Advertisement

A Review of Earthquake Statistics: Fault and Seismicity-Based Models, ETAS and BASS

  • James R. Holliday
  • Donald L. Turcotte
  • John B. Rundle
Chapter
Part of the Pageoph Topical Volumes book series (PTV)

Abstract

There are two fundamentally different approaches to assessing the probabilistic risk of earthquake occurrence. The first is fault based. The statistical occurrence of earthquakes is determined for mapped faults. The applicable models are renewal models in that a tectonic loading of faults is included. The second approach is seismicity based. The risk of future earthquakes is based on the past seismicity in the region. These are also known as cluster models. An example of a cluster model is the epidemic type aftershock sequence (ETAS) model. In this paper we discuss an alternative branching aftershock sequence (BASS) model. In the BASS model an initial, or seed, earthquake is specified. The subsequent earthquakes are obtained from statistical distributions of magnitude, time, and location. The magnitude scaling is based on a combination of the Gutenberg-Richter scaling relation and the modified Båth’s law for the scaling relation of aftershock magnitudes relative to the magnitude of the main earthquake. Omori’s law specifies the distribution of earthquake times, and a modified form of Omori’s law specifies the distribution of earthquake locations. Unlike the ETAS model, the BASS model is fully self-similar, and is not sensitive to the low magnitude cutoff.

Keywords

Large Earthquake Main Shock Aftershock Sequence Large Aftershock Probabilistic Seismic Hazard Assessment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Båth, M. (1965), Lateral inhomogeneities in the upper mantle, Tectonophysics 2, 483–514.CrossRefGoogle 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, 24359–24372.CrossRefGoogle Scholar
  3. Bufe, C.G. and Varnes, D.J. (1993), Predictive modeling of the seismic cycle of the greater San Francisco Bay region, J. Geophys. Res. 98, 9871–9883.CrossRefGoogle Scholar
  4. Console, R. and Murru, M. (2001), A simple and testable model for earthquake clustering, J. Geophys. Res. 106, 8699–8711.CrossRefGoogle Scholar
  5. Console, R., Murru, M., and Catalli, F. (2006), Physical and stochastic models of earthquake clustering, Tectonophysics 417, 141–153.CrossRefGoogle Scholar
  6. Console, R., Murru, M., and Lombardi, A.M. (2003), Refining earthquake clustering models, J. Geophys. Res. 108, 2468.CrossRefGoogle Scholar
  7. Ellsworth, W.L., Mathews, M.V., Nadeau, R.M., Nishenko, S.P., Reasenberg, P.A., and Simpson, R.W. (1999), A physically-based earthquake recurrence model for estimation of long-term earthquake probabilities, Open-File Report 99-522, US Geological Survey.Google Scholar
  8. Felzer, K.R., Becker, T.W., Abercrombie, R.E., Ekstrom, G., and Rice, J.R. (2002) Triggering of the 1999 m w 7.1 hector mine earthquake by aftershocks of the 1992 m w 7.3 landers earthquake, J. Geophys. Res. 107(B9), 2190.CrossRefGoogle Scholar
  9. Felzer, K.R., Abercrombie, R.E., and Ekstrom, G. (2003), Secondary aftershocks and their importance for aftershock forecasting. Bull. Seismol. Soc. Am. 93(4), 1433–1448.CrossRefGoogle Scholar
  10. Felzer, K.R., Abercrombie, R.E., and Ekstrom, G. (2004), A common origin for aftershocks, foreshocks, and multiplets, Bull. Seismol. Soc. Am. 94, 88–98.CrossRefGoogle Scholar
  11. Felzer, K.R. and Brodsky, E.E. (2006), Decay of aftershock density with distance indicates triggering by dynamic stress, Nature, 441, 735–738.CrossRefGoogle Scholar
