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Investigating the time-clustering properties in seismicity of Umbria–Marche region (central Italy)
Chaos, Solitons and Fractals 18 (2003) 203–217 www.elsevier.com/locate/chaos
Investigating the time-clustering properties in seismicity of Umbria–Marche region (central Italy) Luciano Telesca a
a,*
, Vincenzo Lapenna a, Maria Macchiato
b
Institute of Methodologies for Environmental Analysis, National Research Council, C de S. Loja, Tito Scalo PZ I-85050, Italy b Dipartimento di Scienze Fisiche, Universit a ‘‘Federico II’’, Naples, Italy
Abstract We investigated by fractal tools the temporal patterns of 1996–1999 seismicity of the Umbria–Marche region, central Italy, that was struck by a strong event ðMS ¼ 5:9Þ on September 26, 1997. Clustering behaviour has been evidenced by using different fractal tools, Fano factor analysis, Allan factor analysis and detrended fluctuation analysis, that show power-law shaped statistics, whose scaling exponents describe the degree of clusterization in the earthquake sequence. Furthermore, the scaling exponents evolve in time showing an increase of the clustering features in correspondence to larger events. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction The dynamical variability of many complex geophysical phenomena can be described by means of the concepts of scale-invariance, fractals and multifractals [32], which represent useful tools to characterize and interpret the inhomogeneity, which features that variability. As example of inhomogeneous dynamics, seismic processes have been recently analysed in order to reveal their time-scaling properties, which can be very informative concerning the geodynamical mechanisms underlying the presence of scaling laws in earthquake statistics. The presence of time-scaling behaviours suggests that the seismic process is clusterized in time, and the scaling exponents give a quantitative measure of the degree of such clusterization. The analysis of the earthquake catalogues have shown that seismic processes are characterized by scaling behaviour concerning the distributions of magnitudes, epicentres and time-occurrences. Gutenberg and Richter [21] found that there is no a characteristic rupture size and that the magnitude distribution follows a self-similar relation. Kagan and Jackson [22] and Kagan [23] showed that clustering behaviours feature the time-occurrence sequences of earthquakes on the long-term timescale as well as on the short-term scale, due to the presence of aftershocks and foreshocks, which characterize large shallow earthquakes [24]. The spatial distribution of earthquakes was found to be clusterized on several scales of observation [25,44]. The self-organized criticality (SOC) theory can explain the scale-invariance phenomenology typical of earthquake sequences and it has been first applied to seismic processes by Bak and Tang [2]. The observational evidence of the presence of scale-invariant behaviour in earthquake dynamics has led to the development of a wide variety of physical models of seismogenesis. As shown in Sornette and Sornette [40] and Sornette et al. [41], earthquakes are an important part of the relaxation mechanism of the crust which is submitted to inhomogeneous increasing stresses accumulating at continental-plate borders; and the SOC idea implies that earthquakes in turn organize the crust both in the spatial and temporal domains. Seismicity patterns have been explained using theoretical
*
Corresponding author. Tel.: +39-971-427-206; fax: +39-971-427-271. E-mail address: [email protected] (L. Telesca).
0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(02)00654-9
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models for earthquake generation and propagation [9,13,26,47]. Identifying the time-occurrence distribution in earthquake sequences is an important task in the context of studies related to the seismic hazard. The Poisson distribution of the interevent times implies that each seismic event is independent from the time elapsed since the previous event, thus leading to a memoryless process, which is characterized by uncorrelation among the events. Although a simple discrete Poisson distribution provides a close fit for the sequence of large events [6], in recent studies it has been shown that earthquake occurrence is characterized by temporal clustering properties with both short and long timescales [22], and this implies that an earthquake sequence can be modelled by means of correlated processes. In correlated processes, it is an important task to identify the type of the temporal fluctuations, which are connected with the power spectral density Sðf Þ of the process itself, which, for scaling processes, is a power-law function of the frequency f , Sðf Þ / 1=f a , with a the most characterizing parameter of the time-clustering of the process. Several methods can be used to estimate the power spectral density of seismic processes. Bittner et al. [4] and Lapenna et al. [29] used the variance–time curve (VTC) method, that for spectra with power-law (fractal) shape furnishes a relation with the spectral exponent a. The Fano factor (FF) and the Allan factor (AF) have been shown to have a power-law behaviour for processes with scaling properties [43–45]. The numerical value of a quantifies the degree of time-clusterization of the seismic process [30], revealing the presence of Poissonian ða 0Þ or clusterized ða 6¼ 0Þ temporal fluctuations. In this paper, we analyzed the earthquake sequence from 1996 to 1999 in the Umbria–Marche region, central Italy, by means of fractal tools in order to qualify and quantify the presence of dynamical clusterization.
2. Mathematical background Point process describe sequences of events occurring randomly in time [15]. We can represent such processes by a finite sum of DiracÕs delta functions centred on the occurrence times ti yðtÞ ¼
n X
dðt ti Þ:
ð1Þ
i¼1
The process can be described equivalently by the set of the interevent times or by the associated counting process. The second representation is given by dividing the time axis into equally spaced contiguous counting windows of duration s, and producing a sequence of counts fZk ðsÞg, with Zk ðsÞ denoting the number of earthquakes in the kth window: Z tk X n Zk ðsÞ ¼ dðt tj Þ dt: ð2Þ tk1 j¼1
The different representations lead to different methods to reveal and analyze the presence of clustering features in a seismic process. A commonly used measure to evaluate the clustering behaviour of a seismic sequence has been the coefficient of variation CV , defined as CV ¼ rT =hT i;
ð3Þ
where hT i is the mean interevent time and rT is its standard deviation [22]: a Poissonian process (completely random) has a CV ¼ 1, but a clusterized process is characterized by a CV > 1. This coefficient does not give information about the timescale ranges where the process can be reliably characterized as a clustered process. Nevertheless, a complex phenomenon can be deeply known only if the different timescales governing its dynamics are well understood. The detrended fluctuation analysis (DFA) performs the clustering analysis on the series of the interevent intervals. It was proposed by Peng et al. [35], and avoids spurious detection of correlations that are artifacts of nonstationarity, that often affects experimental data. Such trends have to be well distinguished from the intrinsic fluctuations of the system in order to find the correct scaling behaviour of the fluctuations. Very often we do not know the reasons for underlying trends in collected data and we do not know the scales of underlying trends. DFA is a method for determining the scaling behaviour of data in the presence of possible trends without knowing their origin and shape. The methodology operates on the time series xðiÞ, where i ¼ 1; 2; . . . ; N and N is the length of the series. With xave we indicate the mean value xave ¼
N 1 X xðkÞ: N k¼1
ð4Þ
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The signal is first integrated yðkÞ ¼
k X ½xðiÞ xave :
ð5Þ
i¼1
Next, the integrated time series is divided into boxes of equal length, n. In each box a least-squares line is fit to the data, representing the trend in that box. The y-coordinate of the straight line segments is denoted by yn ðkÞ. Next we detrend the integrated time series yðkÞ by subtracting the local trend yn ðkÞ in each box. The root-mean-square fluctuation of this integrated and detrended time series is calculated by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X ½yðkÞ yn ðkÞ 2 : F ðnÞ ¼ t ð6Þ N k¼1 Repeating this calculation over all box sizes, we obtain a relationship between F ðnÞ, that represents the average fluctuation as a function of box size, and the box size n. If F ðnÞ behaves as a power-law function of n, data present scaling F ðnÞ / nd :
ð7Þ
Under these conditions the fluctuations can be described by the scaling exponent d, representing the slope of the line fitting log F ðnÞ to log n. For a white noise process, d ¼ 0:5. If there are only short-range correlations, the initial slope may be different from 0.5 but will approach 0.5 for large window sizes. 0:5 < d < 1:0 indicates the presence of persistent long-range correlations, meaning that a large (compared to the average) value is more likely to be followed by large value and vice versa. 0 < d < 0:5 indicates the presence of antipersistent long-range correlations, meaning that a large (compared to the average) value is more likely to be followed by small value and vice versa. d ¼ 1 indicates flicker-noise dynamics, typical of systems in a self-organized critical state. d ¼ 1:5 characterizes processes with Brownian-like dynamics. The Allan factor analysis (AFA) concerns the calculation of the AF, that is a measure which can be used to distinguish fractal from Poissonian temporal fluctuations in point processes [1]. This factor is defined as the variance of successive counts for a specified counting time s divided by twice the mean number of events in that counting time AFðsÞ ¼
hðZkþ1 ðsÞ Zk ðsÞÞ2 i : 2hZk ðsÞi
The AF of a fractal point process varies with the counting time s with a power-law form a s AFðsÞ ¼ 1 þ : s1
ð8Þ
ð9Þ
The monotonic power-law increase is representative of the presence of fluctuations on many timescales [31]; s1 is the socalled fractal onset time, and marks the lower limit for significant scaling behaviour in the AF [42], so that for s s1 the clustering property becomes negligible within these timescales. For Poissonian processes the AF assumes approximately values near or below unity for all counting times s. From Eq. (9), the calcualtion of a can be performed by estimating the slope of the straight line that fits in a least square sense the AF curve, plotted in log–log scales. Of course, only the linear part of the curve will be considered to calculate the fractal exponent. Another method is the Fano factor analysis (FFA). It is based on the FF, that is defined as the variance of the number of events in a specified counting time s divided by the mean number of events in that counting time; that is FFðsÞ ¼
hNk2 ðsÞi hNk ðsÞi2 ; hNk ðsÞi
where h i denotes the expectation value. The FF of a fractal point process with 0 < a < 1 varies as a function of counting time s as a s FFðsÞ ¼ 1 þ ; s0 where s0 is the fractal onset time, similar to s1 .
