Cellular automata to describe seismicity: A review

Acta Geophysica

Volumn 61, Issue 6, 2013, Pages 1325-1350

  Jiménez, Abigail ^a,b  

^a UNIVERSITY OF ALMERÍA   (Spain)

^b UNIVERSITY OF ULSTER   (United Kingdom)

Author keywords

Cellular Automata; complex systems; discrete models; seismicity

Indexed keywords

GEOPHYSICS; LARGE SCALE SYSTEMS;

DISCRETE MODELS; SEISMICITY; SPRING-BLOCK MODEL;

CELLULAR AUTOMATA;

CELLULAR AUTOMATON; COMPLEXITY; NUMERICAL MODEL; SEISMICITY;

EID: 84884812495     PISSN: 18956572     EISSN: 18957455     Source Type: Journal    
DOI: 10.2478/s11600-013-0144-y     Document Type: Article

References (150)

1. Adamatzky, A., (1994), Identification of Cellular Automata, Taylor and Francis Ltd., London.
2. Akishin, P. G., M. V. Altaisky, I. Antoniou, A. D. Budnik, and V. V. Ivanov (1998), Simulation of earthquakes with cellular automata, Discrete Dyn. Nat. Soc. 2, 4, 267-279, DOI: 10. 1155/S1026022698000247.
3. Aladjev, V. Z. (1971), Computability in the Homogeneous Structures, VINITI Press, Moscow.
4. Amoroso, S., and Y. N. Patt (1972), Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures, J. Comp. Syst. Sci. 6, 5, 448-464, DOI: 10. 1016/S0022-0000(72)80013-8.
5. Bak, P., and K. Chen (1995), Fractal dynamics of earthquakes, In: C. C. Barton and P. R. La Pointe (eds.), Fractals in the Earth Sciences, Plenum Press, New York, 227-236.
6. Bak, P., and C. Tang (1989), Earthquakes as a self-organized critical phenomenon, J. Geophys. Res. 94, B11, 15635-15637, DOI: 10. 1029/JB094iB11p15635.
7. Bak, P., C. Tang, and K. Wiesenfeld (1987), Self-organized criticality: An explanation of the 1/f noise, Phys. Rev. Lett. 59, 4, 381-384, DOI: 10. 1103/Phys RevLett. 59. 381.
8. Bak, P., C. Tang, and K. Wiesenfeld (1988), Self-organized criticality, Phys. Rev. A38, 1, 364-374, DOI: 10. 1103/PhysRevA. 38. 364.
9. Bak, P., K. Chen, and C. Tang (1990), A forest-fire model and some thoughts on turbulence, Phys. Lett. A147, 5-6, 297-300, DOI: 10. 1016/0375-9601(90)90451-S.
10. Barriere, B., and D. L. Turcotte (1991), A scale-invariant cellular-automata model for distributed seismicity, Geophys. Res. Lett. 18, 11, 2011-2014, DOI: 10. 1029/91GL02415.
11. Barriere, B., and D. L. Turcotte (1994), Seismicity and self-organized criticality, Phys. Rev. E49, 2, 1151-1160, DOI: 10. 1103/PhysRevE. 49. 1151.
12. Belubekian, M. E., and A. S. Kiremidjian (1997), A stochastic automata network for earthquake simulation and hazard estimation, Report No. 124, The John A. Blume Earthquake Engineering Center, Stanford University, USA.
13. Białecki, M. (2012a), Motzkin numbers out of Random Domino Automaton, Phys. Lett. A376, 45, 3098-3100, DOI: 10. 1016/j. physleta. 2012. 09. 022.
14. Bialecki, M. (2012b), An explanation of the shape of the universal curve of the Scaling Law for the Earthquake Recurrence Time Distributions, arXiv: 1210. 7142 [physics. geo-ph].
15. Białecki, M., and Z. Czechowski (2013), On one-to-one dependence of rebound parameters on statistics of clusters: Exponential and inverse-power distributions out of Random Domino Automaton, J. Phys. Soc. Jpn. 82, 1, 014003-014003-9, DOI: 10. 7566/JPSJ. 82. 014003.
16. Białecki, M., and A. Doliwa (2005), Algebro-geometric solution of the discrete KP equation over a finite field out of a hyperelliptic curve, Commun. Math. Phys. 253, 1, 157-170, DOI: 10. 1007/s00220-004-1207-3.
17. Bowman, D. D., and G. C. P. King (2001), Accelerating seismicity and stress accumulation before large earthquakes, Geophys. Res. Lett. 28, 21, 4039-4042, DOI: 10. 1029/2001GL013022.
