Models for Volcano Avalanches
Constructing Risk Map for Pyroclastic Flows:
Combining simulations and data to predict rare events
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
Image Workshop
May 21-23, 2007
Volcan Colima, Mexico
Montserrat
Modeling (applied math)
Postulate governing Physics
Observe flow of material
Predictions
Abstract differential
equations
Model Uncertainty in
DEs
Modeling (statistical)
Observe flow of material
Predictions
Model data
Introduce Data
Uncertainty, model error
Montserrat
Soufriere Hills Volcano
After centuries of dormancy,
January 92 brings the start of
earthquake swarms in southern
Montserrat
Model Topography and
Equations(2D)
z  s( x, y, t )
Upper free surface
Fs(x,t) = s(x,y,t) – z = 0,
geophysical mass
ground
h  s b
z  b ( x, y )
Basal material surface
Fb(x,t) = b(x,y) – z = 0
Kinematic BC:
at F s (x, t )  0 :  t Fs  v  F s  0
z is the direction normal
to the hillside
at Fb (x, t )  0 :  t Fb  v  Fb  es
Elevation data from public and private DEMs - different sources and
different resolutions.
Simulations
Depth averaging and scaling: Hyperbolic System of
balance laws
h hvx hv y


 es
t
x
y
continuity
x momentum
2
2
hv x  (hv x  .5  k ap g z h ) hv y v x



t
x
y
 g x h  v x es 
1


 v 
1
hg z
  g z   v x2   h  tan bed   sgn  x   hk ap
sin int 
2
2


y

y
vx  v y 


x

vx
2
3
1. Gravitational driving force
2. Coulomb friction at the base
3. Intergranular Coulomb force due to velocity gradients normal to the
direction of flow
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
internal friction
basal friction
initial mass
initial location
initial velocity – speed and direction
 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.
Total volume ~ α-Stable
(Wolpert & Luna-Gomez)
Total 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 19952007 to obtain predictive distribution for
large flows from 2007-2057
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
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
Ω={V,θ : h(V,θ) ≥ hcrit}
Truth: ψ* and Ω*
Within Ω* a hit, H, has occured
Probability of hit
P( H | E )   I ( H | E , , v) f ( , v)dvd
1
  1
f (v, ) 
 v 1{v  }
2
Eruptions independent
Adjust probability above to account for event
frequency, λ_ε and prediction time interval
(~100 years)
Emulator guided sampling
Want to sample important θs
f  ( , v)  f  ( ) f  (v |  )

1
1   1
f  ( ) 


v
dv

p()  ( ) 2
Integrate directly, plug in ψ(θ)
Draw θs by rejection sampling
Probability estimate
Upper bound on estimate
2
P( H | E )   [
0

I ( H | E ,  , v) f

 

(v |  )dv] f  ( )d
( )
Draw θs as described
MC, can calculate […] exactly
For cartoon, about 10^-8
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
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 and probabilistic v-f models
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
March 29, 2007, from Old Towne