Pure and Applied Geophysics

, Volume 167, Issue 6–7, pp 763–782 | Cite as

Ergodicity and Earthquake Catalogs: Forecast Testing and Resulting Implications

  • K. F. Tiampo
  • W. Klein
  • H.-C. Li
  • A. Mignan
  • Y. Toya
  • S. Z. L. Kohen-Kadosh
  • J. B. Rundle
  • C.-C. Chen


Recently the equilibrium property of ergodicity was identified in an earthquake fault system (Tiampoet al., Phys. Rev. Lett. 91, 238501, 2003; Phys. Rev. E 75, 066107, 2007). Ergodicity in this context not only requires that the system is stationary for these networks at the applicable spatial and temporal scales, but also implies that they are in a state of metastable equilibrium, one in which the ensemble averages can be substituted for temporal averages when studying their behavior in space and time. In this work we show that this property can be used to identify those regions of parameter space which are stationary when applied to the seismicity of two naturally-occurring earthquake fault networks. We apply this measure to one particular seismicity-based forecasting tool, the Pattern Informatics index (Tiampoet al., Europhys. Lett. 60, 481–487, 2002; Rundleet al., Proc. National Acad. Sci., U.S.A., Suppl. 1, 99, 2463, 2002), in order to test the hypothesis that the identification of ergodic regions can be used to improve and optimize forecasts that rely on historic seismicity catalogs. We also apply the same measure to synthetic catalogs in order to better understand the physical process that affects this accuracy. We show that, in particular, ergodic regions defined by magnitude and time period provide more reliable forecasts of future events in both natural and synthetic catalogs, and that these improvements can be directly related to specific features or properties of the catalogs that impact the behavior of their spatial and temporal statistics.


Earthquake forecasting ergodic behavior PI method Thirumalai–Mountain metric Canadian seismicity Taiwanese seismicity 


