Addressing snow model uncertainty for hydrologic prediction

Streamflow forecasting methods are moving towards probabilistic approaches that quantify the uncertainty associated with the various sources of error in the forecasting process. Multi-model averaging methods which try to address modeling deficiencies by considering multiple models are gaining much popularity. We have applied the Bayesian Model Averaging method to an ensemble of twelve snow models that vary in their heat and melt algorithms, parameterization, and/or albedo estimation method. Three of the models use the temperature-based heat and melt routines of the SNOW17 snow accumulation and ablation model. Nine models use heat and melt routines that are based on a simplified energy balance approach, and are varied by using three different albedo estimation schemes. Finally, different parameter sets were identified through automatic calibration with three objective functions. All models use the snow accumulation, liquid water transport, and ground surface heat exchange processes of the SNOW17. The resulting twelve snow models were combined using Bayesian Model Averaging (BMA). The individual models, BMA predictive mean, and BMA predictive variance were evaluated for six SNOTEL sites in the western U.S. The models performed best and the BMA variance was lowest at the colder sites with high winter precipitation and little mid-winter melting. An individual snow model would often outperform the BMA predictive mean. However, observed snow water equivalent (SWE) was captured within the 95% confidence intervals of the BMA variance on average 80% of the time at all sites. Results are promising that consideration of multiple snow structures would provide useful uncertainty information for probabilistic hydrologic prediction.     

Understanding the role of hydrologic model structures on evapotranspiration-driven sensitivity

ABSTRACT

Most conceptual hydrological models do not treat vegetation as a dynamic component. This study focuses on understanding the impact of model structural complexity on the sensitivity of hydrologic models to potential evapotranspiration forcing data. To achieve this, two classes of hydrologic models are examined: (1) lumped, conceptual rainfall–runoff models and (2) eco-hydrologic models. A sample of 57 US catchments, covering eight eco-regions, included in the MOPEX dataset is used. While streamflow simulation performance in complex models did not exhibit increased sensitivity to PET, actual evapotranspiration simulation performance showed greater sensitivity in energy-limited catchments. This analysis warns against using over-simplistic PET estimations in energy-limited catchments for eco-hydrologic models and for more complex conceptual hydrologic models. This is particularly true for streamflow-only calibrations that commonly fail to properly constrain physically based parameters. Ultimately, these results have the potential to inform data collection and model selection efforts to yield the greatest benefit.     

Application of the patient rule induction method to detect hydrologic model behavioural parameters and quantify uncertainty

Finding an operational parameter vector is always challenging in the application of hydrologic models, with over‐parameterization and limited information from observations leading to uncertainty about the best parameter vectors. Thus, it is beneficial to find every possible behavioural parameter vector. This paper presents a new methodology, called the patient rule induction method for parameter estimation (PRIM‐PE), to define where the behavioural parameter vectors are located in the parameter space. The PRIM‐PE was used to discover all regions of the parameter space containing an acceptable model behaviour. This algorithm consists of an initial sampling procedure to generate a parameter sample that sufficiently represents the response surface with a uniform distribution within the “good‐enough” region (i.e., performance better than a predefined threshold) and a rule induction component (PRIM), which is then used to define regions in the parameter space in which the acceptable parameter vectors are located. To investigate its ability in different situations, the methodology is evaluated using four test problems. The PRIM‐PE sampling procedure was also compared against a Markov chain Monte Carlo sampler known as the differential evolution adaptive Metropolis (DREAM_ZS) algorithm. Finally, a spatially distributed hydrological model calibration problem with two settings (a three‐parameter calibration problem and a 23‐parameter calibration problem) was solved using the PRIM‐PE algorithm. The results show that the PRIM‐PE method captured the good‐enough region in the parameter space successfully using 8 and 107 boxes for the three‐parameter and 23‐parameter problems, respectively. This good‐enough region can be used in a global sensitivity analysis to provide a broad range of parameter vectors that produce acceptable model performance. Moreover, for a specific objective function and model structure, the size of the boxes can be used as a measure of equifinality.     

