Assessing Spatio-Temporal Eruption Forecasts in a Monogenetic Volcanic Field

Assessing spatio-temporal eruption forecasts in a monogenetic volcanic field

Mark Bebbington Volcanic Risk Solutions, Massey University Spatio-temporal hazard estimates Monogenetic volcanic fields have multiple volcanoes, each of which erupts once. A new eruption creates a new volcano.

Unfortunately, people will build cities on them ...

For land-use and emergency planning purposes: when and where is the next eruption likely to be?

In short time frame, answered(?) by monitoring What about in a period of repose? data (seismicity, gas, ...)

Probability forecast: estimate the hazard λ(t,x) such that the probability of an event in the time-space window (t,t+ ∆t) x (y:||y-x|| < ∆x) ~ λ(t,x) ∆t ∆x

Look for temporal and spatial We know the spatial patterns (events are more likely to locations of (most) vents occur `near’ previous events?) The Auckland Volcanic Field

50(?) small basaltic volcanoes young (~250,000 years) Most recent eruption ~600 years ago

Data: • Stratigraphy, ~33 vents constrained in at least one direction • Age determinations • Paleomagnetism ~5+ vents • C14, ~13 vents • Tephrostratigraphy, 24+ tephra in 5 locations • Ar-Ar, ~4 vents • Thermoluminesence, 2 vents reliability reliability Decreasing Decreasing • K-Ar, unreliable due to excess Ar • Relative geomorphology or weathering

Also: known vent locations, reasonable volume data (Allen and Smith, 1994) A Monte Carlo sample of age-orderings

Name Mean Age Age Error Min Max (ka) (ka) Order Order By reverse engineering the Onepoto Basin 248.4 27.8 1 7 Albert Park 229.8 39.5 1 7 tephra dispersal, Bebbington ...... and Cronin (2011, 2012) St Heliers 185.0 52.8 2 9 Te Pouhawaiki 152.9 70.3 1 34 constructed an algorithm to ...... produce feasible age- Mt St John 54.8 4.6 10 13 Maungataketake 41.4 0.4 13 15 orderings. Otuataua 41.4 0.4 14 16 Laschamp McLennan Hills 40.1 1.2 13 16 magnetic excursion One Tree Hill 34.9 0.7 16 18 ...... This has no apparent spatio- Hopua Basin 32.3 0.4 19 26 Puketutu 31.9 0.3 22 27 temporal structure , although Wiri Mountain 31.9 0.3 21 28 there is plenty of both temporal Mt Richmond 31.7 0.3 21 28 and spatial structure Taylors Hill 31.7 0.3 21 28 Crater Hill 31.6 0.3 23 28 Mono Lake magnetic North Head 31.2 0.1 27 29 excursion ...... Spatio-temporal point process models

Inversely weight spatial density, by time elapsed (Connor and Hill 1995):

Let ri = distance from x to ith centre, ti = occurrence time of ith centre. Then m λ(x,t) = m πr2 (t − t ) ∑ j= 1 j j where the sum is over the m nearest neighbours according to the π 2 distance metric ri (t- ti).

Alternatively, use independent spatial and temporal terms: λ(x,t) = ν (t) η (x) Temporal Intensities

Poisson process: , ν = ν (t) 0 Excitation process (Bebbington and Cronin 2011): 2 υ ()t− t  ν = µ +2  − j  (t)∑ exp 2 σ π< 2 σ  j:tj t  

Weibull renewal process (Bebbington and Lai 1996): α− ν = αβα − < 1 (t)( t maxt:t{}j j t )

.... Spatial Intensities 1: Ellipses

Minimum Area Ellipse (Sporli and , Eastwood 1997):

Equation of ellipse: (x− c)T A(x − c) = 1 - Find A,c by method of Khachiyan (1996)

Problem: zero likelihood outside boundary. So

2η / 3, x :(x − c)T A(x −≤ c) 1 η(x) =  0 η <−T −≤  0 / 3, x :1 (x c) A(x c) 2

Inner and where the area of the ellipse = 1/ η0 ,or outer ellipses  2η / 3, x :(x −−≤ c)T A(x c ) 1 η =  0 Ellipse plus (x)  − −T −−  ()η2 (x c) A(x c) 1  <−−T exponential decay  20 /3e , x:1(x c)A(x c) (with distance) Spatial Intensities 2: Kernel Densities

Isotropic Gaussian kernel: , 2 1n 1 x− x  η(x) = exp − i  ∑ π 2 2  ni= 1 2 h 2h  Anisotropic Gaussian kernel: n 1 1 T η= −−()−1 − (x)∑ exp( 0.5 x xi H (x x i ) ) n i= 1 2π H Nearest Neighbour Gaussian kernel (spatially varying bandwidth) 2 1m 1 x− x  η = − i  (x)∑ 2 exp 2 m= 2π d 2d  i 1 m m 

Where d m is the distance to the mth nearest centre.

