Seismic energy radiation from dynamic faulting

Raúl MadariagaEcole Normale Supérieure

Laboratoire de Géologie

(from Aochi and Madariaga, BSSA 2003)

1. Slip distributions and ruptures are complex at all scales.

2. Very large variations of stress change.

3. Slip weakening is a substantial fraction of static slip

4. Self-healing rupture (Heaton pulses) is the rule.

5. Energy release rate (Gc) is of the same order as strain energy density U

6. Local control of rupture

7. How about Energy and High frequencies?

Some inferred properties of seismic ruptures

Earthquake energy balance

U

Slip weakening model with healing

This is an average

global model

not a local model

(Rivera and Kanamori, 2004)

All the terms scale with

earthquake size (Aki, 1967)

Event dependent

Es= Gc(qs) – Gc(dyn)

Radiation from a simple circular crack

This

This model has just 3 parameters:Radius R

Stress drop Rupture velocity vr

Plus elasticity

Actually it has only one : R

Gc, vr

Radiated Energy

Displacement field

w

Er ~ R3

Gc ~ R

Etc.

Mo ~ R3

Possible rupture scenarios for the Izmit Earthquake

Possible modelsA seismic (Bouchon)B GPS (Wright)C Spot ImagesD FDM HarrisE Aochi Madariaga

Modelling complex fault geometries

Fault model

Rupture propagation model

Wave propagation model

BIE

FD

SE

M/B

IEM

Bouchon like « smooth » model Harris-like « rough» model

Two reasonable models of the Izmit earthquake

After Aochi and Madariaga (2003)

Model B Model E

The « smooth » fault modeldevelops supershear shocks

The « rough » fault models produces

subshear ruptures

Why? Detailed energy balance

There is an apparent paradox:

Supershear

Little high frequency radiation along the way

Subshear

A lot of high frequency radiation

Es

The higher the speed, the less energy is absorved, the less is radiated

Seismic radiation from a kink in an antiplane fault

At t = tc the crack kinks

Emits a strong highfrequency wave

of ---2 type

(Jump in velocity)

( Adda-Bedia et al, 2003-2005)

Radiation from an antiplane crack moving along a kink

Displacement Shear stress

Analytical solution from Adda-Bedia et al (2003-2005)

Radiation from an antiplane crack moving along a kinkRadiation from an antiplane crack moving along a kink

Shear stress Particle velocity

Energy balance

If rupture propagates very slowly there is no seismic radiation

If rupture does not absorb available strain energy, Rupture accelerates and radiates. Neglecting Kostrov’s term

Is this localizable ?

(Kostrov, Husseini, Freund, etc )

quasistatic dynamic

Constant radiation

Es =Gc(qs)-Gc(Dyn)

Constant ra

diation

How are High Frequencies generated ?

High frequency S wave frontRadiation density

Local strain energy

Along the fault

Solution by spectral elements

Propagation solvedby SEM

(Vilotte, Ampuero, Festa and Komatisch)

Fracture solvedby BIEM-like

boundary conditions

(Cochard,Fukuyama, Aochi, Tada,

Kame,Yamashita)

Typical grid

The in-plane kink

Displacement field for a rupture moving along a kinkWrinkle

Slip discontinuity

Slip is frustrated by the kink

Residual stress concentration

(King, Yamashita, Kame, Polyakov, etc)(Williams, 1952)

X component

Y component

Vorticity of the particle velocity field

Computed by Festa and Vilotte April 2005

Rupture moves along the kinkVelocity along yVelocity along x

CONCLUSIONS

1. High frequencies play a fundamental rôle in energy balance

2. Fault kinks produce radiation so that they reduce available energy

3. Kinks reduce rupture speed

4. Kinks can stop rupture

5. Kinks are the site of residual stress concentrations

Rupture stops rapidly after the kink

P

S

R

Figures show particlevelocity at three

succesive instantsof time

Along x Along y

Radiation from a suddenly starting antiplane crack

Velocity Stress

(Madariaga, 1977)Analytical solution from Madariaga (1977)

(or stopping)

Why ?

Energy Partition into radiation, fracture and Kostrov energies

rupture onset

Simple mode II fault kink model

by Aochi et al, 2004

Stopping phase

Normal displacement.Parallel displacement

Supershear

After Aochi et al (2004)

Rupture stops rapidly after the kinkVertical displacementHorizontal displacement

Rupture moves along the kink

Horizontal displacement Vertical displacement

Seismic energy radiated by an earthquake

Strain energy release>0

Kostrov Termany value

Rupture energy>0

T stress changeT stress change rateu displacementGc energy release rate

.