Toward the next generation of earthquake source models by accounting for model prediction error

1. Toward the next generation of earthquake source models by accounting for model prediction error • Zacharie Duputel • Seismo Lab, GPS division, Caltech Acknowledgements: Piyush Agram, Mark Simons, Sarah Minson, James Beck, Pablo Ampuero, Romain Jolivet, Bryan Riel, Michael Aivasis, Hailiang Zhang.

2. Project : Toward the next generation of source models including realistic statistics of uncertainties SIV initiative • Modeling ingredients • Data: • Field observations • Seismology • Geodesy • ... • Theory: • Source geometry • Earth model • ... • Sources of uncertainty • Observational uncertainty: • Instrumental noise • Ambient seismic noise • Prediction uncertainty: • Fault geometry • Earth model • A posteriori distribution Izmit earthquake (1999) Slip, m Depth, km Slip, m Depth, km Single model Slip, m Depth, km Ensemble of models 2

3. A reliable stochastic model for the prediction uncertainty The forward problem • posterior distribution: Exact theory Stochastic (non-deterministic) theory p(d|m) = δ(d - g( ,m)) p(d|m) = N(d | g( ,m), Cp) Calculation of Cp based on the physics of the problem: A perturbation approach Covariance matrix describing uncertainty in the Earth model parameters Partial derivatives w.r.t. the elastic parameters (sensitivity kernel) 3

4. Prediction uncertainty due to the earth model 1000 stochastic realizations Covariance Cμ Cp

5. Toy model 1: Infinite strike-slip fault μ1 - Data generated for a layered half-space (dobs) - 5mm uncorrelated observational noise (→Cd) - GFs for an homogeneous half-space (→Cp) - CATMIP bayesian sampler (Minson et al., GJI 2013): μ2 Synthetic Data + Noise shallow fault + Layered half-space Inversion: Homogeneous half-space Slip, m Slip, m ? μ1 0.9H 0.9H H H μ2 Depth / H Depth / H μ2 μ2/μ1 =1.4 2H 2H

6. Toy model 1: Infinite strike-slip fault Posterior Mean Model Input (target) model

7. Why a smaller misfit does not necessarily indicate a better solution No Cp (overfitting) Cp Included (larger residuals) Depth / H Depth / H Slip, m Slip, m Displacement, m Displacement, m Distance from fault / H Distance from fault / H

8. Toy Model 2: Static Finite-fault modeling Input (target) model • Finite strike-slip fault • Top of the fault at 0 km • South-dipping = 80° • Data for a layered half-space Slip, m Dist. along Dip, km Dist. along Strike, km Earth model Data Horizontal Disp., m Vertical Disp., m North, km Depth, km Shear modulus, GPa East, km 8

9. Toy Model 2: Static Finite-fault modeling Input (target) model • Finite strike-slip fault • 65 patches, 2 slip components • 5mm uncorrelated noise(→Cd) • GFs for an homogeneous half- space (→Cp) Slip, m Dist. along Dip, km Dist. along Strike, km Earth model Data Horizontal Disp., m Vertical Disp., m North, km Depth, km Model for Data Model for GFs Shear modulus, GPa East, km 9

10. Toy Model 2: Static Finite-fault modeling Input (target) model - 65 patches average • Finite strike-slip fault • 65 patches, 2 slip components • 5mm uncorrelated noise(→Cd) • GFs for an homogeneous half- space (→Cp) Slip, m Dist. along Dip, km Dist. along Strike, km Earth model Posterior mean model, No Cp Slip, m Dist. along Dip, km Dist. along Strike, km Depth, km Posterior mean model, including Cp Uncertainty on the shear modulus Slip, m Dist. along Dip, km Dist. along Strike, km Shear modulus, GPa 10

11. Conclusion and Perspectives • Improving source modeling by accounting for realistic uncertainties • 2 sources of uncertainty • Observational error • Modeling uncertainty • Importance of incorporating realistic covariance components • More realistic uncertainty estimations • Improvement of the solution itself • Accounting for lateral variations • Improving kinematic source models

12. Application to actual data: Mw 7.7 Balochistan earthquake Jolivet et al., submitted to BSSA AGU Late breaking session on Tuesday

13. Toy Model 2: Static Finite-fault modeling Posterior mean model, including Cp • Finite strike-slip fault • 65 patches, 2 slip components • 5mm uncorrelated noise(→Cd) • GFs for an homogeneous half- space (→Cp) Slip, m Dist. along Dip, km Dist. along Strike, km Earth model Covariance with respect to xr CpEast(xr), m2 x 104 Depth, km North, km Uncertainty on the shear modulus xr Shear modulus, GPa East, km 13

14. Toy Model 2: Static Finite-fault modeling Posterior mean model, including Cp • Finite strike-slip fault • 65 patches, 2 slip components • 5mm uncorrelated noise(→Cd) • GFs for an homogeneous half- space (→Cp) Slip, m Dist. along Dip, km Dist. along Strike, km Earth model Covariance with respect to xr CpEast(xr), m2 x 104 Depth, km North, km xr Log(μi / μi+1) East, km 14

15. Toy model 1: prior: U(-0.5,20) Posterior Mean Model Input (target) model

16. Toy model 1: prior: U(0,20) Posterior Mean Model Input (target) model

17. Toy model including a slip step

18. Toy model including a slip step

19. Evolution of m at each beta step

20. Evolution of Cp at each beta step

21. Covariance Cμ 1000 realizations

22. Covariance Cp 1000 realizations

23. On the importance of Prediction uncertainty • Observational error: • Measurements dobs : single realization of a stochastic variable d* which can be described by a probability density p(d*|d) = N(d*|d, Cd) • Prediction uncertainty: whereΩ = [ μT , φT ]T • Ωtrue is not known and we work with an approximation • The prediction uncertainty: • scales with the with the magnitude of m • can be described by p(d|m) = N(d | g( ,m), Cp) • A posteriori distribution: • In the Gaussian case, the solution of the problem is given by: Measurements Displacement field Earth model Source geometry Prior information Prediction errors Measurement errors D: Prediction space