Article Outline
Glossary
Definition of the Subject
Introduction
Historical Overview
Stochastic Models for Earthquake Mechanisms
Models for Paleoseismological and Historical Earthquakes
Point Process Models for Regional Catalogues
Stochastic Models with Precursors
Further Topics
Future Directions
Acknowledgments
Bibliography
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Abbreviations
- Stochastic:
-
occurring by chance;
- Stochastic process:
-
physical or other process evolving in time governed in part by chance.
- Earthquake mechanism:
-
physical processes causing the occurrence of an earthquake.
- Independent events:
-
events not affecting each other's probability of occurrence.
- Branching process:
-
process of ancestors and offspring, as in the model of nuclear fission.
- Point process:
-
stochastic process of point‐events in time or space.
- Probability forecast:
-
prediction of the probability distribution of the time and other features of some future event, as distinct from a forecast for the time (etc.) of the event itself.
- Model test:
-
a statistical test for the extent to which a stochastic model is supported by the relevant data.
- Precursory signal:
-
observed quantity which affects the occurrence probability of a future event (earthquake).
Bibliography
Ambraseys NN, Melville CP (1982) A History of Persian Earthquakes. Cambridge University Press, Cambridge
Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical Models Based on Counting Processes. Springer, New York
Bak P, Tang C (1989) Earthquakes as a self‐organized critical phenomenon. J Geophys Res 94:15635–15637
Bebbington M, Harte DS (2003) The linked stress release model for spatio‐temporal seismicity: formulations, procedures and applications. Geophys J Int 154:925–946
Bebbington M, Vere-Jones D, Zheng X (1990) Percolation theory: a model for earthquake faulting? Geophys J Int 100:215–220
Ben-Zion Y (1996) Stress, Slip and earthquakes in models of complex single‐fault systems incorporating brittle and creep deformations. J Geophys Res 101:5677–5706
Ben-Zion Y, Dahmen K, Lyakhowsky V, Ertas D, Agnon A (1999) Self‐driven mode‐switching of earthquake activity on a fault system. Earth Planet. Sci Lett 172:11–21
Ben-Zion Y, Eneva M, Liu Y (2003) Large earthquake cycles and intermittent criticality on heterogeneous faults due to evolving stress and seismicity. J Geophys Res 108:2307V. doi:10.1029/2002JB002121
Ben-Zion Y, Lyakhovsky V (2002) Accelerated seismic release and related aspects of seismicity patterns on earthquake faults. Pure Appl Geophys 159:2385–2412
Ben-Zion Y, Rice J (1995) Slip patterns and earthquake populations along different classes of faults on elastic solids. J Geophys Res 100:12959–12983
Borovkov K, Vere-Jones D (2000) Explicit formulae for stationary distributions of stress release processes. J Appl Prob 37:315–321
Brémaud P, Massoulié L (2001) Hawkes branching processes without ancestors. J App Prob 38:122–135
Brillinger DR (1981) Time Series: Data Analysis and Theory, 2nd edn. Holden Day, San Francisco
Burridge R, Knopoff L (1967) Model and theoretical seismicity. Bull Seismol Soc Am 57:341–371
Chelidze TL, Kolesnikov YM (1983) Modelling and forecasting the failure process in the framework of percolation theory. Izvestiya Earth Phys 19:347–354
Chong FS (1983) Time-space‐magnitude interdependence of upper crustal earthquakes in the main seismic region of New Zealand. J Geol Geophys 26:7–24, New Zealand
Console R, Lombardi AM, Murru M, Rhoades DA (2003) Båth's Law and the self‐similarity of earthquakes. J Geophys Res 108(B2):2128V. doi:10.1029/2001JB001651
Cox DR (1972) Regression models and life tables (with discussion). Roy J Stat Soc Ser B 34:187–220
Dahmen K, Ertas D, Ben-Zion Y (1998) Gutenberg‐Richter and characteristic earthquake behavior in simple mean-field models of heterogeneous faults. Phys Rev E 58:1494–1501
Daley DJ, Vere-Jones D (2003) An Introduction to the Theory of Point Processes, 2nd edn, vol I. Springer, New York
Davison C (1938) Studies on the Periodicity of Earthquakes. Murthy, London
