Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

BET: a probabilistic tool for Eruption BET: a probabilistic tool for Eruption Forecasting and Volcanic Hazard AssessmentForecasting and Volcanic Hazard Assessment

W. Marzocchi, L. Sandri, J. Selva

INGV-Bologna

BET: a probabilistic tool for Eruption BET: a probabilistic tool for Eruption Forecasting and Volcanic Hazard AssessmentForecasting and Volcanic Hazard Assessment

W. Marzocchi, L. Sandri, J. Selva

INGV-Bologna

Project INGV-DPC V4: “Innovative techniques to study active Project INGV-DPC V4: “Innovative techniques to study active volcanoes” volcanoes” (W.Marzocchi, INGV-Bo, A. Zollo, Univ. of Naples)(W.Marzocchi, INGV-Bo, A. Zollo, Univ. of Naples)Project INGV-DPC V4: “Innovative techniques to study active Project INGV-DPC V4: “Innovative techniques to study active volcanoes” volcanoes” (W.Marzocchi, INGV-Bo, A. Zollo, Univ. of Naples)(W.Marzocchi, INGV-Bo, A. Zollo, Univ. of Naples)

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

PART I: BET modelPART I: BET model

PART I: BET modelPART I: BET model

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

What is BET?What is BET?What is BET?What is BET?

BET (Bayesian Event Tree)BET (Bayesian Event Tree) is a new statistical codestatistical code to estimate and visualize short- to long-term eruption forecasting (BET_EF)eruption forecasting (BET_EF) and volcanic hazard (BET_VH)(BET_VH) and relative uncertainties (epistemicepistemic and aleatoryaleatory)

BET (Bayesian Event Tree)BET (Bayesian Event Tree) is a new statistical codestatistical code to estimate and visualize short- to long-term eruption forecasting (BET_EF)eruption forecasting (BET_EF) and volcanic hazard (BET_VH)(BET_VH) and relative uncertainties (epistemicepistemic and aleatoryaleatory)

BET Output:BET Output: Time and space evolution of the probability function of each specific event in which we are interested in.

BET Output:BET Output: Time and space evolution of the probability function of each specific event in which we are interested in.

BET Input:BET Input: Volcanological data, models, and/or expert opinion. These data are provided by the end-user.

BET Input:BET Input: Volcanological data, models, and/or expert opinion. These data are provided by the end-user.

BET transforms these information into BET transforms these information into probabilitiesprobabilities

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

The method is based on three basic stepsThe method is based on three basic stepsThe method is based on three basic stepsThe method is based on three basic steps

1. Design of a generic Bayesian Event Tree

BibliographyBibliography

• Newhall and Hoblitt, Bull. Volc. 2002 (for step 1)• Marzocchi et al., JGR 2004 (for steps 2 and 3)• Marzocchi et al., 2006; IAVCEI volume on statistics in Volcanology (for steps 2 and 3)• Marzocchi et al., 2007, Bull. Volcan., in press (full description of BET_EF, available online)

BibliographyBibliography

• Newhall and Hoblitt, Bull. Volc. 2002 (for step 1)• Marzocchi et al., JGR 2004 (for steps 2 and 3)• Marzocchi et al., 2006; IAVCEI volume on statistics in Volcanology (for steps 2 and 3)• Marzocchi et al., 2007, Bull. Volcan., in press (full description of BET_EF, available online)

How BET works?How BET works?How BET works?How BET works?

2. Estimate the conditional probability at each node

3. Combine the probabilities of each node to obtain probability distribution of any relevant event

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

The probability The probability of the SELECTED PATH is the product of conditional of the SELECTED PATH is the product of conditional

probability probability ii at ALL SELECTED BRANCHES: at ALL SELECTED BRANCHES:

11]] • [• [22]] • [• [33]] • [• [44]] • [• [55]] • …• …

The probability The probability of the SELECTED PATH is the product of conditional of the SELECTED PATH is the product of conditional

probability probability ii at ALL SELECTED BRANCHES: at ALL SELECTED BRANCHES:

11]] • [• [22]] • [• [33]] • [• [44]] • [• [55]] • …• …

BET Structure & ProbabilityBET Structure & ProbabilityBET Structure & ProbabilityBET Structure & Probability