Reservoir performance under uncertainty in hydrologic impacts of climate change

Relatively few studies have addressed water management and adaptation measures in the face of changing water balances due to climate change. The current work studies climate change impact on a multipurpose reservoir performance and derives adaptive policies for possible future scenarios. The method developed in this work is illustrated with a case study of Hirakud reservoir on the Mahanadi river in Orissa, India, which is a multipurpose reservoir serving flood control, irrigation and power generation. Climate change effects on annual hydropower generation and four performance indices (reliability with respect to three reservoir functions, viz. hydropower, irrigation and flood control, resiliency, vulnerability and deficit ratio with respect to hydropower) are studied. Outputs from three general circulation models (GCMs) for three scenarios each are downscaled to monsoon streamflow in the Mahanadi river for two future time slices, 2045–65 and 2075–95. Increased irrigation demands, rule curves dictated by increased need for flood storage and downscaled projections of streamflow from the ensemble of GCMs and scenarios are used for projecting future hydrologic scenarios. It is seen that hydropower generation and reliability with respect to hydropower and irrigation are likely to show a decrease in future in most scenarios, whereas the deficit ratio and vulnerability are likely to increase as a result of climate change if the standard operating policy (SOP) using current rule curves for flood protection is employed. An optimal monthly operating policy is then derived using stochastic dynamic programming (SDP) as an adaptive policy for mitigating impacts of climate change on reservoir operation. The objective of this policy is to maximize reliabilities with respect to multiple reservoir functions of hydropower, irrigation and flood control. In variations to this adaptive policy, increasingly more weightage is given to the purpose of maximizing reliability with respect to hydropower for two extreme scenarios. It is seen that by marginally sacrificing reliability with respect to irrigation and flood control, hydropower reliability and generation can be increased for future scenarios. This suggests that reservoir rules for flood control may have to be revised in basins where climate change projects an increasing probability of droughts. However, it is also seen that power generation is unable to be restored to current levels, due in part to the large projected increases in irrigation demand. This suggests that future water balance deficits may limit the success of adaptive policy options.     

A framework for dealing with uncertainty due to model structure error

Although uncertainty about structures of environmental models (conceptual uncertainty) is often acknowledged to be the main source of uncertainty in model predictions, it is rarely considered in environmental modelling. Rather, formal uncertainty analyses have traditionally focused on model parameters and input data as the principal source of uncertainty in model predictions. The traditional approach to model uncertainty analysis, which considers only a single conceptual model, may fail to adequately sample the relevant space of plausible conceptual models. As such, it is prone to modelling bias and underestimation of predictive uncertainty.     

Multisource hydrologic modeling uncertainty analysis using the IBUNE framework in a humid catchment

Hydrologic cycle is a complex system associated with both certain and uncertain constituents. The propagation of confidence bounds from different uncertainty sources to model output is of great significance for hydrologic modeling. In this paper, we applied the integrated bayesian uncertainty estimator to quantify the effects of parameter, input and model structure uncertainty on hydrologic modeling progressively. Two hydrologic models (Xinanjiang model and TOPMODEL) were applied to a humid catchment under three scenarios. Case I: the shuffled complex evolution metropolis (SCEM-UA) algorithm was conducted to determine the posterior parameter distribution of hydrologic models and analyze the corresponding forecast uncertainty. Case II: input uncertainty was also considered by assuming rain depth bias follows a normal distribution, and integrated with SCEM-UA. Case III: Simulations from two models were combined by the Bayesian model averaging to fully quantify multisource uncertainty effects. Results suggested that, from Case I to II, the containing ratio (percentage of observed streamflow enveloped by 95% confidence interval) obviously increased by an average magnitude of 10% for the study period 2000–2006. Besides, it also found that the width of 95% confidence interval became wider and narrower for Xinanjiang model and TOPMODEL, respectively, from Case I to II. This may indicate that the uncertainty of TOPMODEL results was more remarkable than Xinanjiang model in Case I. By combining results from two models, model structure uncertainty was also considered in Case III. The accuracy of uncertainty bounds further improved with the containing ratio of 95% confidence interval >95%. In addition, the optimized deterministic results from the uncertainty analysis showed that the average Nash–Sutcliffe coefficient increased continually from Case I to II and III (0.82, 0.84 and 0.90, respectively) for the study period. The analysis demonstrated the improvement of modeling accuracy when extra uncertainty sources were also quantified, and this finding also proved the applicability of IBUNE framework in hydrologic modeling.     

