Introduction

The investigation of the earthquake source properties in active volcanic regions offers insights into magmatic activity, high-temperature rock mechanics, and hydrothermal fluid circulation that drive the earthquake production during the volcanic activity1. Among the parameters controlling the earthquake source mechanics, the rupture velocity, i.e., the speed at which earthquake ruptures propagate along faults, is intricately linked to dynamic friction properties, stress distribution and drop, energy partitioning (e.g., seismic radiated energy vs. dissipative processes associated with faulting), fault geometry (presence of step-overs, etc.) and the physical properties of rocks (e.g., elastic moduli and fracture toughness2). Measuring and monitoring the rupture velocity and its spatial and temporal changes can offer valuable information about external effects on the earthquake dynamic rupture processes, such as the triggering effect of fluids, changes in pore pressure and in rock strength properties.

Despite the uncertainty associated with rupture velocity estimation, a variety of methods are used to determine this parameter for moderate to large earthquakes (M 5.5+), such as back-projection, waveform inversion, and dynamic modeling3,4,5,6,7. However, when dealing with small earthquakes (M < 4), the rupture velocity is generally assumed uniform and known from theoretical models and set to 90% the value of the shear wave velocity in methods that analyze seismic signals in the frequency- or time-domain8,9,10,11,12.

In these approaches it is generally assumed that a circular rupture model and the rupture velocity is the parameter allowing to determine the fault radius through an inverse relationship with the displacement spectrum’s corner frequency8,13 or by a forward relationship with the body-wave pulse time duration9.

The measurement of stress drop—the difference between the initial stress and the minimum (dynamic stress drop) or final (static stress drop) shear stress—carries significant uncertainty for small magnitude earthquakes. The latter, primarily originates from the lack of resolution of the seismic data, inaccurate estimates of the rupture length and associated fault area14. Investigating how stress drop connects to other source parameters such as fault slip, and rupture velocity provides a comprehensive understanding of seismic phenomena as illustrated by theoretical studies and laboratory experiments9,15,16,17. In these studies, the rupture velocity is intimately related to the stress-drop during the seismic faulting process. Laboratory measurements indicate a positive correlation between average rupture velocity and stress drop16,18,19,20,21, which is explained as a faster propagating fracture being favored by the more available energy related to high stress release during slip. In contrast, several authors have found a negative correlation relationship22,23,24,25 when measurements are performed at the moderate to large earthquake scale. Kanamori and Rivera26 show that, in a wide magnitude range, the product \(\Delta \sigma {{V}_{R}}^{3}\) (\(\Delta \sigma\) is the static stress drop and \({V}_{R}\) is the rupture velocity) scales with a power function of log[seismic moment], that can lead, under specific conditions, to \(\Delta \sigma\) inversely related with \({V}_{R}\).

However, other phenomena implying energy dissipation during the rupture propagation that initiates at a high stress drop level, like a secondary slip front generation or the off-fault plastic damage would slow down the rupture velocity, as caused by the concurrent reduction of available energy for fracture27,28,29,30.

In this study, by combining P- and S-wave measurements, we propose an approach (see “Methods” section) for obtaining estimates of seismic moment and rupture velocity that in turn allow us to infer the fault radius and static stress drop for small to moderate magnitude earthquakes (duration magnitude, 3 ≤ Md ≤ 4.4) that has occurred during the ongoing seismic crisis at Campi Flegrei caldera (Italy). Furthermore, we perform two independent time-domain estimates of stress release, the apparent stress [proportional to the ratio of radiated seismic energy to seismic moment, ref. 31] and Arms stress-drop [proportional to the root-mean square acceleration, refs. 32, 33], that are compared with the static stress drop and rupture velocity to infer characteristic rupture dynamic properties.

Campi Flegrei is a volcanic caldera originated from two huge eruptions that occurred about 39 ky and 15 ky ago and located in the western part of the highly urbanized metropolitan area of Naples (Italy)34. During the past 10ky more than 70 eruptions have been identified in the geological record, the last of which occurred in 1538, after about 4000 years of quiescence, giving rise to a 170 m high, volcanic cone, called Monte Nuovo34. The secular, meter-size, slow, vertical ground up- and down-lift of the inner part of the caldera is the prominent unrest phenomenon at Campi Flegrei and is accompanied by intense seismic activity mostly related to accelerated uplift episodes35. During the past five decades, the main ground uplift phenomena occurred in 1970–72, 1982–84, and 2014–ongoing, reaching about 1 m and 1.8 m of vertical uplift, respectively, as measured at the town of Pozzuoli, the apex of the ground deformation area36. After a period of slow subsidence, since 2011, the phase of ground uplift has restarted reaching about 1.3 m in January 2024, when it proceeded at a rate of 1–2 cm/month37.

Over the past decade, more than 10,000 earthquakes have occurred along a near-ellipsoidal pattern following the inner resurgence area within the caldera ring structure38,39. High-precision earthquake location techniques combined with high-resolution 3D tomography of the shallow caldera structure allowed these authors to image key features of the caldera’s structure as a gas-rich reservoir below 2 km depth which is confined by a deformed caprock at 1–2 km depth and overlying a basement below 3.5 km depth. Most of the seismicity is concentrated between 1 and 4 km depth, with the shallow events (depth smaller than 1–2 km) being located beneath the Solfatara crater and the city of Pozzuoli. In the onshore area, low-magnitude earthquakes predominantly occur within the caprock seal layer at depths of 1–2 km, while beneath Pozzuoli, the gas-enriched reservoir is imaged between 2 and 4 km depth39.

