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Chapter 1 Volcanic Processes as a Source of Statistical Data Heidy M Mader There are currently about 1500 active volcanoes on Earth (Tilling 1989). Eruptive activity presents many different styles, ranging from highly explosive eruptions through 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 due to 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 100’s to 1000’s 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 chapter is to give a general introduction to volcanic eruption processes, data, and modelling as well as an overview of the volume as a whole. The volume 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. For a review of ‘phreatomagmatic’ volcanism, which results from the interaction of magma and water, see Zimanowski (1998). 1 1.1 Spatial And Temporal Distributions Volcanoes occur where molten rock, or ‘magma’, reaches the Earth’s surface. A cursory glance at Figure 1.1 shows that the distribution of the world’s volcanoes is not random; most of them are located along the edges of the tectonic plates that make up the Earth’s lithosphere. In fact, about twothirds of the world’s active subaerial volcanoes are located in the plate margins that surround the Pacific Ocean forming the so-called ‘Ring of Fire’. At these margins, two plates converge and the oceanic plate sinks, or ‘subducts’, into the mantle below the continental plate (Figure 1.2). Hence these subduction zones are also called ‘convergent’ or ‘destructive’ plate margins. The descending plate is dehydrated and the escaping fluids (mainly water) rise into the overlying mantle wedge thereby lowering the solidus (the temperature at which a solid begins to melt) of the Earth’s mantle. Melting occurs producing water-rich magmas that rise to the surface. Decompression of material as it rises from depth is an alternative mechanism that causes melting. This occurs along the ‘divergent’, ‘constructive’ plate margins, indicated by the mid-ocean ridges (e.g. Iceland), where the tectonic plates are created by upwelling of material. Decompression is also the melting mechanism at localized, intraplate ‘hot-spots’ (e.g. Hawaii), where it is hypothesized that material is rising in the form of a localized plume in response to convective processes deep in the mantle. Significant data also exist on the temporal distributions of volcanic eruptions on a global scale. Figure 1.3 shows the record of historicallyrecorded eruptions with magnitude greater than 4M for the last 2000 years. This is a stochastic dataset that suffers from the problem that recording is likely to be less reliable, especially for smaller magnitude eruptions, as one goes back in time. Coles and Sparks (this volume) apply extreme value methods to this dataset, that take into account historical bias in recording, to show how one might obtain quantitative estimates of the likelihood of future extreme volcanic events. Similar spatial and temporal data exist on more local scales. Our ability to produce reliable stochastic analyses of such data can be extremely important. For example, Connor et al. (2000) investigate the likelihood of a volcanic eruption at the location of a proposed high-level nuclear waste repository in the volcanically-active Yucca Mountain region of Nevada, USA. In this volume Connor et al., Jaquet, Varley et al. and Weller et al. present studies of stochastic forecasting for spatial and temporal eruptive behaviour. Their chapters, along with that of Coles and Sparks, demonstrate that probabilistic predictions of long-term activity depend heavily on our understanding of the spatial and temporal controls on volcanoes and volcanic eruption behaviour. Such models are greatly enhanced by strong 2 links between physical and statistical models. It is essential then to acquire some knowledge of the underlying physical processes operating in volcanic eruptions. 1.2 Explosive Versus Effusive Activity Figure 1.4 shows schematically the generic architecture of a terrestrial volcano and the range of surface flows that might occur as a consequence of an eruption. Magma rises from some source region, usually located at the base of the Earth’s crust or in the upper mantle, and ponds at depth within the volcanic edifice in a magma chamber (geologic studies of eroded volcanoes reveal chamber volumes 1 − 106 km3 with the top at a few km depth or more) which is connected to the vent by a conduit (usually of order 1 to 100 m in diameter). The magma consists of a pure liquid silicate melt containing dissolved volatiles (primarily H2 O vapour with lesser amounts of CO2 and S) and possibly crystals or fragments of older rock. Magma chamber dynamics is complex. Bubble and crystal nucleation and growth occur and convective flows, due to both thermal and compositional density gradients, are possible in magma chambers (e.g. Huppert and Woods 2002; Woods and Huppert 2003). To understand what gives rise to a volcanic eruption, we need to consider the process of bubble nucleation and growth within magmas. The amount of volatiles that can be dissolved in the magma is a function of pressure. The pressure at depth is generated by the weight of the overlying rock and an overpressure due to the input/output magma budget within the volcanic edifice. If the pressure is such that an appreciable volatile supersaturation is reached, then bubbles will nucleate above that depth (which could be either in the magma chamber or in the conduit). The chapter by Diez (this volume) explores simple models of magma ascent and flow in conduits and the impact of these processes on eruption columns in the atmosphere. Nucleation can occur prior to an eruption, such that a population of pre-existing bubbles exists at the onset of an eruption. Alternatively, nucleation can be initiated by an unloading event, for example a landslide at the surface, that leads to the propagation of a decompression wave down through the volcanic edifice. Once formed, the bubble nuclei grow by diffusion of more dissolved volatiles into the bubbles from the surrounding melt and by decompression as the bubbly magma expands up the conduit. It is this process of exsolution or ‘degassing’, possibly in addition to continued input of fresh magma from depth, that drives terrestrial eruptions by causing the growth of a gas phase internally within the magma at depth. The different eruption styles reflect the physico-chemical conditions and 3 processes experienced by the magma during its transit from the chamber to deposition at the surface. 