Dmitri Kavetski

Bayesian analysis of input uncertainty in hydrological modeling: 2. Application

The Bayesian total error analysis (BATEA) methodology directly addresses both input and output errors in hydrological modeling, requiring the modeler to make explicit, rather than implicit, assumptions about the likely extent of data uncertainty. This study considers a BATEA assessment of two North American catchments: (1) French Broad River and (2) Potomac basins. It assesses the performance of the conceptual Variable Infiltration Capacity (VIC) model with and without accounting for input (precipitation) uncertainty. The results show the considerable effects of precipitation errors on the predicted hydrographs (especially the prediction limits) and on the calibrated parameters. In addition, the performance of BATEA in the presence of severe model errors is analyzed. While BATEA allows a very direct treatment of input uncertainty and yields some limited insight into model errors, it requires the specification of valid error models, which are currently poorly understood and require further work. Moreover, it leads to computationally challenging highly dimensional problems. For some types of models, including the VIC implemented using robust numerical methods, the computational cost of BATEA can be reduced using Newton-type methods.

Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory

Parameter estimation in rainfall-runoff models is affected by uncertainties in the measured input/output data (typically, rainfall and runoff, respectively), as well as model error. Despite advances in data collection and model construction, we expect input uncertainty to be particularly significant (because of the high spatial and temporal variability of precipitation) and to remain considerable in the foreseeable future. Ignoring this uncertainty compromises hydrological modeling, potentially yielding biased and misleading results. This paper develops a Bayesian total error analysis methodology for hydrological models that allows (indeed, requires) the modeler to directly and transparently incorporate, test, and refine existing understanding of all sources of data uncertainty in a specific application, including both rainfall and runoff uncertainties. The methodology employs additional (latent) variables to filter out the input corruption given the model hypothesis and the observed data. In this study, the input uncertainty is assumed to be multiplicative Gaussian and independent for each storm, but the general framework allows alternative uncertainty models. Several ways of incorporating vague prior knowledge of input corruption are discussed, contrasting Gaussian and inverse gamma assumptions; the latter method avoids degeneracies in the objective function. Although the general methodology is computationally intensive because of the additional latent variables, a range of modern numerical methods, particularly Monte Carlo analysis combined with fast Newton-type optimization methods and Hessian-based covariance analysis, can be employed to obtain practical solutions.

Critical evaluation of parameter consistency and predictive uncertainty in hydrological modeling: A case study using Bayesian total error analysis

The lack of a robust framework for quantifying the parametric and predictive uncertainty of conceptual rainfall-runoff (CRR) models remains a key challenge in hydrology. The Bayesian total error analysis (BATEA) methodology provides a comprehensive framework to hypothesize, infer, and evaluate probability models describing input, output, and model structural error. This paper assesses the ability of BATEA and standard calibration approaches (standard least squares (SLS) and weighted least squares (WLS)) to address two key requirements of uncertainty assessment: (1) reliable quantification of predictive uncertainty and (2) reliable estimation of parameter uncertainty. The case study presents a challenging calibration of the lumped GR4J model to a catchment with ephemeral responses and large rainfall gradients. Postcalibration diagnostics, including checks of predictive distributions using quantile-quantile analysis, suggest that while still far from perfect, BATEA satisfied its assumed probability models better than SLS and WLS. In addition, WLS/SLS parameter estimates were highly dependent on the selected rain gauge and calibration period. This will obscure potential relationships between CRR parameters and catchment attributes and prevent the development of meaningful regional relationships. Conversely, BATEA provided consistent, albeit more uncertain, parameter estimates and thus overcomes one of the obstacles to parameter regionalization. However, significant departures from the calibration assumptions remained even in BATEA, e.g., systematic overestimation of predictive uncertainty, especially in validation. This is likely due to the inferred rainfall errors compensating for simplified treatment of model structural error.

