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 DuputelSeismo 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)

Dep

th,

kmD

ep

th,

kmD

ep

th,

km

Slip

, m

Slip

, m

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), Cp)p(d|m) = δ(d - g( ,m))

Calculation of Cp based on the physics of the problem: A perturbation approach

Covariance

Cμ

Cp

Prediction uncertainty due to the earth model

1000 stochastic realizations

?

Slip, m

H

Dep

th /

H

2H

μ1

μ2

μ2/μ1 =1.4

0.9H

- 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):

Toy model 1: Infinite strike-slip fault

Slip, m

H

Dep

th /

H

2H

μ2

0.9H

Synthetic Data + Noiseshallow fault + Layered half-

space

Inversion:Homogeneous half-space

μ1

μ2

Toy model 1: Infinite strike-slip fault

Input (target) model

Posterior Mean Model

Slip, m

Slip, m

Depth

/

H

Dis

pla

cem

en

t, m

Distance from fault / H

No Cp (overfitting)

Cp Included (larger residuals)

Depth

/

H

Why a smaller misfit does not necessarily indicate a better solution

Distance from fault / H

Dis

pla

cem

en

t, m

8

Toy Model 2: Static Finite-fault modeling

Dist. along Strike, km

Dis

t. a

long D

ip,

km

East, km

Nort

h,

km

Shear modulus, GPa

Depth

, km

Horizontal Disp., m

Vertical Disp., m

Slip, m

Input (target) model

Earth model

Data

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

Dis

t. a

long D

ip,

km

East, km

Nort

h,

km

Shear modulus, GPa

Depth

, km

Horizontal Disp., m

Vertical Disp., m

Slip, m

Input (target) model

Earth model

Data

Model for

Data

Model forGFs

Finite strike-slip fault‣ 65 patches, 2 slip components‣ 5mm uncorrelated noise

(→Cd)‣ GFs for an homogeneous half- space (→Cp)

10

Toy Model 2: Static Finite-fault modeling

Dist. along Strike, km

Dis

t. a

long D

ip,

km

Shear modulus, GPa

Depth

, km

Slip, mFinite strike-slip fault‣ 65 patches, 2 slip components‣ 5mm uncorrelated noise

(→Cd)‣ GFs for an homogeneous half- space (→Cp)

Input (target) model - 65 patches average

Earth model

Dist. along Strike, km

Dis

t. a

long D

ip,

km

Slip, m

Posterior mean model, No Cp

Dist. along Strike, km

Dis

t. a

long D

ip,

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 BSSAAGU 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

(→Cd)‣ GFs for an homogeneous half- space (→Cp)

Earth model

Uncertainty on the shear

modulus

Dist. along Strike, km

Dis

t. a

long D

ip,

km

Slip, m

Posterior mean model, including Cp

CpEast(xr), m2

x 104

East, km

Nort

h,

km

Covariance with respect to xr

xr

14

Toy Model 2: Static Finite-fault modeling

Log(μi / μi+1)

Depth

, km

Finite strike-slip fault‣ 65 patches, 2 slip components‣ 5mm uncorrelated noise

(→Cd)‣ GFs for an homogeneous half- space (→Cp)

Earth model

Dist. along Strike, km

Dis

t. a

long D

ip,

km

Slip, m

Posterior mean model, including Cp

CpEast(xr), m2

x 104

East, km

Nort

h,

km

xr

Covariance with respect to xr

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

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

Measurement

errors

Prediction

errors

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: where Ω = [ 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:

Earthmode

l

Sourcegeometr

y

Measurements

Displacement field

Prior information

On the importance of Prediction uncertainty

D: Prediction space