ð10Þ
ð11Þ
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It is related to the AF by AFðsÞ ¼ 2FFðsÞ FFð2sÞ
ð12Þ
and it assumes unitary value for Poissonian processes. The exponent a conveys the information about the temporal fluctuations of the process, because it is the exponent of the power spectrum, that for fractal processes decays as a power-law function of the frequency f , Sðf Þ / f a with a measuring the strength of the clusterization [46].
3. Tectonic settings The Umbria–Marche region, central Italy, has been subjected to a great scientific interest, after the strong September 26, 1997 seismic event ðMS ¼ 5:9Þ that struck the area, and many studies have been performed regarding its geodynamical features [5,7,49,36], active tectonics [14,20,33], spatio-temporal seismic distribution [10,16,17,38], induced geophysical effects [18,19,37]. The central Apennines are made up of several tectonic units put straight since the Oligocene as a results of convergence and collision between the continental margins of the Corsica-Sardinia block and the Adriatic block [12]. The main compressive phase started in the Tortonian and the lack of Pliocene-Pleistocene marine deposits prove that after the Miocene the area was definitively uplifted [36]. The compressive structures were dissected by normal faults during the Quaternary, and, according to the most recent studies [8,28], these are related to the crustal thinning processes occurring in the Tyrrhenian Tuscan area. The Quaternary normal faults led to the formation of intramountain basins, and the seismicity of the area is mainly related to the activity of these faults. The Umbria–Marche region is part of the east-verging Neogene thrust and fold belt of central Italy [3,8,34]. Middle-late Quaternary faults and intermontane tectonic depressions, overprinted onto the thrust structure, are the preferential loci of strong to moderate earthquakes, as shown by historical and instrumental data and geological evidence [11,48,49]. The Umbria–Marche region is characterized by a well-documented historical and instrumental seismicity, mainly confined within the upper part of the crust (<16 km) [28]. West of the Tiber Basin, the seismic activity is the lowest. It increases remarkably within the Apennine area, where small to moderate ð4 < M < 6Þ earthquakes are frequent, while largest earthquakes ð6 < M < 7Þ have long recurrence-intervals [5]. Four highly damaging earthquakes (1279 Camerino, 1328 Norcia, 1703 Norcia and 1751 Gualdo Tadino) have occurred in the last centuries within the Citt a di Castello–Gubbio–Gualdo Tadino–Norcia seismic band. In the same area the largest instrumental earthquakes of the last 20 years (M5.9 1979 Norcia, M5.2 1984 Gubbio, M5.9 1997 Colfiorito) are located [5]. The analysed earthquake data were extracted by the Bulletin of Osservatorio Geofisico Sperimentale of Macerata (OGSM), available on the Internet site at http://www.geofisico.mc.it, and cover the period 1996–1999.
Fig. 1. Map of the Umbria–Marche region (central Italy). The stars indicate the stronger events in the area during the seismic crisis of September–October 1997 (. . .).
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Fig. 2. Gutenberg-Richter analysis of the earthquake catalogue. The magnitude of completeness is estimated as 2.2 with b 1.
4. Results Now we analyze the earthquake activity in the Umbria–Marche area (central Italy) (Fig. 1). We performed the Gutenberg–Richter analysis (Fig. 2) by least square method over the cumulative distribution of the events, to obtain the minimum magnitude from which the catalogue could be considered complete [39]: The catalogue is complete for events with magnitude M P 2:2 with b 1. The analyzed sequence contains 4672 events with magnitude M P 2:2. We computed the FFðsÞ and AFðsÞ for counting times of duration of 10 s to P /10 s, where P is the period of the catalogue. The FF and AF plots (Figs. 3 and 4) give clear indication of the fractal behaviour. Both the FF and AF are seen to increase with linear form in log–log scales, and this indicates the presence of range-correlated structures. The early flat behaviour can be interpreted as the presence of Poissonian dynamics for short timescales s. The slope of the fitting straight line of the FF and AF curves in the linear range gives the estimation of the fractal exponent a of the point process modelling the earthquake sequence in our study. In particular, we estimated aFF 0:84 and aAF 1:02 (for earthquakes with M P 2:2); the difference in the numerical value is due to a saturation effect that FF method suffers [46].
Fig. 3. Results of the FFA performed on the sequence of events with M P 2:2. The power-law behaviour is clearly visible from approximately 6 103 to 1:3 106 s, with the estimate of the scaling exponent aFF 0:84.
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Fig. 4. Results of the AFA performed on the sequence of events with M P 2:2. The power-law behaviour is clearly visible from approximately 3 104 to 3 106 s, with the estimate of the scaling exponent aAF 1:02.
Fig. 5. Interevent time distribution for the sequence of events with M P 2:2.
We analyzed the dynamics of the interevent time series calculating the coefficient of variation and performing the detrended fluctuation analysis (DFA) of the interspike sequence. Fig. 5 shows the distribution of the interevent intervals; the very low amplitude of the interevent series in the medium part of the plot corresponds to the seismic sequence initiated by the M5.9 earthquake, occurred on September 26, 1997. For this series we calculated CV 3:08, indicating a clustering effect. Fig. 6 shows the DFA performed on the interevent time series for earthquakes with M P 2:2. Two scaling regions are evidenced by the analysis; the first region, corresponding to short timescales, is characterized by the scaling exponent dDFA 0:68, which indicates a slightly persistent dynamics; in the second region, corresponding to long timescales, the scaling behaviour is revealed by the scaling coefficient dDFA 1:13, that denotes a flicker-noise dynamics. The results are consistent with the estimates of the scaling exponents performed by means of the AF and FF methods. In order to analyze the nature of the fractal structure of the seismic point process, we performed the FFA on several shuffled versions of the original sequence, thus preserving the interevent time and variance of the series. We calculated the FF curves, and then we averaged them obtaining the shuffled mean FF curve (SFF). In Fig. 7, we plotted the original FF and the SFF curves ðMth ¼ 2:2Þ. We can observe that the SFF curve presents a lower value of the scaling exponent a, approximately 0.4, and this indicates that the fractal behaviour of the seismic sequence depends on the ordering of the interevent intervals [46]. In Figs. 8 and 9, we reported analogous plots regarding the AFA and DFA, which clearly show the lowering effect on the clustering behaviour induced by the shuffling procedure.