1.2.1 Explosive Eruptions Explosive eruptions (Figures 1.4 (a) and 1.5) are often associated with magmas that have a high silica (SiO2 ) content, such as rhyolites. These magmas tend to have high water contents (typically 6% to 8% by weight) and so a high potential expansion of the magma on degassing. Rhyolites also tend to have high viscosities (typically 108 to 1010 Pas when fully degassed) and low gas diffusivities. Thus, the mobility of the gas phase is low. A sudden decompression event imposed on a rhyolite tends to result in widespread bubble nucleation and the magma column expands maximally as very little gas can escape from the highly viscous magma. The gas expansion in an explosive eruption is sufficiently violent that at some depth (the fragmentation surface) the magma is torn apart or fragmented into many small pieces called pyroclasts. Fragmentation marks the transition from a bubbly flow, where the continuous phase is a liquid, to a flow of vesicular (bubbly) pyroclasts carried upwards by a hot gas. The mixture is erupted into the atmosphere in the form of a momentum-controlled jet (exit velocities 100’s ms−1 ; total mass discharged 1011 − 1014 kg, equivalent to 0.1 − 150 km3 of dense rock, at discharge rates of 103 − 105 m3 s−1 ). What happens at the surface depends critically on the buoyancy of the gas-pyroclast mixture. If the bulk density of the mixture is less than atmospheric on eruption the material will rise as a convecting, buoyant plume called a Plinian eruption column (heights of 25 km or more) (Figure 1.5 (a)). The plume will continue to rise until it reaches its height of neutral buoyancy. At this height it spreads out laterally to form an umbrella cloud from which pyroclastic material rains down on the surrounding area or can be transported around the globe. On the other hand, if the bulk density of the gas-pyroclast mixture is greater than atmospheric pressure on eruption, the column will collapse (Figure 1.5 (b)) and flow down the side of the volcano as a pyroclastic flow (Figures 1.5 (c) and (d)). Pyroclastic flows are amongst the most devastating and lethal volcanic events for populations close to volcanoes, primarily because of their extreme mobility; they can reach speeds of 100’s kmh−1 and can travel as much as 100 km or more from the source vent. An example of the fearsome impact of these flows occurred on the Caribbean island of Martinique in 1902 when a pyroclastic flow was generated by an explosion of Mount Pelée volcano and descended on the city of St Pierre, killing some 29,000 people in a matter of about two minutes; the detonations leading to the flow were heard at 07:50 and the clock of the Military Hospital was found with its hands stopped 4 at 07:52. The cause of death as a result of these flows is primarily due to suffocation by the dust rather than the effects of temperature (of order 1000 ◦ C). There were only two survivors left in St Pierre. One of these was Augustus Ciparis, who was a prisoner in the town jail. His cell was below ground level and had a small grating above the door but no window, thereby providing effective protection from the dust of the flow. He was found four days after the flow had passed, still locked in his cell and badly burnt and shocked. In common with many of the dead at St Pierre, Ciparis’ clothes were unscathed; the temperature in the flow was intense but short-lived and the heat generated was enough to burn skin but not to ignite fabric. 1.2.2 Effusive Eruptions At the other end of the spectrum is effusive activity (Figures 1.4 (b) and (c) and 1.6). Effusive eruptions tend to occur in magmas with low silica (SiO2 ) content, such as basalts. Basalts have low water contents (typically just a few weight %), low viscosities (typically 102 to 103 Pas when fully degassed) and high diffusivities. In these magmas, fewer nuclei are formed and rapid diffusion into them leads to large bubbles that can rise up through the low-viscosity magma, often collecting more bubbles as they go (Figure 1.6 (c)). Thus, much of the gas escapes from the magma, thereby reducing the driving force for a major explosion. Typical activity can range from slow bubbling in vent-bound lava lakes through to intense lava ‘fire fountains’ (Figure 1.6 (a)) that can feed lava flows (Figure 1.6 (b)). The ease of gas segregation can allow metre-sized bubbles to be formed that ascend from depth as gas slugs and burst at the surface (Figure 1.6 (c)). Lava flows typically have discharge rates in the range 1 − 103 m3 s−1 and total volumes of 10−2 − 20 km3 . Lava dome eruptions (Figures 1.4 (c) and 1.6 (d)) are perhaps the least explosive form of volcanic activity. These eruptions are characterized by slow extrusion (1-100 m3 s−1 ) of high-viscosity lava such that a dome is formed over the vent (up to 1 km high and several kilometres across). The viscosity