  12. Field, E.H. (2007a). Overview of the working group for the development of regional earthquake likelihood models (RELM), Seis. Res. Lett. 78, 7–16.CrossRefGoogle Scholar
  13. Field, E.H. (2007b). A summary of previous working groups on California earthquake probabilities, Seismol. Soc. Am. Bull. 97, 1033–1053.CrossRefGoogle Scholar
  14. Frankel, A.F., Muller, C., Barnhard, T., Perkins, D., Leyendecker, E.V., Dickman, N., Hanson, S., and Hopper, M. (1996), National seismic hazard maps, Open-File Report 96-532, US Geological Survey.Google Scholar
  15. Gabrielov, A., Newman, W.I., and Turcotte, D.L. (1999), Exactly soluble hierarchical clustering model: Inverse cascades, self-similarity, and scaling, Phys. Rev. E, 60, 5293.CrossRefGoogle Scholar
  16. Gersterberger, M. Wiemer, S., and Jonese, L. (2004), Real-time forecasts of tomorrow’s earthquakes in California: A new mapping tool, Open-File Report 2004-1390, US Geological Survey.Google Scholar
  17. Gersterberger, M.C., Wiemer, S., Jones, L.M., and Reasenberg, P.A. (2005), Real-time forecasts of tomorrow’s earthquakes in California, Nature 435, 328–331.CrossRefGoogle Scholar
  18. Goes, S.D.B. and Ward, S.N. (1994), Synthetic seismicity for the San Andreas fault, Annali Di Geofisica 37, 1495–1513.Google Scholar
  19. Gross, S. and Rundle, J.B. (1998), A systematic test of time-to-failure analysis, Geophys. J. Int. 133, 57–64.CrossRefGoogle Scholar
  20. Guo, Z.Q. and Ogata, Y. (1997), Statistical relations between the parameters of aftershocks in time, space, and magnitude, J. Geophys. J. Res. 102(B2), 2857–2873.CrossRefGoogle Scholar
  21. Gutenberg, B. and Richter, C.F., Seismicity of the Earth and Associated Phenomena (Princeton University Press, Princeton, NJ 1954).Google Scholar
  22. Helmstetter, A. (2003), Is earthquake triggering driven by small earthquakes? Phys. Rev. Let. 91, 0585014.Google Scholar
  23. Elmstetter, A. and Sornette, D. (2002a), Diffusion of epicenters of earthquake aftershocks, Omori’s law, and generalized continuous-time random walk models, Phys. Rev. E 66(6), 061104.CrossRefGoogle Scholar
  24. Helmstetter, A. and Sornette, D. (2002b), Subcritical and supercritical regimes in epidemic models of earthquake aftershocks, J. Geophys. Res. 107(B10), 2237.CrossRefGoogle Scholar
  25. Helmstetter, A. and Sornette, D. (2003a), Foreshocks explained by cascades of triggered seismicity, J. Geophys. Rev, 108(B10), 2457.CrossRefGoogle Scholar
  26. Helmstetter, A. and Sornette, D. (2003b), Predictability in the epidemic-type aftershock sequence model of interacting triggered seismicity, J. Geophys. Rev. 108(B10), 2482.CrossRefGoogle Scholar
  27. Helmstter, A.S. and Sornette, D. (2003c), Båth’s law derived from the Gutenberg-Richter law and from aftershock properties, Geophys. Res. Lett. 30(20), 2069.CrossRefGoogle Scholar
  28. Helmstter, A.S. and Sornette, D. (2003d), Importance of direct and indirect triggered seismicity in the ETAS model of seismicity. Geophys. Res. Lett. 30(11), 1576.CrossRefGoogle Scholar
  29. Helmstter, A.S. and Sornette, D., and Grasso, J.R. (2003a), Mainshocks are aftershocks of conditional foreshocks: How do fore shock statistical properties emerge from aftershock laws, J. Geophys. Res. 108(B1, 2046).CrossRefGoogle Scholar
  30. Helmstter, A.S., Ouillon, G., and Sornette, D. (2003b), Are aftershocks of large California earthquakes diffusing? J. Geophys. Res. 108(B10):2483.CrossRefGoogle Scholar
  31. Helmstter, A., Hergarten, S., and Sornette, D. (2004), Properties of foreshocks and aftershocks of the non-conservative self-organized critical Olami-Feder-Christensen model, Phys. Rev. E 70, 046120.CrossRefGoogle Scholar