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Fig. 6. Results of the DFA performed on the sequence of events with M P 2:2. Two scaling regions are visible, with different value of the scaling coefficient, dDFA 0:68 and dDFA 1:13, respectively for the small timescale and the long timescale ranges.
Fig. 7. Comparison between the original FF and the SFF curves ðMth ¼ 2:2Þ. The SFF curve presents a lower value of the scaling exponent ahSFFi 0:4.
In order to evidence possible variation of the temporal behaviour of the earthquake process under investigation with the lower threshold magnitude, we performed FFA, AFA and DFA for increasing values of Mth . Since the number of events decreases rapidly when Mth increases, the analysis has been stopped at the magnitude Mth ¼ 3:2 (604 events). Fig. 10 shows the FF curves related to the seismic sequences with varying threshold magnitude Mth . Varying the threshold magnitude, the behavioural trend of the curves does not change, although it is clear a variation of the slope of the linear part, corresponding to the intermediate timescale range, and a flattening effect, that becomes more apparent at long timescales and higher threshold magnitudes. This is probably due to a limited-number effect, visible increasing the threshold magnitude. We observe that the slopes, corresponding to the linear range of the curves, indicated by the arrows, seem to decrease with the increase of the threshold magnitude, and this denotes a decrease of the clustering effect. Fig. 12 shows the variation of the scaling exponent aFF ðMth Þ, estimated as the slope of the lines fitting each curve in the temporal range between approximately 6 103 and 1:3 106 s, that is the range of timescales commonly involved in the power-law behaviour of all the curves. Furthermore, the figure shows the variation with Mth of other two parameters: haSFF i, that is the average of the FF scaling exponents calculated for the FF curves of shuffled versions of the
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Fig. 8. Comparison between the original AF and the SAF curves ðMth ¼ 2:2Þ. The SAF curve presents a lower value of the scaling exponent ahSAFi 0:36.
Fig. 9. Comparison between the original DFA and the SDFA curves ðMth ¼ 2:2Þ. The SDFA curve presents a lower value of the scaling exponent dhSDFAi 0:62, involving all the box sizes.
earthquake sequence, and ahSFFi , that is the FF scaling exponent calculated for the SFF; both the parameters are calculated in the range between 103 and 5 105 s, that seem to be the common linear range of all the shuffled curves (Fig. 11). We observe that the aFF decrease with Mth , indicating a decrease of the clusterization of the process; the values of haSFF i and ahSFFi are almost coincident and approximately constant with varying Mth , and, furthermore, they fluctuate around the value of 0.38–0.4. From this result, we can conclude that for all the threshold magnitudes the scaling behaviour depends on the ordering of the interevent intervals; and this means that the clustering lies in the particular temporal structure of the earthquake series. Fig. 13 shows the AF curves with variable threshold magnitude Mth . Similarly to the FF case, the slopes of the linear part of the curves seem to decrease with the increase of the threshold magnitude. Fig. 14 presents the mean shuffled AF (SAF) curves for each Mth ; all the curves show an approximately linear behaviour in the common timescale range between 2:7 103 and 2:3 106 s. Fig. 15 shows the variation of the scaling exponents: aAF ðMth Þ, estimated for each AF curve in the temporal range between approximately 3 104 and 3 106 s, as indicated by the arrows in Fig. 13, haSAF i and ahSAFi , obtained analogously to haSFF i and ahSFFi . We observe that the values of aAF decrease with Mth , denoting a
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Fig. 10. FF curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2. The arrows indicate the timescale linear range that involves all the curves.
Fig. 11. SFF curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2 The arrows indicate the timescale linear range that involves all the curves.
Fig. 12. Variation with the threshold magnitude 2:2 6 Mth 6 3:2 of: aFF , estimated for the FF curves; haSFF i, obtained averaging the scaling exponents of the shuffled FF curves, and ahSFFi , estimated from SFF curves. All the estimates are performed by means of the least square method.
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Fig. 13. AF curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2. The arrows indicate the timescale linear range that involves all the curves.
Fig. 14. SAF curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2. The arrows indicate the timescale linear range that involves all the curves.
Fig. 15. Variation with the threshold magnitude 2:2 6 Mth 6 3:2 of: aAF , estimated for the AF curves; haSAF i, obtained averaging the scaling exponents of the shuffled AF curves, and ahSAFi , estimated from SAF curves. All the estimates are performed by means of the least square method.
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Fig. 16. DFA curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2. The arrows indicate the two linear regions, that are approximately present in all the curves.
Fig. 17. SDFA curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2.
decrease of the clusterization of the process; the values of haSAF i and ahSAFi assume almost the same value, that is approximately constant. Figs. 16–18 concern DFA. In particular, Fig. 16 shows the fluctuations F ðnÞ n with increasing the threshold magnitude Mth ; we can observe the existence of two linear regions for the most of curves: one corresponding to small box sizes (small timescales) and another corresponding to large box sizes (long timescales). We calculated the DFA scaling exponent dDFA in the range of large box sizes; the results are plotted in Fig. 18, that shows a slight decrease of the exponent with the increase of Mth . Fig. 17 presents the mean shuffled fluctuation (SDFA) curves for each Mth ; all the curves are characterized to explain linear behaviour at all box sizes. Fig. 18 shows the variation of the scaling exponents: dDFA ðMth Þ, hdSDFA i and dhSDFAi , obtained analogously to haSFF i and ahSFFi ; contrarily to dDFA , hdSDFA i and dhSDFAi are almost constant, fluctuating around 0.6, denoting a lowering effect in the long-range correlation structure of the seismic sequences operated by the shuffling procedure. In order to probe the existence of locally correlated sequences, we constructed a so-called observation window (a probe) of 150 events size placed at the beginning of the seismic sequence, and we calculated the AF scaling exponent aAF , the DFA scaling exponent dDFA , the coefficient of variation CV and the mean interevent time hT i for the data in that window. The window size was chosen in somewhat sense arbitrarily, but enough to evaluate the slope of the local
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Fig. 18. Variation with the threshold magnitude 2:2 6 Mth 6 3:2 of dDFA , estimated for the DFA curves in the linear region corresponding to the large box sizes; hdSDFA i, obtained averaging the scaling DFA exponents of the shuffled SDFA curves, and dhSDFAi , estimated from SDFA curves. All the estimates are performed by means of the least square method.
Fig. 19. Temporal variation of aAF (sequence of events with M P 2:2); in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows.
AF curve [44], and sufficient to estimate the slope of the local F ðnÞ n [27]; this approach evidences global scaling relations between events of the catalogue. The shift between successive windows was set at one event, permitting sufficient smoothing among the values of the exponents. Each calculated value is associated with the time of the last event in the window. We performed the analysis only for the sequence of events with M P 2:2. Fig. 19 shows the time evolution of aAF , in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows. It is clearly visible that aAF sharply increases in correspondence to the stronger events: in particular the maximum value of aAF after the first event ðM ¼ 5:0Þ could be due to the aftershock activation of this strong event, or, maybe, to the foreshock activation before the occurrence of the strongest event ðM ¼ 5:9Þ on September 26, 1997. Fig. 20 shows the time variation of dDFA ; the evolution of this parameter is very similar to the evolution of the AF scaling exponent, with marked high values in relation with the events with higher magnitude. Fig. 21 presents the time evolution of the CV , that assumes the maximum value in correspondence to the strongest event in the sequence; other larger events are associated with larger values of CV . Fig. 22 presents the time evolution of the mean interevent time hT i, calculated averaging the interevent
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Fig. 20. Temporal variation of dDFA (sequence of events with M P 2:2); in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows.
Fig. 21. Temporal variation of CV (sequence of events with M P 2:2); in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows.
time intervals in each window; we can observe that this parameter tends to increase approaching an asymptotic value, corresponding to the mean rate of the background seismicity, but a little before the time of occurrence of a strong event, it sharply decreases, maybe due to the presence of foreshocks.
