Assimilating Data into Earthquake Assimilating Data into Earthquake SimulationsSimulations

Michael Sachs, J.B. Rundle, D.L. TurcotteMichael Sachs, J.B. Rundle, D.L. TurcotteUniversity of California, DavisUniversity of California, Davis

Andrea DonnellanAndrea DonnellanJet Propulsion LaboratoryJet Propulsion Laboratory

y j f (zk ) j

Data = Model + Errors

Data Assimilation, Model Steering, Model TuningData Assimilation, Model Steering, Model Tuning

Linearize

y j k

A jkzk j

Matrix of Partial Derivatives

A jk

ApplicationsApplications

Climate ModelsNumerical weather forecastingGlobal Ocean Data Assimilation Experiment

http://www.usgodae.org/DART (NCAR): Community DAta & Research Testbed)

http://www.image.ucar.edu/DAReS/DART/Land Data Assimilation Systems

http://ldas.gsfc.nasa.gov/Satellite Data Assimilation

http://www.jcsda.noaa.gov/Snow Data Assimilation System

http://nsidc.org/data/g02158.htmlOrbit correctionFinancial markets and trading modelsEconometricsEngineering control systems

Kalman Filter From Wikipedia

Kalman Filter From Wikipedia

Kalman FilterSolution

Kalman FilterSolution

SCIDAC Tools for Computation, Data Assimilation, & Model SteeringSCIDAC Tools for Computation, Data Assimilation, & Model Steering

Data Assimilation and Model Steering UsingAutomated Numerical Differentiators (e.g. ADIFOR -- FORTRAN)

http://www.mcs.anl.gov/research/projects/adifor/

Data Assimilation and Model Steering UsingAutomated Numerical Differentiators (e.g. ADIFOR -- FORTRAN)

http://www.mcs.anl.gov/research/projects/adifor/

Data Assimilation and Model Steering UsingAutomated Numerical Differentiators (e.g. ADIC – ANSI C)

http://www.mcs.anl.gov/research/projects/adic/

Data Assimilation and Model Steering UsingAutomated Numerical Differentiators (e.g. ADIC – ANSI C)

http://www.mcs.anl.gov/research/projects/adic/

9

Other Standard MethodsOther Standard MethodsOther Standard MethodsOther Standard Methods

1. Linear Programming

2. Simulated Annealing

3. Genetic Algorithms & Evolutionary Programming

4. Monte Carlo Search

5. Simulations + Data Scoring

Data Assimilation via ScoringData Assimilation via ScoringData Assimilation via ScoringData Assimilation via Scoring

MethodCompare Virtual California simulation data with

historical seismic recordPick simulation times whose

history is most similar to the historic data

Use “future simulation times”to generate probabilities offuture large events.

J. Van Aalsburg et al., PEPI, 163, 149 (2007)J. Van Aalsburg et al., PAGEOPH, 167, 967 (2010)

Data SetsData Sets

Virtual California768 fault boundary elements in model

1.5 million events

200,000 years

Paleoseismic Data119 events

20 sites

J. Van Aalsburg et al., PEPI, 163, 149 (2007)J. Van Aalsburg et al., PAGEOPH, 167, 967 (2010)

Assimilation (Scoring) AlgorithmAssimilation (Scoring) AlgorithmAssociate VC segments with paleo sites

single-site pair (nearest-neighbor)specified radius (long-range neighborhood)

Select scoring method and generate scoring functions

“Score” the simulation dataWe use a “unit area Gaussian” scoring function

0.0

1.0

Paleo Date Std. Dev.Simulation Event

ScoreArea Under Gaussian Curve = 1

0.0

1.0

Paleo Date Std. Dev.Simulation Event

ScoreArea Under Gaussian Curve = 1

Simulation Time (years)

Sco

re

Gaussian Scoring Function

Gaussian Scoring Gaussian Scoring

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Plots from Weldon (2005)

2000

Yea

rs o

f Ela

psed

Tim

e Log [ 1-CFF(x,t) ] Color Cycle

NSA

F

Cree

ping

SSAF

Gar

lock

Tim

e (Y

r)

Space (Distance, km)

Stress Dynamics and the Optimization of Numerical Forecasts using “Data-Scoring” Time-space plot of

the dynamics: Coulomb failure stress (colors), earthquakes (horizontal lines) for all segments in the model.

Evaluation Window

Forecast Window

Which epochs of simulation data are most like the observed data? Using only the intervals following these epochs will allow us to optimize forecast statistics.

Dat

a Sc

ore

Time

Optimal Forecast Interval

High Scoring EventLow Scoring Event

High and Low Scoring Events:High and Low Scoring Events:Virtual California - PaleoseismologyVirtual California - Paleoseismology

Spatial & Temporal PDFsSpatial & Temporal PDFs

Determine magnitude threshold (magnitude > m)

Use m = 7.0 for temporal pdf

Use m = 6.5 and m = 7.0 for spatial pdf

Set a decision threshold (approx. 1% of simulation data)

Temporal: Starting at these “high scoring” years compute the time until the next large event having m > 7.0

Spatial: For each “high scoring” year, determine boundary elements that participate in the next m > 6.5 and m > 7.0 events

Temporal Waiting Time Statistics:Temporal Waiting Time Statistics:Starting at these “high scoring” years compute the time until the next large event having m > 7.0 Then find the most likely locations for these events

Waiting Time Distribution

Spatial Probability Densityfor next event m > 6.5

Peak value = 0.253

Spatial Probability Densityfor next event m > 7.0

Peak value = 0.214

Spatial Probability Spatial Probability Density Functions:Density Functions:

For each “high scoring” year, determine the boundary elements that tend to participate in the next m > 6.5 and m > 7.0 events

ResultsResults

Temporal: 50% probability that the next large event with m >

7.0 will occur within ~ 8 yearsProbability distribution is nearly Poisson due to

incoherent stacking of data from many fault elements

Spatial: Next event having m > 6.5 most likely to occur on

Calaveras faultNext event having m > 7.0 most likely to occur on

either Carrizo plain segment of San Andreas fault, northern San Andreas, or Garlock faults