  32. Helmstter, A., Kagan, Y.Y., and Jackson, D.D. (2006). Comparison of short-term and time-independent earthquake forecast models for southern California, Bull. Seismol. Soc. Am. 96, 90–106.CrossRefGoogle Scholar
  33. Holliday, J.R., Chen, C.C., Tiampo, K.F., Rundle, J.B., Turcotte, D.L., and Donnellan, A. (2007), A RELM earthquake forcast based on pattern informatics, Seis. Res. Lett. 78(1), 87–93.CrossRefGoogle Scholar
  34. Holliday, J.R., Nanjo, K.Z., Tiampo, K.F., Rundle, J.B., and Turcotte, D.L. (2005), Earthquake forecasting and its verification, Nonlinear Processes in Geophysics, 12, 965–977.Google Scholar
  35. Holliday, J.R., Rundle, J.B., Tiampo, K.F., Klein, W., and Donnellan, A. (2006a), Modification of the pattern informatics method for forecasting large earthquake events using complex eigenvectors, Tectonophys. 413, 87–91.CrossRefGoogle Scholar
  36. Holliday, J.R., Rundle, J.B., Tiampo, K.F., Klein, W., and Donnwllan, A. (2006b), Systematic procedural and sensitivity analysis of the pattern informatics method for forecasting large(M τ-5) earthquake events in southern California Pure Appl. Geophys.Google Scholar
  37. Kagan, Y.Y. and Knopoff, L. (1981), Stochastic synthesis of earthquake catalogs, J. Geophys. Res 86(4), 2853–2862.CrossRefGoogle Scholar
  38. Keilis-Borok, V.I. (1990), The lithosphere of the earth as a nonlinear system with implications for earthquake prediction, Rev. Geophys. 28, 19–34.CrossRefGoogle Scholar
  39. Keilis-Borok, V. (2002), Earthquake predictions: State-of-the-art and emerging possibilities, An. Rev. Earth Planet. Sci. 30:1–33.CrossRefGoogle Scholar
  40. Keilis-Borok, V., Shebalin, P., Gabrielov, A., and Turcotte, D. (2004), Reverse tracing of short-term earthquake precursors, Phys. Earth Planet. Int. 145, 75–85.CrossRefGoogle Scholar
  41. Kossobokov, V.G., Keilis-Borok, V.I., Turcotte, D.L., and Malamud, B.D. (2000), Implications of a statistical physics approach for earthquake hazard assessment and forecasting, Pure Appl. Geophys. 157, 2323–2349.CrossRefGoogle Scholar
  42. Lepiello, E., Godano, C., and De Arcangelis, L. (2007), Dynamically scaling in branching models for seismicity Phys. Rev. Lett. 98, 098501.CrossRefGoogle Scholar
  43. Main, I. (1996), Statistical physics, seismogenesis, and seismic hazard Rev. Geophys. 34, 433–462.CrossRefGoogle Scholar
  44. Main, I.G. (1999), Applicability of time-to-failure analysis to accelerated strain before earthquakes and volcanic eruptions, Geophys. J. Int 139, F1–F6.CrossRefGoogle Scholar
  45. Newman, W.I., Turcotte, D.L., and Gabrielov, A.M. (1997), Fractal trees with side branching, Fractals 5, 603–614.CrossRefGoogle Scholar
  46. Ogata, Y. (1988), Statistical models for earthquake occurrences and residual analysis for point processes, J. Am. Stat. Assoc. 83, 9–27.CrossRefGoogle Scholar
  47. Ogata, Y. (1989), Statistical model for standard seismicity and detection of anomalies by residual analysis, Tectonophysics 169, 159–174.CrossRefGoogle Scholar
  48. Ogata, Y. (1992), Detection of precursory relative quiescence before great earthquakes through a statistical model, J. Geophys Res. 97, 19845–19871.CrossRefGoogle Scholar
  49. Ogata, Y. (1998), Space-time point process models for earthquake occurrences, Ann. Inst. Statist. Math. 50, 379–402.CrossRefGoogle Scholar
  50. Ogata, Y. (1999), Seismicity analysis through point-process modeling: A review, Pure Appl. Geophys. 155, 471–507.CrossRefGoogle Scholar