5. Conclusions In the present study, we performed a deep fractal analysis of the 1996–1999 Umbria–Marche region (central Italy) seismicity. The AFA, FFA and DFA, using different representation of the same sequence (counting and interevent interval representations, respectively) have highlighted the following features: (i) the Umbria–Marche seismic process scales in time; (ii) the scaling exponents, that quantify the time-clusterization of the earthquake sequence, seem to decrease with increasing the threshold magnitude; (iii) the scaling exponents evolve with time, showing a clear correspondence between maxima and the larger events occurred in the area investigated.
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Fig. 22. Temporal variation of hT i (sequence of events with M P 2:2); in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows.
References [1] Allan DW. Statistics of atomic frequency standards. Proc IEEE 1966;54:221–30. [2] Bak P, Tang C. Earthquakes as a self-organized critical phenomenon. J Geophys Res 1989;94:15635–7. [3] Bally AW, Burbi L, Cooper C, Ghelardoni R. Balanced sections and seismic reflection profiles across the Central Apennines. Mem Soc Geol It 1988;35:257–310. [4] Bittner HR, Tosi P, Braun C, Maeesmann M, Kniffki KD. Counting statistics of f –b fluctuations: a new method for analysis of earthquake data. Geol Rundsch 1996;85:110–5. [5] Boncio P, Lavecchia G. A structural model for active extension in central Italy. J Geodyn 2000;29:233–44. [6] Boschi E, Gasperini P, Mulargia F. Forecasting where larger crustal earthquakes are likely to occur in Italy in the near future. Bull Seism Soc Am 1995;85:1475–82. [7] Calamita F, Coltorti M, Piccinini D, Pierantoni PP, Pizzi A, Ripepe M, et al. Quaternary faults and seismicity in the UmbriaMarchean Apennines (central Italy): evidences from the 1997 Colfiorito earthquake. J Geodyn 2000;29:245–64. [8] Calamita F, Cello G, Deiana G, Paltrinieri W. Structural styles, chronology-rates of deformation and time space relationships in the Umbria-Marche thrust system (Central Apennines, Italy). Tectonics 1994;13:873–81. [9] Carlson JM, Langer JS. Properties of earthquakes generated by fault dynamics. Phys Rev Lett 1989;62:2632. [10] Cattaneo M, Augliera P, De Luca G, Gorini A, Govoni A, Marcucci S, et al. The 1997 Umbria-Marche (Italy) earthquake sequence: analysis of the data recorded by the local and temporary networks. J Seismol 2000;4:401–14. [11] Cello G, Mazzoli S, Tondi E, Turco E. Active tectonics in the Central Apennines and possible implications for seismic hazard analysis in peninsular Italy. Tectonophys 1997;272:43–68. [12] Cello G, Turco E, Zuppetta A. Cinematica del settore centrale del Mediterraneo ed implicazioni sulla struttura dellÕarco calabro. In: Bonardi G, DeVivo B, Gasparini P, Vallario A, editors. CinquantÕanni di attivita didattica del Professor Felice Ippolito. Napoli: Liguori Editore; 1995. p. 293. [13] Chen K, Bak P, Obukhov SP. Self-organized criticality in a crack-propagation model of earthquakes. Phys Rev A 1991;43:625–30. [14] Cinti FR, Cucci L, Marra F, Montone P. The 1997 Umbria-Marche earthquakes (Italy): relation between the surface tectonic breaks and the area of deformation. J Seismol 2000;4:333–43. [15] Cox DR, Isham V. In: Point processes. London: Chapman and Hall; 1980. [16] Deschamps A, Courboulex F, Gaffet S, Lomax A, Virieux J, Amato A, et al. Spatio-temporal distribution of seismic activity during the Umbria-Marche crisis 1997. J Seismol 2000;4:377–86. [17] Di Giovambattista R, Tyupkin YS. Spatial and temporal distribution of seismicity before the Umbria-Marche September 26, 1997 earthquakes. J Seismol 2000;4:589–98. [18] Esposito E, Porfido S, Simonelli AL, Mastrolorenzo G, Iaccarino G. Landslides and other surface effects induced by the 1997 Umbria-Marche seismic sequence. Eng Geol 2000;58:353–76. [19] Favara R, Italiano F, Martinelli G. Earthquake-induced chemical changes in the thermal waters of the Umbria region during the 1997–1998 seismic swarm. Terra Nova 2001;13:227–33. [20] Galadini F, Galli P, Leschiutta I, Monachesi G, Stucchi M. Active tectonics and seismicity in the area of the 1997 earthquake sequence in central Italy: a short review. J Seismol 1999;3:167–75. [21] Gutenberg B, Richter CF. Frequency of earthquakes in California. Bull Seism Soc Am 1944;34:185–8. [22] Kagan YY, Jackson DD. Long-term earthquake clustering. Geophys J Int 1991;104:117–33. [23] Kagan YY. Observational evidence for earthquakes as a nonlinear dynamic process. Phys D 1994;77:160–92.
L. Telesca et al. / Chaos, Solitons and Fractals 18 (2003) 203–217 [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
[43] [44] [45] [46] [47] [48] [49]
217
Kagan YY, Knopoff L. Statistical short-term earthquake prediction. Science 1987;236:1563. Hirata T. A correlation between the b value and the fractal dimension of earthquakes. J Geophys Res 1989;94:7507–14. Ito K, Matsuzaki M. Earthquakes as a self-organized critical phenomena. J Geophys Res 1990;95:6853–60. Ivanova K, Ausloos M. Application of the detrended fluctuation analysis (DFA) method for describing cloud breaking. Phys A 1999;274:349–54. Lavecchia G, Brozzetti F, Barchi M, Menichetti M, Keller JVA. Seismotectoning zoning in East-central Italy deduced from an analysis from the Neogene to present deformations and related stress fields. Geol Soc Am Bull 1994;106:1107–20. Lapenna V, Macchiato M, Telesca L. 1=f b fluctuations and self-similarity in earthquake dynamics: observational evidences in southern Italy. Phys Earth Planet Int 1998;106:115–27. Lowen SB, Teich MC. Estimating the dimension of a fractal point process. In: 64/SPIE chaos in biology and medicine, vol. 2036; 1993. Lowen SB, Teich MC. Estimation and simulation of fractal stochastic point processes. Fractals 1995;3:183–210. Mandelbrot BB. In: The fractal geometry of Nature. San Francisco: Freeman Press; 1982. Morelli A, Ekstr€ om G, Olivieri M. Source properties of the 1997–98 central Italy earthquake sequence from inversion of longperiod and broad-band seismograms. J Seismol 2000;4:365–75. Patacca E, Sartori R, Scandone P. Tyrrhenian basin and Apennine arcs: kinematics relations since late Tortonian times. Mem Soc Geol It 1992;3:1–8. Peng C-K, Havlin S, Stanley HE, Goldberger AL. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 1995;5(1):82–7. Pergalani F, Romeo R, Luzi L, Petrini V, Pugliese A, San o T. Seismic microzoning of the area struck by Umbria-Marche (central Italy) Ms 5.9 earthquake of 26 September 1997. Soil Dyn Earthq Eng 1999;18:279–96. Quattrocchi F, Pik R, Pizzino L, Guerra M, Scarlato P, Angelone M, et al. Geochemical changes at the Bagni di Triponzo thermal spring during the Umbria-Marche 1997–1998 seismic sequence. J Seismol 2000;4:567–87. Ripepe M, Piccinini D, Chiaraluce L. Foreshock sequence of the September 26th, 1997 Umbria-Marche earthquakes. J Seismol 2000;4:387–99. Shi Y, Bolt BA. The standard error of the magnitude-frequency b value. Bull Seism Soc Am 1982;72:1677–87. Sornette A, Sornette D. Self-organizing criticality and earthquakes. Europhys Lett 1989;9(2):197. Sornette A, Dubois J, Cheminee JL, Sornette D. Are sequence of volcanic eruptions deterministrically chaotic? J Geophys Res 1991;96(B7):11931–45. Teich MC, Heneghan C, Lowen SB, Turcott RG. Estimating the fractal exponent of point processes in bilogical systems using wavelet- and Fourier-transform methods. In: Aldroubi A, Unser M, editors. Wavelets in medicine and biology. Boca Raton, FL: CRC Press; 1996. p. 383–412. Telesca L, Cuomo V, Lapenna V, Macchiato M. Analysis of the time-scaling behaviour in the sequence of the aftershocks of the Bovec (Slovenia) April 12, 1998 earthquake. Phys Earth Planet Int 2000;120:315–26. Telesca L, Cuomo V, Lapenna V, Macchiato M. Identifying space-time clustering properties of the 1983–1997 Irpinia-Basilicata (southern Italy) seismicity. Tectonophysics 2001;330:93–102. Telesca L, Lapenna V, Vallianatos F. Monofractal and multifractal approaches in investigatine scaling properties in temporal patterns of the 1983–2000 seismicity in the western Corinth graben, Greece. Phys Earth Planet Int 2002;131:63–79. Thurner S, Lowen SB, Feurstein MC, Heneghan C, Feichtinger HG, Teich MC. Analysis, synthesis, and estimation of fractal-rate stochastic point processes. Fractals 1997;5:565–96. Turcotte DL. Fractals and chaos in geology and geophysics. Cambridge University Press; 1995. Vittori E, Maschio L, Ferreli L, Michetti AM, Serva L. Carta e base di dati delle faglie capaci per lÕItalia centro-meridionale: presentazione e stato di avanzamento del progetto ITHACA. Il Quatern 1997;10(2):305–21. Vittori E, Deiana G, Esposito E, Ferreli L, Marchegiani L, Mastrolorenzo G, et al. Ground effects and surface faulting in the September–October 1997 Umbria-Marche (central Italy) seismic sequence. J Geodyn 2000;29:535–64.
