1. From geodesy to magma physics Bayesian joint inversions of diverse observations using multiphysical volcano models Kyle Anderson kranderson@usgs.gov ?

2. Volcanoes are complex: fundamental questions remain Pavlov Volcano (March 27, 2016) What triggers eruptions? What drives effusive/explosive transitions, or cyclic behavior? What is the balance between rates of supply and eruption? How much magma is stored beneath volcanoes? etc. etc. etc.

3. X Volume of intruded or removed magma X Absolute volume, pressure, properties of the magma X Time-evolution / forecasting X Relationship to other types of observations A magma reservoir resolved by ground deformation ΔVΔV MAGMA PHYSICS Locations, geometries, volume changes Even with perfect data and a perfect model, geodetic data alone can only get us so far. To go further, we need magma physics.

4. Fluid-dynamical, physics-based volcano models fd Conservation of mass and momentum; constitutive laws; often highly nonlinear Often computationally expensive Directly predict relatively few observables

5. Magma/eruption physics are the glue holding the data together Deformation Pressures/tractions MAGMA PHYSICS Kawaguchi and Nishimura 2015 Relatively well understood problem We have lots of simple analytical and sophisticated numerical models Much more complex if deformation influences magma flow

6. Magma/eruption physics are the glue holding the data together Deformation Gas emissions Gravity changes Petrology/ Geochemistry Seismicity Eruption rate Flux Stress changes in host rock Chemical/physical properties Volatile content and flux Mass/density Pressures/tractions MAGMA PHYSICS The Inverse Some links are more difficult than others

7. GOAL: Estimate model parameter values consistent with observed data, independent a priori information, and uncertainties The inverse: constraint by both data and prior information PRIOR INFORMATION Directly constrain model parameters Results of previous studies As objective as possible, but may have to deal with conflicting information DATA Anything that the model can predict (ground deformation, gas emissions...) MODEL DATA PRIOR: Direcly constrains model parameter m = [m 1, m 2,... m n ] PRIOR This is no longer a geodesy problem! This is a volcano inverse problem constrained in key ways by geodetic data.

8. Formulation of the inverse problem using Bayes’ Theorem Bayes’ Theorem Prior Information Posterior PDF Data Likelihood m : model vector; d : data vector; G: forward model; P : probability; r : residual; K : number of independent data sets; : data covariance matrix Model Parameters Forward Model P is a function of the fit of model predictions to data, and the consistency of those model parameters with what we already know about them Posterior PDFs evaluated (sampled) using a Markov Chain Monte Carlo (MCMC) algorithm; this requires running the forward model millions of times

9. Previous work: Mount St. Helens (2004—2008) Dome-forming eruption of ~100 Mm 3 Co-eruptive reservoir recharge? Properties of reservoir magma? Volume of magma storage?

10. Model: dome-forming eruption Geometry Predicted Observable Model Parameter Nonlinear differential equations solved numerically in ~1-2 seconds Anderson and Segall (2011, 2013), JGR; c.f. Mastin (2008) Multiphase ascent of magma from a deflating reservoir through a conduit to the surface. Parameterized at t=0. Ground deformation by coupling magma pressures and tractions to pre-computed finite-element solutions; erupted volume by integrating flowrate

11. Kinematic geodetic inversion vs. multiphysical approach Kinematic model using only net GPS displacements Physics-based model using erupted volume time series and full time-evolution of GPS data Influx into chamber: likely below 0.01 m 3 /sec Melt water content! Anderson and Segall (2013), JGR XXXX Chamber volume

12. Bayesian, multiphysical modeling at Kīlauea Volcano (Hawaii) (1) Magma supply rate (MSR) drives eruptive hazard. Evidence that Kilauea’s MSR waxes and wanes – but deformation data alone cannot yield MSR. (2) Episodic deformation indicates depressurization cycles. Geodetic data yield reservoir location – but what can we say about reservior volume or magma composition? Ground tilt 11 days 0

13. A simple multiphysical model of Kīlauea Volcano to estimate MSR Summit East Rift Zone SO 2 rate Ground deformation Lava effusion rate Previous GPS inversions Reservoir location (x,y,z), magma storage rate (Q s ) Primary S (χ S ) Magma storage rate in deep rift (Q r ) Lava effusion rate (Q o ), residual sulfur in lava (χ O S ) Magma density (ρ), compressibility (β) Magma supply rate (Q i ) Steady-state mass conservation (magma, sulfur) Summit ground deformation computed from storage rate and compressibility Rate of magma storage in the deep rift related to geodetic models of rift opening rate Gas emissions related to magma flow rates and sulfur composition