  1. Adams, J. E. and Basham, P. W. (1989), The seismicity and seismotectonics of Canada east of the Cordillera, Geoscience Canada 16, 3–16.Google Scholar
  2. Bowman, D. D., Ouillon, G., Sammis, C. G., Sornette, A., and Sornette, D. (1998), An observational test of the critical earthquake concept, J. Geophys. Res. 103, 24,359–24,372.Google Scholar
  3. Brehm, D. J. and Braile, L. W. (1998), Intermediate-term earthquake prediction using precursory events in the New Madrid Seismic Zone, Bull. Seismol. Soc. Am. 88, 564–580.Google Scholar
  4. Bufe, C. G. and Varnes, D. J. (1993), Predictive modeling of the seismic cycle of the greater San Francisco Bat region, J. Geophys. Res. 98, 9871–9883.Google Scholar
  5. Dieterich, J. (1994), A constitutive law for rate of earthquake production and its application to earthquake clustering, J. Geophys. Res. 99, 2601–2618.Google Scholar
  6. Dieterich, J. H., Cayol, V., and Okubo, P. (2000), The use of earthquake rate changes as a stress meter at Kilauea volcano, Nature 408, 457–460.Google Scholar
  7. Dodge, D. A., Beroza, G. C., and Ellsworth, W. L. (1996), Detailed observations of California foreshock sequences: Implications for the earthquake initiation process, J. Geophys. Res. 101, 22371–22392.Google Scholar
  8. Evison, F. F. and Rhoades, D. A. (2004), Demarkation and scaling of long-term seismogenesis, Pure Appl. Geophys. 161, 21–45.Google Scholar
  9. Felzer, K. R., Abercrombie, R. E., and Ekström, G. (2004), A common origin for aftershocks, foreshocks, and multiplets, Bull. Seismol. Soc. Am. 94, 88–98.Google Scholar
  10. Ferguson, C. D., Klein, W., and Rundle, J. B. (1999), Spinodals, scaling, and ergodicity in a threshold model with long-range stress transfer, Phys. Rev. E 60,1359–1374.Google Scholar
  11. Guo, Z. and Ogata, Y. (1997), Statistical relations between the parameters of aftershocks in time, space, and magnitude, J. Geophys. Res. 102, 2857–2873.Google Scholar
  12. Gutenberg, B. and Richter, C. F. (1944), Frequency of earthquakes in California, Bull. Seismol. Soc. Am. 34, 185–188.Google Scholar
  13. Habermann, R. E. (1983), Teleseismic detection in the Aleutian Island Arc, J. Geophys. Res. 88, 5056–5064.Google Scholar
  14. Habermann, R. E. (1987), Man-made changes of seismicity rates, Bull. Seism. Soc. Am. 77, 141–159.Google Scholar
  15. Helmstetter, A., Kagan, Y. Y., and Jackson, D. D. (2005), Importance of small earthquakes for stress transfers and earthquake triggering, J. Geophys. Res. 110, doi:10.1029/2004JB003286.
  16. Helmstetter, A., Ouillon, G., and Sornette, D. (2003), Are aftershocks of large Californian earthquakes diffusing? J. Geophys. Res. 108, doi: 10.1029/2003JB002503.
  17. Holliday, J., Chen, C., Tiampo, K. F., Rundle, J. B., and Turcotte, D. L. (2007), A RELM earthquake forecast based on pattern informatics, Seis. Res. Lt. 78, 87–93.Google Scholar
  18. Holliday, J. R., Nanjo, K. Z., Tiampo, K. F., Rundle, J. B., Tiampo, K. F., and Turcotte, D. L. (2005), Earthquake forecasting and its verification, Nonlinear Processes in Geophys. 12, 965–977.Google Scholar
  19. Holliday, J. R., Rundle, J. B., Tiampo, K. F., Klein, W., and Donnellan, A. (2006a), Systematic procedural and sensitivity analysis of the Pattern Informatics method for forecasting large (M > 5) earthquake events in southern California, Pure Appl. Geophys., doi:10.1007/s00024-006-0131-1.
  20. Holliday, J. R., Rundle, J. B., Tiampo, K. F., and Turcotte, D. L. (2006b), Using earthquake intensities to forecast earthquake occurrence times, Nonlinear Processes in Geophysics 13, 585–593.Google Scholar
  21. Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge, University Press, U.K. 1996).Google Scholar
  22. Jiménez, A., Tiampo, K. F., Levin, S., and Posadas, A. (2005), Testing the persistence in earthquake catalogs: The Iberian Peninsula , Europhys. Lett., doi:10.1209/epl/i2005-10383-8.
