Response to comment by Keith Beven on “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.     

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.     

Uncertainty and sensitivity in a bank stability model: implications for estimating phosphorus loading

Eutrophication of aquatic ecosystems is one of the most pressing water quality concerns in the United States and around the world. Bank erosion has been largely overlooked as a source of nutrient loading, despite field studies demonstrating that this source can account for the majority of the total phosphorus load in a watershed. Substantial effort has been made to develop mechanistic models to predict bank erosion and instability in stream systems; however, these models do not account for inherent natural variability in input values. To quantify the impacts of this omission, uncertainty and sensitivity analyses were performed on the Bank Stability and Toe Erosion Model (BSTEM), a mechanistic model developed by the US Department of Agriculture – Agricultural Research Service (USDA‐ARS) that simulates both mass wasting and fluvial erosion of streambanks. Generally, bank height, soil cohesion, and plant species were found to be most influential in determining stability of clay (cohesive) banks. In addition to these three inputs, groundwater elevation, stream stage, and bank angle were also identified as important in sand (non‐cohesive) banks. Slope and bank height are the dominant variables in fluvial erosion modeling, while erodibility and critical shear stress had low sensitivity indices; however, these indices do not reflect the importance of critical shear stress in determining the timing of erosion events. These results identify important variables that should be the focus of data collection efforts while also indicating which less influential variables may be set to assumed values. In addition, a probabilistic Monte‐Carlo modeling approach was applied to data from a watershed‐scale sediment and phosphorus loading study on the Missisquoi River, Vermont to quantify uncertainty associated with these published results. While our estimates aligned well with previous deterministic modeling results, the uncertainty associated with these predictions suggests that they should be considered order of magnitude estimates only. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple objective evaluation of a simple phosphorus transfer model

A simple phosphorus (P) transfer model of the Welland catchment, UK, is evaluated against multiple objective functions using a Monte Carlo approach that combines calibration, identifiability, sensitivity and uncertainty analysis. The model is based on simple conceptual rainfall‐runoff and river routing components, combined with estimates of the daily non‐point source load derived from annual landuse‐based export coefficients, disaggregated as a function of the runoff. The model has limited data requirements, consistent with data availability, and is parsimoneous with respect to the number of parameters identified through inverse modelling. The best performing parameter sets capture the main aspects of the observed flow and total P (TP) concentrations and provide a suitable basis for a decision‐support tool. However, a trade‐off is evident between matching the observed flow peaks, flow recessions and TP concentrations simultaneously, highlighting some limitations of the model structure and/or calibration data. Model analysis indicates that daily non‐point source load cannot be described as a function of near‐surface runoff and land use alone, but that other influences, including seasonality, are important. However, further model development to improve performance is likely to introduce additional complexity (in terms of parameter numbers), and hence additional problems of parameter identifiability and output uncertainty, which in turn raises issues of the information content of the available data. Copyright © 2004 John Wiley & Sons, Ltd.     

The integrated effects of climate and hydrologic uncertainty on future flood risk assessments

This work examines future flood risk within the context of integrated climate and hydrologic modelling uncertainty. The research questions investigated are (1) whether hydrologic uncertainties are a significant source of uncertainty relative to other sources such as climate variability and change and (2) whether a statistical characterization of uncertainty from a lumped, conceptual hydrologic model is sufficient to account for hydrologic uncertainties in the modelling process. To investigate these questions, an ensemble of climate simulations are propagated through hydrologic models and then through a reservoir simulation model to delimit the range of flood protection under a wide array of climate conditions. Uncertainty in mean climate changes and internal climate variability are framed using a risk‐based methodology and are explored using a stochastic weather generator. To account for hydrologic uncertainty, two hydrologic models are considered, a conceptual, lumped parameter model and a distributed, physically based model. In the conceptual model, parameter and residual error uncertainties are quantified and propagated through the analysis using a Bayesian modelling framework. The approach is demonstrated in a case study for the Coralville Dam on the Iowa River, where recent, intense flooding has raised questions about potential impacts of climate change on flood protection adequacy. Results indicate that the uncertainty surrounding future flood risk from hydrologic modelling and internal climate variability can be of the same order of magnitude as climate change. Furthermore, statistical uncertainty in the conceptual hydrological model can capture the primary structural differences that emerge in flood damage estimates between the two hydrologic models. Copyright © 2014 John Wiley & Sons, Ltd.     

