Featuring contributions from… Mark Simons James L. Beck Junle Jiang Francisco H. Ortega Asaf Inbal Susan E. Owen Anthony Sladen

Embrace Uncertainty  One of the geophysical community’s failures before the Tohoku earthquake was not emphasizing what we don’t know. To whatever extent the Tohoku earthquake was unexpected, it was because the uncertainties of the rupture history and plate coupling were not understood To avoid falling into this trap, we adopt a Bayesian approach to kinematic rupture modeling

Bayes’ Theorem (1763)  For inverse problems: PosteriorPriorDataLikelihood

Why choose Bayesian analysis? OptimizationBayesian One solutionDistribution of solutions Converges to one minimumMulti-peaked solution spaces OK Regularized (lots of decisions)No a priori regularization required Limited choice of a priori constraintsGeneralized a priori constraints Error analysis hard for nonlinear problems Error analysis comes free with solution Sensitive to model parameterization (model covariance leads to trade-offs) Insensitive to model parameterization (if model covariance is estimated)

Why not choose Bayesian analysis?  Huge numbers of samples required for high- dimensional problems “Curse of Dimensionality”  Requires evaluation of billions of forward models NASA Pleiades supercomputer

Cascading Adaptive Transitional Metropolis In Parallel: CATMIP  Transitioning AKA Tempering AKA Simulated Annealing * Dynamic cooling schedule **  Parallel Metropolis  Simulation adapts to model covariance **  Simulation adapts to rejection rate ***  Resampling **  Cascading * Marinari and Parisi (1992) ** Ching and Chen (2007) *** Matt Muto StaticKinematic

Example: Mixture of Gaussians  Target distribution:

CATMIP 1. Sample P( θ ) 2. Calculate β 3. Resample 4. Metropolis algorithm in parallel 5. Collect final samples 6. Go back to Step 2, lather, rinse, and repeat until cooling is achieved

CATMIP

Model Prediction Errors  Assumed errors control posterior model errors  For geophysical data, ε D is often small Seismograms are accurate, InSAR phases are accurate  For slip models, errors on ε G(θ) can be large Percent error on G scales with source Seismograms have background noise, InSAR has atmospheric blobs ○ These “data errors” represent unmodeled signals  If ε=ε D is assumed, total error is under-estimated

 We can sample for ε G(θ) just like any other variable

Black is “data” Red is same slip model with half-space GF

 Posterior mean with model prediction error is not better than mean without prediction error Bad model = can’t recover true solution  But the posterior 95% confidence with prediction error more accurately represents the real uncertainty

Methodology  Choose a finite discretized planar fault model (strike constant, dip increasing from 3°-29°), hypocenter and elastic structure This is our model design  Broad priors  Use Bayesian inference to determine posterior PDFs of slip distributions which fit available static data (GPS, DART, seafloor pressure gauges, seafloor geodesy)  Use posterior PDF as prior for joint static- kinematic source rupture model using 1 Hz GPS 1-D Green’s functions STF is triangle Each patch can only rupture once Rupture is causal (calculated via Fast Sweeping algorithm) Assume:

Average posterior slip

Posterior mean (one statistic of many) Slip duration Rupture velocity

Tsunami data

1 Hz GPS East North

Consistent with previously published preliminary solution Kinematic solution Simons et al. (Science, 2011)

All models Slip is less well constrained as rupture evolves But total moment magnitude is well constrained

One sigma Slip is less well constrained as rupture evolves But total moment magnitude is well constrained

Lay and Kanamori (2011) Posterior moment rate

Along-strike integrated slip  We fit both GPS, seafloor observations, and both near- field and far-field tsunami observations  Both our preliminary model and this model find significant near-trench fault slip but, on average, peak slip is not at the trench  This decrease in slip amplitude near trench is recoverable because the model is not regularized  However, localized zones of high slip may exist near trench

Conclusions  Fully Bayesian kinematic rupture model for the Tohoku-Oki earthquake Fast preliminary slip model is consistent with full kinematic solution Significant slip near trench, but peak large- scale slip feature is not at trench  A wide range of data are very well-fit Model prediction error is essential to estimating uncertainty in solution ○ Uncertainty in slip model is essential to understanding the subduction zone

Earthquake monitoring using GPS  Dense real-time 1 Hz GPS network  Utilize GPS with seismic data to identify and analyze events for which GPS data contribute the most Hazardous earthquakes Slow slip events

Rapid response vs. early warning  Rapid response Real-time event detection On-demand source analysis  Real-time Bayesian magnitude determination for EEW Proper alarm criterion is “Mw > threshold with x% confidence”