A method to reduce the computational requirement while assessing uncertainty of complex hydrological models

The quantification of uncertainty in the simulations from complex physically based distributed hydrologic models is important for developing reliable applications. The generalized likelihood uncertainty estimation method (GLUE) is one of the most commonly used methods in the field of hydrology. The GLUE helps reduce the parametric uncertainty by deriving the probability distribution function of parameters, and help analyze the uncertainty in model output. In the GLUE, the uncertainty of model output is analyzed through Monte Carlo simulations, which require large number of model runs. This induces high computational demand for the GLUE to characterize multi-dimensional parameter space, especially in the case of complex hydrologic models with large number of parameters. While there are a lot of variants of GLUE that derive the probability distribution of parameters, none of them have addressed the computational requirement in the analysis. A method to reduce such computational requirement for GLUE is proposed in this study. It is envisaged that conditional sampling, while generating ensembles for the GLUE, can help reduce the number of model simulations. The mutual relationship between the parameters was used for conditional sampling in this study. The method is illustrated using a case study of Soil and Water Assessment Tool (SWAT) model on a watershed in the USA. The number of simulations required for the uncertainty analysis was reduced by 90 % in the proposed method compared to existing methods. The proposed method also resulted in an uncertainty reduction in terms of reduced average band width and high containing ratio.     

Hydrologic data assimilation using particle Markov chain Monte Carlo simulation: Theory,concepts and applications

During the past decades much progress has been made in the development of computer based methods for parameter and predictive uncertainty estimation of hydrologic models. The goal of this paper is twofold. As part of this special anniversary issue we first shortly review the most important historical developments in hydrologic model calibration and uncertainty analysis that has led to current perspectives. Then, we introduce theory, concepts and simulation results of a novel data assimilation scheme for joint inference of model parameters and state variables. This Particle-DREAM method combines the strengths of sequential Monte Carlo sampling and Markov chain Monte Carlo simulation and is especially designed for treatment of forcing, parameter, model structural and calibration data error. Two different variants of Particle-DREAM are presented to satisfy assumptions regarding the temporal behavior of the model parameters. Simulation results using a 40-dimensional atmospheric “toy” model, the Lorenz attractor and a rainfall–runoff model show that Particle-DREAM, P-DREAM_(VP) and P-DREAM_(IP) require far fewer particles than current state-of-the-art filters to closely track the evolving target distribution of interest, and provide important insights into the information content of discharge data and non-stationarity of model parameters. Our development follows formal Bayes, yet Particle-DREAM and its variants readily accommodate hydrologic signatures, informal likelihood functions or other (in)sufficient statistics if those better represent the salient features of the calibration data and simulation model used.     

Application of distributed hydrological models for predictions in ungauged basins: a method to quantify predictive uncertainty

Stream flow predictions in ungauged basins are one of the most challenging tasks in surface water hydrology because of nonavailability of data and system heterogeneity. This study proposes a method to quantify stream flow predictive uncertainty of distributed hydrologic models for ungauged basins. The method is based on the concepts of deriving probability distribution of model's sensitive parameters by using measured data from a gauged basin and transferring the distribution to hydrologically similar ungauged basins for stream flow predictions. A Monte Carlo simulation of the hydrologic model using sampled parameter sets with assumed probability distribution is conducted. The posterior probability distributions of the sensitive parameters are then computed using a Bayesian approach. In addition, preselected threshold values of likelihood measure of simulations are employed for sizing the parameter range, which helps reduce the predictive uncertainty. The proposed method is illustrated through two case studies using two hydrologically independent sub‐basins in the Cedar Creek watershed located in Texas, USA, using the Soil and Water Assessment Tool (SWAT) model. The probability distribution of the SWAT parameters is derived from the data from one of the sub‐basins and is applied for simulation in the other sub‐basin considered as pseudo‐ungauged. In order to assess the robustness of the method, the numerical exercise is repeated by reversing the gauged and pseudo‐ungauged basins. The results are subsequently compared with the measured stream flow from the sub‐basins. It is observed that the measured stream flow in the pseudo‐ungauged basin lies well within the estimated confidence band of predicted stream flow. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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.     

