereby reducing errors that arise from hydrologic simulations. To assess the predictive uncertainty (total uncertainty) of the coupled models, the method makes use of a bivariate meta-gaussian probability density function. The latter allows estimating the probability distribution of the integrated model errors conditioned by the predicted river discharge values. The proposed methodology is applied to the case study of the Alzette river located in the Grand Duchy of Luxembourg. Confidence limits are computed for various lead times of prediction and compared with observations of river discharge.

Understanding predictive uncertainty in hydrologic modeling: The challenge of identifying input and structural errors

Meaningful quantification of data and structural uncertainties in conceptual rainfall‐runoff modeling is a major scientific and engineering challenge. This paper focuses on the total predictive uncertainty and its decomposition into input and structural components under different inference scenarios. Several Bayesian inference schemes are investigated, differing in the treatment of rainfall and structural uncertainties, and in the precision of the priors describing rainfall uncertainty. Compared with traditional lumped additive error approaches, the quantification of the total predictive uncertainty in the runoff is improved when rainfall and/or structural errors are characterized explicitly. However, the decomposition of the total uncertainty into individual sources is more challenging. In particular, poor identifiability may arise when the inference scheme represents rainfall and structural errors using separate probabilistic models. The inference becomes ill‐posed unless sufficiently precise prior knowledge of data uncertainty is supplied; this ill‐posedness can often be detected from the behavior of the Monte Carlo sampling algorithm. Moreover, the priors on the data quality must also be sufficiently accurate if the inference is to be reliable and support meaningful uncertainty decomposition. Our findings highlight the inherent limitations of inferring inaccurate hydrologic models using rainfall‐runoff data with large unknown errors. Bayesian total error analysis can overcome these problems using independent prior information. The need for deriving independent descriptions of the uncertainties in the input and output data is clearly demonstrated.

Parameter optimization and uncertainty analysis for plot-scale continuous modeling of runoff using a formal Bayesian approach

This study reports on the use the recently developed Differential Evolution Adaptative Metropolis algorithm (DREAM) to calibrate in a Bayesian framework the continuous, spatially distributed, process-based and plot-scale runoff model described by Laloy and Bielders (2008). The calibration procedure, relying on 2 years of daily runoff measurements, accounted for heteroscedasticity and autocorrelation. This resulted in a realistic estimation of parameter uncertainty and its impact on model predictions. The calibrated model reproduced the observed hydrograph satisfactorily during calibration. The model validation on an independent 1-year series of measurements showed reasonable values for the Nash-Sutcliffe efficiency criterion. This was equal to 0.76 and 0.78 for, respectively, the Upper and lower bounds of the 95% uncertainty interval associated with parameter uncertainty. However, the fact that this interval was always very narrow but often did not bracket the observations during the validation period indicates that, assuming observational errors to be negligible, model structure has to be improved in order to achieve more accurate predictions. Overall the methodology allowed to efficiently discriminate between parameter and total predictive uncertainty and to highlight scope for model improvement. (C) 2009 Elsevier B.V. All rights reserved.

Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?

In recent years, a strong debate has emerged in the hydrologic literature regarding what constitutes an appropriate framework for uncertainty estimation. Particularly, there is strong disagreement whether an uncertainty framework should have its roots within a proper statistical (Bayesian) context, or whether such a framework should be based on a different philosophy and implement informal measures and weaker inference to summarize parameter and predictive distributions. In this paper, we compare a formal Bayesian approach using Markov Chain Monte Carlo (MCMC) with generalized likelihood uncertainty estimation (GLUE) for assessing uncertainty in conceptual watershed modeling. Our formal Bayesian approach is implemented using the recently developed differential evolution adaptive metropolis (DREAM) MCMC scheme with a likelihood function that explicitly considers model structural, input and parameter uncertainty. Our results demonstrate that DREAM and GLUE can generate very similar estimates of total streamflow uncertainty. This suggests that formal and informal Bayesian approaches have more common ground than the hydrologic literature and ongoing debate might suggest. The main advantage of formal approaches is, however, that they attempt to disentangle the effect of forcing, parameter and model structural error on total predictive uncertainty. This is key to improving hydrologic theory and to better understand and predict the flow of water through catchments.