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Modeling earthquake dynamics

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Abstract

In this paper, we investigate questions arising in Parsons and Geist (Bull Seismol Soc Am 102:1–11, 2012). Pseudo causal models connecting magnitudes and waiting times are considered, through generalized regression. We do use conditional model (magnitude given previous waiting time, and conversely) as an extension to joint distribution model described in Nikoloulopoulos and Karlis (Environmetrics 19: 251–269, 2008). On the one hand, we fit a Pareto distribution for earthquake magnitudes, where the tail index is a function of waiting time following previous earthquake; on the other hand, waiting times are modeled using a Gamma or a Weibull distribution, where parameters are functions of the magnitude of the previous earthquake. We use those two models, alternatively, to generate the dynamics of earthquake occurrence, and to estimate the probability of occurrence of several earthquakes within a year or a decade.

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References

  • Aki K (1956) Some problems in statistical seismology. J Seismol Soc Jpn 8:205–227

    Google Scholar 

  • Atkinson P, Jiskoot J, Massari R, Murray T (1998) Generalized linear modelling in geomorphology. Earth Surf Process Landf 23(13):1185–1195

    Article  Google Scholar 

  • Bak P, Tang C (1989) Earthquakes as a self-organized critical phenomenon. J Geophys Res 94 (15):635–637

    Google Scholar 

  • Bak P, Christensen K, Danon L, Scanlon T (2002) Unified scaling law for earthquakes. Phys Rev Lett 88:178501

    Article  Google Scholar 

  • Bakun WH, Lindh AG (1985) The Parkfield, California, earthquake prediction experiment. Science 229:619–624

    Article  Google Scholar 

  • Benioff H (1951) Earthquakes and rock creep. Bull Seismol Soc Am 41(1):31–62

    Google Scholar 

  • Bell AF, Naylor M, Heap MJ, Main IG (2011) Forecasting volcanic eruptions and other material failure phenomena: an evaluation of the failure forecast method. Geophys Res Lett 38:L15304

    Google Scholar 

  • Bufe CG, Perkins DM (2005) Evidence for a global seismic-moment release sequence. Bull Seismol Soc Am 95:833–843

    Article  Google Scholar 

  • Carbone V, Sorriso-Valvo L, Harabaglia P, Guerra I (2005) Unified scaling law for waiting times between seismic events. Europhys Lett 71(6):1036–1042

    Article  Google Scholar 

  • Christensen K, Danon L, Scanlon T, Bak P (2002) Unified scaling law for earthquakes. Proc Natl Acad Sci U S A 99:2509–2513

    Article  Google Scholar 

  • Corral A (2003) Local distributions and rate fluctuations in a unified scaling law for earthquakes. Phys Rev E 68:035102

    Article  Google Scholar 

  • Corral A (2004) Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes. Phys Rev Lett 92

  • Corral A (2006) Dependence of earthquake recurrence times and independence of magnitudes on seismicity history. Tectonophysics 424:177–193

    Article  Google Scholar 

  • Corral A (2007) Statistical features of earthquake temporal occurrence. Lect Notes Phys 705:191–221

    Google Scholar 

  • Corral A, Christensen K (2006) Comment on earthquakes descaled: on waiting time distributions and scaling laws. Phys Rev Lett 96:109801

    Article  Google Scholar 

  • Davidsen J, Goltz C (2005) Are seismic waiting time distributions universal? Geophys Res Lett 31:L21612

    Article  Google Scholar 

  • Davis PM, Jackson DD, Kagan YY (1989) The longer it has been since the last earthquake, the longer the expected time till the next? Bull Seismol Soc Am 79:1439–1456

    Google Scholar 

  • Davis SD, Frohlich C (1991) Single-link cluster analysis of earthquake aftershocks—decay laws and regional variations. J Geophys Res 96:6335–6350

    Article  Google Scholar 

  • Enescu B, Struzik Z, Kiyono K (2008) On the recurrence time of earthquakes: insight from Vrancea (Romania) intermediate-depth events. Geophys J Int 172(1):395–404

    Article  Google Scholar 

  • Gardner JK, Knopoff L (1974) Is the sequence of earthquakes in Southern California, with aftershocks removed Poissonian? Bull Seismol Soc Am 64:1363–1367

    Google Scholar 

  • Gilbert GK (1909) Earthquake forecasts. Science 29:121–138

    Article  Google Scholar 

  • Godano C, Pingue F (2000) Is the seismic moment-frequency relation universal? Geophys J Int 142:193–198

