Applying statistical analysis to understanding the dynamics of volcanic explosions N. VARLEY 1 , J. JOHNSON 2 , M. RUIZ 3 , G. REYES 4 & K. MARTIN 5 1 Facultad de Ciencias, Universidad de Colima, Colima, Mexico (e-mail: nick@ucol.mx) 2 Department of Earth Sciences, University of New Hampshire, Durham, NH 03824, USA (e-mail: jeff.johnson@unh.edu) 3 Instituto Geofisico, Escuela Polite ´cnica Nacional, Quito, Box 17-01-2759, Ecuador (e-mail: mruiz@igepn.edu.ec) 4 RESCO, Universidad de Colima, Colima, Mexico (e-mail: gabrielr@cgic.ucol.mx) 5 Department of Geology, University of South Florida, Tampa, FL 33620-5201, USA (e-mail: Ktmartin@mail.usf.edu) An erupting volcano is a complex system controlled by nonlinear dynamics and hence is difficult to model numerically. Statistical methods can be applied to explain behaviour or to aid the forecast- ing of future activity. The majority of previous studies have considered large-scale events: large explosive or effusive eruptions, with intervening long periods of repose. This has severely limited the size of the datasets and hence the significance of statistical results. In previous cases a simple Poisson model was applied, but often more sophis- ticated analysis methods are necessary to model the data. In this study, several statistical techniques are used to describe the data for smaller-scale events from four volcanoes. In each case study the events are relatively frequent explosions; this means that the datasets are large and thus allow a robust stat- istical analysis. First, time-series analysis is used to identify the presence of clustering or trends in the data. For stationary periods, the data are mod- elled in a probabilistic fashion, taking the survival function for increasing repose intervals and fitting different distributions to the data. Different types of events are identified, whose repose intervals have different distributions. This implies variation in the physics of the processes involved in the cau- sation of the events. It is shown that activity can be divided into different periods based on the statistics, which can greatly aid in the construction of a model to explain the temporal evolution of eruptive activity. Contrasts between the volcanoes are high- lighted, reflecting a variation in certain character- istics of their systems and the processes that dominate their activity. Volcanic activity can be regarded as chaotic with no apparent form, but the use of statistical analysis can reveal hidden structure in behaviour patterns. Applied either spatially or temporally, it can be a powerful tool leading to future forecasting or improved models of the system. Denlinger & Hoblitt (1999) recognized and modelled cyclic behaviour in silicic volcanoes, which can be obvious, identified by seismicity or tilt measure- ments. However, longer-term patterns are often dif- ficult to distinguish. The first stage of a probabilistic analysis is to determine if the data are stationary or if they are characterized by periodicity, clustering or a trend. The simple stream of events can be ana- lysed as a time series. Next, a probability density function can be fitted to the repose interval data; the resulting model can provide important infor- mation on the processes that are generating the event. A dataset might be the historical record of a certain type of event, it could be a single variable, such as the occurrence of an event in time (e.g. eruptions of Volcanic Explosivity Index (VEI) . 2) or it could include another variable such as the magnitude. When considering the probability of volcanic events, the first question to ask is whether events represent a stochastic time series, meaning that events can be represented as a random variable, indexed in time. In contrast, a deterministic model is characterized by no randomness in model parameters. Theoretically, by completely specifying a deterministic model, events may be forecast at any point in the future. Such models appear to be unachievable in natural systems. From:MADER, H. M., COLES, S. G. & CONNOR, C. B. (eds) 2006. Statistics in Volcanology. Special Publications of IAVCEI, 1, 57–76. Geological Society, London. 1750-8207/06/$15.00 # IAVCEI 2006.