Abstract
The Alto Tiberina Fault system, located in Central Italy, is an active structure about 60 km long composed of a principal lowangle normal fault and several minor synthetic and antithetic splays. The system is monitored by a dense seismic network, giving us the opportunity to construct highdefinition seismic catalogs with a low completeness magnitude. We analyze the clustering properties of the 20102015 seismicity by using a 3D stochastic declustering algorithm that also includes the earthquakes’ depth. We demonstrate that the earthquake size distribution is strongly correlated with the clustering of seismic events and their depth; in particular, the principal fault and secondary faults show an opposite behavior both in terms of earthquake size distribution and clustering properties.
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Introduction
Statistical seismologists usually characterize seismicity by describing two principal features: the clustering behavior, i.e. the tendency of the events to aggregate both in time and space; and the earthquake size distribution, which rules the magnitude of the events^{1,2,3}. Both features are deeply investigated at a global, regional, and local scale. Earthquakes can occur either individually or in sequence: in the case of several events that occur individually, we have a low degree of clustering; in the other case of multiple events that occur in sequence (e.g. the aftershocks sequences after strong events^{4}), we have a high degree of clustering.
The dependence of the clustering behavior on the physical properties of the Earth’s crust is still debated: clustering properties could change with different tectonics around the world^{5}, or clustering is essentially similar in distinct crustal regions^{6}. Recent studies on specific zones suggest that microseismicity clustering could be related to transient slow slip^{7} or background stress^{8}. Liu et al.^{9} using a long earthquake catalog (about 35 years) showed that in the San Andreas fault clustering is inversely correlated with the creep rate.
The earthquake size distribution is instead dominated by a physical mechanism. Such a distribution can be well described by an exponential distribution with one parameter^{10}, called “bvalue” in the context of the GutenbergRichter law^{2}. The inverse proportionality of the bvalue with the differential stress was largely confirmed in laboratory events^{11,12} and in natural events^{13,14,15}, showing that a physical mechanism rules this parameter.
With this study, we contribute to the investigation of a possible correlation between the clustering properties and the bvalue. The detection of a common physical mechanism influencing both the clustering and the bvalue would improve our general knowledge of seismicity. In particular, the understanding of the relation between these two quantities is also important for seismicity forecasting, because the large majority of the models used for shortterm earthquake forecasting (ETASlike models^{16,17}) assume a uniform bvalue for all the events.
The clustering of seismicity can be studied by using algorithms able to separate the background/independent part of the seismicity (in some methods called mainshocks) from the triggered one (in some methods called aftershocks).
The majority of the algorithms used to split independent and triggered events lead to a biased estimation of the bvalue for these two subsets^{18}. The stochastic declustering method^{19,20} does not suffer from this biased estimation problem for the bvalue, thanks to its probabilistic approach in the identification of independent and triggered events^{18}. Other studies already suggested a relation between the clustering and bvalue^{7,8,9}; in this work we also focus our attention on the proper (and in some cases sophisticated) statistical techniques to distinguish independent and triggered seismicity, to estimate their bvalues avoiding bias, and to evaluate the possible correlation.
We investigate the clustering properties and the earthquake size distribution of the Alto Tiberina Fault (ATF) system, using a highresolution earthquake catalog (Fig. 1)^{21} and a 3D version of a stochastic declustering algorithm able to take into account also the earthquakes’ depth^{22}. In the case of the ATF system, composed of a principal lowangle normal fault (LANF; dip < 30°) and several minor synthetic and antithetic higherangle normal faults^{23}, the inclusion of depth is fundamental to properly understanding the possible connection between mechanical properties of the fault system and the characteristics of the seismicity.
The ATF case is particularly interesting for this kind of analysis. LANF have in fact been proposed as key structures for accommodating crustal extension, despite standard reactivation theory predicts that slip on these structures is extremely unlikely to happen^{24}, consistently with the absence worldwide of large earthquakes on such low angle faults^{25}. In detail, Anderson–Byerlee frictional fault reactivation theory, where the extending crust is characterized by vertical σ1 and faults possessing friction in the Byerlee^{26} range 0.6–0.85, predicts that it is easier to form a new optimally oriented fault (dip about 60°) instead of reactivating an existing one dipping less than 30°^{27}. Thus, our study aims to better constrain the spacetimesize distribution of the seismicity between mis (e.g. low) and welloriented (e.g. higherangle) normal faults.
