Natural Hazards and Earth System Science www.nat-hazards-earth-syst-sci.net 1561-8633 1684-9981 9 2 2009 10.5194/nhess-9-425-2009 http://www.nat-hazards-earth-syst-sci.net/9/425/2009/ http://www.nat-hazards-earth-syst-sci.net/9/425/2009/nhess-9-425-2009.html http://www.nat-hazards-earth-syst-sci.net/9/425/2009/nhess-9-425-2009.pdf 425 431 2009-03-19 A mixture of exponentials distribution for a simple and precise assessment of the volcanic hazard A. T. Mendoza-Rosas S. De la Cruz-Reyna Posgrado en Ciencias de la Tierra, Instituto de Geofísica, Universidad Nacional Autónoma de México, Ciudad Universitaria, México 04510 D. F., México Instituto de Geofísica, Universidad Nacional Autónoma de México, Ciudad Universitaria, México 04510 D. F., México The assessment of volcanic hazard is the first step for disaster mitigation. The distribution of repose periods between eruptions provides important information about the probability of new eruptions occurring within given time intervals. The quality of the probability estimate, i.e., of the hazard assessment, depends on the capacity of the chosen statistical model to describe the actual distribution of the repose times. In this work, we use a mixture of exponentials distribution, namely the sum of exponential distributions characterized by the different eruption occurrence rates that may be recognized inspecting the cumulative number of eruptions with time in specific VEI (Volcanic Explosivity Index) categories. The most striking property of an exponential mixture density is that the shape of the density function is flexible in a way similar to the frequently used Weibull distribution, matching long-tailed distributions and allowing clustering and time dependence of the eruption sequence, with distribution parameters that can be readily obtained from the observed occurrence rates. Thus, the mixture of exponentials turns out to be more precise and much easier to apply than the Weibull distribution. We recommended the use of a mixture of exponentials distribution when regimes with well-defined eruption rates can be identified in the cumulative series of events. As an example, we apply the mixture of exponential distributions to the repose-time sequences between explosive eruptions of the Colima and Popocatépetl volcanoes, México, and compare the results obtained with the Weibull and other distributions. Aspinall, W. P., Carniel, R., Jaquet, O., Woo, G., and Hincks, T.: Using hidden multi-state Markov models with multi-parameter volcanic data to provide empirical evidence for alert level decision-support, J. Volcanol. Geoth. Res., 153, 112–124, 2006. Bebbington, M. S. and Lai, C. D.: On nonhomogeneous models for volcanic eruptions, Math. Geol., 28, 585–600, 1996a. Bebbíngton, M. S. and Lai, C. D.: Statistical analysis of New Zealand volcanic occurrence data, J. Volcanol. Geoth. Res., 74, 101–110, 1996b. Bebbington, M. S.: Identifying volcanic regimes using hidden Markov models, Geophys. J. Int., 171, 921–942, 2007. Carta, S., Figari, R., Sartoris, G., Sassi, E., and Scandone, R.: A statistical model for Vesuvius and its volcanological implications, Bull. Volcanol., 44, 129–151, 1981. Cox, D. R. and Lewis, P. A. W.: The Statistical Analysis of Series of Events, Methuen and Co., London, UK, 285~pp., 1966. De la Cruz-Reyna, S.: Poisson-Distributed Patterns of Explosive Activity, Bull. Volcanol., 54, 57–67, 1991. De la Cruz-Reyna, S.: Random patterns of occurrence of explosive eruptions at Colima Volcano, México, J. Volcanol. Geotherm. Res., 55, 51–68, 1993. De la Cruz-Reyna, S.: Long- Term Probabilistic Analysis of Future Explosive Eruptions, in: Monitoring and Mitigation of Volcanic Hazards, edited by: Scarpa, R. and Tilling, R. I., Springer, Berlin, Germany, 