# R9-Probabilistic Hazard Analysis for Tsunamis Generated by Subaqueous Volcanic Explosions in the Campi Flegrei Caldera, Italy

Anita Grezio, Raphaël Paris, Jacopo Selva

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Anita Grezio, Raphaël Paris, Jacopo Selva. R9-Probabilistic Hazard Analysis for Tsunamis Generated by Subaqueous Volcanic Explosions in the Campi Flegrei Caldera, Italy. Sørensen M.B. Probabilistic Tsunami Hazard and Risk Analysis. A cookbook, 14, Springer, pp.459-466, 2026, Mathematics and Planet Earth Series, 10.1007/978-3-031-98115-9_18. hal-05530581

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Chapter 18

# R9—Probabilistic Hazard Analysis for Tsunamis Generated by Subaqueous Volcanic Explosions in the Campi Flegrei Caldera, Italy

Anita Grezio ID, Raphaël Paris ID, and Jacopo Selva ID

## 18.1 Introduction

An eruption event tree is developed to explore the natural variability of potential submarine volcanic explosions on the basis of past volcanic activity and to quantify the tsunami hazard conditional upon the occurrence of an underwater eruption at Campi Flegrei, Italy.

Very few events of this kind are documented, but both instrumental and observational data are available for two case-studies (Myojin-sho 1952 and Karymsky Lake 1996, in Chap. 3).

This recipe requires a good knowledge on past eruptions (sizes of the vents and energy of explosions, inferred from geological data), as well as a good bathymetry of the study area (a minimum resolution of 50 m is recommended). The initialization of the waves is static, i.e. instantaneous, water surface deformation, as explained in Chap. 4. It is recommended to use a depth-averaged weakly dispersive tsunami propagation model.

<footnote>A. Grezio (✉)
Istituto Nazionale di Geofisica e Vulcanologia, Bologna, Italy
e-mail: anita.grezio@ingv.it</footnote>

<footnote>R. Paris
CNRS, IRD, OPGC, Laboratoire Magmas et Volcans, Université Clermont Auvergne, Aubière, France
e-mail: raphael.paris@uca.fr</footnote>

<footnote>J. Selva
Dipartimento di Scienze della Terra, dell'Ambiente e delle Risorse (DiSTAR), Università di Napoli Federico II, Naples, Italy
e-mail: jacopo selva@unina.it</footnote>
This recipe is presented in Paris et al. (2019). Considering that there are no other tsunami studies using this workflow, we may refer to studies of volcanic hazard with a similar approach but related to tephra and pyroclastic density currents (Tierz et al. 2018; Grezio et al. 2020).

The present recipe includes the following ingredients:

* A fully simulation-based hazard framework with a large number of scenarios and comprehensive uncertainty treatment—FW1A or FW1B of Sect. 2.2

* Source selection for volcanic sources—Sect. 3.4

* Source modelling for volcanic sources—Sect. 4.4

* Tsunami modelling—Chap. 5.

## 18.2 Framework Used in Recipe

The framework is fully probabilistic based on a sufficiently large set of scenarios and defined through a Bayesian Event Tree, as commonly adopted for volcanic hazards (Marzocchi et al. 2008, 2010). The analysis is limited to hazard quantification, and provides both individual scenario results and hazard curves. Thus, this recipe produces a hazard quantification compatible to both frameworks FW1A (Fully simulation-based risk assessment) or FW1B (Simulation-based hazard convoluted with vulnerability and loss using numerical integration). The method applied is summarized in Fig. 18.1. Nodes 1–2–3 of the event tree (Fig. 18.1d) can be neglected, since for tsunami-generating explosive eruptions, probabilities are conditioned upon the occurrence of an eruption (Orsi et al. 2009). Node 4 for the vent location assumes that the relative probability distribution is not uniform, eruptions in the middle of the gulf being most probable due to the presence in the main NE-SW trending fault system of the caldera. Node 5 for the eruption size (here, the energy of the explosion) presents the distribution parameters with a decreasing probability for increasing explosion size and links the tsunamigenic capability of explosions with the size of emerged past craters for which estimation of eruption magnitudes and volumes were available. Node 6 for the tsunami generation assumes that effusive eruptions certainly do not produce tsunamis (conditional probability equal to 0 and no epistemic uncertainty) and explosive eruptions certainly produce tsunamis if the vent is located at a depth larger than 10 m (conditional probability equal to 1). Nodes 7–8 for the tsunami intensity and hazard curves are respectively defined by the maximum wave amplitude (Hmax) at 1 m depth in the numerical simulations and by the probability to overcome the tsunami intensity measures in the 421 sites, from Cuma to Positano.