Modelling the hydrologic effects of dynamic land‐use change using a distributed hydrologic model and a spatial land‐use allocation model

This study develops a novel approach for modelling and examining the impacts of time–space land‐use changes on hydrological components. The approach uses an empirical land‐use change allocation model (CLUE‐s) and a distributed hydrological model (DHSVM) to examine various land‐use change scenarios in the Wu‐Tu watershed in northern Taiwan. The study also uses a generalized likelihood uncertainty estimation approach to quantify the parameter uncertainty of the distributed hydrological model. The results indicate that various land‐use policies—such as no change, dynamic change and simultaneous change—have different levels of impact on simulating the spatial distributions of hydrological components in the watershed study. Peak flow rates under simultaneous and dynamic land‐use changes are 5·71% and 2·77%, respectively, greater than the rate under the no land‐use change scenario. Using dynamic land‐use changes to assess the effect of land‐use changes on hydrological components is more practical and feasible than using simultaneous land‐use change and no land‐use change scenarios. Furthermore, land‐use change is a spatial dynamic process that can lead to significant changes in the distributions of ground water and soil moisture. The spatial distributions of land‐use changes influence hydrological processes, such as the ground water level of whole areas, particularly in the downstream watershed. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian uncertainty analysis in hydrological modeling associated with watershed subdivision level: a case study of SLURP model applied to the Xiangxi River watershed,China

Uncertainty analysis in hydrological modeling would help to better implement decision-making related to water resources management, which relies heavily on hydrologic simulations. However, an important concern will be raised over the uncertainty associated with watershed subdivision broadly applied in distributed/semi-distributed hydrological models since scale issues would significantly affect model performance, and thus, lead to dramatic variations in simulations. To fully understand the uncertainty associated with watershed subdivision level, however, is still a tough work confronting researchers because of complex modeling processes and high computation requirements. In this study, we analyzed this uncertainty within a formal Bayesian framework using a Markov Chain Monte Carlo method based on Metropolis–Hastings algorithm. In a case study using the semi-distributed land use-based runoff processes hydrologic model in the Xiangxi River watershed, results showed that the variation in the simulated discharges due to parameter uncertainty was much smaller than that due to parameter and model uncertainty under different watershed subdivision levels defined using aggregated simulation areas (ASAs). However, the posterior probability distribution of model parameters varied in response to subdivision levels, and four parameters (i.e. maximum infiltration rate, retention constant for slow store, maximum capacity for slow store, and retention constant for fast store) were identified with smaller uncertainty. Although the uncertainty in the simulated discharge due to parameter and model uncertainty varied little across subdivisions, the simulation uncertainty only due to parameter uncertainty was found to be reduced through increasing the subdivisions. In addition, the coarsest subdivision level (7 ASAs) was not sufficient for obtaining satisfying simulations in the Xiangxi River watershed, but inappreciable improvement was achieved through increasing the level among finer subdivisions. Moreover, it was demonstrated that increasing subdivision level would have no advantage of improving the reliability of hydrological simulations beyond the threshold (45 ASAs). The findings of this research may shed light on the design of operational hydrological forecasting in the Three Gorges Reservoir region with profound socio-economic implications.     

Assessing uncertainties in a conceptual water balance model using Bayesian methodology / Estimation bayésienne des incertitudes au sein d’une modélisation conceptuelle de bilan hydrologique

Abstract

Abstract The aim of this study was to estimate the uncertainties in the streamflow simulated by a rainfall–runoff model. Two sources of uncertainties in hydrological modelling were considered: the uncertainties in model parameters and those in model structure. The uncertainties were calculated by Bayesian statistics, and the Metropolis-Hastings algorithm was used to simulate the posterior parameter distribution. The parameter uncertainty calculated by the Metropolis-Hastings algorithm was compared to maximum likelihood estimates which assume that both the parameters and model residuals are normally distributed. The study was performed using the model WASMOD on 25 basins in central Sweden. Confidence intervals in the simulated discharge due to the parameter uncertainty and the total uncertainty were calculated. The results indicate that (a) the Metropolis-Hastings algorithm and the maximum likelihood method give almost identical estimates concerning the parameter uncertainty, and (b) the uncertainties in the simulated streamflow due to the parameter uncertainty are less important than uncertainties originating from other sources for this simple model with fewer parameters.     