599–629, 1996. De la Cruz Reyna, S., and Tilling, R.: Scientific and public responses to the ongoing volcanic crisis at Popocatépetl Volcano, México: importance of an effective hazards warning system, J. Volcanol. Geoth. Res., 170, 121–134, 2008. Everitt, B. S. and Hand, D. J.: Finite mixture distributions. Monographs on Applied Probability and Statistics, Chapman and Hall, London, UK, 143~pp., 1981. Feldmann, A. and Whitt, W.: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models, Perform. Evaluation, 31, 245–279, 1998. Gibbons, J. P.: Nonparametric Methods for quantitative Analysis, Holt, Rinehart and Winston, New York, USA, 1976. Ho, C.-H.: Bayesian analysis of volcanic eruptions, J. Volcanol. Geoth. Res., 43, 91–98, 1990. Ho, C.-H.: Time trend analysis of basaltic volcanism for the Yucca Mountain site, J. Volcanol. Geoth. Res., 46, 61–72, 1991. Ho, C.H.: Sensitivity in Volcanic Hazard Assessment for the Yucca Mountain High-Level Nuclear Waste Repository Site: The Model and the Data, Math. Geol., 27, 239–258, 1995. Ho, C. H., Smith, E. I., and Keenan, D. L.: Hazard area and probability of volcanic disruption of the proposed high-level radioactive waste repository at Yucca Mountain, Nevada, USA, Bull. Volcanol., 69, 117–123, 2006. Jaquet, O. and Carniel, R.: Estimation of volcanic hazard using geostatistical models, in: Statistics in Volcanology, edited by: Mader, H. M., Coles, S. G., Connor, C. B., and Connor, L. J., IAVCEI Publications n.1., Geological Society, London, UK, 89–103, 2006. Jaquet, O., Löw, S., Martinelli, B., Dietrich, V., and Gilby, D.: Estimation of volcanic hazards based on Cox stochastic processes, Phys. Chem. Earth, 25, 571–579, 2000. Johnson, N. L. and Kotz, S.: Distributions in Statistics: Continuous Univariate Distributions, Houghton Mifflin Company, USA, 300~pp., 1953. Klein, F. W.: Patterns of Historical eruptions at Hawaiian Volcanoes, J. Volcanol. Geoth. Res., 12, 1–35, 1982. Marzocchi, W., Sandri, L., and Selva, J.: BET_EF: a probabilistic tool for long- and short-term eruption forecasting, Bull. Volcanol., 70, 623–632, 2008. Mendoza-Rosas A. T. and De la Cruz-Reyna, S.: A statistical method linking geological and historical eruption time series for volcanic hazard estimations: Applications to active polygenetic volcanoes, J. Volcanol. Geoth. Res., 176, 277–290, 2008. Newhall, C. G. and Self, S.: The Volcanic Explosivíty Index (VEI): an estimate of explosive magnitude for historical volcanism, J. Geophys. Res., 87(C2), 1231–1238, 1982. Newhall, C. G. and Hoblitt, R. P.: Constructing event trees for volcanic crises, Bull. Volcanol., 64, 3–20, 2002. Reyment, R. A.: Statistical analysis of some volcanologic data. Regarded as series of point events, Pageoph., 74(3), 57–77, 1969. Rider, P. R.: The method of moments applied to a mixture of two exponential distributions, Annals. Math. Stats., 32, 143–147, 1961. Solow, A. R.: An empirical Bayes analysis of volcanic eruptions, Math. Geol., 33(1), 95–102, 2001. Sum S. T. and Oommen, B. J.: Mixture decomposition for distributions from the exponential family using a generalized method of moments, IEEE transactions on systems, man and cybernetics, 25(7), 11–39, 1995. Turner, M. B., Cronin, S. J., Bebbington, M. S., and Platz, T.: Developing probabilistic eruption forecasts for dormant volcanoes: a case study from Mt Taranaki, New Zealand, Bull. Volcanol., 70, 507–515, 2008. Wickman, F. E.: Repose period patterns of volcanoes, 5. General discussion and a tentative stochastic model, Ark. Mineral. Geol., 4, 351–367, 1965. Wickman, F. E.: Markov models of repose-period patterns of volcanoes, in: Random Processes in Geology, edited by: Merriam, D. F., Springer-Verlag, Berlin, Germany, 135–161, 1976.