Pursuing the method of multiple working hypotheses for hydrological modeling

Ambiguities in the representation of environmental processes have manifested themselves in a plethora of hydrological models, differing in almost every aspect of their conceptualization and implementation. The current overabundance of models is symptomatic of an insufficient scientific understanding of environmental dynamics at the catchment scale, which can be attributed to difficulties in measuring and representing the heterogeneity encountered in natural systems. This commentary advocates using the method of multiple working hypotheses for systematic and stringent testing of model alternatives in hydrology. We discuss how the multiple-hypothesis approach provides the flexibility to formulate alternative representations (hypotheses) describing both individual processes and the overall system. When combined with incisive diagnostics to scrutinize multiple model representations against observed data, this provides hydrologists with a powerful and systematic approach for model development and improvement. Multiple-hypothesis frameworks also support a broader coverage of the model hypothesis space and hence improve the quantification of predictive uncertainty arising from system and component nonidentifiabilities. As part of discussing the advantages and limitations of multiple-hypothesis frameworks, we critically review major contemporary challenges in hydrological hypothesis-testing, including exploiting different types of data to investigate the fidelity of alternative process representations, accounting for model structure ambiguities arising from major uncertainties in environmental data, quantifying regional differences in dominant hydrological processes, and the grander challenge of understanding the self-organization and optimality principles that may functionally explain and describe the heterogeneities evident in most environmental systems. We assess recent progress in these research directions, and how new advances are possible using multiple-hypothesis methodologies.

Elements of a flexible approach for conceptual hydrological modeling: 1. Motivation and theoretical development

This paper introduces a flexible framework for conceptual hydrological modeling, with two related objectives: (1) generalize and systematize the currently fragmented field of conceptual models and (2) provide a robust platform for understanding and modeling hydrological systems. In contrast to currently dominant “fixed” model applications, the flexible framework proposed here allows the hydrologist to hypothesize, build, and test different model structures using combinations of generic components. This is particularly useful for conceptual modeling at the catchment scale, where limitations in process understanding and data availability remain major research and operational challenges. The formulation of the model architecture and individual components to represent distinct aspects of catchment-scale function, such as storage, release, and transmission of water, is discussed. Several numerical strategies for implementing the model equations within a computationally robust framework are also presented. In the companion paper, the potential of the flexible framework is examined with respect to supporting more systematic and stringent hypothesis testing, for characterizing catchment diversity, and, more generally, for aiding progress toward more unified hydrological theory at the catchment scale.

A unified approach for process-based hydrologic modeling: 1. Modeling concept

This work advances a unified approach to process-based hydrologic modeling to enable controlled and systematic evaluation of multiple model representations (hypotheses) of hydrologic processes and scaling behavior. Our approach, which we term the Structure for Unifying Multiple Modeling Alternatives (SUMMA), formulates a general set of conservation equations, providing the flexibility to experiment with different spatial representations, different flux parameterizations, different model parameter values, and different time stepping schemes. In this paper, we introduce the general approach used in SUMMA, detailing the spatial organization and model simplifications, and how different representations of multiple physical processes can be combined within a single modeling framework. We discuss how SUMMA can be used to systematically pursue the method of multiple working hypotheses in hydrology. In particular, we discuss how SUMMA can help tackle major hydrologic modeling challenges, including defining the appropriate complexity of a model, selecting among competing flux parameterizations, representing spatial variability across a hierarchy of scales, identifying potential improvements in computational efficiency and numerical accuracy as part of the numerical solver, and improving understanding of the various sources of model uncertainty.

Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards equation

Abstract Adaptive time stepping with embedded error control is applied to the mixed form of Richards equation. It is the first mathematically based adaptive scheme applied to this form of Richards equation. The key to the method is the approximation of the local truncation error of the scheme in terms of the pressure head, although, to enforce mass conservation, the principal time approximation is based on the moisture content. The time stepping scheme is closely related to an implicit Thomas–Gladwell approximation and is unconditionally stable and second-order accurate. Numerical trials demonstrate that the new algorithm fully automates stepsize selection and robustly constrains temporal discretisation errors given a user tolerance. The adaptive mechanism is shown to improve the performance of the non-linear solver, providing accurate initial solution estimates for the iterative process. Furthermore, the stepsize variation patterns reflect the adequacy of the spatial discretisation, here accomplished by linear finite elements. When sufficiently dense spatial grids are used, the time step varies smoothly, while excessively coarse grids induce stepsize oscillations.

