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
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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:
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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
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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
European Geosciences Union General Assembly 2010, Vienna, May 2010
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
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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?
European Geosciences Union General Assembly 2010, Vienna, May 2010
C id i th t d l t b d fi d b tt th th i
MCP: MULTIMCP: MULTI‐‐VARIATE APPROACHVARIATE APPROACH
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
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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
MCP: MULTIMCP: MULTI‐‐VARIATE APPROACHVARIATE APPROACH
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:
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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:
European Geosciences Union General Assembly 2010, Vienna, May 2010
Gridded hourly precipitation andtemperature data
Observed hourly discharge at Eldon
MCP: APPLICATIONMCP: APPLICATION
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?
European Geosciences Union General Assembly 2010, Vienna, May 2010
MCP: APPLICATIONMCP: APPLICATION
Observed threshold exceedingExceeding Probability
1 MODEL:TOPKAPI
Observed Discharge
5% 95% Quantiles
Model forecast (T0+6h) PU Expected Value
Observed Discharge
5%, 95% Quantiles
European Geosciences Union General Assembly 2010, Vienna, May 2010
MCP: APPLICATIONMCP: APPLICATION
Observed threshold exceedingExceeding Probability
1 MODEL:TETIS
Observed Discharge
5% 95% Quantiles
Model forecast (T0+6h)PU Expected Value
Observed Discharge
5%, 95% Quantiles
European Geosciences Union General Assembly 2010, Vienna, May 2010
MCP: APPLICATIONMCP: APPLICATION
Observed threshold exceedingExceeding Probability
1 MODEL:ANN
Observed Discharge
5% 95% Quantiles
Model forecast (T0+6h)PU Expected Value
Observed Discharge
5%, 95% Quantiles
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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
3 MODELS:TETIS + TOPKAPI
+ ANN
European Geosciences Union General Assembly 2010, Vienna, May 2010
MCP: APPLICATIONMCP: APPLICATIONObserved threshold exceedingexceeding2 MODELS Exceeding Probability
3 MODELS Exceeding Probability
2 MODELS:TETIS + TOPKAPI
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
European Geosciences Union General Assembly 2010, Vienna, May 2010
MCP
MCP: APPLICATIONMCP: APPLICATION
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
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0.60
MCP: APPLICATIONMCP: APPLICATION
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
European Geosciences Union General Assembly 2010, Vienna, May 2010
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
European Geosciences Union General Assembly 2010, Vienna, May 2010
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
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used.