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We studied the whole and aftershock-depleted shallow (depth ≤ 30 km) Taiwanese seismicity from 1990 to 2017 by using several independent methods. The coefficient of variations and the Allan Factor have revealed that time-clustering characterizes not only the time distribution of the whole seismic catalogue, but also that of the aftershock-depleted one, although this displays time-clustering at larger timescales. In particular the Allan Factor shows that at small timescales the seismic process of the whole as well as the aftershock-depleted one can be well described by a Poisson process. The detrended fluctuation analysis clearly evidence persistence in the interevent times and weak persistence or randomness in the magnitudes. The singular spectrum analysis of the daily counts in the aftershock-depleted catalogue reveals the existence of quasi-monthly and bi-weekly periodicities. Furthermore, quasi-annual and quasi-semestral periodicities have been identified that strengthen the link between seismic activity in Taiwan and Earth’s tides.

The intraplate and interplate seismic catalogues of Iquique (Chile) area were investigated by using seismological (b value of the Gutenberg–Richter law), fractal (Allan Factor and detrended fluctuation analysis) and topological (Horizontal Visibility Graph) methods. The two catalogues show different stress state indicated by the different b value, larger for the intraplate seismicity. The interplate seismicity shows significant time-clustering behavior of the earthquake occurrence times at large time scales, while the time distribution of the intraplate earthquakes seems to be featured by random fluctuations without any significant sign of time-clustering. Different role seems to be played by aftershocks in the two investigated seismic zones, revealed by the different time distribution of the aftershock-depleted catalogue of the intraplate zone, which remains Poissonian for most of the time scales, and the interplate zone that still keeps its clustering phenomenon but at higher timescales, thus shortening its clustering time scale range. Furthermore, the interevent times of the interplate catalogue are characterized by persistent dynamics even after removing the aftershocks, while uncorrelated fluctuations characterize the behavior of interevent intervals of the whole as well as aftershock-depleted intraplate catalogue. Additionally, the magnitudes of both catalogues are characterized by time-reversibility before and after the aftershock removal; while the interevent times of intraplate catalogue change their status from time-irreversibility to time-reversibility after removing the aftershocks. The obtained results seem to evidence a different role played by the aftershocks in the intraplate and interplate zones, that needs to be further explored.

The historical and instrumental catalog of the Absheron-Prebalkhan region in the Caspian Sea area was analyzed in order to reveal the existence of temporal clustering in the time dynamics of the seismicity. The timespan of the catalog is from 1842 to 2012 and the magnitude of the events ranges from 2.5 to 6.8. The Gutenberg-Richter analysis indicates 4.0 as the completeness magnitude of the catalog. The temporal clustering analysis was performed over the sequence of events with magnitude M ≥ 4 by using the methods of the Allan Factor and the coefficient of variation. Both the methods have revealed the presence of time-clusterized structures in the time dynamics of large events in the Absheron-Prebalkhan region. Such findings, which suggest a non-Poissonian behavior of the seismicity of the investigated area, could contribute to a deeper knowledge of the time dynamics of the seismicity and to a better assessment of the relative seismic hazard.

The analysis of time-clustering behaviour of the sub-crustal seismicity (depth larger than 60 km) of the Vrancea region has been performed. The time span of the analyzed catalogue is from 1978 to 2008, and only the events with a magnitude of Mw ≥ 3 have been considered. The analysis, carried out on the full and aftershock-depleted catalogues, was performed using the Allan Factor (AF) that allows the identificatiion and quantification of correlated temporal structures in temporal point processes. Our results, whose significance was analysed by means of two methods of generation of surrogate series, reveal the presence of time-clustering behaviour in the temporal distribution of seismicity data of the full catalogue. The analysis performed on the aftershock-depleted catalogue indicates that the time-clustering is associated mainly to the aftershocks generated by the two largest events occurred on 30 August 1986 (Mw = 7.1) and 30 May 1990 (Mw = 6.9).