18. Brender, R. F. (1970), A programming system for the simulation of cellular spaces, Tech. Rep. 25, CONCOMP, University of Michigan, Ann Arbor, USA.
19. Bufe, C. G., and D. J. Varnes (1993), Predictive modeling of the seismic cycle of the greater San Francisco Bay region, J. Geophys. Res. 98, B6, 9871-9883, DOI: 10. 1029/93JB00357.
20. Burks, A. W. (1970), Essays on Cellular Automata, University of Illinois Press, Urbana.
21. Burridge, R., and L. Knopoff (1967), Model and theoretical seismicity, Bull. Seismol. Soc. Am. 57, 3, 341-371.
22. Carlson, J. M., and J. S. Langer (1989a), Mechanical model of an earthquake fault, Phys. Rev. A40, 11, 6470-6484, DOI: 10. 1103/PhysRevA. 40. 6470.
23. Carlson, J. M., and J. S. Langer (1989b), Properties of earthquakes generated by fault dynamics, Phys. Rev. Lett. 62, 22, 2632-2635, DOI: 10. 1103/PhysRev-Lett. 62. 2632.
24. Carlson, J. M., J. S. Langer, and B. E. Shaw (1994), Dynamics of earthquake faults, Rev. Mod. Phys. 66, 2, 657-670, DOI: 10. 1103/RevModPhys. 66. 657.
25. Castellaro, S., and F. Mulargia (2001), A simple but effective cellular automaton for earthquakes, Geophys. J. Int. 144, 3, 609-624, DOI: 10. 1046/j. 1365-246x. 2001. 01350. x.
26. Castellaro, S., and F. Mulargia (2002), What criticality in cellular automata models of earthquakes?, Geophys. J. Int. 150, 2, 483-493, DOI: 10. 1046/j. 1365-246X. 2002. 01709. x.
27. Chen, K., P. Bak, and S. P. Obukhov (1991), Self-organized criticality in a crackpropagation model of earthquakes, Phys. Rev. A43, 2, 625-630, DOI: 10. 1103/PhysRevA. 43. 625.
28. Christensen, K., and Z. Olami (1992), Variation of the Gutenberg-Richter b values and nontrivial temporal correlations in a spring-block model of earthquakes, J. Geophys. Res. 97, B6, 8729-8735, DOI: 10. 1029/92JB 00427.
29. Christensen, K., L. Danon, T. Scanlon, and P. Bak (2002), Unified scaling law for earthquakes, PNAS99, Suppl. 1, 2509-2513, DOI: 10. 1073/pnas. 012581099.
30. Church, A. (1936), A Note on the Entscheidungsproblem, J. Symbolic Logic1, 1, 40-41, DOI: 10. 2307/2269326.
31. Cohen, S. C. (1977), Computer simulation of earthquakes, J. Geophys. Res. 82, 26, 3781-3796, DOI: 10. 1029/JB082i026p03781.
32. Czechowski, Z., and M. Białecki (2012a), Three-level description of the domino cellular automaton, J. Phys. A: Math. Theor. 45, 15, 155101, DOI: 10. 1088/1751-8113/45/15/155101.
33. Czechowski, Z., and M. Białecki (2012b), Ito equations out of domino cellular automaton with efficiency parameters, Acta Geophys. 60, 3, 846-857, DOI: 10. 2478/s11600-012-0021-0.
34. Delorme, M. (1998), An introduction to cellular automata, Research Rep. no. 98-37, École Normale Supérieur de Lyon, Lyon, France.
35. Dewdney, A. K. (1989), Simulated evolution: wherein bugs learn to hunt bacteria, Sci. Amer. 260, 5, 138-141.
36. Dieterich, J. H. (1972), Time-dependent friction as a possible mechanism for aftershocks, J. Geophys. Res. 77, 20, 3771-3781, DOI: 10. 1029/JB077i020p03771.
37. Dieterich, J. H., and K. B. Richards-Dinger (2010), Earthquake recurrence in simulated fault systems, Pure Appl. Geophys. 167, 8-9, 1087-1104, DOI: 10. 1007/s00024-010-0094-0.
38. Doliwa, A., M. Białecki, and P. Klimczewski (2003), The Hirota equation over finite fields: algebro-geometric approach and multisoliton solutions, J. Phys. A: Math. Gen. 36, 17, 4827-4839, DOI: 10. 1088/0305-4470/36/17/309.
