1. Long-term forecasting of volcanic explosivity
Mark Bebbington
IFS(Statistics) & Volcanic Risk Solutions, Massey University

2. 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?

3. In the next ~25 minutes Motivation – why this matters
Size-predictability
Regression models – lack of power
Aggregate volcanoes
VEI
Data (volcanoes/eruptions) selection
Probability distributions for VEI
Bayesian hierarchical generalized linear models
Results

4. How far back should I stand?
A perceptive forecast:
“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”
- Crandell et al., “Mt St Helens volcano: Recent and future behavior”. Science 187: , 1975
Mt St Helens erupted in 1980, having been quiescent since 1857
Unfortunately, the prediction involved only timing, not size

5. Mt St Helens: Volcanic Record
Tendency for large(r) eruptions after prolonged quiesence?

6. 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 t t
H fixed : repose α prev. volume
(Time Predictable Model)
L fixed : next vol. α repose
(Size Predictable Model)
H and L variable

7. Regression Methods – Individual Volcanoes
There are a handful of volcanoes with extensive eruptive volume records (Etna, Vesuvius, Kilauea, Mauna Loa ...)
Time predictable, positive correlation
Repose times and volumes for flank eruptions of Mt Etna, with best fitting regression lines. (Bebbington 2008)
If repose length ri is ended by volume vi
log ri+1 = a + b log vi (time-predictable)
log vi = a + b log ri (size-predictable)
(Not) size predictable, no correlation
Repose times (subject to error) and volumes for large eruptions of Mt Taranaki (Turner et al. 2011)
Size-predictable, negative correlation
Repose times and volumes for eruptions of Mauna Loa, with best fitting regression lines. (Bebbington 2008)
Time predictable, positive correlation
(Not) time predictable, no correlation
(Not) size predictable, no correlation

8. Volcanic Explosivity Index (VEI)
VEI 2 is a ‘default’ in the absence of other information
Even large (e.g. Kilauea 1983-present, ~4km3) effusive eruptions are VEI 1

9. 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)

10. 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
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

11. 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

12. 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

13. 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
Aggregate data from the 26 volcanoes considered in this study. (a) VEI versus onset date. (b) VEI versus repose length. (c) VEI versus previous VEI. (d) VEI versus mean repose length for the volcano. All VEI data have been jittered to improve visibility. Circles indicate open conduits, squares closed conduits.

14. A Probability Distribution for VEI
Or, normalizing for VEI = 2,3,4 or 5
Monte Carlo test:
The gi are significantly different.

15. A Parametric Model Generalized linear model
kth eruption at jth volcano
Generalized linear model
Individual volcano baseline
Time trend (larger VEI earlier?)
Hierarchical Bayes
Size-predictability
Characteristic time scale
Reference priors
Volume-volume effect

16. Results *** P(q1 ) > 0 = 0.867 VEI increases with time
VEI increases with repose (open conduit)
P(q2 ) > 0 = 0.833
VEI increases with repose (closed conduit)
P(q3 ) > 0 = 0.999
***
VEI independent of previous VEI
P(q4 ) > 0 = 0.439
P(q5 ) > 0 = 0.920
VEI increases with av. repose

17. Model Validation
Separation into open/closed conduit justified Hetroscedastic (unequal variances) model not justified Hierarchical model justified. Insensitive to data priors.

18. Forecasts
Closed conduit
Open conduit

19. Is VEI a power-law?
2-parameter distribution based on Beta distn. - Much more flexible shape

20. 2-parameter VEI fits
Monte Carlo simulation/refit: Parameters a,b may be common to all
-- non-hierarchical model

21. Why such variability in a,b?
The majority of the information is contained in the ratio b/a

22. 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

23. 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

24. 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 (PL) or (2param) per 10 yr -> 0.7 to 5 % increase in volume / yr of repose
No correlation with time-predictability or susceptibility to earthquake triggering