Assessment of design rainfall uncertainty for hydrologic engineering applications in Hong Kong

Hydrologic engineering designs and analyses often require the specification of design storm which involves rainfall amount, duration and hyetograph. In practice, the determination of design rainfall in hydrologic engineering applications involves the frequency analysis of extreme rainfalls of different durations and the establishment of rainfall hyetograph for the design event under consideration. Sampling errors exist in the estimation of rainfall depth (or intensity) quantiles from frequency analysis, which will be transmitted in the process of determining the design rainfall hyetograph. This paper presents a practical methodological framework based on the bootstrap resampling scheme to assess the uncertainty features associated with the magnitude of estimated rainfall depth/intensity quantiles and the corresponding design hyetographs. The procedure is implemented to quantify uncertainty of design rainfall hyetograph following the Stormwater Drainage Manual of Hong Kong involving the use of rainfall intensity–duration–frequency (IDF) model. Of particular interesting is that the bootstrap resampling scheme implemented herein is modified to handle unequal record period of annual maximum rainfall data series of different durations and to account for their intrinsic correlations. According to the adopted rainfall IDF model, the design rainfall hyetograph is a function of the IDF model coefficients. Due to the correlation among rainfall quantiles of different durations, the IDF coefficients are found to be strongly related in a nonlinear fashion which should not be ignored in the establishment of the design hyetographs.     

Analyzing hydrologic uncertainty and its impact upon decision making in water resources

Bayesian Inference and Decision Theory tools are applied to the problem of synthetic hydrology when model and parameter uncertainty exist. Issues such as optimal parameter estimation, use of synthetic generation in design problems, and the effects of parameter uncertainty on statistical estimation are discussed and applied to the problem of reservoir slorage-yield analysis.     

Estimating the 4DVAR analysis error of GODAE products

We explore the ocean circulation estimates obtained by assimilating observational products made available by the Global Ocean Data Assimilation Experiment (GODAE) and other sources in an incremental, four-dimensional variational data assimilation system for the Intra-Americas Sea. Estimates of the analysis error (formally, the inverse Hessian matrix) are computed during the assimilation procedure. Comparing the impact of differing sea surface height and sea surface temperature products on both the final analysis error and difference between the model state estimates, we find that assimilating GODAE and non-GODAE products yields differences between the model and observations that are comparable to the differences between the observation products themselves. While the resulting analysis error estimates depend on the configuration of the assimilation system, the basic spatial structures of the standard deviations of the ocean circulation estimates are fairly robust and reveal that the assimilation procedure is capable of reducing the circulation uncertainty when only surface data are assimilated.     

Estimating landslide susceptibility areas considering the uncertainty inherent in modeling methods

