Warner Marzocchi, Laura Sandri INGV, Via D. Creti 12, 40128 Bologna, Italy [email protected]...

Warner Marzocchi, Laura Sandri

INGV, Via D. Creti 12, 40128 Bologna, [email protected]

Istituto Nazionale di Geofisica e Vulcanologia

Considerazioni sulla stima del rischio Considerazioni sulla stima del rischio associato a eventi naturaliassociato a eventi naturali

Considerazioni sulla stima del rischio Considerazioni sulla stima del rischio associato a eventi naturaliassociato a eventi naturali

1. Definition of RISK associated with a natural event

2. Probability and risk assessment

3. A Probabilistic approach for risk assessment: the Event Tree

Outline of the Course

HAZARD: Probability of any particular area being affected by a destructive volcanic event within a given time interval.

VALUE: # of human lives at stake, the capital value, and the productive capacity exposed to the destructive event.

VULNERABILITY: Proportion of the value which is likely to be lost as a result of a given event.

RISK = HAZARD * VULNERABILITY * VALUE

1. Risk associated with a natural event

INDIVIDUAL RISK = HAZARD * VULNERABILITY


Why is risk assessment important?

For land use purposes (long-term risk mitigation)

For emergency management

(mid- to short-term risk mitigation)

a. Risk assessment needs multidisciplinary competences

b. Hazard assessment is often the most diffucult to address. We need to deal with the problem from an “engineeristic” point of view c. Individual risk and hazard assessments consist of providing quantitative estimates of the probability of each event

d. Hazard assessment has to use all the available information



a. Risk assessment needs multidisciplinary competences

HAZARD: scientists

VULNERABILITY: engineers, architects, doctors…

VALUE: economists, politicians…


b. Hazard assessment has to be (often) addressed from an “engineeristic” point of view

Often, our state of knowledge and available data are (very) scarce, but we must do something, as best as we can.


c. Hazard assessment consists of providing quantitative estimates of the probability of each event (individual risk)

The eruptive process seems to be a typical “complex system”, therefore it is intrinsically unpredictable in a deterministic way. Therefore, we have to use a probabilistic model.

The probability estimation is fundamental for decision making


d. Hazard assessment has to consider all the available information

1. Our theoretical a priori knowledge (theoretical models, state of the system, etc…)

2. Past data from the system and/or from other systems with similar behavior

3. Data from the “real-time” monitoring


About decision making…

The comparison between risks relative to different hazards

Definition of “acceptable risk” (costs/benefits balance)


The probability is the best way to describe

quantitatively the occurrence of an aleatoric event

(i.e., individual risk)

An aleatoric event is an event that cannot be predicted

deterministically.

Almost all the natural events of interests are

aleatoric events


The probability of an aleatoric event E is indicated

by P(E) and it is a number (or a function)

between 0 and 1.

The probability is defined in different ways (classical,

frequentist, assiomatic, Bayesian).

Often, the probability estimation is very problematic.


The Bayesian approach is particularly useful in practical

problems characterized by few data and scarce

theoretical knowledge

The Bayesian approach implies that the probability

is not a single value but it is a probability distribution.

The probability distribution has an average (the best

guess of the probability) and a standard deviation.

These two parameters estimates the aleatoric and

epistemic uncertainties.


Bayes theorem…

ii

ii

i

pp

pp

p

)(

)(

A posteriori Probability A priori Probability

Likelihood

i = Prob. of event Ei

= Observations


We do not use a single value but a distribution of probability. In this way we can account for aleatoric and epistemic uncertainties.


11 )1()()(

)()(

p

)(E1

1)](1)[()var(

EE

and are the parameters of the distribution.

Their values depend on the a priori theoretical knowledge, and on thedata from the past and from possible “real-time” monitoring of thesystem.

An example: the Beta distribution

nNn

n

Nnp

1)(

Update of a Beta distribution with

new observed data… Bayesian approach

Bernoulli

(N trials

n successes)

1'1'''

''

)1()()(

)()(

npupdated

where:

’=+n

’=+N-n



… about the ELLSBERG PARADOX

URN

50 red balls

50 blue balls

URN

100 unknown

Combination of

red and blue balls

0 10.5

Probability

Dirac’s function

0 0.5

Probability

Uniform distribution

1

Average


… about the ELLSBERG PARADOX


Some example of annual risk of death:

- Smoking 5 x 10-3

- Cardiovascular disease 3.5 x 10-3

- Cancer 2 x 10-3

- Workers in construction 1.5 x 10-4

- Car accidents 1.5 x 10-4

- All accidents (US) 3.4 x 10-4

- Eruptions (world) 1 x 10-7

3. A probabilistic approach for risk assessment: the Event Tree

The Event Tree is a tree-like representation of events in which

branches are logical steps from a general priori event through

increasingly specific subsequent events (intermediate outcomes)

to final outcomes. In this way, ET shows the most relevant possible

outcomes of the system at a progressively higher degree of detail.

