A Bayesian Approach to Earthquake Source Studies

Sarah Minson

Caltech

February 22, 2011LLNL

The people who make this possible

Mark Simons (Caltech)James Beck (Caltech)

The big picture questions (or what I won’t be presenting)

Things we want to knowDo regions of co-seismic slip overlap with areas of post-seismic or inter-seismic slip?How do hypocenter locations relate to co-/post-/inter-seismic slip?Do earthquakes have smooth slip distributions and short slip durations? Vice versa?Do earthquakes rupture at super-shear velocities?

The answers to these questions come from finite fault earthquake source models…

2007 Mw 7.7Tocopilla,Chile

Teleseismic Strong motion Joint

km

Delouis et al. (2009)

Loveless et al. (2010)

Seismic + Static

Motivation

Seismology:To get to earthquake physics, we need better source models

Math:Many geophysical problems are under-determined

Ill-posed inverse problems require regularizationRoughness of the slip distribution (or whatever quantity is being regularized) cannot be identified because it was set a piori.

What if you didn’t have to evaluate the inverse problem?

Bayes’ Theorem (1763)

For inverse problems:

Model

Data

)|P()P()|P(

D

DD

Posterior PriorData

Likelihood

GdCGd dT

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2

1

Advantages of Bayesian analysisOptimization Bayesian

One solution Distribution of solutions

Converges to one minimum Multi-peaked solution spaces OK

Regularization may be required No a priori regularization required

Limited choice of a priori constraints Generalized 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)

Problems

Calculating posterior PDF generally requires Monte Carlo simulation

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

Solution

Efficient parallel sampling algorithmMetropolis algorithm and Markov Chain Monte Carlo (MCMC) are serial

Must efficiently sample highly anti-correlated model parametersMust share information between worker CPUs

Cascading Adaptive Tempered Metropolis In Parallel: CATMIP

Tempering (A.K.A. Simulated Annealing)*

Dynamic cooling schedule**

Resampling**

Parallel MetropolisSimulation adapts to model covariance** Simulation adapts to rejection rate***

Cascading

* Marinari and Parisi (1992)** Ching and Chen (2007)*** Matt Muto

10)|P()P()|P( DD

10,1

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),|P()|P()P()P()|P(

kskssks DDD

CATMIP

1. Sample P(θ)2. Calculate β3. Resample4. Metropolis algorithm in parallel5. Collect final samples6. Go back to Step 2, lather, rinse, and repeat until

cooling is achieved

CATMIP

1. Sample P(θ)2. Calculate β3. Resample4. Metropolis algorithm in parallel5. Collect final samples6. Go back to Step 2, lather, rinse, and repeat until

cooling is achieved

Target

CATMIP Target

Marginal PDF based on 400,000 samples of a 10-dimensional mixture of Gaussians

The seismic model

For each patch on a regularly gridded fault plane

Two components of slipRise timeRupture velocity

Also may solve for ramp components of InSARTotal number of parameters:

4*Npatches (+ ramp)

Prior distribution

SlipRotated coordinate system relative to teleseismic rake direction

Slip parallel to rake has a positivity constraintForbidden from back-slipping more than 1 m

Prior distribution on slip perpendicular to rake is a zero-mean GaussianPrior distribution is populated from Dirichlet distribution

Prior distribution on rupture velocity and rise time is uniform

Forward model for static data

dpredicted=G*slip

Forward model for kinematic dataGreen’s functions are pre-computed

These are then convolved with all possible source time durations and stored in memory

Convolution is too slow to be done at evaluation timeFinal predicted waveforms are linear combination of each point source (with the appropriate pre-convolved source-time functions) scaled by slip, time-shifted according to results of Fast Sweeping

Fast Sweeping Algorithm (Zhao, 2005) is used to calculate initial rupture time at each source from Vr on each patch

Source parameters from each patch are interpolated onto a fine grid of point sources

Computation

Run parameters:Markov chains = 500,000Steps per Markov chain = 1002001 cores on CITerra Beowulf cluster

Results:Cooling steps = 39

Forward model evaluations = 1.931 billionRun time = 10.6 hoursTotal computation time = 76.3 million CPU seconds

How parallel is it?

Total computation time = 76.3 million CPU secondsEach model evaluation = 0.035 sec

Static forward model = 0.005 secKinematic forward model = 0.03 sec

Total model evaluation time = 68 million CPU seconds~90% of time spent on parallel model evaluation

In the future…

Caltech Center for Advanced Computing Research (CACR) is working on a GPU version of CATMIP

Michael AivazisMartin Michaelson

As of now, they have a first cut version with partial utilization of GPU

~35x speed-up relative to CPU versionNeed to move more of code onto GPU

Mw 7.7 Tocopilla, Chile Earthquake

The Data

Static GPS displacements1 Hz GPS time series6 interferograms

Cascading Static Posterior/Kinematic Prior

Kinematic Posterior

Static Prior

Static vs. Joint Slip Distributions

Rise Time vs. Rupture Velocity

InSAR fits

GPS fits

Source parameters (now with uncertainties)

SourceProperties

Peak Slip: 3.015 ± 0.2733 m

Slip Heterogeneity

SmootherRougher

Visualization

How do you represent an N-dimensional PDF?People like looking at slip models…

…But individual models can be very misleading…

Confidence bounds

Individual models (bad)Mean

MAP

Median

Even the mean and median tend to accentuate roughness…

SummaryCATMIP algorithm allows the sampling of very high-dimensional problems

Also useful for low-dimension problems with expensive forward modelsWide variety of uses in geophysics

Fully Bayesian finite fault earthquake source modeling

Resolution of the slip distribution and rupture propagationUncertainties on derived source propertiesDetermine which source characteristics are constrained and which are not

Future workData errors and model prediction errors are importantAssumed data errors control posterior model errors

Posterior distribution is also affected by model prediction errors (the failings of the forward model)

The next step is to estimate these errors

GG2

1 1

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Thank you!