    Article  Google Scholar 

  • Gutenberg B, Richter CF (1941) Seismicity of the earth. Geol Soc Am Spec Pap 34:1–131

    Google Scholar 

  • Hagiwara Y (1974) Probability of earthquake occurrence as obtained from a Weibull distribution analysis of crustal strain. Tectonophysics 23:323–318

    Article  Google Scholar 

  • Hainzl S, Scherbaum F, Beauval C (2006) Estimating background activity based on interevent-time distribution. Bull Seismol Soc Am 96:313–320

    Article  Google Scholar 

  • Hainzl S, Marsan D (2008) Dependence of the Omori-Utsu law parameters on main shock magnitude: observations and modeling. J Geophys Res 113

  • Hasumi T, Akimoto T, Aizawa Y (2009a) The Weibull–Log Weibull distribution for interoccurrence times of earthquakes. Phys A 388:491–498

    Article  Google Scholar 

  • Hasumi T, Akimoto T, Aizawa Y (2009b) The Weibull–Log Weibull transition of the interoccurrence time statistics in the two-dimensional Burridge-Knopoff earthquake model. Phys A 388:483

    Article  Google Scholar 

  • Hasumi C, Chen T, Akimoto, Aizawa Y (2010) The Weibull-log Weibull transition of interoccurrence time for synthetic and natural earthquakes. Tectonophysics 485:9

    Article  Google Scholar 

  • Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat. 3:1163–1174

  • Holschneider M, Zöller G, Hainzl S (2011) Estimation of the maximum possible magnitude in the framework of a doubly truncated Gutenberg-Richter model. Bull Seismol Soc Am 101(4):1649–1659

    Article  Google Scholar 

  • Huc M, Main IG (2003) Anomalous stress diffusion in earthquake triggering: correlation length, time dependence, and directionality. J Geophys Res 108:2324

    Google Scholar 

  • Jensen HJ (1998) Self-organized criticallity. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Kagan YY (1991) Seismic moment distribution. Geophys J Int 106:123–134

    Article  Google Scholar 

  • Kagan YY (1993) Statistics of characteristic earthquakes. Bull Seismol Soc Am 83:7–24

    Google Scholar 

  • Kagan YY (2002a) Seismic moment distribution revisited: I. statistical results. Geophys J Int 148(3):520–541

    Article  Google Scholar 

  • Kagan YY (2010) Earthquake size distribution: power-law with exponent β = 1/2? Tectonophysics 490:103–114

    Article  Google Scholar 

  • Kagan YY (2011) Random stress and Omori’s law. Geophys J Int 186(3):1347–1364

    Article  Google Scholar 

  • Kagan YY, Jackson DD (1991) Seismic gap hypothesis: ten years after. J Geophys Res 96 (21):419–421

    Google Scholar 

  • Kagan YY, Jackson DD (1995) New seismic gap hypothesis: five years after. J Geophys Res 100:3943–3959

    Article  Google Scholar 

  • Kagan YY, Knopoff L (1987a) Statistical short-term earthquake prediction. Science 236:1563–1567

    Article  Google Scholar 

  • Kagan YY, Knopoff L (1987b) Random stress and earthquake statistics: time dependence. Geophys J R Astron Soc- A 88:723–731

    Article  Google Scholar 

  • Kanamori H (1977) The energy release in great earthquakes. J Geophys Res 82:2981–2987

    Article  Google Scholar 

  • Kerr RA (2004) Parkfield keeps secrets after a long-awaited quake. Science 306:206–207

    Article  Google Scholar 

  • Kijko A (2004) Estimation of the maximum earthquake magnitude, m max. Pure Appl Geophys 161:1–27

    Article  Google Scholar 

  • Knopoff L, Kagan YY (1977) Analysis of the theory of extremes as applied to earthquake problems. J Geophys Res 82:5647–5657

    Article  Google Scholar 

  • Langenbruch C, Dinske C, Shapiro SA (2011) Inter event times of fluid induced earthquakes suggest their poisson nature. Geophys Res Lett 38

  • Lennartz S, Livina VN, Bunde A, Havlin S (2008) Long-term memory in earthquakes and the distribution of interoccurrence times. Europhys Lett 81

  • Leonard T, Papsouliotis O, Main IG (2001) A Poisson model for identifying characteristic size effects in frequency data: application to frequency-size distributions for global earthquakes, “starquakes” and fault lengths. J Geophys Res 106(13):473–484