The main advantage of the ET scheme consists of its intrinsic

simplicity and of providing a quantitative estimation of any kind

of hazard and individual risk.

Long

-term

fore

cast

ing

& h

azar

dLo

ng-te

rm fo

reca

stin

g &

haz

ard

Stop

VulnerabilityVulnerability& individual risk& individual risk

Short-term forecastingShort-term forecasting

?

Newhall & Hoblitt (Bull. Volcanol., 64, 3-20, 2002)


To estimate the general structure of the tree The structure of the tree has been shown in the previous slide

To estimate the probability at each node The probability at each node is estimated through a statistical distribution (to

take into account the uncertainties)

To combine the probabilities of the nodes to calculate the probability of any possible event

The probabilities are combined through the Bayes theorem

What do we need for estimating the individual risk?


At each node we assign a probability distribution with

the average and standard deviation that depend on

1. Our a priori knowledge

(theoretical models, and state of the system)

2. The past data

(from the system and from systems with similar behavior)

3. The data from the “real-time” monitoring of the

system


What is the probability to kill one person that lives at 10 km North of the volcano, by a pyroclastic flow from a VEI 3 eruption?

P(deathVEI 3, pf) = P(1) P(2|1) P(3|2) P(4|3) P(5|4VEI 3) P(6N|5pf) P(710|6N) P(8|710) P(9|8)

If we want to calculate the probability to kill the same person by a pyroclastic flow, regardless the VEI of the eruption we have

P(deathpf)=P(deathVEI 2, pf)+P(deathVEI 3, pf)+P(deathVEI 4+, pf)

Those probabilities can be summed because the VEI categories are mutually exclusive

In principle, we can calculate the probability of any event

Combining the probability of the nodes…


The event tree allows assigning “dinamically” the probability for each kind of possible event (quantitative hazard assessment) long-term: useful for land use planning of the territory, and for comparing the hazard with other different kind of hazards short-term:useful during emergency to help managing of short-term actions aimed to reduce risk (e.g., evacuation)

The scheme considers all of the available information.

The scheme takes properly into account the epistemic and aleatoric uncertainties.

The procedure highlights what we know and what we do not know about the system, indicating future possible works to improve the scheme.

Estimating the volcanic hazard at Mount Vesuvius…

P(1) = Long-term probability of unrest (Restlessness)

P(2|1) = Prob. of presence of magma, given unrest (Genesis)

P(3|2) = Prob. of eruption, given presence of magma (Outcome)

P(4|3) = Prob. of a particular VEI, given the eruption occurs (Magnitude)

P(5|4) = Prob. of a particular phenomenon, given a VEI (Phenomena)

P(6|5) = Prob. of a particular sector, given a specific phenomena (Sector)

P(7|6) = Prob. of a particular distance, given a specific sector (Distance)

P(8|7) = Prob. of presence of a structure and/or human beings, given a specific phenomena in a specific sector (Exposure)

P(9|8) = Prob. of death or destruction, given the above hazard (Vulnerability)

P* = Prob. to have an eruption

What is unrest? (The definition depends on the present state of the volcano considered)

If

The seismicity is significantly different from the background.

Any inflation of the volcano.

Presence of a significant amount of SO2.

The CO2 flux is significantly different from the background.

The Temperature of the fumaroles inside the crater is significantly larger than the background.

Prob. that the volcano will become restlessP(1):

1. A priori knowledge: none Uniform distribution

2. Past data: 32 years of no unrests Beta distribution

3. If seismicity, ground deformation and/or gasses emission monitored deviate from the background p(1)=1

Prob. that the volcano will become restless

We consider two sets of thresholds: for the most conservative we have: We consider two sets of thresholds: for the most conservative we have:

P(1):

- # earthquakes 100 /month - Md 4.1 - # low-frequency events 3/month - Presence of significant SO2 flux - CO2 flux 5 kg m-2 d-1 - Strain rate > 0 d-1

- T > 105 oC

Prob. that the volcano will become restlessP(1):

1. A priori knowledge: none Uniform distribution (==1)

2. Past data: none we do not use this information

3. Monitoring:

- Average spectral frequency <> 3 Hz - Strain rate 10-5 d-1

- Temperature of the fumaroles > 105 oC - < 0.40 (ratio between average and dispersion of the depth of earthquakes) - Presence of SO2 flux

Prob. that, given unrest, the unrest is caused by magmatic intrusionP(2|1):

