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International audience; This study identified and analysed the sources of error in runoff estimation via rating curves. This was undertaken by applying the BATEA framework to the rating curve model used to transform river stage to runoff for the Mahurangi catchment a heavily instrumented experimental catchment in New Zealand. The systematic errors due to rating curve uncertainty were the dominant source of uncertainty. This rating curve uncertainty is due to a limited number of uncertain runoff gaugings. The contribution of the random error component was small. This contradicts the common assumption of random runoff measurement error used in hydrological model calibration. The impact of rating curve uncertainty was incorporated into hydrological model calibration for the hourly Mahurangi data using the BATEA framework. Comparing these results to the typical standard least squares approach (with fixed rating curve) illustrated that incorporating rating curve uncertainty improved the reliability of the hydrological model predictions and caused a shift in the parameter distributions. These results have implications for hydrological model calibration. Runoff data are typically assumed to be observed but rating curve uncertainty induces significant systematic runoff errors which can have a significant impact on hydrological model predictions.

International audience; The quantification of uncertainties in hydrological predictions is a key step in improving the reliability and usefulness of flood forecasts. In this study, we investigate the impact of uncertainty from streamflow data used in flood forecasting models for real-time updating of hydrological ensemble predictions. The GRP hydrological forecasting model, a lumped rainfall-runoff model, is used over a selected number of catchments in France, for which gaugings and rating curves are available. Rating curve uncertainty is estimated for each catchment using a Bayesian approach. In the first step, the uncertainty in individual gaugings is quantified. It depends on the gauging method (e.g. velocity-area method, tracer dilution, ADCP, surface velocity measurements, etc.) and the operational characteristics of the gauging (e.g. spatial sampling of velocity and depth throughout the cross-section, unsteadiness of the flow, etc.). These uncertain gaugings are then used to estimate the rating curve and related uncertainties, using hydraulics-based priors on the rating curve coefficients. In the second step, the GRP model is used to produce streamflow forecast series. The model uses the last observed discharge and its related uncertainty to update the state of the routing reservoir at the time of forecasting. Errors in streamflow data are considered jointly with errors in precipitation forecasts. For the latter, the hydrological model is driven by the 10-day ECMWF deterministic and ensemble (51 members) precipitation forecasts for a period of about 3.5 years (March 2005 to September 2008). Ensemble streamflow predictions are evaluated against observed discharges (and judged in terms of bias, reliability, spread, ability to detect exceedances of critical discharges or water levels in advance, etc.) and as a function of forecast lead-time and warning thresholds. The results obtained by enabling/disabling the treatment of streamflow uncertainty are then compared and discussed, contributing to a better understanding of the role of streamflow errors in the total predictive uncertainty.

This study explores the decomposition of predictive uncertainty in hydrological modeling into its contributing sources. This is pursued by developing data-based probability models describing uncertainties in rainfall and runoff data, and incorporating them into the Bayesian Total Error Analysis methodology (BATEA). A case study based on the Yzeron catchment (France) and the conceptual rainfall-runoff model GR4J is presented. It exploits a calibration period where dense raingauge data is available to characterize the uncertainty in the catchment-average rainfall using geostatistical conditional simulation. The inclusion of information about rainfall and runoff data uncertainties overcomes ill-posedness problems and enables simultaneous estimation of forcing and structural errors as part of the Bayesian inference. This yields more reliable predictions than approaches that ignore or lump different sources of uncertainty in a simplistic way (e.g., standard least squares). It is shown that independently-derived data quality estimates are needed to decompose the total uncertainty in the runoff predictions into the individual contributions of rainfall, runoff and structural errors. In this case study, the total predictive uncertainty appears dominated by structural errors. Although further research is needed to interpret and verify this decomposition, it can provide strategic guidance for investments in environmental data collection and/or modeling improvement. More generally, this study demonstrates the power of the Bayesian paradigm to improve the reliability of environmental modeling using independent estimates of sampling and instrumental data uncertainties.