Since 2019, seismicity has increased significantly in number, maximum magnitude, and spatial extent, with offshore events occurring along fault structures that align with the near-elliptical inner caldera ring, bordering the reservoir. Based on the INGV catalog, three events with Md 4+ have occurred at Campi Flegrei till the date of this paper. Notably, larger-magnitude earthquakes—including two events along an ∼E–W-trending fault near Baia–Bacoli and a ∼N–S-trending, subvertical fault near La Pietra—have followed similar rupture trends38,39. However, the largest reported event, the Md 4.6 of March 13, 2025, was in fact a compound sequence consisting of two closely timed shocks (Mw 3.3 and Mw 4.0) and thus does not represent a single large rupture in the same sense40.

The spatial and temporal distribution of seismicity supports a model where earthquakes are primarily driven by changes in loading conditions on pre-existing faults around the caldera39,41. Under the area of maximum uplift at Pozzuoli, the interaction between the gas-rich reservoir and the overlying caprock seal appears to control the migration of fluids and the triggering of earthquakes. Swarms of earthquakes are commonly observed beneath the Solfatara–Pisciarelli vent system, where fluids preferentially ascend through highly fractured rock volumes and induce seismicity42.

Most of the identified causative faults are normal, high-angle, or nearly vertical, consistent with the regional uplift observed across the caldera centre38. The temporal evolution of seismicity is linked to progressive reservoir depressurization, which modifies the stress conditions along the caprock and on faults bordering the caldera ring39. These observations collectively reinforce our mechanistic interpretations of the ongoing unrest at Campi Flegrei.

With the main focus on the larger magnitude events (Md 3+), produced by several hundred-meter radius faults, we investigate the source properties of the earthquakes associated with the unrest phenomena at Campi Flegrei caldera in the period 2020–2025. We determine and interpret the scaling of the seismic source parameters, in particular of the stress drop (static and dynamic) and rupture velocity that also control the earthquake magnitude and maximum ground shaking. For this purpose, we use the data (acceleration and velocity waveforms) provided by 17 seismic stations belonging to three networks: OV-INGV (Osservatorio Vesuviano, Istituto Nazionale di Geofisica e Vulcanologia), RAN-Italian Strong Motion Network of DPC (Dipartimento della Protezione Civile), and ISNet (Irpinia Seismic Network) of the University of Naples Federico II (Fig. 1). The selected waveform data set includes three-component velocity and acceleration records of 56 earthquakes, with duration magnitudes \({M}_{D}=3+\) (moment magnitude \({M}_{w}=2.3-3.6\)), event-station epicentral distances \({R}_{{epi}}=0-10\) km and depths \(z=2-4\) km.

Fig. 1: Map of the Campi Flegrei (Italy) including the locations of the events (2020–2025) and stations belong to three networks of INGV (Istituto Nazionale di Geofisica e Vulcanologia), Italian Strong Motion Network (Rete Accelerometrica Nazionale—RAN), and ISNet (Irpinia Seismic Network).
figure 1

The simplified caldera boundaries79 are shown with black solid lines. Main ground uplift phenomena, up to 1.8 m in the period 2014–2024, are measured at the town of Pozzuoli where most epicenters are located.

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Our method is based on a parametric modeling, time-domain technique named the “LPDT-method” (logarithm of the P-wave displacement amplitude vs time) that has already been validated in several applications using a wide range of small- to moderate-magnitude earthquakes and strong motion datasets worldwide43,44,45,46. In this work, we applied it to both P- and S-waves, resulting in the availability of joint estimates of the rupture velocity, radius, and static stress drop.

Results

This study investigates the rupture characteristics of small-to-moderate magnitude earthquakes occurring during the ongoing uplift phase at Campi Flegrei caldera. By applying a joint analysis of P- and S-waveforms to a dataset of 56 events (Table S1), we determine the source parameters under the assumption of a circular rupture model with uniform propagation speed. To estimate the source parameters of each analyzed earthquake—including seismic moment \({M}_{0}\), rupture radius \(a\), average slip \(\bar{D}\), rupture velocity \({V}_{r}\), and static stress drop \(\Delta \sigma\) —we applied the joint P- and S-wave approach described in detail in the “Methods” section and reported in Table S2.

In addition to the static stress drop \(\Delta \sigma\), we independently calculate two other stress metrics: the apparent stress \({\tau }_{a}\), and the aRMS stress drop. The apparent stress reflects the radiated energy efficiency and is calculated from the total radiated energy (based on both P and S waves) and the seismic moment. The aRMS stress drop, instead, is derived from the root-mean-square acceleration. Both parameters are proxies of the dynamic stress drop according to Boatwright11. These additional estimates allow for cross-validation and a more complete picture of static and dynamic stress release.

In this regard, the instrument response is removed from the raw records in the frequency domain, and a 0.1 Hz high-pass Butterworth filter is applied to displacement waveforms to mitigate low-frequency noise and baseline drift. The norm of the displacement vector (DV-norm) is computed at each station and used for both P- and S-phase analysis (see Fig. S3). The manual phase picking is performed for stations within 10 km of the hypocenter, and low-SNR traces are excluded based on a logarithmic SNR threshold. The initial part of the seismic waves is used to compute the SNR using the logarithmic decibel scale and the squared amplitude ratio.

