Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 14 1 2007 10.5194/npg-14-5-2007 http://www.nonlin-processes-geophys.net/14/5/2007/ http://www.nonlin-processes-geophys.net/14/5/2007/npg-14-5-2007.html http://www.nonlin-processes-geophys.net/14/5/2007/npg-14-5-2007.pdf 5 15 2007-01-19 An Ising model for earthquake dynamics A. Jiménez ajimene@uwo.ca K. F. Tiampo A. M. Posadas Department of Earth Sciences Biological and Geological Sciences, University of Western Ontario, London, Canada Department of Applied Physics, University of Almería, Spain This paper focuses on extracting the information contained in seismic space-time patterns and their dynamics. The Greek catalog recorded from 1901 to 1999 is analyzed. An Ising Cellular Automata representation technique is developed to reconstruct the history of these patterns. We find that there is strong correlation in the region, and that small earthquakes are very important to the stress transfers. Finally, it is demonstrated that this approach is useful for seismic hazard assessment and intermediate-range earthquake forecasting. Aki, K.: Correlogram analyses of seismograms by means of a simple automatic computer, J. Phys. Earth, 4, 71–79, 1956. Allègre, C J. and Le Mouel, J L.: Introduction of Scaling Technique in Brittle Failure of Rocks, Phys. Earth Planet Inter., 87, 85–93, 1994. Allègre, C J., Le Mouel, J L., and Provost, A.: Scaling Rules in Rock Fracture and Possible Implications for Earthquake Predictions, Nature, 297, 47–49, 1982. Anghel, M., Klein, W., Rundle, J B., and Martins, J S S.: Scaling in a cellular automaton model of earthquake faults, ArXiv Condensed Matter e-prints, 2000. Bak, P. and Tang, C.: Earthquakes as a Self-Organized Critical Phenomenon, J. Goephys. Res., 94, 15 635–15 637, 1989. Bak, P., Tang, C., and Wiesenfeld, K.: Are Earthquakes, Fractals, and 1/f Noise Self-Organized Critical Phenomena, Cooperative Dynamics in Complex Systems, edited by: Takayama, H., Springer-Verlag, Tokyo, 1988. Barriere, B. and Turcotte, D L.: A scale-invariant cellular automaton model for distributed seismicity, Geophys. Res. Lett., 18, 2011–2014, 1991. Boettcher, S. and Hartmann, A K.: Reduction of Two-Dimensional Dilute Ising Spin Glasses, Phys. Rev. B, 72, 014 429, prefixhttp://arxiv.org/abs/cond-mat/0503486, 2005. Bowman, D D. and King, G. C P.: Accelerating seismicity and stress accumulation before large earthquakes, Geophys. Res. Lett., 28, 4039–4042, 2001. Bufe, C G. and Varnes, D J.: Predictive Modeling of the Seismic Cycle of the Greater San Francisco Bay Regiony, J. Geophys. Res., 98, 9871–9883, 1993. Burridge, R. and Knopoff, L.: Model and theoretical seismicity, Bull. Seism. Soc. Am., 57, 341–371, 1967. Burton, P W.: Dicing with earthquakes, Geophys. Res. Lett., 23, 1379–1382, 1996. Burton, P W., Xu, Y., Tselentis, G., Sokos, E., and Aspinall, W.: Strong ground acceleration seismic hazard in Greece and neighboring regions, Soil Dyn. Earthqu. Eng., 23, 159–181, 2003. Burton, P W., Xu, Y., Qin, C., Tselentis, G., and Sokos, E.: A catalogue of seismicity in Greece and the adjacent areas for the twentieth century, Tectonophysics, 390, 117–127, 2004a. Burton, P W., Qin, C., Tselentis, G., and Sokos, E.: Extreme earthquake and earthquake perceptibility study in Greece and its surrounding area, Nat. Hazards, 32, 277–312, 2004b. Carlson, J M. and Langer, J S.: A mechanical model of an earthquake fault, Phys. Rev. A, 40, 6470–6484, 1989a. Carlson, J M. and Langer, J S.: Properties of earthquake generated by fault dynamics, Phys. Rev. Lett., 62, 2632–2635, 1989b. Carpinteri, A. and Pugno, N.: Are scaling laws on strength of solids related to mechanics or to