39. Drossel, B., and F. Schwabl (1992a), Self-organized critical forest-fire model, Phys. Rev. Lett. 69, 11, 1629-1632, DOI: 10. 1103/PhysRevLett. 69. 1629.
40. Drossel, B., and F. Schwabl (1992b), Self-organized criticality in a forest-fire model, Physica A191, 1-4, 47-50, DOI: 10. 1016/0378-4371(92)90504-J.
41. Eneva, M., and Y. Ben-Zion (1997), Application of pattern recognition techniques to earthquake catalogs generated by model of segmented fault systems in three-dimensional elastic solids, J. Geophys. Res. 102, B11, 24513-24528, DOI: 10. 1029/97JB01857.
42. Ferguson, C. D., W. Klein, J. B. Rundle, H. Gould and J. Tobochnik (1998), Longrange earthquake fault models, Comput. Phys. 12, 1, 34-40, DOI: 10. 1063/1. 168681.
43. Fisher, D. S., K. Dahmen, S. Ramanatham, and Y. Ben-Zion (1997), Statistics of earthquakes in simple models of heterogeneous faults, Phys. Rev. Lett. 78, 25, 4885-4888, DOI: 10. 1103/PhysRevLett. 78. 4885.
44. Frish, U., B. Hasslacher, and Y. Pomeau (1986), Lattice-gas automata for the Navier-Stokes equation, Phys. Rev. Lett. 56, 14, 1505-1508, DOI: 10. 1103/PhysRevLett. 56. 1505.
45. Gálvez-Coyt, G., A. Muñoz-Diosdado, and F. Angulo-Brown (2007), Some fractal cellular automata models of seismic faults, Fractals15, 3, 207-215, DOI: 10. 1142/S0218348X07003563.
46. Gardner, M. (1970), Mathematical games: The fantastic combinations of John Conway's new solitarire game 'Life', Sci. Amer. 223, 4, 120-123, DOI: 10. 1038/scientificamerican1070-120.
47. Georgoudas, I. G., G. Ch. Sirakoulis, and I. Andreadis (2007a), Modelling earthquake activity features using cellular automata, Math. Comput. Model. 46, 1-2, 124-137, DOI: 10. 1016/j. mcm. 2006. 12. 029.
48. Georgoudas, I. G., G. Ch. Sirakoulis, E. M. Scordilis, and I. Andreadis (2007b), A cellular automaton simulation tool for modelling seismicity in the region of Xanthi, Environ. Modell. Softw. 22, 10, 1455-1464, DOI: 10. 1016/j. envsoft. 2006. 06. 015.
49. Georgoudas, I. G., G. Ch. Sirakoulis, E. M. Scordilis, and I. Andreadis (2011), Parametric optimisation in a 2-D cellular automata model of fundamental seis mic attributes with the use of genetic algorithms, Adv. Eng. Softw. 42, 9, 623-633, DOI: 10. 1016/j. advengsoft. 2011. 04. 003.
50. González, Á., M. Vázquez-Prada, J. B. Gómez, and A. F. Pacheco (2006), A way to synchronize models with seismic faults for earthquake forecasting: Insights from a simple stochastic model, Tectonophysics424, 3-4, 319-334, DOI: 10. 1016/j. tecto. 2006. 03. 039.
51. Greenberg, J. M., and S. P. Hastings (1978), Spatial patterns for discrete models of diffusion in excitable media, SIAM J. Appl. Math. 34, 3, 515-523, DOI: 10. 1137/0134040.
52. Greenberg, J. M., B. D. Hassard, and S. P. Hastings (1978), Pattern formation and periodic structures in systems modeled by reaction-diffusion equations, Bull. Am. Math. Soc. 84, 6, 1296-1327, DOI: 10. 1090/S0002-9904-1978-14560-1.
53. Hainzl, S., G. Zöller, and J. Kurths (1999), Similar power laws for foreshock and aftershock sequences in a spring-block model for earthquakes, J. Geophys. Res. 104, B4, 7243-7253, DOI: 10. 1029/1998JB900122.
54. Hainzl, S., G. Zöller, and J. Kurths (2000a), Self-organization of spatio-temporal earthquake clusters, Nonlinear Proc. Geophys. 7, 1-2, 21-29.
55. Hainzl, S., G. Zöller, J. Kurths, and J. Zschau (2000b), Seismic quiescence as an indicator for large earthquakes in a system of self-organized criticality, Geophys. Res. Lett. 27, 5, 597-600, DOI: 10. 1029/1999GL011000.
