A Bayesian approach to decision-making under uncertainty: An application to real-time forecasting in the river Rhine

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Summary

Enhanced ability to forecast peak discharges remains the most relevant non-structural measure for flood protection. Extended forecasting lead times are desirable as they facilitate mitigating action and response in case of extreme discharges. Forecasts remain however affected by uncertainty as an exact prognosis of water levels is inherently impossible. Here, we implement a dedicated uncertainty processor, that can be used within operational flood forecasting systems.

The processor is designed to support decision-making under conditions of uncertainty. The scientific approach at the basis of the uncertainty processor is general and independent of the deterministic models used. It is based on Bayesian revision of prior knowledge on the basis of past evidence on model performance against observations. The revision of the prior distributions on water levels and/or flow rates leads to posterior probability distributions that are translated into an effective decision support under uncertainty. The processor is validated on the operational real-time river Rhine flood forecasting system.

Introduction

Extreme river runoff events, which include both, high and low-flows, have had large social and economic impact worldwide and continue to pose a regular concern to society. Recent large floods in Europe, such as those that have occurred in the Meuse and Rhine basins in 1995, over large areas of the United Kingdom in 1998 and 2000 and in the Elbe basin in summer 2002, have led to increased interest in research and development on upgrading existing operational flood forecasting systems in Europe.

Following recent conclusions of the fourth IPCC climate assessment report (IPCC, 2007), enhanced meteorological extremes are to be expected during the 21st century. A possible acceleration of the hydrological cycle may lead to large fluctuations in discharges in European river systems. Extreme discharges may become more frequent, calling for structural and non-structural interventions. Significant investments for the installation and upgrading of operational flood forecasting systems are already on the agenda of national hydro-meteorological services. The World Meteorological Organization (WMO, 2006) acknowledges that in many parts of the world forecasting remains the only effective measure which can be realistically implemented to protect life and property in the face of extreme meteorological events.

Real-time forecasting constitutes a non-structural measure by providing warning and issuing alerts ahead of an emergency. Extending the forecasting horizon allows time allocation for mitigating action. A reliable assessment of certainty of event occurrence in a real-time context safeguards operational users from issuing false alarms and institutional decision-makers from calling for unwarranted action. Real-time flood forecasting systems are currently operational in many parts of the world, including the Netherlands, where forecasting systems for the Rhine and Meuse have been installed. Potential flooding caused by those two river systems raises security issues for the low-lying territory in the country.

A single deterministic water level or flow rate forecast several hours ahead via a model simulation is of little value to decision-makers, as there remains inherent uncertainty associated with model output. The sources for the uncertainty in the forecast are manifold (Krzysztofowicz, 1999) and will be addressed in this paper. If an exceptional event is forecasted, evacuation is a last resort intervention to save lives. It is considered in situations, for which safety of the population in risk areas can no longer be guaranteed. It remains a costly action, which causes disruption to a range of societal activities. Institutional bodies generally regard the conclusion to initiate evacuation as a last resort option, which should be based on carefully weighted and objective decision-making. The uncertainty in the forecast at the basis of the decision, needs to be fully incorporated in the decisional process and combined with a cost function (Raiffa and Schlaifer, 1961, De Groot, 1970, Todini, 2007). By means of such a cost function, weighted by the probability of event occurrence, the decision-maker can soundly determine if the total expected damage in issuing an alert is higher than in the case of taking no action (see also Section “Discussion”). Providing an objective measure of the uncertainty associated with a forecast constitutes thus an essential prerequisite for sound decision support.

Research in the past 30 years has addressed many of these aspects, including the development of forecasting systems that output some (partial or approximate) measure of predictive uncertainty. Forecasts specifying the statistics of discharge were produced for short lead times via the Kalman filter applied either to a deterministic hydrological model (Kitanidis and Bras, 1980a, Kitanidis and Bras, 1980b, Szöllösi-Nagy and Mekis, 1988, Georgakakos and Smith, 1990) or to an integrated hydro-meteorological model (Georgakakos, 1987). Forecasts specifying an ensemble of hydrographs were produced via a deterministic hydrological model for short lead times (Lardet and Obled, 1994). While these systems pointed paths to probabilistic forecasting, they have limitations: the first type of system did not output the entire predictive probability distribution function, whereas the second type of system did not account for all sources of uncertainty. The introduction of Ensemble weather predictions in recent years gave way to producing multiple stream-flow forecasts (Schaake and Larsson, 1998). The currently ongoing large-scale international hydrological ensemble prediction experiment (HEPEX, Schaake et al. (2006)) explicitly addresses uncertainty assessment in the context of ensemble stream-flow predictions.

An adequate assessment of the uncertainty has the additional benefit of clearly separating the responsibilities and tasks of the organization, operating the forecasting system, from those of the decision-maker (Krzysztofowicz, 2001). Here, we present an implementation of a probabilistic uncertainty processor which elaborates a forecast in probabilistic terms and offers an measure of the uncertainty of the forecast. The theory of the processor has been developed in the Bayesian forecasting system (BFS) by Krzysztofowicz, 1999, Krzysztofowicz and Kelly, 2000. The processor is executed off-line, after a forecast, and evaluates the probability of occurrence of the predicted flow rate or water level, conditional on all information available in the forecasting process. The statistics performed by the processor take historical model performance against observations into account. Consequently, decisions under conditions of uncertainty can be taken objectively once an acceptable damage level has been defined at the institutional level. In this perspective quantifying the uncertainty contributes to establishing an effective decision support framework for end-users of flood forecasting products.

