APPLICATION OF A BAYESIAN PROCESSORAPPLICATION …lluvia.dihma.upv.es/ES/publi/congres/012_Coccia et...

APPLICATION OF A BAYESIAN PROCESSORAPPLICATION OF A BAYESIAN PROCESSOR FOR PREDICTIVE UNCERTAINTY

ASSESSMENT IN REAL TIME FLOODASSESSMENT IN REAL TIME FLOOD FORECASTING

Gabriele Coccia1, Felix Francés2, Juan Camilo Múnera2 and Ezio Todini1(1) University of Bologna, Bologna, Italyy g , g , y

(2) Polytechnic University of Valencia, Valencia, Spain

European Geosciences Union General Assembly 2010, Vienna, May 2010

DECISION MAKING UNDER UNCERTAINTYDECISION MAKING UNDER UNCERTAINTY

I ti l blIn many operational problems:

• Flood warning; decision makers must take importantd i i d th t i t f• Flood emergency management;

• Reservoir management;• Etc.

decisions under the uncertainty offuture events.

According to the Decision theory, in order to take a rational decision it isnecessary to:1 Define an Utility Function in accordance with the Decision Maker1. Define an Utility Function in accordance with the Decision Maker2. Quantify the probability density of the future event3. Maximize the expected value of the Utility Function

Model deterministic forecastUtility Function

Predictive Uncertainty 0

0UE

xEU

Damages

0xUE

)()( dfUUE


0)()( dxxfxUxUE

PREDICTIVE UNCERTAINTY: DEFINITIONPREDICTIVE UNCERTAINTY: DEFINITION

THE DEFINITION OF PREDICTIVE UNCERTAINTY

Predictive Uncertainty can be defined as the probability of occurrence ofa future value of a predictand (such as water level, discharge or watervolume) conditional on all the information that can be obtained on thevolume) conditional on all the information that can be obtained on thefuture value, which is typically embodied in one or more meteorological,hydrological and hydraulic model forecasts.

Predictive Uncertainty must be quantified in terms of probabilitydi t ib tidistribution.

If the available information is a model forecast, Predictive Uncertaintycan be written and here will be called as:


MODEL MODEL CONDITIONAL PROCESSOR (MCP): CONDITIONAL PROCESSOR (MCP): METHODOLOGY DESCRIPTIONMETHODOLOGY DESCRIPTION

Conditioned CdfJoint Pdf

Conditioned Cdf

Image of Observed1) Conversion

from the real *ˆˆ|Eworld to the Normal Space using the NQT

| yyyE

4) The expected value of the Predictive Uncertainty is

Historical data

Image ofForecasted

Predictive Uncertainty is computed sampling the probability density function in the Normal Space and

Observed

3) Predictive Uncertainty isobtained by the BayesTheorem and its mean and

the Normal Space and reconverting the obtained quantiles by the Inverse NQT:

Forecasted

2) Joint Pdf is assumed to be

*ˆˆ

Theorem and its mean andvariance are:

*1

*

ˆˆ

ˆˆ|

fNQTE

yyyEassumed to be a Normal Bivariate Distribution


2ˆ

2ˆ 1

fNQTEDistribution

MCP: PROBABILITY TO EXCEED A THRESHOLDMCP: PROBABILITY TO EXCEED A THRESHOLD

The knowledge of the future event probability distribution allows to easilyThe knowledge of the future event probability distribution allows to easilyextrapolate the probability to exceed a threshold value, such as an alert level.

This is an important information when dealing with the decision about givingThis is an important information when dealing with the decision about givingor not an alarm in emergency managing.