of the lava is high because the magmas are typically somewhere between basalts and rhyolites in composition, extensive degassing has taken place during ascent causing a major increase in the viscosity of the melt (Section 1.3.3), and extensive crystal growth caused by undercooling may also have occurred. The management of volcanic eruption scenarios is often complicated by the fact that volcanic activity can display dramatic changes in eruption style. For example, it is common for slow, steady lava-dome eruptions to be interrupted suddenly (and repeatedly) by violent explosions. In some cases, such as on Montserrat, a cyclic pattern emerges. This condition has 5 been studied by Sparks (1997) and Voight et al. (1999). Gas exsolution and crystallisation during the slow, extrusive phase increase magma viscosity. This produces a stiff plug of magma which inhibits flow of magma up the conduit. As a result, high pressures start to evolve, especially at the top of the conduit. Eventually, these pressures are sufficient to drive the plug from the conduit. Pressure is released in an explosive event. A fresh batch of magma ascends and a new cycle commences. Connor et al. (2003), developed a log-logistic probability model of repose intervals that attempted to account for these competing processes in the conduit. 1.3 Pre-Eruptive Physical Processes Processes that occur within the conduit during or immediately preceding an eruption are arguably of prime importance for a number of reasons: they are inextricably linked to eruption precursors that are (or could be) used for forecasting; they provide the initial conditions for the surface flows; and they control the sudden, dangerous switches in eruption styles. We will now consider some of these subsurface processes in a little more detail so that we can later consider how they relate to data collected during volcano monitoring. 1.3.1 Nucleation and Growth of Bubbles and Crystals Bubble nucleation and growth provide the primary driving force in most eruptions. But how do bubbles nucleate? Imagine a system with a liquid and gas in contact at a constant pressure. At ‘saturation’, gas and liquid are in equilibrium, the amount of gas dissolved in the liquid is stable. For bubbles to nucleate, the pressure must be dropped until a certain ‘supersaturation’ ∆P is reached. We distinguish between ‘homogeneous’ and ‘heterogeneous’ nucleation (see Figure 1.7) (Hurwitz and Navon, 1994). Homogeneous nucleation occurs in the absence of solid surfaces (Figure 1.7 (a)). The supersaturation required for homogeneous nucleation in a rhyolite (i.e. the high silica, highly explosive type of magma) is high (∆P ≥ 120 − 350 MPa). The supersaturation required for heterogeneous nucleation on crystals (or other solid surfaces) is generally much lower (∆P ≥ 5 − 25 MPa) (Figure 1.7(b)). Whether bubbles nucleate on crystals at all and what the supersaturation is depends on the wetting characteristics of the crystals. A melt that strongly adheres to, or ‘wets’, a crystal will not allow an intervening gas phase to displace it. The supersaturation is a function of the melt-vapour surface tension, the Helmholtz free energy for creating a curved surface, and the strength of wetting, which can be determined from the contact angle 6 θ at the intersection of the three phases (Figure 1.7 (c)). When θ = 0◦ , the melt completely coats the crystal, wetting is strong, nucleation is homogeneous and requires a high supersaturation, i.e. ∆P is a maximum. When θ = 180◦ , the gas completely coats the crystal, there is no wetting, nucleation is heterogeneous and occurs at low supersaturation, i.e. ∆P is a minimum. The composition of the magma and the crystals has a strong effect on the supersaturation needed to initiate nucleation. Mangan et al. (2004) have recently reported that in dacitic magmas (which have a silica content that is intermediate between rhyolite and basalt) homogeneous nucleation occurs at the low supersaturations more commonly associated with heterogeneous nucleation (∆P = 35 ± 5 MPa). As a result, nucleation occurs early in dacitic melts, there is more time for bubble coalescence and other processes related to gas loss, and the likelihood of effusive or only moderately explosive eruptions is increased. This explains why the most explosive eruptions are most commonly associated with rhyolitic magma; only this type of magma has the capacity to generate the necessary high supersaturations. Once bubbles have nucleated they grow by decompression, as the material expands up the conduit, and by diffusion of more volatiles from the melt. The gas concentration in the bubbles is lower than in the surrounding melt and so the volatiles in the melt surrounding the bubbles will diffuse towards the bubble. This process is described by the diffusion equation ∂C ∂t = D∇2 C, ∇2 = where ∂2 ∂2 ∂2 + + , ∂x2 ∂y 2 ∂z 2 (1.1) (1.2) C is the concentration, t is time, and D is the diffusivity. The diffusivity D is a measure of the rate at which the molecules of a particular volatile species move down a concentration gradient in a particular medium. Crystallisation of a silicate melt during magma intrusion or eruption may result from either a decrease in temperature or a change in the concentration of one of its components. Cooling is not the main driving force for crystallization of ascending water-saturated magmas, which may be nearly isothermal over most of the eruption sequence. Rather crystallization is more commonly induced by an increase in the liquidus (the temperature at which the solid and liquid are in equilibrium) as H2 O is lost from the melt by decompression. Figure 1.8 shows the schematic model of Métrich and Rutherford (1998) in which crystallization is driven by decompression and H2 O loss in a basaltic magma which contains 2.5 wt% H2 O at the outset. The model supposes equilibrium and a decompression rate slow enough to allow nucleation and crystal growth. 