ezio todini president of italian hydrological society

Predictions and Uncertainties in Hydrology

Ezio TodiniPresident of Italian Hydrological Society

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Predictions and Uncertainties in Hydrology

In many fields of hydrology (flood warning and evacuation management; flood diversion and detention; real-time reservoir management; etc.), Decision Makers have to take important decisions without perfect knowledge of future events.

Since decisions may have heavy social, economical and environmental consequences, simulation and forecasting models are generally used to complement all available data and information and to predict the future outcomes.

Inevitably, predictive models cannot forecast “exactly” what will happen, but allow the Decision Makers to improve their prior belief on what will actually occur. Nonetheless, given that predictions are not exact, it is essential to assess Predictive Uncertainty in order to correctly estimate the “expected consequences” of decisions in order to increase their reliability and reduce the possibility of wrong ones.

Scope

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The need for Assessing Predictive UncertaintyThe Reservoir Management Case

In the Reservoir Management Problem it is easy to show that Deterministic Forecasts lead to wrong estimates of losses. In this simple example losses occur if the reservoir is overtopped. If the Deterministic Forecast reaches the top level of the reservoir the estimated losses are equal to zero.

Internationales hydrologisches Symposium | 02. – 03. November 2010

Deterministic Forecast

Losses = 0Expected Losses ≠

0

Probabilistic Forecast

Damages

Volume

PU as pdf

This is obviously wrong because the uncertainty in the forecast implies that the “expected value” of losses is not null. The “expected value” of losses can be estimated if and when an assessment of Predictive Uncertainty will be available.

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Predictive Uncertainty : the concept 1/3

A simple question may help at clarifying the Predictive Uncertainty concept:

What is the probability that the river dykes will be overtopped in the next 24 hours?

This seems to be a well-posed question, and certainly a topical one. It is the kind of question a flood emergency manager might ask to his technical staff.

There are two aspects of this question that ought to be highlighted.

- First, the question asks explicitly for a probability; - Second, it asks about the behaviour of the future (real) flood levels.

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The probability mentioned in the previous question descends from a mixture of prior belief and objective assessment of the situation. Our state of knowledge is always a mixture of “what we know”, or better “what we believe we know” (in the sense that we may be wrong), which is a “subjective state of mind” and what we “learn from observations” (which includes data and models), which can be seen as “objective”.

Therefore, a definition of Predictive Uncertainty can be the following:

Predictive uncertainty is the expression of our assessment of the probability of occurrence a future (real) event conditional upon all the knowledge available up to the present and the information we were able to acquire through a learning inferential process.

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The second fundamental aspect of Predictive Uncertainty is that it must relate to the future (real) event, not to the prediction.

This fact is pretty evident when answering the following question:

Will damages be caused by the actual water level overtopping the banks or by the fact that the predicted level is above them?

The obvious reply is: damages will occur when the actual river level will overtop the banks.

Therefore, Predictive Uncertainty must be expressed in terms of the probability distribution of a future unknown real quantity (water level, discharge, volume, etc.) and not in terms of the probability distribution of model predictions, which is a common mistake.

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Validation vs Predictive UncertaintyAs a consequence, we must distinguish between two types of uncertainty, namely:

Validation and Predictive Uncertainty.

Validation Uncertainty represents how well our model(s) reproduce the observations and is affected by all sorts of errors (measurement, model, parameters, initial and boundary conditions).

Predictive Uncertainty represents the probability of the occurrence of a future event given (conditional to) the observations and the model(s) forecasts, where the model predicions are taken as known and not uncertain quantities.

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Validation vs Predictive Uncertainty

Meteorological Ensembles

are a measure of Validation Uncertainty, while

Climatological Distributions or

Extreme Value Distributions

are measures of Predictive Uncertainty, although

non conditional on real time information.

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Validation vs Predictive Uncertainty

Validation Uncertainty

Uncertainty of model predictions knowing (conditional on) the observations

Predictive Uncertainty

Uncertainty of future occurrencesKnowing (conditional on) the model prediction

ty

0ˆ t ty

*ty

* 00

*ˆ

ˆtt t

tt ty yf y yObserved

Values

Model Prediction

ty

0ˆ t ty

0

*ˆ t ty

* 00

*ˆ

ˆt t t

t t ty yf y yObserved

Values

Model Prediction * 1t tProb y y

0 0

*ˆ ˆ 1t t t t

P rob y y

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Reality vs Virtual Reality (Models)

Therefore, model forecasts must not be considered as

REALITYbut only

VIRTUAL REALITY

Nonetheless, model forecasts contain essential information to reduce our uncertainty on what will actually occur in the future, to be incorporated into Predictive Uncertainty.

