The Campi Flegrei caldera volcanic activity dates back to the upper Pleistocene, with the earliest volcanic activity observed in outcrops estimated at approximately 80 kyr (Pappalardo et al., 1999; Scarpati et al., 2013). Recent studies identified widespread tephra layers originated from Campi Flegrei, extending its activity to nearly 200 kyr (e.g., Monaco et al., 2022; Fernandez et al., 2024; Sparice et al., 2024). The first caldera collapse occurred approximately 39 000 years ago with the Campanian Ignimbrite (CI) eruption (Giaccio et al., 2017). A second major eruption occurred around 14 000 years ago, the Neapolitan Yellow Tuff (NYT), possibly causing a second collapse, after which volcanic activity resumed within the caldera (Orsi et al., 2004).

The post-NYT eruptive history is divided into three main epochs (Di Vito et al., 1999; Bevilacqua et al., 2016; Fig. 1). The first epoch comprises at least 33 eruptions, spanning from 14 000 to 10 600 years ago. The eruptive vents align with the caldera boundaries, with the most energetic eruption being that of the “Pomici Principali” around 12 000 years ago (Bevilacqua et al., 2016). The second epoch follows a relatively brief quiescent period and includes nine eruptions dated between 9600 and 9100 years ago, primarily concentrated in the northeastern sector of the caldera. The third epoch encompasses a total of 26 eruptions from approximately 5500 to 3800 years ago. Its activity was predominantly focused in the northeastern part of the caldera (Agnano area), secondarily in the northwestern sector (Averno area), and concluded with peripheral distal eruptions (Nisida, Capo Miseno, and Fossa Lupara, Natale et al., 2025). The most energetic eruption during this period was that of Agnano-Monte Spina around 4500 years ago (Bevilacqua et al., 2016). The long period of quiescence following the third epoch lasted until 1538 AD, when the Monte Nuovo eruption took place in the northwestern sector of the caldera (Di Vito et al., 2016). The caldera has not experienced any eruption since.

After NYT, the caldera experienced significant resurgence, accompanied by seismicity, degassing, and slow ground deformation often referred to as bradyseism (Isaia et al., 2019; Natale et al., 2022). The latter is centred mainly in the Pozzuoli area, which is approximately at the centre of the caldera (Fig. 1), and major movements were tracked at least from the 4th century AD thanks to the ruins of a Roman temple in the port of this city, the Serapeum (Di Vito et al., 2016; Vitale and Natale, 2023). The largest known bradyseismic events are the ones related to the last Campi Flegrei eruption (the 1538 Monte Nuovo eruption, Di Vito et al., 2016). In the last century, three main bradyseismic crises occurred in 1950–52, 1970–72, and 1982–84, characterised by episodes of uplift of almost 2 m interrupting a slow long-lasting subsidence (Del Gaudio et al., 2010). Approximately in 2005, a slow uplift started, accelerating over time, which fully recovered the subsidence in 2021, and now exceeds the uplift peaks observed in the last century (Bevilacqua et al., 2024). This event is still ongoing.

Forecasting vent opening is crucial for any volcanic hazard quantification. Different approaches were adopted at Campi Flegrei through time. Alberico et al. (2002) developed a method identifying crustal weaknesses using geophysical, geological, and geochemical parameters. Their probability map indicates the highest likelihood of vent openings in the central caldera near Pozzuoli (Fig. 1). Selva et al. (2012) utilized a Bayesian approach fed by fewer parameters, focusing on tectonic structures recognized at that time, to track crust weakness, and past vents, shifting the area at higher probability toward the northeastern and northwestern sectors, where post-NYT activity concentrated. Bevilacqua et al. (2015) adopted a method based on Gaussian kernel and accounting for the uncertainty on past vent positions, confirming the northeastern sector (near Agnano and Astroni, Fig. 1) as the most likely area for future eruptions. Both Selva et al. (2012) and Bevilacqua et al. (2015) included also a formal quantification of the epistemic uncertainty in their models, accounting for model uncertainty. More recently, based on past observations and removing multiple eruptions from clustered vents, Charlton et al. (2020) noted that vent opening occurred substantially randomly within a ring area surrounding the caldera centre, corresponding to the caldera rim (Fig. 1), producing a new qualitative indication about potential future vent opening.