Diggle PJ (2003) Statistical Analysis of Spatial Point Patterns. 2nd edn. University Press, Oxford
Ebel JB, Chambers DW, Kafka AL, Baglivo JA (2007) Non‐Poissonian earthquake clustering and the hidden Markov model as bases for earthquake forecasting in California. Seismol Res Lett 78:57–65
Evison F, Rhoades D (2001) Model of long-term seismogenesis. Annali Geofisica 44:81–93
Felzer KR, Abercrombie RE, Ekström G (2004) A common origin for aftershocks, foreshocks and multiplets. Bull Amer Seismol Soc 94:88–98
Fisher RL, Dahmen K, Ramanathan S, Ben-Zion Y (1997) Statistics of earthquakes in simple models of heterogeneous faults. Phys Rev Lett 97:4885–4888
Griffiths AA (1924) Theory of rupture. In: Proceedings 1st Int Congress in Applied Mech, Delft, pp 55–63
Gutenberg B, Richter C (1949) Seismicity of the Earth and Associated Phenomena, 2nd edn. University Press, Princeton
Habermann RE (1987) Man-made changes of seismicity rates. Bull Seismol Soc Am 77(1):141–159
Hainzl S, Ogata Y (2005) Detecting fluid signals in seismicity data through statistical earthquake modelling. J Geophys Res 110. doi:10.1029/2004JB003247
Harte D (2001) Multifractals: Theory and Applications. Chapman and Hall/CRC, Boca Raton
Harte D, Li DF, Vreede M, Vere-Jones D (2003) Quantifying the M8 prediction algorithm: reduction to a single critical variable and stability results. NZ J Geol Geophys 46:141–152
Harte D, Li D-F, Vere-Jones D, Vreede M, Wang Q (2007) Quantifying the M8 prediction algorithm II: model, forecast and evaluation. NZ J Geol Geophys 50:117–130
Harte D, Vere-Jones D (2005) The entropy score and its uses in earthquake forecasting. Pure Appl Geophys 162:1229–1253
Hawkes AG (1971) Spectra of some self‐exciting and mutually exciting point processes. Biometrika 58:83–90
Hawkes AG, Oakes D (1974) A cluster representation of a self‐exciting process. J Appl Prob 11:493–503
Helmstetter A, Sornette D (2002) Subcritical and supercritical regimes in epidemic models of earthquake aftershocks. J Geophys Res 107:2237. doi:10.1029/2001JB001580
Helmstetter A, Sornette D (2003) Båth's law derived from the Gutenberg‐Richter law and from aftershock properties. Geophys Res Lett 103(20):2069. doi:10.1029/2003GL018186
Ishimoto M, Iida K (1939) Bull Earthq Res Inst Univ Tokyo 17:443–478
Iwata T, Young RP (2005) Tidal stress/strain and the b‑values of acoustic emissions at the Underground Research Laboratory. Canada. Pure Appl Geophys 162:(6–7):1291–1308. doi:10.1007/s00024-005-2670-2 (P*1357)
Jackson DD, Kagan YY (1999) Testable earthquake forecasts for 1999. Seismol Res Lett 70:393–403
Jaeger JC, Cook NGW (1969) Fundamentals of Rock Mechanics. Methuen, London
Jaume SC, Bebbington MS (2004) Accelerating seismic moment release from a self‐correcting stochastic model. J Geophys Res 109:B12301. doi:10.1029/2003JB002867
Jeffreys H (1938) Aftershocks and periodicity in earthquakes. Beitr Geophys 53:111–139
Jeffreys H (1939) Theory of Probability, 1st edn (1939), 3rd edn (1961). University Press, Cambridge
Jones LM, Molnar P (1979) Some characteristics of foreshocks and their possible relationship to earthquake prediction and premonitory slip on a fault. J Geophys Res 84:3596–3608
Kagan Y (1973) Statistical methods in the study of the seismic process. Bull Int Stat Inst 45(3):437–453
Kagan Y (1991) Seismic moment distribution. Geophys J Int 106:121–134
Kagan Y (1991) Fractal dimension of brittle fracture. J Non‐linear Sci 1:1–16
Kagan Y (1994) Statistics of characteristic earthquakes. Bull Seismol Soc Am 83:7–24
Kagan Y, Jackson DD (1994) Probabilistic forecasting of earthquakes. Geophys J Int 143:438–453
Kagan Y, Knopoff L (1977) Earthquake risk prediction as a stochastic process. Phys Earth Planet Inter 14:97–108
Kagan Y, Knopoff L (1980) Spatial distribution of earthquakes: the two-point correlation function. Geophys J Roy Astronom Soc 62:303–320
Kagan Y, Knopoff L (1981) Stochastic synthesis of earthquake catalogues. J Geophys Res 86:2853–2862
Kagan Y, Knopoff L (1987) Statistical short-term earthquake prediction. Sci 236:1563–1567
