Assimilating Data into Earthquake Simulations Michael Sachs, J.B. Rundle, D.L. Turcotte University of California, Davis Andrea Donnellan Jet Propulsion Laboratory

Data = Model + Errors Data Assimilation, Model Steering, Model Tuning Linearize Matrix of Partial Derivatives

Applications Climate Models Numerical weather forecasting Global Ocean Data Assimilation Experiment DART (NCAR): Community DAta & Research Testbed) Land Data Assimilation Systems Satellite Data Assimilation Snow Data Assimilation System Orbit correction Financial markets and trading models Econometrics Engineering control systems

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SCIDAC Tools for Computation, Data Assimilation, & Model Steering

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

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

9 Other Standard Methods 1.Linear Programming 2.Simulated Annealing 3.Genetic Algorithms & Evolutionary Programming 4.Monte Carlo Search 5.Simulations + Data Scoring

Data Assimilation via Scoring Method Compare Virtual California simulation data with historical seismic record Pick simulation times whose history is most similar to the historic data Use “future simulation times” to generate probabilities of future large events. J. Van Aalsburg et al., PEPI, 163, 149 (2007) J. Van Aalsburg et al., PAGEOPH, 167, 967 (2010)

Data Sets Virtual California 768 fault boundary elements in model 1.5 million events 200,000 years Paleoseismic Data 119 events 20 sites J. Van Aalsburg et al., PEPI, 163, 149 (2007) J. Van Aalsburg et al., PAGEOPH, 167, 967 (2010)

Assimilation (Scoring) Algorithm Associate 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 data We use a “unit area Gaussian” scoring function Paleo Date Std. Dev. Simulation Event Score Area Under Gaussian Curve = 1

Paleo Date Std. Dev. Simulation Event Score Area Under Gaussian Curve = 1 Simulation Time (years) Score Gaussian Scoring Function Gaussian Scoring

14 Plots from Weldon (2005) 2000 Years of Elapsed Time Log [ 1-CFF(x,t) ] Color Cycle NSAF Creeping SSAF Garlock Time (Yr) 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. Data Score Time Optimal Forecast Interval

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

Spatial & 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: 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 Time (years) Waiting Time Distribution

Spatial Probability Density for next event m > 6.5 Peak value = Spatial Probability Density for next event m > 7.0 Peak value = Spatial Probability 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

Results Temporal: 50% probability that the next large event with m > 7.0 will occur within ~ 8 years Probability 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 fault Next event having m > 7.0 most likely to occur on either Carrizo plain segment of San Andreas fault, northern San Andreas, or Garlock faults