Assessment of change in design flood frequency under climate change using a multivariate downscaling model and a precipitation-runoff model

Precipitation and runoff are key elements in the hydrologic cycle because of their important roles in water supply, flood prevention, river restoration, and ecosystem management. Global climate change, widely accepted to be happening, is anticipated to have enormous consequences on future hydrologic patterns. Studies on the potential changes in global, regional, and local hydrologic patterns under global climate change scenarios have been an intense area of research in recent years. The present study contributes to this research topic through evaluation of design flood under climate change. The study utilizes a weather state-based, stochastic multivariate model as a conditional probability model for simulating the precipitation field. An important premise of this study is that large-scale climatic patterns serve as a major driver of persistent year-to-year changes in precipitation probabilities. Since uncertainty estimation in the study of climate change is needed to examine the reliability of the outcomes, this study also applies a Bayesian Markov chain Monte Carlo scheme to the widely used SAC-SMA (Sacramento soil moisture accounting) precipitation-runoff model. A case study is also performed with the Soyang Dam watershed in South Korea as the study basin. Finally, a comprehensive discussion on design flood under climate change is made.     

13e SESSION DU COMITÉ CONSULTATIF KARACHI 4–15 NOVEMBRE 1957

Abstract

Event-based methods are used in flood estimation to obtain the entire flood hydrograph. Previously, such methods adopted in the UK have relied on pre-determined values of the input variables (e.g. rainfall and antecedent conditions) to a rainfall–runoff model, which is expected to result in an output flood of a particular return period. In contrast, this paper presents a method that allows all the input variables to take on values across the full range of their individual distributions. These values are then brought together in all possible combinations as input to an event-based rainfall–runoff model in a Monte Carlo simulation approach. Further, this simulation strategy produces a long string of events (on average 10 per year), where dependencies from one event to the next, as well as between different variables within a single event, are accounted for. Frequency analysis is then applied to the annual maximum peak flows and flow volumes.

Citation Svensson, C., Kjeldsen, T.R., and Jones, D.A., 2013. Flood frequency estimation using a joint probability approach within a Monte Carlo framework. Hydrological Sciences Journal, 58 (1), 1–20.     

On the treatment of uncertainties in the development of fragility functions for earthquake loss estimation of building portfolios

State‐of‐the‐art methods for the assessment of building fragility consider the structural capacity and seismic demand variability in the estimation of the probability of exceeding different damage states. However, questions remain regarding the appropriate treatment of such sources of uncertainty from a statistical significance perspective. In this study, material, geometrical and mechanical properties of a number of building classes are simulated by means of a Monte Carlo sampling process in which the statistical distribution of the aforementioned parameters is taken into consideration. Record selection is performed in accordance with hazard‐consistent distributions of a comprehensive set of intensity measures, and issues related with sufficiency, efficiency, predictability and scaling robustness are addressed. Based on the appraised minimum number of ground motion records required to achieve statistically meaningful estimates of response variability conditioned on different levels of seismic intensity, the concept of conditional fragility functions is presented. These functions translate the probability of exceeding a set of damage states as a function of a secondary sufficient intensity measure, when records are selected and scaled for a particular level of primary seismic intensity parameter. It is demonstrated that this process allows a hazard‐consistent and statistically meaningful representation of uncertainty and correlation in the estimation of intensity‐dependent damage exceedance probabilities. Copyright © 2016 John Wiley & Sons, Ltd.     

Uncertainty based modeling of rainfall-runoff: Combined differential evolution adaptive Metropolis (DREAM) and K-means clustering

Simulation of rainfall-runoff process in urban areas is of great importance considering the consequences and damages of extreme runoff events and floods. The first issue in flood hazard analysis is rainfall simulation. Large scale climate signals have been proved to be effective in rainfall simulation and prediction. In this study, an integrated scheme is developed for rainfall-runoff modeling considering different sources of uncertainty. This scheme includes three main steps of rainfall forecasting, rainfall-runoff simulation and future runoff prediction. In the first step, data driven models are developed and used to forecast rainfall using large scale climate signals as rainfall predictors. Due to high effect of different sources of uncertainty on the output of hydrologic models, in the second step uncertainty associated with input data, model parameters and model structure is incorporated in rainfall-runoff modeling and simulation. Three rainfall-runoff simulation models are developed for consideration of model conceptual (structural) uncertainty in real time runoff forecasting. To analyze the uncertainty of the model structure, streamflows generated by alternative rainfall-runoff models are combined, through developing a weighting method based on K-means clustering. Model parameters and input uncertainty are investigated using an adaptive Markov Chain Monte Carlo method. Finally, calibrated rainfall-runoff models are driven using the forecasted rainfall to predict future runoff for the watershed. The proposed scheme is employed in the case study of the Bronx River watershed, New York City. Results of uncertainty analysis of rainfall-runoff modeling reveal that simultaneous estimation of model parameters and input uncertainty significantly changes the probability distribution of the model parameters. It is also observed that by combining the outputs of the hydrological models using the proposed clustering scheme, the accuracy of runoff simulation in the watershed is remarkably improved up to 50% in comparison to the simulations by the individual models. Results indicate that the developed methodology not only provides reliable tools for rainfall and runoff modeling, but also adequate time for incorporating required mitigation measures in dealing with potentially extreme runoff events and flood hazard. Results of this study can be used in identification of the main factors affecting flood hazard analysis.     