Major part of the selected events occurred at an average depth of 2.5 km (Table S1), allowing us to adjust the average medium parameters accordingly for homogeneous half-space Earth’ model. According to a smoothed P and S velocity model, drawn from the velocity model used by the seismic laboratory at INGV-Osservatorio Vesuviano47, the average P-wave velocity up to this depth is 3 km/s, resulting in 1.4 km/s for the average S-wave velocity.

To account for anelastic attenuation in our analysis, we use a range of quality factor values (Qp and Qs between 30 and 120) derived from the tomographic model of Calò and Tramelli47. Source parameters are then estimated for each Qp–Qs pair.

Following Zollo et al.43, uncertainty estimates for the source parameters are obtained by propagating the observational errors from fitting the LP(S)DT curves. We quantified the relative uncertainties of the key source parameters across the catalog. Indeed, this approach accounts for both observational variability and the effects of anelastic attenuation (Qp and Qs variability), ensuring that attenuation does not bias the source estimates. For example, variability in Qp and Qs values (ranging from 30 to 120) introduces median uncertainties of ~0.05 and 0.26 log units, respectively, in M₀ estimates. However, after applying the attenuation correction in a post-processing step, the median relative uncertainties are ~6–8% for moment magnitude, and 27–52% for source radius and slip. For stress drop, which exhibits a greater variability, the uncertainty is more appropriately expressed in logarithmic terms, with a standard deviation of 0.82 log units. Finally, uncertainty in Vr, mainly driven by corner time estimation is typically 7–19%. These values were obtained by computing the ratio of each parameter’s uncertainty to its nominal value and summarizing the distribution using the 25th and 75th percentiles. Thus, the reported uncertainties reflect both observational error and the influence of Q variability on the final corrected estimates.

Figure 2 illustrates the scaling relations of the estimated rupture radius, average rupture velocity, and average fault slip as a function of the seismic moment for the analyzed earthquakes. The earthquake ruptures occurring in the brittle and shallow volcanic structure of the Campi Flegrei caldera, propagated at sub-shear average velocities (\({v}_{R}\) = 0.4–0.9 Vs) along 100–600 m radius fault surfaces. As listed in Table S2, all rupture velocities estimated from the joint P and S phase analysis are below 1.2 km/s with no events exceeding the shear wave velocity (\(\left\langle {V}_{S}\right\rangle =1.4-1.6{\rm{km}}/{\rm{s}}\)). Notwithstanding the limited moment magnitude range (Mw between 2 and 4), the average slip (0.05–10 cm) varies over about two orders of magnitude. Even considering the parameter uncertainties, the rupture radius (area) and the average slip do not follow a constant stress drop scaling with seismic moment, which is evidence for the violation of the self-similarity mode of the rupture mechanism of the largest earthquakes occurring in the present seismic crisis.

Fig. 2: Joint P- and S-wave analysis is done to obtain rupture velocity, rupture radius and average fault slip as a function of the seismic moment for the analyzed earthquakes.
figure 2

a Shows the ratio \(\frac{{V}_{R}}{{V}_{S}}\) vs log \({M}_{o}\) and \({M}_{W}\). b, c Illustrate the scaling of source radius and slip vs seismic moment. In the (b, c), the lines represent the theoretical scaling using the Keilis–Borok67 formula and assuming the constant static stress drop values from 0.01 to 10 MPa.

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Our results are broadly consistent with those reported by Iervolino et al.48, though key methodological differences lead to divergent interpretations. Despite analyzing a narrower magnitude range (Md > 3.0, compared to their Md > 2.5), we find close agreement in the moment magnitude–duration magnitude relationship (Text S1; Fig. S1). However, our modeling approach differs in several important aspects. Iervolino et al.48 assumes a rectangular rupture geometry and adopts a fixed rupture velocity of 0.9Vs. In contrast, we use a circular rupture model and estimate rupture velocity (\({Vr}\)) individually for each event using both P- and S-wave measurements. This approach enables us to resolve dynamic rupture characteristics that are not captured in models with constant Vr. Regarding stress drop, Iervolino et al.48 report self-similar scaling (stress drops ranging from 0.1 to 10 MPa), in contrast, our results indicate a breakdown of self-similarity. In summary, while both studies report consistent magnitude scaling and overlapping stress drop ranges, our methodology provides a more detailed, dynamic perspective on rupture processes—including variable Vr and scaling deviations—that are critical for understanding the evolving seismogenic behavior at Campi Flegrei.

A comparison is also made with the source parameters reported by Pino et al.49 for the May 20, 2024, Md 4.4 event, the only one in our dataset for which a formal source inversion has been performed. Our results for moment magnitude and rupture velocity are consistent with those of Pino et al.49. While both studies assume a circular rupture geometry and find comparable moment magnitude (Mw = 3.85  ±  0.05 vs. our Mw = 3.6  ±  0.15) and rupture velocities (~0.6 km/s), our time-domain approach leads to a smaller estimated source radius and, consequently, a higher stress drop. These differences likely reflect the distinct data types and modeling techniques employed, highlighting the methodological sensitivity of source parameter estimates.