Results and discussion
Clustering and earthquake size distribution at ATF system
We started our analysis with an accurate selection of the events in the catalog and a rigorous magnitude of completeness estimation (see Methods for details). Then we applied the stochastic declustering algorithm to identify the degree of independence of each seismic event. This algorithm associates a number between 0 and 1 to each event (i.e. the probability of being independent, PBI): values near 0 are related to the clustered/triggered seismicity, while values near 1 are related to the background/independent seismicity. Figure 2a shows the epicentral map of these probabilities. To investigate the possible relationship between the bvalue and the clustering of seismicity, we estimated the bvalue of the background and triggered components of the seismicity separately, using the recent weighted likelihood method^{21,28}. We found a significantly different bvalue for the two types of seismicity: a high bvalue for background events, and a low bvalue for triggered events (Fig. 2b). This different behavior of the bvalue for background and triggered events is a key result, because until now such a difference in bvalues is interpreted just as a possible bias in the estimation^{18,29}. We underline that in this case the difference is not an artifact due to the declustering of the seismic catalog, because our stochastic declustering approach does not suffer from this biased estimation problem. The different bvalues are a real property of seismicity. Therefore, to better investigate the relation of bvalue and clustering properties with depth, we focused our attention on one vertical section of the catalog (11 km width). We chose this section because of the large number of events available in this part of the fault system. Figure 2c, d shows the PBI and the spatial bvalue on the vertical section. From these figures, it is possible to appreciate that both high PBI and high bvalues are related to the Alto Tiberina Fault, while low PBI and low bvalues are related to the shallower part of the section, where the synthetic and antithetic faults are located. To give a quantitative assessment of this correspondence, we computed the correlation coefficient ρ between PBI and bvalues for the events in the section, obtaining a significant positive value of ρ (Fig. 2e). Different radii are used to select the events in the section, always leading to a significant correlation (see Supplementary Information).
Different faults, different behaviors
The results of our analysis indicate that the major Alto Tiberina Fault produces mainly background events alongside a high bvalue, while the minor synthetic and antithetic faults, located in the ATF hanging wall, mainly produce clustered events with a low bvalue (Fig. 3). The outcomes relative to the bvalue are consistent with previous findings obtained for the same fault system^{23,30}. Here we demonstrate that not only bvalues are different in the ATF and synthetic and antithetic faults, but also the clustering behaviors are different, and strongly correlated with the bvalues.
Degree of clustering of the seismic activity has been already coupled with frictional properties characterizing diverse portions of the faults system. An example is the creeping rate representing the whole spectrum of fault coupling from stable sliding to locked sectors along the San Andreas fault plane^{9}.
Our robust statistical approach, by coupling the clustering properties of the seismicity with the earthquake size distribution, corroborates this idea. Similarly to some creeping patches of the San Andreas fault, we found a low degree of clustering and high bvalues on the ATF. This is consistent with the results of the analysis of the Global Navigation Satellite System (GNSS) data suggesting creeping portions along the ATF^{31}. Therefore, both the San Andreas and the Alto Tiberina faults show high bvalues and low clustering in their creeping portions. The same GNSS data, uncovered in terms of a twodimensional GPS velocity profile constructed through the ATF system, indicated that the lowangle normal fault has a high degree of tectonic coupling with its main antithetic fault, suggesting that creeping along the ATF may control the observed strain localization along these minor segments^{32}.
