The temporal occurrence of volcanic events is one of the key ingredients to evaluate volcanic hazard and risk. In the literature, the statistical modelling of the temporal occurrence at a given volcano has been tackled with different models, fundamentally assuming eruptions as realizations of a point process in time, i.e., neglecting the duration of the eruption, as it is usually much shorter than the inter-event time (IET) between two successive eruptions. For a review of the use of temporal models in volcanology, we remand the reader to .

Mt. Etna in Sicily (Italy) is one of the most active volcanoes in the world. Its activity is mostly divided into summit and flank activity, characterized by different eruptive behaviour. Summit activity occurs from summit craters and it is mostly characterized by Strombolian activity and episodic lava fountains or paroxysms, often associated with lava emission . Paroxysms are typically short-lived (up to several hours) and frequent episodes (on average 7 per year since 1986 CE), producing an eruptive plume that can reach 17–18 km a.s.l. , with total erupted mass values up to ≈ 1010 kg . In contrast, flank eruptions are more rare events (on average less than 2 events every 10 years, in the last 4 centuries), characterized by lava effusion often associated with explosive activity forming an ash plume, lasting from a few days to months .

In this work we model the occurrence in time of summit paroxysms and flank eruptions separately, assuming they occur independently one type from the other. There are several reasons why we model separately the two types of event. First, from a phenomenological point of view, they have very different dynamics, explosivity, erupted magma and characteristic return time, somewhat implying that the processes leading to the two types of events are different. Second, from the point of view of the data, the temporal range covered by the complete part of their catalogs are very different. Flank eruptions, although much more rare than summit paroxysms, likely impact the towns and villages located on the volcano slopes through their lava flows and prolonged tephra emission, and leave a footprint in the geological record: this is why we can reasonably assume that the catalog for flank eruptions published by represents a complete list of this type of events since at least 1760 CE. Paroxysms, on the other hand, had little impact on economy and society until a century ago or so, when air and road traffic was not an issue. Furthermore, transient paroxysms may be missed by human eye in case of cloudy weather, and their footprint in terms of tephra layer at the ground is quickly removed or depleted by wind and rains. For all these reasons, the catalog of summit paroxysms at Etna can be reasonably considered complete since 1986 CE , when the support from monitoring (seismic and remote) has become more and more effective in detecting and characterizing summit activity at Etna.

To model the temporal occurrence of flank and summit paroxysms separately, for each dataset first we compute Inter-Event Times (IETs) as the time intervals between the onset of two subsequent events. Due to the different features of the two datasets, mostly in terms of recurrence and clusterization, the methodology adopted to model time occurrence, and to test the model, is different. For flank eruptions, we first estimate the completeness date, defined as the time after which the eruption catalogue is assumed to be complete, i.e., no eruptions are considered to have been missed. This estimate is obtained under the assumption of stationarity. We then test the resulting catalogue for temporal independence by means of the method reported in, and clusterization (by means of the Coefficient of Variation, CV) of the IET data over the completeness time interval (see Table ). For summit paroxysms, we assume that the catalogue is complete since its first event reported (due to the continuity of monitoring since 1986) and, as for the flank eruptions, we also test stationarity, independence and clusterization.

The models presented show that the occurrence of eruptions at Etna seems to be best described by a Brownian Passage Time model. This model has been previously used to model the occurrence of point processes such as seismic events e.g., or eruptions e.g.,. It is based on the so-called Brownian Relaxation Oscillator, in which a given state variable is subject to a constant loading with superimposed Brownian perturbations. An event occurs (i.e., the point process has a realization) when the state variable reaches a threshold. When this occurs, the state variable relaxes to the background value, and a new “stochastic” cycle begins. This leads to a stochastic load-state process. The load-state process is the Brownian Relaxation Oscillator, whose IETs are described by the Brownian Passage Time distribution (also termed Inverse Gaussian or Wald distribution in the literature). We remand the reader to Fig. 2 in for a visual explanation of the Brownian Relaxation Oscillator. The state variable of this model can be linked to the tectonic loading for earthquake occurrence, or to the pressure in the magma storage system under a volcano . The main feature of this type of renewal model is that, depending on the combination of the values of the 2 model parameters, the model is able to describe from a quasi-periodic occurrence up to a non-homogeneous Poisson process. As shown in Fig. , its hazard function increases in time after an event (with very different characteristic timescale, depending on the model parameters values) until a plateau is reached, which implies then a Poissonian occurrence (for comparison, in Fig. a and c we also provide the Poisson hazard function for the flank and inter-cluster IETs).

For flank eruptions, assuming stationarity to retain the complete part of the catalogue, we consider as complete the eruptive record since 1760 CE. Under this assumption, here we cut out from the catalogue the unusually long IET that follows the largest eruption occurred in 1669 CE. We acknowledge that this long IET might be real and may reflect the “generalized” (i.e., not linear) time-predictable behaviour suggested by e.g. and , although those studies used a different and less revised catalogue. Here, we base our study on the most recent available catalogue for flank eruptions at Etna and the unusually long IET following that large eruption breaks the stationarity assumption. Thus we removed it, suspecting a possible incompleteness in the aftermath of such a large eruption which destroyed Catania layout and may have disrupted the recording of Etna eruptions. In any case, since 1760 CE, based on the value of the CV, no significant clustering is observed so we model the occurrence of these events simply by the preferred renewal model (Brownian Passage Time). The bootstrap version of the KS1 test for the goodness of fit does not reject the model. We compare the performance of the preferred model to the “reference” Poissonian occurrence over the IET data among flank eruptions, by computing the individual probability gain for each event (as we do not know the exact day and month of each eruption, we consider the probability over the year of occurrence of each eruption), and the overall gain . We see in Fig.  that the preferred model outperforms the reference Poissonian occurrence when the IET is “not too short” (approximately larger than 1 year) and “not too long” (i.e., within 6 years since the previous event). Over the whole dataset, the probability gain of the Brownian Passage Time with respect to Poissonian occurrence is positive (value is 0.073).

