Visualisation of spatial uncertainty in hydrological modelling

Hydrologic models use relatively simple mathematical equations to conceptualize and aggregate the complex, spatially distributed, and highly interrelated water, energy, and vegetation processes in a watershed. A consequence of process aggregation is that the model parameters often do not represent directly measurable entities and must therefore be estimated using measurements of the system inputs and outputs. During this process, known as model calibration, the parameters are adjusted so that the behavior of the model approximates, as closely and consistently as possible, the observed response of the hydrologic system over some historical period of time. In practice, however, because of errors in the model structure and the input (forcing) and output data, this has proven to be difficult, leading to considerable uncertainty in the model predictions. This paper surveys the limitations of current model calibration methodologies, which treat the uncertainty in the input‐output relationship as being primarily attributable to uncertainty in the parameters and presents a simultaneous optimization and data assimilation (SODA) method, which improves the treatment of uncertainty in hydrologic modeling. The usefulness and applicability of SODA is demonstrated by means of a pilot study using data from the Leaf River watershed in Mississippi and a simple hydrologic model with typical conceptual components.

Parameter uncertainty in hydrologic modeling is commonly evaluated, but assessing the impact of spatial input data uncertainty in spatially descriptive ‘distributed’ models is not common. This study compares the significance of uncertainty in spatial input data and model parameters on the output uncertainty of a distributed hydrology and sediment transport model, HYdrology Simulation using Time-ARea method (HYSTAR). The Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm was used to quantify parameter uncertainty of the model. Errors in elevation and land cover layers were simulated using the Sequential Gaussian/Indicator Simulation (SGS/SIS) techniques and then incorporated into the model to evaluate their impact on the outputs relative to those of the parameter uncertainty. This study demonstrated that parameter uncertainty had a greater impact on model output than did errors in the spatial input data. In addition, errors in elevation data had a greater impact on model output than did errors in land cover data. Thus, for the HYSTAR distributed hydrologic model, accuracy and reliability can be improved more effectively by refining parameters rather than further improving the accuracy of spatial input data and by emphasizing the topographic data over the land cover data.

Received 19 April 2007 Accepted 24 April 2007 Beven’s (2006) commentary on the uncertainty in hydrological modelling seems to have touched a bit of a nerve in the hydrological community. However, while there may be different definitions for uncertainty (Montanari, 2007), different approaches to quantifying it (Todini and Mantovan, in press), and calls for more effort in evaluating it (Hall et al., 2007), there is no denial that uncertainty is an inherent and unwelcome attribute of all predictions. This conversation about adapting to the rude reality of uncertainty could be misconstrued as an admission that we have reached a limit in hydrology, where improvements in the value of our predictions are less a function of theoretical advances or of further field observations and experimentation and more a function of advances in numeric methods or computing horsepower. Furthermore, young hydrologists overhearing this discussion may be excused if they mistook the reaction to Beven’s call for strategically advancing the science and ethics of hydrology as an argument for devising ever more elegant and sophisticated apologies for shortcomings in our hydrological predictions. This current focus on how to maximize the usefulness of models in the context of uncertainty is pragmatic. However, optimizing the present value of model output does not preclude a vision of the future where models are not only useful but are also truthful. Uncertainty is frustrating hydrologists in a way that is similar to the way that microbes frustrated medical science before the discovery of antibiotics. We need to find ways to move beyond mere treatment of symptoms to resolution of underlying pathologies. Like most hydrologists, I believed that I used parameters responsibly. I discovered my addiction to poorly defined parameter ranges almost by accident. A colleague wrote an R script, which made it easy to produce dotty-plots of my calibrations showing parameter identifiability. My first reaction was denial, that there must be a problem with the objective function, but gradually I have come to the realization that I have a history of parameter-abuse. Accepting that I suffer from equifinality was the first step on the long road to recovery. The signs of addiction are subtle but recognizable. Hydrographs that are just a little too good, where the uncertainty in model predictions is less than the uncertainty in the driving force data. Axis-scaling that is a little too compressed, hiding inconsistencies in the model fit. You may recognize these symptoms in your colleagues but deny that you too, keep a flask of parameter ranges in a desk drawer that you sip generously from when faced with a difficult hydrograph. The wrack and ruin of parameter abuse is wide-spread. We need to be particularly concerned about the effect our example has on young hydrologists who are just now trying to learn how to calibrate responsibly. We need to provide alternatives to parameter abuse. A priori parameter identification (e.g. Wagener and Wheater, 2006) and development of observation systems that will support increased understanding and the development and testing of hydrologic theories and organizing principles (e.g. McDonnell et al., 2007) are the basis for the first of the four-pillars approach of Prevention, Treatment, Harm