Rivalta et al. (2019) studied the physical propagation of magma dykes by modelling the trajectory of potential dykes due to the subsurface stress field and the dike's initial position. Their model accounts for various stresses affecting magma ascent. For calderas, it suggests that eruptive vents are concentrated at specific distances from the centre, influenced by the stress induced by the caldera depression, defining a higher propensity to eruption closer to caldera rims, as noted by Charlton et al. (2020). Considering the caldera depression size, Rivalta et al. (2019) forecasted for Campi Flegrei a potential peak for vent opening at a semi-annular belt located between 2.3 and 4.2 km from the caldera centre. Rivalta et al. (2019) analysed also the effect of caldera unloading, as well as those of topographic peaks, which may break the caldera symmetry. They analysed the case of the Campi Flegrei caldera, explaining the concentration of volcanic activity in the northeastern sector due to the topographic peak of the Camaldoli Hill (Fig. 1), which creates a stress field that may favour magma trajectories in the north-east direction.

The main features of Rivalta et al. (2019) model are (i) that the geometry of the caldera (unloading effect) significantly influences dykes propagation outward, promoting eruptions away from the geometric centre of the caldera, and (ii) topographic asymmetries create localised stress variations in the subsurface, affecting eruption frequency across different angular sectors. While a sufficiently detailed knowledge of the sub-surface stress state is difficult to reach, it is possible to verify if these two main features left a trace on the available record of past vent positions, and to use this empirical signature to define new vent opening probability maps.

Rivalta et al. (2019) found that the path of the dykes feeding eruptions is mainly controlled by the geometry of the caldera, which determines the distance from the centre of the caldera, and by the topographic peaks surrounding the caldera, which establishes preferential directions for propagation. To this end, for Campi Flegrei Rivalta et al. (2019) assumed that the origin at depth of the magma is an oblate volume located at the centre of the caldera, 3 km below the location of the maximum observed uplift (Amoruso et al., 2014a; Rivalta et al., 2019; Buono et al., 2025), and producing mostly lateral propagation with trajectories controlled by the structure of the caldera and the consequent stress distribution within the caldera. In other words, Rivalta et al. (2019) demonstrate that, in an approximately symmetric caldera with a given magma source located around the centre of the caldera, independently of its specific depth and size, we expect a symmetrical radial dyke propagation from the source, and with preferred directions controlled by local factors. The empirical track of these features may be retrieved by studying the distribution of past vents around the caldera centre, being this position not necessarily the origin point for dykes, which may indeed detach from the edge of the source (Gregg et al., 2012; Rivalta et al., 2019), but simply the reference for tracking the radial symmetry.

Assuming a similar magma origin also for future eruptions, these empirical distributions can then be used to set a vent opening probability map. The probability density function in a specific point in the caldera can be calculated in polar coordinates using the following formula: 1fpol(r,θ)=fr(r)fθ(θ), where the term fr(r) is the probability distribution for the distances from the centre of the caldera, and the term fθ(θ) is the angular probability distribution. In this formulation, it is assumed that these two distributions can be considered independent. Indeed, the physical process described in Rivalta et al. (2019) suggests a potential independence between direction of dyke propagation and distance from the caldera centre, as they are controlled by two different features of the caldera, being the distance fundamentally controlled by the size of the nearly circular shape of the caldera, and the direction predominantly controlled by local topographic features. This independence should also leave an empirical track in past vent positions, which can also be formally verified by testing whether the direction and the distances found using past vent positions are correlated to each other.