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International audience; Calibration and prediction in conceptual rainfall-runoff (CRR) modelling is affected by the sampling and measurement uncertainty in the observed input/output data and by the structural error of the model conceptualisation. The Bayesian Total Error Analysis methodology (BATEA) provides the opportunity to directly and comprehensively address these sources of uncertainty. BATEA is based on Bayesian hierarchical methods and uses explicit error models for input/output data and structural errors. However, previous studies demonstrated that simultaneous inference on forcing (e.g. rainfall) and structural errors requires strong prior knowledge of the error mechanisms (e.g., statistical properties of rainfall errors). This paper investigates methods to derive informative prior knowledge on data (input/output) errors. A geostatistical approach is used to estimate the sampling error made in approximating the areal rainfall by averaging point-values from a raingauge network. The geostatistical model generates an ensemble of rainfall fields conditioned on gauged values (conditional simulation). This ensemble is treated as a distribution of the true areal rainfall over the catchment and used as a prior in the BATEA framework. Prior knowledge on runoff errors is derived from a statistical analysis of the gaugings used to derive the rating curve. A comparative case study is then carried out, by comparing several inference schemes (full BATEA, ignoring input errors, ignoring structural errors, ignoring both input/structural errors). Results show that: (i) the inclusion of prior knowledge on rainfall accuracy allows simultaneous estimation of input and structural errors, whereas in the absence of such information the inference is ill-posed; (ii) accounting for all sources of errors leads to a more reliable estimation of predictive uncertainty. These findings demonstrate the key role played by priors on data (rainfall/runoff) accuracy, since they control the well-posedness of the inference and allow decomposing predictive uncertainty into its contributing sources. This paves the way for a meaningful analysis of structural errors affecting hydrologic models, thus enabling meaningful model comparison and improvement.

International audience; Hydrological modeling has traditionally suffered from a variety of errors including input-output data errors, structural errors in the model conceptualization and, finally, numerical errors in the model implementation. These errors have distinctly different origins, behavior and require specialized treatment. Following a brief theoretical review, we present a number of case studies where these errors are identified and treated. First, we demonstrate the occurrence of numerical errors when the model equations are solved using unreliable techniques. Remarkably, the hydrological community has allowed such errors to arise and persist despite the weaknesses of these techniques being well known for decades in the applied mathematics literature, and despite far more robust techniques being widely available. Second, we demonstrate the use of statistical techniques to get useful insights into rainfall and runoff data errors. When exploited within a Bayesian inference, these data insights can be used to estimate structural errors. Third, we demonstrate a methodology for characterizing structural errors using stochastic parameters, and flexible model structures. The case studies move us another step closer towards the elusive goal of reliably understanding and treating the distinct sources of error afflicting hydrological models. Benefits for improved scientific hypothesis-testing and more reliable operational predictions are discussed.

International audience; Confronted with frequently poor model performance, rainfall-runoff modellers have in the past blamed a plethora of sources of uncertainty, including rainfall and runoff errors, non-Gaussianities, model nonlinearities, parameter uncertainty, and just about everything else from Pandorra's box. Moreover, recent work has suggested astonishing numerical artifacts may arise from poor model numerics and confound the Hydrologist. There is a growing recognition that maintaining the lumped nebulous conspiracy of these errors is impeding progress in terms of understanding and, when possible, reducing predictive errors and gaining insights into catchment dynamics. In this study, we take the hydrological bull by its horns and begin disentangling individual sources of error. First, we outline robust and efficient error-control methods that ensure adequate numerical accuracy. We then demonstrate that the formidable interaction between data and structural errors, irresolvable in the absence of independent knowledge of data accuracy, can be tackled using geostatistical analysis of rainfall gauge networks and rating curve data. Structural model deficiencies can then begin being identified using flexible model configurations, paving the way for meaningful model comparison and improvement. Importantly, informative diagnostic measures are available for each component of the analysis. This paper surveys several recent developments along these research directions, summarized in a series of real-data case studies, and indicates areas of future interest.