Hydrologic and water quality models (HWQMs) are increasingly used to support decisions on various environmental issues and policy directions for present and future scenarios, at scales varying from watershed to continental levels. Uncertainty associated with such models may affect the ability of the models to accurately evaluate the response of complex systems, leading to misguided assessments and risk management decisions. Current well-known HWQMs contain numerous input parameters, many of which are not known with certainty, and in other cases model users can hardly recognize the genesis of uncertainty. Uncertainty in data, model structure, and model parameters can propagate throughout model runs, causing the model output to substantially deviate from the expected response of the natural system. Various uncertainty assessment methods have been used with different HWQMs, creating concerns about an adequate approach for handling uncertainty in these models and how such an approach can be implemented across various discretization complexities and scales. In this article, our primary intention is to review uncertainty in the currently used HWQMs and to provide guidance and useful information for researchers and investigators. In this regard, we explore the genesis of uncertainty in hydrologic and water quality modeling (i.e., spatiotemporal scales, model representation, model discretization, model parameterization) and provide strategies for assessing uncertainty in hydrologic and water quality modeling on local and global scales when interpreting the model output.

Meteorological forcing is a major source of uncertainty in hydrological modeling. The recent development of probabilistic large‐domain meteorological data sets enables convenient uncertainty characterization, which however is rarely explored in large‐domain research. This study analyzes how uncertainties in meteorological forcing data affect hydrological modeling in 289 representative cryosphere basins by forcing the Structure for Unifying Multiple Modeling Alternatives (SUMMA) and mizuRoute models with precipitation and air temperature ensembles from the Ensemble Meteorological Data set for Planet Earth (EM‐Earth). EM‐Earth probabilistic estimates are used in ensemble simulation for uncertainty analysis. The results reveal the magnitude, spatial distribution, and scale effect of uncertainties in meteorological, snow, runoff, soil water, and energy variables. There are three main findings. (a) The uncertainties in precipitation and temperature lead to substantial uncertainties in hydrological model outputs, some of which exceed 100% of the magnitude of the output variables themselves. (b) The uncertainties of different variables show distinct scale effects caused by spatial averaging or temporal averaging. (c) Precipitation uncertainties have the dominant impact for most basins and variables, while air temperature uncertainties are also nonnegligible, sometimes contributing more to modeling uncertainties than precipitation uncertainties. We find that three snow‐related variables (snow water equivalent, snowfall amount, and snowfall fraction) can be used to estimate the impact of air temperature uncertainties for different model output variables. In summary, this study provides insight into the impact of probabilistic data sets on hydrological modeling and quantifies the uncertainties in cryosphere basin modeling that stem from the meteorological forcing data.

Physically-based urban rainfall-runoff models are mostly applied without parameter calibration. Given some preliminary estimates of the uncertainty of the model parameters the associated model output uncertainty can be calculated. Monte-Carlo simulation followed by multi-linear regression is used for this analysis. The calculated model output uncertainty can be compared to the uncertainty estimated by comparing model output and observed data. Based on this comparison systematic or spurious errors can be detected in the observation data, the validity of the model structure can be confirmed, and the most sensitive parameters can be identified. If the calculated model output uncertainty is unacceptably large the most sensitive parameters should be calibrated to reduce the uncertainty. Observation data for which systematic and/or spurious errors have been detected should be discarded from the calibration data. This procedure is referred to as preliminary uncertainty analysis; it is illustrated with an example. The HYSTEM program is applied to predict the runoff volume from an experimental catchment with a total area of 68 ha and an impervious area of 20 ha. Based on the preliminary uncertainty analysis, for 7 of 10 events the measured runoff volume is within the calculated uncertainty range, i.e. less than or equal to the calculated model predictive uncertainty. The remaining 3 events include most likely systematic or spurious errors in the observation data (either in the rainfall or the runoff measurements). These events are then discarded from further analysis. After calibrating the model the predictive uncertainty of the model is estimated.