I b di l d f h P di i U i

*ˆˆ

It can be directly computed from the Predictive Uncertainty,as its integral above the threshold a. Joint and Conditioned Pdf

a

yyayP

*

* )ˆˆ|( Image of the observed values

aa

dyyyyfa *ˆˆ

Image of the forecasted values

df

*ˆˆ


a

PROBLEM:The estimate of the correlation coefficient not always well

MCP: IMPROVEMENTMCP: IMPROVEMENT

PROBLEM:The estimate of the correlation coefficient not always wellrepresents the high flow state, which is due to the differentbehaviour of the model in reproducing the low and high flows and toth NQT li it bi d ith th hi h f f d t ithe NQT non-linearity combined with the higher frequency of data inlow flow state than in high flow state.

PROPOSED SOLUTION: in the Normal Space data are divided in twoPROPOSED SOLUTION: in the Normal Space, data are divided in twosamples and each one is supposed to belong to a differentTruncated Normal Distribution. Hence, two Joint TruncatedN l di t ib ti id tifi d th b i f th l


Normal distributions are identified on the basis of the samplesmean, variance and covariance.

MCP: MULTIMCP: MULTI‐‐VARIATE APPROACHVARIATE APPROACH

Usually, a real time flood forecasting system is composed by more than onemodel chain, different from each others for structure and results.

HOW TO DEAL WITH THESE FORECASTS?FORECASTS?

WHICH WEIGHT CAN BE ASSIGNED TO EACH ONE?ASSIGNED TO EACH ONE?


C id i th t d l t b d fi d b tt th th i


Considering that a model cannot be defined better than another one inabsolute terms, the MCP tries to answer these questions combining all theforecasts through a multivariate bayesian analysis.

P di ti U t i t i d fi d th b bilit f th l f t


Predictive Uncertainty is now defined as the probability of the real futureevent conditioned to the forecasts of all the deterministic models

G li i th i d if N f t il bl th lti


Generalizing the previous procedure, if N forecasts are available, the multi-Normal space is composed by N+1 variables; each one is distributed as aStandard Normal and the joint distribution is a Standard Normal (N+1)-Variate,

ˆˆ1

1

N0

with mean and variance:

ˆˆˆ

ˆˆˆˆ 111

N

00

ˆˆ

ˆˆ

11

1

NN

NN

0

T ˆˆ

Predictive Uncertainty hasPredictive Uncertainty has mean and variance:


MCP: APPLICATIONMCP: APPLICATION

BARON FORK RIVER AT ELDON OK USABARON FORK RIVER AT ELDON, OK, USATOPKAPI MODEL

ANN MODEL

H0

RAINFALLX0

T0SNOW

TETIS MODEL

01/10/1995 30/09/200231/05/1997 31/01/1998

ANN: CALIBRATION VERIFIC. VALIDATION

EVAPOTRANSPIRATION

EXCEDENT

INFILTRATION

T1

X3

X2 Hu

H2

D2

T2

H1

Y0

SNOWMELT

X1

D1Y1

DIRECTRUNOFF

Y2

PRECIPITATION

01/10/1995 30/09/200231/05/1997 01/05/2000

MCP: VALIDATION CALIBRATIONUNDERGROUND

LOSSESX5 H4

T4

D4

PERCOLATIONX4

T3

H3

D3

BASE FLOWY4

INTERFLOWY3

Gridded hourly precipitation and Observed hourly discharge

Available data, provided by the NOAA’s National Weather Service, within the DMIP 2 Project:


Gridded hourly precipitation andtemperature data

Observed hourly discharge at Eldon


The aim of the application was to answer the following questions :

1. Does the MCP assess the Predictive Uncertainty?

2. Does the MCP improve the deterministic forecasts?

3. Does the use of the truncated joint distributions improve the MCP behaviour in reproducing flood event?

4. Does the Multivariate approach reduce the Predictive Uncertainty?



Observed threshold exceedingExceeding Probability

1 MODEL:TOPKAPI

Observed Discharge

5% 95% Quantiles

Model forecast (T0+6h) PU Expected Value

Observed Discharge

5%, 95% Quantiles




1 MODEL:TETIS

Observed Discharge

5% 95% Quantiles

Model forecast (T0+6h)PU Expected Value

Observed Discharge

5%, 95% Quantiles




1 MODEL:ANN

Observed Discharge

5% 95% Quantiles

Model forecast (T0+6h)PU Expected Value

Observed Discharge

5%, 95% Quantiles


MCP: APPLICATIONMCP: APPLICATIONObserved threshold exceedingexceeding2 MODELS Exceeding Probability