Discharge and nutrient uncertainty: implications for nutrient flux estimation in small streams

The measurement of discharge is fundamental in nutrient load estimation. Because of our ability to monitor discharge routinely, it is generally assumed that the associated uncertainty is low. This paper challenges this preconception, arguing that discharge uncertainty should be explicitly taken into account to produce robust statistical analyses. In many studies, paired discharge and chemical datasets are used to calculate ‘true’ loads and used as the benchmark to compare with other load estimates. This paper uses two years of high frequency (daily and sub‐hourly) discharge and nutrient concentration data (nitrate‐N and total phosphorus (TP)) collected at four field sites as part of the Hampshire Avon Demonstration Test Catchment (DTC) programme. A framework for estimating observational nutrient load uncertainty was used which combined a flexible non‐parametric approach to characterising discharge uncertainty, with error modelling that allowed the incorporation of errors which were heteroscedastic and temporally correlated. The results showed that the stage–discharge relationships were non‐stationary, and observational uncertainties from ±2 to 25% were recorded when the velocity–area method was used. The variability in nutrient load estimates ranged from 1.1 to 9.9% for nitrate‐N and from 3.3 to 10% for TP when daily laboratory data were used, rising to a maximum of 9% for nitrate‐N and 83% for TP when the sensor data were used. However, the sensor data provided a better representation of the ‘true’ load as storm events are better represented temporally, posing the question: is it more beneficial to have high frequency, lower precision data or lower frequency but higher precision data streams to estimate nutrient flux responses in headwater catchments? Copyright © 2015 John Wiley & Sons, Ltd.     

Capturing temporal variability for estimates of annual hydrochemical export from a first‐order agricultural catchment in southern Ontario,Canada

This 2‐year study (2000, 2001) reports annual nutrient (phosphorus, nitrate) export from a first‐order agricultural watershed in southern Ontario based on an intensive monitoring programme. The importance of storm and melt events in annual export estimates is demonstrated and the temporal variability in nutrient loading during events is related to processes occurring within the catchment. The feasibility of predicting event‐related nutrient export from hydrometric data is explored. The importance of sampling frequency throughout events is also shown. Export of total phosphorus (TP), soluble reactive phosphorus (SRP) and nitrate ( ) for 2000 and 2001 averaged 0·35 kg ha^?1 year^?1, 0·09 kg ha^?1 year^?1, and 35 kg ha^?1 year^?1 (as N) respectively. Approximately 75% of annual TP export, 80% of annual SRP export and 70% of annual export occurred during 28 events per year. A small number of large‐magnitude events (>34 mm) accounted for 18–42% of annual TP export, 0–61% of annual SRP export and 13–33% of annual NO export. Our results show that temporal variability in nutrient export is largely governed by discharge in this basin, and export can be predicted from discharge. SRP and TP export can also be predicted from discharge, but only for events that are not large in magnitude. The sampling interval throughout events is important in obtaining precise estimates of nutrient export, as infrequent sampling intervals may over‐ or under‐estimate nutrient export by ± 45% per event for P. This study improves our understanding of and P export patterns and our ability to predict or model them by relating temporal variability in event nutrient export to discharge and processes occurring within the basin, and also by exploring the significance of sampling interval in the context of the importance of individual events, season and temporal variability during events. Copyright © 2006 John Wiley & Sons, Ltd.     

Parameter estimation and uncertainty analysis of SWAT model in upper reaches of the Heihe river basin

Heihe river basin, the second largest inland river basin in China, has attracted more attention in China due to the ever increasing water resources and eco‐environmental problems. In this article, SWAT (Soil and Water Assessment Tool; http://www.brc.tamus.edu/swat/ ) model was applied to upper reaches of the basin for better understanding of the hydrological process over the watershed. Parameter uncertainty and its contribution on model simulation are the main foci. In model calibration, the aggregate parameters instead of the original parameters in SWAT model were used to reduce the computing effort. The Bayesian approach was employed for parameter estimation and uncertainty analysis because its posterior distribution provides not only parameter estimation but also uncertainty analysis without normality assumption. The results indicated that: (1) SWAT model performs satisfactorily in this watershed as a whole, although some low and high flows were under‐ or overestimated, particularly in dry (e.g. 1991) and wet (e.g. 1996) years; (2) all calibrated parameters were not normally distributed (essentially positively or negatively skewed) and the parameter uncertainties were relatively small; and (3) the contributions of parameter uncertainty on model simulation uncertainty were relatively small. Copyright © 2009 John Wiley & Sons, Ltd.     

