Seismic energy radiation from dynamic faulting Raúl Madariaga Ecole 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 (G c ) 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

E s = G c (qs) – G c (dyn)

Radiation from a simple circular crack This This model has just 3 parameters: Radius R Stress drop  Rupture velocity v r Plus elasticity Actually it has only one : R

G c, v r Radiated Energy Displacement field w   E r ~ R 3 G c ~ R Etc. M o ~ R 3

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 BIE FD SEM/BIEM

Bouchon like « smooth » model Harris-like « rough» model Two reasonable models of the Izmit earthquake After Aochi and Madariaga (2003)

Model BModel E The « smooth » fault model develops 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 = t c the crack kinks Emits a strong high frequency wave of  ---2 type (Jump in velocity) ( Adda-Bedia et al, )

Radiation from an antiplane crack moving along a kink DisplacementShear stress Analytical solution from Adda-Bedia et al ( )

Radiation 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 E s =Gc(qs)-Gc(Dyn) Constant radiation How are High Frequencies generated ? High frequency S wave front Radiation density Local strain energy Along the fault

Solution by spectral elements Propagation solved by SEM (Vilotte, Ampuero, Festa and Komatisch ) Fracture solved by BIEM-like boundary conditions (Cochard,Fukuyam a, Aochi, Tada, Kame,Yamashita) Typical grid The in-plane kink

Displacement field for a rupture moving along a kink Wrinkle 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 kink Velocity along y Velocity 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 particle velocity at three succesive instants of 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 kink Vertical displacementHorizontal displacement

Rupture moves along the kink Horizontal displacement Vertical displacement

Seismic energy radiated by an earthquake Strain energy release >0 Kostrov Term any value Rupture energy >0  T stress change  T stress change rate  u displacement Gcenergy release rate.