The general framework for a Bayesian forecasting system (BFS) presented by Krzysztofowicz (1999) constitutes a significant effort in formalizing the quantification of uncertainty in the flood forecasting process, given arbitrary deterministic models. The BFS consists of three components: (i) an input uncertainty processor, (ii) a hydrologic uncertainty processor and (iii) an integrator of uncertainty. The BFS is based on a complete and consistent theory, which identifies and separates relevant sources of uncertainty. It provides a general reference framework for building the uncertainty processors and the integrator, components which can be readily implemented in a real-time flood forecasting system. Sequel papers have implemented various BFS components as a one-branch processor (precipitation-independent model, Kelly and Krzysztofowicz (2000) and Krzysztofowicz and Kelly (2000)) or two-branch processor (precipitation-dependent model, Krzysztofowicz and Herr (2001)), analyzed their statistical properties and carried out preliminary verifications. One of the most salient features of the BFS theory is the provision of a formal basis for the development and successive revision of its components, whilst preserving internal consistency and statistical properties, which make it appealing for operational use.

In a recent paper (Reggiani and Weerts, 2008), we have presented an application of the input uncertainty processor (IUP) proposed by Krzysztofowicz (2004). The aim of the present paper is an implementation of the hydrological uncertainty processor (HUP) for the operational flood forecasting system of the river Rhine. Given the large river basin size (160,000 km2) and an basin contraction time between 4 and 5 days, flood propagation and travel times in the principal river system are predictable with reasonable accuracy. We present an application of the HUP, whereby we emphasize the importance of an optimal specification of the prior density in the particular Bayesian revision process. In contrast to Krzysztofowicz and Kelly (2000), who assume the river system to behave as a Markov chain stochastic process, we propose and parameterize a prior density on river stages as a linear regression, in which multiple observations at upstream observing stations are taken into consideration.

The paper is structured into four sections: Section “Principles” describes the principles underlying the Bayesian processor, Section “Uncertainty processor” describes the theory, Section “Application” introduces the operational Rhine flood forecasting system and describes the data elaboration steps for the Bayesian processor, Section “Experiments” discusses the numerical experiments and Section “Discussion” draws the conclusions.

Section snippets

Variates

Adopting the notation by Krzysztofowicz (1999), we introduce the following random variates, that describe the forecasting process: The ensembleHn,X=[H1,X,,Hn,X]called the predictand, are water levels recorded at forecasting times 1,,n at location X. These quantities lie in the future with respect to those observed at the same locations, at an arbitrary number of historical times 1,,k up to the time t0=0 at the onset of the forecast:H0-k,X=[H0,X,,H0-k,X]Last,Sn,X=[(S1,X,,Sn,X)]is an

Operational requirements

The uncertainty processor for a water level forecast is provided by an estimator for the conditional probability density function ϕ in (5). We seek a parameterization of the conditional density ϕ in a series of procedural steps, that have been laid out in Krzysztofowicz and Kelly (2000). The estimator will be formulated as Bayesian processor, which revises a prior density on hn,X1 by means of a likelihood function. In the context of a real-time forecasting system, the operational requirements

The river Rhine forecasting system

The HUP is applied to the operational flood forecasting system for the river Rhine with a total surface area of 160,000 km2. The system is embedded in the forecasting platform Delft-FEWS (Flood Early Warning System), an open-architecture data management environment, which facilitates interfacing generic hydrological and hydraulic models with online data streams and a central data base (Werner et al., 2004, Werner and Heynert, 2006). The online data include water levels and precipitation and

Experiments

Equations (15), (16), (17) constitute the working equations for the uncertainty processor. The prior density parameterized with respect to water level observations at Lobith at t0 and at Cologne at t0-k is given by (15). Fig. 4a depicts the prior densities for lead times 24 (continuous), 48 (dash-dotted) and 72 (dotted) hours and water level observations at Cologne 24 h ahead of t0. The water levels have been observed in winter (3-March, black), spring (17-April, magenta), summer (18-Aug, green)

Discussion

One of the principal strengths of the uncertainty processor lies in the fact that it can be executed with minimal computational effort on-line, while the required parameters are evaluated off-line. This characteristic is due to the NQT technique transforming the process variables into the normal space. In the normal space we perform regressions between process variables in the prior and the likelihood distributions. The resulting expressions are then mapped back into the original space,

Summary and conclusions

We have presented an application of a Bayesian processor to asses the predictive uncertainty on water level predictions in the river Rhine flood forecasting system. The processor is applied after the sequential execution of a hydrological and a hydrodynamical model. The meteorological input is provided by deterministic weather predictions in forecast mode and by ground observations. The uncertainty processor is based on Bayesian revision. Prior knowledge on flood propagation is processed in

Acknowledgements

We would like to thank the forecasting office of the Dutch Ministry of Transport and Waterways and the German Federal Office of Hydrology for granting permission for the use of the operational river Rhine Forecasting system in this research.

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