Uncertainty analysis has become an important topic in environmental research. Uncertainty in hydrological modeling, in general, has been studied by investigating mainly the influence of the parameter uncertainty on the uncertainty of the simulated outputs. This paper focuses essentially on the impact of point input uncertainty on fully distributed hydrological modeling and proposes an integrated approach to cope with input and parameter uncertainty. The approach uses Bayesian theory in two steps: first, to compute the uncertainty in input forcing data, and second, to compute the hydrological parameter uncertainty and the uncertainties of a lumped‐error term that include other sources of errors: response errors and model error. The method is applied with a fully distributed model: WaSiM‐ETH. The results show that the methodology proposed here is a valuable tool to assess different sources of uncertainty in hydrological modeling and show the effects of uncertainty in the input forcing when a fully distributed, physically based hydrological model is used.

The traditional treatment of uncertainty in hydrological modelling primarily attributes it to model parameters, but rarely systematically considers meteorological input errors, especially in quantifying the impact of meteorological input errors on parameter uncertainty. This study developed a Bayesian‐based integrated approach to quantitatively investigate uncertainties in meteorological inputs (precipitation and temperature) and model parameters as well as the variation in parameter uncertainty due to meteorological input errors. Additionally, we analysed the propagation from these uncertainties to runoff response in snowmelt and non‐snowmelt periods. The applicability and advantages of this approach were presented by applying of the Soil and Water Assessment Tool to the Shitoukoumen Reservoir Catchment. Differential Evolution Adaptive Metropolis‐Markov Chain Monte Carlo was applied for the straightforward Bayesian inference the uncertainties of meteorological inputs and model parameters. On this basis, multilevel factorial analysis technology was used to quantitatively investigate the specific impact of the model parameters' individual and interactive effects due to meteorological input errors. Finally, the impact of meteorological input errors and model parameter uncertainty on the model performance were analysed and quantified systematically. The results showed that the meteorological input errors could affect the random characteristics of multiple model parameters. Moreover, meteorological input errors could further affect the model parameters' effects on annual average runoff. Overall, the above results have significant implications in enhancing hydrological model to simulate/predict runoff and understanding hydrological processes during different periods.

In this paper, two controllers with a compound disturbance observer are proposed for a two-wheeled inverted robot (TWIR) with model uncertainty and unknown input disturbance. First, an equivalent linear model of the TWIR with uncertainty and input disturbance is proposed using the Taylor series expansion for the nonlinear model of the TWIR at an equilibrium point, in which the nonlinear part of the Taylor series and the model uncertainty are combined with unknown input disturbance as compound input disturbance. Then, the compound input disturbance is estimated by using the Newton method and reference model. As the estimated compound disturbance is used to compensate for the compound disturbance, the equivalent linear system becomes closely definite without compound input disturbance. Finally, two controllers are proposed using the equivalent linear system. Stability analysis of the proposed control methods is also given. To illustrate the proposed methods, some simulations for the TWIR are performed and compared with the existing methods. The main contribution of this work includes the following: (i) simple controllers based on compound input disturbance observer for trajectory tracking and balancing of TWIRs with unknown input disturbance and model uncertainty are proposed; (ii) the stability of proposed closed-loop control systems is proved; (iii) our proposed methods are simulated and compared with the existing methods.

Bayesian methods for model uncertainty analysis with application to future sea level rise.

Diffuse groundwater recharge estimation confronting hydrological modelling uncertainty