Investigating the time-clustering properties in seismicity of Umbria–Marche region (central Italy) Luciano Telesca a
a,*
, Vincenzo Lapenna a, Maria Macchiato
b
Institute of Methodologies for Environmental Analysis, National Research Council, C de S. Loja, Tito Scalo PZ I-85050, Italy b Dipartimento di Scienze Fisiche, Universit a ‘‘Federico II’’, Naples, Italy
Abstract We investigated by fractal tools the temporal patterns of 1996–1999 seismicity of the Umbria–Marche region, central Italy, that was struck by a strong event ðMS ¼ 5:9Þ on September 26, 1997. Clustering behaviour has been evidenced by using different fractal tools, Fano factor analysis, Allan factor analysis and detrended fluctuation analysis, that show power-law shaped statistics, whose scaling exponents describe the degree of clusterization in the earthquake sequence. Furthermore, the scaling exponents evolve in time showing an increase of the clustering features in correspondence to larger events. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction The dynamical variability of many complex geophysical phenomena can be described by means of the concepts of scale-invariance, fractals and multifractals [32], which represent useful tools to characterize and interpret the inhomogeneity, which features that variability. As example of inhomogeneous dynamics, seismic processes have been recently analysed in order to reveal their time-scaling properties, which can be very informative concerning the geodynamical mechanisms underlying the presence of scaling laws in earthquake statistics. The presence of time-scaling behaviours suggests that the seismic process is clusterized in time, and the scaling exponents give a quantitative measure of the degree of such clusterization. The analysis of the earthquake catalogues have shown that seismic processes are characterized by scaling behaviour concerning the distributions of magnitudes, epicentres and time-occurrences. Gutenberg and Richter [21] found that there is no a characteristic rupture size and that the magnitude distribution follows a self-similar relation. Kagan and Jackson [22] and Kagan [23] showed that clustering behaviours feature the time-occurrence sequences of earthquakes on the long-term timescale as well as on the short-term scale, due to the presence of aftershocks and foreshocks, which characterize large shallow earthquakes [24]. The spatial distribution of earthquakes was found to be clusterized on several scales of observation [25,44]. The self-organized criticality (SOC) theory can explain the scale-invariance phenomenology typical of earthquake sequences and it has been first applied to seismic processes by Bak and Tang [2]. The observational evidence of the presence of scale-invariant behaviour in earthquake dynamics has led to the development of a wide variety of physical models of seismogenesis. As shown in Sornette and Sornette [40] and Sornette et al. [41], earthquakes are an important part of the relaxation mechanism of the crust which is submitted to inhomogeneous increasing stresses accumulating at continental-plate borders; and the SOC idea implies that earthquakes in turn organize the crust both in the spatial and temporal domains. Seismicity patterns have been explained using theoretical
*
Corresponding author. Tel.: +39-971-427-206; fax: +39-971-427-271. E-mail address: [email protected] (L. Telesca).
0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(02)00654-9
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models for earthquake generation and propagation [9,13,26,47]. Identifying the time-occurrence distribution in earthquake sequences is an important task in the context of studies related to the seismic hazard. The Poisson distribution of the interevent times implies that each seismic event is independent from the time elapsed since the previous event, thus leading to a memoryless process, which is characterized by uncorrelation among the events. Although a simple discrete Poisson distribution provides a close fit for the sequence of large events [6], in recent studies it has been shown that earthquake occurrence is characterized by temporal clustering properties with both short and long timescales [22], and this implies that an earthquake sequence can be modelled by means of correlated processes. In correlated processes, it is an important task to identify the type of the temporal fluctuations, which are connected with the power spectral density Sðf Þ of the process itself, which, for scaling processes, is a power-law function of the frequency f , Sðf Þ / 1=f a , with a the most characterizing parameter of the time-clustering of the process. Several methods can be used to estimate the power spectral density of seismic processes. Bittner et al. [4] and Lapenna et al. [29] used the variance–time curve (VTC) method, that for spectra with power-law (fractal) shape furnishes a relation with the spectral exponent a. The Fano factor (FF) and the Allan factor (AF) have been shown to have a power-law behaviour for processes with scaling properties [43–45]. The numerical value of a quantifies the degree of time-clusterization of the seismic process [30], revealing the presence of Poissonian ða 0Þ or clusterized ða 6¼ 0Þ temporal fluctuations. In this paper, we analyzed the earthquake sequence from 1996 to 1999 in the Umbria–Marche region, central Italy, by means of fractal tools in order to qualify and quantify the presence of dynamical clusterization.
