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Bulletin of Volcanology

, Volume 74, Issue 2, pp 545–558 | Cite as

A Brownian model for recurrent volcanic eruptions: an application to Miyakejima volcano (Japan)

  • Alexander Garcia-AristizabalEmail author
  • Warner Marzocchi
  • Eisuke Fujita
Research Article

Abstract

The definition of probabilistic models as mathematical structures to describe the response of a volcanic system is a plausible approach to characterize the temporal behavior of volcanic eruptions and constitutes a tool for long-term eruption forecasting. This kind of approach is motivated by the fact that volcanoes are complex systems in which a completely deterministic description of the processes preceding eruptions is practically impossible. To describe recurrent eruptive activity, we apply a physically motivated probabilistic model based on the characteristics of the Brownian passage-time (BPT) distribution; the physical process defining this model can be described by the steady rise of a state variable from a ground state to a failure threshold; adding Brownian perturbations to the steady loading produces a stochastic load-state process (a Brownian relaxation oscillator) in which an eruption relaxes the load state to begin a new eruptive cycle. The Brownian relaxation oscillator and Brownian passage-time distribution connect together physical notions of unobservable loading and failure processes of a point process with observable response statistics. The Brownian passage-time model is parameterized by the mean rate of event occurrence, μ, and the aperiodicity about the mean, α. We apply this model to analyze the eruptive history of Miyakejima volcano, Japan, finding a value of 44.2  (±6.5 years) for the μ parameter and 0.51  (±0.01) for the (dimensionless) α parameter. The comparison with other models often used in volcanological literature shows that this physically motivated model may be a good descriptor of volcanic systems that produce eruptions with a characteristic size. BPT is clearly superior to the Exponential distribution, and the fit to the data is comparable to other two-parameters models. Nonetheless, being a physically motivated model, it provides an insight into the macro-mechanical processes driving the system.

Keywords

Probabilistic models Brownian passage-time distribution Hazard function Miyakejima volcano 

Mathematics Subject Classifications (2010)

86A32 60E05 62H10 

Notes

Acknowledgements

The manuscript was greatly improved by helpful reviews and constructive comments from G. Wadge, J. Phillips, and an anonymous reviewer. Critical and useful comments of an earlier version of the manuscript were made by L. Sandri and J. Selva. A. Garcia thanks the staff of NIED’s Volcano Research Department for their friendly hospitality during his stay in Japan; A. Garcia’s stay in Japan was funded by the Marco Polo program of the Università di Bologna (Italy).