The time dynamics of seismicity of Aswan area (Egypt) from 2004 to 2010 was investigated by means of the (i) Allan Factor, which is a powerful tool allowing the capture of time-clusterized properties of temporal point processes; and the (ii) detrended fluctuation analysis, which is capable of detecting scaling in nonstationary time series. The analysis was performed varying the depth and the magnitude thresholds. The 2004–2010 Aswan seismicity is characterized by significant three-fold time-clustering behaviors with scaling exponents ~0.77 for timescales between 104.16 s and 105.14 s, ~0.34 for timescales between 105.14 s and 106.53 s, and ~1 for higher timescales. The seismic interevent times and distances are characterized by persistent temporal fluctuations for most of the magnitude and depth thresholds.

The seismicity observed in the Aswan area (Egypt) between 1986 and 2003 was deeply investigated by means of time-fractal methods. The time dynamics of the aftershock-depleted seismicity, investigated by means of the Allan Factor, reveals that the time-clustering behavior for events occurred at shallow depths (down to 12.5 km from the ground) as well as for events occurred at larger depths (from 15 km down to 27.5 km) does not depend on the ordering of the interevent times but mainly on the shape of the probability density functions of the interevent intervals. Moreover, deep seismicity is more compatible with a Poissonian dynamics than shallow seismicity that is definitely more super-Poissonian. Additionally, the set of shallow events shows a periodicity at about 402 days, which could be consistent with the cyclic loading/unloading operations of the Lake Naser Dam. Such findings contribute to better characterize the seismicity of the Aswan area, which is one of the most interesting water reservoirs in the world, featured by reservoir-induced earthquakes.

The time-clustering behavior of the 1996–2005 seismicity of Koyna–Warna region (India), a unique site where reservoir-triggered earthquakes have been continuously occurring over the last about 50 year, has been analyzed. The scaling exponent α, estimated by using the Allan Factor method, a powerful tool to investigate clusterization in point processes, shows co-seismic and pre-seismic enhancements associated with the occurrence of the major events.

Temporal dynamical analysis in fire sequences recorded from 1992 to 2005 in a fire-vulnerable area of Patagonia (Argentina) was performed by using the Allan Factor statistics. Three typologies of fires were investigated: forest, steppe and urban. The obtained results show the presence of annual periodicities, superimposed to a time-scaling behaviour, which characterize the point processes of fires as fractal time processes with a rather high degree of time-clusterization of the events.

The series of lightning measured during 2002 and 2003 in two areas of Italy, Lombardia (northern Italy) and Campania (southern Italy), has been deeply investigated to detect and quantify time-clustering structures by using several methods. Our results suggest the existence of time-clustering in the sequence of lightning time-occurrences, characterized by the presence of two scaling regions, respectively related with clustering effect inside a single thunderstorm and among successive thunderstorms.

By using the global and local coefficient of variation and the Allan Factor we investigated the time-clustering properties of the time dynamics of fluid-injection–induced microseismicity. The experiment consists of a microseismic monitoring through a nearly vertical borehole of 12 receivers of a hydraulic fracturing stimulation along a horizontal well separated into more than 20 sections (stages) The main finding of the applied methodology is the discrimination between fault triggering and new fracturing, being the first characterized by a clusterization of the induced microseismic events and the second by a Poissonian behaviour of the generated events.

In this paper, we propose to compare different declustering methods on the basis of the time-correlation and the space-clustering of the residual earthquake catalog after the declustering techniques have been applied. To this aim, we applied two point process clustering measures, the Allan Factor and the Morisita Index, for the identification and quantification of temporal correlation and spatial clustering in point processes, respectively. We used our joint space–time approach to study the earthquake space–time point processes of southern California and Switzerland with surrounding area, declustered by using the method of Gardner and Knopoff (with Grunthal and Uhmhammer window) and that of Reasenberg (with different setting parameters). Our results show that the residual declustered catalog is still characterized by time-correlated structures at long timescales; however, the cutoff timescale that is the lowest timescale above which the time-correlation is visible is higher with the Reasenberg method while is smaller with the Gardner and Knopoff method with Grunthal window. The space-clustering analysis performed by means of the Morisita Index suggests that the declustering technique effectively reduces the spatial clustering of the seismicity of Switzerland, but does not change the spatial properties of the residual seismic catalogue of the southern California.

