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L. Sandri1, W. Marzocchi1, C.G. Newhall2, and P. Gasparini3 1Osservatorio Vesuviano, INGV Bologna, Italy; 2USGS, University of Washington, U.S.A. 3Dip. Fisica, Universita' di Napoli Federico II, Italy

Mt. Vesuvius is located in a densely urbanized area with more than 700,000 people living on its flanks. The enormous social and economic impact of a reactivation of this volcano compels a definition of the levels of alert as accurate as possible. The main goal of this work is to provide a quantitative tool to define the level of alert at Mt. Vesuvius. In our opinion, the definition of the system of alert, in an area as the Mount Vesuvius surroundings, must be considered from an “engineering” point of view. With this term we mean that the assessment of the risk deriving from a reactivation of the volcano cannot be rigorously and satisfactorily solved from a pure scientific point of view, because we have insufficient data to do it, and because our knowledge of the eruptive process is too rudimentary for unequivocal, useful, and testable considerations. However, the devastating potential of Mount Vesuvius explosive activity forces the scientific community to provide a tool to define quantitatively the system of alert for Mount Vesuvius, and to do it as correctly and formally as possible. Such a tool should allow to estimate the most likely scenario for the evolution of a volcanic unrest, and it could help in the decision making during a crisis. Furthermore, it should estimate the probability of any possible event, and consider all the available information. A viable strategy to provide such a tool is statistics, and, in particular, Bayesian statistics. Here, we furnish a technique based on a Bayesian probabilistic event tree to estimate the probabilities associated to each possible eruptive scenario following an unrest. The outcome of the calculation consists of a probability distribution for the non-eruptive event, and for eruptions with different VEI. The visualization of such distributions gives an immediate view of what is the most likely eruptive scenario. The present work might provide a tool to significantly improve the system of alert estimated in the past for Mount Vesuvius.

The available information
The available data used for this study can be divided in: 1. A priori and theoretical beliefs on the eruptive process. 2. Past (historical) data. This type of data can be further divided into two categories: •past data from Mount Vesuvius; •data coming from other volcanoes that are believed to behave similarly to Mount Vesuvius. 3. ”Fresh” measurements taken at Mount Vesuvius regularly. Based on the experience gained from other explosive volcanoes and on the availability of quantities monitored at Mount Vesuvius, we make use of three different measurements: •seismic (such as the daily number of events, their magnitude distribution, their average frequency content and their focal depth); •deformation; •SO2 emission.

The event tree
A possible way to provide probabilities for a set of possible events is to build up a so-called probabilistic event tree, which is a useful frameworks for discussing, from a probabilistic point of view, all the relevant

possible outcomes of a volcanic crisis (Newhall & Hoblitt, 2002). An event tree is a tree-like representation of events in which branches are logical steps from a general prior event to final outcomes, passing through increasingly specific subsequent events (intermediate outcomes). In this way, event trees show the most relevant possible outcomes of volcanic unrests at a progressively higher degree of detail. Event trees have positively been applied to some volcanic crisis in different volcanic areas, such as Mount St Helens, Mount Pinatubo, Soufriere Hills (Monserrat), Popocatepetl, Guagua Pichincha and Tungurahua. In each node of the event tree, an estimate of the probability for the event at that node must be provided. In this way, the probabilities along any path in the tree will allow calculating the probability of the terminal event.

Figure 1. Sketch of the event tree for Mount Vesuvius. The nine steps of estimation progress from general to more specific events, as described in the text. Note that any branch that terminates with Clone is identical to the subsequent central branch. For example, in the column “Sectors”, the clones for branches relative to sectors 1, 2, 3, 4, 6, 7 and 8 are identical to the branch relative to sector 5. The branches terminating with Stop are not discussed here because they cannot develop into very dangerous subsequent events.

It is important to stress that, by means of the event tree structure, we can take into account all the possible events, and provide a probability estimate for their occurrence. Different volcanoes have different behaviours and give rise to different types of event. Event trees allow considering this aspect, and the branching of an event tree depends on the behaviour of the volcano itself, both in the structure and in the probability value at each node. In other words, the structure of the event tree is strictly linked to the eruptive activity of the volcano, to the geographic location of the volcanic system and to the degree of urbanization. The sketch of a suitable event tree for Mount Vesuvius is shown in figure 1, taken and adapted by Newhall and Hoblitt (2002), and it is the one we will refer to. In figure 1, the different nodes (corresponding to different columns) have the following meanings: •Node 1: This node evaluates the probability distribution that the volcano will become restless;

•Node 2: This node evaluates the probability distribution that the unrest is due to magma, provided that there is an unrest; •Node 3: This node evaluates the probability distribution that the magma will reach the surface (i.e., it will erupt), provided that the unrest has a magmatic origin; •Node 4: This node evaluates the probability distribution that the eruption will be of a certain magnitude, provided that there will be an eruption; •Node 5: This node evaluates the probability distribution that a certain phenomenon (lava flow, tephra fall, pyroclastic flow, lahar) will occur, provided that the magnitude of the eruption will be of a certain value; •Node 6: This node evaluates the probability distribution that the phenomenon will move into one of 8 radial sectors of 45 degrees each, provided that the phenomenon will occur; •Node 7: This node evaluates the probability distribution that the phenomenon will reach a certain distance form the vent, provided that the phenomenon will occur and move into a certain radial sector; •Node 8: This node evaluates the probability distribution that an individual or a building will be present at the specified sector and distance from the vent, provided that the phenomenon will arrive there; this is called exposure. •Node 9: This node evaluates the probability distribution that an individual will be killed or a building will be destroyed at the specified sector and distance from the vent, provided that the phenomenon will arrive there and that people and/or building are present; this is called vulnerability. In this presentation, we focus our attention on the first 4 nodes of the tree.