(’=1.3 ’=0.7)

(’=1.7 ’=0.3)

(’=1.88 ’=0.12)

(’=0.5 ’=1.5)

Prob. that, given unrest, the unrest is caused by magmatic intrusionP(2|1):

1. A priori knowledge: none Uniform distribution (==1)

2. Past data: none we do not use this information (analogs?)

3. Monitoring:

Prob. that, given a magmatic intrusion, magma will erupt

- Rate of average spectral frequency, d(<>)/dt < 0 (hours to few weeks)

- Strain acceleration > 0 d-2 (hours to few days)

- < 0.40 (hours to weeks) - Acceleration of seismic energy released (hours to few days)

- Cumulative strain > 10-4 (hours to weeks)

- Sudden reversals of at least one of the parameters (hours to few days)

- Phreatic explosion (days to months)

- Sudden increase of HF or HCl over SO2 (few days)

Time interval considered ONE MONTH: Time interval considered ONE MONTH: but the NEW probability P(3|2) is almost reached in FEW days.but the NEW probability P(3|2) is almost reached in FEW days.

P(3|2):

Experience on the global volcanism and on critical systems show that the magnitude of the eruption probably have some kind of a power-law distribution.

The power-law distribution is probably different in the open and

closed conduit regime.

(Probably, the most critical point)

Prob. that, given a magmatic eruption, it will be of a specified explosive size P(4|3):

1. A priori knowledge


Data from historical activity of Mt. Vesuvius.

Data from the so-called “analogs”

Data from worldwide volcanoes


2. Past data

“Analogs” are chosen by taking into account similarities in

A pronounced different/alternation between open and closed conduit behavior in the volcano’s geological or historical record The viscosity of the magma contained in a range that excludes the most acid (e.g., rhyolites) and the most basic (e.g., basalts) magma.

We have identified 17 “analogs” of Mount Vesuvius

Suwanose-jima, Sakura-jima, Popocatepetl, Shikotsu-Tarumai, Fuji, Alaid,Shiveluch, Trident, Komaga-take, Asama, Colima, Fuego, Mayon, Arenal, Cotopaxi, Tungurahua, Merapi.



The basic point is the role of the “repose time” (i.e., Poisson vs. some kind of “memory” process). We consider different standards…

A4 – Data from worldwide volcanoes with 60<RT<200

B4 – Data from “analogs” with 60<RT<200

C4 – Data from Mt. Vesuvius with 60<RT<200

D4 – Data from worldwide volcanoes with RT>60

E4 – Data from “analogs” with RT>60

F4 – Data from Mt. Vesuvius with RT>60



0.110.110.240.240.650.65F4F4

0.200.200.230.230.570.57E4E4

0.100.100.300.300.600.60D4D4

0.010.010.270.270.720.72C4C4

0.140.140.230.230.630.63B4B4

0.020.020.300.300.680.68A4A4

VEIVEI55VEI=4VEI=4VEI=3VEI=3StandardStandard

60<RT<200

MEE

Average 10%

60<RT


The deformation can suggest the opening of lateral vents

We do not use (yet!) the monitoring data to implement the distribution

The degassing seems very promising to modify the probability of small and large eruptive events (cf. Newhall, 2003)

It seems that the precursors are not indicative of the size of the impending eruption.

3. Monitoring data


A possible future pre-eruptive scenario…

Feb. 1: No unrest observed

Feb. 4: Slight inflation of the volcano (5 x 10-6 /d)

Mar. 1: The slight inflation continues.

Occurrence of a swarm of LF at 2.5 km depth,

with an average spectral frequency of 3 Hz.

Mar. 15: The same as March 1, but the hypocenter of

LF events are at 1 km depth and the average

spectral frequency is 1-2 hz.

P=P(1)*P(2|1)*P(3|2)*P(4|3)P=P(1)*P(2|1)*P(3|2)*P(4|3)

P=P(2|1)*P(3|2)*P(4|3)P=P(2|1)*P(3|2)*P(4|3)

Long-term forecasting

Short-term forecasting

Probability of occurrence of an eruption with a specific VEI

Tephra Fall (TF)

Pyroclastic Flow or Surge (PF)

Lahar (LA)

Lava Flow (LF)

Types of phenomena that we consider:

Prob. that, given an eruption of a specified explosive size, it will generate a phenomenon P(5|4):

For now, we consider these ones

VEIVEI TFTF PF PF LALA LFLF

= 3= 3 1.01.0(Dirac Delta)(Dirac Delta)

0.350.35(=0.7 =1.3)

0.25 0.25 (= 0.5 =1.5)

0.6 0.6 (= 1.2 =0.8)

44 1.01.0 (Dirac Delta)

0.70 0.70 (=1.4 =0.6)

0.55 0.55 (= 1.1 =0.9)

0.450.45 (= 0.9 =1.1)

Data from Newhall and Hoblitt (2002)

We modify the probability distributions for TF and PF

according to their past frequency during Vesuvius eruptions.