    Google Scholar 

  • Lombardi AM, Marzocchi W (2007) Evidence of clustering and nonstationarity in the time distribution of large worldwide earthquakes. J Geophys Res 112

  • Lomnitz C (1994) Fundamentals of earthquake prediction. Wiley, New York

    Google Scholar 

  • Main I (2000) Apparent breaks in scaling in the earthquake cumulative frequency-magnitude distribution: fact or artifact? Bull Seismol Soc Am 90:86–97

    Article  Google Scholar 

  • Matthews MV, Ellsworth WL, Reasenberg PA (2002) A Brownian model for recurrent earthquakes. Bull Seismol Soc Am 92:2233–2250

    Article  Google Scholar 

  • McCann WR, Nishenko SP, Sykes IR, Krause J (1979) Seismic gaps and plate tectonics: seismic potential for major boundaries. Pure Appl Geophys 117:1082–1147

    Article  Google Scholar 

  • Mega MS, Allegrini P, Grigolini P, Latora V, Palatella L, Rapisarda A, Vinciguerra S (2003) Power-law time distribution of large earthquakes. Phys Rev Lett 90(18):188501

    Article  Google Scholar 

  • Molchan G (2005) Interevent time distribution in seismicity: a theoretical approach. Pure Appl Geophys 162:1135–1150

    Article  Google Scholar 

  • Molchan G, Kronrod T (2007) Seismic interevent time: a spatial scaling and multifractality. Pure Appl Geophys 164:75–96

    Article  Google Scholar 

  • Murray J, Segall P (2002) Testing time-predictable earthquake recurrence by direct measurement of strain accumulation and release. Nature 419:287–291

    Article  Google Scholar 

  • Newman W, Turcotte DL, Shcherbakov R, Rundle JB (2005) Why Weibull?. In: Proceedings of the American geophysical union. Fall Meeting 2005, #S43D-07

  • Nikoloulopoulos AK, Karlis D (2008) Fitting copulas to bivariate earthquake data: the seismic gap hypothesis revisited. Environmetrics 19:251–269

    Article  Google Scholar 

  • Nishenko SP, Sykes LR (1993) Comment on ‘Seismic gap hypothesis: ten years after’ by Y.Y. Kagan and D.D. Jackson. J Geophys Res 98:9909–9916

    Article  Google Scholar 

  • Ogata Y (1988a) Statistical models for earthquake occurrence and residual analysis for point processes. J Am Stat Assoc 83:9–27

    Article  Google Scholar 

  • Ogata Y (1998b) Space-time point-process models for earthquake occurrences. Ann Inst Stat Math 50(2):379–402

    Article  Google Scholar 

  • Ogata Y (1999) Seismicity analysis through point-process modeling: a review. Pure Appl Geophys 155:471–507

    Article  Google Scholar 

  • Ogata Y, Katsura K (1993) Analysis of temporal and spatial heterogeneity of magnitude frequency distribution inferred from earthquake catalogues. Geophys J Int 113:727–738

    Article  Google Scholar 

  • Omori F (1894) On aftershocks. J Coll Sci, Imperial University of Tokyo 7:111–200

    Google Scholar 

  • Pacheco JF, Scholz CH, Sykes LR (1992) Changes in frequency-size relationship from small to large earthquakes. Nature 355:71–73

    Article  Google Scholar 

  • Pareto V (1896) Cours d’économie politique, tome 2. F. Rouge, Lausanne

  • Parsons T (2002) Global Omori law decay of triggered earthquakes: large aftershocks outside the classical aftershock zone. J Geophys Res 107:2199

    Article  Google Scholar 

  • Parsons T (2008) Earthquake recurrence on the south Hayward fault is most consistent with a time dependent, renewal process. Geophys Res Lett 35

  • Parsons T, Geist EL (2012) Were global M≥8.3 earthquake time intervals random between 1900—2011? Bull Seismol Soc Am 102:1–11

    Article  Google Scholar 

  • Ramírez-Rojas A, Flores-Marquez EL, Valverde-Esparza S (2012) Weibull characterization of inter-event lags of earthquakes occurred in the Pacific coast of Mexico. Geophys Res Abstr 14:13708

    Google Scholar 

  • Reid HF (1910) The mechanics of the earthquake. Vol. 2 of The California Earthquake of April 18, 1906. Report of the State Earthquake Investigation Commission (Carnegie Institution of Washington Publication 87)

  • Rhoades DA (1996) Estimation of the Gutenberg-Richter relation allowing for individual earthquake magnitude uncertainties. Tectonophysics 258:71–83

