A stochastic methodology for risk assessment of a large earthquake when a long time has elapsed

  • Raúl Fierro
  • Víctor Leiva
Original Paper


We propose a stochastic methodology for risk assessment of a large earthquake when a long time has elapsed from the last large seismic event. We state an approximate probability distribution for the occurrence time of the next large earthquake, by knowing that the last large seismic event occurred a long time ago. We prove that, under reasonable conditions, such a distribution is exponential with a rate depending on the asymptotic slope of the cumulative intensity function corresponding to a nonhomogeneous Poisson process. As it is not possible to obtain an empirical cumulative distribution function of the waiting time for the next large earthquake, an estimator of its cumulative distribution function based on existing data is derived. We conduct a simulation study for detecting scenario in which the proposed methodology would perform well. Finally, a real-world data analysis is carried out to illustrate its potential applications, including a homogeneity test for the times between earthquakes.


Earthquake data analysis Exponential and gamma distributions Maximum-likelihood method Monte Carlo simulation Nonhomogeneous Poisson process 



The authors thank the editors and referees for their constructive comments on an earlier version of this manuscript which resulted in this improved version. This research was partially supported by FONDECYT 1160868 Grant of CONICYT-Chile.


  1. Adelfio G, Chiodi M (2015) Alternated estimation in semi-parametric space-time branching-type point processes with application to seismic catalogs. Stoch Environ Res Risk Assess 29:443–450CrossRefGoogle Scholar
  2. Arkin B, Leemis L (2000) Nonparametric estimation of the cumulative intensity function for a non-homogeneous Poisson process from overlapping realizations. Manag Sci 46:989–998CrossRefGoogle Scholar
  3. Balakrishnan N, Stehlík M (2015) Likelihood testing with censored and missing duration data. J Stat Theory Pract 9:2–22CrossRefGoogle Scholar
  4. Breiman L (1968) Probability. Addison-Wesley, Menlo ParkGoogle Scholar
  5. Brémaud P (1981) Point processes and queues. Martingale dynamics. Springer, New YorkCrossRefGoogle Scholar
  6. Cornell C (1968) Engineering seismic risk analysis. Bull Seismol Soc Am 58:1583–1606Google Scholar
  7. Dacunha-Castelle D, Duflo M (1986) Probability and statistics, vol 2. Springer, New YorkCrossRefGoogle Scholar
  8. Dargahi-Noubary G (1986) A method for predicting future large erthquakes using extreme order statistics. Phys Earth Planet Inter 42:241–245CrossRefGoogle Scholar
  9. Dieterich J (1988) Probability of earthquake recurrence with non-uniform stress rates and time-dependent failure. Pure Appl Geophys 126:589–617CrossRefGoogle Scholar
  10. Fierro R (2015) Functional limit theorems for the multivariate Hawkes process with different exciting functions. Latin Am J Probab Math Stat 12:477–489Google Scholar
  11. Fierro R, Leiva V, Møller J (2015) The Hawkes process with different exciting functions and its asymptotic behavior. J Appl Probab 52:37–54CrossRefGoogle Scholar
  12. Fierro R, Leiva V, Ruggeri F, Sanhueza A (2013) On the Birnbaum-Sauders distribution arising from a non-homogeneous Poisson process. Stat Probab Lett 83:1233–1239CrossRefGoogle Scholar
  13. Fukutan Y, Suppasri A, Imamura F (2015) Stochastic analysis and uncertainty assessment of tsunami wave height using a random source parameter model that targets a tohoku-type earthquake fault. Stoch Environ Res Risk Assess 29:1763–1779CrossRefGoogle Scholar
  14. Gutenberg R, Richter C (1944) Frequency of earthquakes in California. Bull Seismol Soc Am 34:185–188Google Scholar
  15. Henderson S (2003) Estimation for non-homogeneous Poisson processes from aggregated data. Oper Res Lett 31:375–382CrossRefGoogle Scholar
