📄 earth science

Unlocking Volcanic Swarms: Swarm-Informed ETAS and Plastic Rock Deformation

This study introduces the Swarm-Informed ETAS (SIETAS) model, which incorporates exponential temporal cutoffs and sigmoidal spatial kernels to better characterize volcanic earthquake swarms than standard ETAS models, and validates this approach through global seismic data and a frictional two-block model linking the observed decay to transient rock creep.

Original authors: Anna Tramelli, Cataldo Godano, Vincenzo Convertito, Giuseppe Petrillo

Published 2026-10-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anna Tramelli, Cataldo Godano, Vincenzo Convertito, Giuseppe Petrillo

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Volcanoes are not merely mountains of stone; they are living systems where the ground breathes, shifts, and occasionally erupts. Beneath the surface, the movement of magma and hot fluids creates stress that cracks the rock, sending out tremors that scientists call earthquake swarms. Unlike the familiar pattern of a single massive earthquake followed by a long line of smaller aftershocks, these swarms often appear as a chaotic burst of many similar-sized quakes, clustered tightly in time and space. For decades, scientists have tried to predict and understand these events using standard mathematical models designed for tectonic faults. However, these traditional tools often struggle to capture the unique, fluid-driven nature of volcanic activity, leading to a gap between what the models predict and what actually happens on the ground.

A team of researchers from Italy, France, and the United States has now proposed a new way to look at these volcanic tremors. By analyzing seismic records from six major volcanic systems around the world, including Etna in Italy, Yellowstone in the United States, and Bárðarbunga in Iceland, they developed a refined model that better accounts for how stress and fluids move through volcanic rock. Their work suggests that the timing and location of these earthquake swarms are not random but are governed by specific physical processes related to how the rock deforms and how fluids migrate through cracks.

The researchers began by examining detailed catalogs of earthquakes from six distinct volcanic regions: Bárðarbunga, Etna, the Hawaiian islands, Nevado del Ruiz in Colombia, Vesuvius in Italy, and Yellowstone. They compared the standard model, which treats earthquake triggering as a process that fades away slowly over time without a clear end, against their new approach. The standard model assumes that once an earthquake happens, it can trigger others indefinitely, with the likelihood of a new quake dropping off gradually. The new model, which the authors call a "swarm-informed" approach, introduces two key changes. First, it recognizes that the ability of an earthquake to trigger others in a volcanic swarm does not last forever; instead, the triggering power fades out more quickly after a certain period, like a battery running down. Second, it adjusts how the model views the distance between quakes, acknowledging that in volcanic areas, the spread of seismic activity has a specific, organized shape that differs from the spread seen in normal fault lines.

When the team applied their new model to the data, it provided a significantly better fit for five of the six volcanic systems they studied. The model successfully described the rapid rise and fall of seismic activity and the tight clustering of earthquakes in a way the older model could not. For the sixth system, Hawaii, the new model showed promise in how it matched the physical patterns of the data, even though the computer optimization needed further tuning to confirm it was the absolute best statistical fit. The results indicate that the new model captures the essential behavior of volcanic swarms more accurately, suggesting that these events are driven by a self-sustaining but time-limited process.

To understand why this new model works, the researchers looked at the physical mechanics happening deep underground. They proposed a simple mechanical picture involving two blocks of rock: one that is brittle and prone to breaking, and another that is softer and flows slowly, like a thick paste. When the flowing block moves, it pulls on the brittle block, building up stress. As the brittle block cracks and releases energy, it triggers more cracks nearby. The researchers found that the rate at which these cracks appear follows a pattern where the activity starts strong and then decays, but this decay is not endless; it is cut off by a characteristic time limit. This behavior matches the idea that the rock is undergoing a specific type of slow, plastic deformation, where the material creeps and adjusts to pressure in a way that eventually stops the chain reaction of earthquakes.

The spatial aspect of their model also revealed something important about how these swarms are organized. In the new framework, the likelihood of an earthquake happening drops off sharply after a certain distance from the main cluster, rather than fading away slowly over a vast area. This suggests that the fluids and stress driving the swarm are confined to a specific zone, creating a core of intense activity that does not spread as widely as previously thought. For example, in the case of Nevado del Ruiz, the new model showed that while the central area of activity was similar to what older models predicted, the outer edges of the swarm were much more restricted, indicating that the triggering process is highly localized.

This work does not claim to have solved the mystery of volcanic eruptions or to provide a perfect tool for predicting exactly when a volcano will blow. Instead, it offers a more flexible and accurate statistical framework for describing how earthquakes group together in volcanic environments. By incorporating the specific ways that volcanic rock deforms and how fluids move through it, the new model bridges the gap between abstract mathematics and the physical reality of the Earth's crust. The findings suggest that to truly understand volcanic swarms, scientists must move beyond standard assumptions and embrace models that reflect the unique, time-bound, and spatially restricted nature of these events. This deeper understanding is a crucial step toward better monitoring volcanic systems and interpreting the signals they send before an eruption.