7 1.3.2 Open-System Degassing It was stated earlier that rhyolites typically contain high water contents (dissolved as water vapour), that the water is not very mobile, and that degassing is rapid during eruptions, i.e. the water cannot escape during an eruption. This leads to highly explosive activity. Nevertheless, rhyolitic volcanoes often erupt effusive lava flows and domes which are comparatively free of volatiles. One possible explanation of this paradox is that these ‘dry’ lavas derive from dry magma at depth. This is not compatible with information from Fe-Ti oxide geothermometry and observations of mineral assemblages. For example, it is often observed that effusively-erupted rhyolitic lavas contain just ∼ 0.1 wt% H2 O and that they also contain hornblende crystals. However, at least 3 wt% H2 O is necessary to stabilise the hornblende crystals at the temperatures indicated from Fe-Ti oxide geothermometry. The implication is that the magmas were much wetter at depth and that the water must have escaped prior to the eruption, i.e. that the system was ‘open’ in some way. The generally accepted explanation for this paradox is provided by the ‘permeable foam’ model of Eichelberger (1995) (see Figure 1.9). The timescale necessary for a bubble to diffuse across the conduit is too long to explain the observed gas loss. Eichelberger (1995) proposed that the pores must be connected and form a permeable network that enhances gas escape from the magma, both laterally and vertically. In this way, the diffusion length-scale is substantially shortened. Volatiles only have to diffuse to the nearest bubble that is connected to the permeable network, and can then flow out of the system. Thus, the style of volcanic eruptions, extremely explosive Plinian eruption columns or comparatively gentle dome-building eruptions, depends on both the initial physical properties of the magma and processes that alter these properties during magma ascent. 1.3.3 Magma Rheology The viscosity of magma is a key parameter in all considerations of flow behaviour below the fragmentation level. For a simple definition of viscosity consider Figure 1.10 (a). In this arrangement, fluid is contained in the gap between two flat plates with the two horizontal cartesian coordinates as shown and the third coordinate x3 perpendicular to the page. A stress (force per area of plate) is applied to the upper plate to make it move. The fluid in the gap responds by flowing. This stress force is described by the following equation ∂u1 τ12 = η , (1.3) ∂x2 8 where τ12 is the stress in the x1 direction perpendicular to the x2 direction. The flow velocity is in the x1 direction and changes in the x2 direction, so the partial derivative gives the spatial velocity gradient and describes the deformation rate or ‘strain rate’ of the fluid in the gap. η is the viscosity and is a measure of the internal resistance to flow. In the special case where η is independent of the strain rate, i.e. a graph of stress as a function of strain rate produces a straight line, the viscosity is called ‘Newtonian’. The viscosity of pure silicate melts (i.e. magmas that contain no bubbles or crystals) can be considered Newtonian for a wide range of flow conditions. However, the viscosity varies strongly as a function of temperature and dissolved volatile content. The effect of water content on the viscosity of the melt is of particular importance to conduit flow dynamics (see Figure 1.10 (b)). This is a consequence of the polymeric nature of silicate melts, which consist of long chains or rings of Si–O tetrahedra. As water is added to such a melt, the effect is to break-up these chains thereby lowering the viscosity. Conversely, during an eruption, water is lost from the melt to the bubbles, and the viscosity can rise by many orders of magnitude. Whilst the viscosity increase during an eruption is by far the dominant rheological effect, it is important not to neglect the effect of the multiple phases. During bubble formation, a melt undergoes a transition from a single phase to a two-phase bubbly flow (or possibly from a two-phase liquidcrystal mixture to a three-phase flow). This structural change also has a rheological effect, which has recently been investigated by Llewellin et al. (2002) and Rust and Manga (2002). Adding bubbles to a Newtonian liquid causes the mixture to become shear-thinning, i.e. the viscosity drops as the strain-rate increases. Also, the surface tension at the bubble walls provides a restoring force. This introduces a component of elasticity into unsteady flows, i.e. the bubbly flows are ‘viscoelastic’. By increasing the gas content of the flow the viscosity of the mixture tends to reduce, except for slow, steady flow (spherical bubbles with no elastic recovery). 