2. Mathematical background Point process describe sequences of events occurring randomly in time [15]. We can represent such processes by a finite sum of DiracÕs delta functions centred on the occurrence times ti yðtÞ ¼
n X
dðt ti Þ:
ð1Þ
i¼1
The process can be described equivalently by the set of the interevent times or by the associated counting process. The second representation is given by dividing the time axis into equally spaced contiguous counting windows of duration s, and producing a sequence of counts fZk ðsÞg, with Zk ðsÞ denoting the number of earthquakes in the kth window: Z tk X n Zk ðsÞ ¼ dðt tj Þ dt: ð2Þ tk1 j¼1
The different representations lead to different methods to reveal and analyze the presence of clustering features in a seismic process. A commonly used measure to evaluate the clustering behaviour of a seismic sequence has been the coefficient of variation CV , defined as CV ¼ rT =hT i;
ð3Þ
where hT i is the mean interevent time and rT is its standard deviation [22]: a Poissonian process (completely random) has a CV ¼ 1, but a clusterized process is characterized by a CV > 1. This coefficient does not give information about the timescale ranges where the process can be reliably characterized as a clustered process. Nevertheless, a complex phenomenon can be deeply known only if the different timescales governing its dynamics are well understood. The detrended fluctuation analysis (DFA) performs the clustering analysis on the series of the interevent intervals. It was proposed by Peng et al. [35], and avoids spurious detection of correlations that are artifacts of nonstationarity, that often affects experimental data. Such trends have to be well distinguished from the intrinsic fluctuations of the system in order to find the correct scaling behaviour of the fluctuations. Very often we do not know the reasons for underlying trends in collected data and we do not know the scales of underlying trends. DFA is a method for determining the scaling behaviour of data in the presence of possible trends without knowing their origin and shape. The methodology operates on the time series xðiÞ, where i ¼ 1; 2; . . . ; N and N is the length of the series. With xave we indicate the mean value xave ¼
N 1 X xðkÞ: N k¼1
ð4Þ
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The signal is first integrated yðkÞ ¼
k X ½xðiÞ xave :
ð5Þ
i¼1
Next, the integrated time series is divided into boxes of equal length, n. In each box a least-squares line is fit to the data, representing the trend in that box. The y-coordinate of the straight line segments is denoted by yn ðkÞ. Next we detrend the integrated time series yðkÞ by subtracting the local trend yn ðkÞ in each box. The root-mean-square fluctuation of this integrated and detrended time series is calculated by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X ½yðkÞ yn ðkÞ 2 : F ðnÞ ¼ t ð6Þ N k¼1 Repeating this calculation over all box sizes, we obtain a relationship between F ðnÞ, that represents the average fluctuation as a function of box size, and the box size n. If F ðnÞ behaves as a power-law function of n, data present scaling F ðnÞ / nd :
ð7Þ
Under these conditions the fluctuations can be described by the scaling exponent d, representing the slope of the line fitting log F ðnÞ to log n. For a white noise process, d ¼ 0:5. If there are only short-range correlations, the initial slope may be different from 0.5 but will approach 0.5 for large window sizes. 0:5 < d < 1:0 indicates the presence of persistent long-range correlations, meaning that a large (compared to the average) value is more likely to be followed by large value and vice versa. 0 < d < 0:5 indicates the presence of antipersistent long-range correlations, meaning that a large (compared to the average) value is more likely to be followed by small value and vice versa. d ¼ 1 indicates flicker-noise dynamics, typical of systems in a self-organized critical state. d ¼ 1:5 characterizes processes with Brownian-like dynamics. The Allan factor analysis (AFA) concerns the calculation of the AF, that is a measure which can be used to distinguish fractal from Poissonian temporal fluctuations in point processes [1]. This factor is defined as the variance of successive counts for a specified counting time s divided by twice the mean number of events in that counting time AFðsÞ ¼
hðZkþ1 ðsÞ Zk ðsÞÞ2 i : 2hZk ðsÞi
The AF of a fractal point process varies with the counting time s with a power-law form a s AFðsÞ ¼ 1 þ : s1
ð8Þ
ð9Þ
The monotonic power-law increase is representative of the presence of fluctuations on many timescales [31]; s1 is the socalled fractal onset time, and marks the lower limit for significant scaling behaviour in the AF [42], so that for s s1 the clustering property becomes negligible within these timescales. For Poissonian processes the AF assumes approximately values near or below unity for all counting times s. From Eq. (9), the calcualtion of a can be performed by estimating the slope of the straight line that fits in a least square sense the AF curve, plotted in log–log scales. Of course, only the linear part of the curve will be considered to calculate the fractal exponent. Another method is the Fano factor analysis (FFA). It is based on the FF, that is defined as the variance of the number of events in a specified counting time s divided by the mean number of events in that counting time; that is FFðsÞ ¼
hNk2 ðsÞi hNk ðsÞi2 ; hNk ðsÞi
where h i denotes the expectation value. The FF of a fractal point process with 0 < a < 1 varies as a function of counting time s as a s FFðsÞ ¼ 1 þ ; s0 where s0 is the fractal onset time, similar to s1 .
ð10Þ
ð11Þ
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It is related to the AF by AFðsÞ ¼ 2FFðsÞ FFð2sÞ
ð12Þ
and it assumes unitary value for Poissonian processes. The exponent a conveys the information about the temporal fluctuations of the process, because it is the exponent of the power spectrum, that for fractal processes decays as a power-law function of the frequency f , Sðf Þ / f a with a measuring the strength of the clusterization [46].
3. Tectonic settings The Umbria–Marche region, central Italy, has been subjected to a great scientific interest, after the strong September 26, 1997 seismic event ðMS ¼ 5:9Þ that struck the area, and many studies have been performed regarding its geodynamical features [5,7,49,36], active tectonics [14,20,33], spatio-temporal seismic distribution [10,16,17,38], induced geophysical effects [18,19,37]. The central Apennines are made up of several tectonic units put straight since the Oligocene as a results of convergence and collision between the continental margins of the Corsica-Sardinia block and the Adriatic block [12]. The main compressive phase started in the Tortonian and the lack of Pliocene-Pleistocene marine deposits prove that after the Miocene the area was definitively uplifted [36]. The compressive structures were dissected by normal faults during the Quaternary, and, according to the most recent studies [8,28], these are related to the crustal thinning processes occurring in the Tyrrhenian Tuscan area. The Quaternary normal faults led to the formation of intramountain basins, and the seismicity of the area is mainly related to the activity of these faults. The Umbria–Marche region is part of the east-verging Neogene thrust and fold belt of central Italy [3,8,34]. Middle-late Quaternary faults and intermontane tectonic depressions, overprinted onto the thrust structure, are the preferential loci of strong to moderate earthquakes, as shown by historical and instrumental data and geological evidence [11,48,49]. The Umbria–Marche region is characterized by a well-documented historical and instrumental seismicity, mainly confined within the upper part of the crust (<16 km) [28]. West of the Tiber Basin, the seismic activity is the lowest. It increases remarkably within the Apennine area, where small to moderate ð4 < M < 6Þ earthquakes are frequent, while largest earthquakes ð6 < M < 7Þ have long recurrence-intervals [5]. Four highly damaging earthquakes (1279 Camerino, 1328 Norcia, 1703 Norcia and 1751 Gualdo Tadino) have occurred in the last centuries within the Citt a di Castello–Gubbio–Gualdo Tadino–Norcia seismic band. In the same area the largest instrumental earthquakes of the last 20 years (M5.9 1979 Norcia, M5.2 1984 Gubbio, M5.9 1997 Colfiorito) are located [5]. The analysed earthquake data were extracted by the Bulletin of Osservatorio Geofisico Sperimentale of Macerata (OGSM), available on the Internet site at http://www.geofisico.mc.it, and cover the period 1996–1999.
Fig. 1. Map of the Umbria–Marche region (central Italy). The stars indicate the stronger events in the area during the seismic crisis of September–October 1997 (. . .).
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Fig. 2. Gutenberg-Richter analysis of the earthquake catalogue. The magnitude of completeness is estimated as 2.2 with b 1.
4. Results Now we analyze the earthquake activity in the Umbria–Marche area (central Italy) (Fig. 1). We performed the Gutenberg–Richter analysis (Fig. 2) by least square method over the cumulative distribution of the events, to obtain the minimum magnitude from which the catalogue could be considered complete [39]: The catalogue is complete for events with magnitude M P 2:2 with b 1. The analyzed sequence contains 4672 events with magnitude M P 2:2. We computed the FFðsÞ and AFðsÞ for counting times of duration of 10 s to P /10 s, where P is the period of the catalogue. The FF and AF plots (Figs. 3 and 4) give clear indication of the fractal behaviour. Both the FF and AF are seen to increase with linear form in log–log scales, and this indicates the presence of range-correlated structures. The early flat behaviour can be interpreted as the presence of Poissonian dynamics for short timescales s. The slope of the fitting straight line of the FF and AF curves in the linear range gives the estimation of the fractal exponent a of the point process modelling the earthquake sequence in our study. In particular, we estimated aFF 0:84 and aAF 1:02 (for earthquakes with M P 2:2); the difference in the numerical value is due to a saturation effect that FF method suffers [46].
Fig. 3. Results of the FFA performed on the sequence of events with M P 2:2. The power-law behaviour is clearly visible from approximately 6 103 to 1:3 106 s, with the estimate of the scaling exponent aFF 0:84.
