Modeling earthquake dynamics Original Article First Online: 16 April 2015 Received: 06 June 2014 Accepted: 25 February 2015 DOI :
10.1007/s10950-015-9489-9

Cite this article as: Charpentier, A. & Durand, M. J Seismol (2015) 19: 721. doi:10.1007/s10950-015-9489-9
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.

Keywords Duration Earthquakes Generalized linear models Seismic gap hypothesis

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

CrossRef 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

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

CrossRef 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

CrossRef 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

CrossRef 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

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

CrossRef 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

CrossRef 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

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

CrossRef 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

CrossRef 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

CrossRef 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

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

CrossRef 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

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

CrossRef 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

CrossRef 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

CrossRef 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

CrossRef 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

CrossRef 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

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

CrossRef 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

CrossRef Google Scholar Kagan YY (2010) Earthquake size distribution: power-law with exponent

β = 1/2? Tectonophysics 490:103–114

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

CrossRef 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

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

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

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

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

CrossRef Google Scholar Kijko A (2004) Estimation of the maximum earthquake magnitude,

m
_{max} . Pure Appl Geophys 161:1–27

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

CrossRef 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

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

CrossRef 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

CrossRef 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

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

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

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

CrossRef 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

CrossRef 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

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

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

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

CrossRef 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

CrossRef 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

CrossRef 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

CrossRef 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

CrossRef 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

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

CrossRef 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

CrossRef 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

CrossRef 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

CrossRef Google Scholar Stein RS (1995) Characteristic or hazard? Nature 370:443–444

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

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

CrossRef 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

CrossRef 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

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

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

CrossRef 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

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

CrossRef 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

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

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

CrossRef 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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