Bayesian event tree for eruption forecasting (BET_EF) at Vesuvius, Italy: a retrospective forward application to the 1631 eruption

Abstract

Reliable forecasting of the next eruption at Vesuvius is the main scientific factor in defining effective strategies to reduce volcanic risk in one of the most dangerous volcanic areas of the world. In this paper, we apply a recently developed probabilistic code for eruption forecasting to new and independent historical data related to the pre-eruptive phase of the 1631 eruption. The results obtained point out three main issues: (1) the importance of “cold” historical data (according to Guidoboni 2008) related to pre-eruptive phases for evaluating forecasting tools and possibly refining them; (2) the BET_EF code implemented for Vesuvius would have forecasted the 1631 eruption satisfactorily, marking different stages of the pre-eruptive phase; (3) the code shows that pre-eruptive signals that significantly increase the probability of eruption were likely detected more than 2 months before the event.

Appendices

Appendix A: The authors of the five treatises examined

The authors of these five texts were Neapolitan intellectuals, men of the cloth or ecclesiastic, all present at the eruption of Vesuvius in 1631. Here is their concise biography, that is useful in order to frame the cultural ambient those authors belonged to.

Giovanni Domenico Armini. Hardly anything is known about this author, apart from the fact that he was a doctor at the Ospedale degli Incurabili of Naples, as he himself declares in the presentation of the Treatise. So he was a direct witness to what had happened, above all of what was seen in Naples. His treatise contains many theoretical elements on the causes of the eruption, mainly oriented to an almost anthropomorphic interpretation of the volcanic activity.

Gregorio Carafa (1588–1675). Carlo Marcello (later Gregorio) Carafa was born in Naples. He was a philosopher and theologian; he joined the Teatini fathers of Naples, where he held the chair of Philosophy and Theology. It was in those years that he acquired great fame as a preacher and a cultured intellectual. The fame he earned among his contemporaries and his noble origins allowed him to obtain the highest positions in his order. He was also the Bishop of Cassano (Calabria) and Archbishop of Salerno. He held some diplomatic posts: he was the special diplomatic representative of the Emperor Philip IV of Spain with the Pope Innocence X.

Giovanni Battista Mascolo (1583–1648). Born in Naples, he became a Jesuit at a very young age, in 1598. He taught theology and philosophy at the college of his order, of which he was the rector for some time; later he held a school of rhetoric at his home. He was famed for being a good Latin scholar. His treatise De incendio Vesuvii (Naples 1634), his most complex work, was written in his full maturity, probably using some of his previous unpublished writings.

Giulio Cesare Braccini (1570-1632). He was a man of law, later an ecclesiast and abbot, born at Gioviano di Lucca. He was appointed apostolic proto-notary by Urban VIII and between 1629 and 1632 he was in Naples, in touch with many leading political characters of the day. As regards Vesuvius, abbot Braccini published two texts: one was a letter to cardinal Girolamo Colonna, disseminated in the first few days after the eruption, to the extent that already between the end of December 1631 and the beginning of 1632 three editions had been published (Relazione dell’incendio del Vesuvio alli 16 dicembre 1631 in una lettera diretta all’Em.mo e Re. mo Signore Card. Colonna, printed in Naples at S. Roncagliolo); and the treatise examined here, printed in Naples in September 1632.”

Salvatore Varrone (1593–1656). We have very little information about him. He became a Jesuit in 1612; he taught grammar, humanities and rhetoric, and for 6 years scholastic theology and morals. He was famous for being a very learned intellectual. In the period of the eruption of Vesuvius, Varrone was staying at Portici, a village on the slopes of Vesuvius, and therefore was an eyewitness to the whole eruption.

Appendix B: Summarizing tables of the BET_EF rules for Vesuvius

Here we show the rules uploaded in BET_EF in this application. They are exactly the same as used in Marzocchi et al (2008).

A summary of all the rules is provided in Table 4.