Landslides are one of the most dangerous types of natural disasters, and damage due to landslides has been increasing in certain regions of the world because of increased precipitation. Policy decision makers require reliable information that can be used to establish spatial adaptation plans to protect people from landslide hazards. Researchers presently identify areas susceptible to landslides using various spatial distribution models. However, such data are associated with a high amount of uncertainty. This study focuses on quantifying the uncertainty of several spatial distribution models and identifying the effectiveness of various ensemble methods that can be used to provide reliable information to support policy decisions. The area of study was Inje-gun, Republic of Korea. Ten models were selected to assess landslide susceptibility. Moreover, five ensemble methods were selected for the aggregated results of the 10 models. The uncertainty was quantified using the coefficient of variation and the uncertainty map we developed revealed areas with strongly differing values among single models. A matrix map was created using an ensemble map and a coefficient of variation map. Using matrix analysis, we identified the areas that are most susceptible to landslides according to the ensemble model with a low uncertainty. Thus, the ensemble model can be a useful tool for supporting decision makers. The framework of this study can also be employed to support the establishment of landslide adaptation plans in other areas of the Republic of Korea and in other countries.     

Methods used for quantifying the prediction uncertainty of artificial neural network based hydrologic models

Application of artificial neural network (ANN) models has been reported to solve variety of water resources and environmental related problems including prediction, forecasting and classification, over the last two decades. Though numerous research studies have witnessed the improved estimate of ANN models, the practical applications are sometimes limited. The black box nature of ANN models and their parameters hardly convey the physical meaning of catchment characteristics, which result in lack of transparency. In addition, it is perceived that the point prediction provided by ANN models does not explain any information about the prediction uncertainty, which reduce the reliability. Thus, there is an increasing consensus among researchers for developing methods to quantify the uncertainty of ANN models, and a comprehensive evaluation of uncertainty methods applied in ANN models is an emerging field that calls for further improvements. In this paper, methods used for quantifying the prediction uncertainty of ANN based hydrologic models are reviewed based on the research articles published from the year 2002 to 2015, which focused on modeling streamflow forecast/prediction. While the flood forecasting along with uncertainty quantification has been frequently reported in applications other than ANN in the literature, the uncertainty quantification in ANN model is a recent progress in the field, emerged from the year 2002. Based on the review, it is found that methods for best way of incorporating various aspects of uncertainty in ANN modeling require further investigation. Though model inputs, parameters and structure uncertainty are mainly considered as the source of uncertainty, information of their mutual interaction is still lacking while estimating the total prediction uncertainty. The network topology including number of layers, nodes, activation function and training algorithm has often been optimized for the model accuracy, however not in terms of model uncertainty. Finally, the effective use of various uncertainty evaluation indices should be encouraged for the meaningful quantification of uncertainty. This review article also discusses the effectiveness and drawbacks of each method and suggests recommendations for further improvement.     

Stochastic observation error and uncertainty in water quality evaluation

When evaluating water quality, the influence of physical weight of the observed index is normally taken into account, but the influence of stochastic observation error (SOE) is not adequately considered. Using Monte Carlo simulation, combined with Shannon entropy, the Principle of Maximum Entropy (POME) and Tsallis entropy, this study investigates the influence of stochastic observation error (SOE) for two cases of the observed index: small observation error and large observation error. Randomness and fuzziness represent two types of uncertainties that are deemed significant and should be considered simultaneously when developing or evaluating water quality models. To that end, three models are employed here: two of the models, named as model I and model II, consider both the fuzziness and randomness, and another model, considers only fuzziness. The results from three representative lakes in China show that for all three models, the influence of stochastic observation error (SOE) on water quality evaluation can be significant irrespective of whether the water quality index has a small observation error or a large observation error. Furthermore, when there is a significant difference in the accuracy of observations, the influence of stochastic observation error (SOE) on water quality evaluation increases. The water quality index whose SOE is minimum determines the results of evaluation.     

