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Seismic energy radiation from dynamic faulting. Laboratoire de Géologie. Ra ú l Madariaga Ecole Normale Supérieure. (from Aochi and Madariaga, BSSA 2003). Some inferred properties of seismic ruptures. 1. Slip distributions and ruptures are complex at all scales.

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Seismic energy radiation from dynamic faulting

Laboratoire de Géologie

Raúl Madariaga

Ecole Normale Supérieure

(from Aochi and Madariaga, BSSA 2003)

Some inferred properties of seismic ruptures

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 DU

6. Local control of rupture

7. How about Energy and High frequencies?

Slip weakening model with healing

All the terms scale with

earthquake size (Aki, 1967)


This is an average

global model

not a local model

(Rivera and Kanamori, 2004)

Es= Gc(qs) – Gc(dyn)

Radiation from a simple circular crack

This model has just 3 parameters:

Radius R

Stress dropDs

Rupture velocity vr

Plus elasticity


Actually it has only one : R



Radiated Energy

Gc, vr

Er ~ R3

Gc ~ R

Displacement field

Mo ~ R3


Possible rupture scenarios for the Izmit Earthquake

Possible models

A seismic (Bouchon)

B GPS (Wright)

C Spot Images

D FDM Harris

E Aochi Madariaga

Modelling complex fault geometries

Fault model


Rupture propagation model



Wave propagation model

Two reasonable models of the Izmit earthquake

Bouchon like « smooth » model

Harris-like « rough» model

After Aochi and Madariaga (2003)

Model B

Model E

The « smooth » fault model

develops supershear shocks


Detailed energy balance

The « rough » fault models produces subshear ruptures

There is an apparent paradox:

Little high frequency

radiation along the




A lot of high frequency radiation


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

Seismic radiation from a kink in an antiplane fault

( Adda-Bedia et al, 2003-2005)

At t = tc the crack kinks

Emits a strong high

frequency wave

of---2 type

(Jump in velocity)


Shear stress

Radiation from an antiplane crack moving along a kink

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

Energy balance

(Kostrov, Husseini, Freund, etc )

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 ?

How are High Frequencies generated ?

Constant radiation

Constant radiation

Local strain energy

High frequency S wave front

Radiation density

Es =Gc(qs)-Gc(Dyn)

Along the fault

The in-plane kink

Solution by spectral elements

Typical grid

Propagation solved

by SEM

(Vilotte, Ampuero,

Festa and Komatisch)

Fracture solved

by BIEM-like

boundary conditions

(Cochard,Fukuyama, Aochi, Tada, Kame,Yamashita)

Displacement field for a rupture moving along a kink


Slip discontinuity

X component

Slip is frustrated by the kink

Residual stress concentration

Y component

(Williams, 1952)

(King, Yamashita, Kame, Polyakov, etc)

Vorticity of the particle velocity field

Computed by Festa and Vilotte April 2005

Rupture moves along the kink

Velocity along y

Velocity along x


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

Along x

Along y

Figures show particle

velocity at three

succesive instants

of time






Radiation from a suddenly starting antiplane crack

(or stopping)

Analytical solution from Madariaga (1977)

(Madariaga, 1977)

Energy Partition into radiation, fracture and Kostrov energies

Simple mode II fault kink model

by Aochi et al, 2004

Why ?

rupture onset

Normal displacement energies.

Parallel displacement

Stopping phase


After Aochi et al (2004)

Rupture stops rapidly after the kink energies

Horizontal displacement

Vertical displacement

Rupture moves along the kink energies

Vertical displacement

Horizontal displacement

Seismic energy radiated by an earthquake energies

Kostrov Term

any value

Rupture energy


T stress change

T stress change rate

u displacement

Gc energy release rate


Strain energy release