56. Hardy, J., O. De Pazzis, and Y. Pomeau (1976), Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions, Phys. Rev. A13, 5, 1949-1961, DOI: 10. 1103/PhysRevA. 13. 1949.
57. He, J., and M. Li (2010), Cellular automata to study mode-I crack propagation, Proc. Third Int. Symp. on Computer Science and Computational Technology, 14-15 August 2010, Jiaozuo, China, 475-479.
58. Hedlund, G. A. (1969), Endomorphisms and automorphisms of the shift dynamical system, Math. Syst. Theory3, 4, 320-375, DOI: 10. 1007/BF01691062.
59. Hedlund, G. A., K. I. Appel, and L. R. Welch (1963), All onto functions of span less than or equal to five, Tech. Rep., Communications Research Division.
60. Henderson, J. R., I. G. Main, C. Maclean, and M. G. Norman (1994), A fracturemechanical cellular automaton model of seismicity, Pure Appl. Geophys. 142, 3-4, 545-565, DOI: 10. 1007/BF00876054.
61. Henley, C. L. (1993), Statics of a "self-organized" percolation model, Phys. Rev. Lett. 71, 17, 2741-2744, DOI: 10. 1103/PhysRevLett. 71. 2741.
62. Hetherington, A., and S. Steacy (2007), Fault heterogeneity and earthquake scaling, Geophys. Res. Lett. 34, 16, L16310, DOI: 10. 1029/2007GL030365.
63. Hirata, T. (1989), A correlation between the b value and the fractal dimension of earthquakes, J. Geophys. Res. 94, B6, 7507-7514, DOI: 10. 1029/JB094iB06p07507.
64. Hirata, T., and M. Imoto (1991), Multifractal analysis of spatial distribution of microearthquakes in the Kanto region, Geophys. J. Int. 107, 1, 155-162, DOI: 10. 1111/j. 1365-246X. 1991. tb01163. x.
65. Holland, J. H. (1966), Universal spaces: A basis for studies of adaptation. In: E. R. Caianiello (ed.), Automata Theory, Academic Press, New York, 218-230.
66. Huang, Y., H. Saleur, C. Sammis, and D. Sornette (1998), Precursors, aftershocks, criticality and self-organized criticality, Europhys. Lett. 41, 1, 43-48, DOI: 10. 1209/epl/i1998-00113-x.
67. Ito, K., and M. Matsuzaki (1990), Earthquakes as self-organized critical phenomena, J. Geophys. Res. 95, B5, 6853-6860, DOI: 10. 1029/JB095iB05p06853.
68. Iudin, D. I., and A. N. Grigoriev (2003), Cellular automaton model of lithosphere degassing, Nucl. Instrum. Methods A502, 2-3, 736-738, DOI: 10. 1016/S0168-9002(03)00561-8.
69. Jiménez, A., and A. M. Posadas (2006), A Moore's cellular automaton model to get probabilistic seismic hazard maps for different magnitude releases: A case study for Greece, Tectonophysics423, 1-4, 35-42, DOI: 10. 1016/j. tecto. 2006. 03. 030.
70. Jiménez, A., A. M. Posadas, T. Hirata, and J. M. García (2004), Probabilistic seismic hazard maps from seismicity patterns analysis: the Iberian Peninsula case, Nat. Hazard Earth Syst. Sci. 4, 3, 407-416, DOI: 10. 5194/nhess-4-407-2004.
71. Jiménez, A., A. M. Posadas, and J. M. Marfil (2005), A probabilistic seismic hazard model based on cellular automata and information theory, Nonlin. Process. Geophys. 12, 3, 381-396, DOI: 10. 5194/npg-12-381-2005.
72. Jiménez, A., K. F. Tiampo, and A. M. Posadas (2007), An Ising model for earthquake dynamics, Nonlin. Process. Geophys. 14, 1, 5-15, DOI: 10. 5194/npg-14-5-2007.
73. Jiménez, A., A. M. Posadas, and K. F. Tiampo (2008), Describing seismic pattern dynamics by means of Ising Cellular Automata. In: R. V. Donner and S. M. Barbosa (eds.), Nonlinear Time Series Analysis in the Geosciences, Springer, Berlin Heidelberg, 273-290, DOI: 10. 1007/978-3-540-78938-3_12.
74. Kanki, M., J. Mada, and T. Tokihiro (2012), Discrete integrable equations over finite fields, SIGMA8, 054, DOI: 10. 3842/SIGMA. 2012. 054.