The soil-moisture zone in a physically-based hydrologic model

The present paper describes an approach to modelling the unsaturated soil-moisture zone in the framework of an integrated physically-based hydrologic response model. It is supposed that the subsurface flow regime may be viewed as two separate entities — a saturated flow system which may be modelled by standard two-dimensional regional techniques, and a single overlying unsaturated zone in which the flow is essentially vertical. Coupling takes place via the definition of saturation at the lower boundary of the unsaturated zone, and via a conservative water balance. Attention is focused on the computational procedure for the unsaturated zone as a self-contained module. The major difficulties are the definition of the interface between the saturated and unsaturated zones, the nonlinear character of the equation used to describe unsaturated flow, the inclusion of realistic atmospheric boundary conditions, and, the interaction between water uptake by plants and available soil-moisture. Each of these points is discussed, in turn, with the emphasis on mathematically formulating the problem in such a way that the most important physical features are reproduced with a minimal amount of computational effort. The text concludes with a few illustrative examples.     

A low‐dimensional physically based model of hydrologic control of shallow landsliding on complex hillslopes

Hillslopes have complex three‐dimensional shapes that are characterized by their plan shape, profile curvature of surface and bedrock, and soil depth. To investigate the stability of complex hillslopes (with different slope curvatures and plan shapes), we combine the hillslope‐storage Boussinesq (HSB) model with the infinite slope stability method. The HSB model is based on the continuity and Darcy equations expressed in terms of storage along the hillslope. Solutions of the HSB equation account explicitly for plan shape by introducing the hillslope width function and for profile curvature through the bedrock slope angle and the hillslope soil depth function. The presented model is composed of three parts: a topography model conceptualizing three‐dimensional soil mantled landscapes, a dynamic hydrology model for shallow subsurface flow and water table depth (HSB model) and an infinite slope stability method based on the Mohr–Coulomb failure law. The resulting hillslope‐storage Boussinesq stability model (HSB‐SM) is able to simulate rain‐induced shallow landsliding on hillslopes with non‐constant bedrock slope and non‐parallel plan shape. We apply the model to nine characteristic hillslope types with three different profile curvatures (concave, straight, convex) and three different plan shapes (convergent, parallel, divergent). In the presented model, the unsaturated storage has been calculated based on the unit head gradient assumption. To relax this assumption and to investigate the effect of neglecting the variations of unsaturated storage on the assessment of slope stability in the transient case, we also combine a coupled model of saturated and unsaturated storage and the infinite slope stability method. The results show that the variations of the unsaturated zone storage do not play a critical role in hillslope stability. Therefore, it can be concluded that the presented dynamic slope stability model (HSB‐SM) can be used safely for slope stability analysis on complex hillslopes. Our results show that after a certain period of rainfall the convergent hillslopes with concave and straight profiles become unstable more quickly than others, whilst divergent convex hillslopes remain stable (even after intense rainfall). In addition, the relation between subsurface flow and hillslope stability has been investigated. Our analyses show that the minimum safety factor (FS) occurs when the rate of subsurface flow is a maximum. In fact, by increasing the subsurface flow, stability decreases for all hillslope shapes. Copyright © 2008 John Wiley & Sons, Ltd.     

Estimating the uncertainty of coral isochron U–Th ages

Recently, alternative models to estimate the age of diagenetically altered fossil reef corals have been presented based on either redistribution of U or its immediate daughters ²³⁴Th and ²³⁰Th. Here, we present three methods to estimate the uncertainty of ages derived using an amended version of our coral isochron method [Scholz et al., 2004. U-series dating of diagenetically altered fossil reef corals. Earth and Planetary Science Letters 218, 163–178], which is based on addition/loss of U. The obtained uncertainties are substantially larger than those previously published and should, in general, be more reliable. The isochron method yields larger uncertainties than alternative models based on Th redistribution due to -recoil processes. However, comparison of model open-system ages based on such redistribution of U-series daughters for different sub-samples from an individual coral specimen shows that the smaller errors derived with these models cannot account for the observed variability. We recognise that none of the available models is applicable to all corals, probably reflecting different diagenetic processes even in different sub-samples from one coral specimen. To better understand the diagenetic processes and precisely constrain the uncertainties of the ages derived from diagenetically altered corals, the application of all available models is recommended.     

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.     