For each VEI, we count how many eruptions (N) occurred and, of

these, how many (n) produced, for example, TF.

We use these N and n to update the and of the Beta distribution.

’=+n

’=+N-n


EruptionEruption Average conditioned Average conditioned probability of a TFprobability of a TF

Average conditioned probability Average conditioned probability of a PFof a PF

VEI=3VEI=3 11

(Dirac Delta)(Dirac Delta)

0.35 0.35

(we have no past data, so (we have no past data, so ’= and ’=))

VEI=4VEI=4 11


0.85 0.85 (’=1.7 ’=0.3)

VEIVEI55 11


0.880.88(’=1.76 ’=0.24)

Definition of a spatial domainP(6|5) and p(7|6):

Prob. that, given an eruption of a specified explosive size and generating a particularphenomenon, THE NEXT episode will move in acertain sector

P(6|5):

N

NE

W

SW

S

SE

E

NW


P(6|5):

TF: the dispersion of tephra fall depends solely on the wind field at the time of the eruption.

We start by assuming equiprobability on all the sectors, and update the probability distributions according to the past frequency of TF in each sector.

The result is a higher probability in Eastern sectors (in agreement with the wind field statistics, which tell us that the prevailing winds are Eastward).

Prob. that, given an eruption of a specified explosive size and generating a particularphenomenon, THE NEXT TF will move in acertain sector

P(6|5):

TF


P(6|5):

PF:

for VEI=3 eruptions, Mt Somma represents an inviolable barrier N and NE sectors have 0 probability. For the remaining sectors, we start by assuming equiprobability, and update according to the past frequency of PF in each sector.

for larger eruptions, we start by assuming equiprobability on all the sectors, and update according to the past frequency of PF in each sector.

Prob. that, given an eruption of a specified explosive size and generating a particularphenomenon, THE NEXT PF will move in acertain sector

P(6|5):

PF

(No new data)

Prob. that, given an eruption of a specified explosive size and generating THE NEXT particular phenomenon, it will reach acertain distance

P(7|6):

TF: the risk associated to TF is mainly related to roof collapse. In Vesuvius surroundings, the roof maximum load is approximately 200-300 Kg/m2. Up to now we assume that a layer 10 cm thick might cause roof collapse.

TF risk might also be related to traffic jamming. In this case, even few cm layer of TF might cause serious problems.

PF: the risk associated to PF is related to its occurrence, regardless its intensity or the thickness of the deposit left.

Prob. that, given an eruption of a specified explosive size and generating THE NEXT TF, it will reach a certain distance

P(7|6):

We start from Newhall and Hoblitt (2002)

0-5 Km0-5 Km 5-10 Km5-10 Km 10-15 10-15 KmKm

15-20 15-20 KmKm

20-25 20-25 KmKm

VEI=3VEI=3 0.700.70

((=1.4 =0.6)

0.45 0.45

((=0.9 =1.1)

0.350.35

((=0.7 =1.3)

0.250.25

((=0.5 =1.5)

0.200.20

((=0.4 =1.6)

VEIVEI44 0.950.95

((=1.9 =0.1)

0.93 0.93

((=1.86 =0.14)

0.900.90

((=1.8 =0.2)

0.85 0.85

((=1.7 =0.3)

0.80 0.80

((=1.6 =0.4)

TF

TF

Prob. that, given an eruption of a specified explosive size and generating THE NEXT TF, it will reach a certain distance

P(7|6):

P(7|6):Prob. that, given an eruption of a specified explosive size and generating THE NEXT PF, it will reach a certain distance

We start from Newhall and Hoblitt (2002)

0-5 Km0-5 Km 5-10 Km5-10 Km 10-15 10-15 KmKm

15-20 15-20 KmKm

20-25 20-25 KmKm

VEI=3VEI=3 0.800.80

((=1.6 =0.4)

0.200.20

((=0.4 =1.6)

0.010.01

((=0.02 =1.98)

0.010.01

((=0.02 =1.98)

0.010.01

((=0.02 =1.98)

VEIVEI44 0.870.87

((=1.74 =0.26)

0.640.64

((=1.28 =0.72)

0.450.45

((=0.9 =1.1)

0.240.24

((=0.48 =1.52)

0.160.16

((=0.32 =1.68)

PF

P(6|5):

PF

(No new data)

P(7|6):Prob. that, given an eruption of a specified explosive size and generating THE NEXT PF, it will reach a certain distance

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