1.3.4 Fragmentation and the Gas-Particle Flow Regime The final major change in flow regime that can occur within the conduit is that of fragmentation. Prior to fragmentation, the fluid consists of a continuous liquid phase (the silicate melt) that contains growing bubbles (and possibly crystals). If degassing is sufficiently rapid and widespread then at some height in the conduit the continuous liquid phase is disrupted. A suspension of hot, discrete, bubbly magma fragments, the ‘pyroclasts’, in a continuous gas phase is created. Fragmentation is the defining characteristic of explosive eruptions (see Figure 1.4 (a); i.e. the presence or absence of fragmentation is used to distinguish between explosive and effusive volcan9 ism. Figure 1.11 shows a flowchart that describes the processes leading up to fragmentation. A pressure decrease initiates degassing. Bubbles nucleate and start to grow. This causes two important changes. Firstly, water is lost from the melt into bubbles. This causes the viscosity of the melt surrounding the bubbles to increase dramatically (Figure 1.10 (b)). The viscosity increase is maximal just at the bubble walls as here the dehydration is the greatest. Secondly, the growth of the bubbles causes an overall density decrease. If the conduit is open and in the absence of significant gas loss from the magma, the mixture expands, causing a huge increase in the flow velocity (i.e. an acceleration) and hence strain-rate. The combined effect of the increased viscosity and strain rate is that the stress at the bubble walls becomes very great (remember that τ = η γ̇ Figure 1.10 (a)). Fragmentation occurs when the stress at the bubble walls exceeds a critical strength condition. Several mathematical fragmentation criteria have been proposed by Papale (1999), Zhang (1999), and Melnik (2000). Above the fragmentation level, the continuous phase is no longer magma but gas. The frictional and viscous forces are much reduced and the flow becomes turbulent. By the time the flow reaches the vent, it has typically accelerated sufficiently to reach the local speed of sound. It is a standard fluid dynamical result that flow in an enclosed channel, such as a conduit, cannot exceed the local speed of sound. Such flows are termed ‘choked’. Thus, whilst the flow pressure at the vent (usually well in excess of atmospheric pressure) is a consequence of the dynamics of the flow, the choked flow condition provides a strong constraint on the flow velocity. 1.4 Data From Volcano Monitoring The previous section discusses some primary physical processes that occur prior to or during an eruption. More in-depth information about conduit processes is needed in order to forecast the nature of volcanic eruptions. However, these below-ground processes are not accessible to direct observation. This section considers the range of observations of volcanic behaviour that are possible. A wealth of data are routinely collected by volcano observatories all over the world. These data are, in effect, proxies for sub-surface processes and provide a rich source of statistical data. Proper statistical treatment of these data will assist us in drawing accurate inferences about the processes occurring at depth that control eruptive style and duration. 10 1.4.1 Seismic Monitoring Seismological data are measurements of ground vibrations due to earthquakes and volcanic tremor. Seismic monitoring is one of the most common forms of ‘volcano-watching’. Seismometers, being fairly cheap and easy to install, have been used for more than a century to collect seismic data near volcanoes. This seismic monitoring has demonstrated conclusively that volcanic eruptions are most often preceded by some form of seismic unrest. Seismic activity appears to be one of the best methods currently available for forecast eruptions, at least on timescales of a few days to weeks. Numerous successful hazard assessments have relied primarily on seismic data; for example, many thousands of lives were saved in the 1990’s as a result of evacuations of Pinatubo and Mayon volcanoes following seismic precursors to the eruptions. The study of seismic traces and time-series is complex. The method relies on first monitoring the background seismicity. Observers look for changes in this background trace that might herald an eruption. Historically, the interpretation of seismic data has been more of an art than a science. The staff at a particular volcano observatory become familiar with ‘their’ volcano, possibly over very long acquaintance, and come to understand what type of outcome to expect from changes in seismic behaviour, without necessarily being able to link the observations to specific processes at depth. Neuberg, in his chapter, provides an examination of seismic monitoring and modelling using time-series data analysis techniques. Rigorous statistical methods of analysis are being increasingly used and examples are given in the chapters by Alasonati (discrete wavelet transformations and hidden Markov models), Hellweg and Seidl (contingency tables), Jaquet et al. (variogram analysis) and Palacios et al. (the GutenbergRichter law). Recently, there have been attempts to model the physical processes to reproduce the seismic signals and hence explain their origin. Models fall into two camps (Neuberg, this volume). Chouet (1985) proposed that the seismic wave field is produced by resonating cracks filled with bubbly magma. By contrast, Neuberg et al. (1998) postulate that the wave field results from excitation of a volcanic conduit filled with a gas-melt mixture with depth-dependent properties. These models are the first to consider the fundamental processes, and produce synthetic seismic signals that can be compared to natural signals. 