  23. Jolliffe, I. T. and Stephenson, D. B., Forecast Verification: A Practitioner’s Guide in Atmospheric Science (John Wiley, Hoboken, N. J. 2003).Google Scholar
  24. Keilis-Borok, V. I. (1999), What comes next in the dynamics of lithosphere and earthquake prediction? Phys. Earth Planet. Int. 111, 179–185.Google Scholar
  25. Klein, W., Rundle, J. B., and Ferguson, C. D. (1997), Scaling and nucleation in models of earthquake faults, Phys. Rev. Lett. 78, 3793–3796.Google Scholar
  26. Marsan, D. (2003), Triggering of seismicity at short timescales following Californian earthquakes, J. Geophys. Res. 108, 2266, doi:10.1029/2002JB001946.
  27. Matthews, M.V. and Reasenberg, P. (1987), Comment on Habermann’s method for detecting seismicity rate changes, J. Geophys. Res. 92, 9443–9445.Google Scholar
  28. Mignan, A. (2008a), The Non-Critical Precursory Accelerating Seismicity Theory (NC PAST) and limits of the power-law fit methodology, Tectonophysics 452, doi: 10.1016/j.tecto.2008.02.010.
  29. Mignan, A. (2008b), The stress accumulation model: Accelerating moment release and seismic hazard, Adv. Geophys. 49, doi: 10.1016/S0065-2687(07)49002-1.
  30. Mignan, A., King, G. C. P., and Bowman, D. D. (2007), A mathematical formulation of accelerating moment release based on the Stress Accumulation model, J. Geophys. Res. 112, B07308, doi: 10.1029/2006JB004671.
  31. Mignan, A. and Tiampo, K. F. (2010), Testing the Pattern Informatics index on synthetic seismicity catalogues based on the Non-Critical PAST, Tectonophysics 483, 255–268, doi:10.1016/j.tecto.2009.10.023.
  32. Mogi, K. (1969), Some features of recent seismic activity in and near Japan (2), Activity before and after large earthquakes, Bull. Earthquake Res. Inst., Univ. Tokyo 47, 395–417.Google Scholar
  33. Mogi, K. (1977), Seismic activity and earthquake predictions, Proc. Symp. on Earthquake Prediction, Seis. Soc. Japan, 203–214.Google Scholar
  34. Mogi, K. (1979), Two kind of seismic gaps, Pure Appl. Geophys. 117, 1172–1186.Google Scholar
  35. Moore, E. F. (1962), Machine models of self reproduction. In Proc. Fourteenth Symp. Appl. Math., 17–33.Google Scholar
  36. Mori, H. and Kuramoto, Y. Dissipative Structures and Chaos (Springer-Verlag, Berlin 1998).Google Scholar
  37. Mountain, R. M. and Thirumalai, D. (1992), Ergodicity and loss of dynamics in supercooled liquids, Phys. Rev. A 45, 3380–3383.Google Scholar
  38. Ogata, Y. and Zhuang, J. (2006), Space-time ETAS models and an improved extension, Tectonophysics 413, 13–23.Google Scholar
  39. Ogata, Y. (1992), Detection of precursory seismic quiescence before major earthquakes through a statistical model, J. Geophys. Res. 97, 19845–19871.Google Scholar
  40. Okada, Y. (1992), Internal deformation due to shear and tensile faults in a half-space, Bull. Seismol. Soc. Am. 82, 1018–1040.Google Scholar
  41. Orihara, Y., Noda, Y., Nagao, T., and Uyeda, S. (2002), A possible case of SES selectivity at Kozushima island, Japan, J. Geodyn. 33, 425–432.Google Scholar
  42. Papazachos, B. C., Karakaisis, G. F., Papazachos, C. B., and Scordilis, E. M. (2007), Evaluation of the results for an intermediate-term prediction of the 8 January, 2006 M w = 6.9 Cythera earthquake in the Southwestern Aegean, Bull. Seismol. Soc. Am. 97, 347–352, doi: 10.1785/0120060075.
  43. Reasenberg, P. A. (1985). Second-order moment of central California seismicity, J. Geophys. Res. 90, 5479–5495.Google Scholar
  44. Rundle, J. B., Klein, W., Tiampo, K. F., and Gross, S. (2000), Linear pattern dynamics of nonlinear threshold systems, Phys. Rev. E 61, 2418–2432.Google Scholar
  45. Rundle, J. B., Tiampo, K. F., Klein, W., and Sá Martins, J. S. (2002), Self-organization in leaky threshold systems: The influence of near mean field dynamics and its implications for earthquakes, neurobiology and forecasting, Proc. National Acad. Sci., U.S.A., Suppl. 1, 99, 2463.Google Scholar
