# Nonhomogeneous Poisson model for volcanic eruptions

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## Abstract

A simple Poisson process is more specifically known as a homogeneous Poisson process since the rate*λ* was assumed independent of time t. The homogeneous Poisson model generally gives a good fit to many volcanoes for forecasting volcanic eruptions. If eruptions occur according to a homogeneous Poisson process, the repose times between consecutive eruptions are independent exponential variables with mean*θ*=1/*λ*. The exponential distribution is applicable when the eruptions occur “at random” and are not due to aging, etc. It is interesting to note that a general population of volcanoes can be related to a nonhomogeneous Poisson process with intensity factor*λ*(t). In this paper, specifically, we consider a more general Weibull distribution, WEI (*θ, β*), for volcanism. A Weibull process is appropriate for three types of volcanoes: increasing-eruption-rate (*β*>1), decreasing-eruption-rate (*β*<1), and constant-eruption-rate (*β*=1). Statistical methods (parameter estimation, hypothesis testing, and prediction intervals) are provided to analyze the following five volcanoes: Also, Etna, Kilauea, St. Helens, and Yake-Dake. We conclude that the generalized model can be considered a goodness-of-fit test for a simple exponential model (a homogeneous Poisson model), and is preferable for practical use for some nonhomogeneous Poisson volcanoes with monotonic eruptive rates.

## Key words

prediction interval volcanism Weibull distribution## Preview

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## References

- Bain, L. J., 1978, Statistical Analysis of Reliability and Life-Testing Models: Marcel Dekker, New York, 450 p.Google Scholar
- Klein, F. W., 1982, Patterns of Historical Eruptions at Hawaii Volcanoes: Journal of Volcanology and Geothermal Research, v. 12, p. 1–35.Google Scholar
- Mulargia, F., Tinti, S., and Boschi, E., 1985, A Statistical Analysis of Flank Eruptions on Etna Volcano: Journal of Volcanology and Geothermal Research, v. 23, p. 263–272.Google Scholar
- Parzen, E., 1962, Stochastic Processes: Holden-Day, San Francisco, 324 p.Google Scholar
- Simkin, T., Seibert, L., McClelland, L., Bridge, D., Newhall, C., and Latter, J. H., 1981. Volcanoes of the World: Smithsonian Institution and Hutchinson Ross, Stroudsburg, Pennsylvania, 232 p.Google Scholar
- Steel, R. G. D., and Torrie, J. H., 1980. Principles and Procedures of Statistics (2nd ed): McGraw-Hill, New York, 633 p.Google Scholar
- Wickman, F. E., 1966, Repose-Period Patterns of Volcanoes: Ark. Mineral. Geol., v. 4, p. 291–367.Google Scholar
- Wickman, F. E., 1976, Markov Models of Repose-Period Patterns of Volcanoes:
*in*D. F. Merriam (Ed.), Random Processes in Geology: Springer, New York, p. 135–161.Google Scholar