Comparison of parameter uncertainty analysis techniques for a TOPMODEL application

Parameter uncertainty in hydrologic modeling is crucial to the flood simulation and forecasting. The Bayesian approach allows one to estimate parameters according to prior expert knowledge as well as observational data about model parameter values. This study assesses the performance of two popular uncertainty analysis (UA) techniques, i.e., generalized likelihood uncertainty estimation (GLUE) and Bayesian method implemented with the Markov chain Monte Carlo sampling algorithm, in evaluating model parameter uncertainty in flood simulations. These two methods were applied to the semi-distributed Topographic hydrologic model (TOPMODEL) that includes five parameters. A case study was carried out for a small humid catchment in the southeastern China. The performance assessment of the GLUE and Bayesian methods were conducted with advanced tools suited for probabilistic simulations of continuous variables such as streamflow. Graphical tools and scalar metrics were used to test several attributes of the simulation quality of selected flood events: deterministic accuracy and the accuracy of 95 % prediction probability uncertainty band (95PPU). Sensitivity analysis was conducted to identify sensitive parameters that largely affect the model output results. Subsequently, the GLUE and Bayesian methods were used to analyze the uncertainty of sensitive parameters and further to produce their posterior distributions. Based on their posterior parameter samples, TOPMODEL’s simulations and the corresponding UA results were conducted. Results show that the form of exponential decline in conductivity and the overland flow routing velocity were sensitive parameters in TOPMODEL in our case. Small changes in these two parameters would lead to large differences in flood simulation results. Results also suggest that, for both UA techniques, most of streamflow observations were bracketed by 95PPU with the containing ratio value larger than 80 %. In comparison, GLUE gave narrower prediction uncertainty bands than the Bayesian method. It was found that the mode estimates of parameter posterior distributions are suitable to result in better performance of deterministic outputs than the 50 % percentiles for both the GLUE and Bayesian analyses. In addition, the simulation results calibrated with Rosenbrock optimization algorithm show a better agreement with the observations than the UA’s 50 % percentiles but slightly worse than the hydrographs from the mode estimates. The results clearly emphasize the importance of using model uncertainty diagnostic approaches in flood simulations.     

On the applicability of the RETAS model for forecasting aftershock probability in underground mines (Kiirunavaara Mine,Sweden)

Aftershock series of even comparatively small seismic events can pose a risk to the mining operation or the personnel in deep underground mines as the main shocks and some of the aftershocks can cause damage in the rock mass. Stochastic modeling was applied in this study for the analysis of the temporal evolution of aftershock occurrence probability during a M1.85 aftershock sequence in Kiirunavaara Mine, Sweden. The Restricted Epidemic-Type Aftershock Sequence (RETAS) model was chosen for estimation of the aftershock occurrence probability. This model considers all events with magnitude above the magnitude of completeness M₀ and has the advantage of including the Modified Omori Formula (MOF) model and Epidemic-Type Aftershock Sequence (ETAS) model as its end versions, considering also all intermediate models. The model was applied sequentially to data samples covering cumulative periods of time, starting from the first 2 h after the main event and increasing them by 2 h until the period covered the entire 72-h sequence. For each sample, the best-fit RETAS version was identified and the probability of a M?≥?0.5 aftershock for every next 2 h was determined through Monte Carlo simulation. The feasibility of the resulting probability evolution for suspension and re-starting of the mining operations was discussed together with possible prospects for future development of the methodology.

     

Bayesian flood frequency analysis in the light of model and parameter uncertainties

The specific objective of the paper is to propose a new flood frequency analysis method considering uncertainty of both probability distribution selection (model uncertainty) and uncertainty of parameter estimation (parameter uncertainty). Based on Bayesian theory sampling distribution of quantiles or design floods coupling these two kinds of uncertainties is derived, not only point estimator but also confidence interval of the quantiles can be provided. Markov Chain Monte Carlo is adopted in order to overcome difficulties to compute the integrals in estimating the sampling distribution. As an example, the proposed method is applied for flood frequency analysis at a gauge in Huai River, China. It has been shown that the approach considering only model uncertainty or parameter uncertainty could not fully account for uncertainties in quantile estimations, instead, method coupling these two uncertainties should be employed. Furthermore, the proposed Bayesian-based method provides not only various quantile estimators, but also quantitative assessment on uncertainties of flood frequency analysis.     

