Modelling and Managing Uncertainties in Natural Hazards
Daniel Straub
in collaboration with Matthias Schubert & Michael H. Faber
Chair of Risk and SafetyIBK / D-Baug / HazNETHETH Zürich
2nd Swiss Geoscience MeetingLausanne, November 20, 2004
Source: www.planat.ch
Natural hazards and the engineer: example rockfall
• Rockfall protection gallery:
Gallery
Design rock(1m3 - 70m)
70m
Outlook
Three stages in natural hazards risk assessment:
1. Evaluation and description of the uncertainties related to the occurrence of hazards (natural scientist)
2. Propagation of the uncertainties in the (engineering) model
3. Combining the uncertainties with consequences: Appraisal of risks (decision theory)
The uncertainties can be and MUST be addressed explicitly in each stage
Sources of uncertainty
Two fundamentally different types of uncertainty:• Aleatory uncertainties (random processes)• Epistemic uncertainties (related to incomplete knowledge)
Aleatory uncertainties• Some processes are subject to inherent randomness (e.g. the
detachments of rocks)• The prediction of future developments (such as climate)
Epistemic uncertainties in natural hazards:• The (empirical) models are not perfect and thus subject to
scatter• Model parameters are subject to statistical uncertainty• Incomplete knowledge of site specific characteristics
Representing uncertainties by exceedance probabilties & frequencies: Example maximum annual discharge
Messwerte
Gumbel-Verteilung
0.001
0.01
0.1
1 1
10
100
1000
Jährlich
keit[yr]
0 200 400 600 800 1000 1200
Jährlicher Spitzendurchfluss q [m3/s]
Üb
ersc
hre
itu
ng
swah
rsch
ein
lich
keit
1 -
FQ
(Q)
HQ
300HQ
100
HQ
30
Example rockfall: Representing uncertainty in the detachment process
• Information provided originally by the geologist
(Steinschlag)
(Blockschlag)
(Blockschlag)
(Felssturz)
Rock volume [m3]
0.1 1 10 100 1000
102
101
1
10-1
10-2
10-3
10-4
10-5
Exc
eed
ance
Fre
qu
ency
[yr
-1]
Example rockfall: Representing uncertainty in the detachment process
• A simple model (not considering the uncertainties in the estimate)
(Steinschlag)
(Blockschlag)
(Blockschlag)
(Felssturz)
Rock volume [m3]
0.1 1 10 100 1000
102
101
1
10-1
10-2
10-3
10-4
10-5
Exc
eed
ance
Fre
qu
ency
[yr
-1]
Example rockfall: Representing uncertainty in the detachment process
• Modelling the uncertainties in the original estimation
(Steinschlag)
(Blockschlag)
(Blockschlag)
(Felssturz)
Rock volume [m3]
0.1 1 10 100 1000
102
101
1
10-1
10-2
10-3
10-4
10-5
Exc
eed
ance
Fre
qu
ency
[yr
-1]
Example rockfall: Representing uncertainty in the detachment process
• Fitting a parametric model to the estimates
HV (V)=A·V -B(Steinschlag)
(Blockschlag)
(Blockschlag)
(Felssturz)
Rock volume [m3]
0.1 1 10 100 1000
102
101
1
10-1
10-2
10-3
10-4
10-5
Exc
eed
ance
Fre
qu
ency
[yr
-1]
Example rockfall: Representing uncertainty in the detachment process
• Including the uncertainties in the estimate as obtained by a MLE
HV (V)=A·V -B
Rock volume [m3]
0.1 1 10 100 1000
102
101
1
10-1
10-2
10-3
10-4
10-5
Exc
eed
ance
Fre
qu
ency
[yr
-1]
Propagation of uncertainties
• Using stochastic tools such as
– Structural Reliability Analysis– Monte Carlo Simulation– Direct integration
the estimated exposure probabilities can be combined with the models describing the hazard propagation and the performance of defence structures.
• The uncertainties in these models and their parameters can also be accounted for
Propagation of uncertainties: rockfall
E
fE (E|V2)
E
fE (E|V1)
E
fE (E)( )
( ) ( )
( ) ( )0 0
d d
1
d
d
E
E Vf EV h V V E
E
EE
F E e
F Ef E
E
∞
−∫ ∫= −
=Distribution of the annual maximum impact energy:
V
HV (V)
Propagation of uncertainties: protection gallery
• The risk is related to the probability of failure of the gallery, P(F)
• P(F) is determined by the fundamental structural reliability problem:
• For the rockfall case, S is represented by the distribution of E• R is represented by a stochastic model of the gallery
resistance
( ) ( )P F P R S= ≤
)(),( sfrf SR
sr ,
Load S
Resistance R
x
)(),( sfrf SR
sr ,
)(),( sfrf SR
s
Risk & Optimisation
• The risk is determined by combining (multiplying) the consequences with the probabilities of occurrence, here the failure of the gallery
• Taking into account the number of people and objects exposed
• The risk is decisive for the decision on the optimal measure/action to take
Cost of actions / measures
Safety
Exp
ect
ed
co
st
Risk: Expected damage P(F) x C(F)
Total cost
Optim
al risk level /
Safety
GalleryNo action Net
Including additional information in the analysis
• When additional information (evidence) is available, this will reduce the uncertainty in the model.
• This additional information can be accounted for by updating the probabilistic model according to Bayes’ rule.
• Example: The number of stones observed on and below the gallery at T=10yr.
Prior
Posterior
Rock volume [m3]
0.1 1 10 100 1000
Observed stones on the gallery10
0
Rock volume [m3]
0.1 1 10 100 1000
102
101
1
10-1
10-2
10-3
10-4
10-5
Exc
eed
ance
Fre
qu
ency
[yr
-1]
Including additional information in the analysis
• When additional information (evidence) is available, this will reduce the uncertainty in the model.
• This additional information can be accounted for by updating the probabilistic model according to Bayes’ rule.
• Example: The number of stones observed on and below the gallery at T=10yr, modified
Rock volume [m3]
0.1 1 10 100 1000
Observed stones on the gallery10
0
Prior
Posterior
Rock volume [m3]
0.1 1 10 100 1000
102
101
1
10-1
10-2
10-3
10-4
10-5
Exc
eed
ance
Fre
qu
ency
[yr
-1]
Conclusions
• Uncertainties must be addressed to identify optimal decision in natural hazards management
• The outlined procedures allow for dealing with the uncertainties in a consistent and formalised manner
• All available information can be integrated in the model by means of Bayes’ rule
• Care is needed when choosing the probabilistic models: Are the models really representing the considered (physical or other) processes?
The end
Thank you for your attention!