Figure 3 shows the variation of the static stress drop, together with apparent stress (\({{\rm{\tau }}}_{{\rm{a}}}\)) and Arms stress drop (\({\Delta {\rm{\sigma }}}_{{\rm{aRMS}}}\)) versus the seismic moment. \({{\rm{\tau }}}_{{\rm{a}}}\) and \({\Delta {\rm{\sigma }}}_{{\rm{aRMS}}}\) are independent estimates of the static stress drop (Boatwright11 and discussion below). In the narrow-explored moment magnitude range (Mw 2–4), the static stress drop, and \({\Delta {\rm{\sigma }}}_{{\rm{aRMS}}}\) vary significantly (0.05–10 MPa), while \({{\rm{\tau }}}_{{\rm{a}}}\) varies in a smaller range. Typical uncertainties associated with these parameters are also evaluated assuming a log-normal distribution. The standard deviation in log10 space is ~0.49 for \({{\rm{\tau }}}_{{\rm{a}}}\) and 0.65 for \({\Delta {\rm{\sigma }}}_{{\rm{aRMS}}}\). For comparison, the stress drop (\(\Delta {\rm{\sigma }}\)) shows greater variability, with a log10 standard deviation of 0.82. These estimates reflect the combined effects of variability in energy estimation, acceleration peak measurement, and methodological assumptions, and are consistent with the broader range of uncertainties observed in stress drop studies14.

Fig. 3: Stress parameters of Campi Flegrei earthquakes.
figure 3

Three independent measured stress released parameters, a apparent stress \({\tau }_{a}\), b Arms stress drop \({\varDelta \sigma }_{{aRMS}}\), and c static stress drop \(\Delta \sigma\) versus the seismic moment (bottom axis) and moment magnitude (top axis). d comparison of the bin-averaged values for \({\tau }_{a}\), \({\varDelta \sigma }_{{aRMS}}\), and \(\Delta \sigma\). Freedman–Diaconis rule51 is used to bin the data.

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Figure S6 compares static stress drop (\({\mathbf{\Delta }}{\boldsymbol{\sigma }}\)) with the apparent stress (\({{\boldsymbol{\tau }}}_{{\boldsymbol{a}}}\)) and the aRMS stress drop (\({{\mathbf{\Delta }}{\boldsymbol{\sigma }}}_{{\boldsymbol{aRMS}}}\)). Both \({{\boldsymbol{\tau }}}_{{\boldsymbol{a}}}\) and \({{\mathbf{\Delta }}{\boldsymbol{\sigma }}}_{{\boldsymbol{aRMS}}}\) systematically fall below \({\mathbf{\Delta }}{\boldsymbol{\sigma }}\) values, typically by one to two orders of magnitude. Apparent stress values are particularly low, suggesting low radiation efficiency and significant energy loss during rupture. aRMS stress drop is somewhat closer to \({\mathbf{\Delta }}{\boldsymbol{\sigma }}\) than \({{\boldsymbol{\tau }}}_{{\boldsymbol{a}}}\) but still systematically lower, reflecting moderately abrupt but relatively inefficient ruptures at high frequencies. Since both apparent stress and RMS stress drop can be considered proxies of the dynamic stress drop11, their systematic underestimation of static stress drop is consistent with a positive overshoot mechanism50 possibly indicating additional after-slip fracture energy dissipation. This is further evidence for the fraction of radiation energy being much smaller than the amount spent on new fracture creation. Despite the data scatter, \({{\boldsymbol{\tau }}}_{{\boldsymbol{a}}}\) and \({{\boldsymbol{\Delta }}{\boldsymbol{\sigma }}}_{{\boldsymbol{aRMS}}}\) show a co-variation with \({\boldsymbol{\Delta }}{\boldsymbol{\sigma }}\) that suggests a constant dynamic overshoot mechanism over about three orders of magnitude. Thus, the corresponding dynamic overshoot, calculated as \({\bf{0}}{\boldsymbol{.}}{\bf{5}}-\frac{{{\boldsymbol{\tau }}}_{{\boldsymbol{a}}}}{{\boldsymbol{\Delta }}{\boldsymbol{\sigma }}}\)50\(,\) yields a value of ~0.46, suggesting that an additional dynamic stress release amounting approximately to 46% of the static stress drop occurs during rupture process.

Despite the measure variabilities, the \({\varDelta \sigma }_{{aRMS}}\) and \({\tau }_{a}\) reveal an inverse log-linear scaling with rupture velocity (Fig. 4). A statistical analysis using the Pearson correlation coefficient confirms that log[\({\varDelta \sigma }_{{aRMS}}\)] and log[\({\tau }_{a}\)] are anti-correlated with Vr at a statistical significance level of 5%. To find the underlying trend between rupture velocity and stress parameters, we binned the data using the Freedman–Diaconis rule51, which selects bin width based on data spread and sample size, ensuring statistically and physically meaningful aggregation. Despite the significant scatter in \({\tau }_{a}\) and \({\varDelta \sigma }_{{aRMS}}\), primarily due to parameter uncertainties and rupture complexity, both stress measures display a statistically significant inverse correlation with Vr (Pearson correlation coefficient ~ –0.6 to –0.7, p < 0.05). This inverse trend is robust within the reported uncertainty bounds and suggests a consistent relationship between radiation efficiency and rupture velocity.

Fig. 4: Relationship between rupture velocity and stress parameters.
figure 4

Rupture velocity versus a, c apparent stress and b, d ARMS stress drop estimates. The black line represents the best-fit relationship, applied to the bin-averaged stress release values (displayed by boxplots). Data were binned using the Freedman–Diaconis rule51, which selects bin width based on data spread and sample size. a, b Include all events, while c, d show the results after excluding earthquakes with M > 3.2. This comparison highlights that the observed anti-correlation is not only controlled by the largest events.

