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Models for Volcano Avalanches A Risk Map for Pyroclastic Flows: Combining simulations and data to predict rare events. Bruce Pitman, Robert Wolpert, Elaine Spiller The University at Buffalo, Duke University, and SAMSI. SAMSI Transition Workshop May 14-16, 2007. Goal: A Hazard Map.

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Models for volcano avalanches a risk map for pyroclastic flows

Models for Volcano Avalanches

A Risk Map for Pyroclastic Flows:

Combining simulations and data to predict rare events

Bruce Pitman, Robert Wolpert, Elaine Spiller

The University at Buffalo, Duke University, and SAMSI

SAMSI Transition Workshop

May 14-16, 2007


Goal a hazard map
Goal: A Hazard Map


The questions
The Questions

  • What is the frequency-volume distribution?

  • How can one develop a hazard map?

  • How does one perform enough simulations or evaluation of emulators to develop the map?

  • What about regions where probability of flow is very small?


Predictive distribution
Predictive distribution

Reflects:

  • Uncertainty about α, λ

  • Stochastic nature of system

    Problem:

  • Pareto has heavy tails => probability of at least one very large flow event over decades-long period


Hazard map
Hazard map

Idea

  • Sample from predictive v-f distribution

  • Monte-Carlo (MC) to find flow probability contours (i.e. hazard map)

  • Simulations with TITAN software

    Problem

  • Cannot tell us about very small probability events --- (hopefully) significant flow in populated areas is a rare event


A first problem
A first problem:

  • Consider one interesting location, i.e., center of town, proposed school location

  • Find probability that max flow height exceeds critical height over, say, 100 years.

  • Equivalent to finding most likely combination of initial volume and flow angles that generates flows where max height > critical height.


Plan of attack
Plan of attack

1. Course grid

  • Begin with course grid over volume/initiation angle design space

  • Run flow simulations and collect max flow height at location of interest

  • Emulate max height surface

    Goal

  • Identify “interesting” region of design space to narrow search

    Bonus

  • Might suggest useful regression functions


Emulator
Emulator

  • Inputs (…for now)

    -volume, v, and angle, θ

  • Output

    -height h(v, θ) (or some reasonable metric)

  • Interesting region

    -interested in contour where h(v, θ)=hcrit

    -ψ(θ)=v => h(ψ(θ) , θ)=hcrit


Models for volcano avalanches a risk map for pyroclastic flows
Plan

  • Build emulator on sub-design space

  • Identify ψ(θ) and reasonable volume bounds from confidence interval

  • Error on side of smaller volumes producing hits

  • Use ψ(θ) and predictive volume/flow distribution to calculate probability of catastrophic pyroclastic event hitting target


Models for volcano avalanches a risk map for pyroclastic flows

  • Ω={V,θ : h(V,θ) ≥ hcrit}

  • Truth: ψ* and Ω*

  • Within Ω* a hit, H, has occured


Probability of hit
Probability of hit

  • Eruptions independent

  • Adjust probability above to account for event frequency, λ_ε and prediction time interval (~100 years)


Emulator guided sampling
Emulator guided sampling

  • Want to sample important θs

  • Integrate directly, plug in ψ(θ)

  • Draw θs by rejection sampling


Probability estimate
Probability estimate

  • Upper bound on estimate

  • Draw θs as described

  • MC, can calculate […] exactly

  • For cartoon, about 10^-8


Plan to do better
Plan to do better

  • Draw θs as before

  • For each θ, draw a v from f(v| θ)

  • If (θ,v) in thatched area, run simulator to see if hit occurred. If so, update probability estimate

  • Update confidence bands based on new simulator runs

  • iterate


Conclusions remarks
Conclusions/remarks

  • Proposed a method to combine data, simulation, and emulation for calculating probabilities of rare events

  • Probability calculations are “free” once we have a decent grasp on ψ(θ)

  • Gives us some flexibility to redo calculations for a range of flow-volume parameters


Future directions
Future directions

  • Implement plan – run simulator, build emulate, define ψ(θ), calculate probabilities

  • Include other input parameters – initiation velocity, friction angles

  • Validation


Tar river valley may 3 2007
Tar River Valley May 3, 2007

March 29, 2007, from Old Towne