Abstract There are currently about 1500 active volcanoes on Earth ( Tilling 1989 ). Eruptive activity presents in many different styles, ranging from highly explosive eruptions to non-explosive or effusive behaviour, which vary greatly in the hazard that they pose. Currently, millions of people are at risk from volcanic hazards. The average annual death toll as a result of volcanic hazards is rising because more people are living in close proximity to active volcanoes. Our understanding of the physical processes and parameters involved in the generation and evolution of volcanic flows is now advanced, and sophisticated process-oriented numerical models exist that describe eruptive processes well. There are hundreds to thousands of eruptions each year on Earth and many volcanoes are monitored around the clock by dedicated observatories. Thus, volcanology is rich in statistical data and statistical modelling is an emergent and rapidly growing area of interest. This volume is aimed at presenting the current state of statistical modelling within volcanology. The purpose of this paper is to give a general introduction to volcanic eruption processes, data and modelling, as well as an overview of the volume as a whole. This Special Publication is restricted to terrestrial volcanism (i.e. on land) and in the absence of large volumes of water, such as groundwater, sea or lake water, or snow. ‘Phreatomagmatic’ volcanism, which results from the interaction of magma and water, has been reviewed by Zimanowski (1998) . This Special Publication is restricted to terrestrial volcanism (i.e. on land) and in the

Abstract When a potentially dangerous volcano becomes restless, civil authorities invariably turn to scientific specialists to help them anticipate what the volcano will do next, and to provide them with guidance as to the likely threats. Although it is usually possible to discern the earliest signs of unrest, the science of forecasting the course and timing of eruptions remains inexact. In this paper, recent volcanic crises in the eastern Caribbean are recounted in order to trace the emergence of a need for volcanologists to formalize the way they present scientific advice in such circumstances. The discussion then moves on to the concepts and principles of eliciting expert opinion, and structured elicitation within a mathematical framework, before describing in more detail a specific performance-based procedure for eliciting opinions that relies on proper scoring rules. Ways in which this procedure and its scoring basis have been adapted for use in the recent Montserrat volcanic crisis are discussed, and the purposes for which the formalized procedure has been used during that eruption, in application to hazard and risk management, are described. Finally, a few general observations are offered on the benefits and limitations of using a structured procedure for eliciting scientific opinion in the unique and special circumstances of a volcanic eruption crisis.

Abstract Volcanic hazard assessment is a basic ingredient for risk-based decision-making in land-use planning and emergency management. Volcanic hazard is defined as the probability of any particular area being affected by a destructive volcanic event within a given period of time ( Fournier d'Albe 1979 ). The probabilistic nature of such an important issue derives from the fact that volcanic activity is a complex process, characterized by several and usually unknown degrees of freedom that are often linked by nonlinear relationships (e.g. Bak et al. 1988 ). Except in sporadic cases, the result of this complexity is an intrinsic, and perhaps unavoidable, unpredictability of the time evolution of the volcanic system from a deterministic point of view. In reality, current volcanic hazard assessment is even more entangled by scarce data and relatively poor knowledge of the physical processes. Cumulatively, these difficulties prevent a solution of the hazard or risk problem from a rigorous scientific perspective. Nevertheless, the potential for extreme risk pushes us to be pragmatic and to attempt to solve the problem from an ‘engineering’ point of view. Because of the devastating potential of volcanoes close to urbanized areas, the scientific community must address the issue as accurately and precisely as possible with the currently available methods and based on our current understanding of volcanic systems. This assessment is best done by treating scientific uncertainty in a fully structured manner. In this respect, Bayesian statistics provides a suitable framework for producing a volcanic hazard or risk assessments in

Abstract One of the most difficult problems we face in assessing volcanic hazards is that of evaluating the potential activity of volcanoes with little or no record of Holocene eruptions. Is there some minimum period of inactivity after which we can safely rule out a future eruption of large magnitude? Or, failing that, can we say how likely it is that such a volcano will return to activity within a particular span of time? Violent explosive eruptions are uncommon during the youthful stage of active growth. They are confined almost entirely to large mature volcanoes. Here, we use the global record of volcanic activity ( Simkin & Siebert 1994 ) to evaluate the duration of repose intervals preceding such explosive volcanic eruptions. This analysis indicates that the hazard rate for explosive eruptions is not constant with time, but depends on the time since last eruption and that explosive eruptions may occur at volcanoes that have been quiescent for 10 ka or more. The techniques we employ are common to a class of problems in survival analysis ( Cox & Oakes 1984 ; Woo 1999 ), and can be applied to a variety of hazard problems on volcanoes (e.g. Hill et al. 1998 ; Connor et al. 2003 ; Calder et al. 2005 ). One of the major lessons of this type of analysis is that applied statistical methods can teach us much about the time scales of volcanic activity, and about the underlying physical mechanisms governing these.