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Fig. 4. Results of the AFA performed on the sequence of events with M P 2:2. The power-law behaviour is clearly visible from approximately 3 104 to 3 106 s, with the estimate of the scaling exponent aAF 1:02.
Fig. 5. Interevent time distribution for the sequence of events with M P 2:2.
We analyzed the dynamics of the interevent time series calculating the coefficient of variation and performing the detrended fluctuation analysis (DFA) of the interspike sequence. Fig. 5 shows the distribution of the interevent intervals; the very low amplitude of the interevent series in the medium part of the plot corresponds to the seismic sequence initiated by the M5.9 earthquake, occurred on September 26, 1997. For this series we calculated CV 3:08, indicating a clustering effect. Fig. 6 shows the DFA performed on the interevent time series for earthquakes with M P 2:2. Two scaling regions are evidenced by the analysis; the first region, corresponding to short timescales, is characterized by the scaling exponent dDFA 0:68, which indicates a slightly persistent dynamics; in the second region, corresponding to long timescales, the scaling behaviour is revealed by the scaling coefficient dDFA 1:13, that denotes a flicker-noise dynamics. The results are consistent with the estimates of the scaling exponents performed by means of the AF and FF methods. In order to analyze the nature of the fractal structure of the seismic point process, we performed the FFA on several shuffled versions of the original sequence, thus preserving the interevent time and variance of the series. We calculated the FF curves, and then we averaged them obtaining the shuffled mean FF curve (SFF). In Fig. 7, we plotted the original FF and the SFF curves ðMth ¼ 2:2Þ. We can observe that the SFF curve presents a lower value of the scaling exponent a, approximately 0.4, and this indicates that the fractal behaviour of the seismic sequence depends on the ordering of the interevent intervals [46]. In Figs. 8 and 9, we reported analogous plots regarding the AFA and DFA, which clearly show the lowering effect on the clustering behaviour induced by the shuffling procedure.
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Fig. 6. Results of the DFA performed on the sequence of events with M P 2:2. Two scaling regions are visible, with different value of the scaling coefficient, dDFA 0:68 and dDFA 1:13, respectively for the small timescale and the long timescale ranges.
Fig. 7. Comparison between the original FF and the SFF curves ðMth ¼ 2:2Þ. The SFF curve presents a lower value of the scaling exponent ahSFFi 0:4.
In order to evidence possible variation of the temporal behaviour of the earthquake process under investigation with the lower threshold magnitude, we performed FFA, AFA and DFA for increasing values of Mth . Since the number of events decreases rapidly when Mth increases, the analysis has been stopped at the magnitude Mth ¼ 3:2 (604 events). Fig. 10 shows the FF curves related to the seismic sequences with varying threshold magnitude Mth . Varying the threshold magnitude, the behavioural trend of the curves does not change, although it is clear a variation of the slope of the linear part, corresponding to the intermediate timescale range, and a flattening effect, that becomes more apparent at long timescales and higher threshold magnitudes. This is probably due to a limited-number effect, visible increasing the threshold magnitude. We observe that the slopes, corresponding to the linear range of the curves, indicated by the arrows, seem to decrease with the increase of the threshold magnitude, and this denotes a decrease of the clustering effect. Fig. 12 shows the variation of the scaling exponent aFF ðMth Þ, estimated as the slope of the lines fitting each curve in the temporal range between approximately 6 103 and 1:3 106 s, that is the range of timescales commonly involved in the power-law behaviour of all the curves. Furthermore, the figure shows the variation with Mth of other two parameters: haSFF i, that is the average of the FF scaling exponents calculated for the FF curves of shuffled versions of the
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Fig. 8. Comparison between the original AF and the SAF curves ðMth ¼ 2:2Þ. The SAF curve presents a lower value of the scaling exponent ahSAFi 0:36.
Fig. 9. Comparison between the original DFA and the SDFA curves ðMth ¼ 2:2Þ. The SDFA curve presents a lower value of the scaling exponent dhSDFAi 0:62, involving all the box sizes.
earthquake sequence, and ahSFFi , that is the FF scaling exponent calculated for the SFF; both the parameters are calculated in the range between 103 and 5 105 s, that seem to be the common linear range of all the shuffled curves (Fig. 11). We observe that the aFF decrease with Mth , indicating a decrease of the clusterization of the process; the values of haSFF i and ahSFFi are almost coincident and approximately constant with varying Mth , and, furthermore, they fluctuate around the value of 0.38–0.4. From this result, we can conclude that for all the threshold magnitudes the scaling behaviour depends on the ordering of the interevent intervals; and this means that the clustering lies in the particular temporal structure of the earthquake series. Fig. 13 shows the AF curves with variable threshold magnitude Mth . Similarly to the FF case, the slopes of the linear part of the curves seem to decrease with the increase of the threshold magnitude. Fig. 14 presents the mean shuffled AF (SAF) curves for each Mth ; all the curves show an approximately linear behaviour in the common timescale range between 2:7 103 and 2:3 106 s. Fig. 15 shows the variation of the scaling exponents: aAF ðMth Þ, estimated for each AF curve in the temporal range between approximately 3 104 and 3 106 s, as indicated by the arrows in Fig. 13, haSAF i and ahSAFi , obtained analogously to haSFF i and ahSFFi . We observe that the values of aAF decrease with Mth , denoting a
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Fig. 10. FF curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2. The arrows indicate the timescale linear range that involves all the curves.
Fig. 11. SFF curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2 The arrows indicate the timescale linear range that involves all the curves.
Fig. 12. Variation with the threshold magnitude 2:2 6 Mth 6 3:2 of: aFF , estimated for the FF curves; haSFF i, obtained averaging the scaling exponents of the shuffled FF curves, and ahSFFi , estimated from SFF curves. All the estimates are performed by means of the least square method.
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Fig. 13. AF curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2. The arrows indicate the timescale linear range that involves all the curves.
Fig. 14. SAF curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2. The arrows indicate the timescale linear range that involves all the curves.
Fig. 15. Variation with the threshold magnitude 2:2 6 Mth 6 3:2 of: aAF , estimated for the AF curves; haSAF i, obtained averaging the scaling exponents of the shuffled AF curves, and ahSAFi , estimated from SAF curves. All the estimates are performed by means of the least square method.
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Fig. 16. DFA curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2. The arrows indicate the two linear regions, that are approximately present in all the curves.
Fig. 17. SDFA curves related to the seismic sequences with varying threshold magnitude 2:2 6 Mth 6 3:2.
decrease of the clusterization of the process; the values of haSAF i and ahSAFi assume almost the same value, that is approximately constant. Figs. 16–18 concern DFA. In particular, Fig. 16 shows the fluctuations F ðnÞ n with increasing the threshold magnitude Mth ; we can observe the existence of two linear regions for the most of curves: one corresponding to small box sizes (small timescales) and another corresponding to large box sizes (long timescales). We calculated the DFA scaling exponent dDFA in the range of large box sizes; the results are plotted in Fig. 18, that shows a slight decrease of the exponent with the increase of Mth . Fig. 17 presents the mean shuffled fluctuation (SDFA) curves for each Mth ; all the curves are characterized to explain linear behaviour at all box sizes. Fig. 18 shows the variation of the scaling exponents: dDFA ðMth Þ, hdSDFA i and dhSDFAi , obtained analogously to haSFF i and ahSFFi ; contrarily to dDFA , hdSDFA i and dhSDFAi are almost constant, fluctuating around 0.6, denoting a lowering effect in the long-range correlation structure of the seismic sequences operated by the shuffling procedure. In order to probe the existence of locally correlated sequences, we constructed a so-called observation window (a probe) of 150 events size placed at the beginning of the seismic sequence, and we calculated the AF scaling exponent aAF , the DFA scaling exponent dDFA , the coefficient of variation CV and the mean interevent time hT i for the data in that window. The window size was chosen in somewhat sense arbitrarily, but enough to evaluate the slope of the local
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Fig. 18. Variation with the threshold magnitude 2:2 6 Mth 6 3:2 of dDFA , estimated for the DFA curves in the linear region corresponding to the large box sizes; hdSDFA i, obtained averaging the scaling DFA exponents of the shuffled SDFA curves, and dhSDFAi , estimated from SDFA curves. All the estimates are performed by means of the least square method.
