BET_VH: a probabilistic tool for longterm volcanic hazard assessment
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Abstract
In this paper, we illustrate a Bayesian Event Tree to estimate Volcanic Hazard (BET_VH). The procedure enables us to calculate the probability of any kind of longterm hazardous event for which we are interested, accounting for the intrinsic stochastic nature of volcanic eruptions and our limited knowledge regarding related processes. For the input, the code incorporates results from numerical models simulating the impact of hazardous volcanic phenomena on an area and data from the eruptive history. For the output, the code provides a wide and exhaustive set of spatiotemporal probabilities of different events; these probabilities are estimated by means of a Bayesian approach that allows all uncertainties to be properly accounted for. The code is able to deal with many eruptive settings simultaneously, weighting each with its own probability of occurrence. In a companion paper, we give a detailed example of application of this tool to the Campi Flegrei caldera, in order to estimate the hazard from tephra fall.
Keywords
Volcanic hazard Probabilistic assessment Bayesian event treeIntroduction
Volcanic hazard studies have a prominent impact on society and volcanology itself, being an area where the “rubber hits the road”, that is, where science is applied to an important societal problem. Despite its importance, volcanic hazard assessment is still commonly presented in many different ways, ranging from maps of past deposits of the volcano to more quantitative probabilistic assessment (e.g., Scandone et al. 1993; Newhall and Hoblitt 2002; Marzocchi et al. 2004; Martin et al. 2004; Neri et al. 2008; Marti et al. 2008). The latter, being quantitative, has remarkable advantages: (1) it allows comparisons among different volcanoes and with other natural and nonnatural hazards, (2) its reliability can be tested through statistical procedures, and (3) it provides a basic component for rationale decision making (e.g., Marzocchi and Woo 2007, 2009; Woo 2008).
In order to unambiguously distinguish the quantitative approach from others which are more qualitative, it has been suggested to call it probabilistic volcanic hazard assessment (PVHA; see Marzocchi et al. 2007). The term “probabilistic” means that the extreme complexity, nonlinearities, limited knowledge, and the large number of degrees of freedom of a volcanic system result in difficult, if not impossible, deterministic prediction of the evolution of volcanic processes (see, e.g., Marzocchi 1996; Sparks 2003). In other words, volcanic systems are stochastic, and hazardous volcanic phenomena involve so many uncertainties that a probabilistic approach is needed.
That being said, we note that full PVHA is still quite rare (Magill et al. 2006; Ho et al. 2006; Neri et al. 2008; Marti et al. 2008 are among the few exceptions). Generally, hazard assessment represents, at best, a conditional probability of one specific hazard conditioned to the occurrence of one specific event (for instance, the most likely event; Cioni et al. 2003; Macedonio et al. 2008), or it is focused on one specific aspect of volcanic hazard, such as vent opening (Martin et al. 2004; Jaquet et al. 2008; Selva et al. 2010). In other cases, as mentioned above, hazard assessment merely consists of maps of volcanic deposits of past events. By contrasts, a full PVHA requires the assessment of the impact of hazardous phenomena associated with every possible “eruptive setting” (ES) and eventually the merging of all ESs, each of them weighted with its own probability of occurrence. Hereafter, with the terms “eruptive setting”, we mean the occurrence of an eruption of a specific size or type from a specific vent.
Another basic feature of PVHA is that it needs to account for all relevant sources of uncertainty. Indeed, the great importance of PVHA is due to its practical implications for society. In this perspective, it is fundamental that PVHA is “accurate” (i.e., without significant biases) because a biased estimation would be useless in practice. On the other hand, PVHA may have a low “precision” (i.e., a large uncertainty) that would reflect limited knowledge of some physical processes involved, from the preparation of an impending eruption to the derived impact on the surrounding area.
In the following sections, we describe the structure of the event tree for BET_VH, and the basic rules to estimate the probability distributions at each node. Finally, we show how a full PVHA is achieved, combining the probabilities of each node. A tutorial application of these concepts is outlined in the companion paper (Selva et al. 2010), where a PVHA for ashfall at Campi, Flegrei, Italy, is reported.
The Bayesian Event Tree scheme for longterm volcanic hazard

Node 1–2–3: there is an eruption, or not, in the time interval (t _{0},t _{0} + τ], where t _{0} is the present time and τ is the forecasting time window. This node condenses the probabilities of nodes 1 (unrest), 2 (presence of magma given an unrest), and 3 (eruption given a magmatic unrest) in BET_EF by Marzocchi et al. (2008) as regards the nonmonitoring part.