75. Kari, J. (1994), Reversibility and surjectivity of problems of cellular automata, J. Comp. Syst. Sci. 48, 1, 149-182, DOI: 10. 1016/S0022-0000(05)80025-X.
76. Kleene, S. C. (1956), Representation of events in nerve nets and finite automata. In: C. E. Shannon and J. McCarthy (eds.), Automata Studies, Annals of Mathematics Studies, Princeton University Press, Princeton, 3-42.
77. Klein, W., J. B. Rundle, and C. D. Ferguson (1997), Scaling and nucleation in models of earthquake faults, Phys. Rev. Lett. 78, 19, 3793-3796, DOI: 10. 1103/PhysRevLett. 78. 3793.
78. Kossobokov, V. G., V. I. Keilis-Borok, D. L. Turcotte, and B. D. Malamud (2000), Implications of a statistical physics approach for earthquake hazard assessment and forecasting, Pure Appl. Geophys. 157, 11-12, 2323-2349, DOI: 10. 1007/PL00001086.
79. Lahaie, F., and J. R. Grasso (1998), A fluid-rock interaction cellular automaton of volcano mechanics: Application to the Piton de la Fournaise, J. Geophys. Res. 103, B5, 9637-9649, DOI: 10. 1029/98JB00202.
80. Leamy, M. J. (2008), Application of cellular automata modeling to seismic elastodynamics, Int. J. Solids Struct. 45, 17, 4835-4849, DOI: 10. 1016/j. ijsolstr. 2008. 04. 021.
81. Leduc, T. (1997), One-dimensional discrete computer model of the subduction erosion phenomenon (plate tectonics process). In: Proc. CESA′98, Symposium in Applied Mathematics and Optimization, April 1998, Nabeul-Hammamet, Tunisia.
82. Lee, Y.-T., C.-C. Chen, C.-Y. Lin, and S.-C. Chi (2012a), Negative correlation between power-law scaling and Hurst exponents in long-range connective sandpile models and real seismicity, Chaos Soliton. Fract. 45, 2, 125-130, DOI: 10. 1016/j. chaos. 2011. 10. 009.
83. Lee, Y.-T., L. Telesca, and C.-C. Chen (2012b), Negative correlation between frequencymagnitude power-law exponent and Hurst coefficient in the Long-Range Connective Sandpile model for earthquakes and for real seismicity, Europhys. Lett. 99, 2, 29001, DOI: 10. 1209/0295-5075/99/29001.
84. Lomnitz-Adler, J. (1993), Automaton models of seismic fracture: Constraints imposed by the magnitude-frequency relation, J. Geophys. Res. 98, B10, 17745-17756, DOI: 10. 1029/93JB01390.
85. Main, I. (1996), Statistical physics, seismogenesis, and seismic hazard, Rev. Geophys. 34, 4, 433-462, DOI: 10. 1029/96RG02808.
86. Main, I. G., P. G. Meredith, J. R. Henderson, and P. R. Sammonds (1994), Positive and negative feedback in the earthquake cycle: the role of pore fluids on states of criticality in the crust, Ann. Geofis. 37, 6, 1461-1479.
87. McCulloch, W. S., and W. Pitts (1943), A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys. 5, 4, 115-133, DOI: 10. 1007/BF02478259.
88. McGinnis, S. A. (2001), On the effects of geometry in discrete element numerical earthquake simulations, Ph. D. Thesis, University of Colorado, Boulder.
89. Miranda, E. R. (1994), Music composition using cellular automata, Lang. Design2, 105-107.
90. Moore, E. F. (1962), Machine models of self-reproduction. In: Mathematical Problems in the Biological Sciences, Proceedings of Symposia in Applied Mathematics, Vol. 14, American Mathematical Society, 17-33.
91. Morein, G., D. L. Turcotte, and A. Gabrielov (1997), On the statistical mechanics of distributed seismicity, Geophys. J. Int. 131, 3, 552-558, DOI: 10. 1111/j. 1365-246X. 1997. tb06599. x.
92. Mori, T., and H. Kawamura (2005), Simulation study of spatiotemporal correlations of earthquakes as a stick-slip frictional instability, Phys. Rev. Lett. 94, 5, 058501, DOI: 10. 1103/PhysRevLett. 94. 058501.