    Article  Google Scholar 

  • Rikitake T (1974) Probability of an earthquake occurrence as estimated from crustal strain. Tectonophysics 23:299–312

    Article  Google Scholar 

  • Rundle JB, Rundle PB, Shcherbakov R, Turcotte DL (2005) Earthquake waiting times: theory and comparison with results from virtual California simulations. Geophys Res Abstr 7:2477

    Google Scholar 

  • Savage JC (1994) Empirical earthquake probabilities from observed recurrence intervals. Bull Seismol Soc Am 84(1):219–221

    Google Scholar 

  • Shcherbakov R, Yakovlev G, Turcotte DL, Rundle JB (2005) Model for the distribution of aftershock interoccurrence times. Phys Rev Lett 95:218501

    Article  Google Scholar 

  • Shlien S, Toksöz MN (1970) A clustering model for earthquake occurrences. Bull Seismol Soc Am 60:1765–1787

    Google Scholar 

  • Sieh K (1996) The repetition of large-earthquake ruptures. Proc Natl Acad Sci U S A 93:3764–3771

    Article  Google Scholar 

  • Sieh K, Stuiver M, Brillinger D (1989) A more precise chronology of earthquakes produced by the San Andreas fault in Southern California. J Geophys Res 94:603–623

    Article  Google Scholar 

  • Stein RS (1995) Characteristic or hazard? Nature 370:443–444

    Article  Google Scholar 

  • Stein RS (2002) Parkfield’s unfulfilled promise. Nature 419:257–258

    Article  Google Scholar 

  • Thatcher W (1989) Earthquake recurrence and risk assessment in circum-Pacific seismic gaps. Nature 341:432–434

    Article  Google Scholar 

  • Udías A, Rice J (1975) Statistical analysis of microearthquake activity near San Andreas geophysical observatory, Hollister, California. Bull Seismol Soc Am 65:809–828

    Google Scholar 

  • Utsu T (1984) Estimation of parameters for recurrence models of earthquakes. Bull Earthq Res Inst, Univ Tokyo 59:53–66

    Google Scholar 

  • Utsu T (1999) Representation and analysis of the earthquake size distribution: a historical review and some new approaches. Pure Appl Geophys 155:509–535

    Article  Google Scholar 

  • Vere-Jones D (1970) Stochastic models for earthquake occurrence (with discussion). J R Stat Soc Ser B 32:1–62

    Google Scholar 

  • Vere-Jones D (1975) Stochastic models for earthquake sequences. Geophys J R Astron Soc 42:811–826

    Article  Google Scholar 

  • Vere-Jones D (1976) A branching model for crack propagation. Pure Appl Geophys 114:711–725

    Article  Google Scholar 

  • Vere-Jones D (1977) Statistical theories of crack propagation. Math Geol 9:455–481

    Article  Google Scholar 

  • Vere-Jones D, Robinson R, Yang W (2001) Remarks on the accelerated moment release model: problems of model formulation, simulation and estimation. Geophys J Int 144(3):517–531

    Article  Google Scholar 

  • Wang JH, Kuo C-H (1998) On the frequency distribution of interoccurrence times of earthquakes. J Seismol 2: 351–358

    Article  Google Scholar 

  • Wang Q, Jackson DD, Zhuang J (2010) Missing links in earthquake clustering models. Geophys Res Lett 37

  • Weldon R, Scharer K, Fumal T, Biasi G (2004) Wrightwood and the earthquake cycle: what a long recurrence record tells us about how faults work. GSA Today 14(9):4–10

    Article  Google Scholar 

  • Wyss M (1973) Towards a physical understanding of the earthquake frequency distribution. Geophys J R Astron Soc 31:341–359

    Article  Google Scholar 

  • Zhuang J, Ogata Y, Vere-Jones D (2002) Stochastic declustering of space-time earthquake occurrences. J Am Stat Assoc 97:369–380

    Article  Google Scholar 

  • Zhuang J, Ogata Y, Vere-Jones D (2004) Analyzing earthquake clustering features by using stochastic reconstruction. J Geophys Res 109

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  1. UQAM, 201, avenue du Président-Kennedy, Montréal, Quebec, H2X 3Y7, Canada

    Arthur Charpentier & Marilou Durand

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  1. Arthur Charpentier
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Charpentier, A., Durand, M. Modeling earthquake dynamics. J Seismol 19, 721–739 (2015). https://doi.org/10.1007/s10950-015-9489-9

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