To account for both the aleatory variability of the source space and the intrinsic uncertainty in the tsunami generation and propagation models, we assume that each simulation result is associated with a conditional hazard curve (conditional upon a scenario in that vent location and that size class) described by a log-normal distribution with a sigma equal to 1 (Davies et al. 2017; Selva et al. 2022).
<figure><img src="image_3.png" /><figcaption>**Fig. 18.1** a Physical parameters controlling the initial displacement of the water surface and subsequent tsunami (E: explosion energy; d: explosion depth; D: total water depth around the volcanic cone; R: eruptive vent radius; η₀: initial water surface displacement). b: Locations of the 17 vents for tsunami simulations. c: Craters of the past eruptions at Campi Flegrei (the size of the craters -circles-determine the 3 classes of radii of the eruptive vents). d: Bayesian Event Tree for Volcanic Hazard (light gray: nodes that are not quantified, due to the choice of conditioning the probabilities upon the occurrence of subaqueous explosions). e: Conditional mean hazard maps for the probability threshold = 0.05 (map on the left) and conditional hazard curve (mean and 10th–90th percentiles confidence interval) in one point (black square) within the city of Naples (red lines on the right)</figcaption></figure>

Tsunami generation is controlled by a range of physical parameters: water depth, size of eruptive vent, energy and depth of explosion, and complex magma-water interactions which determine the explosion itself. The Bayesian Event Tree quantifies and propagates this natural source variability uncertainty through the conditional probabilities and provides the epistemic uncertainty in each conditional probability.
The main advantage of this event tree method is the fully probabilistic hazard quantification, decoupling the uncertainties in volcanic phenomena and tsunami generation and propagation issues. Moreover, the Bayesian formulation allows a simultaneous quantification of both aleatory and epistemic uncertainty. A limitation is represented by the fact that the results do not consider absolute unconditional probabilities, which may be obtained by multiplying the conditional hazard by the mean annual rate of eruptions, even if the quantification of this mean annual rate is not trivial as volcanic eruptive sequences are not stationary in time.

## 18.3 Source Identification and Characterization Used in Recipe

Submarine explosion sources (Fig. 18.1b) are positioned in vents located at the geometrical centre of cells generated by a regular grid (1500 m × 1500 m) in order to decrease the computational cost of the subsequent tsunami simulations. Cells are limited to the areas with non-zero conditional probability and water depth capable of generating tsunamis (D > 10 m). Thus, the centres of 17 cells are identified as potential vent locations, with the explosion depth (d) similar to water depth (D) in the range 23–111 m.

Four representative eruption classes are estimated by the past explosive caldera activity in three different periods: 34 explosive eruptions occurred 15–9.5 ka before present (BP), 6 explosive eruptions in 8.6–8.2 ka BP, and 22 explosive and 3 effusive eruptions in 4.8–3.8 ka BP. The four classes are related to different explosion energies and are linked to standard eruptive sizes for which specific probabilistic studies were available. Also, on the basis of the corresponding crater characteristics, we selected 3 crater radii (200 m, 650 m, and 900 m) assuming that both submerged and emerged areas have similar explosive processes.

Important sources of epistemic uncertainty related to the eruption areas exist, due to several major faults in the submerged part of the caldera. In fact, the last eruption of Campi Flegrei (Monte Nuovo, 1538 CE) occurred along these structures. Due to computational cost, we only explored the aleatory variability of the vent location and its crater radius. This limitation was compensated by introducing a rather large uncertainty into simulation results.

We considered only tsunamis generated by explosive eruptions (Fig. 18.1a), specifically at Campi Flegrei, fully accounting for the corresponding source variability. We do not include other volcanic processes that may generate tsunami waves in the same area, such as flank collapses or pyroclastic density currents entering the sea. Given the characteristics of Campi Flegrei volcanism, the contribution of the other sources at Campi Flegrei is probably negligible. However, the contributions of other volcanoes (e.g. Mt. Vesuvius and Ischia) are surely not negligible for the local hazard but are not considered here. By following the “representative scenario” approach, each combination of the 17 vent locations and 3 crater radii (Fig. 18.1b,c)
represents a single scenario with representative parameters. In this way, the intra-
class variability is assumed negligible with respect to the inter-class variability, as a
reasonable first order approximation (Sandri et al. 2016).

**18.4 Source Mechanics Modelling Used in Recipe**

The volcanic explosions are located in the water column with D > 10 m (potential cavities) and an instantaneous upward displacement is formed in each water column, shaping a crater and a steep cone of water in the centre which collapses and turns to a second cylindrical bore (as demonstrated by experimental explosions and numerical simulations).

The energy of the volcanic explosion is estimated following the empirical rela-
tionship E = 3.56 10⁷ R³ given by Sato and Taniguchi (1997) where E is the explosion
energy in Joules and R the radius of the eruptive vent in meters. Using the empirical
formula of Le Méhauté and Wang (1996), the initial surface displacement generated
by the explosion is a parabolic cavity with a vertical steep water rim. The size of the
water cavity (R_c, in meters) is estimated based on explosion energy by the relation-
ship R_c = 0.0361 E⁰.²⁵. The initial maximum vertical water surface displacement η₀
of the cavity rim is a function of the explosion energy at a given depth and is based
on the empirical formula: η₀ = c E⁰.²⁴ where c = 0.0143 if 6.15 × 10⁻⁴ < d/E¹/³ <
1.85 × 10⁻² or c = 0.0291 if d/E¹/³ < 6.15 × 10⁻⁴.