The probability distribution of Eq. (1) can be transformed into Cartesian coordinates as follows: 2fxy(x,y)=1rfpol(r,θ)=1rfr(r)fθ(θ), where the term fpol(r,θ) is again factorized in its two terms using Eq. (1). Defining an application grid, Eq. (2) can be used to establish a vent probability map by substituting fr(r) and fθ(θ) with the appropriate distributions of potential distance-from-centre and azimuth. Depending on the available data, such distributions may be estimated empirically, looking at the distribution of past vents, including all the data that reflect the present state of the caldera, or analysing the topographical peaks surrounding the caldera.

At Campi Flegrei, an oblate central pressurized melt zone located at the center of the caldera is compatible with many geophysical evidences (e.g. Barberi et al., 1991; Capuano et al., 2013; Castaldo et al., 2021; De Landro et al., 2025). Inflation of a caldera-centered oblate spheroidal magma chamber at a depth of ∼ 3.5 km is consistent with the deformation in the last ∼ 600 years at least (Di Vito et al., 2016; D'Auria et al., 2015; Amoruso et al., 2007, 2014b), and a similar source is likely active at least in the last 5 ka (Di Vito et al., 2016; Rivalta et al., 2019). The position of the caldera centre is here assumed at the point of maximum uplift, here set 800 m south of the GNSS station of Rite at coordinates Easting 426355 and Northing 4518743 (UTM WGS84, zone 33N; Bevilacqua et al., 2024, Fig. 1). To establish an empirical model, we study the empirical distribution of azimuth (angle to the North of the line connecting the centre of the caldera and the vent, hereinafter azimuth) and of the distance from the centre of the caldera (hereinafter distance) of past eruptions. We consider the 71 vent positions of the post-NYT activity (Bevilacqua et al., 2016). The stability of these empirical distributions on the specific location of the caldera center is tested by sampling other possible centers in boxes of 1 and 2 km around the selected centre (Fig. S1 in the Supplement). Past vent positions are also affected by significant uncertainty. To evaluate its potential impact on all statistical analyses, we consider the uncertainty bounds defined in Bevilacqua et al. (2016), accounting for the uncertainty by randomly sampling 1000 alternative synthetic positions uniformly distributed within the defined uncertainty bounds (Fig. S1).

Comparing these results with the main probability maps for Campi Flegrei discussed in the literature – Alberico et al. (2002), Selva et al. (2012), and Bevilacqua et al. (2015), hereinafter indicated as A02, S12 and B15, respectively – interesting coincidences and some significant differences emerge. As already noted, the map produced by A02 differs widely from S12 and B15 maps, having maximum values in the centre of the caldera, which is in contrast with the empirical evidence of recent past vents. In this, M1 and M2 are closer to S12 and B15, providing lower probabilities close to the caldera centre and larger probabilities in the NE area of Astroni and Agnano, where most of past vents concentrate.

To better highlight further similarities and differences, in Fig. 4A–F we report the maps of the relative differences in probability between M1, M2 and A02, S12, and B15, rescaling all of them to the same grid and, for S12 and B15, considering the mean of the epistemic uncertainty (Fig. S8). These maps highlight that the main differences are connected to the two ring regions with relatively higher probability discussed above, which were not present in previous studies.

The outer ring of relatively high probability, at 4 km, which coincides with the area identified by Charlton et al. (2020), includes the highest probability area of S12 and B15, located in the Agnano area to the NE of the caldera. However, also in this area, both M1 and M2 provide larger probabilities (Fig. 4, panels B, C, E, F). This ring also includes the area of Monte Nuovo (toward N, last eruption at Campi Flegrei), and especially M1 provides larger probabilities than previous studies. Also the secondary peak identified in the Averno area (toward NW) in S12 is found in both M1 and M2, but shifted inward and eastward, in the direction of Gauro. The outer ring also generates several other secondary probability peaks at sea in the W direction, in the direction of the topographic peaks of Baia/Bacoli, especially in M1. These peaks are larger than in literature maps. On the other hand, in the areas of Nisida (toward SE) and Capo Miseno (toward SW), the probability values obtained here are relatively lower than the ones in the maps published in literature.