International audience; Efforts to improve hydrological predictions are often thwarted by several sources of error, including (i) data uncertainty, e.g, input (rainfall) errors and output (runoff) errors; and (ii) structural uncertainty in the hydrological model. The Bayesian total error analysis methodology was introduced to overcome these challenges and provides a framework which can explicitly account for each source of error. While several studies have shown that accounting for rainfall error notably improves hydrological predictions, the runoff error models used are based on unduly simplistic assumptions (e.g 10% random error). This study will use streamflow rating curve data to develop probabilistic model of streamflow measurement error prior to the calibration of the hydrological model. Several diagnostic tools aiding the runoff error model development are presented. Different types of runoff error models (with both random and systematic errors) are developed using the rating curve data. These runoff error models are incorporated into hydrological model calibration using the BATEA methodology and a comparative analysis of the calibration results (predictive time series, model parameters and state variables) is presented, thoroughly scrutinizing its assumptions and comparing them to commonly error model assumptions. Several catchment case studies are compared to evaluate the benefits under diverse hydrological regimes. The implications for improving hydrological predictions via disentangling structural errors from data errors is discussed.

Hydrological calibration and prediction using conceptual models is affected by forcing/response data uncertainty and structural model error. The Bayesian Total Error Analysis methodology (BATEA) uses a hierarchical framework to represent individual sources of uncertainty. However, it is shown that standard multi-block Metropolis-within-Gibbs samplers commonly used in traditional Bayesian hierarchical Markov Chain Monte Carlo (MCMC) are exceedingly computationally expensive when applied to hydrologic models based on recursive numerical solution of coupled nonlinear differential equations describing the evolution of catchment states such as soil and groundwater storages. This note develops a limited-memory algorithm for accelerating multi-block MCMC sampling from the posterior distributions of such models using low-dimensional jump distributions. The new algorithm exploits the decaying memory of hydrological systems to provide accurate tolerance-based approximations of traditional fullmemory MCMC methods and is orders of magnitude more efficient than the latter.

International audience; Calibration and prediction in conceptual rainfall-runoff (CRR) modelling is affected by the sampling and measurement uncertainty in the forcing/response data and by the structural error of the model conceptualisation. These errors significantly affect the calibration procedure, leading to biased estimates of CRR model parameters and unreliable assessment of predictive uncertainty. The Bayesian Total Error Analysis methodology (BATEA) provides the opportunity to directly and comprehensively address these sources of uncertainty. BATEA is built on Bayesian hierarchical methods, constructing explicit error models for forcing/response data and structural errors. This communication provides a general presentation of such error models, and studies their identifiability using two synthetic data sets. The first synthetic data set is affected by forcing and response errors, but encompasses no structural errors. In this case, BATEA is able to retrieve the actual errors corrupting the input data, resulting in unbiased parameter estimates and reliable predictive uncertainty quantification. In a second, more realistic, case study, the synthetic data set also encompasses structural errors. Results show that when these structural errors are ignored by BATEA, the estimated errors in input data overestimate the actual errors. Moreover, identifiability issues arise when both input and structural errors are to be estimated from the data. These results illustrate the difficulty in identifying the different sources of errors affecting the calibration procedure. This is likely to be due to the insufficient information content of the data to identify several distinct error processes. A possibility to overcome this limitation is to introduce in the calibration procedure additional sources of information (e.g. radar data providing a rough estimate of the rainfall errors due to spatial averaging).

