Longterm forecasting of volcanic explosivity Mark Bebbington IFSStatistics
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 • • 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) ? 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 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 = 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 1983 present, ~4 km 3) 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 • • 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(q 1 ) > 0 = 0. 867 VEI increases with time P(q 2 ) > 0 = 0. 833 VEI increases with repose (open conduit) P(q 3 ) > 0 = 0. 999 VEI increases with repose (closed conduit) P(q 4 ) > 0 = 0. 439 VEI independent of previous VEI P(q 5 ) > 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 ∝ 10 VEI, then observed ΔVEI of 0. 03 --0. 22 (PL) or 0. 04 --0. 18 (2 param) per 10 yr -> 0. 7 to 5 % increase in volume / yr of repose • No correlation with time-predictability or susceptibility to earthquake triggering
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