The inner ring of relatively high probability includes the high probability area of Solfatara, present in both S12 and B15, but also in this case both M1 and M2 provide larger probabilities (Fig. 4B, C, E, F). In addition, the inner ring introduces new peaks in probability within the bay, toward W, and in the coastal area of Accademia and La Pietra, toward E, both less pronounced in previous studies.

Also in the outer area of the caldera, external to the outer ring of relatively high probability, M1 and M2 are different than previous studies. Here, M1 and M2 do not assume that vent opening may occur only inside the caldera border, as in S12 and B15, but an outer limit is established only by the maximum observed distance from the centre of the caldera in Epoch 3, which is used to truncated the empirical distribution (Sect. 3.1). This cuts out the most external areas, mainly toward E, NW and S, where both M1 and M2 foreseen smaller probabilities than previous models.

The vent opening maps derived in the study are based on assumptions radically different from the ones in literature. The approach is inspired from the dyke propagation model proposed in Rivalta et al. (2019), which suggests a substantial independence between the radial and azimuthal distributions of dykes propagating from a finite source located at the centre of the caldera. The results confirm some of the features already highlighted in previous studies, but also it introduces important differences. In particular, the most striking difference is that the vent probability peaks in two circles at around 2 and 4 km from the caldera centre, with significant modulations in the different directions, while smaller probabilities are found in the peripheral areas of the caldera. These rings of relatively higher probability, particularly evident in model M1 but present also in model M2, are compatible with in the range forecasted by Rivalta et al. (2019) when adopting only Epoch 3 data (Fig. 4G). However, the propagation distance distribution here produces two distinct peaks at 2 and 4 km, more than a continuous distribution in this range. The inner circle essentially coincides with an area characterized by high seismic activity recorded in the ongoing unrest episode (black dots in Fig. 4H, I), while the outer circle is instead closer to the position of the ring faults surrounding the inner caldera (red lines in Fig. 4H–I, Natale et al., 2024, 2025). These observations are completely independent and seem to spot privileged paths for magma ascent around the inner caldera border. This may be the subject of future studies.

It is important to stress that M1 and M2 are produced using the empirical distributions which are considered relevant for forecasting future occurrences, that is Epoch 3 for distances, and either all epochs or topographic peaks for azimuth. Indeed, while the distribution of the azimuth does not change significantly across epochs, the analysis of the distances with respect to the centre of the caldera has shown that Epoch 1 differs significantly from Epoch 3 in terms of distances. This may be sign of a change in the volcanic source and/or in the active stress field, in agreement with the findings of Rivalta et al. (2019), and with Orsi et al. (2004), who concluded that the last change in stress regime occurred prior to onset of the Epoch 3, and suggested that only the past 5 ka should be considered as reference for the present state of the caldera (Orsi et al., 2009). This assumption was adopted also in Selva et al. (2012).

Regarding the azimuth distribution, we found that the correlation between azimuth and topographic peaks is instead stable across epochs and is maximized considering the topographic peaks within 7 km from the caldera centre, demonstrating that the topography surrounding the caldera may be a good proxy for vent forecasting. In the future, this relationship may be better explored at different scales in space and time, studying for example the potential effects of the varying spatio-temporal morphology due to eruption history, or the impact of different positions of the magma source.

Finally, even if we develop two maps, we prefer not to quantify the epistemic uncertainty on the proposed approach, differently from what was done in previous studies like Selva et al. (2012) and Bevilacqua et al. (2015). The reason is that several case studies recently demonstrated that the effective epistemic uncertainty on a target physical process (here vent opening) is better estimated by combining radically alternative approaches (e.g., defining weighted ensembles of alternative models), rather than by exploring the epistemic uncertainty inherent to one specific approach (Selva et al., 2018; Marzocchi et al., 2021; Meletti et al., 2021, among the others). Consequently, while the epistemic uncertainty on a given approach may be of relative interest, the very development of an alternative approach like the one presented here may be a significant added value to future quantification of epistemic uncertainty in the process of vent opening at Campi Flegrei, via multi-model ensembles or equivalent approaches that combine all the scientifically grounded approaches available in literature (e.g. SSHAC, 1997; Marzocchi et al., 2015, 2021).