International audience; Meaningful characterization of uncertainties affecting conceptual rainfall-runoff (CRR) models remains a challenging research area in the hydrological community. Numerous methods aimed at quantifying the uncertainty in hydrologic predictions have been proposed over the last decades. In most cases, the outcome of such methods takes the form of a predictive interval, computed from a predictive distribution. Regardless of the method used to derive it, it is important to notice that the predictive distribution results from the assumptions made during the inference. Consequently, unsupported assumptions may lead to inadequate predictive distributions, i.e. under- or over-estimated uncertainties. It follows that the estimated predictive distribution must be thoroughly scrutinized (validated); as discussed by Hall et al. [2007] Without validation, calibration is worthless, and so is uncertainty estimation. The aim of this communication is to study diagnostic tools aimed at assessing the reliability of uncertainty estimates. From a methodological point of view, this requires diagnostic approaches that compare a time-varying distribution (the predictive distribution at all times t) to a time series of observations. This is a much more stringent test than validation methods currently used in hydrology, which simply compare two time series (observations and optimal simulations). Indeed, standard goodness-of-fit assessments (e.g. using the Nash-Sutcliff statistic) can not check if the predictive distribution is consistent with the observed data.

Meaningful quantification of data and structural uncertainties in conceptual rainfall-runoff modeling is a major scientific and engineering challenge. This paper focuses on the total predictive uncertainty and its decomposition into input and structural components under different inference scenarios. Several Bayesian inference schemes are investigated, differing in the treatment of rainfall and structural uncertainties, and in the precision of the priors describing rainfall uncertainty. Compared with traditional lumped additive-error approaches, the quantification of the total predictive uncertainty in the runoff is improved when rainfall and/or structural errors are characterized explicitly. However, the decomposition of the total uncertainty into individual sources is more challenging. In particular, poor identifiability may arise when the inference scheme represents rainfall and structural errors using separate probabilistic models. The inference becomes ill-posed unless sufficiently precise prior knowledge of data uncertainty is supplied; this ill-posedness can often be detected from the behavior of the Monte Carlo sampling algorithm. Moreover, the priors on the data quality must also be sufficiently accurate if the inference is to be reliable and support meaningful uncertainty decomposition. Our findings highlight the inherent limitations of inferring inaccurate hydrologic models using rainfall-runoff data with large unknown errors. Bayesian total error analysis (BATEA) can overcome these problems using independent prior information. The need for deriving independent descriptions of the uncertainties in the input and output data is clearly demonstrated.

International audience; Uncertainty in the rainfall inputs, which constitute a primary forcing of hydrological systems, considerably affects the calibration and predictive use of hydrological models. In a recent paper, Ajami et al. [2007] proposed the Integrated Bayesian Uncertainty Estimator (IBUNE) to quantify input, parameter and model uncertainties. This comment analyzes two interpretations of the IBUNE method and compares them to the Bayesian Total Error Analysis (BATEA) method [Kavetski et al., 2002, 2006a]. It is shown that BATEA and IBUNE are based on the same hierarchical conceptualization of the input uncertainty. However, in interpretation A of IBUNE, the likelihood function, and hence the posterior distribution, are random functions of the inferred variables, which violates a standard requirement for probability density functions (pdf). A synthetic study shows that IBUNE-A inferences are inconsistent with the correct parameter values and model predictions. In the second interpretation, IBUNE-B, it is shown that a specific implementation of IBUNE is equivalent to a special Metropolis-Hastings sampler for the full Bayesian posterior, directly including the rainfall multipliers as latent variables (but not necessarily storing their samples). Consequently, IBUNE-B does not reduce the dimensionality of the sampling problem. Moreover, the jump distribution for the latent variables embedded in IBUNE-B is computationally inefficient and leads to prohibitively slow convergence. Modifications of these jump rules can cause convergence to incorrect posterior distributions. The primary conclusion of this comment is that, unless the hydrological model and the structure of data uncertainty allow specialized treatment, Bayesian hierarchical models invariably lead to high-dimensional computational problems, whether working with the full posterior (high-dimensional sampling problem) or with the marginal posterior (highdimensional integration problem each time the marginal posterior is evaluated).

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.