Application and evaluation of a snowmelt runoff model in the Tamor River basin,Eastern Himalaya using a Markov Chain Monte Carlo (MCMC) data assimilation approach

Previous studies have drawn attention to substantial hydrological changes taking place in mountainous watersheds where hydrology is dominated by cryospheric processes. Modelling is an important tool for understanding these changes but is particularly challenging in mountainous terrain owing to scarcity of ground observations and uncertainty of model parameters across space and time. This study utilizes a Markov Chain Monte Carlo data assimilation approach to examine and evaluate the performance of a conceptual, degree‐day snowmelt runoff model applied in the Tamor River basin in the eastern Nepalese Himalaya. The snowmelt runoff model is calibrated using daily streamflow from 2002 to 2006 with fairly high accuracy (average Nash–Sutcliffe metric ~0.84, annual volume bias < 3%). The Markov Chain Monte Carlo approach constrains the parameters to which the model is most sensitive (e.g. lapse rate and recession coefficient) and maximizes model fit and performance. Model simulated streamflow using an interpolated precipitation data set decreases the fractional contribution from rainfall compared with simulations using observed station precipitation. The average snowmelt contribution to total runoff in the Tamor River basin for the 2002–2006 period is estimated to be 29.7 ± 2.9% (which includes 4.2 ± 0.9% from snowfall that promptly melts), whereas 70.3 ± 2.6% is attributed to contributions from rainfall. On average, the elevation zone in the 4000–5500 m range contributes the most to basin runoff, averaging 56.9 ± 3.6% of all snowmelt input and 28.9 ± 1.1% of all rainfall input to runoff. Model simulated streamflow using an interpolated precipitation data set decreases the fractional contribution from rainfall versus snowmelt compared with simulations using observed station precipitation. Model experiments indicate that the hydrograph itself does not constrain estimates of snowmelt versus rainfall contributions to total outflow but that this derives from the degree‐day melting model. Lastly, we demonstrate that the data assimilation approach is useful for quantifying and reducing uncertainty related to model parameters and thus provides uncertainty bounds on snowmelt and rainfall contributions in such mountainous watersheds. Copyright © 2013 John Wiley & Sons, Ltd.     

A computationally efficient method for uncertainty analysis of SWAT model simulations

The physically based distributed hydrological models are ideal for hydrological simulations; however most of such models do not use the basic equations pertaining to mass, energy and momentum conservation, to represent the physics of the process. This is plausibly due to the lack of complete understanding of the hydrological process. The soil and water assessment tool (SWAT) is one such widely accepted semi-distributed, conceptual hydrological model used for water resources planning. However, the over-parameterization, difficulty in its calibration process and the uncertainty associated with predictions make its applications skeptical. This study considers assessing the predictive uncertainty associated with distributed hydrological models. The existing methods for uncertainty estimation demand high computational time and therefore make them challenging to apply on complex hydrological models. The proposed approach employs the concepts of generalized likelihood uncertainty estimation (GLUE) in an iterative procedure by starting with an assumed prior probability distribution of parameters, and by using mutual information (MI) index for sampling the behavioral parameter set. The distributions are conditioned on the observed information through successive cycles of simulations. During each cycle of simulation, MI is used in conjunction with Markov Chain Monte Carlo procedure to sample the parameter sets so as to increase the number of behavioral sets, which in turn helps reduce the number of cycles/simulations for the analysis. The method is demonstrated through a case study of SWAT model in Illinois River basin in the USA. A comparison of the proposed method with GLUE indicates that the computational requirement of uncertainty analysis is considerably reduced in the proposed approach. It is also noted that the model prediction band, derived using the proposed method, is more effective compared to that derived using the other methods considered in this study.     

A field‐scale infiltration model accounting for spatial heterogeneity of rainfall and soil saturated hydraulic conductivity

This study first explores the role of spatial heterogeneity, in both the saturated hydraulic conductivity K_s and rainfall intensity r, on the integrated hydrological response of a natural slope. On this basis, a mathematical model for estimating the expected areal‐average infiltration is then formulated. Both K_s and r are considered as random variables with assessed probability density functions. The model relies upon a semi‐analytical component, which describes the directly infiltrated rainfall, and an empirical component, which accounts further for the infiltration of surface water running downslope into pervious soils (the run‐on effect). Monte Carlo simulations over a clay loam soil and a sandy loam soil were performed for constructing the ensemble averages of field‐scale infiltration used for model validation. The model produced very accurate estimates of the expected field‐scale infiltration rate, as well as of the outflow generated by significant rainfall events. Furthermore, the two model components were found to interact appropriately for different weights of the two infiltration mechanisms involved. Copyright © 2005 John Wiley & Sons, Ltd.     