geometry?, Nature Materials, 4, 421–423, 2005. Christensen, K. and Olami, Z.: Variation of the Gutemberg-Richter b values and nontrivial temporal correlations in a spring-block model of earthquakes, J. Geophys. Res., 97, 8729–8735, 1992. Colella, V S., Klopfer, E., and Resnick, M.: Adventures in Modeling: Exploring Complex Dynamic Systems with Starlogo, Teachers College Press, 2001. Cover, T. and Thomas, J.: Elements of information theory, Wiley and Sons, 1991. Daley, D J. and Vere-Jones, D.: Scoring probability forecasts for point processes: the entropy score and information gain, J. Appl. Probab., 41A, 297–312, 2004. Geophys. Lab. A.U.TH. and ITSAK: Collection and Procession of Seismological data and computation of seismic hazard map of Greece according to the current Greek Seismic Code and Eurocode, Tech. rep., Geophysical Laboratory of Aristotle University of Thessaloniki and Institute of Engineering Seismology and Earthquake Engineering (ITSAK), 2002. Gould, H. and Tobochnik, J.: Thermal and Statistical Physics, available at http://stp.clarku.edu/notes/, 2004. Gutenberg, B. and Richter, C.: Earthquake magnitude, intensity, energy, and acceleration, Bull. Seism. Soc. Am., 46, 105–145, 1956. Hirata, T.: A correlation between the b value and the fractal dimension of earthquakes, J. Geophys. Res., 94, 7507–7514, 1989. Hirata, T. and Imoto, M.: Multifractal analysis of spatial distribution of microearthquakes in the Kanto region, Geophys. J. Int., 107, 155–162, 1991. Hirata, T. and Imoto, M.: A probabilistic cellular automaton approach for a spatiotemporal seismic activity pattern, Zisin, 2(49), 441–449, 1997. Jiménez, A. and Posadas, A M.: A Moore's cellular automaton model to get probabilistic seismic hazard maps for different magnitude releases: A case study for Greece, Tectonophysics, 423, 35–42, 2006. Jiménez, A., Posadas, A M., and Marfil, J M.: A Probabilistic Seismic Hazard Model based on Cellular Automata and Information Theory, Nonlin. Processes Geophys., 12, 381–396, 2005. Jiménez, A., Tiampo, K F., Levin, S., and Posadas, A M.: Testing the persistence in earthquake catalogs: The Iberian Peninsula, Europhys. Lett., 73 (2), 171–177, 2006. Klein, W., Rundle, J B., and Ferguson, C D.: Scaling and Nucleation in Models of Earthquake Faults, Phys. Rev. Lett., 78, 3793–3796, 1997. Klein, W., Gould, H., Tobochnik, J., Alexander, F J., Anghel, M., and Johnson, G.: Clusters and Fluctuations at Mean-Field Critical Points and Spinodals, Phys. Rev. Lett., 85, 1270–1273, 2000. Kroll, D M. and Gompper, G.: Wetting in fcc Ising antiferromagnets and binary alloys, Phys. Rev. B, 36, 7078–7090, 1987. Mai, P M. and Beroza, G C.: Source Scaling Properties from Finite-Fault-Rupture Models, Bull. Seism. Soc. Am., 90, 604–615, 2000. Makropoulos, K C. and Burton, P W.: A catalogue of seismicity in Greece and adjacent areas, Geophys. J. R. Astron. Soc., 65, 741–762, 1985. Makropoulos, K C., Drakopoulos, J K., and Latousakis, J B.: A revised and extended earthquake catalogue for Greece since 1900, Geophys. J. Int. Res. Note, 98, 391–394, 1989. Nakanishi, H.: Cellular-automaton model of earthquakes with deterministic dynamics, Phys. Rev. A, 41, 7068–7089, 1990. Nakanishi, H.: Statistical properties of the cellular-automaton model for earthquakes, Phys. Rev. A, 43, 6613–6621, 1991. Olami, Z., Feder, H. J S., and Christensen, K.: Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes, Phys. Rev. Lett., 68, 1244–1247, 1992. Omori, F.: On the aftershocks of earthquakes, J. Coll. Sci., 7, 111–200, 1895. Otsuka, M.: A simulation of earthquake occurrence, Phys. Earth Planet. Inter., 6, 311–315, 1972. Papaioannou, C A. and Papazachos, B C.: Time-independent and time-dependent seismic hazard in Greece based on seismogenic sources, Bull. Seismol. Soc. Am., 90, 22–33, 2000. Papazachos, B C. and Comninakis, P E.: Geophysical and tectonic features of the Aegean arc, J. Geophys. Res., 76, 8517–8533, 1971. Papazachos, B C. and Papazachou, C.: The Earthquakes of Greece, Editions Ziti, Thessaloniki, 1997. Papazachos, B C., Karakaisis, G F., and Hatzidimitriou, P M.: Further information on the transform fault of the Ionian sea, XXIV Gen. Ass., Europ. Seism. Com., Athens, 19–24 September, 1, 377–384, 1994. Peña, J., Vidal, F., Posadas, A M., Morales, J., Alguacil, G., De Miguel, F., Ibáñez, J., Romacho, M., and López-Linares, A.: Space clustering properties of the Betic-Alboran earthquakes in the period 1962–1989, Tectonophysics, 221, 125–134, 1993. Polimenakos, L C.: Shallow seismicity in the area of Greece: its character as seen by means of a stochastic model, Nonlin. Processes Geophys., 2, 136–146, 1995. Posadas, A M., Hirata, T., Vidal, F., and Correig, A.: Spatiotemporal seismicity patterns using mutual information application to southern Iberian peninsula (Spain) earthquakes, Phys. Earth Planet. Inter., 122, 269–276, 2000. Rundle, J B.: A physical model for earthquakes, 2. Application to Southern California, J. Geophys. Res., 93, 6255–6274, 1988. Rundle, J B.: Magnitude-frequency relations for earthquakes using a statistical mechanical approach, J. Geophys. Res., 98, 21 943–21 949, 1993. Rundle, J B., Turcotte, D L., Shcherbakov, R., Klein, W., and Sammis, C.: Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems, Rev. Geophys., 41, 1019–1049, 2003. Rundle, P B., Rundle, J B., Klein, W., Martins, J. S S., Tiampo, K F., Donnellan, A., and Kellogg, L H.: GEM plate boundary simulations for the plate boundary observatory: Understanding the physics of earthquakes on complex fault systems, Pure. Appl. Geophys., 159, 2357–2381, 2002. Ryan, M J. and Frater, M R.: Communications and information systems, Argos Press, 2002. Sammis, C G. and Smith, S W.: Seismic Cycles and the Evolution of Stress Correlation in Cellular Automaton Models of Finite Fault Networks, Pure Appl. Geophys., 155, 307–334, 1999. Schaap, H G.: Ising models and neural networks, Ph.D. thesis, University of Groningen, 2005. Smalley, R F., Chatelain, J L., Turcotte, D L., and Prevot, R.: A fractal approach to the clustering of earthquakes: applications to the seismicity of the New Hebrides, Bull. Seism. Soc. Amer., 77, 1368–1381, 1987. Sornette, A. and Sornette, D.: Self-organized criticality and earthquakes, Europhys. Lett., 9, 197–202, 1989. Sornette, D. and Sammis, C G.: Complex Critical Exponent from Renormalization Group Theory of Earthquakes: Implications for Earthquake Predictions, J. Phys. I France, 5, 607–619, 1995. Tiampo, K F., Rundle, J B., McGinnis, S., Gross, S J., and Klein, W.: Mean-field threshold systems and phase dynamics: an application to earthquake fault systems, Europhys. Lett., 60, 481–488, 2002a. Tiampo, K F., Rundle, J B., McGinnis, S A., and Klein, W.: Pattern Dynamics and Forecast Methods in Seismically Active Regions, Pure Appl. Geophys., 159, 2429–2467, 2002b. Toussaint, R. and Pride, S R.: Interacting damage models mapped onto Ising and percolation models, ArXiv Condensed Matter e-prints, 2004. Vicsek, T.: Fractal growth phenomena, World Scientific, 1992. Wells, D L. and Coppersmith, K J.: New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement, Bull. Seism. Soc. Am., 84, 974–1002, 1994. Wong, S K.: A cursory study of the thermodynamic and mechanical properties of Monte-Carlo simulations of the Ising model, Ph.D. thesis, University of Notre Dame, Indiana, 2005.