Statistically downscaled probabilistic multi‐model ensemble projections of precipitation change in a watershed

This paper presents the development of a probabilistic multi‐model ensemble of statistically downscaled future projections of precipitation of a watershed in New Zealand. Climate change research based on the point estimates of a single model is considered less reliable for decision making, and multiple realizations of a single model or outputs from multiple models are often preferred for such purposes. Similarly, a probabilistic approach is preferable over deterministic point estimates. In the area of statistical downscaling, no single technique is considered a universal solution. This is due to the fact that each of these techniques has some weaknesses, owing to its basic working principles. Moreover, watershed scale precipitation downscaling is quite challenging and is more prone to uncertainty issues than downscaling of other climatological variables. So, multi‐model statistical downscaling studies based on a probabilistic approach are required. In the current paper, results from the three well‐reputed statistical downscaling methods are used to develop a Bayesian weighted multi‐model ensemble. The three members of the downscaling ensemble of this study belong to the following three broad categories of statistical downscaling methods: (1) multiple linear regression, (2) multiple non‐linear regression, and (3) stochastic weather generator. The results obtained in this study show that the new strategy adopted here is promising because of many advantages it offers, e.g. it combines the outputs of multiple statistical downscaling methods, provides probabilistic downscaled climate change projections and enables the quantification of uncertainty in these projections. This will encourage any future attempts for combining the results of multiple statistical downscaling methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Impact of the numbers of observations and calibration parameters on equifinality,model performance,and output and parameter uncertainty

The level of model complexity that can be effectively supported by available information has long been a subject of many studies in hydrologic modelling. In particular, distributed parameter models tend to be regarded as overparameterized because of numerous parameters used to describe spatially heterogeneous hydrologic processes. However, it is not clear how parameters and observations influence the degree of overparameterization, equifinality of parameter values, and uncertainty. This study investigated the impact of the numbers of observations and parameters on calibration quality including equifinality among calibrated parameter values, model performance, and output/parameter uncertainty using the Soil and Water Assessment Tool model. In the experiments, the number of observations was increased by expanding the calibration period or by including measurements made at inner points of a watershed. Similarly, additional calibration parameters were included in the order of their sensitivity. Then, unique sets of parameters were calibrated with the same objective function, optimization algorithm, and stopping criteria but different numbers of observations. The calibration quality was quantified with statistics calculated based on the ‘behavioural’ parameter sets, identified using 1% and 5% cut‐off thresholds in a generalized likelihood uncertainty estimation framework. The study demonstrated that equifinality, model performance, and output/parameter uncertainty were responsive to the numbers of observations and calibration parameters; however, the relationship between the numbers, equifinality, and uncertainty was not always conclusive. Model performance improved with increased numbers of calibration parameters and observations, and substantial equifinality did neither necessarily mean bad model performance nor large uncertainty in the model outputs and parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

MODELLING THE IMPACT OF LAND USE CHANGE ON WATER QUALITY IN AGRICULTURAL CATCHMENTS

Export coefficient modelling was used to model the impact of agriculture on nitrogen and phosphorus loading on the surface waters of two contrasting agricultural catchments. The model was originally developed for the Windrush catchment where the highly reactive Jurassic limestone aquifer underlying the catchment is well connected to the surface drainage network, allowing the system to be modelled using uniform export coefficients for each nutrient source in the catchment, regardless of proximity to the surface drainage network. In the Slapton catchment, the hydrological pathways are dominated by surface and lateral shallow subsurface flow, requiring modification of the export coefficient model to incorporate a distance–decay component in the export coefficients. The modified model was calibrated against observed total nitrogen and total phosphorus loads delivered to Slapton Ley from inflowing streams in its catchment. Sensitivity analysis was conducted to isolate the key controls on nutrient export in the modified model. The model was validated against long-term records of water quality, and was found to be accurate in its predictions and sensitive to both temporal and spatial changes in agricultural practice in the catchment. The model was then used to forecast the potential reduction in nutrient loading on Slapton Ley associated with a range of catchment management strategies. The best practicable environmental option (BPEO) was found to be spatial redistribution of high nutrient export risk sources to areas of the catchment with the greatest intrinsic nutrient retention capacity. © 1997 by John Wiley & Sons, Ltd.     