Table 4 BET_EF settings

At each node k of the Bayesian Event Tree we compute two different probability distributions, one by using only monitoring data and the other one by using only other kinds of data, e.g. models, past occurrence, and expert opinions. These two probability distributions are indicated, respectively, by [θ_k ^(M)] and [θ_k ^(NM)], where the index k stands for the kth node and the square brackets denote a probability distribution. In order to compute the actual probability distribution at node k, we linearly combine [θ_k ^(M)] and [θ_k ^(NM)] with a relative weight that is function of the state of unrest. Both [θ_k ^(M)] and [θ_k ^(NM)] are computed through the Bayes theorem, i.e., by starting from a prior distribution (based on models and beliefs for [θ_k ^(NM)], and on present monitoring for [θ_k ^(M)]). The prior distribution is characterized by a mean (Θ_k ^(NM) and Θ_k ^(M) in the two cases), representing our best prior guess on the probability at node k, and by a measure of the variance that we call “equivalent number of data” (Λ_k ^(NM) and Λ_k ^(M) in the two cases) because intuitively it translates the confidence we have in our prior guess in terms of number of data. The minimum possible “equivalent number of data” is 1 and it represents the maximum variance allowed, implying a very low confidence on our prior guess, while there is no upper limit to its maximum value. The prior distribution is then transformed into the posterior distribution through the Bayes theorem, i.e., by multiplying it by the likelihood function, based on past frequencies of occurrence for [θ_k ^(NM)] and on past monitoring (if any) for [θ_k ^(M)].

An important aspect of BET is the way it deals with monitoring measurements. Through a fuzzy approach, the measured values are translated into degrees of anomaly, from whom the mean of the monitoring probability distribution (Θ_k ^(M)) is derived (Marzocchi et al. 2008). In practice, for each monitoring parameter, we define a lower and an upper threshold, and an order relationship, used to infer the degree of anomaly for every specific measured value.

In the following, we give a detailed account of the choices on all BET parameters for Mt. Vesuvius (see also Marzocchi et al. 2004, 2008), as frozen before this retrospective application. In Marzocchi et al. (2008) the reader can find a deeper discussion on BET structure (the nodes), general rules and related concepts.

B.1 First node: Unrest

B.1.1 Non-monitoring part: [θ₁ ^(NM)]

There is no theoretical model or expert belief to be used for assessing the probability of a volcano entering an unrest phase. Thus here we set up a maximum ignorance prior distribution with mean Θ₁ ^(NM) = 0.5 and Λ₁ ^(NM) = 1.

Regarding past data, we know that OVO seismic station (see Zollo et al. 2002) has been monitoring Mt. Vesuvius continuously since 1972. We assume that there has been no episode of unrest ever since. Thus, we have 0 unrest episodes out of 420 months (i.e., 35 years).

B.1.2 Monitoring part: [θ₁ ^(M)]

From now on, for the monitoring part at each node we give the list of selected monitoring variables used in Marzocchi et al. (2004). In particular, for node 1, as summarized in Table 5, they are (from now on, for monitoring parameters we denote in braces the order relationship and the upper and lower thresholds used to define the degree of anomaly of the measured parameter):

• n_e {>;23;150} number of seismic events per month with M_d≥1.9 recorded at OVO station; the thresholds have been chosen on the basis of the monthly distribution of the number of seismic events observed at OVO, and they represent, respectively, the 55th and the 95th percentiles

• M_d {>;3.4;4.3} largest duration magnitude of the earthquakes recorded during last month at OVO station; the thresholds have been chosen on the basis of the monthly distribution of the magnitude of seismic events observed at OVO, and they represent, respectively, the 55th and the 95th percentiles

• n_LF {>;1;3} number of low-frequency (LF) events deeper than 1 km per month; since only sporadic and temporally isolated LF events have been observed so far, even a small swarm is indicative of unrest

• Π_SO2 {=;1;1} significant presence SO₂ (0, no; 1, yes)

• Φ_CO2 {>;5;30 kg m^–2 d^–1} daily CO₂ emission rate; the thresholds represent, respectively, about twice and ten times the background value

• dɛ/dt {>;0;0 d^–1} strain rate (inflation), assuming that any detected uplift implies unrest

• T {>;98;105°C} temperature of the fumaroles inside the crater; the thresholds represent, respectively, about 3% and 10% higher than the value observed since 1990 (95°C).