Abstract Extreme value theory sits alone in the statistical sciences. It is the branch of statistics devoted to the inference of extreme events in random processes, and differs from most areas of statistics in modelling rare, rather than typical, behaviour. A common aim is to estimate what future extreme levels of a process might be expected based on a historical series of observations. As such, the methodology is widely used in engineering applications that need an assessment of extreme environmental conditions: for example, sea levels, wind speeds or river flow ( Coles 2001 , chapter 1). Another recent area of application is to financial markets, for which calculations on the plausibility of large returns (postitive or negative) are helpful, and even a statutory requirement for many banks (e.g. Embrechts et al. 1998 ). Volcanic events are among the most explosive on Earth. A recent study ( Mason et al. 2004 ) has compared the potential damage from large volcanoes with that of other potentially catastrophic events, such as a meteor strike. At least in an informal sense therefore, they are extreme events, and it seems reasonable to hope that extreme value theory can make a useful contribution to their analysis. In this paper we explore this possibility. We take the recent history of large eruptive volcanoes, and apply extreme value techniques to obtain estimates of the probability of future extreme eruption events at different levels of magnitude. Such inferences are of general volcanologi-cal interest, but they are also directly .

Abstract 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 forecasting 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 sophisticated 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 statistical analysis. First, time-series analysis is used to identify the presence of clustering or trends in the data. For stationary periods, the data are modelled 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 causation 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 highlighted, reflecting a variation in certain characteristics of their.

Abstract Active volcanoes can generate natural hazards with potentially catastrophic consequences for society. With the growth of population worldwide, more exposed zones are occupied and more critical industrial facilities are being constructed, which require siting in areas of very low geological hazard. Because of these expanding societal demands, the need for risk assessments of volcanic activity has grown over recent decades and will continue to do so. According to the definitions established by UNDRO (1979) , volcanic risk is a measure of the expected number of lives lost, persons injured, damage to property and disruption of economic activity as a result of a particular volcanic event. It is defined as the product of volcanic hazard, vulnerability and elements at risk. Volcanic hazard represents the probability of occurrence of a potentially damaging volcanic event within a specific period of time in a given area. Vulnerability indicates the degree of loss to a given element at risk or set of such elements resulting from the occurrence of a given volcanic event and is expressed on a scale from zero (no damage) to one (total loss). Finally, elements at risk denote the population, buildings and civil engineering works, economic activities, public services, utilities and infrastructure, etc. at risk in a given area. Because of the anthropogenic nature of the vulnerability and elements at risk, mitigation can usually be achieved with appropriate technical measures. The volcanic hazard, however, is intrinsically linked to the volcano and, except for specific hazards such as lava flows

Abstract As in other areas of geophysics or meteorology, the observations and data collected at volcanoes are the result of experiments in which we cannot control the variables we wish to study. Thus, statistical analysis is an extremely important step in the data processing. Variations in the experimental parameters must be controlled through the choice of samples and through the hypotheses chosen for testing. The evaluation of the samples is possible only through the application of the proper statistical methods, especially multivariate statistics. Volcanic activity is the manifestation of complex dynamic processes and interactions within the volcano. It dependson the movement of fluids as well as on the thermodynamics of the magma and gases within a branched network of conduits and cavities. The dynamic processes generate various geophysical signals, as well as visible phenomena at the volcano’s surface. State-of-the-art techniques for monitoring at volcanoes now include continuous and concurrent recording of a variety of both quantitative and qualitative observations using a multi-parameter station in the near-field of thecrater. Such a station has been installed at Galeras volcano in Colombia, and has been operating for several years ( Seidl et al. 2003 ). Presumably, the signals and phenomena observed at the surface of a volcano have acommon source in terms of the strong interactions between various internal processes. Thus, data from different measurements should show a significant correlation. Contingency tables are a powerful statistical method for investigating such multi-dimensional correlations between quantitative and qualitative data. The pattern of signals and phenomena, aswell