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Discussion

A key outcome of the analysis is the inverse relationship observed between rupture velocity and stress drop—reflected in independent estimates of aRMS stress drop and apparent stress. This trend may point to dynamic interactions between energy release and rupture propagation within the complex mechanical setting of a volcanic caldera. The event-specific estimates of rupture velocity (Vr) in this study reveal significant variability even among relatively small-magnitude earthquakes. This observation supports the notion that rupture velocity is not constant or uniform across events, in agreement with previous studies of small earthquakes that report complex rupture processes and directivity effects52,53,54.

As the magnitude of the vertical stress increases with hypocentral depth, and the earthquake ruptures occurred within the shallow brittle volcanic layers (max depth of 3–5 km), the observed relatively low (<1 MPa) stress drop estimates could be a result of a lower vertical stress. Moreover, the presence of a vigorous hydrothermalism, with intense hot fluid circulation in the very shallow volcanic layers call in cause the mechanisms of pore-pressure increase (effective stress reduction) and/or fault lubrication, that can favor the occurrence of fault reactivation under low differential stress levels and possibly causing low stress drops. Fluid effects go beyond pore pressure, as the fluid permeating the fault zone can also affect stress drop. In the presence of fluids, theoretical models and rock deformation experiments predict faults to weaken by thermal and mechanical pressurization of fault fluids55,56 thus facilitating fault slip and contributing to seismicity production. Furthermore, the type of fluid, its physical state, viscosity, and the degree of saturation in the fault zone all contribute to stress drop variations57,58,59.

Boatwright11 concluded that the \({\varDelta \sigma }_{{aRMS}}\) is strongly correlated with the Brune stress drop, an estimate of the average dynamic stress drop, similar to \({\tau }_{a}\) that is proportional to the dynamic stress drop8. Following Boatwright11 definitions, we can consider \({\varDelta \sigma }_{{aRMS}}\) and \({\tau }_{a}\) as proxies of the dynamic stress drop that is compared with the LPDT static stress-drop estimate (\(\Delta \sigma\)), which grounds on reliable estimates of the source radius through precise rupture velocity measurements. The bin-averaged plot (Fig. 3d) shows a near-parallel increasing trend with seismic moment for three stress release quantities, with \({\tau }_{a}\) significantly smaller than other two ones. This is evidence for a dominant dynamic overshoot process (i.e., the final stress is lower than the dynamic coseismic shear strength, see ref. 60) during the rupture process in the Campi Flegrei, that implies very low radiation efficiency (\({\eta }_{{SW}}\)), as it is shown in Fig. 5. The ratio of apparent stress to static stress drop is as a proxy for Savage-Wood seismic radiation efficiency (\({\eta }_{{SW}}=\frac{{2\tau }_{a}}{\Delta \sigma }\))61, offering valuable insight into the proportion of energy radiated during earthquake stress release. The observed radiation efficiency is low with a median value of about 0.1 and a confidence interval of [0.02, 0.3], suggesting that only a small portion of the total energy goes into seismic radiation, with the majority dissipated by friction- and fracture-related processes in the fault zone (fault core + damage zone sensu62,63). These coseismic processes contribute to the development of the fault damage zone64 and to changes in its physical properties, including stiffness65,66. Such changes affect the dynamics of individual seismic ruptures (e.g., rupture velocity66,67) and the longer-term seismic cycle (fault healing and sealing68).

Fig. 5: Energy partitioning and radiation efficiency of Campi Flegrei earthquakes.
figure 5

The “Savage-Wood” seismic radiation efficiency (\({\eta }_{{SW}}=\frac{{2\tau }_{a}}{\Delta \sigma }\)), is plotted as a function of seismic moment in a log-log scale, with a histogram of observed values. The dashed line at \(\eta =0.5\) represents the threshold distinguishing undershoot from overshoot in dynamic weakening mechanisms. Black circles represent the bin-averaged values.

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Savage and Wood69 suggest that seismic radiation efficiency may range between 0 and 0.5 for this parameter. The observed low radiation efficiency indicates that substantial dynamic overshoot (elevated dynamic shear strength) likely plays a central role in driving microearthquake fracturing in the studied area.

Conversely, the observed inverse scaling relationship between rupture velocity and dynamic stress parameters in the earthquakes at the Campi Flegrei caldera (Figs. 3, 4) also points to a mechanism of significant energy dissipation during seismic faulting. Off-fault damage formation can play a crucial role in this process by reducing the energy available for rupture propagation23,70,71. The anticorrelation between stress drop and rupture velocity can be explained by the coseismic expansion of plastic off-fault damage, particularly in highly stressed wall rocks, which limits the energy available for rupture propagation.

As the off-fault damage zone expands, it causes a range of effects that ultimately influence fault rupture processes66,67,72,73. First, it dissipates the available elastic strain energy in the surrounding rock, thereby reducing the overall capacity for high-intensity rupture71. Additionally, this damage lowers the stiffness of the fault zone, which leads to slower rupture speeds66,71. Furthermore, expansion of the damage zone triggers a rotation of the principal stresses, altering the stress field around the fault65,67. In the volcanic environment of the Campi Flegrei caldera, energy dissipation associated with off-fault damage appears to be the primary driver of the inverse relationship between rupture velocity and stress drop (Figs. 3, 4), as well as the observed low radiation efficiency and overshoot dynamic weakening (Fig. 5). These processes lead to more extensive off-fault damage in higher stress-drop events, which can hinder the development of long ruptures thereby limiting the maximum achievable earthquake magnitude.

