1. Earthquake Energy Balance
Raúl Madariaga
Laboratoire de Géologie
Ecole Normale Supérieure de Paris
Spice training network EU & ANR cattel

2. Modern test of earthquake scaling law
collapsed
Circular crack model fp
Universal Spectral Shape fs
Test by Prieto et al JGR, 2004

3. Consequence of the scaling law
(Savage and Wood,1971, Madariaga, 1976, Boatwright, )
Ide et al 2004
Apparent stress is a fraction of stress drop
App sa
Where C2 = for Brune’s spectrum
C2= for Boatwright’s Ds
Ds and sa are almost scale independent

4. es = Dw - Gc A simple shear crack (earthquake) moving at fast speed vr
Local energy balance
es = Dw - Gc
friction
peak
external
ini Gc Dw
residual
slip Dc

5. More than 1/3 of the stress drop is used to move the rupture!!
Energy balance
Seismic energy sa
Fracture energy Ds
Self-energy change
slip
More than 1/3 of the stress drop is used to move the rupture!!
(used by Abercombrie and Rice, 2005)
Gc scales with slip !

6. What controls rupture propagation?
= Dw/Gc
-1 = es/Gc
Kunlun EQ
vr > vs
0.2
1.4
sa=0.33Ds
vr= 0 1
Tsunami EQ
How to reduce speed?  increase Gc

7. Far field radiation from circular crack
Displacement pulse
Spectrum
0.25 Hz 4s
w-2 decay is controled by the stopping phases not by the duration

8. slip weakening friction
Slip rate
Slip
Stress change
Elliptical crack
dynamics
Fully spontaneous
rupture propagation
under
slip weakening friction
Stopping phase (S wave)

9. Far field radiation from an elliptical fault
Radiated signals are very variable
Spectra are not

10. Isochrones and stopping phases
1st Stopping phase 1 2
2nd Stopping phase

11. Stopping phases in the near field
In plane (P-SV) section
Antiplane (SH) section

12. Conclusions
Current spectral models contain information about fracture and friction
Naive interpretation is that rupture speed is related to apparent stress
Gc scales with slip
Rough energy release rate estimates Gc can be done directly from apparent stress (Abercombrie et Rice, 2005)
Actually spectral model only sees energy release rate changes!
Modern numerical methods simulate stopping phases very well

13. Rupture process for a circular crack
Slip rate
Slip
P st.phase
Rayleigh S
The rupture process is controlled by wave propagation!

14. Global Energy Balance L L = length scale S = surface
Es = radiated energy
Ws = strain energy change
Gc = surface energy
Kt = Kostrov's term
Ws
Seismologists mesure Es directly (still difficult)
Estimate Wc from seismic moment (easily measured)

15. Energy flow and rupture speed
Radiated energy density es v
Husseini et al, 1978
Static energy density
g(v) v b
From Kostrov, Eshelby, Husseini, Freund

16. Radiation from an antiplane crack with a kink
Velocity z
Stress zy
Stresszx S   
kink S wave
(-2 )
Rupture front
vr < vs
Corner stresses
Starting asperity

17. Scaling of energy Landers Gc ~ Es ~ DW ~ 106 J/m2 W~15 km
Sumatra Gc ~ Es ~ DW ~ 107 J/m2
W~150 km