1.4.2 Ground Deformation Prior to an eruption, there is an increase in pressure at depth. This causes the volcanic edifice to swell. Therefore, studies of ground deformation are one of the most useful indicators of existing volcanic activity and an im11 pending volcanic eruption. Ground deformation is measured by conventional surveying techniques, GPS (global positioning system), SAR (synthetic aperture radar) interferometry by satellite, or tiltmeters that can be installed either on the flanks of a volcano or in a borehole. These techniques aim to obtain repeated measurements of relative vertical height and/or horizontal distance. Changes in ground deformation often indicate an alteration in sub-surface activity. Interpreting the data and so making reliable forecasts is problematic. The paper by Mogi (1958) was the first attempt to infer sub-surface processes from surface deformation. He modelled the magma chamber as a sphere within an semi-infinite, elastic medium with a flat surface. He calculated the topographic change of the flat surface due to a pressure change in the sphere. Mogi successfully applied this theory to Sakurajima volcano, Japan, and Kilauea volcano, Hawaii. His theory has been extended to allow for more realistic topographies and magma chamber shapes. That these theories manage to predict general ground deformation as well as they do, given that volcanoes are not strictly elastic bodies, is perhaps surprising. Brittle fracture occurs at volcanoes often indicated by an abrupt occurrence of ground deformation at the onset of an eruption. This contrasts with ground deformation being modelled as a gradual, precursive, change. 1.4.3 Gas Monitoring Volatile degassing provides the primary driving force for explosive eruptions. The quantity of volatiles dissolved in magma prior to an eruption determines to a large part whether an eruption will be explosive or not. Since volatiles play such an important role, studies of gas emissions at volcanoes are used to provide insights into the sources of magmas, how magmas move up through the volcanic system, the impact of hydrothermal systems, and eruptive styles. Volcanic gases are released through ‘fumaroles’ and hot springs located on the flanks of volcanoes. The gases can be sampled directly by hand, by ground-based UV (ultraviolet) or IR (infrared) spectrometers, or more recently by space-based TOMS (total ozone mapping spectrometer) which measures sulphur dioxide erupted by volcanoes into the stratosphere. 1.4.4 Other Monitoring Data A host of other types of data are collected on active volcanoes. It is beyond the scope of this chapter to describe them all in detail. Commonly collected data include, gravity, magnetic, electrical and thermal measurements. Visual observations are made of plume heights, flow propagation 12 (e.g. runout, speed, direction) eruption style and duration. Observations of the volcanic rocks left behind by volcanism are also important in establishing physicochemical properties and so providing important constraints on modelling. 1.5 Modelling Approaches During the past few decades, an important focus of physical volcanological research has been the numerical modelling of flow processes. The goal of process modelling is to understand the fundamental controls that drive the dynamics of volcanic eruptions. Ultimately, these models may provide a forecasting tool. Models now exist for a wide range of different flow processes. A new generation of physically and mathematically sophisticated models often include 2 or (pseudo) 3 dimensions, independent sets of equations for different phases, unsteady flow conditions, and complex descriptions of key physical parameters (e.g. viscosity, diffusivity, equation of state, solubility laws etc.). These models capture the primary features of the dynamics of the sub-surface flows in the magma chamber and conduit and the surface flows in Plinian eruption columns and pyroclastic flows. Several examples of such modelling are presented in this volume. Sparks and Melnik present a model of magma flow up a volcanic conduit, whereas Neuberg discusses the wave field produced by excitation of a conduit filled with a melt-bubble mixture. These models produce synthetic time series that can be compared against natural time series. A key feature of synthetic time series models is that they provide insight into the statistical structure of observed volcano time series. Likewise, the potential exists to validate (or calibrate) these models by comparing (or matching) the statistics of the natural time series with those of the synthetic time series. The chapters by Alasonati et al., Jaquet et al., Nason, and Young review contrasting statistical approaches for analysing such time series data. The chapter by Bonadonna considers probabilistic forecasting of fallout from Plinian eruption columns. Her approach uses a process-based numerical model that allows for atmospheric diffusion, wind transport, and particle fall with stochastic sampling of inputs to produce a probability map of tephra accumulation. As the complexity of models increases, so do the computational difficulties associated with implementing them. The chapters by Connor and Connor, and Diez address some of these issues. Diez describes computational approaches for implementing a coupled conduit flow and eruption column model. Connor and Connor discuss data inversion using parallel computing techniques. Parallel computing is increasingly used in probabilistic modelling, such as that presented by Bonadonna, where the full 13 parameter-space of a complex process-oriented model has to be explored. The inversion technique presented by Connor and Connor optimizes a nonlinear model to estimate eruption parameters using geologic observations (in their case, measurements of a tephra deposit). So far this chapter has recognized volcanic hazards as physico-chemical processes having the potential to cause loss of life or damage to property and infrastructure. Hazard management and mitigation involves determining the risk associated with these hazards, i.e. the probability and extent to which a particular hazardous event will cause loss of life or damage to property or infrastructure. The distinction between hazard and risk is critical to any sensible decision- or policy-making process. Determining risk requires scientists to draw together many disparate strands of information. These include data from monitoring, calculated outputs from process-oriented models, and the formal incorporation of expert opinion. Outputs from such compilations include probabilistic event trees and hazard maps, both of which attempt to identify and evaluate risk and define regions for possible evacuation and exclusion. The chapters by Aspinall, Marzocchi et al., and Bonnadonna discuss approaches to this problem. 