To further evaluate the influence of the few largest earthquakes (M > 3.2) on the observed anti-correlation between rupture velocity and stress parameters, we repeated the analysis after excluding those events (Fig. 4c, d). The inverse trend remains visible, although it weakens slightly for rupture velocity versus apparent stress: the correlation coefficient decreases from –0.6 (all events) to –0.3 (excluding M > 3.2). In contrast, the anti-correlation with ARMS stress drop remains essentially unchanged. This indicates that, while larger events tend to contribute to the higher end of apparent stress values, the overall inverse relationship is not an artifact of a few outliers. Instead, it reflects a systematic property of the dataset across the studied magnitude range.

Conclusions

In this study, we analyzed 56 earthquakes (Md 3.0–4.4) at the Campi Flegrei caldera, revealing critical insights into their source dynamics (Table S1, Supplementary Material). Using circular rupture models and joint P and S waveform analyses, we identified an inverse relationship between rupture velocity and stress drop, possibly influenced by the unique volcanic structure (Fig. 4). Shallow hypocentral depths (3–5 km) and hydrothermal activity likely contribute to low stress drops (<1 MPa), with fluid effects—such as pore-pressure increase and thermo-mechanical pressurization—further facilitating fault slip at reduced yield stress levels.

We observed low seismic radiation efficiency (median ηSW ≈ 0.1, Fig. 5), suggesting most energy dissipates through friction and fracture processes, especially associated with the formation and expansion of fault damage zones, rather than seismic radiation. We observe a slight decrease in radiation efficiency with increasing moment magnitude (Fig. 5), suggesting that larger magnitude events at Campi Flegrei dissipate a relatively greater portion of their energy budget through non-radiative mechanisms—such as frictional heating, and inelastic deformation in the fault core and especially in the fault damage zone (off-fault damage)—rather than seismic radiation. Since stress release estimates show a general increase with magnitude within the explore magnitude range (Fig. 3) we interpret the apparent decreasing seismic efficiency with magnitude as related to fracture energy dissipation mechanism through the production of a larger off-fault damage for high-stress drop/high magnitude earthquakes. Dynamic overshoot and off-fault damage were key drivers of energy dissipation, particularly in high stress-drop events, limiting rupture velocity and maximum earthquake magnitude. The comparison between static stress drop, apparent stress, and aRMS stress drop (Fig. 3 and S6) further confirms the inefficiency of seismic radiation, with apparent and aRMS stress values systematically lower than static stress drop values. These findings emphasize the critical role of energy dissipation mechanisms in volcanic settings and highlight how local geological structures and hydrothermal conditions at Campi Flegrei significantly influence rupture behavior and seismic hazard. The process of more extensive off-fault damage and fracture energy dissipation during high stress-drop events, can prevent the development of long ruptures, thus imposing a limit to the maximum achievable earthquake magnitude, that during the ongoing crisis has never exceeded a duration magnitude 4.4 (or Mw 3.6 ± 0.2).

Method

Source time function (STF)

Assuming an elastic heterogeneous, half-space Earth’ model, a point-source approximation describes a seismic source of small to moderate earthquakes whose Source Time Function (STF) has a triangular shape. The overall shape of this triangular STF, presumed generally to be a scalene, strongly depends on the rupture radius, speed and wave-type (Fig. S4). Figure S4 shows that the rupture velocity is a significant determining factor in the definition of time to the peak amplitude and total duration of the STF as the two main parameters to adjust the shape of the scalene triangular STF.

If rupture propagates symmetrically within a circular fault (radius of \(a\)) with uniform rupture velocity (\({V}_{r}\)) and average slip of \(\bar{D}=\frac{{M}_{o}}{\mu \Sigma }\) in which \(\mu =\rho {V}_{s}^{2}\) is rigidity and \(\Sigma =\pi {a}^{2}\) is fault surface, the average time to the peak amplitude (\({\bar{T}}_{\max }^{{ph}}\)) and total duration (\({\bar{T}}_{D}^{{ph}}\)) of the rupture across all possible directions of ray propagation are given by refs. 44, 74:

$${\bar{T}}_{\max }^{{ph}}=a\left(\frac{1}{{V}_{r}}-\frac{2}{\pi {V}_{{ph}}}\right)$$
(1)
$${\bar{T}}_{D}^{{ph}}=a\left(\frac{1}{{V}_{r}}+\frac{2}{\pi {V}_{{ph}}}\right)$$
(2)

where “\({ph}\)” stands for the seismic phase chosen for the analysis i.e., P- or S-wave, \({V}_{{ph}}\) is phase velocity at the source location, and \(a\) is the source radius of the circular crack which can be computed from Keilis-Borok75 expression for a given static stress drop value (\(\Delta \sigma\)):

$$a={\left(\frac{7}{16}\frac{{M}_{0}}{\Delta \sigma }\right)}^{1/3}$$
(3)

Since the STF represents the history of moment release during an earthquake, its area is equal to the scalar seismic moment \(({M}_{0})\), which can be determined using far-field ground displacement \(({u}_{ph})\) equation at the fault surface for a couple of source and receiver located at \(\xi\) and \(x\) respectively74:

$${M}_{0}=\frac{4\pi }{{F}_{s}{{\mathcal{R}}}_{{ph}}^{\theta \varphi }\left(\xi ,x\right)}{\rho (x)}^{1/2}{\rho (\xi )}^{1/2}{{V}_{{ph}}(x)}^{1/2}{{V}_{{ph}}(\xi )}^{5/2}G\left(\xi ,x\right)\int {u}_{{ph}}\left(t\right){dt}$$
(4)

where \({F}_{s}\) is free surface factor, \({{\mathcal{R}}}_{{ph}}^{\theta \varphi }\left(\xi ,x\right)\) is radiation pattern, depending on the fault mechanism and direction from source to receiver, \(\rho (x)\) and \(\rho (\xi )\) are density at the receiver and source respectively, and \(\int {u}_{{ph}}\left(t\right){dt}\) is the area beneath the displacement pulse, e.g., half-product of maximum amplitude (\({\overline{{Pd}}}^{{ph}}\), height) and duration (\({\bar{T}}_{D}^{{ph}}\), base) of the assumed scalene triangular STF. \(G(\xi ,x)\) is a geometrical spreading correction, defined as \(\sqrt{\frac{\rho (\xi ){V}_{{ph}}(\xi )}{\rho (x){V}_{{ph}}(x)}}\frac{X}{\sin {j}_{h}}\), in which \(X\) is epicentral distance and \({j}_{h}\) is the take-off angle.