93. Muñoz-Diosdado, A., F. Angulo-Brown, and J. L. del Río-Correa (2004), Multifractal analysis of a spring-block seismic fault. In: Proc. 13th World Conf. on Earthquake Engineering, 1-6 August 2004, Vancouver, Canada, paper no. 525.
94. Muñoz-Diosdado, A., A. H. Rudolf-Navarro, and F. Angulo-Brown (2012), Simulation and properties of a non-homogeneous spring-block earthquake model with asperities, Acta Geophys. 60, 3, 740-757, DOI: 10. 2478/s11600-012-0027-7.
95. Myhill, J. (1963), The converse of Moore's Garden-of-Eden theorem, Proc. Am. Math. Soc. 14, 685-686.
96. Nakanishi, H. (1990), Cellular-automaton model of earthquakes with deterministic dynamics, Phys. Rev. A41, 12, 7086-7089, DOI: 10. 1103/PhysRevA. 41. 7086.
97. Nakanishi, H. (1991), Statistical properties of the cellular-automaton model for earthquakes, Phys. Rev. A43, 12, 6613-6621, DOI: 10. 1103/PhysRevA. 43. 6613.
98. Newman, M. E. J. (2011), Complex systems: A survey, Am. J. Phys. 79, 800-810, DOI: 10. 1119/1. 3590372.
99. Nishiyama, A., and T. Tokihiro (2011), Construction of an isotropic cellular automaton for a reaction-diffusion equation by means of a random walk, J. Phys. Soc. Jpn. 80, 5, 054003, DOI: 10. 1143/JPSJ. 80. 054003.
100. Ohta, J., and I. Matsuba (1999), Analysis of earthquakes based on a dissipative cellular-automata model, Electron. Comm. Jpn. Pt. III82, 2, 20-27, DOI: 10. 1002/(SICI)1520-6440(199902)82: 2〈20:: AID-ECJC3〉3. 0. CO;2-S.
101. Olami, Z., H. J. S. Feder, and K. Christensen (1992), Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes, Phys. Rev. Lett. 68, 8, 1244-1247, DOI: 10. 1103/PhysRevLett. 68. 1244.
102. Omori, F. (1895), On the aftershocks of earthquakes, J. Coll. Sci. Imper. Univ. Jpn7, 2, 111-200.
103. Otsuka, M. (1972), A simulation of earthquake occurrence, Phys. Earth Planet. Inter. 6, 4, 311-315, DOI: 10. 1016/0031-9201 (72)90015-5.
104. Paczuski, M., and S. Boettcher (1996), Universality in sandpiles, interface depinning, and earthquake models, Phys. Rev. Lett. 77, 1, 111-114, DOI: 10. 1103/PhysRevLett. 77. 111.
105. Preston, E. F., J. de sa Martins, J. B. Rundle, M. Anghel, and W. Klein (2000), Models of earthquake faults with long-range stress transfer, IEEE Comput. Sci. Eng. 2, 3, 34-41, DOI: 10. 1109/5992. 841794.
106. Preston, K., and M. Duff (eds.) (1984), Modern Cellular Automata: Theory and Applications, Plenum Press, New York.
107. Ramos, O. (2010), Criticality in earthquakes. Good or bad for prediction?, Tectonophysics485, 1-4, 321-326, DOI: 10. 1016/j. tecto. 2009. 11. 007.
108. Reid, H. F. (1910), The California earthquake of April 18, 1906: The mechanics of the earthquake, Report of the State Earthquake Investigative Committee, Carnegie Institute, Washington D. C.
109. Richardson, D. (1972), Tessellations with local transformations, J. Comp. Syst. Sci. 6, 5, 373-388, DOI: 10. 1016/S0022-0000 (72)80009-6.
110. Rundle, J. B. (1988), A physical model for earthquakes. 2. Application to southern California, J. Geophys. Res. 93, B6, 6255-6274, DOI: 10. 1029/JB093iB 06p06255.
111. Rundle, J. B., and D. D. Jackson (1977), Numerical simulation of earthquake sequences, Bull. Seismol. Soc. Am. 67, 5, 1363-1377.
112. Rundle, J. B., and W. Klein (1993), Scaling and critical phenomena in a cellular automaton slider-block model for earthquakes, J. Stat. Phys. 72, 1/2, 405-412, DOI: 10. 1007/BF01048056.
