Models for Volcano Avalanches A Risk Map for Pyroclastic Flows:

1. 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

2. Goal: A Hazard Map

3. 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?

4. 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

5. 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

6. 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.

7. 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

9. 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

11. 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

12. Ω={V,θ : h(V,θ) ≥ hcrit} • Truth: ψ* and Ω* • Within Ω* a hit, H, has occured

13. Probability of hit • Eruptions independent • Adjust probability above to account for event frequency, λ_ε and prediction time interval (~100 years)

14. Emulator guided sampling • Want to sample important θs • Integrate directly, plug in ψ(θ) • Draw θs by rejection sampling

16. Probability estimate • Upper bound on estimate • Draw θs as described • MC, can calculate […] exactly • For cartoon, about 10^-8

17. 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

18. 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

19. Future directions • Implement plan – run simulator, build emulate, define ψ(θ), calculate probabilities • Include other input parameters – initiation velocity, friction angles • Validation

20. Tar River Valley May 3, 2007 March 29, 2007, from Old Towne