1.6 Concluding Remarks This chapter summarises our current understanding and ability to model the fundamental processes operating during volcanic eruptions. Predicting these processes will allow us to manage volcanic emergencies more effectively. The processes are inherently stochastic, so predictions need to be probabilistic. Years of direct observation and monitoring of individual volcanoes has produced a wealth of data which can be analysed statistically. These analyses will ultimately link monitoring observations with unobserved processes at depth, infer significance from change, and help volcanologists assign probabilities to expected outcomes. This volume provides a snapshot of statistical ideas and methods that are currently of interest within volcanology. The following chapters describe applications of statistical methods to specific volcanological problems, and cultivate didactic links between process-oriented volcanology and statistical modelling. Further Reading For general reviews of eruption processes and models see the volumes edited by Gilbert and Sparks (1998) and Freundt and Rosi (1998). Monitoring and hazard 14 mitigation are covered by the volumes edited by Scarpa and Tilling (1996) and McGuire, Kilburn and Murray (1995). An excellent general text covering many aspects of volcanology, including monitoring and hazard assessment but excluding statistics, is the Encyclopaedia of Volcanology edited by Sigurdsson et al. (2000). Acknowledgements This chapter has benefited from comments by my co-editors Stuart Coles and Chuck Connor. My attendance at the “Statistics in Volcanology” workshop was sponsored as part of the Environmental Mathematics and Statistics Programme funded jointly by NERC/EPSRC, UK. 15 Bibliography [1] Blower, J.D., Keating, J.P., Mader, H. M. & Phillips, J. C. 2002. The evolution of bubble size distributions in volcanic eruptions. Journal of Volcanology and Geothermal Research, 120, 1-23. [2] Chouet, B. 1985. Excitation of a buried magmatic pipe: A seismic source model for volcanic tremor. Journal of Geophysical Research 90(B2), 1881-1893. [3] Connor, C.B., Stamatakos, J.A., Ferrill, D.A., Hill, B.E., Ofoegbu, G.O., Conway, M.F., Sagar, B. & Trapp, J. 2000. Geologic factors controlling patterns of small-volume basaltic volcanism: Application to a volcanic hazards assessment at Yucca Mountain, Nevada. Journal of Geophysical Research 105(1), 417-432. [4] Connor, C.B., Sparks, R.S.J., Mason, R.M., Bonadonna, C. & Young, S.R. 2003. Exploring links between physical and probabilistic models of volcanic eruptions: The Soufrière Hills Volcano, Montserrat. Geophysical Research Letters, 30(13), 1701, doi:10.1029/2003GL017384. [5] Eichelberger, J.C. 1995. Silicic volcanism: Ascent of viscous magmas from crustal reservoirs. Annual Reviews of Earth and Planetary Science, 23, 41-63. [6] Gilbert, J.S. & Sparks, R.S.J. (eds) 1998. The physics of explosive volcanic eruptions. Geological Society, London, Special Publications, 145. [7] Freundt, A. & Rosi, M. (eds) 1998. From magma to tephra: Modelling physical processes of explosive volcanic eruptions. Developments in Volcanology, 4, Elsevier. [8] Huppert, H.E. & Woods, A.W. 2002. The role of volatiles in magma chamber dynamics. Nature, 420(6915), 493-495. 16 [9] Hurwitz, S. & Navon, O. 1994. Bubble nucleation in rhyolitic melts: Experiments at high pressure, temperature and water content. Earth and Planetary Science Letters, 122, 267-280. [10] Llewellin, E.W. , Mader, H.M. & Wilson, S.D.R. 2002. The rheology of a bubbly liquid. Proceedings of the Royal Society of London, Series A 458, 987-1016. [11] Mangan, M.T., Sisson, T.W. and Hankins, W.B. 2004. Decompression experiments identify kinetic controls on explosive silicic eruptions. Geophysical Research Letters, 31, L08605, doi:190.1029/2004GL019509. [12] McGuire, W.J., Kilburn, C.R.J. & Murray, J. (eds) 1995. Monitoring active volcanoes. UCL Press, London. [13] Melnik, O. 2000. Dynamics of two-phase conduit flow of high-viscosity gas-saturated magma: Large variations of sustained explosive eruption intensity. Bulletin of Volcanology, 62(3), 153-170. [14] Métrich, N. & Rutherford, M.J. 1998. Low pressure crystallization paths of H2 O-saturated basaltic-hawaiitic melts from Mt Etna: Implications for open-system degassing of basaltic volcanoes. Geochimia Cosmochima Acta, 62, 1195-1205. [15] Mogi, K. 1958. Relations between the eruptions of various volcanoes and the deformation of the ground surfaces between them. Bulletin of the Earthquake Research Institute (University of Tokyo), 36, 99-134. [16] Murase, T. 1962. Viscosity and related properties of volcanic rocks at 800 ◦ C to 1400 ◦ C. Hokkaido University Faculty of Science Journal Series, 7(1), 487-584. [17] Neuberg, J., Baptie, B., Luckett, R. & Stewart, R. 1998. Results from the broad-band seismic network on Montserrat. Geophysical Research Letters, 25(19), 3661-3664. [18] Papale, P. 1999. Strain-induced magma fragmentation in explosive eruptions. Nature, 397(6718), 425-428. [19] Rust, A.C. & Manga, M. 2002. Effects of bubble deformation on the viscosity of dilute suspensions. Journal of Non-Newtonian Rheology, 104, 53-63. [20] Scarpa, R. and& Tilling, R.I. (eds) 1996. Monitoring and mitigation of volcanic hazards. Springer, Berlin/Heidelberg. 