The average far-field radiation patterns (\({{\mathcal{R}}}_{{ph}}^{\theta \varphi }\)) can be simply assumed as values of 0.52 and 0.63 for P- and S-wave, respectively76. Alternatively, they can be estimated as an average value of all individuals at each receiver using the Aki and Richards74 formulae which require parametric information on the rupture focal mechanism (i.e., strike, dip, rake, hypocentral depth), epicentral distance, azimuth of the station concerning event epicenter, and 1D velocity model to calculate the take-off angles.

Single phase analysis to build up the corresponding STF, SP-STF

To construct the STF from each body wave independently, we applied a time-domain, average-based method that builds a proxy for the STF, called the LPDT curve. This approach has previously been used to estimate source parameters from vertical P-wave displacement signals recorded by local accelerometers and velocimeters43,44,45,46. Instead, in the current work to have more complete view of the earthquake source, we jointly analyzed P- and S-phases using the norm of the displacement vector (hereafter referred to as DV-norm) i.e., \(\sqrt{{E}^{2}+{N}^{2}+{Z}^{2}}\) (Figure S3).

As illustrated in Fig. 1 of Nazeri and Zollo45, the method follows a clear sequence of steps, which are summarized in the following paragraphs. Once the DV-norm signals are aligned concerning the first arrival time of the P- or S-wave, they are simply multiplied by the corresponding R, the hypocentral distance, to remove any geometric attenuation (hereinafter, distance-corrected DV-norm). Then, an informative curve is constructed by stacking and then keeping the maximum amplitude of the logarithm of the distance-corrected DV-norm signals (shown with black and red curves in Fig. 6), i.e., LPDT or LSDT depending on using the P- or S- phases.

$${LP}(S){DT}\left(t\right)=\max {\left\{\left\langle \log ({\rm{distance}}-{\rm{corrected\; DV}}-{\rm{norm\; signals}})\right\rangle \right\}}_{{t}_{0}}^{t}$$
(5)
Fig. 6: Time-domain analysis of DV-norm signals and source time function fitting.
figure 6

a Initial part of the DV-norm signals for both P- and S-waves sorted by hypocentral distance, b logarithm of the distance corrected DV-norm signals (gray curves) and the average ones mentioned as LPDT and LSDT (black solid line) curves. Red curves keep the maximum amplitudes of the curves in the expanding time window, used to find the best fits. c LP(S)DT curves and their best fit (blue curve). In the subplots, the SP-STF has been constructed using two rupture velocity values, 1. \({V}_{r}=0.9{V}_{S}\), the dotted STF, and 2. \({V}_{r}\) obtained directly from joint phases analysis (solid lines).

Full size image

Nazeri et al.43 demonstrated (see Fig. 2 in this paper) that this specific curve—characterized by a smooth, ramp-like shape—can serve as a reliable proxy for the STF. In this approach, the corner time (\({T}_{c}\)) and plateau level (\({P}_{L}\)) of the curve are linked to the parameters of the STF: the maximum amplitude (\({\overline{{Pd}}}^{{ph}}={10}^{{P}_{L}}\)) and its relevant time (\({\bar{T}}_{\max }^{{ph}}={T}_{c}\)).

The corner time (\({T}_{c}\)) and plateau level (\({P}_{L}\)) are estimated by analyzing the curvature of the fitted function; specially, \({T}_{c}\) corresponds to the point where the curvature tends toward zero (indicated by black circles in Fig. 6c).

To model these curves, we apply the empirical function introduced by Nazeri and Zollo45:

$$y\left(t\right)=y(0)+{P}_{L}\left[1-\sqrt{\frac{1}{1+{\left(\frac{t}{{T}_{L}}\right)}^{\gamma }}}\right]$$
(6)

where \(y(0)\) represents the initial point of the curve, \({P}_{L}\), \({T}_{L}\), and \(\gamma\) are the fitting parameters representing the plateau level, characteristic time, and shape exponent, respectively (blue curves, Fig. 6c).

Finally, to retrieve the attenuation-corrected, scalene triangular STF, the parameters \({\overline{{Pd}}}^{{ph}}\) and \({\bar{T}}_{\max }^{{ph}}\) are corrected for the anelastic attenuation effect assuming a frequency- independent Q model, resulting in \({\overline{{Pd}}}_{Q}^{{ph}}\) and \({\bar{T}}_{\max -Q}^{{ph}}\) via a post-processing procedure described in Zollo et al.44. This correction is done following a forward-modeling strategy based on the linear Q model of Kjartansson77. First, we simulate the effect of Q-filtering by convolving triangular STFs with attenuation operators. Then. a global search algorithm identifies the best-fitting triangle parameters that match the observed LP(S)DT curves. This approach avoids instability from deconvolution and enables robust correction for attenuation.

