Modeling Earthquakes Using Fractal Circular Patch Models with Lessons from the 2011 Tohoku-Oki Earthquake
Satoshi Ide* and Hideo Aochi**
*Department of Earth and Planetary Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
**Bureau de Recherches Géologiques et Miniéres, 3 Avenue Claude Guillemin, 45060 Orléans Cedex 2, France
Earthquakes occur in a complex hierarchical fault system, meaning that a realistic mechanically-consistent model is required to describe heterogeneity simply and over a wide scale. We developed a simple conceptual mechanical model using fractal circular patches associated with fracture energy on a fault plane. This model explains the complexity and scaling relation in the dynamic rupture process. We also show that such a fractal patch model is useful in simulating longterm seismicity in a hierarchal fault system by using external loading. In these studies, an earthquake of any magnitude appears as a completely random cascade growing from a small patch to larger patches. This model is thus potentially useful as a benchmarking scenario for evaluating probabilistic gain in probabilistic earthquake forecasts. The model is applied to the real case of the 2011 Tohoku-Oki earthquake based on prior information from a seismicity catalog to reproduce the complex rupture process of this very large earthquake and its resulting ground motion. Provided that a high-quality seismicity catalog is available for other regions, similar approach using this conceptual model may provide scenarios for other potential large earthquakes.
-  C. H. Scholz, “The Mechanics of Earthquake and Faulting, 2nd ed.,” Cambridge Univ. Press, New York, 2002.
-  P. G. Okubo and K. Aki, “Fractal geometry in the San Andreas fault system,” J. Geophys. Res., Vol.92, pp. 345-355, 1987.
-  S. R. Brown and C. H. Scholz, “Broad bandwidth study of the topography of natural rock surfaces, ” J. Geophys. Res., Vol.90, pp. 12,575-12,582, 1985.
-  W. L. Power, T. E. Tullis, S. R. Brown, G. N. Boitnott, and C. H. Scholz, “Roughness of natural fault surfaces,” Geophys. Res. Lett., Vol.14, pp. 29-32, 1987.
-  A. Sagy, E. E. Brodsky, and G. J. Axen, “Evolution of fault-surface roughness with slip,” Geology, Vol.35, pp. 283-286, 2007.
-  T. Candela, F. Renard, Y. Klinger, K. Mair, J. Schmittbuhl, and E. E. Brodsky, “Roughness of fault surfaces over nine decades of length scales,” J. Geophys. Res., Vol.117, B08409, doi:10.1029/2011JB009041, 2012.
-  M. Naoi, M. Nakatani, S. Horiuchi, Y. Yabe, J. Philipp, T. Kgarume, G. Morema, S. Khambule, T. Masakale, L. Ribeiro, K. Miyakawa, A.Watanabe, K. Otsuki, H.Moriya, O.Murakami, H. Kawakata, N. Yoshimitsu, A.Ward, R. Durrheim, and H. Ogasawara, “Frequencymagnitude distribution of –3.7 ≤ MW ≤ 1 mining-induced earthquakes around a mining front and b value invariance with post-blast time,” PAGEOPH, doi:10.1007/s00024-013-0721-7, 2013.
-  C. H. Scholz, “The frequency-magnitude relation of microfracturing in rock and its relation to earthquake,” Bull. Seismol. Soc. Am., Vol.58, pp. 399-415, 1968.
-  H. Kanamori and D. L. Anderson, “Theoretical basis of some empirical relations in seismology,” Bull. Seismol. Soc. Am., Vol.65, pp. 1073-1095, 1975.
-  P. Bernard, A. Herrero, and C. Berge-Thierry, “Modeling directivity of heterogeneous earthquake ruptures,” Bull. Seism. Soc. Am., Vol.86, pp. 1149-1160, 1996.
-  M. Mai and G. C. Beroza, “Source Scaling Properties from Finite-Fault-Rupture Models,” Bull. Seismol. Soc. Am., Vol.90, pp. 604-615, 2000.