Abstract The magnitude of an earthquake is a measure related to its energy. If a geographical region is selected for research, many earthquakes with different energies may appear during the study interval. An analysis of the distribution of magnitude values can be useful to understand some spatial and temporal characteristics of the region. Gutenberg & Richter (1944) proposed a statistical relation between the magnitude and the number of the seismic events. In this paper we review this relationship, analysing its assumptions and comparing them with observed data from selected volcanic and tectonic regions of Ecuador. The Gutenberg-Richter (GR) Law is an empirical relation between the magnitude x of some seismic event and N(x), the number of events with magnitudes higher than x. Ishimoto & Iida (1939) and Gutenberg & Richter (1944) proposed the following linear relation:

Abstract Time-series analysis is about the study of data collected through time. The field of time series is a vast one that pervades many areas of science and engineering, particularly statistics and signal processing: this short paper can only be an advertisement. Hence, the first thing to note is that there are several excellent texts on time-series analysis. Most statistical books concentrate on stationary time series and some texts have good coverage of ‘globally non-stationary’ series such as those often used in financial time series. For a general, elementary introduction to time-series analysis the author highly recommends the book by Chatfield (2003) . The core of Chatfield’s book is a highly readable account of various topics in time-series including time-series models, forecasting, time series in the frequency domain and spectrum estimation, and also linear systems. More recent editions contain useful, well-presented and well-referenced information on important new research areas. Of course, there are many other books: ones the author finds useful are those by Priestley (1983) , Diggle (1990) , Brockwell & Davis (1991) and Hamilton (1994) . The book by Hannan (1960) is concise (but concentrated) and that by Pole et al. (1994) is a good introduction to a Bayesian way of doing time-series analysis. There are undoubtedly many more books.This paper is a brief survey of several kinds of time-series model and analysis. The ï—rst section covers stationary time series, which, loosely speaking, are those whose statistical properties

Abstract The literature on geophysical time-series analysis is so extensive that to review even one topic, such as volcanic tremor series, is a major task (e.g. Konstantinou & Schlindwein 2002 ). The purpose of this paper is not to attempt such a review but rather to outline some new tools for nonstationary and nonlinear time-series analysis that have been developed and used successfully in other areas of the environmental sciences and appear to have good potential for application in a volcanological or wider geophysical context. These stochastic methods of time-series analysis have the advantage that they all exploit powerful recursive (sequential updating) methods of estimation that facilitate the analysis of data generated by nonstationary and nonlinear systems (e.g. Young 1984 ). This paper starts by reviewing briefly some of the model-based methods of time-series analysis, signal extraction and forecasting that have appeared in the statistical and time-series analysis literature over many years and then proceeds to describe in more detail one approach that has attracted considerable interest over the past two decades. This is based on the concept of an ‘unobserved component’ model and it exploits recursive estimation for the purposes of estimating time-variable parameters in nonstationary systems. It is shown that such an approach allows for various, practically useful procedures in time-series analysis: signal extraction; interpolation over gaps in time series; and forecasting or backcasting. It then goes on to outline the basic aspects of input-output time series modelling, considering both discrete-time and continuous-time ‘transfer function’ models that.

Abstract The classification of seismic signals of volcanic origin (SSVOs) is an important task in the context of monitoring active volcanoes. The number and the size of certain types of seismic events usually increase before periods of volcanic crisis and are a key index of forthcoming activity. However, the task of classifying SSVOs is in most cases still carried out manually during daily routine work. The implementation of an automatic classification system not only would allow the processing of large amounts of data in short time, but would also have the advantage of providing a consistent, objective and time-invariant classification. Techniques for automatic detection and classification of seismic events have been of great interest to the seismological community since the introduction of digital seismic monitoring. Nowadays, automatic detection or picking of impulsive transients in earthquake seismic signals can be efficiently achieved by short-time-average to longtime-average ratio (STA/LTA) trigger algorithms ( Bormann et al. 2002 ; Trnkoczy 2002 ). Moreover, some algorithms have also been proposed for the automatic classification of earthquake seismic signals ( Joswig 1996 ; Gendron et al. 2000 ). However, when working with SSVOs additional problems are encountered. From the signal analysis point of view, SSVOs can be very different from earthquake seismic signals. As already noted (e.g. by Wassermann 2002 ), they vary from earthquakelike transients to long-lasting and continuous tremor signals. Moreover, the signal-to-noise ratio of SSVOs is usually rather low. For these reasons, the automatic classification of SSVOs is still an open question.