Technical Summary: Unlocking Volcanic Swarms: Swarm-Informed ETAS and Plastic Rock Deformation

Problem Statement
Earthquake swarms on volcanoes are critical precursors to eruptions, reflecting complex interactions between rock deformation, fluid migration, and fault activation. Unlike classical mainshock-aftershock sequences driven by stress redistribution, volcanic swarms often lack a single dominant event and exhibit distinct temporal and spatial clustering. Standard Epidemic-Type Aftershock Sequence (ETAS) models frequently fail to accurately replicate these sequences because their scale-free triggering kernels lack the intrinsic time-bound and spatially restricted characteristics typical of volcanic environments. Specifically, standard temporal kernels (modified Omori law) lack an exponential taper, and standard spatial kernels constrain the transition between the central region and the power-law tail, limiting their ability to reproduce the observed "cored" and sigmoidal geometry of volcanic swarms.

Methodology
The authors introduce the Swarm-Informed ETAS (SIETAS) model, a modified framework designed to better capture volcanic seismicity. The model retains the uniform spatial background and isotropic triggering of standard ETAS but introduces two key modifications to the triggering kernels:

Temporal Kernel: The standard power-law decay is modified to include an exponential cutoff governed by a characteristic relaxation time ( $\tau$ ). The kernel takes the form $g_S(t) \propto \exp(-t/\tau)(t+c)^{-p}$ . This allows the model to behave as a power law at short-to-intermediate times while being exponentially suppressed at long times, making it integrable even for $p \leq 1$ .
Spatial Kernel: Motivated by empirical observations of stacked volcanic swarms showing a uniform short-distance regime followed by rapid decay, the authors introduce a normalized, generalized isotropic spatial kernel based on a powered-logistic profile in logarithmic distance. The density is given by $f_S(r) \propto [1 + (r/L_i)^b]^{-\nu}$ , where $L_i$ is the magnitude-dependent characteristic length, and $b$ and $\nu$ are shape parameters. This formulation allows for a "cored" structure and flexible far-field attenuation, recovering the standard ETAS spatial kernel as a special case.

To provide a physical basis for the temporal kernel, the authors propose a frictional two-block model. In this mechanical interpretation, a brittle block is elastically coupled to a creeping block driven by external loading. The triggering rate is derived as the product of a transient creep response (Andrade-type, providing the power-law factor) and a time-dependent susceptibility that decays exponentially (representing the progressive loss of triggering capacity).

The models were applied to comprehensive seismic catalogs from six volcanic systems: Bárðarbunga, Etna, Hawaii (Mauna Loa–Kīlauea), Nevado del Ruiz, Vesuvius, and Yellowstone. Model performance was evaluated using negative log-likelihood (NLL), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC), with parameters optimized via the Nelder-Mead algorithm.

Key Results

Statistical Performance: For five of the six volcanic systems (Bárðarbunga, Etna, Nevado del Ruiz, Vesuvius, and Yellowstone), the SIETAS model yielded lower NLL, AIC, and BIC values compared to the standard ETAS model, indicating a superior statistical fit.
Hawaii Case: For the Hawaii catalog, the current SIETAS optimization returned a higher NLL than standard ETAS. However, the authors note that because standard ETAS is a limiting case of SIETAS, this result likely indicates a local numerical optimum rather than a true model preference. They conclude that the Hawaii result is inconclusive pending further multistart optimization.
Temporal Decay: The fitted relaxation times ( $\tau$ ) were generally long, suggesting that within the fitted intervals, the exponential taper is weak, but the functional form remains robust.
Spatial Distribution: In the case of Nevado del Ruiz, the SIETAS fit produced a large value for the derived parameter $\nu$ (22.904) coupled with a small $b$ . The authors clarify that this does not imply an anomalously large physical swarm radius; rather, the combined parameters result in a spatial distribution with a central range comparable to standard ETAS but a significantly suppressed extreme spatial tail.
Visual Agreement: Despite statistical nuances in the Hawaii case, visual comparisons of the experimental earthquake occurrence rate against the theoretical SIETAS rate showed excellent agreement across all six systems, including Hawaii.

Significance and Claims
The paper claims that the SIETAS framework provides a more flexible statistical description of earthquake clustering in volcanic environments by explicitly accounting for swarm-like behaviors that standard ETAS formulations miss. The strong agreement between the analytical mechanical formulation (based on transient creep and decreasing susceptibility) and empirical results supports the hypothesis that swarm seismicity is governed by a deformation-controlled, self-exciting triggering process linked to Andrade-type transient creep.

The authors maintain a modest stance regarding the practical application of these findings. They assert that while SIETAS offers a better statistical fit and a physically motivated structure, a better fit does not automatically equate to improved predictive skill. They emphasize that SIETAS should currently be viewed as a flexible statistical framework for characterizing and investigating volcanic-earthquake clustering, rather than as an operational model for eruption forecasting or monitoring. Future work involving temporal and spatial residual analyses and rigorous out-of-sample testing is required to determine if these gains translate into improved forecasting performance.

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