The time-scaling behavior of the 1970-2003 seismicity of southern Aegean area has been investigated by means of the Allan Factor analysis, which allows the detection and the quantification of time-clustering structures in point processes. The analysis of the time variation of the scaling exponent, defined by means of this statistics, has revealed precursory patterns related with the largest event (M = 6.1) of the sequence investigated. (c) 2005 Elsevier Ltd. All rights reserved.

Lightning data, measured from April to October 2002 in an area of northern Italy, have been analyzed by using the Allan Factor statistics. The sequence of lightning time-occurrences is characterized by two scaling regions, which reveal intra-cluster (inside an individual thunderstorm) and inter-cluster (among successive thunderstorms) time-clustering properties of lightning data. We investigated intra- and inter-cluster correlations for three classes of lightning: (i) the whole series, (ii) the sequence of only negative flashes and (iii) that of only positive flashes. Furthermore, we performed the analysis of the scaling behavior of the three classes of lightning varying the current intensity threshold.

Temporal regimes in fire data recorded from 1997 to 2003 in an area of central Italy have been detected by using the Allan Factor statistics. Our findings reveal the presence of meteo-climatic-related periodicities; furthermore time-scaling behaviour characterizes the point process of the fire sequence as a fractal process with a high degree of time-clusterization of the events.

Two years of lightning data, measured during 2002 and 2003 in an area of southern Italy, have been analyzed in order to reveal scaling behaviour in their time dynamics. We used the Allan Factor method to evidence the presence of time-clustering in the lightning data. We found that i) the sequence of lightning time-occurrences is characterized by two scaling regions, which reveal intra-cluster (inside an individual thunderstorm) and inter-cluster (among successive thunderstorms) time-clustering properties of lightning data. A crossover timescale is identified, which is connected with the mean duration of a thunderstorm. We investigated intra- and inter-cluster time-dynamics for the whole sequence as well as the sequence of only positive and only negative flashes. The negative lightning show a time-clustering feature stronger than that of the positive lightning.

We performed the Allan factor analysis on the temporal distribution of seismicity of Italy from 1983 to 2002. Significant power-law temporal behaviour has been identified analyzing both the full and the aftershock-depleted seismic catalogues of Italy; this suggests that the correlated behaviour is due not only to the generation of aftershocks, following the occurrence of large shocks, but is a feature of the background seismicity.

The temporal distribution of aftershock sequences can be analyzed by means of new parameters other than the Omori's p-value. A deep analysis of the temporal properties of six aftershock sequences occurred in Greece from 1980 to 1997 has been carried out by means of fractal tools. We performed the Omori's law, the R/S and the Allan Factor analyses in order to estimate the p-value of the aftershock rate, the Hurst exponent (H) of the interevent-time series, and the spectral exponent (α) respectively. Our results show a significant variability of the p-value from 0.89 to 1.48, of H from 0.78 to 1.186 and of α from 0.49 to 1.44. The variation of the normalized α with the threshold magnitude for all the aftershock sequences examined has displayed almost the same decreasing behaviour. The temporal fractality of aftershock sequences is primarily due to the power-law decay of the aftershock rate.

Fractal stochastic point processes are described by power-law statistics, and therefore, are suited to model the time-occurrence sequence of seismic events, that are generally characterized by a time-clustering phenomenon. In this paper we quantified the temporal clustering in earthquake sequences by means of the Fano factor and the Allan factor, that can be used to reveal the fractality of such processes. These methods are applied to model the fractality of the 1983–1997 seismicity of one of the most seismically active area of southern Apennine Chain (southern Italy), that has shown the presence of temporal clusterization.

Time regimes in fire data recorded from 1992 to 2007 in a fire vulnerable area of Patagonia (Argentina) were revealed by using the Allan Factor statistics. The obtained results show the presence of seasonal periodicities, superimposed to three time-scaling regimes, which characterize the point process of the fire sequence as a fractal time process with a rather high degree of time-clusterization of the events.

