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

2 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 Project : Toward the next generation of source models including realistic statistics of uncertainties Izmit earthquake (1999) Depth, km Slip, m Single model Ensemble of models SIV initiative

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

Covarianc e CμCμ CpCp Prediction uncertainty due to the earth model 1000 stochastic realizations

? Slip, m H Depth / H 2H2H μ1μ1 μ2μ2 μ 2 /μ 1 = H - Data generated for a layered half-space (d obs ) - 5mm uncorrelated observational noise (→C d ) - GFs for an homogeneous half-space (→C p ) - CATMIP bayesian sampler (Minson et al., GJI 2013): Toy model 1: Infinite strike-slip fault Slip, m H Depth / H 2H2H μ2μ2 0.9 H Synthetic Data + Noise shallow fault + Layered half-space Inversion: Homogeneous half-space μ1μ1 μ2μ2

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

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

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

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

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

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

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

13 Toy Model 2: Static Finite-fault modeling Shear modulus, GPa Depth, km Finite strike-slip fault ‣ 65 patches, 2 slip components ‣ 5mm uncorrelated noise (→C d ) ‣ GFs for an homogeneous half- space (→C p ) Earth model Uncertainty on the shear modulus Dist. along Strike, km Dist. along Dip, km Slip, m Posterior mean model, including Cp C p East (x r ), m 2 x 10 4 East, km North, km Covariance with respect to x r xrxr

14 Toy Model 2: Static Finite-fault modeling Log(μ i / μ i+1 ) Depth, km Finite strike-slip fault ‣ 65 patches, 2 slip components ‣ 5mm uncorrelated noise (→C d ) ‣ GFs for an homogeneous half- space (→C p ) Earth model Dist. along Strike, km Dist. along Dip, km Slip, m Posterior mean model, including Cp C p East (x r ), m 2 x 10 4 East, km North, km xrxr Covariance with respect to x r

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

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

Toy model including a slip step

Evolution of m at each beta step

Evolution of C p at each beta step

Covariance C μ 1000 realizations

Covariance C p 1000 realizations

Measurement errors Prediction errors Observational error: ‣ Measurements d obs : single realization of a stochastic variable d * which can be described by a probability density p(d*|d) = N(d*|d, C d ) Prediction uncertainty: where Ω = [ μ T, φ T ] T ‣ Ω true 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), C p ) A posteriori distribution: ‣ In the Gaussian case, the solution of the problem is given by: Earth model Source geometry Measurement s Displacement field Prior information On the importance of Prediction uncertainty D : Prediction space