Transcript Mark Bebbington

Long-term forecasting of volcanic explosivity
Mark Bebbington
IFS(Statistics) & Volcanic Risk Solutions, Massey University
Probabilistic Volcanic Hazard Analysis
• Many statistical models exist for the time to the next
eruption onset.
• Some of them even seem to work!
• For long-term hazard, more important to forecast
eruption size
• This is not currently done well.
• Current practice is to forecast size independently
• Is it even possible?
In the next ~25 minutes
• Motivation – why this matters
• Size-predictability
• Regression models – lack of power
• Aggregate volcanoes
• VEI
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Data (volcanoes/eruptions) selection
Probability distributions for VEI
Bayesian hierarchical generalized linear models
Results
How far back should I stand?
A perceptive forecast:
- Crandell et al., “Mt St Helens volcano:
Recent and future behavior”. Science
187:438-441, 1975
“The repetitive nature of the eruptive activity at
Mt St Helens during the last 4000 years, with
dormant intervals typically of a few centuries or
less, suggests that the current quiet period will not
last a 1000 years. Instead, an eruption is likely
within the next hundred years, possibly before the
end of this century”
• Mt St Helens erupted in
1980, having been
quiescent since 1857
• Unfortunately, the
prediction involved only
timing, not size
Mt St Helens: Volcanic Record
Tendency for
large(r) eruptions
after prolonged
quiesence?
Size- (and time-) predictability
General load and discharge model (De la Cruz-Reyna 1991, Bull Volcanol)
• magma inflow at constant rate
• eruption occurs when stored amount V(t) exceeds threshold H
• eruption continues until stored amount V(t) is depleted below threshold L
V(t)
V(t)
?
?
V(t)
t
H and L variable
t
H fixed : repose α prev. volume
(Time Predictable Model)
t
L fixed : next vol. α
repose
(Size Predictable
Model)
Regression Methods – Individual Volcanoes
There are a handful of volcanoes with
extensive eruptive volume records
(Etna, Vesuvius, Kilauea, Mauna Loa ...)
If repose length ri is ended by volume vi
log ri+1 = a + b log vi
log vi = a + b log ri
Time predictable,
positive correlation
(Not) size predictable,
no correlation
(time-predictable)
(size-predictable)
Time predictable,
positive correlation
(Not) size predictable,
no correlation
Repose times
and volumes
for eruptions
of Mauna
Loa, with
best fitting
regression
lines.
(Bebbington
2008)
Repose
times
(subject to
error) and
volumes for
large
eruptions of
Mt Taranaki
(Turner et al.
2011)
Repose times
and volumes
for flank
eruptions of
Mt Etna, with
best fitting
regression
lines.
(Bebbington
2008)
Size-predictable,
negative correlation
(Not) time predictable,
no correlation
Volcanic Explosivity Index (VEI)
VEI 2 is a ‘default’ in
the absence of other
information
Even large (e.g.
Kilauea 1983present, ~4km3)
effusive eruptions
are VEI 1
Regression Methods – Groups of Volcanoes
More data required
• use Volcanic Explosivity Index (VEI)
• combine volcano records
Time predictable (left) and size predictable (right) models (Marzocchi & Zaccarelli 2006, J Geophys Res)
Not doing too well so far ...
Problems with regression
• Individual volcanoes: t-p significant, s-p not.
• Hence s-p is a much weaker effect than t-p
• Aggregations: inhomogeneity
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•
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Open/closed conduits
‘Cycles’
Incompleteness
Time-scaling
• Need to carefully construct aggregation
• VEI: regression assumptions (normal errors w. constant
variance) dubious
• Parametric (Gen. Lin. Model), Bayesian (hierarchical) , approach
Completeness
Globally, the observance probability rises from 10% in 1500 to 100% in 1980
(assumed).
BUT
– some volcanoes
are much better
observed
- big eruptions are
much better
observed
Data (Indonesia)
• Well-reported since
1800 (earlier for
certain volcanoes).
• Homogeneous
compositions
(Basaltic-Andesitic)
• VEI ranges 2-5
(exclude volcanoes
with no VEI > 2)
• 531 eruptions from
26 volcanoes
A little EDA …
Aggregate data from the 26
volcanoes:
(a) VEI versus onset date.
(b) VEI versus repose.
(c) VEI versus prev. VEI.
(d) VEI versus mean repose.
All VEI data have been jittered.
Circles are open conduits,
squares closed conduits
A Probability Distribution for VEI
Or, normalizing for VEI = 2,3,4 or 5
Monte Carlo
test:
The gi are
significantly
different.
A Parametric Model
kth eruption at jth volcano
Generalized linear model
Individual volcano baseline
Time trend (larger VEI earlier?)
Hierarchical Bayes
Size-predictability
Characteristic time scale
Volume-volume effect
Reference priors
Results
P(q1 ) > 0 = 0.867
VEI increases with time
P(q2 ) > 0 = 0.833
VEI increases with
repose (open conduit)
P(q3 ) > 0 = 0.999
VEI increases with
repose (closed conduit)
P(q4 ) > 0 = 0.439
VEI independent
of previous VEI
P(q5 ) > 0 = 0.920
VEI increases with av.
repose
***
Model Validation
Separation into
open/closed conduit
justified
Hetroscedastic (unequal
variances) model not
justified
Hierarchical model
justified.
Insensitive to data priors.
Forecasts
Closed conduit
Open conduit
Is VEI a power-law?
2-parameter distribution
based on Beta distn.
- Much more flexible shape
2-parameter VEI fits
Monte Carlo simulation/refit:
Parameters a,b may be common to all
-- non-hierarchical model
Why such variability in a,b?
The majority of
the information
is contained in
the ratio b/a
Non-hierarchical model
Model is now additive, not
multiplicative, as parameters need
not be positive.
Hence use exp(VEI) instead of VEI, r
instead of log r, etc.
Reference priors as before
Non-monotonic results; summary
a, and hence VEI, increases with repose for closed conduits: P(q > 0) = 1
(open conduits: P(q > 0) = 0.907)
b decreases, and hence VEI increases, for long average reposes: P(q < 0) = 0.933
Closed conduit
Open conduit
Conclusions
• Consistent size-predictable effect for closed conduit volcanoes
• Insensitive to VEI distribution, volcano-specific or common
• Independent of date -> catalogs complete
• No dependence on previous VEI
• Open/closed conduit -> condition at end of previous eruption is control
• dynamically updated as repose increases, with prediction intervals.
• Easily included in event tree, explicitly includes prior data from volcano, and a
suite of suitable analogs.
• If volume ∝ 10VEI, then observed ΔVEI of 0.03--0.22 (PL) or 0.04--0.18
(2param) per 10 yr -> 0.7 to 5 % increase in volume / yr of repose
• No correlation with time-predictability or susceptibility to earthquake
triggering