Changes in hydrology and streamflow as predicted by a modelling experiment forced with climate models

Changes in precipitation and temperature have direct effects on crop water use, water stress, crop yield, evapotranspiration, water nutrient dynamics and other indicators. This study, built on a modelling framework with the Soil and Watershed Assessment Tool (SWAT) model for the Raccoon River Watershed in central Iowa, a typical US Midwestern agricultural watershed, examines the watershed response to changes in meteorological inputs from an ensemble of ten global climate models under the A1B scenario. Changes in climate were directly applied to observations (the delta change method) assuming that the estimates of climate change are reliable even if the simulated current climate may be biased. The ensemble average for the mid‐century (1946–1965) predicted 0.7% increase in daily precipitation (monthly variation from ?11.3% to +19.5%) and 2.78 °C increase in average temperature over the entire watershed. These predictions were translated through a well‐calibrated SWAT modelling setup into 22% decrease in snowfall, 16% decrease in surface runoff, 18% decrease in baseflow, 8% increase in evapotranspiration and 17% decrease in total water yield. The spatial impact at the subwatershed level revealed a wide variation (but no defined trend) with decrease in water yield that ranged from 10% to 23%. Flow near the watershed outlet (Van Meter, Iowa) is expected to decline by 17% on an average annual basis with the highest impact occurring during summer months with a maximum 39% reduction in August. Changes in climate were found to have a clear and significant impact signal of decreasing streamflow at the watershed outlet with far‐reaching implication for drinking water supplies for the central Iowa communities. Copyright © 2013 John Wiley & Sons, Ltd.     

An un-mixing model to study watershed erosion processes

An un-mixing model is formulated within a Bayesian Markov Chain Monte Carlo framework for use within land-use fingerprinting to study watershed erosion processes. The model has two new components: (1) An equation and erosion process parameter are used to weight tracer signatures from each erosion process within a land-use. (2) An extra tracer distribution and episodic erosion parameter are used to represent soil eroded throughout the sampling duration and thus include the episodic nature of erosion. To test specification of these new parameters, the un-mixing model is applied in the 15 km² Jerome Creek Watershed in the Palouse Region of Northwestern Idaho. Erosion processes include surface erosion upon mountain slopes due to logging in the forest land-use and rill/interrill erosion on cultivated slopes and headcut erosion in riparian floodplains of the agricultural land-use (winter wheat/peas rotation and hay pasture). Episodic erosion occurs for the event where the model is applied. A sensitivity analysis shows that the smallest Bayesian credible set results when the new parameters are specified using hydrologic data and process-based models. The un-mixing model predicts that 90% of the eroded-soil originated from the agricultural land-use and 10% originated from the forest land-use. A comparative study is performed that estimates 90.5% and 9.5% of eroded-soil originated from the agricultural and forest land-uses. Successful performance of the un-mixing model highlights future application as a standalone probabilistic tool to monitor watershed erosion processes that exhibit non-equilibrium conditions and provide calibration data for process-based watershed models.     

Coupling basin- and site-scale inverse models of the Española aquifer

Large-scale models are frequently used to estimate fluxes to small-scale models. The uncertainty associated with these flux estimates, however, is rarely addressed. We present a case study from the Espa?ola Basin, northern New Mexico, where we use a basin-scale model coupled with a high-resolution, nested site-scale model. Both models are three-dimensional and are analyzed by codes FEHM and PEST. Using constrained nonlinear optimization, we examine the effect of parameter uncertainty in the basin-scale model on the nonlinear confidence limits of predicted fluxes to the site-scale model. We find that some of the fluxes are very well constrained, while for others there is fairly large uncertainty. Site-scale transport simulation results, however, are relatively insensitive to the estimated uncertainty in the fluxes. We also compare parameter estimates obtained by the basin- and site-scale inverse models. Differences in the model grid resolution (scale of parameter estimation) result in differing delineation of hydrostratigraphic units, so the two models produce different estimates for some units. The effect is similar to the observed scale effect in medium properties owing to differences in tested volume. More important, estimation uncertainty of model parameters is quite different at the two scales. Overall, the basin inverse model resulted in significantly lower estimates of uncertainty, because of the larger calibration dataset available. This suggests that the basin-scale model contributes not only important boundary condition information but also improved parameter identification for some units. Our results demonstrate that caution is warranted when applying parameter estimates inferred from a large-scale model to small-scale simulations, and vice versa.     