Once the pair of attenuation-corrected parameters (\({\overline{{Pd}}}_{Q}^{{ph}}\) and \({\bar{T}}_{\max -Q}^{{ph}}\)) is available for each phase, the full set of source parameters i.e., duration \({\bar{T}}_{D}^{{ph}}\), seismic moment \({M}_{0}\), average slip \(\bar{D}\), static stress drop (\(\Delta \sigma\)), and rupture radius \(a\) can be consequently calculated using the relevant equations. The average source parameters can be then computed from the individual phase measurements. In this method, the correction for a constant-Q, anelastic attenuation effect is performed using pre-existing estimations of the P and S quality factor for the investigated propagation medium.

Joint phase analysis, rupture velocity estimation

The advantage of using both seismic phases with this straightforward time domain technique is that it enables the estimation of rupture velocity—an important source parameter that is often difficult to be determined without relying on more complex methods such as full waveform inversion. Since, applying this method, from the specific curve \({\bar{T}}_{\max -Q}^{{ph}}\) is the main observation, in the following, expressions 1 is re-written for both phases to build up a system consist of two linear equations and 2 unknown variables of \(a\) and \({V}_{r}\):

$$\left\{\begin{array}{c}{\bar{T}}_{\max -Q}^{P}=a\left(\frac{1}{{v}_{r}}-\frac{2}{\pi {v}_{P}}\right)\\ {\bar{T}}_{\max -Q}^{S}=a\left(\frac{1}{{v}_{r}}-\frac{2}{\pi {v}_{S}}\right)\end{array}\right.$$
(7)

Solving this system leads us to the following equations for \(a\) and \({V}_{r}\):

$$\left\{\begin{array}{c}a=\frac{\pi }{2}\left(\frac{{V}_{P}{V}_{s}}{{V}_{P}-{V}_{S}}\right)\left({\bar{T}}_{\max -Q}^{P}-{\bar{T}}_{\max -Q}^{S}\right) \hfill \\ {v}_{r}=\pi {V}_{P}{V}_{s}{\left[\left({V}_{P}-{v}_{S}\right)\left(\frac{{\bar{T}}_{\max -Q}^{P}+{\bar{T}}_{\max -Q}^{S}}{{\bar{T}}_{\max -Q}^{P}-{\bar{T}}_{\max -Q}^{S}}\right)+\left({V}_{P}+{V}_{S}\right)\right]}^{-1}\end{array}\right.$$
(8)

It is obvious from these equations that to have a reasonable solution from the joint analysis, \({\bar{T}}_{\max -Q}^{P}\) should be greater than \({\bar{T}}_{\max -Q}^{S}\).

Stress released

Alongside the static stress drop (\(\Delta \sigma\)), the variability in stress changes is assessed by calculating two additional stress parameters of apparent stress (\({\tau }_{a}\)), and root mean square (\({aRMS}\)) stress drop (\({\varDelta \sigma }_{{aRMS}}\)). Apparent stress (\({\tau }_{a}\)) quantifies the efficiency of energy radiation during an earthquake, calculated as the ratio of radiated seismic energy to seismic moment, scaled by the rigidity of the surrounding rock. The standard method of Wyss and Brune31 is followed using the definition of:

$${\tau }_{a}=\mu \frac{E}{{M}_{0}}$$
(9)

where \(E\) represents the radiated seismic energy, defined based on average energy of both P and S waves as \(E=\left\langle {E}_{P}\right\rangle +\left\langle {E}_{S}\right\rangle\). According to Boatwright and Fletcher78, radiated energy from a single phase recorded at a station (\(i\)) located at hypocentral distance of \({R}_{i}\) is obtained in time domain by:

$${E}_{{Phase},i}=4\pi \rho {V}_{{Ph}}{{C}_{\theta \varphi }}^{2}{R}_{i}^{2}\int {v}_{i}^{2}\left(t\right){dt}$$
(10)

where \({C}_{\theta \varphi }\) is a correction for radiation pattern coefficient and free-surface amplification, and \({v}_{i}\) is ground velocity corrected for anelastic attenuation (see Text S4).

Arms stress drop (\({\Delta \sigma }_{{aRMS}}\)) measures the average reduction in shear stress on the fault, reflecting the intensity of stress released per unit area during fault slip, calculated from Baltay et al.33, formula, independent to the corner frequency of the seismograms by:

$${\Delta \sigma }_{{aRMS}}={\left({a}_{{RMS}}\frac{106 \, \rho \, R}{2\,{R}_{\theta \varphi }{\left(2\pi \right)}^{2}}\right)}^{\frac{6}{5}}{\left(\,\frac{{V}_{S}}{{f}_{\max }}\right)}^{\frac{3}{5}}{\left(\frac{1}{8.47{M}_{0}}\right)}^{\frac{1}{5}}$$
(11)

where \({a}_{{RMS}}\) is root mean square acceleration, defined by the formula below; \({V}_{S}\) is the shear-wave velocity of the medium; and \({f}_{\max }\) is the maximum observable frequency, up to which the acceleration spectrum remains flat (Fig. S6).

$${a}_{{RMS}}=\sqrt{\frac{1}{{T}_{d}}{\int }_{0}^{{T}_{d}}{\left|a(t)\right|}^{2}{dt}}$$
(12)

here \({{\boldsymbol{T}}}_{{\boldsymbol{d}}}\) is the duration of the shaking (see Text S3, Fig. S2) and a(t) is the acceleration time series.