Bandwidths are estimated by least squares cross-validation (LSCV) and, in the case of the anistropic kernel by SAMSE (Duong and Hazelton 2003). Description v. prediction: Are the forecasts any good?

“I can call spirits from the vasty deep.”

“Why, so can I, or so can any man; But will they come when you do call for them?”

Henry The Fourth, Part I Act 3, scene 1, 52–58

Methods such as residual analysis exist (Schoenberg, 2003) to determine if a spatio-temporal model is a good description of the data to which it is fitted. But this is a different question to whether it predicts future data. Forward likelihood-based predictive criterion

, (Chiodi and Adelfio, 2011):

Score models on the likelihood of the ( i+1)th event, using the model fitted to the first i events.

Sum over all events after some initial number k

n− 1  t+1  =λ ≤−λ ≤  FLP∑ log() xi1i1jj+ ,t + (x,t),j i∫ ∫ () x,t(x,t),j jj idxdt  i= k  ti X 

Here k = 17 (out of 51) Results Over 1000 Monte Carlo age-order, realizations:

-217.22 -272.29 = -489.51 Spatio-Temporal logL

, m=9 | m=17 No. of nearest neighbours

m=i/4 | m=i/2

Problem: Spatial component (integrates to 1) is not identifiable. So where is boundary? Example: Mt Eden location Gaussian (LSCV) ,

Elliptical (LSCV)

Elliptical (SAMSE)

Note: Intensity integrates to 1 Kernel parameter estimates

, Spatial `regimes’? Forecasts within a regime much more accurate...

i = 22 i = 23 i = 35

i = 36 Conclusions

• Forecasting ‘skill’ of competing spatio-temporal, hazard models assessed on monogenetic volcanic fields.

• The Auckland Volcanic Field (elliptical boundary, highly variable temporal rate) best modelled by: 1. independent flat spatial and adaptive temporal terms, or 2. spatio-temporal model with hyperbolic decay in both space and time.

• Possible structure in the temporal evolution of the spatial bandwidth • during steady-state periods, more elaborate spatial models outperform uniform models • this is outweighed by penalties for inaccurate forecasts coinciding with changes in the pattern. • can this be harnessed to produce more accurate spatial forecasts? References

• Allen SR, Smith IEM (1994) Eruption styles and volc, anic hazard in the Auckland Volcanic Field, New Zealand. Geosci Rep Shizuoka Uni 20: 5-14. • Bebbington M, Cronin SJ (2011) Spatio-temporal hazard estimation in the Auckland Volcanic Field, New Zealand, with a new event-order model. Bull Volcanol 73: 55-72 • Bebbington M, Cronin SJ (2012) Paleomagnetic and geological updates to an event-order model for the Auckland Volcanic Field. In: Proc. 4th International Maar Conference, Geoscience Soc. of New Zealand Miscellaneous Publication 131A, pp. 5-6. • Bebbington MS, Lai CD (1996) On nonhomogeneous models for volcanic eruptions. Math Geol 28: 585-600. • Chiodi M, Adelfio G (2011) Forward likelihood-based predictive approach for space-time point processes. Environmetrics 22: 749 -757. • Connor CB, Hill BE (1995) Three nonhomogeneous Poisson models for the probability of basaltic volcanism: application to the Yucca Mountain region, Nevada. J Geophys Res 100: 10,107-10,125. • Duong T, Hazelton ML (2003) Plug-in bandwidth selectors for bivariate kernel density estimation. J Nonparametric Statist 15: 17-30. • Khachiyan LG (1996) Rounding of polytopes in the real number model of computation. Math Oper Res 21: 307–320. • Schoenberg FP (2003) Multidimensional residual analysis of point process models for earthquake occurrence. J Amer Statist Assoc 98: 789-795. • Sporli K, Eastwood VR (1997) Elliptical boundary of an intraplate volcanic field, Auckland, New Zealand. J Volcanol Geotherm Res 79: 169-179.