Sensitivity analysis of prior model probabilities and the value of prior knowledge in the assessment of conceptual model uncertainty in groundwater modelling

A key point in the application of multi‐model Bayesian averaging techniques to assess the predictive uncertainty in groundwater modelling applications is the definition of prior model probabilities, which reflect the prior perception about the plausibility of alternative models. In this work the influence of prior knowledge and prior model probabilities on posterior model probabilities, multi‐model predictions, and conceptual model uncertainty estimations is analysed. The sensitivity to prior model probabilities is assessed using an extensive numerical analysis in which the prior probability space of a set of plausible conceptualizations is discretized to obtain a large ensemble of possible combinations of prior model probabilities. Additionally, the value of prior knowledge about alternative models in reducing conceptual model uncertainty is assessed by considering three example knowledge states, expressed as quantitative relations among the alternative models. A constrained maximum entropy approach is used to find the set of prior model probabilities that correspond to the different prior knowledge states. For illustrative purposes, a three‐dimensional hypothetical setup approximated by seven alternative conceptual models is employed. Results show that posterior model probabilities, leading moments of the predictive distributions and estimations of conceptual model uncertainty are very sensitive to prior model probabilities, indicating the relevance of selecting proper prior probabilities. Additionally, including proper prior knowledge improves the predictive performance of the multi‐model approach, expressed by reductions of the multi‐model prediction variances by up to 60% compared with a non‐informative case. However, the ratio between‐model to total variance does not substantially decrease. This suggests that the contribution of conceptual model uncertainty to the total variance cannot be further reduced based only on prior knowledge about the plausibility of alternative models. These results advocate including proper prior knowledge about alternative conceptualizations in combination with extra conditioning data to further reduce conceptual model uncertainty in groundwater modelling predictions. Copyright © 2009 John Wiley & Sons, Ltd.     

Uncertainty issues of a conceptual water balance model for a semi‐arid watershed in north‐west of China

Hydrological models are useful tools for better understanding the hydrological processes and performing the hydrological prediction. However, the reliability of the prediction depends largely on its uncertainty range. This study mainly focuses on estimating model parameter uncertainty and quantifying the simulation uncertainties caused by sole model parameters and the co‐effects of model parameters and model structure in a lumped conceptual water balance model called WASMOD (Water And Snow balance MODeling system). The validity of statistical hypotheses on residuals made in the model formation is tested as well, as it is the base of parameter estimation and simulation uncertainty evaluation. The bootstrap method is employed to examine the parameter uncertainty in the selected model. The Yingluoxia watershed at the upper reaches of the Heihe River basin in north‐west of China is selected as the study area. Results show that all parameters in the model can be regarded as normally distributed based on their marginal distributions and the Kolmogorov–Smirnov test, although they appear slightly skewed for two parameters. Their uncertainty ranges are different from each other. The model residuals are tested to be independent, homoscedastic and normally distributed. Based on such valid hypotheses of model residuals, simulation uncertainties caused by co‐effects of model parameters and model structure can be evaluated effectively. It is found that the 95% and 99% confidence intervals (CIs) of simulated discharge cover 42.7% and 52.4% of the observations when only parameter uncertainty is considered, indicating that parameter uncertainty has a great effect on simulation uncertainty but still cannot be used to explain all the simulation uncertainty in this study. The 95% and 99% CIs become wider, and the percentages of observations falling inside such CIs become larger when co‐effects of parameters and model structure are considered, indicating that simultaneous consideration of both parameters and model structure uncertainties accounts sufficient contribution for model simulation uncertainty. Copyright © 2012 John Wiley & Sons, Ltd.