Abstract Volcanic eruptions are commonly characterized by time series of events, such as earthquakes and explosions, which can be analysed by statistical techniques to interpret physical mechanisms of eruption and to be applied to forecasting. We apply geostatistical methods ( Chiles & Delfiner 1999 ) to a time series of Vulcanian explosions that occurred at the Soufriére Hills volcano, Montserrat (Fig. 1 ) in the period 22 September to 21 October 1997. These techniques can be used to detect correlations in occurrences of volcanic processes. Such correlations indicate that the processes are capable of remembering their past activities and can be used to detect memory effects in dynamic systems. The sequence of 75 Vulcanian explosions at Soufriére Hills followed a collapse of the andesite lava dome on 21 September 1997 ( Druitt et al. 2002 ). The mean repose interval was 9.6 h with a minimum of 2.8 h and a maximum of 33.7 h ( Connor et al. 2003 ). The explosions were shortlived (tens of seconds), impulsive and energetic ( Druitt et al. 2002 ) with column heights between 5 and 15 km above sea level and individual ejecta volumes up to 6.6 x 10 5 m 3 . The explosion time series is complete and precisely timed by seismic signals, allowing us to apply a stochastic approach using the variogram ( Chiles & Delfiner 1999 ; Jaquet & Carniel 2001 ) to characterize the statistical behaviour. The data consist of the date and time of an explosion (Tm), the time interval

Abstract The flow of magma in a volcanic conduit and the flow of the mixture of gas, solid and liquid through a Plinian eruption column are highly complex physical processes governed by a large number of mechanisms operating at different spatial and temporal scales. Nevertheless, under certain conditions, steady-state homogeneous fluid mechanical models provide volcanologists with leading order approximations of such complex phenomena. These types of models, for conduit flow ( Wilson et al. 1980 ; Buresti Casarosa 1989 ; Mastin 1995 ; Woods 1995 ) and for Plinian eruption columns ( Wilson et al. 1978 ; Sparks 1986 ; Wilson Walker 1987 ; Woods 1988 , 1995 ) are welldeveloped in the volcanological literature and help volcanologists to gain insight into the important parameters governing real volcanic eruptions. A good knowledge of these parameters is essential not only in characterizing ancient eruptions but also in improving forecasts of future eruptions. This parameter exploration is usually accomplished through a parametric sensitivity analysis, in which the sensitivity of the models to various changes in parameter values is studied. A parametric sensitivity analysis can also be used to determine how well output parameters, such as eruption column height, can be determined from input parameter ranges that are poorly constrained, and to investigate the originof shapes of parameter distributions observed in nature (e.g. log-normal distributions of eruption column heights). This can be accomplished by introducing input parameters in conduit and eruption column models as probability density functions.

Abstract Magma flow in volcanic conduits involves complicated physico-chemical transformations during ascent, including gas exsolution, bubble nucleation and growth, gas escape from the magma, and magma fragmentation (in the case of explosiveeruptions). These changes are accompanied by major changes in the rheological properties of magma. The structure of the flow can change from homogeneous liquid flow at depth to gas-particle dispersion flow in the upper part of a conduit.There are two distinct zones of the flow: the zone where the liquid is a continuous phase and flow is mainly controlled by viscous resistance, and the zone with continuous gas phase where the flow is dominated by inertia. These zones are separated by a fragmentation front, whose position must be determined during the solution of the flow dynamics. This makes modelling of conduit flows a difficult problem and requires strict constraints on the accuracy and stability of the numerical method. The literature on the modelling of conduit flow processes during explosive eruptions contains many tens of papers.Good overviews include those by Woods (1995) , Sparks et al. (1997) , Papale (1998) , Melnik (2000) , Slezin (2003) , Sahagian (2005) , Mader (2006) . Most of the models presented in the literature describe conduit flow during volcanic eruptions as a 1D steady-state process based on the assumptions that the length of the conduit is much largerthan its radii and the time scale of parameter variations is much longer than the residence time of an individual