Table 5 Monitoring parameters at NODE 1

No past monitoring is available. Thus, the posterior for [θ₁ ^(M)] is equal to the prior.

B.2 Second node: Magma given unrest

B.2.1 Non-monitoring part: [θ₂ ^(NM)]

There is no model to be used for assessing the probability of a volcano unrest being due to magma. Thus here we set up a maximum ignorance distribution with mean Θ₂ ^(NM) = 0.5 and Λ₂ ^(NM) = 1.

Regarding past data, we have no unbiased past data to infer the origin of past unrest at Vesuvius, since we know much better the magmatic unrest episodes (at least those that yielded an eruption), compared to the hydrothermal ones. Thus, the posterior for [θ₂ ^(NM)] remains equal to the prior.

B.2.2 Monitoring part: [θ₂ ^(M)]

As summarized in Table 6, from Marzocchi et al. (2004), the selected monitored parameters at node 2 are:

• Π_SO2 {=;1;1} significant presence SO₂ (0, no; 1, yes)

• dɛ/dt {>;5 × 10^–6; 5 × 10^–5 d^–1} strain rate (inflation), considering that a rapid and localized positive strain of the volcanic edifice is usually indicative of rising magma

• ν {<;2.5;3.5 Hz} the dominant spectral frequency of earthquakes; if it is around the frequency specified by the thresholds, there are probably LF events or tremor, that might be indicative of magma motion

• ξ_e {<;0.3;0.4] ratio between average and dispersion of the depth of the earthquakes during unrest. The present value of ξ_e for Mt. Vesuvius from seismic events recorded in the last 30 years is about 0.4. The thresholds are chosen in order to fuzzify this value, assuming that earthquakes coming closer to the surface and/or occurring in a larger range of depths may indicate an upward migration of magma or a coalescence of small fractures that may facilitate the magma uprising. At the same time, we do not consider shallow earthquakes concentrated in a very small range of depths to be indicative of magma, as they could also be due to hydrothermal activity

• T {>;98;105°C} the temperature of the fumaroles inside the crater; the thresholds are chosen as for node 1.

Table 6 Monitoring parameters at NODE 2

Remarkably, we assume that the parameters Π_SO2 and T have a weight twice as much as the other parameters in determining the magmatic origin for the unrest.

Past monitoring is also not available here. Thus, the posterior for [θ₂ ^(M)] is equal to the prior.

B.3 Third node: Eruption given magmatic unrest

B.3.1 Non-monitoring part: [θ₃ ^(NM)]

There is no model to be used for assessing the probability of a volcano erupting given that there is an unrest of magmatic origin. Thus here we set up a maximum ignorance distribution with mean Θ₃ ^(NM) = 0.5 and Λ₃ ^(NM) = 1.

Regarding past data, we have no unbiased past data to infer the frequency of eruptions following past magmatic unrest at Vesuvius, since we know much better the magmatic unrest episodes that yielded an eruption with respect to those that died out without eruption. Thus, the posterior for [θ₃ ^(NM)] remains equal to the prior.

