Advertisement

Mathematical Geology

, Volume 28, Issue 5, pp 585–600 | Cite as

On nonhomogeneous models for volcanic eruptions

  • M. S. Bebbington
  • C. D. Lai
Article

Abstract

Recently, a special nonhomogeneous Poisson process known as the Weibull process has been proposed by C-H. Ho for fitting historical volcanic eruptions. Revisiting this model, we learn that it possesses some undesirable features which make it an unsatisfactory tool in this context. We then consider the entire question of a nonstationary model in the light of availability and completeness of data. In our view, a nonstationary model is unnecessary and perhaps undesirable. We propose the Weibull renewal process as an alternative to the simple (homogeneous) Poisson process. For a renewal process the interevent times are independent and distributed identically with distribution function F where, in the Weibull renewal process, F has the Weibull distribution, which has the exponential as a special situation. Testing for a Weibull distribution can be achieved by testing for exponentiality of the data under a simple transformation. Another alternative considered is the lognormal distribution for F. Whereas the homogeneous Poisson process represents purely random (memoryless) occurrences, the lognormal distribution corresponds to periodic behavior and the Weibull distribution encompasses both periodicity and clustering, which aids us in characterizing the volcano. Data from the same volcanoes considered by Ho were analyzed again and we determined there is no reason to reject the hypothesis of Weibull interevent times although the lognormal interevent times were not supported. Prediction intervals for the next event are compared with Ho's nonhomogeneous model and the Weibull renewal process seems to produce more plausible results.

Key words

prediction interval volcanism renewal process Weibull distribution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ascher, S., 1990, A survey of tests for exponentiality: Communications in Statistics, Theory and Methods, v. 19, no. 5, p. 1811–1825.Google Scholar
  2. Bain, L. J., and Engelhart, M., 1991, Statistical analysis of reliability and life-testing models (2nd ed.): Marcel Dekker, New York, 496 p.Google Scholar
  3. Conover, W. J., 1980, Practical nonparametrie statistics (2nd ed.): John Wiley & Sons, New York, 493 p.Google Scholar
  4. Guttorp, P., and Thompson, M. L., 1991, Estimating second-order parameters of volcanicity from historical data: Jour. Am. Stat. Assoc., v. 86, no. 415, p. 578–583.Google Scholar
  5. Ho, C.-H., 1991, Nonhomogeneous Poisson model for volcanic eruptions: Math. Geology, v. 23, no. 2, p. 167–173.Google Scholar
  6. Klein, F. W., 1982, Patterns of historical eruptions at Hawaiian volcanoes: Jour. Volcanology and Geothermal Research, v. 12, p. 1–35.Google Scholar
  7. Mulargia, F., Tinti, S., and Bouchi, E., 1985, A statistical analysis of flank eruptions on Etna volcano: Jour. Volcanology and Geothermal Research, v. 23, p. 263–272.Google Scholar
  8. Rohlf, F. J., and Sokal, R. R., 1981, Statistical tables (2nd ed.): W. H. Freeman, San Francisco, 219 p.Google Scholar
  9. Siegel, S., 1956, Nonparametric statistics: McGraw-Hill Book Co., New York, 312 p.Google Scholar
  10. Simkin, T., Seibert, L., McCleeland, L., Bridge, D., Newhall, C., and Latter, J. H., 1981, Volcanoes of the world: Hutchinson Ross, Stroudsburg, Pennsylvania, 232 p.Google Scholar
  11. Wickman, F. E., 1966, Repose-period patterns of volcanoes: Arkiv för Mineralogi och Geologi, v. 4, no. 7, p. 291–367.Google Scholar
  12. Wickman, F. E., 1976, Markov models of repose-period patterns of volcanoes,in Merriam, D. F., ed., Random processes in geology: Springer-Verlag, New York, p. 135–161.Google Scholar

Copyright information

© International Association for Mathematical Geology 1996

Authors and Affiliations

  • M. S. Bebbington
    • 1
  • C. D. Lai
    • 1
  1. 1.Department of StatisticsMassey UniversityPalmerston NorthNew Zealand

Personalised recommendations