  51. Ogata, Y. (2001a), Exploratory analysis of earthquake clusters by likelihood-based trigger models, J. Appl. Probab. 38A, 202–212.CrossRefGoogle Scholar
  52. Ogata, Y. (2001b), Increased probability of large earthquakes near aftershock regions with relative quiescence, J. Geophys. Res. 106, 8729–8744.CrossRefGoogle Scholar
  53. Ogata, Y. (2004), Space-time model for regional seismicity and detection of crustal stress changes, J. Geophys. Res. 109, B06308.CrossRefGoogle Scholar
  54. Ogata, Y., Matsuura, R.S., and Katusura, K. (1993), Fast likelihood computation of epidemic type aftershock-sequence model, Geophys. Res. Lett. 20, 2143–2146.CrossRefGoogle Scholar
  55. Ogata, Y., Jones, L.M., and Toda, S. (2003), When and where the aftershock activity was depressed: Contrasting decay patterns of the proximate large earthquakes in southern California, J. Geophys. Res. 108, 2318.CrossRefGoogle Scholar
  56. Ogata, Y. and Zhuang, J. (2006), Space-time ETAS models and an improved extension, Tectonophysics 413, 13–23.CrossRefGoogle Scholar
  57. Ossadnik, P. (1992), Branch order and ramification analysis of large diffusion limited aggregation clusters, Phys. Rev. A 45, 1058–1066.CrossRefGoogle Scholar
  58. Peckham, S.D. (1995), New results for self-similar trees with applications to river networks, Water Resour. Res. 31, 1023–1029.CrossRefGoogle Scholar
  59. Pelletier, J.D. (1999), Self-organization and scaling relationships of evolving river networks, J. Geophys. Res. 104, 7359–7375.CrossRefGoogle Scholar
  60. Reasenberg, P.A. (1999), Foreshock occurrence rates before large earthquake worldwide, Pure. Appl. Geophys. 155, 355–379.CrossRefGoogle Scholar
  61. Reasenberg, P.A. and Jones, L.M. (1989), Earthquake hazard after a mainshock in California, Science 243(4895), 1173–1176.CrossRefGoogle Scholar
  62. Rikitake, T., Earthquake Forecasting and Warning, (D. Reidel Publishing Co, Dordrecht. 1982)Google Scholar
  63. Robison, R. and Benites, R. (1995), Synthetic seismicity models Of multiple interacting faults, J. Geophys. Res. 100, 18229–18238.CrossRefGoogle Scholar
  64. Robison, R. and Benites, R. (1996), Synthetic seismicity models for the Wellington Region, New Zealand: Implications for the temporal distribution of large events, J. Geophys. Res. 101, 27833–27844.CrossRefGoogle Scholar
  65. Rundle, J.B., Tiampo, K.F., Klein, W., and Martins, J.S.S. (2002), Self-organization in leaky threshold systems: The influence of near-mean field dynamics and its implications for earthquakes, neurobiology, and forecasting, Proc. NatL. Acad. Sci. U.S.A. 99, 2514–2521, Suppl. 1.CrossRefGoogle Scholar
  66. Rundle, J.B., Turcotte, D.L., Shcherbakov, R., Klein, W., and Sammis, C. (2003), Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems, Rev. Geophys. 41(4), 1019.CrossRefGoogle Scholar
  67. Rundle, J.B., Rundle, P.B., Donnellan, A., and Fox, G. (2004), Gutenberg-Richter statistics in topologically realistic system-level earthquake stress-evolution simulations, Earth, Planets and Space 55(8), 761–771.Google Scholar
  68. Rundle, J.B., Rundle, P.B. and Donnellan, A. (2005), A simulation-based approach to forecasting the next great San Francisco earthquake, Proc. Natl. Acad. Sci. 102(43), 15363–15367.CrossRefGoogle Scholar
  69. Rundle, P.B., Rundle, J.B., Tiampo, K.F., Donnellan, A., and Turcotte, D.L. (2006), Virtual California: Fault model, frictional parameters, applications, Pure AppL. Geophys. 163, 1819–1846.CrossRefGoogle Scholar
  70. Saichev, A., Helmstetter, A., and Sornette, D. (2005), Power-law distributions of offspring and generation numbers in branching models of earthquake triggering, Pure Appl. Geophys. 162, 1113–1134.CrossRefGoogle Scholar