A unified approach for process-based hydrologic modeling: 2. Model implementation and case studies

This work advances a unified approach to process-based hydrologic modeling, which we term the “Structure for Unifying Multiple Modeling Alternatives (SUMMA).” The modeling framework, introduced in the companion paper, uses a general set of conservation equations with flexibility in the choice of process parameterizations (closure relationships) and spatial architecture. This second paper specifies the model equations and their spatial approximations, describes the hydrologic and biophysical process parameterizations currently supported within the framework, and illustrates how the framework can be used in conjunction with multivariate observations to identify model improvements and future research and data needs. The case studies illustrate the use of SUMMA to select among competing modeling approaches based on both observed data and theoretical considerations. Specific examples of preferable modeling approaches include the use of physiological methods to estimate stomatal resistance, careful specification of the shape of the within-canopy and below-canopy wind profile, explicitly accounting for dust concentrations within the snowpack, and explicitly representing distributed lateral flow processes. Results also demonstrate that changes in parameter values can make as much or more difference to the model predictions than changes in the process representation. This emphasizes that improvements in model fidelity require a sagacious choice of both process parameterizations and model parameters. In conclusion, we envisage that SUMMA can facilitate ongoing model development efforts, the diagnosis and correction of model structural errors, and improved characterization of model uncertainty.

Elements of a flexible approach for conceptual hydrological modeling: 2. Application and experimental insights

In this articles companion paper, flexible approaches for conceptual hydrological modeling at the catchment scale were motivated, and the SUPERFLEX framework, based on generic model components, was introduced. In this article, the SUPERFLEX framework and the “fixed structure” GR4H model (an hourly version of the popular GR4J model) are applied to four hydrologically distinct experimental catchments in Europe and New Zealand. The estimated models are scrutinized using several diagnostic measures, ranging from statistical metrics, such as the statistical reliability and precision of the predictive distribution of streamflow, to more process-oriented diagnostics based on flow-duration curves and the correspondence between model states and groundwater piezometers. Model performance was clearly catchment specific, with a single fixed structure unable to accommodate intercatchment differences in hydrological behavior, including seasonality and thresholds. This highlights an important limitation of any “fixed” model structure. In the experimental catchments, the ability of competing model hypotheses to reproduce hydrological signatures of interest could be interpreted on the basis of independent fieldwork insights. The potential of flexible frameworks such as SUPERFLEX is then examined with respect to systematic and stringent hypothesis-testing in hydrological modeling, for characterizing catchment diversity, and, more generally, for aiding progress toward a more unified formulation of hydrological theory at the catchment scale. When interpreted in physical process-oriented terms, the flexible approach can also serve as a language for dialogue between modeler and experimentalist, facilitating the understanding, representation, and interpretation of catchment behavior.

Comparison of joint versus postprocessor approaches for hydrological uncertainty estimation accounting for error autocorrelation and heteroscedasticity

The paper appraises two approaches for the treatment of heteroscedasticity and autocorrelation in residual errors of hydrological models. Both approaches use weighted least squares (WLS), with heteroscedasticity modeled as a linear function of predicted flows and autocorrelation represented using an AR(1) process. In the first approach, heteroscedasticity and autocorrelation parameters are inferred jointly with hydrological model parameters. The second approach is a two-stage “postprocessor” scheme, where Stage 1 infers the hydrological parameters ignoring autocorrelation and Stage 2 conditionally infers the heteroscedasticity and autocorrelation parameters. These approaches are compared to a WLS scheme that ignores autocorrelation. Empirical analysis is carried out using daily data from 12 US catchments from the MOPEX set using two conceptual rainfall-runoff models, GR4J, and HBV. Under synthetic conditions, the postprocessor and joint approaches provide similar predictive performance, though the postprocessor approach tends to underestimate parameter uncertainty. However, the MOPEX results indicate that the joint approach can be nonrobust. In particular, when applied to GR4J, it often produces poor predictions due to strong multiway interactions between a hydrological water balance parameter and the error model parameters. The postprocessor approach is more robust precisely because it ignores these interactions. Practical benefits of accounting for error autocorrelation are demonstrated by analyzing streamflow predictions aggregated to a monthly scale (where ignoring daily-scale error autocorrelation leads to significantly underestimated predictive uncertainty), and by analyzing one-day-ahead predictions (where accounting for the error autocorrelation produces clearly higher precision and better tracking of observed data). Including autocorrelation into the residual error model also significantly affects calibrated parameter values and uncertainty estimates. The paper concludes with a summary of outstanding challenges in residual error modeling, particularly in ephemeral catchments.