17 [21] Sigurdsson, H., Houghton, B., NcNutt, S.R., Rymer, H. and Stix, J. (eds) 2000. Encyclopaedia of Volcanoes. Academic Press. [22] Simkin, T. & Siebert, L. 1994. Volcanoes of the World, 2nd Edition. Geoscience Press, Tucson. [23] Sparks, R.S.J. 1997. Causes and consequences of pressurisation in lava dome eruptions. Earth and Planetary Science Letters, 150(3-4), 177189. [24] Tilling, R.I. 1989. Volcanic hazards and their mitigation - Progress and problems. Reviews of Geophysics, 27(2), 237-269. [25] Voight, B. et al. 1999. Magma flow instability and cyclic activity at Soufrière Hills Volcano, Montserrat, British West Indies. Science, 283(5405), 1138-1142. [26] Woods, A.W. & Huppert, H.E. 2003. On magma chamber evolution during slow effusive eruption. Journal of Geophysical Research, 108(B8), 2403, doi:10.1029/2002JB002019. [27] Zimanowski, B. 1998. Phreatomagmatic explosions. In: Freundt, A. & Rosi, M. (eds) From magma to tephra: Modelling physical processes of explosive volcanic eruptions. Elsevier, Developments in Volcanology, 4, chapter 2, 225-53. [28] Zhang, Y. X. 1999. A criterion for the fragmentation of bubbly magma based on brittle failure theory. Nature, 402(6762), 648-650. 18 60˚ “Ring of Fire” 30˚ 0˚ -30˚ -60˚ 270˚ 315˚ 0˚ 45˚ 90˚ 135˚ 180˚ 225˚ 270˚ Figure 1.1: The spatial distribution of volcanoes (dots) relative to tectonic plates (dotted-lines). Locations of volcanoes active in the Holocene (last 10,000 years). (Ex: Simkin and Seibert, 1994) Oceanic-continental convergence Figure 1.2: Melt generation in subduction zones, e.g. the ‘Ring of Fire’ shown in Fig. 1.1. The lithosphere and the asthenosphere are layers of the upper mantle. Courtesy: USGS. 19 7.0 7.5 6.5 6.0 5.5 Magnitude 5.0 4.5 4.0 0 500 1000 1500 2000 Year Figure 1.3: Historical record of eruptions with magnitude greater than 4M. The magnitude of an eruption is given by M = logm − 7 where m is the mass released in kg. (Ex: Coles and Sparks, this volume) umbrella cloud fire fountain Plinian eruption column lava dome pyroclastic flow fragmentation bubbly flow (crystal growth?) increasing pressure bubbly flow (crystal growth?) nucleation magma chamber increasing pressure nucleation magma chamber (a) magma input from depth lava flow lava lake (b) magma input from depth bubbly flow (crystal growth?) increasing pressure nucleation magma chamber (c) magma input from depth Figure 1.4: Sketches of the basic physical process for (a) explosive eruptions, (b) effusive eruptions, and (c) lava dome eruptions. 20 (a) (b) (c) (d) Figure 1.5: Explosive eruptions. (a) Plinian eruption column. Lascar volcano, Chile, 1993. (b) Collapsing eruption column, Soufrière volcano, Montserrat, 20th October 1997. Courtesy: E. Calder. (c) and (d) Pyroclastic flow towards Shimbara city, Unzen volcano, Japan, 24th June 1993. Note size of houses for scale. Courtesy: Volcanological Society of Japan. 21 (a) (b) (c) (d) Figure 1.6: Effusive eruptions. (a) Fire fountain on Pu’u ’O’o, Kilauea, 2nd June 1986. Note helicopter in white circle for scale. Courtesy: C. Heliker, USGS. (b) Lava fountain feeding lava flows, Fernandina, Galápagos, 1978. Courtesy: Tom Simkin, Smithsonian Institution (photo by Marc Orbach). (c) Large bubble ∼ 5 m across on lava pond, Mauna Ulu, 1969. Courtesy: J. B. Judd, USGS. (d) Lava dome, Mt St Helens, USA, 13th September 1984. Courtesy: L. Topinka, USGS. bubble q crystal melt (a) (c) (b) Figure 1.7: Nucleation processes. (a) Homogeneous bubble nucleation in the absence of solid surfaces. (b) Heterogeneous bubble nucleation on crystals. (c) Heterogeneous nucleation dominates for contact angles θ > 68◦ 22 Summit craters 0 Melt composition wt% Fraction (wt%) of solid .8 .6 .4 .2 0 ol. + cpx. + plag. + mgt. 0 0 1 2 3 4 200 -740 m ol. + cpx. 400 600 -2200 m ol. 800 Basaltic magma with 2.5 wt% H²0 1000 K²0 H²0 PH²0 Figure 1.8: Crystallisation processes. In the sketch on the left, a basaltic magma containing 2.5 wt% H2 O at 2200 m depth rises up an inclined conduit to erupt on the flank of a volcano. The graphs on the right show the increase in the crystal content and magma composition as the material rises up the conduit. As the magma rises, water is steadily lost due to decompression. The liquidus increases and different crystals are nucleated at different depths. The solid fraction of crystals increases and the chemical composition of the remaining melt (see K2 O curve) is changed. Specific crystals identified are: ol = olivine, cpx = clinopyroxene, plag = plagioclase, mgt = magnetite. (Ex: Métrich and Rutherford, 1998) 23 (a) (b) Figure 1.9: The ‘permeable foam’ model. (a) Volatiles escape by diffusing towards bubbles that are connected to a permeable network. (b) Flow through the network can be lateral (volatiles escape via cracks and fissures in the conduit walls), or vertical, (volatiles escape via the vent). 24 stress=force/area = τ12 [Pa] u1+du1 dx2 u1 velocity gradient du1/dx2 [s-1] = strainrate [s-1] x2 x1 (a) 1011 Viscosity [Pa s] 109 107 105 103 800°C 1000°C 1200°C 1400°C 101 2 4 6 8 10 12 H2O content [wt%] (b) Figure 1.10: The viscosity of magmas. (a) Flow of a viscous fluid between a fixed (bottom) and a moving (top) plate. The strain-rate is identical to the spatial velocity gradient. (b) The Newtonian viscosity of rhyolitic magmas as a function of temperature and gas content. Note the logarithmic y-axis. (Ex: Murase, 1962) 25 density decrease velocity increase strainrate increase fragmentation pressure decrease gas volume fraction increase & dissolved water decrease viscosity increase Figure 1.11: Flowchart of processes leading to fragmentation. Courtesy: P. Papale. 26