Keilis-Borok VI, Kossobokov VG (1990) Premonitory activation of the earthquake flow: algorithm M8. Phys Earth Planet Inter 61:73–83
Kiremidjian AS, Anagnos T (1984) Stochastic slip predictable models for earthquake occurrences. Bull Seismol Soc Am 74:739–755
Knopoff L (1971) A stochastic model for the occurrence of main sequence earthquakes. Rev Geophys Space Phys 9:175–188
Kossobokov VG (1997) User manual for M8. In: Algorithms for Earthquake Statistics and Prediction. IASPEI Softw Ser 6:167–221
Kossobokov VG (2005) Earthquake prediction: principles, implementation, perspectives. Part I of Computational Seismology 36, “Earthquake Prediction and Geodynamic Processes.” (In Russian)
Kossobokov VG (2006) Testing earthquake prediction methods: The West Pacific short-term forecast of earthquakes with magnitude MwHRV ≥ 5.8. Tectonophysics 413:25–31
Libicki E, Ben-Zion Y (2005) Stochastic branching models of fault surfaces and estimated fractal dimensions. Pure Appl Geophys 162:1077–1111
Lombardi A (2002) Probabilistic interpretation of Båth's law. Ann Geophys 45:455–472
Lomnitz CA (1974) Plate Tectonics and Earthquake Risk. Elsevier, Amsterdam
Lomnitz‐Adler J (1985) Asperity models and characteristic earthquakes Geophys. J Roy Astron Soc 83:435–450
Lomnitz‐Adler J (1985) Automaton models of seismic fracture: constraints imposed by the frequency‐magnitude relation. J Geophys Res 95:491–501
Lomnitz‐Adler J (1988) The theoretical seismicity of asperity models; an application to the coast of Oaxaca. Geophys J 95:491–501
Liu J, Chen Y, Shi Y, Vere-Jones D (1999) Coupled stress release model for time dependent earthquakes. Pure Appl Geophys 155:649–667
Loève M (1977) Probability Theory I, 4th edn. Springer, New York
Lu C, Vere-Jones D (2001) Statistical analysis of synthetic earthquake catalogs generated by models with various levels of fault zone disorder. J Geophys Res 106:11115–11125
Lu C, Harte D, Bebbington M (1999) A linked stress release model for Japanese historical earthquakes: coupling among major seismic regions. Earth Planet. Science 51:907–916
Macdonald II, Zucchini W (1997) Hidden Markov and Other Models for Discrete‐Valued Time Series. Chapman and Hall, London
Main IG, Burton PW (1984) Information theory and the earthquake frequency‐magnitude distribution. Bull Seismol Soc Am 74:1409–1426
Mandelbrot BB (1977) Fractals: Form, Chance and Dimension. Freeman, San Francisco
Mandelbrot BB (1989) Multifractal measures, especially for the geophysicist. Pure Appl Geophys 131:5–42
Martínez VJ, Saar E (2002) Statistics of the Galaxy Distribution. Chapman & Hall/CRC, Boca Raton
Matsu'ura RS (1986) Precursory quiescence and recovery of aftershock activities before some large aftershocks. Bull Earthq Res Inst Tokyo 61:1–65
Matsu'ura RS, Karakama I (2005) A point process analysis of the Matsushiro earthquake swarm sequence: the effect of water on earthquake occurrence. Pure Appl Geophys 162 1319–1345. doi:10.1007/s00024-005-2762-0
Matthews MV, Ellsworth WL, Reasenberg PA (2002) A Brownian model for recurrent earthquakes. Bull Seism Soc Amer 92:2232–2250
Merrifield A, Savage MK, Vere-Jones D (2004) Geographical distributions of prospective foreshock probabilities in New Zealand. J Geol Geophys 47:327–339, New Zealand
Michael A (1997) Test prediction methods: earthquake clustering versus the Poisson model. Geophys Res Lett 24:1891–1894
Mogi K (1962) Study of elastic shocks caused by the fracture of heterogeneous materials and its relation to earthquake phenomena. Bull Earthq Res Inst Tokyo Univ 40:125–173
Mogi K (1985) Earthquake Prediction. Academic Press, Tokyo
Molchan GM (1990) Strategies in strong earthquake prediction. Phys Earth Plan Int 61:84–98
Molchan GM, Kagan YY (1992) Earthquake prediction and its optimization. J Geophys Res 106:4823–4838
Ogata Y (1988) Statistical models for earthquake occurrence and residual analysis for point processes. J Amer Stat Soc 83:9–27
Ogata Y (1998) Space-time point process models for earthquake occurrences. Annals Inst Stat Math 50:379–402