Abstract Modern physical volcanology is situated between two different research approaches: multi-disciplinary data acquisition in field and laboratory settings, and analytical and computer-based multi-parameter modelling. Ideally, any data should be interpreted with reference to a physical model; on the other hand, modelling attempts should be constrained by data. Unfortunately, this has not always been the case and some of the reasons for the difficulties encountered willbe analysed in this study. Problems in data analysis can often be traced back to the widespread misunderstanding that a time series (or spatial data profile) of acquired data represents the natural phenomenon that a research team set out to study. Magma movement in the volcanic plumbing system, for example, may lead to a pressure increase in a certain region of a magmatic conduit. The conduit walls may be deformed in an elastic or plastic manner depending on the magnitude of the pressure change as well as the material properties involved. This deformation will propagate, but not necessarily instantaneously,all the way to the flank of the volcanic edifice, where an instrument such as a tilt meter, a broadband seismometer ora strainmeter will detect a corresponding signal. This signal, however, is superimposed on the response of the nearby topography to the internal deformation, as well as all other signals generally referred to as noise. Depending on the topography, an inflation at shallow depth for example, can lead, counter-intuitively, to a subsidence of a part of thevolcanic flank. Furthermore, any instrument will only pick up that particular part of the ground displacement that theinstrument was designed to detect. A tiltmeter will measure the (rigid) rotation of the flank and record the corresponding angle; the broadband seismometer will record the time-derivative (velocity) of the displacement in a certain frequency band corresponding to its frequency characteristics (very much like human ears), typically between 50 Hz and0.008 Hz, or 120 s period; a strainmeter will directly record the displacement, but only the horizontal components, and hear only the long period part. Finally, this signal is digitized in certain time intervals, and appears in a recording medium as a time series. This ‘bunch of numbers’ has little to do with the original, localized deformation of the conduit wall caused by the pressurization of the volcanic system. Intensive data processing is necessary to retrieve the original ground displacement, at the instrument location, from the recorded time series. Intensive modelling is then necessary to link the ground displacement at the surface of the volcanic edifice to the physical processesat the depth that caused it.

Abstract Volcanologists increasingly rely on numerical simulations to better understand the dynamics of erupting volcanoes. Mathematical models are often used to explain the geological processes responsible for eruption deposits found in the geological record, and to better characterize possible hazards from future volcanic activity. Examples of models include the finite-element flow and transport codes used to simulate pyroclastic flows, lahars, and volcanic debris avalanches ( Iverson 1997 ; Patra et al. 2005 ), analytical solutions or finite-difference approximations to the advection-diffusion equation that are used to model tephra dispersion ( Suzuki 1983 ) and gas emissions from quiescentvolcanoes ( Delmelle et al. 2001 ), and cellular automata algorithms that model the advance of lava ( Barca et al. 1994 ). Commonality among these examples involves the fact that the parameters to be estimated are related to the dynamics of volcanic activity derived from field observations. For instance, how well can the magnitude of an eruption be estimated from measurements made of tephra deposits? One solution to this question lies in coupling numerical simulations to inversion methods that search for an optimal set of parameters that explain the physical observations. For example, volcanologists make observations of tephra thickness and variations in particle size to help estimate the parameters that describe the dynamics of the volcanic eruption that created the deposit. These parameters include eruption volume, eruption column height, and wind velocity as a function of height above the ground. The difficulty in this approach is that volcanologists must deduce multiple parameters that characterize

Abstract Depending on their magnitude and location, volcanic eruptions have the potential to become major social and economic disasters (e.g. Tambora, Indonesia, 1815; Vesuvius, Italy, AD 79; Soufriere Hills Volcano, Montserrat, 1995-present).One of the challenges for the volcanology community is to improve our understanding of volcanic processes so as to achieve successful assessments and mitigation of volcanic hazards, which are traditionally based on volcano monitoring and geological records. Geological records are crucial to our understanding of eruptive activity and history of a volcano, but often do not provide a comprehensive picture of the variation of volcanic processes and their effects on the surrounding area. The geological record is also typically biased towards the largest events, as deposits from smaller eruptions are often removed by erosion. Numerical modelling and probability analysis can be used to complement direct observations and to explore a much wider range of possible scenarios. As a result, numerical modelling and probabilistic analysis have become increasingly important in hazard assessment of volcanic hazards (e.g. Barberi et al. 1990 ; Heffter & Stunder 1993 ; Wadge et al. 1994 , 1998 ; Hill et al. 1998 ; Iverson et al. 1998 ; Searcy et al. 1998 ; Canuti et al. 2002 ). Reliable and comprehensive hazard assessments of volcanic This paper offers a detailed review of common approaches for hazard assessments of tephra dispersion. First, the main characteristics of tephra dispersion and tephra hazards are recounted. A critical use of field data for a