  71. Saichev, A. and Sornette, D. (2004), Anomalous power law distribution of total lifetimes of branching processes: Application to earthquake aftershock sequences, Phys. Rev. E 70(4), 046123.CrossRefGoogle Scholar
  72. Saichev, A. and Sornette, D. (2005a), Distribution of the largest aftershocks in branching models of triggered seismicity: Theory of the universal Båth’s law, Phys. Rev. E 71(5), 056127.CrossRefGoogle Scholar
  73. Saichev, A. and Sornette, D. (2005b), Vere-Jonesself-similar branching model, Phys. Rev. E 72, 056122.CrossRefGoogle Scholar
  74. Saichev, A. and Sornette, D. (2006a), Power-law distribution of seismic rates: theory and data analysis, Eur. Phys. J. B49, 377–401.Google Scholar
  75. Saichev, A. and Sornette, D. (2006b), Renormalization of branching models of triggered seismicity from total to observed seismicity, Eur. Phys. J. B51, 443–459.Google Scholar
  76. Saichev, A. and Sornette, D. (2006c), “Universal” distribution of interearthquake times explained, Phys. Rev. Lett. 97, 078501.CrossRefGoogle Scholar
  77. Saichev, A. and Sornette, D. (2007a), Power-law distributions of seismic rates, Tectonophysics 431, 7–13.CrossRefGoogle Scholar
  78. Saichev, A. and Sornette, D. (2007b), Theory of Earthquake recurrence times, J. Geophys. Res. 112, B04313.CrossRefGoogle Scholar
  79. Sammis, C.G., Bowman, D.D., and King, G. (2004), Anomalous seismicity and accelerating moment release preceding the 2001–2002 earthquakes in northern Baha California, Mexico, Pure Appl. Geophys 161, 2369–2378.CrossRefGoogle Scholar
  80. Shcherbakov, R. and Turcotte, D.L. (2004), A modified form of Båth’s law, Bull. Seismol. Soc. Am. 94, 1968–1975.CrossRefGoogle Scholar
  81. Shcherbakov, R., Turcotte, D.L., and Rundle, J.B. (2004), A generalized Omori’s law for earthquake aftershock decay, Geophys. Res. Lett. 31, L11613.CrossRefGoogle Scholar
  82. Shcherbakov, R., Turcotte, D.L., and Rundle, J.B. (2005), Aftershock statistics, Pure. Appl. Geophys. 162, 1051–1076.CrossRefGoogle Scholar
  83. Shebalin, P., Keilis-Borok, V., Zaliapin, I., Uyeda, S., Nagao, T., and Tsybin, N. (2004), Advance short-term prediction of the large Tokachi-oki earthquake, September 25, M = 8.1: A case history, Earth Planets Space 56, 715–724.Google Scholar
  84. Sornette, D. and Helmstetter, A. (2002), Occurrence of finite-time singularities in epedemic models of rupture, earthquakes, and starquakes, Phys. Rev. Lett. 89(15), 158501.CrossRefGoogle Scholar
  85. Sornette, D. and Werner, M.J. (2005a), Apparent clustering and apparent background earthquakes biased by undetected seismicity, J. Geophys. Res. 110, B09303.CrossRefGoogle Scholar
  86. Sornette, D. and Werner, M.J. (2005b), Constraints on the size of the smallest triggering earthquake from the epidemic-type aftershock sequence model, Bath’s law, and observed aftershock sequences, J. Geophys. Res. 110(B8), B08304.CrossRefGoogle Scholar
  87. Tiampo, K.F., Rundle, J.B., McGinnis, S., Gross, S.J., and Klein, W. (2002a), Eigenpatterns in southern California seismicity, J. Geophys. Res. 107(B12), 2354.CrossRefGoogle Scholar
  88. Tiampo, K.F., Rundle, J.B., McGinnis, S., and Klein, W. (2002b), Pattern dynamics and forecast methods in seismically active regions, Pure Appl. Geophys. 159, 2429–2467.CrossRefGoogle Scholar
  89. Tokunaga, E. (1978), Consideration on the composition of drainage networks and their evolution, Geographical Rep. Tokya Metro. Univ. 13, 1–27.Google Scholar