Eruption Forecasting: we Eruption Forecasting: we

focus on…focus on… Eruption Forecasting: we Eruption Forecasting: we

focus on…focus on…

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

kk(M)(M)MONITORING PARTMONITORING PART

Monitoring Data & ModelsMonitoring Data & Models

kk(M)(M)MONITORING PARTMONITORING PART

Monitoring Data & ModelsMonitoring Data & Models

kk(NM)(NM)NON-MONITORING PARTNON-MONITORING PART

Non-monitoring Data, Geological & Non-monitoring Data, Geological & Physical ModelsPhysical Models

kk(NM)(NM)NON-MONITORING PARTNON-MONITORING PART

Non-monitoring Data, Geological & Non-monitoring Data, Geological & Physical ModelsPhysical Models

CONDITIONAL PROBABILITY AT THE NODE:CONDITIONAL PROBABILITY AT THE NODE:

kk = = kk(M)(M) + (1-+ (1- kk

(NM)(NM)

CONDITIONAL PROBABILITY AT THE NODE:CONDITIONAL PROBABILITY AT THE NODE:

kk = = kk(M)(M) + (1-+ (1- kk

(NM)(NM)

MONITORING DATAMONITORING DATA

State of unrest State of unrest at at tt00through a FUZZY through a FUZZY approachapproach

MONITORING DATAMONITORING DATA

State of unrest State of unrest at at tt00through a FUZZY through a FUZZY approachapproach

Conditional Probability [Conditional Probability [KK] (Node k)] (Node k)Conditional Probability [Conditional Probability [KK] (Node k)] (Node k)

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

MODELS Prior

MODELS Prior

DATALikelihood

… … each part each part kk(.)(.)(monitoring and non-monitoring)(monitoring and non-monitoring)… … each part each part kk(.)(.)(monitoring and non-monitoring)(monitoring and non-monitoring)

POSTERIOR PDF

k = k(.) [H(.)|k

(.)H(.)

(no epistemic uncertainty)

In each factor, at each node, we account for:1. Models + data2. Epistemic and aleatoric uncertainities

In each factor, at each node, we account for:1. Models + data2. Epistemic and aleatoric uncertainities

Bayes theoremBayes theoremBayes theoremBayes theorem

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

A priori information: a probability (guess) and its weight in terms of number of equivalent data (p and ). If no information are available BET starts from maximum ignorance (uniform distribution)

Past data information: total number of cases and the number of “successes” (N and n)

What does BET accept in input? What does BET accept in input?

Non-monitoring infoNon-monitoring infoNon-monitoring infoNon-monitoring info

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

Node 5: probability of a specific size(3 sizes: VEI 3, VEI 4, VEI 5+)

Node 5: probability of a specific size(3 sizes: VEI 3, VEI 4, VEI 5+)

Non-monitoring: ExampleNon-monitoring: ExampleNon-monitoring: ExampleNon-monitoring: Example

A priori information: We assume a power law. Our initial guess will be: P(VEI 3) = 0.60 P(VEI 4) = 0.30 P(VEI 5+) = 0.10The weight assigned is =1. This means that our a priori belief has the same weight f 1 single datum. Few data can change our estimation.

Past data information: The eruptive catalog. We need to put in input N = total number of eruptions n(VEI 3) = number of VEI 3 eruptions n(VEI 4) = number of VEI 4 eruptions n(VEI 5+) = number of VEI 5+ eruptions

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

A priori information: list of monitored parameters relevant at the node considered, with lower and upper thresholds, and possibly the weight of each parameter. (NOTE: the parameters have to be measured frequently at the volcano)

Past data information: Total number of past monitored cases (N). For each case, BET requires the values of the monitored parameters, and the “successfulness” of the considered case.

What does BET accept in input? What does BET accept in input?

Monitoring infoMonitoring infoMonitoring infoMonitoring info

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

Node 2: probability of a “magmatic” unrestNode 2: probability of a “magmatic” unrest

Monitoring: ExampleMonitoring: ExampleMonitoring: ExampleMonitoring: Example

A priori information: List of “indicators” of a magmatic unrest. Presence of magmatic gases (e.g., SO2) [>;1;1] Number of LP events deeper than 5 km per day [>;0;5] Largest magnitude M [>;3.6;4.5] Uplift rate d/dt [>;10;30 cm/month]

Past data information: We need to put in input N = total number of monitored eruptions The values of the parameters for each monitored unrest The nature of the unrest (magmatic or not)