113. Rundle, J. B., W. Klein, and S. Gross (1996), Dynamics of a travelling density wave model for earthquakes, Phys. Rev. Lett. 76, 22, 4285-4288, DOI: 10. 1103/PhysRevLett. 76. 4285.
114. Rundle, J. B., W. Klein, and S. Gross (1999), Physical basis for statistical patterns in complex earthquake populations: models, predictions and tests, Pure Appl. Geophys. 155, 2-4, 575-607, DOI: 10. 1007/s000240050278.
115. Rundle, J. B., W. Klein, K. Tiampo, and S. Gross (2000a), Linear pattern dynamics in nonlinear threshold systems, Phys. Rev. E61, 3, 2418-2431, DOI: 10. 1103/PhysRevE. 61. 2418.
116. Rundle, J. B., W. Klein, D. L. Turcotte, and B. D. Malamud (2000b), Precursory seismic activation and critical-point phenomena, Pure Appl. Geophys. 157, 11-12, 2165-2182, DOI: 10. 1007/PL00001079.
117. Rundle, J. B., P. B. Rundle, W. Klein, J. de sa Martins, K. F. Tiampo, A. Donnellan, and L. H. Kellogg (2002), GEM plate boundary simulations for the plate boundary observatory: A program for understanding the physics of earthquakes on complex fault networks via observations, theory and numerical simulation, Pure Appl. Geophys. 159, 10, 2357-2381, DOI: 10. 1007/s00024-002-8739-2.
118. Rundle, J. B., D. L. Turcotte, R. Shcherbakov, W. Klein, and C. Sammis (2003), Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems, Rev. Geophys. 41, 4, 1019-1049, DOI: 10. 1029/2003RG000135.
119. Rundle, P. B., J. B. Rundle, K. F. Tiampo, J. de sa Martins, S. McGinnis, and W. Klein (2001), Nonlinear network dynamics on earthquake fault systems, Phys. Rev. Lett. 87, 14, 148501, DOI: 10. 1103/PhysRevLett. 87. 148501.
120. Sammis, C. G., and S. W. Smith (1999), Seismic cycles and the evolution of stress correlation in cellular automaton models of finite fault networks, Pure Appl. Geophys. 155, 2-4, 307-334, DOI: 10. 1007/s000240050267.
121. Sarlis, N. V., E. S. Skordas, and P. A. Varotsos (2011), The change of the entropy in natural time under time-reversal in the Olami-Feder-Christensen earthquake model, Tectonophysics, 513, 1-4, 49-53, DOI: 10. 1016/j. tecto. 2011. 09. 025.
122. Smalley, R. F., J.-L. Chatelain, D. L. Turcotte, and R. Prévot (1987), A fractal approach to the clustering of earthquakes: Applications to the seismicity of the New Hebrides, Bull. Seismol. Soc. Am. 77, 4, 1368-1381.
123. Smith, A. R. III (1969), Cellular automata theory, Tech. Rep. 2, Stanford Electronics Labs, Stanford University, USA.
124. Sornette, A., and D. Sornette (1989), Self-organized criticality and earthquakes, Europhys. Lett. 9, 3, 197-202, DOI: 10. 1209/0295-5075/9/3/002.
125. Sornette, D., and C. G. Sammis (1995), Complex critical exponents from renormalization group theory of earthquakes: Implications for earthquake predictions, J. Phys. I France5, 5, 607-619, DOI: 10. 1051/jp1: 1995154.
126. Sornette, D., P. Davy, and A. Sornette (1990), Structuration of the lithosphere in plate tectonics as a self-organized critical phenomenon, J. Geophys. Res. 95, B11, 17353-17361, DOI: 10. 1029/JB095iB11p17353.
127. Steacy, S. J., and J. McCloskey (1999), Heterogeneity and the earthquake magnitudefrequency distribution, Geophys. Res. Lett. 26, 7, 899-902, DOI: 10. 1029/1999GL900135.
128. Steacy, S. J., and C. G. Sammis (1991), An automaton for fractal patterns of fragmentation, Nature353, 6341, 250-252, DOI: 10. 1038/353250a0.
129. Takahashi, D., and J. Satsuma (1990), A soliton cellular automaton, J. Phys. Soc. Jpn. 59, 10, 3514-3519, DOI: 10. 1143/JPSJ. 59. 3514.