-  S. Ide, “Scaling relations for earthquake source process,” Zishin 2, Vol.61, pp. S329-S338, 2009.
-  T. Yamada, J. J. Mori, S. Ide, H. Kawakata, Y. Iio, and H. Ogasawara, “Radiation efficiency and apparent stress of small earthquakes in a South African gold mine,” J. Geophys. Res., Vol.110, B01305, 2005.
-  Y. Fukao and M. Furumoto, “Hierarchy in earthquake distribution,” Phys. Earth Planet. Inter., Vol.37, pp. 149-168, 1985.
-  T. Seno, “Fractal asperities, invasion of barriers, and interplate earthquakes,” Earth Planets Space, Vol.55, pp. 649-665, 2003.
-  K. Otsuki and T. Dilov, “Evolution of hierarchical self-similar geometry of experimental fault zones: Implications for seismic nucleation and earthquake size,” J. Geophys. Res., Vol.110, B03303, doi:10.1029/2004JB003359, 2005.
-  H. Aochi and S. Ide, “Numerical study on multi-scaling earthquake rupture,” Geophys. Res. Lett., Vol.31, L02606, doi:10.1029/2003GL018708, 2004.
-  S. Ide and H. Aochi, “Earthquakes as multiscale dynamic rupture with heterogeneous fracture surface energy,” J. Geophys. Res., Vol.110, B11303, doi:10.1029/2004JB003591, 2005.
-  H. Aochi and S. Ide, “Complexity in earthquake sequences controlled by multiscale heterogeneity in fault fracture energy,” J. Geophys. Res., Vol.114, B03305, doi:10.1029/2008JB006034, 2009.
-  H. Aochi and S. Ide, “Conceptual multi-scale dynamic rupture model for the 2011 Tohoku earthquake,” Earth Planets and Space, Vol.63, pp. 761-765, doi:10.5047/eps.2011.05.008, 2011.
-  S. Ide and H. Aochi, “Historical seismicity and dynamic rupture process of the 2011 Tohoku-Oki earthquake,” Tectonophysics, Vol.600, pp. 1-13, doi:10.1016/j.tecto.2012.10.018, 2013.
-  Y. Ida, “Cohesive force across the tip of a longitudinal-shear crack and Griffith’s specific surface energy,” J. Geophys. Res., Vol.77, pp. 3796-3805, doi:10.1029/JB077i020p03796, 1972.
-  B. V. Kostrov, “Selfsimilar problems of propagation of shear cracks,” PMM, Vol.28, pp. 889-898, 1964.
-  M. Ohnaka, “A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intact rock, and earthquake rupture,” J. Geophys. Res., Vol.108, 2080, doi:10.1029/2000JB000123, 2003.
-  S. Ide and M. Takeo, “Determination of constitutive relations of fault slip based on seismic wave analysis,” J. Geophys. Res., Vol.102, pp. 27379-27391, 1997.
-  G. C. Beroza and P. Spudich, “Linearized inversion for fault rupture behavior: Application to the 1984 Morgan Hill, California, earthquake,” J. Geophys. Res., Vol.93, pp. 6275-6296, 1988,
-  D. J. Andrews, “Rupture velocity of plane strain shear cracks,” J. Geophys. Res., 81, pp. 5679-5687, 1976.
-  M. Bouchon and M. Vallée, “Observation of long supershear rupture during the M = 8.1 Kunlunshan earthquake,” Science, Vol.301, pp. 824-826, 2003.
-  E. M. Dunham and R. J. Archuleta, “Evidence for a Supershear Transient during the 2002 Denali Fault Earthquake,” Bull. Seismol. Soc. Am., Vol.94. pp. S256-S268, 2004.
-  Y. Iio, “Observations of the slow initial phase generated by microearthquakes: Implications for earthquake nucleation and propagation,” J. Geophys. Res., Vol.100, pp. 15333-15349, doi:10.1029/95JB01150, 1995.
