The main purpose of this article is to emphasize the
importance of clarifying the probabilistic framework adopted for volcanic
hazard and eruption forecasting. Eruption forecasting and volcanic hazard
analysis seek to quantify the deep uncertainties that pervade the modeling
of pre-, sin-, and post-eruptive processes. These uncertainties can be
differentiated into three fundamental types: (1) the natural variability of
volcanic systems, usually represented as stochastic processes with
parameterized distributions (

Hazards associated with major eruptions and their consequences are highly uncertain owing to the stochastic behaviors of volcanic systems (aleatory variability) as well as our lack of knowledge about these behaviors (epistemic uncertainty). Because such complexity hampers the deterministic prediction of hazards, our goal is to describe them probabilistically (Sparks, 2003; Marzocchi and Bebbington, 2012; Poland and Anderson, 2020). Here we use the term probabilistic volcanic hazard analysis (PVHA) to indicate the probabilistic forecast of any volcanological event of interest (eruption occurrence, ashfall loading, arrival of a pyroclastic flow, etc).

PVHA outcome may be generally described by the exceedance probability of a
positive random variable

The way in which we use the set of hazard curves

Most practitioners recognize that distinguishing between different types of uncertainties can be useful in the interpretation of hazard estimates (e.g., Abrahamson and Bommer, 2005; IAEA, 2012; Rougier, 2013), but the confusion surrounding the topic is evident in the wide variety of schemes proposed for classifying uncertainties: shallow and deep (Stein and Stein, 2013), intra- and inter-model, external and internal (Rougier and Beven, 2013), inter- and intra-model (Selva et al., 2013), value and structural uncertainty (e.g., Solomon et al., 2007), quantified measure of uncertainty and confidence on the validity of a finding (IPCC, 2013), model parameters and initial/boundary conditions, and many others (Reilly et al., 2021). It has been unclear whether these classifications are profound categories that must be reflected in the probabilistic framework or merely convenient, model-based divisions (e.g., NRC, 1997; Rougier and Beven, 2013).

The subject of this paper is to underline the importance of the probabilistic framework in PVHA and its practical implications. This framework must be able to (1) establish a coherent and clear hierarchy of different kinds of uncertainty (aleatory variability, epistemic uncertainty, and ontological error), (2) assimilate subjective expert judgment into probabilistic models, and (3) unconditionally test complete probabilistic models against data. In the next sections, through a toy example and an application to the Campi Flegrei tephra-fall hazard, we describe the unified probabilistic framework developed by Marzocchi and Jordan (2014, 2017, 2018), which satisfies these requirements. To facilitate the reading and comprehension of the paper, we describe the notation and the new terminology both within the paper and in a glossary at the end of the paper.

We consider the case in which

The unified probabilistic framework is rooted in the definition of an
experimental concept, which allows us to define a hierarchy of
uncertainties. Here we introduce it with a tutorial example, which may be
easily generalized to more complex cases. We collect the sequence of annual
observations for a particular value of the tephra-fall loading

A theorem by de Finetti (1974) states that a set of events that is judged to
be exchangeable (i.e., the events may or may not come from different unknown
distributions) can be modeled as identical and independently distributed
random variables with a well-defined frequency of occurrence,

In our example,

In contrast to the Bayesian framework, for which all models are “wrong” and
model validation is pointless (Lindley, 2000), the unified framework allows
model validation. Specifically, we can define an

In practice, collecting sufficient data for this kind of model validation is
only feasible for specific sites surrounding active volcanoes with a high
frequency of eruptions. For a specific site near a high-risk volcano with a
low eruptive frequency, the data are usually insufficient for formal
ontological testing. In probabilistic seismic hazard analysis (PSHA), the
problem is overcoming trading time with space, i.e., considering many sites
simultaneously for one or more time windows. This approach requires that
exceedances recorded at different sites can be considered statistically
independent; although this may become attainable in PSHA under some specific
considerations, it clearly does not hold for sites surrounding a single
volcano. In volcanology, trading time with space is useful in validating
models for global PVHA (e.g., Jenkins et al., 2015), which consider the
eruptive activity of all volcanoes of a specific type. In this case, it is
possible to select sites that are far enough to consider the observed
exceedances conditionally independent from each other, and the
exchangeable sequence

