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!