130. Tejedor, A., J. B. Gómez, and A. F. Pacheco (2009), Earthquake size-frequency statistics in a forest-fire model of individual faults, Phys. Rev. E79, 4, 046102, DOI: 10. 1103/PhysRevE. 79. 046102.
131. Toffoli, T. (1984), Cellular automata as an alternative to (rather than an approximation of) differential equations in modeling physics, Physica D10, 1-2, 117-127, DOI: 10. 1016/0167-2789 (84)90254-9.
132. Toffoli, T., and N. Margolus (1987), Cellular Automata Machines: A New Environment for Modelling, 3rd ed., MIT Press, 259 pp.
133. Tokihiro, T., D. Takahashi, J. Matsukidaira, and J. Satsuma (1996), From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett. 76, 18, 3247-3250, DOI: 10. 1103/PhysRevLett. 76. 3247.
134. Turcotte, D. L. (1997), Fractals and Chaos in Geology and Geophysics, 2nd ed., Cambridge University Press, Cambridge, 398 pp.
135. Turcotte, D. L. (1999a), Self-organized criticality, Rep. Prog. Phys. 62, 1377-1429, DOI: 10. 1088/0034-4885/62/10/201.
136. Turcotte, D. L. (1999b), Seismicity and self-organized criticality, Phys. Earth Planet. In. 111, 3-4, 275-93, DOI: 10. 1016/S0031-9201 (98)00167-8.
137. Turing, A. M. (1965), On computable numbers, with an application to the Entscheidungsproblem. In: M. Davis (ed.), The Undecidable, Raven Press, Hewlett, 115-153 (reprint from Proc. London Mathematical Society, ser. 2, vol. 42 (1936), pp. 230-265; corr. ibid, vol. 43 (1937), pp. 544-546), http://www. abelard. org/turpap2/tp2-ie. asp.
138. Ulam, S. (1952), Random processes and transformations. In: L. M. Graves, E. Hille, P. A. Smith, and O. Zariski (eds.), Proc. International Congress of Mathematicians, Providence, Rhode Island, 1950, Vol. 2, 264-275.
139. Upper, D. R. (1997), Theory and algorithms for hidden Markov models and generalized hidden Markov models, Ph. D. Thesis, University of California at Berkeley, USA.
140. Vere-Jones, D. (2009), Earthquake occurrence and mechanisms, stochastic models for. In: R. A. Meyers (ed.), Encyclopedia of Complexity and Systems Science, Springer, 2555-2581, DOI: 10. 1007/978-0-387-30440-3_155.
141. Von Neumann, J. (1966), Theory of Self-Reproducing Automata, University of Illinois Press, Urbana.
142. Wang, Y.-C., X.-C. Yin, F.-J. Ke, M.-F. Xia, and K.-Y. Peng (2000), Numerical simulation of rock failure and earthquake process on mesoscopic scale, Pure Appl. Geophys. 157, 11-12, 1905-1928, DOI: 10. 1007/PL00001067.
143. Weatherley, D. (2006), Recurrence interval statistics of cellular automaton seismicity models, Pure Appl. Geophys. 163, 9, 1933-1947, DOI: 10. 1007/s00024-006-0105-3.
144. Weatherley, D., S. C. Jaumé, and P. Mora (2000), Evolution of stress deficit and changing rates of seismicity in cellular automaton models of earthquake faults, Pure Appl. Geophys. 157, 11-12, 2183-2307, DOI: 10. 1007/PL00001080.
145. Weimar, J. R. (1998), Simulation with Cellular Automata, Logos Verlag, Berlin.
146. Wolfram, S. (1983), Cellular automata, Los Alamos Science9, 2-27.
147. Wolfram, S. (1986), Theory and Applications of Cellular Automata, World Scientific.
148. Wolfram, S. (1993), Cellular Automata and Complexity: Collected Papers, Perseus Publishing.
149. Yakovlev, G., J. D. Gran, D. L. Turcotte, J. B. Rundle, J. R. Holliday, and W. Klein (2010), A damage-mechanics model for fracture nucleation and propagation, Theor. Appl. Frac. Mech. 53, 3, 180-184, DOI: 10. 1016/j. tafmec. 2010. 06. 002.
150. Zuse, K. (1969), Rechnender Raum, Tech. Rep. AZT-70-164-GEMIT, MIT Project MAC (translated as "Calculating space").