Effects of uncertainty in rock-physics models on reservoir parameter estimation using seismic amplitude variation with angle and controlled-source electromagnetics data

This paper investigates the effects of uncertainty in rock-physics models on reservoir parameter estimation using seismic amplitude variation with angle and controlled-source electromagnetics data. The reservoir parameters are related to electrical resistivity by the Poupon model and to elastic moduli and density by the Xu-White model. To handle uncertainty in the rock-physics models, we consider their outputs to be random functions with modes or means given by the predictions of those rock-physics models and we consider the parameters of the rock-physics models to be random variables defined by specified probability distributions. Using a Bayesian framework and Markov Chain Monte Carlo sampling methods, we are able to obtain estimates of reservoir parameters and information on the uncertainty in the estimation. The developed method is applied to a synthetic case study based on a layered reservoir model and the results show that uncertainty in both rock-physics models and in their parameters may have significant effects on reservoir parameter estimation. When the biases in rock-physics models and in their associated parameters are unknown, conventional joint inversion approaches, which consider rock-physics models as deterministic functions and the model parameters as fixed values, may produce misleading results. The developed stochastic method in this study provides an integrated approach for quantifying how uncertainty and biases in rock-physics models and in their associated parameters affect the estimates of reservoir parameters and therefore is a more robust method for reservoir parameter estimation.     

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.     

Using a fluctuating tracer to estimate hyporheic exchange in restored and unrestored reaches of the Truckee River,Nevada, USA

The goal of this research was to compare hyporheic activity in recently restored and adjacent un‐restored reaches of the Truckee River downstream from the Reno/Sparks metropolitan area. The installation of rocky riffles and raised channel bed elevations in the restored reaches may have increased the degree of surface–subsurface interaction. A fluctuating chloride concentration signal served as the tracer, induced by the variable influx of higher salinity water several miles upstream from the study reach. The solute transport model, OTIS, was used in conjunction with the hydrodynamic model, DYNHYD5, to estimate transient storage parameters under unsteady flow conditions. The model was calibrated to chloride concentrations measured over a period of three days at six in‐stream locations representing restored and un‐restored reaches. An automated parameter estimation algorithm (SCE‐UA) was used to optimize parameters for multiple reaches simultaneously and generate a distribution of parameter estimates. Results suggest that the transient storage zone cross‐sectional area (A_s) is larger in the restored reaches than in the unrestored reaches, but the exchange coefficient (α) is smaller, leading to increased hyporheic residence time and hydrologic retention in the vicinity of channel reconstructions. Scenarios were used to simulate the potential effects of increased subsurface residence time on denitrification and in‐stream NO₃‐N concentrations. Monte Carlo analysis was performed to assess uncertainty in the simulation results and show the potential for greater nutrient retention in the lower Truckee River as a result of channel restoration. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessing the uncertainty in distributed model predictions using observed binary pattern information within GLUE

In this paper we extend the generalized likelihood uncertainty estimation (GLUE) technique to estimate spatially distributed uncertainty in models conditioned against binary pattern data contained in flood inundation maps. Untransformed binary pattern data already have been used within GLUE to estimate domain‐averaged (zero‐dimensional) likelihoods, yet the pattern information embedded within such sources has not been used to estimate distributed uncertainty. Where pattern information has been used to map distributed uncertainty it has been transformed into a continuous function prior to use, which may introduce additional errors. To solve this problem we use here ‘raw’ binary pattern data to define a zero‐dimensional global performance measure for each simulation in a Monte Carlo ensemble. Thereafter, for each pixel of the distributed model we evaluate the probability that this pixel was inundated. This probability is then weighted by the measure of global model performance, thus taking into account how well a given parameter set performs overall. The result is a distributed uncertainty measure mapped over real space. The advantage of the approach is that it both captures distributed uncertainty and contains information on global likelihood that can be used to condition predictions of further events for which observed data are not available. The technique is applied to the problem of flood inundation prediction at two test sites representing different hydrodynamic conditions. In both cases, the method reveals the spatial structure in simulation uncertainty and simultaneously enables mapping of flood probability predicted by the model. Spatially distributed uncertainty analysis is shown to contain information over and above that available from global performance measures. Overall, the paper highlights the different types of information that may be obtained from mappings of model uncertainty over real and n‐dimensional parameter spaces. Copyright © 2002 John Wiley & Sons, Ltd.