  46. Savage, J. C. (1983), A dislocation model of strain accumulation and release at a subduction zone, J. Geophys. Res. 88, 4984–4996.Google Scholar
  47. Shcherbakov, R., Turcotte, D. L., and Rundle, J. B. (2005), Aftershock statistics, Pure Appl. Geophys. 162, 1051–1076.Google Scholar
  48. Thirumalai, D., Mountain, R. D., and Kirkpatrick, T. R. (1989), Ergodic behavior in supercooled liquids and in glasses, Phys. Rev. A 39, 3563–3574.Google Scholar
  49. Thirumalai, D. and Mountain, R. D. (1993), Activated dynamics, loss of ergodicity, and transport in supercooled liquids, Phys. Rev. E 47, 479–489.Google Scholar
  50. Tiampo, K. F., Rundle, J. B., Klein, W., Holliday, J., Sá Martins, J. S., and Ferguson, C. D. (2007), Ergodicity in natural earthquake fault networks, Phys. Rev. E 75, 066107.Google Scholar
  51. Tiampo, K. F., Rundle, J. B., and Klein, W. (2006a), Premonitory seismicity changes prior to the Parkfield and Coalinga earthquakes in southern California, Tectonophysics 413, 77–86.Google Scholar
  52. Tiampo, K. F., Rundle, J. B., and Klein, W. (2006b), Stress shadows determined from a phase dynamical measure of historic seismicity , Pure Appl. Geophys. doi:10.1007/200024-006-0134-y.
  53. Tiampo, K. F., Rundle, J. B., Klein, W., Sá Martins, J. S., and Ferguson, C. D. (2003), Ergodic dynamics in a natural threshold system, Phys. Rev. Lett. 91, 238501.Google Scholar
  54. Tiampo, K. F., Rundle, J. B., McGinnis, S., Gross, S., and Klein, W. (2002), Mean-field threshold systems and phase dynamics: An application to earthquake fault systems, Europhys. Lett. 60, 481–487.Google Scholar
  55. Toda, S., Stein, R. S., and Sagiya, T. (2002), Evidence from the AD 2000 Izu Islands earthquake swarm that stressing rate governs seismicity, Nature 419, 58–61.Google Scholar
  56. Toya, Y., Tiampo, K.F., Rundle, J.B., Chen, C., Li, H., and Klein, W. (2009), Pattern Informatics approach to earthquake forecasting in 3-D, Concurrency and Computation: Practice and Experience, doi: 10.1002/cpe.1531.
  57. Utsu, T. (1961), A statistical study on the occurrence of aftershocks, Geophys. Mag. 30, 521–605.Google Scholar
  58. Wiemer, S. (2005), A software package to analyze seismicity: ZMAP. (http://www.earthquake.ethz.ch/software/zmap).
  59. Wiemer, S. and Wyss, M. (2000), Minimum magnitude of completeness in earthquake catalogs: Examples from Alaska, the Western United States, and Japan, Bull. Seismol. Soc. Am. 90, 859–869.Google Scholar
  60. Wyss, M., and Wiemer, S. (1997), Two current seismic quiescences within 40 km of Tokyo, Geophys. J. Int. 128, 459–473.Google Scholar
  61. Wu, Y.-M., Chang, C.-H., Zhao, L., Teng, T.-L., and Nakamura, M., (2008), A comprehensive relocation of earthquakes in Taiwan from 1991 to 2005, Bull Seismol. Soc. Am. 98, 1471–1481.Google Scholar
  62. Zöller, G., Hainzl, S., and Kurths, J. (2000), Observation of growing correlation length as an indicator for critical point behavior prior to large earthquakes, J. Geophys. Res. 106, 2167–2175.Google Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  • K. F. Tiampo
    • 1
  • W. Klein
    • 2
  • H.-C. Li
    • 3
  • A. Mignan
    • 4
  • Y. Toya
    • 1
  • S. Z. L. Kohen-Kadosh
    • 1
  • J. B. Rundle
    • 5
  • C.-C. Chen
    • 3
  1. 1.Department of Earth SciencesUniversity of Western OntarioLondonCanada
  2. 2.Department of PhysicsBoston UniversityBostonUSA
  3. 3.Institute of GeophysicsNational Central UniversityJhongliTaiwan, ROC
  4. 4.Risk Management Solutions, Science and Technology ResearchLondonUK
  5. 5.Center for Computational Science and EngineeringUniversity of CaliforniaDavisUSA

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