Statistics in Volcanology: Uncertainty in Volcanology Data and Models

Transcription

1 : Data and Models School of Geosciences University of South Florida February 2017

2 Volcanic s... Eruption of Shinmoe-dake volcano, Kirishima volcano complex, Japan.

3 Major research questions Tolbachik, Kamchatka, Russia 2013 Forecast the onset, size, duration and hazard of s by integrating observations with quantitative models of magma dynamics. Quantify the life cycles of volcanoes globally and overcome our biased understanding. Develop a coordinated volcano science community to maximize scientific returns from any volcanic event.

4 How volcanologists learn about volcanoes... Our schema dictates that we come upon key science questions with a set of prejudgments: an idea of what the problem is, what type of information we are looking for, and what will count as an answer. See Bob Frodeman, 2014 Hermeneutics in the field: The philosophy of geology.

5 Old volcanism on Mars distribution of vents and lava flows in the crater of

6 Overlapping lava flows show age relationships geologists classify images and interpret the relative ages Steno s Law

7 A directed graph of age relationships Ages estimated (with high uncertainty!) from crater density

8 MC simulation of event rate Randomly sample ages of all events using directed graph (M = times), Volcano i of total N formed by event ê i, For each set of age estimates, j, for N volcanoes, the cumulative distribution is: N X j (T ) = P [ê i,j, t < T ] i=1 where P [ê i,j, t < T ] = 0 if T < ê i,j and P [ê i,j, t < T ] = 1 if T ê i,j E(X) = 1 M M X j (T ) j=1 R(X) = E(X) t

9 MC simulation of event rate Based on Monte Carlo simulation using age estimates and stratigraphic information

10 Estimated distribution of event rate Age distribution of events improved by using directed graph with Monte Carlo simulation

11 stops air traffic from North america to Europe...at a cost of 1 billion euros. How often does this happen?

12 Volcanic ash preserved in bogs

13 Estimated rate of known events At least the known events are stationary in time

14 Models suggest 44 ± 7 yr Kaplan Meier estimate of the survivor function using data from last 1000 yr with fits for various statistical distributions (Weibull, log-logistic, exponential).

15 Monte Carlo simulation of longer data set Activity seems to cluster in time over last 7000 yr, average over last 1000 yr may not be representative of true uncertainty.

16 Missing events in the geologic record Kiyosugi et al., 2015, How many explosive s are missing from the geologic record? Analysis of the quaternary record of large magnitude explosive s in Japan, Journal of Applied

17 Missing events in the geologic record Kiyosugi et al., 2015, How many explosive s are missing from the geologic record? Analysis of the quaternary record of large magnitude explosive s in Japan, Journal of Applied

18 Tephra sedimentation model Model goal: Estimate the mass erupted Forward problem: Estimate the accumulation of tephra expected, given volcanic activity. Inverse problem: Given a tephra deposit, what were the parameters that produced this deposit? Tolbachik volcano, Russia, 1975

19 Tephra sedimentation model

20 Tephra sedimentation model Model algorithm Tephra2 Model Basics the implementation

21 Tephra sedimentation model Advection diffusion equation Single partial differential equation expresses tephra sedimentation C j t + w C j x x + w C j y y v C j l,j z = K 2 C j x 2 Expressed dimensionally: + K 2 C j y 2 M L 3 T + L M T L 4 + L M T L 4 L M T L 4 = L2 M T L 5 + L2 M T L 5 + M L 3 T + Φ

22 Tephra sedimentation model ADE Closed form Eulerian solution to the Advection-Diffusion equation (Suzuki, Macedonio, Lim) where and f i,j (x, y) = 1 2πσi,j 2 exp [ H i x i,j = x 0 + (x x i,j) 2 + (y ȳ i,j ) 2 2σ 2 i,j k=0 H i ȳ i,j = y 0 + k=0 w x,k z k v j, k w y,k z k v j, k ]

23 Tephra sedimentation model ADE Variable Reynold s number for particle settling (Bonadonna et al., 1998) ρ j gd 2 j 18µ if laminar, Re < 6, [ 4g v j = d 2 ρ 2 j j if intermediate, 6 Re < 500, [ 3.1ρj gd j 225µρ a ] 1/3 ρ a ] 1/2 if turbulent, Re 500,

24 Tephra sedimentation model Plume Geometry σ 2 i,j = t i,j = H i k=0 z k v j t i = [ 0.2h 2 ] 2/5 i { 4K(t i,j + t i ) if t i,j < τ, 8C 5 (t i,j + t i )5/2 if t i,j τ,

25 Tephra sedimentation model Example forward model results Maximum Column Height: m Total Mass: kg Median Grain Size: 0 φ STD Grain Size: 1 φ Wind from NOAA reanalysis

26 Kirishima 2011 PEST Model Inversion try modeling 1992 stages (A and B), and using grain size data collected from each sample pit Use the PEST inversion method to interpret the 2011 Kirishima source parameters: Singular value decomposition with Tikhonov regularization Bayesian procedure - specify prior information and output pdf of parameter model Using the open source, open access PEST code (Parameter Estimation, SVD, and Tikhonov)

27 PEST Model Inversion

28 PEST Model Inversion

29 PEST Model Inversion Prior (dashed) versus posterior (shaded) parameter estimates

30 PEST Model Inversion source term parameters seems to be reduced using the PEST inversion and Tephra2 forward model.

31 A logic tree for forecasts

32 Volcano science is about using observations (data) to improve our models of the timing of volcanic s, their magnitudes, and potential impacts. High uncertainty because volcanoes are difficult to observe, erupt infrequently and exhibit a huge range of behaviors. Great opportunity for the application of statistical methods in interpretation of past events and forecasting future events.

33 PEST Model Inversion estimated model

34 PEST Model Inversion model residuals

35 PEST Model Inversion estimated parameters

36 Missing events in the geologic record Kiyosugi et al., 2015, How many explosive s are missing from the geologic record? Analysis of the quaternary record of large magnitude explosive s in Japan, Journal of Applied

37 A Volcanic Event Age Model (VEAM) Age estimate of one lava flow in the Cima Volcanic field Wilson, Richardson and others

38 A Volcanic Event Model (VERRM) An age model for volcanic events in the Cima Volcanic Field Wilson, Richardson and others

39 A Volcanic Event Model (VERRM) A reccurrence rate model for the Cima volcanic field, emphasizing uncertainty in current event rates Wilson, Richardson and others