14. What do we know? Data and priors ENVISAT and Cosmo-SkyMed interferograms at the summit Results of a GPS+leveling study of summit deformation (Cervelli and Miklius, 2003) Results of previous geodetic studies constraining the rate of magma storage in the deep rift (e.g., Owen et al. 2000) TanDEM-X eruption rate (Poland, 2014) SO 2 emissions (Elias and Sutton, 2007) Lots of independent a priori information to constrain magma composition 0.08 wt%; Moore and Fabbi [1971] 0.12 wt%; Anderson and Brown [1983] 0.11 wt%; Fornari et al. [1979] 0.11 wt%; Garcia et al. [1989] 0.13 wt%; Harris and Anderson [1983] 0.09 wt%; Greenland et al. [1985] 0.09 wt%; Dixon et al. [1991] 0.10 wt%; Clague et al. [1995] ENVISAT (2006)

15. Fit to geodetic, gas, TanDEM-X data 20062012 Data Model

16. Estimated magma supply rates 2001 2006 2012 2001-2012 2001-2006 2006-2012 2001- 2012 2001- 2006 Significant sub-decadal changes in MSR Surge in 2006, then reduction to 2012 (waning of eruption, response to 2006 surge, or random fluctuation?) Magma Supply Rate (km 3 /yr) Probability Probability of exceedance Relative change in supply rate 2006- 2012 Probability 0 0.1 0.2 0.3 0.4 Change in Magma Supply Rate -100% -50% 0% 50% 100% 150% 1 0.75 0.50 0.25 0

17. Episodic depressurization transients at Kīlauea Here’s what we observe Here’s what we get from geodesy only Anderson et al., (2015)

18. But these events appear in more than just geodetic data Eruption pause height Summit ground tilt Lava lake surface height Patrick et al (2015), Geology Poland (2014), JGR

19. A coupled, time-dependent model Simple time-dependent forward model (elastic balloon) Data: Ground tilt, eruption rate, lava lake height, zeroflux Priors: Active lava lake, in magmastatic equilibrium with chamber Eruptive flux, at rate that varies with summit pressure Magma supply, reduced during deflations Reservoir, filled with compressible magma Time Pressure Model Data Chamber volume (km 3 ) log(Magma compressibility) Conduit radius (m) Preliminary Results

20. Many advantages to relating volcano deformation data to magma physics, modeling with other observations, and inverting using a Bayesian approach together with independent prior information An approach to understanding volcanic systems: Bayesian inversions using multiphysical (physics-based) models Advantages/opportunities Quantitatively link geodesy with other disciplines Use all available information Uncertainty quantification Improved constraint on fundamental properties of volcanic systems Much more involved and time- consuming than a kinematic inversion Still difficult to quantitatively predict some observations; models must be relatively fast Magma physics can be poorly understood; how to account for this ‘model error’? Challenges/disadvantages

21. This is the last slide (really) Mount St. Helens: Large reservoir of tens of km 3 Kīlauea: Shallow reservoir with a volume of ~1 km 3 ; magma supply rate varied from ~4 to ~7 to ~3 m 3 /s in 2001, 2006, and 2012 Anderson, K. and Michael Poland (in review), Bayesian estimation of magma supply, storage, and eruption rates using a multiphysical volcano model: Kīlauea Volcano, 2000—2012. EPSL Anderson, K., Matthew Patrick, Michael Poland, and Asta Miklius (2015), Time-variable magma pressure at Kīlauea Volcano yields constraint on the volume and volatile content of shallow magma storage, AGU Fall Meeting Anderson, K. and Paul Segall (2013), Bayesian inversion of data from effusive volcanic eruptions using physics-based models: Application to Mount St. Helens 2004–2008. JGR Anderson, K. and Paul Segall (2011), Physics‐based models of ground deformation and extrusion rate at effusively erupting volcanoes, JGR. References and co-authors:

22. Time-variable rate of magma supply to Kīlauea Volcano Supply rate = storage rate + eruption rate Lots of data and prior constraints, all associated with uncertainties. How to account for them in estimated MSR? From geodetic data, IF we know magma density and compressibility. Storage in volcanic rift zones may be difficult to quantify. Can infer from gas emissions if magma volatile composition is known. Radar-derived flow field volumes are fantastic, but rarely available. Magma supply rate (MSR) dominantly controls volcanic activity but must be inferred indirectly:

23. Some equations Gas emissions Volume change rate of reservoirs Summit deformation Conservation of mass