Future applications may also take advantage of the fact that the
exchangeability judgment can be generalized beyond the stationarity of the
process (implicit in our example) to more complex situations. For example,
we may distinguish ashfall exceedances in the winter and summer seasons,
because ashfall loading may be markedly affected by the seasonally dominant
winds blowing in different directions. In this case, the data-generating
process provides two sequences,

In this section we apply the probabilistic framework outlined in Sect. 2 to the tephra-fall PVHA at Campi Flegrei. Although the low eruption frequency of this volcano makes model validation unrealistic in the human time frame, the probabilistic framework has the advantage of providing a full description of the PVHA, accounting for all uncertainties; this may be of particular importance for decision-makers because, for example, they can immediately appreciate the level of uncertainty over the probabilistic assessment made by volcanologists.

Most (if not all) of the studies available for tephra-fall PVHA at Campi
Flegrei are based on event trees (ETs; see Newhall and Hoblitt, 2002;
Marzocchi et al., 2004, 2008, 2010; Marti et al., 2008; Sobradelo and Marti,
2010). The ET is a popular tree graph representation of events in which
individual branches at each node point to different possible events, states,
or conditions through increasingly specific subsequent events (intermediate
outcomes) to final outcomes; in this way, an ET shows all relevant possible
outcomes of volcanic unrest at progressively higher degrees of detail. The
probability of each outcome is calculated combining the conditional
probability of each branch belonging to the path from the first node to the
final outcome through classical probability theorems. By construction, the ET is
meant to describe only the intrinsic variability of the process (aleatory
variability) and not the epistemic uncertainty; hence it may produce only
one single probabilistic assessment

The BET approach fits quite well with the unified probabilistic framework that we
advocate in this paper. If we consider an experimental concept given by an
exchangeable sequence

The BET approach has been widely investigated for tephra-fall PVHA at Campi
Flegrei, adopting different choices, hypotheses, and models (Selva et al.,
2010, 2018; Sandri et al., 2016). For instance, Selva et al. (2018) show the
outcomes of five different BET configurations (Fig. 1) for one specific
site inside the caldera (Fig. 2), which differ in the implementation of
the tephra-fall dispersion model (aggregation and granulometry). In Fig. 3
we show the PoE distribution

Panel

The location reference site for this study (red star) along with
the hazard map relative to the probability of 5 % of exceedance, conditional
upon the occurrence of one eruption of whatever size and from whatever vent
at Campi Flegrei (mean of the epistemic uncertainty). The figure has been
obtained modifying the Fig. 11 of the corrigendum to the paper Selva
et al. (2018)
(available at

EED of each BET configuration relative to a tephra load threshold
of 300

When all EEDs are significantly overlapping (as for many points inside the Campi Flegrei caldera), it means that each BET configuration describes the epistemic uncertainty in a consistent manner. Instead, for the site in Fig. 2, we infer that inconsistent BET outcomes (Fig. 3) may be due to an underestimation of the epistemic uncertainty in each EED. For this reason, in the example of Fig. 3 we consider only the weighted average of each EED, and then we build a new EED which more satisfactorily describes the overall epistemic uncertainty given by the five BET configurations. This is equivalent to the case of using alternative implementations of the classical ET (Newhall and Pallister, 2015), which produces a set of hazard curves like in the upper panels of Fig. 1. In the specific case of Fig. 3, the reason for which the epistemic uncertainty is underestimated in each EED may be due to the BET setup and/or to limitations of the BET model to handle some sources of epistemic uncertainty.