References

  1. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Automat Contr AC 19:716–723CrossRefGoogle Scholar
  2. Bain LJ (1978) Statistical analysis of reliability and life-testing models, theory and methods. Marcel Dekker, New YorkGoogle Scholar
  3. Bebbington MS, Lai CD (1996a) On nonhomogeneous models for volcanic eruptions. Math Geol 28:585–600CrossRefGoogle Scholar
  4. Bebbington MS, Lai CD (1996b) Statistical analysis of new zealand volcanic occurrence data. J Volcanol Geotherm Res 74:101–110CrossRefGoogle Scholar
  5. Bowers N, Gerber H, Hickman J, Jones D, Nesbitt C (1997) Actuarial mathematics, 2nd edn. Society of ActuariesGoogle Scholar
  6. Burt ML, Wadge G, Scott WA (1994) Simple stochastic modelling of the eruption history of a basaltic volcano: Nyamuragira, Zaire. Bull Volcanol 56:87–97Google Scholar
  7. Coles S, Sparks R (2006) Extreme value methods for modeling historical series of large volcanic magnitudes. In: Mader H, Coles S, Connor C, Connor L (eds) Statistics in volcanology, IAVCEI Publications. Geological Society, London, pp 47–56Google Scholar
  8. Connor CB, Sparks RSJ, Mason RM, Bonadonna C (2003) Exploring links between physical and probabilistic models of volcanic eruptions: the soufriere hills volcano, montserrat. Geophys Res Lett 30(13). doi: 10.1029/2003GL017384 Google Scholar
  9. Cox DR, Lewis PAW (1966) The statistical analysis of series of events. Mathuen, New YorkGoogle Scholar
  10. De la Cruz-Reyna S (1991) Poisson-distributed patterns of explosive eruptive activity. Bull Volcanol 54:57–67CrossRefGoogle Scholar
  11. Ellsworth WL, Matthews MV, Nadeau RM, Nishenko SP, Reasenberg PA, Simpson RW (1999) A physically based earthquake recurrence model for estimation of long-term earthquake probabilities. US Geol Surv Open-File Rept, pp 99–522Google Scholar
  12. Ho C (1991) Nonhomogeneous Poisson model for volcanic eruptions. Math Geol 23(2):167–173CrossRefGoogle Scholar
  13. Ho CH (1996) Volcanic time-trend analysis. J Volcanol Geotherm Res 74(3–4):171–177CrossRefGoogle Scholar
  14. Klein F (1982) Patterns of historical eruptions at Hawaiian volcanoes. J Volcanol Geotherm Res 12(1–2):1–35. doi: 10.1016/0377-0273(82)90002-6 CrossRefGoogle Scholar
  15. Marzocchi W, Zaccarelli L (2006) A quantitative model for time-size distribution of eruptions. J Geophys Res 111(B04204). doi: 10.1029/2005JB003709 Google Scholar
  16. Matthews M, Ellsworth W, Reasenberg P (2002) A Brownian model for recurrent earthquakes. Bull Seismol Soc Am 92(6):2233–2250. doi: 10.1785/0120010267 CrossRefGoogle Scholar
  17. Mulargia F, Tinti S (1985) Seismic sample areas defined from incomplete catalogs: an application to the italian territory. Phys Earth Planet Inter 40(4):273–300CrossRefGoogle Scholar
  18. Mulargia F, Tinti S, Boschi E (1985) A statistical analysis of flank eruptions on Etna volcano. J Volcanol Geotherm Res 23(3–4):263–272CrossRefGoogle Scholar
  19. Mulargia F, Gasperini P, Tinti E (1987) Identifying different regimes in eruptive activity: an application to Etna volcano. J Volcanol Geotherm Res 34(1–2):89–106CrossRefGoogle Scholar
  20. Nakada S, Nagai M, Kaneko T, Nozawa A, Suzuki-Kamata K (2005) Chronology and products of the 2000 eruption of Miyakejima volcano, Japan. Bull Volcanol 67(3):205–218. http://www.springerlink.com/content/cj0d4tkcfamr7f32 CrossRefGoogle Scholar
  21. Newhall C, Self S (1982) The Volcanic Explosivity Index (VEI): an estimate of explosive magnitude for historical volcanism. J Geophys Res 87(C2):1231–1238CrossRefGoogle Scholar
  22. Ogata Y (1999) Estimating the hazard of ruptire using uncertain occurrence times of paleoearthquakes. J Geophys Res 104(B8):17995–18014CrossRefGoogle Scholar
  23. Sandri L, Marzocchi W, Gasperini P (2005) Some insights on the occurrence of recent volcanic eruptions of Mount Etna volcano (Sicily, Italy). Geophys J Int 163(3):1203–1218CrossRefGoogle Scholar
  24. Simkin T, Siebert L (2002) Volcanoes of the World: an illustrated catalog of Holocene volcanoes and their eruptions. Smithsonian Institution, Global Volcanism Program Digital Information Series, GVP-3. http://www.volcano.si.edu/world/ (onwards)
  25. Tsukui M, Suzuki Y (1998) Eruptive history of Miyakejima volcano during the last 7000 years. Bull Volcanol Soc Japan 43:149–166Google Scholar
  26. Ueda H, Fujita E, Ukawa M, Yamamoto E, Irwan M, Kimata F (2005) Magma intrusion and discharge process at the initial stage of the 2000 activity of Miyakejima, Central Japan, inferred from tilt and GPS data. Geophys J Int 161(3):891–906. doi: 10.1111/j.1365-246X.2005.02602.x CrossRefGoogle Scholar
  27. Watt SFL, Mather TA, Pyle DM (2007) Vulcanian explosion cycles: patterns and predictability. Geology 35(9):839–842. doi: 10.1130/G23562A.1 CrossRefGoogle Scholar
  28. Wickman FE (1976) Markov models of repose-period patterns of volcanoes. In: Random process in geology. Springer, New York, pp 135–161CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Alexander Garcia-Aristizabal
    • 1
    Email author
  • Warner Marzocchi
    • 2
  • Eisuke Fujita
    • 3
  1. 1.Istituto Nazionale di Geofisica e VulcanologiaBolognaItaly
  2. 2.Istituto Nazionale di Geofisica e VulcanologiaRomaItaly
  3. 3.National Research Institute for Earth Science and Disaster PreventionTsukubaJapan

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