Jmuvahfvolcanology andgeothermalreseamh zyxwvut ELSEVIER Journal of Volcanologyand Geothermal Research74 (1996) 101-l 10 Statistical analysis of New Zealand volcanic occurrence data M.S. Bebbington *, CD. Lai zyxwvutsrqponmlkjihgfedcbaZYXWVU Department zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of Statistics, Massey University Private Bag 11222, Palmerston North, New Zealand Received6 November 1995;accepted27 June 1996 Abstract The Poisson process is considered to provide a good tit to many volcanoes for forecasting eruptions. However, several exceptions exist, usually where the intensity is non-stationary. The nonhomogeneous Poisson model (the Weibull process) caters for a monotonically varying intensity, but often results in physically unrealistic behaviour. A model which allows for more general behaviour than the Poisson process is the Weibull renewal model. This paper considers data from two New Zealand volcanoes, Mt. Ruapehu and Mt. Ngauruhoe, and fits the Poisson and Weibull renewal models. We conclude that a simple Poisson process fits Ngauruhoe very well. The behaviour of Mt. Ruapehu is considerably more complex, although quite reasonable forecasts can still be obtained from the renewal models. An interesting feature of our analysis is that there seems to be no correlation between the observed eruption sequences of these two closely neighbouring volcanoes. Keywords: prediction nterval; renewal process: Weibull distribution; correlation 1. Introduction New Zealand is an actively volcanic country. During the last two centuries, there have been a large number of volcanic eruptions in the central North Island. The dates of eruptions and estimated sizes (erupted volumes) have been recorded for the period 1850-1984 (see Latter, 1985) for Mt. Ruapehu and Mt. Ngauruhoe (for erupted volumes greater than IO5 m3). It is generally believed that a simple Poisson process gives a good fit to many volcanoes for forecasting volcanic eruptions (particularly for larger events, see De la Cruz-Reyna, 1991; De la Cruz-Re- yna, 1993). If the probability of an event in a small time interval (t, t + At) is A(t) then in a Poisson * Correspondingauthor. process, the rate of occurrence h(t) is a constant independent of time t. This rate is also known as the intensity function in the applied probability litera- ture. However, Wickman found in his study (Wick- man, 1966, 19761, that the intensity function A(t) could be modeled as an increasing step function, that is, that the volcanic activity mainly changed step- wise. This prompted Ho (1991) to consider a nonho- mogeneous Poisson process to model volcanic activ- ity. In particular, Ho assumed that eruptions occurred as a Weibull process (see, for example, Bain and Engelhart, 1991, chapter 91, that is, in a process with time-varying intensity h(t) = pKPtP- ‘, for some positive constants j3 and 8, where t is the time since some arbitrary origin. Thus this model is non-sta- tionary (unless p = 1) since there is a monotonic increase or decrease in the intensity. Bebbington and Lai (1996) revisit this nonhomogeneous model and find that it possesses some undesirable features which 0377-0273/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SO377-0273(96)00050-9