The assessment of sediment yield from reservoir siltation requires knowledge of the reservoir's sediment trap efficiency (TE). Widely used approaches for the estimation of the long‐term mean TE rely on the ratio of the reservoir's storage capacity (C) to its catchment size (A) or mean annual inflow (I). These approaches have been developed from a limited number of reservoirs (N ≤ 40), most of them located in temperate climate regions. Their general applicability to reservoirs receiving highly variable runoff such as in semi‐arid areas has been questioned. Here, we examine the effect of ephemeral inflow on the TE of 10 small (≤ 280 × 103 m3), intermittently dry reservoirs located in the Kruger National Park. Fieldwork was conducted to determine the storage capacity of the reservoir basins. The frequency and magnitude of spillage events was simulated with the daily time step Pitman rainfall–runoff model. Different runoff scenarios were established to cope with uncertainties arising from the lack of runoff records and imperfect input data. Scenarios for the relationship between water and sediment discharge were created based on sediment rating curves. Taking into account uncertainties in hydrological modelling, uncertainties of mean TE estimates, calculated from all scenarios (N = 9), are moderate, ranging from ±6 to ±11% at the 95% confidence level. By comparison, estimating TE from the storage capacity to catchment area (C/A) ratio induces high uncertainty (ranges of 35 to 65%), but this uncertainty can be confined (15 to 33%) when the latter approach is combined with hydrological modelling. Established methods relying on the storage capacity to mean annual inflow (C/I) ratio most probably lead to an overestimation of the TE for the investigated reservoirs. The approach presented here may be used instead to estimate the TE of small, intermittently dry reservoirs in semi‐arid climate regions. Copyright © 2017 John Wiley & Sons, Ltd.

Non-stationarity has long been recognized as a fundamental challenge in hydrological modelling, as climate change and human activities continuously alter catchment properties and the boundary conditions under which hydrological systems operate. However, translating this long-standing recognition into systematic model evaluation remains challenging, as suitable large-sample hydrological datasets that explicitly represent temporal change are still scarce. Most existing datasets adopt a static design, with time-invariant catchment attributes and hydro-meteorological time series limited to retrospective observations, which constrains the systematic testing of hypotheses and models targeting non-stationary hydrological behaviour. Here, we introduce Changing-CAMELS, a pan-European, CAMELS-style dataset explicitly designed to support non-stationary hydrological modelling. Building on the strengths of existing datasets such as CAMELS, Caravan, and EStreams, the developed dataset moves beyond static representations by incorporating time-varying catchment attributes and future climate forcing on a European scale.In particular, dynamic land-use and land-cover changes are derived from the European LUCAS dataset, providing annual updates to vegetation and land-cover fractions. Simultaneously, the inclusion of both raw and bias-corrected daily-resolution regional and global climate model datasets extends hydro-meteorological forcing beyond the historical period. This integration enables consistent analyses of evolving land cover, shifting climate regimes, and their combined impacts on hydrological responses. Changing-CAMELS covers 4,575 catchments across Europe and harmonizes observations and attributes across national boundaries. By providing both retrospective and prospective information within a unified framework, the dataset allows researchers to systematically evaluate competing hypotheses, compare stationary and non-stationary model formulations, and assess the robustness and uncertainty of hydrological models under climate and land-use change.

An analysis of hydrological response to a multi-model approach based on an ensemble of seven snow models (SM; degree-day and mixed degree-day/energy balance models) coupled with three hydrological models (HM) is presented for a snowmelt-dominated basin in Canada. The present study aims to compare the performance and the reliability of different types of SM-HM combinations at simulating snowmelt flows over the 1961–2000 historical period. The multi-model approach also allows evaluating the uncertainties associated with the structure of the SM-HM ensemble to better predict river flows in Nordic environments. The 20-year calibration shows a satisfactory performance of the ensemble of 21 SM-HM combinations at simulating daily discharges and snow water equivalents (SWEs), with low streamflow volume biases. The validation of the ensemble of 21 SM-HM combinations is conducted over a 20-year period. Performances are similar to the calibration in simulating the daily discharges and SWEs, again with low model biases for streamflow. The spring-snowmelt-generated peak flow is captured only in timing by the ensemble of 21 SM-HM combinations. The results of specific hydrologic indicators show that the uncertainty related to the choice of the given HM in the SM-HM combinations cannot be neglected in a more quantitative manner in simulating snowmelt flows. The selection of the SM plays a larger role than the choice of the SM approach (degree-day versus mixed degree-day/energy balance) in simulating spring flows. Overall, the snow models provide a low degree of uncertainty to the total uncertainty in hydrological modeling for snow hydrology studies.

Propagation of uncertainty in nasal spray in vitro performance models using Monte Carlo simulation: Part I. model prediction using Monte Carlo Simulation