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Dynamic rupture propagation on a 2D fault with fractal frictional properties


We study dynamic rupture propagation on a 2D fault with a spatially heterogeneous slip-weakening friction law under a homogeneous stress condition. The slip-weakening distance Dc is determined based on fault topography, characterized by the fractal dimension σ and a normalized slip-weakening distance Dc0. Assuming different combination of these parameters, we carry out a large number of dynamic rupture simulations to discuss event size statistics and the scaling relations of macroscopic parameters such as seismic moment, rupture velocity, seismic energy, and radiation efficiency. All stopped events obey an approximately power-law size-number relation. However, statistically self-similar rupture propagation is observed only for selfsimilar fault topography with σ = 1, where the average rupture velocity is controlled by Dc0. This suggests that a power-law size-number relation does not simply mean the self-similarity of rupture process. Both upper and lower fractal limits in fault topography disturb the power-law statistics and the lower fractal limit results in excess of events of the characteristic size. The facts that slow ruptures can radiate large seismic energy and that fast ruptures are not always efficient suggest the importance of local acceleration, deceleration, and arrest of rupture. We also show that the spatial regularity of slip-weakening distance is essential to make the above results.


  1. Andrews, D. J., Rupture velocity of plane strain shear cracks, J. Geophys. Res., 81, 5679–5687, 1976.

  2. Aochi, H. and E. Fukuyama, Three-dimensional nonplanar simulation of the 1992 Landers earthquake, J. Geophys. Res., 107, doi:10.1029/ 2000JB000061, 2002.

  3. Aochi, H. and S. Ide, Numerical study on multi-scaling earthquake rupture, Geophys. Res. Lett., 31, doi:10.1029/2003GL018708, 2004.

  4. Beroza, G. and P. Spudich, Linearized inversion for fault rupture behavior: Application to the 1984 Morgan Hill, California, earthquake, J. Geo-phys. Res., 93, 6275–6296, 1988.

  5. Brown, S. R. and C. H. Scholz, Broad bandwidth study of the topography of natural rock surfaces, J. Geophys. Res., 90, 12,575–12,582, 1985.

  6. Cochard, A. and R. Madariaga, Dynamic faulting under Rate-dependent friction, PAGEOPH, 142, 419–445, 1994.

  7. Day, S., Three dimensional simulation of spontaneous rupture: the effect of nonuniform prestress, Bull. Seismol. Soc. Am., 72, 1881–1902, 1982.

  8. Freund, L. B., Dynamic Fracture Mechanics, Cambridge Univ. Press, New York, 1998.

  9. Guatteri, M., P. M. Mai, G. C. Beroza, and J. Boatwright, Strong-ground motion prediction from stochastic-dynamic source models, Bull. Seis-mol. Soc. Am., 93, 301–313, 2003.

  10. Ide, S., Estimation of Radiated energy of finite-source earthquake models, Bull. Seismol. Soc. Am., 92, 2294–3005, 2002.

  11. Ide, S., On fracture surface energy of natural earthquakes from viewpoint of seismic observations, Bull. Earthquake Res. Inst., 78, 1–120, 2003.

  12. Ide, S. and H. Aochi, Earthquakes as multiscale dynamic rupture with heterogeneous fracture surface energy, J. Geophys. Res., 110, doi:10. 1029/2004JB003591, 2005.

  13. Ide, S. and G. C. Beroza, Does apparent stress vary with earthquake size?, Geophys. Res. Lett., 28, 3349–3352, 2001.

  14. Kanamori, H. and D. L. Anderson, Theoretical basis of some empirical relations in seismology, Bull. Seismol. Soc. Am., 65, 1073–1095, 1975.

  15. Kostrov, B. V., Seismic moment and energy of earthquakes and seismic flow of rock, Izv. Earth Phys., 1, 23–40, 1974.

  16. Matsu’ura, M., H. Kataoka, and B. Shibazaki, Slip-dependent friction law and nucleation processes in earthquake rupture, Tectonophysics, 211, 135–148, 1992.

  17. Ohnaka, M., A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intact rock, and earthquake rupture, J. Geophys. Res., 108, doi:10.1029/2000JB000123, 2003.

  18. Okubo, P. G. and K. Aki, Fractal geometry in the San Andreas fault system, J. Geophys. Res., 92, 345–355, 1987.

  19. Olsen, K. B., R. Madariaga, and R. J. Archuleta, Three-dimensional dynamic simulation of the 1992 Landers earthquake, Science, 278, 834–837, 1997.

  20. Power, W. L., T. E. Tullis, S. R. Brown, G. N. Boitnott, and C. H. Scholz, Roughness of natural fault surfaces, Geophys. Res. Lett., 14, 29–32, 1987.

  21. Renard, F., C. Voisin, D. Marsan, and J. Schmittbuhl, High resolution 3D laser scanner measurements of a strike-slip fault quantify its morphological anisotropy at all scales, Geophys. Res. Lett., 33, doi:10.1029/2005GL025038, 2006.

  22. Ripperger, J., P. M. Mai, and J. P. Ampuero, Earthquake source characteristics from dynamic rupture with constrained stochastic fault stress, J. Geophys. Res., 112, doi:10.1029/2006JB004515, 2007.

  23. Sagy, A., E. E. Brodsky, and G. J. Axen, Evolution of fault-surface roughness with slip, Geology, 35, 283–286, 2007.

  24. Stirling, M. W., S. G. Wesnousky, and K. Shimazaki, Fault trace complexity, cumulative slip, and the shape of the magnitude-frequency distribution for strike-slip faults: a global study, Geophys. J. Int., 124, 833–868, 1996.

  25. Tada, T., Boundary integral equations for the time-domain and time-independent analyses of 2D non-planar cracks, PhD Thesis, University of Tokyo, 1996.

  26. Uenishi, K. and J. R. Rice, Universal nucleation length for slip-weakening rupture instability under nonuniform fault loading, J. Geophys. Res., 108, doi:10.1029/2001JB001681, 2003.

  27. Walter, W. R., K. Mayeda, R. Gok, and A. Hofstetter, The scaling of seismic energy with moment; Simple models compared with observations, in Earthquakes: Radiated energy and the physics of faulting, AGU geophysical monograph 170, edited by R. Abercrombie, A. McGarr, H. Kanamori, G. DiToro, 25–41, 2006.

  28. Yamada, T., J. J. Mori, S. Ide, R. E. Abercrombie, H. Kawakata, M. Nakatani, Y. Iio, and H. Ogasawara, Stress drops and radiated seismic energies of microearthquakes in a South African gold mine, J. Geophys. Res., 112, doi:10.1029/2006JB004553, 2007.

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Correspondence to Satoshi Ide.

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Ide, S. Dynamic rupture propagation on a 2D fault with fractal frictional properties. Earth Planet Sp 59, 1099–1109 (2007).

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Key words

  • Dynamic rupture
  • fractal topography
  • self-similarity
  • seismic energy
  • rupture propagation velocity