Course Unit 11
Handling Uncertainty H.P. Nachtnebel
Dept. of Water-Atmosphere-Environment Univ. of Natural Resources
and Life Sciences
hans_peter.nachtnebel@boku.ac.at
Sources of Uncertainty
Data gaps, limited sample size
Measurement errors
Stochastic component in nature (climate, hydrology,…..)
Modelling errors (limited scope, simplified equations, unknown boundaries and initial conditions, parameters)
Changing preferences in our society
Unknown development of technologies….
Quantifying Uncertainties
Graphical display of a time series
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Quantifying Uncertainties
Graphical display of a time series
Histogram of a data set
H(Q)
Q [m³/s]
Quantifying Uncertainties
Graphical display of a time series
Histogram of a data set
Fitting a pdf to a data set
model and data uncertainty
parameter estimation (moments, likelihood ?)
Gewässer: Donau Pegel: Kienstock (bei Krems) Beobachtungszeitraum: 1976 - 1999
0 10 20 30 40 50 60 70 80 90 100
250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 4250
Q [m³/s]
[%]
Unterschreitungswahrscheinlichkeit Ueberschreitungswahrscheinlichkeit
v x c s x, ,
Course unit 11: Handling Uncertainty H.P. Nachtnebel
How to handle uncertainty and risk ?
QIN f(QIN)
HP f(HP)
Water Resources Planning and DM: Unit 10 Uncertainty and Risk H.P.
Nachtnebel
Handling Uncertainties
All the outcomes of a project, the rankings are subjected to uncertainty: they have a range, a pdf
Handling Uncertainties
Sensitivity analysis
Simulation
Simulation
For each variable a pdf is assumed
Interest rate Probability
Investment costs Probability
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Simulation
For each variable a pdf is assumed
Hundreds of simulations
A pdf for the efficiency criterion is obtained
1 BCR Probability
Simulation
For each variable a pdf is assumed
Hundreds of simulations
A pdf for the efficiency criterion is obtained
1 BCR Probability
Failure probability
The project will not be economically justified f(BCR)
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Handling Uncertainties
Sensitivity analysis
Simulation:
We can estimate the failure probability
We could prespecify a certain minimum probability of success.
E.g. I would accept a project with a 90 % probability of sucess:
PS >0.9
F
S f x dx P
P ( ) 1
1
1
0
) (x dx f
PF
Example: Water Shortage
0
*)) ( ( ) (
*)
(Q f QD D QD Q dQD
R
t Q(t)
Q* Required amount Water deficits QDi
QD (m3) Damage D (€)
i
i
t
t
i Q Q t dt
QD
1
)) ( ( *
Example: Water Shortage
Objetive: reduce the risk
Manage the demand (improve water use efficiency) Manage the supply (increase the supply rate by
building/enlarging a reservoir)
Storage capacity S
Costs C Costs C
Water Use Efficiency WUE
Example: Water Shortage
What is the decision criterion ?
R ( Q *) C ( S )
Min
R ( Q *) C ( WUE )
Min
Course unit 11: Handling Uncertainty H.P. Nachtnebel
What happens after building a levee ?
f (Q)
Q
Potential D (Q)
Q X*
old new
PDFs and Risk curves
Risk curves
Flood probability f(Q)
Q
Damage potential D(Q)
Q
X*(T)
Damage potential D(Q) Prob(Damage>D)
1/T*
Risc curve
Increased damage potential
Design level X*
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Consequences
The expected damages may be larger after flood implementation of flood protection measures
Land management and development strategies are required
Safety of levees ?
Comparison of different risk curves
• Comparison of two hazards with quite different consequences
• A1 very low probability of occurrence but extreme consequences
• A2 high probability of occurrence but lower consequences
• E.g. A1 nuclear power station and A2 thermal power station
Cumulative probability
Damages
A1 and A2
A1 has a low mean value but is highly skewed A2 has a higher mean but an upper limit
1
0.5
Which alternative would you prefer ?
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Comparing risks
Two alternatives, A1 and A2, exhibit annual net benefits (k€) with a certain probability. The outcomes are Aik
Which one should be chosen ?
Comparing two Uncertain Outcomes
Possible Decision Criteria Max { wi NBik}
Max {Max(NBik)}
Max {Min (NBik)}
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Decision techniques
Bernoulli criterion: choose the one where K1 is better:
K1 = max {K1,i} = max { wk Aik} K1,1 = 4 333 k€/a
K1,2 = 4 266 k€/a
Decision techniques
Neumann-Morgenstern criterion: try to avoid losses or take a risk averse position
K3 = max{K3,i} = max{min(Aik) for wk >p0}
Choose A2 because the worst outcome is 3 600 k€/a which is better than the outcome of A1
Is a useful criterion for public investments, safe decision
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Comparing two Uncertain Outcomes
Two alternatives with uncertain net benefits Which alternative is better
A1 A2
w1=33% 6 200 4 900
w2=33% 4 100 4 300
w3=33% 2 500 3 600
Perception of risk
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Comparing two Uncertain Alternatives
Damages Social Costs
Comparing two Uncertain Alternatives
The mean value of A1 < A2
There is high probability that the damages vom A1 are smaller than from A2
But it may happen that the damages from A1 are >>>
than from A2
Being risk adverse select A2
Course unit 11: Handling Uncertainty H.P. Nachtnebel
Summary and Conclusions
Uncertainty is essential in DM
All our decisions are subjected to uncertainty
Several performance indicators were defined failure rate, reliability, risk
Several strategies for risk management were discussed Max average
Min losses (risk averse) Max max (risk prone)