Node 4: the eruptive vent will open in a specific location, provided there is an eruption.

Node 5: the eruption will be of a certain size or type, provided that there is an eruption in a given location.

Node 6: a particular hazardous phenomenon will be generated, or not, given that an eruption of a specific size or type occurs. Several hazardous phenomena can be uploaded in BET_VH at a time; volcanic hazard due to different phenomena can be juxtaposed and compared.

Node 7: a selected area around the volcano will be reached, provided the occurrence of a specific hazardous phenomenon produced by an eruption of given size or type and location.

Node 8: a specific intensity threshold will be overcome, or not, provided that the area selected at node 7 is reached by a specific hazardous phenomenon generated by an eruption with a given size or type and location.
Note that since the definition of event tree is mainly driven by its practical utility, the branches at each node point to the whole set of different possible events, regardless of their probabilistic features. In other words, the events at each node need not be mutually exclusive (see upper part of Fig. 1). This makes the combination of nodes a little bit more complicated, but it keeps a more logical and comprehensible structure (see Marzocchi et al. 2004, 2006, 2008 for more details).
In the following section, we describe how the probability distribution at each node is assigned. We will try to make use of the same symbols and terminology already published in Marzocchi et al. (2008) and in the Electronic supplementary material of that paper.
Estimating the probability distribution at each node
BET_VH focuses on longterm PVHA only. Because of this choice, the information to be used is related to geological and/or physical models and past data. We do not account for monitoring measures. The latter, in fact, are related to shortterm variations in the state of the volcano. This is one of the basic differences with BET_EF by Marzocchi et al. (2008), and it implies a simpler formalism for the probability computation at the nodes of the event tree in common with BET_EF. On the other hand, BET_VH has a more complex structure than BET_EF because it accounts also for the impact of hazardous phenomena on the territory; this leads to a more complex dependence on the selected path, i.e., on the events selected at the previous nodes. In order to keep the notation as simple as possible, hereinafter, we set specific indexes for specific selected outcomes at the different nodes: i indicates the vent location, j the eruption size or type, p the hazardous phenomenon, k the area, and s the threshold related to the phenomenon p. Despite the description of the following nodes should be selfconsistent, we recommend to refer to Marzocchi et al. (2008) for more detailed description of the statistical distributions used (Beta and Dirichlet distributions).
Node 1–2–3
Node 4
At this node, BET_VH estimates the spatial probability of vent opening, given that an eruption occurs. The basic assumption here is that only one vent at a time will erupt. Because of this, this node has I _{4} possible and mutually exclusive outcomes, corresponding to the number of possible vent locations defined by the user.
Node 5
Here, we examine the probabilities related to the size or type of the eruption. The magnitude can be represented either by the type of the eruption (explosive, effusive, phreatomagmatic, and so on), or by the size (e.g., VEI), or by groups of types or sizes that for the user’s purposes are considered homogeneous (e.g., VEI ≥4). In the following, we use the term size class, meaning anyone of these parametrizations. Thus, the number of possible outcomes at this node is J _{5}, corresponding to the number of possible size classes defined.
At this node, we estimate the probability of a specific size class of the eruption, given its occurrence and given that the vent opens in a specific location. Here, we propose a substantial improvement over Marzocchi et al. (2008) because in BET_VH (and in the version 2.0 of BET_EF that can be downloaded from the website: http://www.bo.ingv.it/bet), we allow the probability of each size class to vary as a function of the vent location. In this way, it is possible to account for potential differences among vent locations. For example, for a certain volcano, we might want to assign a higher probability of a hydromagmatic type of eruption if the vent is located under shallow water than on land. Similarly, with this improvement, we can take into account flank instability in case the vent opens in a radial sector of a central, steep flank volcano rather than in a central crater. Note that J _{5} does not depend on vent location since only the probabilities of the specific size classes do.
Node 6
Nodes 7 and 8
These two nodes have a similar structure. First, the surrounding of the volcano is divided into a number (K _{7}) of areas (not necessarily equal and equally spaced). For each area k, at node 7, we have two possible outcomes: area k is reached or area k is not reached by the hazardous phenomenon p generated by an eruption of size class j and location i. At node 8, the two outcomes are: the selected threshold s is overcome or the selected threshold s is not overcome, considering an area k reached by the hazardous phenomenon p, generated from an eruption of size class j and location i. Both probabilities are assumed to be homogeneous all over the area k.