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

Through expert opinion and/or looking at “analogs” (need of WOVOdat!), the user defines INTERVAL OF THRESHOLD for each “indicator”

• We assure smooth transitions (for small changes) and uncertainty on the definition of the state of anomaly (three sets: surely not anomalous, uncertain, surely anomalous)

… … going into some detailsgoing into some details: including monitoring: including monitoring… … going into some detailsgoing into some details: including monitoring: including monitoring

degree of anomaly zi

measure

State of unrest State of unrest

A priori model [k(1)]A priori model [k(1)]

sure

ly N

OT

AN

OM

ALO

US

sure

ly A

NO

MA

LOU

S

Thresholds are processed through FUZZY SET theory…Thresholds are processed through FUZZY SET theory…Thresholds are processed through FUZZY SET theory…Thresholds are processed through FUZZY SET theory…

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

… … going into some details: going into some details: from from monitoring to probabilitymonitoring to probability… … going into some details: going into some details: from from monitoring to probabilitymonitoring to probability

k(M)|H] = k

(1)[H(1)|k(1)Hk

(M)|H] = k(1)[H(1)|k

(1)H

Z(k) = i wi zi degree of anomaly at the node

<k(1)>1 - exp(-(k)) Average of k

(1)<k(1)>1 - exp(-(k)) Average of k

(1)

Monitoring part

zi degree of anomaly of i-th parameter

BET computes:

The user: input measures

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

Cost/Benefit analysisCost/Benefit analysisCost/Benefit analysisCost/Benefit analysis

Some useful considerations…Some useful considerations…

“Eruption forecasting” means to estimate probabilities

Typical requirement from end-users: YES or NOT (but the Nature seems not to much interested in playing deterministically)

How to interpret and to use probabilities? COMPARING THEM WITH MORE USUAL EVENTS

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

P x L > CP x L > CP x L > CP x L > C

Let’s make the example of an evacuation (SIMPLIFIED!!!)L: cost of human lives lost due to an eruptionC: cost of an evacuationP: prob. of the deadly event (i.e., prob. of a pyroclastic flow)

If

the cost of human lives “probably” lost exceeds the cost of an evacuation. Therefore, the evacuation might be called when

P > C / LP > C / LP > C / LP > C / L

The evacuation will be called when the probability of The evacuation will be called when the probability of the deadly event will overcome a threshold defined a the deadly event will overcome a threshold defined a priori by Civil Protectionpriori by Civil Protection

The evacuation will be called when the probability of The evacuation will be called when the probability of the deadly event will overcome a threshold defined a the deadly event will overcome a threshold defined a priori by Civil Protectionpriori by Civil Protection

Cost/Benefit analysisCost/Benefit analysisCost/Benefit analysisCost/Benefit analysis

Eruption Forecasting through the Bayesian Event Tree: the software package BET_EFEruption Forecasting through the Bayesian Event Tree: the software package BET_EFINGVINGVINGVINGV

More on BET:

CoV5:-Oral 12-O-11, Nov. 22 (Thu) Hall A, 1450-1510Integrating Eruption Forecasting and Cost/benefit Analysis for decision making During an Emergency: the Case of BET_EF Applied to Vesuvius in the MESIMEX Experiment-poster 21b-P-18, Nov. 22 (Thu.), 1640-1800 The Bayesian Event Tree for short- and long-term eruption forecasting at Campi Flegrei, Italy,

Other…- http://www.bo.ingv.it/bet

- Marzocchi, W., Sandri, L., Selva J., BET_EF: a probabilistic tool for long- and sort-term eruption forecasting, Bull. Volcanol., DOI 10.1007/s00445-007-0157-y

More on BET:

CoV5:-Oral 12-O-11, Nov. 22 (Thu) Hall A, 1450-1510Integrating Eruption Forecasting and Cost/benefit Analysis for decision making During an Emergency: the Case of BET_EF Applied to Vesuvius in the MESIMEX Experiment-poster 21b-P-18, Nov. 22 (Thu.), 1640-1800 The Bayesian Event Tree for short- and long-term eruption forecasting at Campi Flegrei, Italy,

Other…- http://www.bo.ingv.it/bet

- Marzocchi, W., Sandri, L., Selva J., BET_EF: a probabilistic tool for long- and sort-term eruption forecasting, Bull. Volcanol., DOI 10.1007/s00445-007-0157-y