Models for Volcano Avalanches Constructing Risk Map for Pyroclastic Flows: Combining simulations and data to predict rare events Image Workshop May 21-23, 2007 Elaine Spiller Bruce Pitman, Robert Wolpert, Eliza Calder, Abani Patra, Simon Luna-Gomez, Keith Dalbey, Susie Bayarri, Jim Berger The University at Buffalo, Duke University, and SAMSI

Volcan Colima, Mexico

Montserrat

Modeling (applied math) Postulate governing Physics Observe flow of material Model Uncertainty in DEs Abstract differential equations Predictions

Modeling (statistical) Observe flow of material Introduce Data Uncertainty, model error Model data Predictions

Goal: A Hazard Map

Montserrat Soufriere Hills Volcano After centuries of dormancy, January 92 brings the start of earthquake swarms in southern Montserrat

Model Topography and Equations(2D) Upper free surface F s (x,t) = s(x,y,t) – z = 0, Basal material surface F b (x,t) = b(x,y) – z = 0 Kinematic BC: Elevation data from public and private DEMs - different sources and different resolutions. z is the direction normal to the hillside ground geophysical mass

Depth averaging and scaling: Hyperbolic System of balance laws continuity x momentum 1.Gravitational driving force 2.Coulomb friction at the base 3.Intergranular Coulomb force due to velocity gradients normal to the direction of flow Simulations 132

TITAN2D  Large scale computations to produce realistic simulations of mass flows  Integrated with GIS to obtain terrain data  High performance techniques for efficiency

Uncertain features uinternal friction ubasal friction uinitial mass uinitial location uinitial velocity – speed and direction u topography

An Aside on friction

Sample of Data (Calder)

An Aside on M (Sheridan) It is those rare very large flows that cause enormous damage and loss of life.

The Questions uWhat is the frequency-volume distribution? uHow can one develop a hazard map? uHow does one perform enough simulations or evaluation of emulators to develop the map? uWhat about regions where probability of flow is very small?

Total volume ~ α -Stable (Wolpert & Luna-Gomez) uTotal volume of all flows in t years α- St(a,1, λ t)  α uncertain (-slope α = 0.5)  λ uncertain (rate of flows)  Learn about α and λ from data to obtain predictive distribution for large flows from

Predictive distribution Reflects:  Uncertainty about α, λ uStochastic nature of system Problem: uPareto has heavy tails => probability of at least one very large flow event over decades-long period

Hazard map Idea uSample from predictive v-f distribution u Monte-Carlo (MC) to find flow probability contours (i.e. hazard map) uSimulations with TITAN software Problem uCannot tell us about very small probability events --- (hopefully) significant flow in populated areas is a rare event

A first problem: uConsider one interesting location, i.e., center of town, proposed school location uFind probability that max flow height exceeds critical height over, say, 100 years. uEquivalent to finding most likely combination of initial volume and flow angles that generates flows where max height > critical height.

Plan of attack 1. Course grid uBegin with course grid over volume/initiation angle design space uRun flow simulations and collect max flow height at location of interest uEmulate max height surface Goal uIdentify “interesting” region of design space to narrow search Bonus uMight suggest useful regression functions

Emulator uInputs (…for now) -volume, v, and angle, θ uOutput -height h(v, θ) (or some reasonable metric) u Interesting region -interested in contour where h(v, θ)=hcrit -ψ(θ)=v => h(ψ(θ), θ)=hcrit

Plan uBuild emulator on sub-design space uIdentify ψ(θ) and reasonable volume bounds from confidence interval uError on side of smaller volumes producing hits uUse ψ(θ) and predictive volume/flow distribution to calculate probability of catastrophic pyroclastic event hitting target

uΩ={V,θ : h(V,θ) ≥ hcrit} uTruth: ψ* and Ω* uWithin Ω* a hit, H, has occured

Probability of hit uEruptions independent uAdjust probability above to account for event frequency, λ_ε and prediction time interval (~100 years)

Emulator guided sampling uWant to sample important θs uIntegrate directly, plug in ψ(θ) uDraw θs by rejection sampling

Probability estimate uUpper bound on estimate uDraw θs as described uMC, can calculate […] exactly uFor cartoon, about 10^-8

Plan to do better uDraw θs as before uFor each θ, draw a v from f(v| θ) uIf (θ,v) in thatched area, run simulator to see if hit occurred. If so, update probability estimate uUpdate confidence bands based on new simulator runs uiterate

Conclusions/remarks uProposed a method to combine data, simulation, and emulation for calculating probabilities of rare events uProbability calculations are “free” once we have a decent grasp on ψ(θ) uGives us some flexibility to redo calculations for a range of flow-volume parameters and probabilistic v-f models

Future directions uImplement plan – run simulator, build emulate, define ψ(θ), calculate probabilities uInclude other input parameters – initiation velocity, friction angles uValidation

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