Point processes, like other geophysical phenomena, such as earthquakes or avalanches, are well suited to model rain event sequences. Time-clustering analysis has been performed over rainfall sequences by using the statistical approaches of Fano Factor and Allan Factor. The obtained results suggest the presence of clustering in the time distribution of the sequence of rain events, with scaling exponent a ranging between similar to 0.35 and similar to 0.42.

The time-clustering behavior of the neutron star SGR1806-20 burst sequence (detected by the International Cometary Explorer, ICE) has been analyzed. The presence of two different timescaling regions, related to the "foreshock" and "aftershock" phases, separated by the largest event occurred on November 16, 1983, 8:34 GMT, E = 5.17 · 1041 erg , has been evidenced. The results highlight that starquake foreshocks involve medium timescales from about 30 minutes to approximately 5 days, while starquake aftershocks involve longer timescales ranging up 4 months.

We performed the Allan factor analysis on the 1986-2001 seismicity of the 80 seismogenetic zones in which the Italian territory has been subdivided by the Gruppo Nazionale Difesa dai Terremoti (GNDT), in order to investigate the presence of time-correlated behaviour. Power-law time-correlated areas have been identified analyzing the aftershock-depleted seismicity. Strong variability in the scaling properties has been found, suggesting that different dynamical mechanisms govern the seismic activity of the Italian seismogenetic zones.

Investigating the time dynamics of forest-fires is a challenge in the environmental sciences. Time-scaling scale-invariant approaches have been used to characterize the temporal distribution of forest-fire sequence detected in Gargano area (southern Italy), which is characterized by high level of wildfire density. Our findings reveal that the point process of the fire sequence can be considered as a fractal process with a high degree of time-clusterization of the events. The time-clustering phenomenon is visible from timescales of order of few days. Furthermore, the fire process tends to be less time-clusterized with the increase of the burned area.

Significant power-law long-range correlated structures have been identified in the Italian seismicity from 1983 to 2003. We performed the Allan Factor Analysis and the Detrended Fluctuation Analysis on both the full and the aftershock-depleted seismic data, extracted by the INGV (National Institute of Geophysics and Volcanology) instrumental catalog.Our findings suggest that (i) the time-scaling behaviour characterizes mainly the Apennine chain; (ii) the time-correlated behaviour of the seismicity is persistent, and this implies that the dynamics underlying the seismic phenomenology is characterized by the presence of "positive feedback mechanisms"; (iii) significant time correlation is not simply related with the mainshock/aftershock mechanisms.

Rain events can be described in terms of point processes, analogously to other geophysical phenomena, such as earthquakes or avalanches. Power-law behaviors have been identified in the distribution of sizes, of rain event durations and drought durations, with scaling exponents 1.4, 2.4 and 1.3, respectively. Time-fractal approaches, like Fano Factor and Allan Factor, well suited to reveal scaling behavior in point processes, allow to detect power-law behavior in rainfall data, pointing out to the presence of clustering in the time distribution of the sequence of rain events.

Much recent work has been focused on understanding the statistical properties of time-occurrence series of earthquakes. Investigating into the patterns of seismic sequences reveals evidence of scaling features in temporal behavior. This is shown in the clustering properties of seismicity of the western Corinth graben, Greece from 1983 to 2000. Power-law behavior has been found by means of Allan factor analysis and Fourier spectra for the earthquake sequence, with consistent values for the scaling exponents, that decrease with the threshold magnitude. The analysis of the temporal variation of the scaling exponent, performed with different threshold magnitudes, reveals an enhancing of the clusterization in correspondence to larger events. The multifractal analysis of the temporal distribution of the events has shown a decrease of the intermittent character with the threshold magnitude.

Temporal clustering structures were identified and quantified in fire sequences recorded from 1980 to 2005 in Continental Portugal, by using the Allan Factor statistics, a statistical tool suited to reveal clustering behaviour in point processes. The obtained results show the presence of daily and annual periodicities, superimposed onto a scaling behaviour, which features the sequence of wildfires as a fractal time process with a rather high degree of time-clusterization of the events.