Estimating PVHA
In the previous chapters, we have described the general features of the probability distributions at each node. Their combination allows a full and complete PVHA to be determined. In order to accomplish that, we have to still explore in detail three issues: (1) how model output can be used to set prior distributions for the nodes described before, (2) how to combine the conditional probability at each node to get absolute probabilities, and (3) how to account for different ESs. The following three subsections are devoted to describe these three issues. The last subsection reports the kind of outputs provided by the code.
Numerical models to define prior distribution
As we have seen so far, the setup of prior probability distributions is mainly based on models, through the best guess Θ_{*} and the number of equivalent data Λ_{*} that represents a sort of confidence on our best guess (see Marzocchi et al. 2008). For the prior distribution of nodes 1 to 5, we refer to the estimation of the nonmonitoring part in Marzocchi et al. (2008). For nodes 6, 7, and 8, we can use results from numerical models that are available for most of the hazardous phenomena related to volcanic eruptions (e.g., Favalli et al. 2005 for lava flows; Neri et al. 2007 for pyroclastic density currents; Pfeiffer et al. 2005 and Costa et al. 2006 for ash dispersion). For a more detailed discussion about this point, see (Selva et al. 2010). Here, we highlight the basic philosophy behind models’ usage in PVHA.
 1.
Intrinsic stochasticity of the process (the socalled aleatory uncertainty)
 2.
Epistemic uncertainty in the model parameters and in the boundary conditions at the time and during the eruption (e.g., for an ash fall model, the uncertainties in wind conditions)
 3.
Epistemic uncertainty in the (volcanological) input parameters (e.g., for an ash fall model, the uncertainties in the relevant eruption parameters given a specific eruption size class)
 4.
Any model is always a simplification of the reality, leading to unavoidable uncertainties into the forecasting.
If different models can be used, BET_VH may use the output of each one of them recursively. In the companion paper (Selva et al. 2010), we propose a general scheme to introduce models’ results from a large number of runs and from more than one model, presenting also a practical application for tephra fall hazard estimation around the Campi Flegrei caldera.
Combination of the probabilities to obtain PVHA
The probability distributions at each node are conditional on the selection of a welldefined path (see Section “The Bayesian Event Tree scheme for longterm volcanic hazard”). In general, PVHA, as well as the evaluation of probability of each event for which we may be interested, requires their combination. With BET_VH, it is possible to compute the probability associated to a single ES (an eruption of size class and location specified) or to a combination of possible ESs. In the latter case, at nodes 4 and 5, we can select more than one branch at the same time (i.e., a set \(\mathcal{J}\) of J possible eruption size classes and/or a set \(\mathcal{I}\) containing I possible vent locations; see also Fig. 1b for a snapshot of the main window of BET_VH code, where it is possible to see that more than one location and/or size class can be selected). This feature of the code is quite remarkable because these new combined probabilities are usually very important for practical purposes and for a full PVHA (Selva et al. 2010). In the following, we will carefully describe how to obtain meaningful probabilities for ES combinations.
Absolute probability
Conditional probability
Beyond the absolute probabilities, in many practical applications, some conditional probabilities are particularly important and useful, like the conditional probabilities that can be obtained by the combination of different ESs (see Selva et al. 2010). To this purpose, the code BET_VH gives the possibility to average, with proper weights, the conditional probabilities for different size classes, and/or locations.
 At node 6, the conditional probability [φ _{ b }] of a specific phenomenon p, given the occurrence of an eruption within the selected combination of ESs is$$ \label{eq:multscenarios_cond6} [\phi_b] =\frac {\sum_{i \in \mathcal{I}} \left( \left[\theta_{4}^{(i)}\right] \sum_{j \in \mathcal{J}} \left[\theta_{5,i}^{(j)}\right] \left[\theta_{6,j}^{(p)}\right] \right)} {\phi_{\rm ES}} $$(19)
 At node 7, the conditional probability [φ _{ c }] of a specific area k being reached by the phenomenon p, given the occurrence of an eruption within the selected combination of ESs is$$ \label{eq:multscenarios_cond7} [\phi_c]=\frac {\sum_{i \in \mathcal{I}} \left( \left[\theta_{4}^{(i)}\right] \sum_{j \in \mathcal{J}} \left[\theta_{5,i}^{(j)}\right] \left[\theta_{6,j}^{(p)}\right] \left[\theta_{7,i,j,p}^{(k)}\right] \right)} {\phi_{\rm ES}} $$(20)
 At node 8, the conditional probability [φ _{ d }] of overcoming the selected threshold s in a specific area k reached by the phenomenon p, given the occurrence of an eruption within the selected combination of ESs is$$ \label{eq:multscenarios_cond8} [\phi_d]=\frac {\sum_{i \in \mathcal{I}} \left( \left[\theta_{4}^{(i)}\right]\! \sum_{j \in \mathcal{J}} \! \left[\theta_{5,i}^{(j)}\right] \!\left[\theta_{6,j}^{(p)}\right]\! \left[\theta_{7,i,j,p}^{(k)}\right]\! \left[\theta_{8,i,j,p,k}^{(s)}\right] \right)} {\phi_{\rm ES}} $$(21)
PVHA output maps

