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

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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.

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
- ...

- Data:
- Sources of uncertainty
- Observational uncertainty:
- Instrumental noise
- Ambient seismic noise

- Prediction uncertainty:
- Fault geometry
- Earth model

- Observational uncertainty:
- A posteriori distribution

Izmit earthquake (1999)

Slip, m

Depth, km

Slip, m

Depth, km

Single model

Slip, m

Depth, km

Ensemble of

models

2

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

Prediction uncertainty due to the earth model

1000 stochastic realizations

Covariance

Cμ

Cp

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

Toy model 1: Infinite strike-slip fault

Posterior Mean Model

Input (target) model

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

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

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

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

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

- 2 sources of uncertainty

Application to actual data: Mw 7.7 Balochistan earthquake

Jolivet et al., submitted to BSSA

AGU Late breaking session on Tuesday

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

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

Toy model 1: prior: U(-0.5,20)

Posterior Mean Model

Input (target) model

Toy model 1: prior: U(0,20)

Posterior Mean Model

Input (target) model

Toy model including a slip step

Toy model including a slip step

Evolution of m at each beta step

Evolution of Cp at each beta step

Covariance Cμ

1000 realizations

Covariance Cp

1000 realizations

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)

- 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