Weighing the importance of model uncertainty against parameter uncertainty in earthquake loss assessments

Epistemic uncertainties can be classified into two major categories: parameter and model. While the first one stems from the difficulties in estimating the values of input model parameters, the second comes from the difficulties in selecting the appropriate type of model. Investigating their combined effects and ranking each of them in terms of their influence on the predicted losses can be useful in guiding future investigations. In this context, we propose a strategy relying on variance-based global sensitivity analysis, which is demonstrated using an earthquake loss assessment for Pointe-à-Pitre (Guadeloupe, France). For the considered assumptions, we show: that uncertainty of losses would be greatly reduced if all the models could be unambiguously selected; and that the most influential source of uncertainty (whether of parameter or model type) corresponds to the seismic activity group. Finally, a sampling strategy was proposed to test the influence of the experts’ weights on models and on the assumed coefficients of variation of parameter uncertainty. The former influenced the sensitivity measures of the model uncertainties, whereas the latter could completely change the importance rank of the uncertainties associated to the vulnerability assessment step.     

Assessing the uncertainties of hydrologic model selection in climate change impact studies

The uncertainties associated with atmosphere‐ocean General Circulation Models (GCMs) and hydrologic models are assessed by means of multi‐modelling and using the statistically downscaled outputs from eight GCM simulations and two emission scenarios. The statistically downscaled atmospheric forcing is used to drive four hydrologic models, three lumped and one distributed, of differing complexity: the Sacramento Soil Moisture Accounting (SAC‐SMA) model, Conceptual HYdrologic MODel (HYMOD), Thornthwaite‐Mather model (TM) and the Precipitation Runoff Modelling System (PRMS). The models are calibrated based on three objective functions to create more plausible models for the study. The hydrologic model simulations are then combined using the Bayesian Model Averaging (BMA) method according to the performance of each models in the observed period, and the total variance of the models. The study is conducted over the rainfall‐dominated Tualatin River Basin (TRB) in Oregon, USA. This study shows that the hydrologic model uncertainty is considerably smaller than GCM uncertainty, except during the dry season, suggesting that the hydrologic model selection‐combination is critical when assessing the hydrologic climate change impact. The implementation of the BMA in analysing the ensemble results is found to be useful in integrating the projected runoff estimations from different models, while enabling to assess the model structural uncertainty. Copyright © 2011 John Wiley & Sons, Ltd.     

Assessment of regional estimation error by canonical residual kriging

Various regional flood frequency analysis procedures are used in hydrology to estimate hydrological variables at ungauged or partially gauged sites. Relatively few studies have been conducted to evaluate the accuracy of these procedures and estimate the error induced in regional flood frequency estimation models. The objective of this paper is to assess the overall error induced in the residual kriging (RK) regional flood frequency estimation model. The two main error sources in specific flood quantile estimation using RK are the error induced in the quantiles local estimation procedure and the error resulting from the regional quantile estimation process. Therefore, for an overall error assessment, the corresponding errors associated with these two steps must be quantified. Results show that the main source of error in RK is the error induced into the regional quantile estimation method. Results also indicate that the accuracy of the regional estimates increases with decreasing return periods. Copyright © 2010 John Wiley & Sons, Ltd.     

Study on uncertainty of inversion and resolution limit in the presence of noise

This study analyzed the uncertainty of inversion and the resolution limit in the presence of noise by means of statistical experiments. The exhaustive method is adopted to obtain the global optimal solution in each experiment. We found that even with small level of noise, solutions fluctuate in a large range for the thin bed. The distribution of solutions in the presence of noise is closely related to the spread of the cost function in the absence of noise. As a result, the area of a certain neighborhood around the true solution on the spread of the cost function in the absence of noise is used to evaluate the uncertainty of inversion and the resolution limit in the presence of noise. In the case that the SNR (signal-to-noise ratio) is 5 in this study, solutions focus around the true solution with a very small uncertainty only when the bed thickness is greater than the reciprocal of the double predominant frequency of the convoluting wavelet.