Ogata Y (1999) Estimating the hazard of rupture using uncertain occurrence times of paleoearthquakes. J Geophys Res 104:17995–18014
Ogata Y (2005) Detection of anomalous seismicity as a stress change sensor. J Geophys Res 110(B5):B05S06. doi:10.1029/2004JB003245
Ogata Y, Utsu T, Katsura K (1996) Statistical discrimination of foreshocks from other earthquake clusters. Geophys J Int 127:17–30
Ogata Y, Jones L, 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(B6):2318. doi:10.1029/2002JB002009
Omori F (1894) On aftershocks of earthquakes. J Coll Sci Imp Acad Tokyo 7:111–200
Otsuka M (1972) A chain reaction type source model as a tool to interpret the magnitude‐frequency relation of earthquakes. J Phys Earth 20:35–45
Pietavolo A, Rotondi R (2000) Analyzing the interevent time distribution to identify seismicity patterns: a Bayesian non‐parametric approach to the multiple change‐point problem. Appl Stat 49:543–562
Pisarenko DV, Pisarenko VF (1995) Statistical estimation of the correlation dimension. Phys Lett A 197:31–39
Reasenberg PA (1999) Foreshock occurrence before large earthquakes. J Geophys Res 104:4755–4768
Reasenberg PA, Jones LM (1989) Earthquake hazard after a mainshock in California. Sci 243:1173–1176
Reid HF (1911) The elastic‐rebound theory of earthquakes. Bull Dept Geol Univ Calif 6:413–444
Renyi A (1959) On the dimension and entropy of probability distributions. Acta Math 10:193–215
Rhoades DA (2007) Application of the EEPAS model to forecasting earthquakes of moderate magnitude in Southern California. Seismol Res Lett 78:110–115
Rhoades DA, Evison FF (2004) Long-range earthquake forecasting with every event a precursor according to scale. Pure Appl Geophys 161:147–171
Rhoades DA, Evison FF (2005) Test of the EEPAS forecasting model on the Japan earthquake catalogue. Pure Appl Geophys 162:1271–1290
Rhoades DA, Van Dissen RJ (2003) Estimation of the time‐varying hazard of rupture of the Alpine Fault of New Zealand, allowing for uncertainties. NZ J Geol Geophys 40:479–488
Ripley BD (1988) Statistical Inference for Spatial Processes. University Press, Cambridge
Robinson R (2000) A test of the precursory accelerating moment release model on some recent New Zealand earthquakes. Geophys J Int 140:568–576. doi:10.1046/j.1365-246X2000.00054.x
Robinson R, Benites R (1995) Synthetic seismicity models for the Wellington region of New Zealand: implications for the temporal distribution of large events. J Geophys Res 100:18229–18238. doi:10.1029/95JB01569
Rundle JB, Klein W, Tiampo K, Gross S (2000) Dynamics of seismicity patterns in systems of earthquake faults. In: Geocomplexity and the Physics of Earthquakes. Geophysical Monograph 120, American Geophysical Union
Saito M, Kikuchi M, Kudo M (1973) An analytical solution of: Go-game model of earthquakes. Zishin 26:19–25
Scholz CH (1968) The frequency‐magnitude relation of microfaulting in rock and its relation to earthquakes. Bull Seism Soc Am 58:399–415
Scholz CH (1990) The Mechanics of Earthquakes and Faulting. Cambridge University Press, New York
Schorlemmer D, Gerstenberger MC, Wiemer S, Jackson DD, Rhoades DA (2007) Earthquake likelihood model testing. Seismol Res Lett 78:17–29
Schuster A (1897) On lunar and solar periodicities of earthquakes. Proc Roy Soc London 61:455–465
Schwartz DP, Coppersmith K (1984) Fault behavior and characteristic earthquakes: examples from the Wasatch and San Andreas Faults. J Geophys Res 89:5681–5698
Shi YL, Liu J, Chen Y, Vere-Jones D (1999) Coupled stress release models for time‐dependent seismicity. J Pure Appl Geophys 155:649–667
Shi Y, Liu J, Zhang G (2001) An evaluation of Chinese annual earthquake predictions, 1990–1998. J Appl Prob 38A:222–231
Shimazaki K, Nakata T (1980) Time‐predictable recurrence model for large earthquakes. Geophys Res Lett 7:179–282
Smith WD (1986) Evidence for precursory changes in the frequency‐magnitude b‑value. Geophys J Roy Astron Soc 86:815–838
Smith WD (1998) Resolution and significance assessment of precursory changes in mean earthquake magnitude. Geophys J Int 135:515–522
Stoyan D, Stoyan H (1994) Fractals, Random Shapes and Point Fields. Wiley, Chichester