-  W. L. Ellsworth and G. C. Beroza, “Seismic evidence for an earthquake nucleation phase,” Science, Vol.268, pp. 851-855, 1995.
-  E. L. Olson and R. M. Allen, “The deterministic nature of earthquake rupture,” Nature, Vol.438, pp. 212-215, doi:10.1038/nature04214, 2005.
-  T. Yamada and S. Ide, “Limitation of the Predominant-Period Estimator for Earthquake Early Warning and the Initial Rupture of Earthquakes,” Bull. Seismol. Soc. Am., Vol.98, pp. 2739-2745, 2008.
-  W. H. Bakun, B. Aagaard, B. Dost, W. L. Ellsworth, J. L. Hardebeck, R. A. Harris, C. Ji, M. J. Johnston, J. Langbein, J. J. Lienkaemper, A. J. Michael, J. R. Murray, R. M. Nadeau, P. A. Reasenberg, M. S. Reichle, E. A. Roeloffs, A. Shakal, R. W. Simpson, and F. Waldhauser, “Implications for prediction and hazard assessment from the 2004 Parkfield earthquake,” Nature, Vol.437, 9690974, doi:10.1038/nature04067, 2005.
-  N. Uchida, T. Matsuzawa, W. L. Ellsworth, K. Imanishi, T. Okada, and A. Hasegawa, “Source parameters of a M4.8 and its accompanying repeating earthquakes off Kamaishi, NE Japan: Implications for the hierarchical structure of asperities and earthquake cycle,” Geophys. Res. Lett., Vol.34, L20313, doi:10.1029/2007GL031263, 2007.
-  S. Ide, A. Baltay, and G. C. Beroza, “Shallow dynamic overshoot and energetic deep rupture in the 2011 Mw9.0 Tohoku-Oki earthquake,” Science, Vol.332, pp. 1426-1429, doi:10.1126/science.1207020, 2011.
-  Y. Mitsui, Y. Iio, and Y. Fukahata, “A scenario for the generation process of the 2011 Tohoku earthquake based on dynamic rupture simulation: role of stress concentration and thermal fluid,” Earth Planets Space, Vol.64, pp. 1177-1187, 2012.
-  H. Aochi and S. Ide, “Ground motions characterized by a multiscale heterogeneous earthquake model,” Earth Planets and Space, 2014 (in press).
-  A. A. Gusev, “High-frequency radiation from an earthquake fault: A review and a hypothesis of fractal rupture front geometry,” PAGEOPH, Vol.170, pp. 65-93, 2013.
-  H. Aochi , E. Fukuyama, and R. Madariaga, “Constraints of Fault Constitutive Parameters Inferred from Non-planar Fault Modeling,” Geichemistry, Geophysics, Geosystems, Vol.4, No.2, doi:10.1029/2001GC000207, 2003.
-  H. Aochi, M. Cushing, O. Scotti, and C. Berge-Thierry, “Estimating rupture scenario likelihood based on dynamic rupture simulations: the example of the segmented Middle Durance fault, southeastern France,” Geophys. J. Int., Vol.165, pp. 436-446, doi:10.1111/j.1365-246X.2006.0284.x, 2006.
-  F. Nanayama, K. Satake, R. Furukawa, K. Shimokawa, B. F. Atwater, K. Shigeno, and S. Yamaki, “Unusually large earthquakes inferred from tsunami deposits along the Kuril trench,” Nature, Vol.424, pp. 660-663, 2003.
-  S. Ide, “Dynamic rupture propagation on a 2D fault with fractal frictional properties,” Earth Planets Space, Vol.59, pp. 1099-1109, 2007.
-  H. Noda, M. Nakatani, and T. Hori, “Large nucleation before large earthquakes is sometimes skipped due to cascade-up – Implications from a rate and state simulation of faults with hierarchical asperities,” J. Geophys. Res. Solid Earth, Vol.118, pp. 2924-2952, doi:10.1002/jgrb.50211, 2013.