By means of the preferred model identified and based on BPT, we compute the probability of flank eruption over different time windows (see Fig. b and Table ). Considering the uncertainty on the model parameters, we compute uncertainty intervals on this best estimate probability. Even considering the lower limit of this interval, we see that flank eruptions have a probability above 95 % over a window of 30 years, considering the current censoring time.

Summit paroxysms at Etna clearly show a clustered behaviour, so we model separately the inter-event between cluster onsets and the IETs of events within clusters. Again both preferred models are the Brownian Passage Time, with different parameters' values. The testing of this model is not as straightfoward as in the case of flank eruptions. To perform the test, we use the preferred models to generate synthetic catalogs and then we test how they capture some basic features of the real dataset of summit paroxysms. Synthetically-generated catalogs well capture the real observations for summit paroxysms in terms of (i) distribution of the number of events over three time windows (1, 3 ad 5 years, Fig. ) and their means (Fig. ), and (ii) annual rates of events and of clusters (Fig. ). So we judge the two Brownian Passage Time models good in describing the occurrence of Etna summit paroxysms in clusters, and we compute their expected number over 1, 3 and 5 years (Table ). The best guess in 1 year is 6 paroxysms, with a 90 % confidence interval from 0 to 28.

In order to present how the latter model for summit paroxysms can be used in practical applications, we provide an illustrative example. To this end, we use the synthetic catalogs to simulate the temporal component of a Probabilistic Volcanic Hazard Assessment (PVHA) for the tephra fallout caused by Etna summit paroxysms. Since we cannot simulate wind field from the future, we pre-run 3500 simulations of tephra fallout from Etna summit craters with TEPHRA2 model by sampling random dates (day, month, year, hour and minute) over the period 2007–2019, thus covering 13 years, for which the wind field is available from INGV-OE daily weather data . Each TEPHRA2 run had associated a set of Eruption Source Parameters (Mass Eruption Rate and Duration of fallout phase) that was sampled from Empirical Cumulative Distribution Function of the real observations for the events for which they are available,. The computational domain of TEPHRA2 simulations covers 14.1960 to 16.148° E and 37.041 to 38.578° N, at 500 m resolution.

We then took 1000 synthetic catalogs of summit paroxysms from the temporal model analysis, and cut them over the period 2024–2048, thus covering 25 years in the future (at the time of the analysis, made in 2023–2024). To each synthetic paroxysm (with a date from 2024 to 2048) we associated the deposit of the pre-simulated eruption that was closest in time among the set of pre-run simulations, after subtracting 17 or 30 from the year of the synthetic event respectively form synthetic years from 2024 to 2036 and 2037 to 2048. In this way we preserve seasonality and, for clusters of synthetic events occurring close in time, we preserve the natural persistence or evolution of the wind field in time, as in reality. An example of this procedure for a specific synthetic catalog is provided in Fig. . The deposits corresponding to the synthetic events were then combined to achieve, on the grid points of TEPHRA2 computational domain, hazard curves of the accumulated tephra at the ground during a paroxysm, during a cluster, and in time windows of 1, 3 and 5 years (for example Fig. a). By cutting the hazard curves at selected tephra load or exceedance probability thresholds, we obtain respectively probability and hazard maps see for example Fig. b and c.

In this study we have modelled the occurrence in time of relevant eruptive events at Etna volcano, the most active volcano in Europe located North of the metropolitan area of Catania (Sicily, Italy). We have separately accounted for flank eruptions and for summit paroxysms. The former events are more rare, but are capable of producing lava flows that can reach inhabited areas, and of generating tephra emissions over periods of months as in 2002–2003;. The latter type of activity is very frequent (around 260 events have occurred in the period of time analyzed here, 1986–2022) and generates short-lived plumes (up to few hours long) that have reached up to 14–15 km above the summit of the volcano, at 3360 m a.s.l. approximately. These plumes can threaten local mobility by accumulating tephra on the roads, as well as air traffic in the area. Moreover, occurring in clusters during which these events are typically separated by few days up to few weeks, they can represent a problem in terms of clean up operations e.g.,.

By considering the most recent catalog of flank eruptions at Etna and assuming stationarity, we consider as complete the data since 1760. The flank eruptions recorded from that date do not show any tendency to clustering. Their inter-event times are best modelled by a Brownian Passage Time distribution. The preferred model, with its maximum-likelihood parameters and their uncertainty, well capture the cumulative distribution of the observed inter-flank-eruptions times.

For summit paroxysms, due to their evident clusterization, we first apply a k-means algorithm to the dataset of summit paroxysms (see Supplement), to objectively find the best partitioning into k clusters. We find that k=12 yields to the best partition, according to the mean silhouette coefficient. We then model separately the Inter-Cluster onset times and the inter-event times between successive paroxysms belonging to clusters. We find that both subsets of data are best described again by a Brownian Passage Time distribution, with very different parameter values. In order to judge the capability of these models in reproducing the basic features of the real inter-paroxysms times, we use the two preferred models to generate synthetic catalogs, that we then analyze. We find that the synthetic catalogs overall capture the real observed and mean number of paroxysms over different time windows (1, 3 and 5 years), and annual rate of clusters and of paroxysms. Finally, we have given an example on how to use in practice these two preferred model to introduce the temporal variability in hazard assessment. In particular, we have provided an illustrative example on how the synthetic catalogs can be coupled to a set of pre-run tephra dispersion simulations in order to quantify the ground load hazard for a paroxysm, for a cluster, or over a time window of 1, 3 and 5 years.