The vent radii R of 200 m, 650 m and 900 m corresponding respectively to
explosion energies E of 2.9 × 10¹⁴ J, 9.8 × 10¹⁵ J and 2.6 × 10¹⁶ J create initial
displacements of the water surface η₀ of 85 m, 200 m and 250 m, respectively.

Also, we assumed that the explosion occurs at the sea floor or at the rim of a
pre-existing eruptive vent. However, the evolution of the vent geometry during the
eruption (crater excavation, growth of the ejecta ring) and location of the explosion
inside the vent might have an influence on tsunami generation.

A limitation comes from the explosion model. In fact, the size of the crater reflects the energy of the most powerful explosion at shallow depth, regardless of the total number of explosions and post-eruptive erosion. In fact, the morphology of the crater might result from multiple vertically- and horizontally-migrating explosions. Thus, assuming that the size of the crater corresponds to the size of the explosion might lead to an overestimation of the explosion energy in some cases.

**18.5 Tsunami Modelling**

The tsunami numerical simulations were performed using the FUNWAVE-TVD code (Shi et al. 2012) solving the fully nonlinear Boussinesq-type wave equations in spherical coordinates. The Boussinesq model includes the effects of amplitude nonlinearity and frequency dispersion in the case of intermediate-to-deep water waves
neglecting the effect of bottom friction. The third-order Runge–Kutta method is
used to advance the model in time and an adaptive time step is chosen following the
Courant-Friedrichs-Lewy criterion (=0.5) to ensure the stability of the scheme for
all simulations. A fourth-order spatial discretization is employed.

In order to prevent reflections from the end-walls at the open sea, we impose a
3 km wide sponge layer at the limit of the computational domain. The spherical grid
covers a bathymetric area of about 3000 km² with a resolution of 39 m in longitude
and 50 m in latitude and is obtained by integrating printed nautical charts with the
available digital elevation and bathymetric models. Wave heights at the coast were
computed at virtual tide gauge stations at the coast. The propagation of the tsunami
in the entire Gulf of Naples takes only 15 min and Pozzuoli is affected in less than
2 min.

The uncertainty in the tsunami propagation model is treated with a log-normal
distribution centred on simulation results, following a common approach adopted in
literature (Choi et al. 2002; Grezio et al. 2017).

Tsunami waves generated by the explosion propagate radially away from the
explosion centre, all over the computational area, with a significant influence of the
bathymetry (e.g. higher wave heights due to shallow banks) and coastal morphology
(e.g. effects of wave refraction around the islands and capes). So, a high horizontal
and vertical resolution of the bathymetry and a well-defined position of the vent is
essential (≤50 m). As the distance from the explosion increases, the arrival of longer-
period waves might predate the short-period waves because they travel faster. This
is a direct effect of frequency dispersion. Wave refraction creates local amplification
that is well reproduced by the numerical models. Thus, the inclusion of tsunami
dispersion is also essential.

One of the main advantages of the adopted framework is that it allows a reasonable
exploration of vent positions, accounting for its uncertainty, and to couple potential
vent position with dispersive tsunami models. However, tsunami wave behaviour
at the coast (e.g. wave breaking, interaction with coastal structures) and inundation
inland were not simulated, because it would require high-resolution terrain models
(typically 1 m). Future simulations could be conducted at local scale, i.e. for areas
of interest (e.g. harbours) using simulations at the larger scale as input.

**18.6 Discussion**

This analysis represents a pioneering study on full PTHA for volcanic sources. The main results of the PTHA are the probabilities (and relative uncertainties) of exceeding given tsunami intensity thresholds (i.e., wave amplitudes at the coast), given the occurrence of a subaqueous eruption. Final conditional hazard curves and the relative epistemic uncertainty are provided in each of the coastal sites of the Gulf of Naples.

The main limitation is the estimation of the conditioning probabilities at eruptions,
which are highly non-stationary in time. The quantification of long-term conditional
hazard is common in volcanic hazard studies and its extension to absolute (unconditional) PTHA may be obtained by multiplying the conditional hazard by the mean annual rate or the probability of occurrence of volcanic eruptions (Selva et al. 2022).

Another limitation is related to the exclusion of other nearby sources of volcanic tsunamis, like pyroclastic flows at Mt. Vesuvius (Grezio et al. 2020) and flank collapses at Ischia.

The adopted approach, being based on standard volcanic hazard quantification methods, reached a quite advanced level of maturity. However, the parametrization of tsunami-related sources is still suffering a significant lack of local data and/or global analogues, increasing the epistemic uncertainty in hazard estimations.

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