B.3.2 Monitoring part: [θ₃ ^(M)]

As summarized in Table 7, from Marzocchi et al. (2004), the selected monitored parameters at node 3 are:

• PE {=;1;1} presence of phreatic explosion (0, no; 1, yes)

• dν/dt {<;0;0 Hz d^–1} rate of change of the average spectral frequency content of earthquakes

• ξ_e {<;0.3;0.4} ratio between average and dispersion of the depth of the earthquakes during unrest; the thresholds are chosen as for node 2

• d²E/dt² {=;1;1} acceleration of seismic energy release (0, no; 1, yes)

• d²ɛ/dt² {=;1;1} acceleration of inflation (0, no; 1, yes)

• ε  {> 5 × 10^–5; 5 × 10^–4} cumulative strain since the beginning of the unrest; the thresholds are just a little bit less than a reasonable value for the maximum prefracture strain

• dρ/dt {=;1;1} change of the ratios HCl/SO₂ and/or HF/SO₂

• REV {=;1;1} sudden reversal of at least one of the above parameters (0, no; 1, yes)

Table 7 Monitoring parameters at NODE 3

For most of these parameters, the thresholds are based on a “yes/no” choice. This is because we are mainly interested in the time trend of the parameters.

We assume that the parameters d²ɛ/dt², PE and REV have a weight twice as much as the other parameters in determining the occurrence of the eruption. Past monitoring is not available. Thus, the posterior for [θ₃ ^(M)] is equal to the prior.

B.4 Forth node: Location of vent given that there is an eruption

B.4.1 Non-monitoring part: [θ₄ ^(NM)]

Since Mt. Vesuvius is a central volcano, and its activity has been mainly concentrated in the crater, we assume that there is a very high probability of vent opening in the crater area, and a very small one outside it. With this idea in mind, we divide the volcanic edifice into five areas: the crater area, and four outer, equal-sized areas. We assume that next eruption will take place in one and only one of them. Because of these assumptions, we set up a Dirichlet distribution (see Marzocchi et al. 2008 for further details on such distribution) with mean Θ₄ ^(NM)(1) = 0.99 in the crater area (area 1) and Θ₄ ^(NM)(i) = 0.0025 (i = 2,…5) for the surrounding areas (i.e., Areas 2, 3, 4 and 5). Since we are very confident on this assumption, we set Λ₄ ^(NM) = 50, which is an (arbitrarily) high equivalent number of data.

B.4.2 Monitoring part: [θ₄ ^(M)]

BET_EF allows to localize some or all of the monitored measurements used at previous nodes to assess the probability of vent opening in different areas, according to monitoring (see Marzocchi et al. 2008 for further details on this issue).

B.5 Fifth node: Size of the eruption given that there is an eruption

We parametrize the possible sizes with VEI. Since Mt. Vesuvius has been dormant for over 60 years now, it is presumably in a closed-conduit regime. Therefore, we assume that the next eruption will be at least of VEI = 3 in order to have sufficient energy to re-open the system. For practical purposes, we define three classes of possible sizes, according to this idea: VEI = 3, VEI = 4 and VEI ≥ 5.

B.5.1 Non-monitoring part: [θ₅ ^(NM)]

We use the observation that the worldwide log(frequency)-size relationship for volcanoes is a straight line, implying that the most frequent (and therefore more likely) eruptions are the smaller ones, with a power law relationship. We translate this information by setting up a Dirichlet distribution for the three size classes. The mean of each class is set equal to the probability for that class given by the above-mentioned worldwide relationship. In this way, we have the following means: Θ₅ ^(NM)(1) = 0.83 for VEI = 3, Θ₅ ^(NM)(2) = 0.14 for VEI = 4 and Θ₅ ^(NM)(3) = 0.03 for larger eruptions. We also assume the maximum variance on this model, thus Λ₅ ^(NM) = 1.

For past data, we use the catalog of Mt. Vesuvius eruptions with a repose time larger or equal to 60 years, so that we are sure to consider only past eruptions occurred in closed-conduit regime, as it is now the case. We have seven such eruptions in the catalog, in particular four eruptions of VEI = 3, two of VEI = 4 and one larger.

B.5.2 Monitoring part: [θ₅ ^(M)]

Up to the present, there is no reliable precursor heralding the size of eruptions. Therefore, we do not use monitoring data at this node.