  90. Turcotte, D.L., Holliday, J.R., and Rundle, J.B. (2007), BASS, an alternative to ETAS, Geophys. Res. Lett. 34, L12303.CrossRefGoogle Scholar
  91. Turcotte, D.L. and Tewman, W.I. (1996), Symmetries in geology and geophysics Proc. Natl. Acad. Sci. 93, 14295–14300.CrossRefGoogle Scholar
  92. Turcotte, D.L., Pelletier, J.D., and Newman, W.I. (1998), Networks with side branching in biology, J. Theor. BioL. 193, 577–592.CrossRefGoogle Scholar
  93. Utsu, T. (1984), Estimation of parameters for recurrence models of earthquakes, Earthq. Res. Insti.-Univ. Tokyo, 59, 53–66.Google Scholar
  94. Vere-Jones, D. (1969), A note on the statistical interpretation of Båth’s law, Bull. SEismol. Soc. Am. 59, 1535–1541.Google Scholar
  95. Vere-Jones, D. (2005), A class of self-similar random measure, Advan. AppLi. Probab. 37, 908–914.CrossRefGoogle Scholar
  96. Ward S.N. (1992), An application of synthetic seismicity in earthquake statistics: The Middle America trench, J. Geophys. Res. 97(B5), 6675–6682.CrossRefGoogle Scholar
  97. Ward S.N. (1996), A synthetic seismicity model for southern California: cycles, probabilities, and hazard, J. Geophys. Res. 101(B10), 22393–22418.CrossRefGoogle Scholar
  98. Ward S.N. (2000), San Francisco Bay Area earthquake simulations: a step toward a standard physical earthquake model, Bull. Scismol. Soc. Am. 90(2), 370–386.CrossRefGoogle Scholar
  99. Working Group on California Earthquake Probabilities (1988), Probabilities of large earthquakes occurring in California on the San Andreas fault, Open-File Report 88-398, US Geological Survey.Google Scholar
  100. Working Group on California Earthquake Probabilities (1990), Probabilities of large earthquakes in the San Francisco Bay region, California, Circular 1053, US Geological Survey.Google Scholar
  101. Working Group on California Earthquake Probabilities (1995), Scismic hazards in southern California: probable earthquakes, 1994–2024, SEis. Soc. Am. Bull. 85, 379–439.Google Scholar
  102. Working Group on California Earthquake Probabilities (2003) Earthquake probabilities in the San Francisco Bay Region, 2002–2031, Open-File Report 2003-214, US Geological Survey.Google Scholar
  103. Yakovlev, G., Turcotte, D.L., Rundle, J.B., and Rundle, P.B. (2006), Simulation-based distributions of earthquake recurrence times on the San Andreas fault system, Bull. Seismol. Soc. Am. 96, 1995–2007.CrossRefGoogle Scholar
  104. Yamanaka, Y. and Shimazaki, K. (1990), Scaling relationship between the number of aftershocks and the size of the main shock, J. Phys. Earth 38(4), 305–324.Google Scholar
  105. Zhuang, J. and Ogata, Y. (2006), Properties of the probability distribution associated with the largest event in an earthquake cluster and their implications to foreshocks, Phys. Rev. E 73, 046134.CrossRefGoogle Scholar
  106. Zhuang, J., Ogata, Y., and Vere-Jones, D. (2002), Stochastic declustering of space-time earthquake occurrences, J. Am. Stat. Assoc. 97, 369–380.CrossRefGoogle Scholar
  107. Zhuang, J., Ogata, Y., and Vere-Jones, D. (2004), Analyzing earthquake clustering features by using stochastic reconstruction, J. Geophys. Res. 109, B05301.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 2008

Authors and Affiliations

  • James R. Holliday
    • 1
    • 2
  • Donald L. Turcotte
    • 3
  • John B. Rundle
    • 1
    • 2
    • 3
  1. 1.Center for Computational Science and EngineeringUniversity of CaliforniaDavis
  2. 2.Department of PhysicsUniversity of CaliforniaDavis
  3. 3.Department of GeologyUniversity of CaliforniaDavis

Personalised recommendations