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The use of Predictive Uncertainty

Decision Theory teaches us how to make use of Predictive Uncertainty.

When one needs to take decisions under uncertainty he must:

1) Describe the uncertainty, namely by assessing the Predictive Uncertainty in terms of a probability distribution function;

2) Define a utility function, which may range from a simple description of the decision maker propension at risk to more complex losses or benefit functions;

3) Marginalise the effect of uncertainty by integrating the product of the probability times the utility function. In other words compute the expected value of the utility function.

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Predictive Uncertainty ProcessorsSeveral Hydrological Uncertainty Processors are today available to asses Predictive Uncertainty.

1) The Hydrological Uncertainty ProcessorsKrzysztofowicz, 1999; Krzysztofowicz and Kelly, 2000

2)The Quantile Regression Koenker and Basset, 1978; Koenker, 2005

3) The Bayesian Model Averaging Raftery et al., 2003

4) The Model Conditional Processor Todini, 2008.

5) ………………………………….

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The Forecasting – Decision Chain

Meteorological, Hydrological, Hydraulic and/orData Driven (Statistical, ANN, etc.) modelchains are used to assess PredictiveUncertainty with the objective of takingreliable and robust decisions, lessaffected by random occurrences.

One must bear in mind that in operation the final goal is not a better model, but rather a better decision.


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The multi-model approach

The Uncertainty Processors also allow to sinthesize several models and ensemble forecasts

Niiyyf ..1,ˆ

The use of Predictive UncertaintyTo improve management of the Aswan Reservoir in Egypt

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Managing the Aswan Reservoir by using Predictive Uncertainty

Predictive Uncertainty can be used to improve management of a large reservoir such as the Aswan Reservoir , by comparing real losses due to actual releases of water to future expected losses estimated usingPredictive Uncertaintyre-computed at each step in time as a function of a forecasting model prediction.

2.7 Billion m3/year of wateron average can be saved if this Predictive Uncertainty based approach is used.


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The use of Predictive UncertaintyTo improve management of the Lake Como in Italy

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Managing the Lake Como by using Predictive UncertaintyResults obtained by simulating 15 years of operations from January 1st, 1981 to December 31st, 1995

Water Level Number of Days

Historical Optimized<-40 cm 214 0

120 cm 133 54

140 cm 71 32

173 cm 35 11

Water Deficit 890.27 106 m3 694.49 106 m3

Energy Production increased by 3%

The use of Predictive UncertaintyFor Real Time Flood Forecasting in the River Po basin

Pontelagoscuro

Basin size = 70,000 km2

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Using Probabilistic Thresholds

Model forecast(s)

Model A Model B

Model forecast(s)

Model A Model B

Threshold basedon expected predicted value

Threshold basedon probability ofovertopping

Instead of expected value of prediction

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Reduction of Missed AlarmsUsing the probability of overtopping the dykes instead of model forecast

Threshold

(h*)Observed LevelMCP Expected ValueHydraulic Model

Forecast

P(h>h*) ObservedP(h>h*) MCP ForecastMCP Binary Response

36 hour in advance forecasts at Pontelagoscuro

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Reduction of False Alarms Using the probability of overtopping the dykes instead of model forecast

Threshold (h*)Observed LevelMCP Expected ValueHydraulic Model Forecast

P(h>h*) ObservedP(h>h*) MCP ForecastMCP Binary Response

36 hour in advance forecasts at Pontelagoscuro

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Conclusions

The new probabilistic approaches make use of the forecasting models as incremental pieces of information aimed at allowing decision makers to reliably take correct decisions.

These approaches, and in particular the use of Predictive Uncertainty Processors have produced several successful operational real time flood forecasting and management systems.

Nonetheless, additional research work is required to deal with multiple precipitation fields as provided by the meteorological ensembles and non- stationarity in data.

Finally, to fully exploit the Predictive Uncertainty potential, operational flood managers must make a substantial effort to change their way of thinking: from the nineteen century deterministic approaches to the more reliable and economically rewarding twentieth and twenty first century probabilistic ones.

ezio todini president of italian hydrological society

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