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

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

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A posteriori distribution:

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 including realistic statistics of uncertainties

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 including realistic statistics of uncertainties

1000 stochastic realizations

Covariance

Cμ

Cp

Toy model 1: Infinite strike-slip fault including realistic statistics of uncertainties

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 including realistic statistics of uncertainties

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 solution

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 solution

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 solution

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 solution

- 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 solution

Jolivet et al., submitted to BSSA

AGU Late breaking session on Tuesday

Toy Model 2: Static Finite-fault modeling solution

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 solution

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 including a slip step solution

Toy model including a slip step solution

Evolution of solutionm at each beta step

Evolution of solutionCp at each beta step

Covariance solutionCμ

1000 realizations

Covariance solutionCp

1000 realizations

On the importance of Prediction uncertainty solution

- 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

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