The probability at each node
The estimation of the probability at each node is, together with the set up of a suitable tree structure for the volcano studied, the key point in order to build up a correct event tree for the volcano. In this study, we decide to estimate a distribution at each node, rather than a probability value. The estimation of a probability distribution allows us taking into account the uncertainties, i.e., errors and variability due to incompleteness of the data, lack of theoretical knowledge, inevitable measurement errors, and so on. In this perspective, the probability for each event θj at a specific node k is a random variable characterised by a distribution pk(θj). The choice of a single probability value, instead of a probability distribution, would be equivalent to assume a Dirac δ distribution (see figure 2) for the probability at each node, i.e., a perfect knowledge of the probability of such an event. On the opposite, a complete ignorance on the eruptive process and its possible precursory activity, or a complete lack of any kind of data, would not allow selecting any “preferred” probability value for the nodes, therefore it can be described by a uniform probability distribution on the interval [0,1]. For all the other intermediate cases, we can use a probability distribution that lies in between these two extreme cases (Dirac and uniform distributions). Such a distribution has to be unimodal and characterized by a variance depending on the degree of knowledge of each node. Moreover, it has to be defined on the domain [0,1], because the random variable is a probability. A suitable distribution, matching these requirements, is the beta distribution (an example is shown in the figure 2, see also Gelman and others, 1995, p. 476). Furthermore, a beta distribution with appropriate parameters is equivalent to the uniform distribution, so we can use it even in those cases for which the a priori information and past data are insufficient to identify a preferred value for the distribution peak. Notably, when the available information allows reducing the variance to zero, the beta distribution tends to a Dirac distribution. The probability density function of the beta distribution is set up, at each node of the event tree, by taking into account our theoretical beliefs, past data from Mount Vesuvius and/or other similar volcanoes, and measurements coming from monitoring of the volcano (as requested in the introductory section). The way of merging these information is different from each node, and it is not discussed here for reasons of space. However, we stress that, at each node, a volcanological model and statistical considerations are

necessary to combine these information. It is worth to point out that the choice of a beta distribution has a certain degree of subjectivity. However, any possible bias introduced by this subjective choice is certainly less than assuming an exact value for the probability, i.e., a Dirac δ distribution. Actually, the latter is a much more subjective choice.

Figure 2. This figure shows three different examples of possible probability distributions, all defined on the domain [0,1]. The solid line represents a Dirac δ distribution, with equation p(θ)=δ(θ-0.33). The dotted line represents a uniform distribution, with equation p(θ)=1. The dashed line represents a beta distribution, in this specific case with parameters α=2 and β=4, corresponding to a mean of 1/3 and variance equal to 2/63. The parameters of the distributions shown here have been chosen randomly, with no reference to the event tree, just to show the different shapes of these three distributions.

Calculating the probability of different events
Since each node of the tree is conditional to the previous and more general node, the natural way of dealing statistically with the estimation of probability distributions is to make use of Bayes theorem. According to this theorem, the probability distribution p(n|n-1), that is the probability distribution at node n given the event at the previous node n-1, is computed as:

p(n | n − 1 ) =

p(n − 1 | n)p(n) p(n − 1 )

In this case, p(n-1|n) is always a Dirac distribution centred around 1, since it is virtually impossible that the event (n-1) does not occur if the subsequent event n is occurred. For this reason, the Bayes theorem can be written as

p(n) = p( 1 )p( 2 | 1 )... p(n | n − 1 )
As an example, we can compute the probability to have a VEI<4 eruption as

pVEI < 4 = pVEI =0 + pVEI =1, 2 + pVEI =3 = p (1) p (2 | 1) p (3 | 2) p (4VEI =0 | 3) + p (1) p (2 | 1) p (3 | 2) p (4VEI =1, 2 | 3) + p (1) p (2 | 1) p (3 | 2) p (4VEI =3 | 3)
In this case, the events of node 4 (eruption of VEI=0, eruption of VEI=1,2, eruption of VEI=3) are mutually exclusive.

Discussion and Conclusions
In this study we have delineated the relevant information about the methodology adopted to provide a quantitative tool to define the system of alert at Mount Vesuvius. In particular, we have reported the structure of the event tree (reported in figure 1), and the general strategy to estimate the probability at each node of the tree, with a specific attention to the first 4 nodes. Major advantages of the probabilistic event tree approach are the possibilities to take into account all the available data and information, and to give probabilities for any possible event. Furthermore, the event scheme highlights what we know and what we do not know about pre-eruptive processes, indicating future possible works to improve the scheme. Finally, the probabilities provided at each node of the tree are in the form of a probability distribution, rather than a probability value, properly taking into account the uncertainties. When the probability distribution will be defined for each node of the tree quantitatively, this procedure will allow estimating the probability associated to any kind of event, such as the probability that human beings living in a particular area of the volcano will be reached and killed by pyroclastic flows, lahars, lava, and so on; or to estimate the probability that a building, hospital, airport, and so on, will be seriously damaged by the volcanic products. Furthermore, these probabilities will be given with an appropriate degree of uncertainty.

Gelman A., Carlin J.B., Stern H.S., and RubinD.B., 1995, Bayesian Data Analysis: Chapman and Hall/CRC, Boca Raton, London, New York, Washington DC, pp.526. Newhall, C. G., and Hoblitt, R. P., 2002, Constructing event trees for volcaninc crises: Bulletin of Volcanology, v. 64, p. 3-20.