Nowadays many observations are consistent with the occurrence of mainly aseismic deformation along the ATF plane reducing the availability of strain energy, thus somewhat lowering the seismic hazard associated to the ATF^{23,30,31,32,33}, a fault that based on empirical relations between fault dimension and maximum magnitude, could host up to a M_{W} 7.2 event. Such a behavior that we can call dominant, due to geometrical (e.g. misoriented fault) or mechanical (e.g. presence of velocity strengthening material along the fault plane) reasons, is not necessarily the only one. Recent studies documented that any given patch of a fault can creep, nucleate slow earthquakes, and host large earthquakes (e.g. Iquique earthquake^{34}, Tohoku earthquake^{35}, and Parkfield^{36}). A seismic rupture can in fact nucleate in a small and locally welloriented locked fault portion of the ATF and then propagate through creeping portions even if embedded within velocitystrengthening materials. During these processes, high slip velocities may favor a switch from creeping to seismic behavior, as observed for the Tohoku 2011 earthquake, which accumulated its largest seismic slip in the area that had been assumed to be creeping^{37,38}. The reasons why a fault patch would switch from one mode of slip to another is still under investigation, together with the interaction between creep, slow, and regular earthquakes. Thus, our evidence cannot be used to evaluate the maximum expected magnitude, understood as the amount of the fault plane on the ATF that can be activated during a single strong event.
Our findings are also particularly important for the current debate on the possible bvalue variations during aftershock sequences. While some works declared strong bvalue variations during aftershock sequences^{39,40}, other works show that these variations could be caused by incompleteness in the seismic catalog^{41,42,43}. Here we demonstrate that triggered events (i.e. aftershocks) have a bvalue statistically different from background events (i.e. mainshocks), adding a fundamental contribution to the controversy on bvalue variations.
Moreover, it is also interesting to note that both Alto Tiberina and synthetic/antithetic faults produce seismic events with prevalent normal focal mechanisms^{30}, but the bvalues of these faults are significantly different. This observation proves that the rake of focal mechanism is just one of the features that can influence the bvalue^{13,15}: in the ATF system, also the dip of focal mechanism plays an important role. Synthetic/antithetic faults show higher dip angles and lower bvalues, while the ATF shows lower dip and higher bvalues.
A limitation of this work is the relatively short temporal length of the catalog (about 5 years), and the fact that it is not possible to include in the ETAS model the uncertainties of earthquake locations. A future longer catalog should be used to additionally check the findings of this study. However, although obtained using a short catalog, the results of this work are robust and statistically significant.
Conclusions
In summary, our results indicate that complex fault systems, like the Alto Tiberina zone, can have an apparently intricate behavior both in terms of clustering and earthquake size distributions. Here we have shown that these two features of seismicity are strongly correlated and correspond to different seismic behavior of the main fault and the secondary synthetic/antithetic faults: high bvalue and low clustering for ATF, low bvalue and high clustering for synthetic/antithetic faults. These different behaviors shed light on the relations of background and triggered seismicity with the earthquake size distributions.
Methods
Seismic catalog
The catalog we used consists of ∼50 K highquality located earthquakes that occurred in the study region from April 2010 to December 2015, with local magnitude (M_{L}) ranging from −2.5 to 3.8 (see details in Pastoressa et al.^{21}).
Event selection and completeness estimation
From the starting catalog composed by ∼50 K earthquakes, we selected the events with a hypocentral depth between 0.5 and 15.0 km inside the area shown in Fig. 1, which contains most of the seismicity that occurred along the ATF system. We excluded the events in the first 500 m of the crust in order to avoid possible contamination by humaninduced events (e.g. quarry blasts and explosions). We determined the completeness magnitude, Mc, by using a robust and stringent approach, i.e. the Lilliefors test method^{44,45}, obtaining an Mc = 0.5 (and 6531 events with Mc ≥ 0.5, for details see Pastoressa et al.^{21}). Possible Short Term Aftershock Incompleteness (STAI) periods are carefully checked, using the approach suggested by Zhuang et al.^{46}; we found no STAI above the Mc = 0.5 threshold.
Stochastic declustering
The ETAS 3D model is described in Console et al.^{22}. The parameters of this model are the same as in a 2D ETAS model, but in this case the distances between events are hypocentral distances (i.e. also depth is included in the computation). The goal of a stochastic declustering algorithm is to assign to each event in the catalog the probability of being independent (PBI). To compute these probabilities, we have to first fit the ETAS parameters. The calculation of the best fit ETAS parameters was carried out following these two steps:

1.
choice of the optimal correlation distance for the 3D spatial smoothing kernel by the maximum likelihood (ML) of half of the catalog on the grid of smoothed distribution of the other half;

2.
iterative search of the maximum likelihood ETAS parameters by computing (a) initial ML best fit of the ETAS parameters, (b) creation of a new catalog where a probability of being independent is associated with every earthquake, (c) creation of a new smoothed grid making use of the probability obtained in (b), (d) iteration of the process starting from (a) with the new smoothed grid and again until the optimal data set is obtained.