3 MODELS Exceeding Probability

2 MODELS:TETIS + TOPKAPI

Observed Discharge

VS5%, 95% Quantiles2 MODELS PU Expected ValueObserved Discharge

VS5%, 95% Quantiles3 MODELS PU Expected Value


+ ANN


MCP: APPLICATIONMCP: APPLICATIONObserved threshold exceedingexceeding2 MODELS Exceeding Probability

3 MODELS Exceeding Probability


2 MODELS PU Expected Value

VS 5%, 95% Quantiles

2 MODELS PU Expected ValueObserved Discharge

3 MODELS PU

3 MODELS:5%, 95% Quantiles

Expected Value

TETIS + TOPKAPI + ANN


MCP


NASH-SUTCLIFFE Coefficient

CP T+MCP

NN+M

CP

ANN+M

CP

K+TET+ANN+M

1.00

WITHOUT

NASH-SUTCLIFFE Coefficient

TPK

K+MCP

TET

T+MCP

ANN

ANN+M

C

TPK+

TET

TPK+

A

TET+ TPK

0.80

0.90WITHOUT truncated

distributions

P

TP TET

0.60

0.70

+MCP

+MCP

N+M

CP

ET+A

NN+M

CP

1.00TP

K

TPK+

MCP

TET

TET+MCP

ANN

ANN+M

CP

TPK+

TET+

TPK+

ANN

TET+AN

TPK+

TE

0 80

0.90WITH truncated

distributions

0 60

0.70

0.80distributions


0.60


WITH the Truncated Joint DistributionERROR STANDARD DEVIATION

K ET+M

CP

CP12 0

14.0

TPK

TET

ANN

ANN+M

CP

TPK+

TE

PK+A

NN+M

CP

T+ANN+M

CP

+TET+A

NN+M

C

10.0

12.0

1 ) A

TP TET

TPK+

6.0

8.0

Q (m

3 s‐1

2.0

4.0

0.0


1) The choice of the threshold using the Joint Truncated Distributions.FUTURE DEVELOPMENTSFUTURE DEVELOPMENTS

1) The choice of the threshold using the Joint Truncated Distributions.

Is it possible to find an objective rule, related tothe forecast cdf gradient, to identify thisg ythreshold?

Can the use of a Quantile Regression lead to aCan the use of a Quantile Regression lead to amore realistic uncertainty assessment?

2) A good model fit of the marginal cdf tails is very important:

For which probabilities should tails be used?

Which is the best curve?Which is the best curve?

Would be better the use of tails or to identify a probability model for the


ou d be be e e use o a s o o de y a p obab y ode o ewhole series?

Even if improvements are still required the presented application of the

CONCLUSIONSCONCLUSIONS

Even if improvements are still required, the presented application of theMCP shows that this methodology:

ll t ti t th P di ti U t i t i i l t ti l• allows to estimate the Predictive Uncertainty, requiring low computationalcosts;

• allows to combine different models forecast, reconciling physically basedand data driven models gaining from the benefits of both approaches.

Furthermore,

• the use of the truncated distributions allows to better reproduce the floodthe use of the truncated distributions allows to better reproduce the floodevents;

• the assessment of the probability to exceed an alert level allows to deal in• the assessment of the probability to exceed an alert level allows to deal inprobabilistic terms with the emergency management and it can lead toidentify probability thresholds instead of the deterministic ones commonlyused


used.

APPLICATION OF A BAYESIAN PROCESSORAPPLICATION …lluvia.dihma.upv.es/ES/publi/congres/012_Coccia et...

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