The way in which we can build a single EED from a set of point forecasts

Standard methods are available for the induction of the EED

Although ensemble modeling does not prescribe any specific procedure to
estimate

The overall EED obtained from the average of the five
distributions of Fig. 3. The EED is a beta distribution with parameters
given by Eqs. (2) and (3). The histogram (right

In this paper we have described a unified probabilistic framework which allows volcanologists to provide a complete description of PVHA, to define a clear taxonomy of uncertainties (aleatory variability, epistemic uncertainty, and ontological errors), and to account for experts' judgments preserving the possibility to unconditionally test PVHA against data, at least for volcanoes with high eruptive rate, or for global forecasting models. Although in this paper we focus entirely on PVHA, we think that this approach may potentially inspire other scientists working on different natural hazards.

One remarkable and distinctive feature of this probabilistic framework is
that the mathematical description of PVHA is given by a distribution of
probability (see, e.g., Marzocchi et al., 2004, 2008, 2010; Neri et al.,
2008; Sobradelo and Marti, 2010; Bevilacqua et al., 2015) or, equivalently,
through a bunch of hazard curves

Making explicit the probabilistic framework in PVHA is important. In the
past, loose definitions of the probabilistic framework have provoked
critiques of natural hazard analysis (see, e.g., Castanos and Lomnitz, 2002;
Mulargia et al., 2017). For example, a vague definition of the nature of
uncertainties and the role of subjective judgments brought some scientists
to assert that (Stark, 2017)

what appears to be impressive `science' is in fact an artificial amplification of the opinions and ad hoc choices built into the model, which has a heuristic basis rather than a tested (or even testable) scientific basis.

This criticism is implicitly rooted in the (false) syllogism: science is objective, and natural hazard analysis relies on subjective experts' judgment; hence natural hazard analysis is not science.The unified probabilistic framework proposed here emphasizes the importance of model validation (at least in principle) as a cornerstone of science; pure objectivity is a myth even in science, and the presence of unavoidable subjectivity in PVHA cannot be used to dismiss its scientific nature. In an extreme case, experts can behave like “models” expressing their subjective measure of the frequency of one defined experimental concept; mutatis mutandis, the same applies to the famous case of farmers who subjectively guess the weight of an ox (Galton, 1907), whose similarities and differences with natural hazard analysis have been discussed in Marzocchi and Jordan (2014). Conversely, the Bayesian framework, which is a full legitimate probabilistic framework to be used in PVHA, does not allow model validation; in this framework all models are wrong (hence, why waste time to validate them?), and we can only evaluate the relative forecasting performance of one model against the others (Lindley, 2000; Jaynes, 2003).

Besides the scientific aspects, the use of a PoE distribution has remarkable practical merits, because it shows to the decision makers both our best guess and the associated uncertainty. In plain words, if two PVHAs have the same average, but with quite different variance, this may significantly affect the way in which PVHA could be used by decision-makers. For example, let us consider a case in which there is a critical threshold in PVHA that triggers a specific mitigation action when overcome (this is just a simplified example, because the decision-making has to be based on risk, not on hazard); both averages may be lower than such a critical threshold (hence both suggesting no action), but, when considering the variance, one of the EEDs shows a significant part of the distribution above the threshold (suggesting taking action). In this case, the decision makers may take into consideration the epistemic uncertainty deciding, for the sake of precautionary reasons, to use one specific high percentile of the EED, instead of the average; for example, the Ministry of Civil Defence & Emergency Management in New Zealand (MCDEM, 2008) uses the 84th percentile of the tsunami hazard analysis as a threshold for taking actions.

As a final consideration, owing to the social implications, we think that only adopting a clear probabilistic framework to get a complete PVHA is the best way to defend probabilistic assessments against future scrutiny and criticism and to use these assessments in the most profitable way.

This glossary contains the statistical notation used in this paper and the definitions of new and uncommon terms.

Codes are available upon request to the corresponding author.

The data used in this paper are available upon request to the corresponding author.

All authors contributed, at different levels, to the conceptualization of the research. WM and JS carried out the formal analysis and numerical investigation. WM wrote the first draft of the paper with contributions from all the other authors. All authors reviewed and edited the final version.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Thomas H. Jordan was supported by a grant from the W. M. Keck Foundation.

This paper was edited by Giovanni Macedonio and reviewed by two anonymous referees.