Tiampo KF, Rundle JB, Klein W, Ben-Zion Y, McGinnis SA (2004) Using eigenpattern analysis to constrain seasonal signals in Southern California. Pure Appl Geophys 16:19–10, 1991 V2003. doi:10.1007/s00024-004-2545-y
Turcotte DL (1992) Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge
Utsu T (1961) A statistical study on the properties of aftershocks. Geophys Mag 30:521–605
Utsu T, Ogata Y (1997) IASPEI Softw Libr 6:13–94
Utsu T, Ogata Y, Matu'ura RS (1995) The centenary of the Omori formula for a decay law of aftershock activity. J Phys Earth 43:1–33
Vere-Jones D (1969) A note on the statistical interpretation of Båth's law. Bull Seismol Soc Amer 59:1535–1541
Vere-Jones D (1970) Stochastic models for earthquake occurrence. J Roy Stat Soc B 32:1–62
Vere-Jones D (1977) Statistical theories for crack propagation. Pure Appl Geophys 114:711–726
Vere-Jones D (1978) Space‐time correlations of microearthquakes – a pilot study. Adv App Prob 10:73–87, supplement
Vere-Jones D (1978) Earthquake prediction: a statistician's view. J Phys Earth 26:129–146
Vere-Jones D (1995) Forecasting earthquakes and earthquake risk. Int J Forecast 11:503–538
Vere-Jones D (1999) On the fractal dimension of point patterns. Adv Appl Prob 31:643–663
Vere-Jones D (2003) A class of self‐similar random measures. Adv Appl Prob 37:908–914
Vere-Jones D, Davies RB (1966) A statistical analysis of earthquakes in the main seismic region of New Zealand. J Geol Geophys 9:251–284
Vere-Jones D, Ozaki T (1982) Some examples of statistical inference applied to earthquake data. Ann Inst Stat Math 34:189–207
Vere-Jones D, Robinson R, Yang W (2001) Remarks on the accelerated moment release model for earthquake forecasting: problems of simulation and estimation. Geophys J Int 144:515–531
von Bortkiewicz L (1898) Das Gesetz der kleinen Zahlen. Teubner, Leipzig
Weibull W (1939) A statistical theory of the strength of materials. Ingvetensk Akad Handl no 151
Working Group on Californian Earthquake Probabilities (1990) Probabilities of earthquakes in the San Francisco Bay region of California. US Geological Survey Circular 153
Yin X, Yin C (1994) The precursor of instability for non‐linear systems and its application to the case of earthquake prediction – the load‐unload response ratio theory. In: Newman WI, Gabrielov AM (eds) Nonlinear dynamics and Predictability of Natural Phenomena. AGU Geophysical Monograph 85:55–66
Zheng X, Vere-Jones D (1994) Further applications of the stress release model to historical earthquake data. Tectonophysics 229:101–121
Zhuang J (2000) Statistical modelling of seismicity patterns before and after the 1990 Oct 5 Cape Palliser earthquake, New Zealand. NZ J Geol Geophys 43:447–460
Zhuang J, Yin X (2000) The random distribution of the loading and unloading response ratio under the assumptions of the Poisson model. Earthq Res China 14:38–48
Zhuang J, Ogata Y, Vere-Jones D (2004) Analyzing earthquake clustering features by using stochastic reconstruction. J Geophys Res 109(B5):B05301. doi:10.1029/2003JB002879
Zhuang J, Vere-Jones D, Guan H, Ogata Y, Ma L (2005) Preliminary analysis of precursory information in the observations on the ultra low frequency electric field in the Beijing region. Pure Appl Geophys 162:1367–1396. doi:10.10007/s00024-004-2674-3
Acknowledgments
I am very grateful to friends and colleagues, especially David Harte, Mark Bebbington, David Rhoades and Yehuda Ben-Zion, for helpful discussions, correcting errors and plugging gaps.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag
About this entry
Cite this entry
Vere-Jones, D. (2011). Earthquake Occurrence and Mechanisms, Stochastic Models for. In: Meyers, R. (eds) Extreme Environmental Events. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7695-6_21
Download citation
DOI: https://doi.org/10.1007/978-1-4419-7695-6_21
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-7694-9
Online ISBN: 978-1-4419-7695-6
eBook Packages: Earth and Environmental ScienceReference Module Physical and Materials ScienceReference Module Earth and Environmental Sciences