Sarah Minson Mark Simons James Beck

TeleseismicStrong motionJoint km Delouis et al. (2009) Loveless et al. (2010) Seismic + Static

For inverse problems: PosteriorPosteriorPriorPriorDataLikelihoodDataLikelihood

OptimizationBayesian RegularizedNo a priori regularization required One solutionDistribution of solutions Converges to one minimumMulti-peaked solution spaces OK Limited choice of a priori constraintsGeneralized a priori constraints Error analysis hard for nonlinear problemsError analysis comes free with solution Computational efficiency is sensitive to model parameterization (model covariance leads to trade-offs) Computational efficiency is insensitive to model parameterization (if model covariance is estimated)

Huge numbers of samples required for high- dimensional problems “Curse of Dimensionality” Sampling can be inefficient

Tempering (A.K.A. Annealing) * Dynamic cooling schedule ** Resampling ** Simulation adapts to model covariance ** Simulation adapts to rejection rate *** Parallel Metropolis Cascading * Marinari and Parisi (1992) ** Ching and Chen (2007) *** Matt Muto

Target distribution:

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

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

Static GPS displacements 1 Hz GPS time series 6 interferograms

For one of our fault patches Static Posterior/ Kinematic Prior Static Posterior/ Kinematic Prior Kinematic Posterior Static Prior

StaticStaticKinematicKinematic

Fully Bayesian finite fault earthquake source modeling Resolution of the slip distribution and rupture propagation Uncertainties on derived source properties Determine which source characteristics are constrained and which are not CATMIP allows sampling high-dimensional problems Also useful for low-dimension problems with expensive forward models Wide variety of potential uses in geophysics

U┴U┴U┴U┴ U┴U┴U┴U┴ U║U║U║U║ U║U║U║U║ τrτrτrτr τrτrτrτr VrVrVrVr VrVrVrVr

VsVs VsVs