At the end of the abovementioned process, we obtain an earthquake catalog where the probability of independence is associated with each event. The first events in the catalog were removed, since they all have a high PBI simply due to their temporal position at the beginning of the catalog (burnout period, see Console et al.^{22} for details).
bvalue estimation
We employed two methods to calculate the bvalue, the classical Maximum Likelihood Estimation (MLE) for Fig. 2d, and Weighted Likelihood Method (WLM) for Fig. 2b. In the first case, the MLE^{10} is corrected for magnitude binning and for an unbiased estimation^{41}:
Where \(N\) is the total number of events, \(\bar{M}\) is the mean of magnitudes, \({M}_{c}\) is the magnitude of completeness of the catalog, and \(\varDelta M\) is the binning of the magnitude.
In detail, to construct Fig. 2d, we considered a dense grid spaced 1.0 km both horizontally and in depth, and a cylindershaped sampling volume with a fixed radius of 3.5 km containing at least 50 events (as in Murru et al.^{47}). The bvalues estimation errors σ were computed as in Aki^{10}, \(\hat{\sigma }=\frac{\hat{b}}{\sqrt{N}}\). Supplementary Figs. S1 and S2 display: the estimation of the bvalue for different Mc values, the estimation of σ relative to the bvalue of Fig. 2d, and the bvalue estimation made with at least 100 events. All these results are coherent with the ones shown in Fig. 2d. For Fig. 2b, with the aim to evaluate the possible influence of changes in completeness magnitude on the bvalue estimation, we computed bvalues and their 95% confidence intervals as a function of completeness magnitude both for background and clustered seismicity using the WLM^{28}. As in Pastoressa et al.^{21}, to properly compute the bvalue for the background and triggered seismicity we used as weights the probabilities of each event to be independent (\({\varphi }_{i}\)) and their complementary (\({\rho }_{i}=1{\varphi }_{i}\)) respectively, achieved from the stochastic declustering based on the 3D ETAS model. In this way, the formulas to estimate bvalues relating to background (\({\hat{b}}_{{back}}\)) and triggered seismicity (\({\hat{b}}_{{trig}}\)) are:
Supplementary Fig. S3 displays the correlation between the mean PBI and the bvalues obtained with different radii for the circular search (4, 3, and 2.5 km), showing results very similar to Fig. 2e.
Data availability
The dataset used in this paper is freely available at: https://zenodo.org/records/10810647.
Code availability
The code used in this paper is freely available at: https://zenodo.org/records/10810647.
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Acknowledgements
We thank Gian Maria Bocchini and one anonymous reviewer for providing insightful comments and suggestions which greatly improve the value of this work. This work was supported by the 2020–2024 Istituto Nazionale di Geofisica e Vulcanologia (INGV) Department Strategic Project named MUSE (Multiparametric and mUltiscale Study of Earthquake preparatory phase in the central and northern Apennines), and by the Centro di Pericolosità Sismica (CPS) of INGV.
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M.T., R.C., C.M., M.M., and G.F. outlined the project; M.T., R.C., M.M., and G.F. performed the statistical analyses; M.T., C.M., and A.E.P. prepared the figures; L.C. (expert of ATF system) helped interpret the results; all authors wrote the manuscript.
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Communications Earth and Environment thanks Gian Maria Bocchini and the other anonymous reviewer for their contribution to the peer review of this work. Primary Handling Editor: Joe Aslin. A peer review file is available.
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Taroni, M., Console, R., Montuori, C. et al. Statistically significant difference between earthquake size distributions of independent and triggered seismicity. Commun Earth Environ 5, 193 (2024). https://doi.org/10.1038/s4324702401367x
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DOI: https://doi.org/10.1038/s4324702401367x
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