Spatial probability of vent opening (conditional on eruption occurrence), which is related to the same probability of the susceptibility map proposed by Felpeto et al. (2007); see Fig. 2 for a snapshot example from the code

Absolute spatial probability of a specific size class or a specific phenomenon (in the code, it is called absolute map); this map shows, for each area, the absolute probability of a specific eruption size class, or the absolute occurrence probability of a specific phenomenon; see Fig. 3 for a snapshot example from the code.

ES conditional probability (conditional on eruption occurrence): this map (called sizes map in the code) displays, for each area (or group of areas), the probability of a specific eruption size class, weighted with the spatial probability of vent opening, conditional on eruption occurrence in that area (or group of areas). In practice, if one class is selected, the map displays [φ _{ a }] (see Eq. 17); if more size classes are selected, the map shows the sum of [φ _{ a }] for all size classes belonging to \({j \in \mathcal{J}}\). Figure 4 displays a snapshot example of the code.

Absolute or conditional probability of the phenomenon impacting the territory, in terms of “reaching” proximal or distal areas (node 7; [φ _{ c }]), or in terms of overcoming a specific threshold (node 8; [φ _{ d }]) in proximal or distal areas; this is called outcome map in the code; see Fig. 5 for a snapshot example from the code
The probabilities reported in each map are represented as distributions. The average value represents the “best guess” of such a probability; the dispersion around it gives the uncertainty about this guess.
Discussion and final remarks

It provides PVHA in a userfriendly and transparent way (it is not a black box).

It estimates almost all probabilities useful for hazard and risk applications. In particular, it calculates and visualizes ES maps, as well as weighted combinations of all possible ESs.

It is based on the Bayesian approach; this enables us to take into account different sources of information, such as models output, field data, and relevant geological and historical information. Moreover, the Bayesian approach allows aleatory and epistemic uncertainties to be estimated and visualized.

The outputs are provided in different formats (maps in GoogleEarth, GIS, gif formats), in order to make easy the use of the results.
In the companion paper (Selva et al. 2010), an extensive application of BET_VH to ash hall hazard at Campi Flegrei, Italy, is reported.
Notes
Acknowledgements
Laura Sandri wishes to thank Alex GarciaAristizabal for his help with the figures of this paper. We also thank Thea Hincks whose suggestions help us to improve the manuscript.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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