  16. Hristopulos D, Mouslopoulou V (2013) Strength statistics and the distribution of earthquake inter-event times. Physica A 392:485–496CrossRefGoogle Scholar
  17. Kagan Y (1997) Are eathquakes predictable? Geophys J Int 131:505–525CrossRefGoogle Scholar
  18. Kamat R (2015) Planning and managing earthquake and flood prone towns. Stoch Environ Res Risk Assess 29:527–545CrossRefGoogle Scholar
  19. Karr A (1991) Point processes and their statistical inference. Marcel Dekker, New YorkGoogle Scholar
  20. Kim D, Kim B, Lee S, Cho Y (2014) Best-fit distribution and log-normality for tsunami heights along coastal lines. Stoch Environ Res Risk Assess 28:881–893CrossRefGoogle Scholar
  21. Knopoff L, Kagan Y (1977) Analysis of the theory of extremes as applied to earthquake problems. J Geophys Res 82:5647–5657CrossRefGoogle Scholar
  22. Leemis L (1991) Nonparametric estimation of the cumulative intensity function for a non-homogeneous Poisson process. Manag Sci 37:886–900CrossRefGoogle Scholar
  23. Leemis L (2004) Nonparametric estimation and variate generation for a non-homogeneous Poisson process from event count data. IIE Trans 36:1155–1160CrossRefGoogle Scholar
  24. Lewis P, Shedler G (1979) Simulation of non-homogeneous Poisson process by thinning. Nav Res Logist 26:403–413CrossRefGoogle Scholar
  25. Nicolis O, Chiodi M, Adelfio G (2015) Windowed ETAS models with application to the Chilean seismic catalogs. Spat Stat 14:151–165CrossRefGoogle Scholar
  26. Nishenko S, Buland R (1987) A generic recurrence interval distribution for earthquake forecasting. Bull Seismol Soc Am 77:1382–1399Google Scholar
  27. Ogata Y (1988) Statistical models for earthquake occurrences and residual analysis for point processes. J Am Stat Assoc 83:9–27CrossRefGoogle Scholar
  28. Omori F (1895) On the after-schoks of earthquakes. J Coll Sci Imp Univ 7:111–200Google Scholar
  29. Panthi A, Shander D, Singh H, Kumar A, Paudyal H (2011) Time-predictable model applicability for earthquake occurrence in northeast India and vicinity. Nat Hazards Earth Syst Sci 39:993–1002CrossRefGoogle Scholar
  30. Reid H (1910) The mechanics of the earthquake, the California earthquake of April 18-1906. Technical report 2. Carnegie Institution of Washington, WashingtonGoogle Scholar
  31. Rikitake T (1976) Recurrence of great earthquakes at subduction zones. Tectonophysics 35:335–362CrossRefGoogle Scholar
  32. Roussas G (1997) A course in mathematical statistics. Academic Press, BurlingtonGoogle Scholar
  33. Rubinstein J, Ellsworth W, Beeler N, Kilgore B, Lockner D, Savage H (2012) Fixed recurrence and slip models better predict earthquake behavior than the time- and slip-predictable models 2: Laboratory earthquakes. J Geophys Res 117:B02307Google Scholar
  34. Shimazaki K, Nakata T (1980) Time-predictable recurrence model for large earthquakes. Geophys Res Lett 7:279–282CrossRefGoogle Scholar
  35. Sturges H (1926) The choice of a class interval. J Am Stat Assoc 21:65–66CrossRefGoogle Scholar
  36. Tucker HG (1967) A graduate course in probability. Academic Press, New YorkGoogle Scholar
  37. Wesnousky S, Scholz C, Shimazaki K, Matsuda T (1984) Integration of geological and seismological data for mthe analysis of seismic hazard: a case of study in Japan. Bull Seismol Soc Am 74:687–708Google Scholar
  38. Yakovlev G, Turcotte D, Rundle J, Rundle P (2007) Simulation-based distributions of earthquake recurrence times on the San Andreas fault system. Bull Seismol Soc Am 96:1995–2007CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad de ValparaísoValparaísoChile
  2. 2.Instituto de MatemáticasPontificia Universidad Católica de ValparaísoValparaísoChile
  3. 3.Facultad de Ingeniería y CienciasUniversidad Adolfo IbáñezViña del MarChile

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