Fig. 19. Temporal variation of aAF (sequence of events with M P 2:2); in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows.
AF curve [44], and sufficient to estimate the slope of the local F ðnÞ n [27]; this approach evidences global scaling relations between events of the catalogue. The shift between successive windows was set at one event, permitting sufficient smoothing among the values of the exponents. Each calculated value is associated with the time of the last event in the window. We performed the analysis only for the sequence of events with M P 2:2. Fig. 19 shows the time evolution of aAF , in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows. It is clearly visible that aAF sharply increases in correspondence to the stronger events: in particular the maximum value of aAF after the first event ðM ¼ 5:0Þ could be due to the aftershock activation of this strong event, or, maybe, to the foreshock activation before the occurrence of the strongest event ðM ¼ 5:9Þ on September 26, 1997. Fig. 20 shows the time variation of dDFA ; the evolution of this parameter is very similar to the evolution of the AF scaling exponent, with marked high values in relation with the events with higher magnitude. Fig. 21 presents the time evolution of the CV , that assumes the maximum value in correspondence to the strongest event in the sequence; other larger events are associated with larger values of CV . Fig. 22 presents the time evolution of the mean interevent time hT i, calculated averaging the interevent
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Fig. 20. Temporal variation of dDFA (sequence of events with M P 2:2); in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows.
Fig. 21. Temporal variation of CV (sequence of events with M P 2:2); in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows.
time intervals in each window; we can observe that this parameter tends to increase approaching an asymptotic value, corresponding to the mean rate of the background seismicity, but a little before the time of occurrence of a strong event, it sharply decreases, maybe due to the presence of foreshocks.
5. Conclusions In the present study, we performed a deep fractal analysis of the 1996–1999 Umbria–Marche region (central Italy) seismicity. The AFA, FFA and DFA, using different representation of the same sequence (counting and interevent interval representations, respectively) have highlighted the following features: (i) the Umbria–Marche seismic process scales in time; (ii) the scaling exponents, that quantify the time-clusterization of the earthquake sequence, seem to decrease with increasing the threshold magnitude; (iii) the scaling exponents evolve with time, showing a clear correspondence between maxima and the larger events occurred in the area investigated.
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Fig. 22. Temporal variation of hT i (sequence of events with M P 2:2); in the bottom the larger earthquakes ðM P 4:5Þ are indicated by vertical arrows.
References [1] Allan DW. Statistics of atomic frequency standards. Proc IEEE 1966;54:221–30. [2] Bak P, Tang C. Earthquakes as a self-organized critical phenomenon. J Geophys Res 1989;94:15635–7. [3] Bally AW, Burbi L, Cooper C, Ghelardoni R. Balanced sections and seismic reflection profiles across the Central Apennines. Mem Soc Geol It 1988;35:257–310. [4] Bittner HR, Tosi P, Braun C, Maeesmann M, Kniffki KD. Counting statistics of f –b fluctuations: a new method for analysis of earthquake data. Geol Rundsch 1996;85:110–5. [5] Boncio P, Lavecchia G. A structural model for active extension in central Italy. J Geodyn 2000;29:233–44. [6] Boschi E, Gasperini P, Mulargia F. Forecasting where larger crustal earthquakes are likely to occur in Italy in the near future. Bull Seism Soc Am 1995;85:1475–82. [7] Calamita F, Coltorti M, Piccinini D, Pierantoni PP, Pizzi A, Ripepe M, et al. Quaternary faults and seismicity in the UmbriaMarchean Apennines (central Italy): evidences from the 1997 Colfiorito earthquake. J Geodyn 2000;29:245–64. [8] Calamita F, Cello G, Deiana G, Paltrinieri W. Structural styles, chronology-rates of deformation and time space relationships in the Umbria-Marche thrust system (Central Apennines, Italy). Tectonics 1994;13:873–81. [9] Carlson JM, Langer JS. Properties of earthquakes generated by fault dynamics. Phys Rev Lett 1989;62:2632. [10] Cattaneo M, Augliera P, De Luca G, Gorini A, Govoni A, Marcucci S, et al. The 1997 Umbria-Marche (Italy) earthquake sequence: analysis of the data recorded by the local and temporary networks. J Seismol 2000;4:401–14. [11] Cello G, Mazzoli S, Tondi E, Turco E. Active tectonics in the Central Apennines and possible implications for seismic hazard analysis in peninsular Italy. Tectonophys 1997;272:43–68. [12] Cello G, Turco E, Zuppetta A. Cinematica del settore centrale del Mediterraneo ed implicazioni sulla struttura dellÕarco calabro. In: Bonardi G, DeVivo B, Gasparini P, Vallario A, editors. CinquantÕanni di attivita didattica del Professor Felice Ippolito. Napoli: Liguori Editore; 1995. p. 293. [13] Chen K, Bak P, Obukhov SP. Self-organized criticality in a crack-propagation model of earthquakes. Phys Rev A 1991;43:625–30. [14] Cinti FR, Cucci L, Marra F, Montone P. The 1997 Umbria-Marche earthquakes (Italy): relation between the surface tectonic breaks and the area of deformation. J Seismol 2000;4:333–43. [15] Cox DR, Isham V. In: Point processes. London: Chapman and Hall; 1980. [16] Deschamps A, Courboulex F, Gaffet S, Lomax A, Virieux J, Amato A, et al. Spatio-temporal distribution of seismic activity during the Umbria-Marche crisis 1997. J Seismol 2000;4:377–86. [17] Di Giovambattista R, Tyupkin YS. Spatial and temporal distribution of seismicity before the Umbria-Marche September 26, 1997 earthquakes. J Seismol 2000;4:589–98. [18] Esposito E, Porfido S, Simonelli AL, Mastrolorenzo G, Iaccarino G. Landslides and other surface effects induced by the 1997 Umbria-Marche seismic sequence. Eng Geol 2000;58:353–76. [19] Favara R, Italiano F, Martinelli G. Earthquake-induced chemical changes in the thermal waters of the Umbria region during the 1997–1998 seismic swarm. Terra Nova 2001;13:227–33. [20] Galadini F, Galli P, Leschiutta I, Monachesi G, Stucchi M. Active tectonics and seismicity in the area of the 1997 earthquake sequence in central Italy: a short review. J Seismol 1999;3:167–75. [21] Gutenberg B, Richter CF. Frequency of earthquakes in California. Bull Seism Soc Am 1944;34:185–8. [22] Kagan YY, Jackson DD. Long-term earthquake clustering. Geophys J Int 1991;104:117–33. [23] Kagan YY. Observational evidence for earthquakes as a nonlinear dynamic process. Phys D 1994;77:160–92.
L. Telesca et al. / Chaos, Solitons and Fractals 18 (2003) 203–217 [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
[43] [44] [45] [46] [47] [48] [49]
217
Kagan YY, Knopoff L. Statistical short-term earthquake prediction. Science 1987;236:1563. Hirata T. A correlation between the b value and the fractal dimension of earthquakes. J Geophys Res 1989;94:7507–14. Ito K, Matsuzaki M. Earthquakes as a self-organized critical phenomena. J Geophys Res 1990;95:6853–60. Ivanova K, Ausloos M. Application of the detrended fluctuation analysis (DFA) method for describing cloud breaking. Phys A 1999;274:349–54. Lavecchia G, Brozzetti F, Barchi M, Menichetti M, Keller JVA. Seismotectoning zoning in East-central Italy deduced from an analysis from the Neogene to present deformations and related stress fields. Geol Soc Am Bull 1994;106:1107–20. Lapenna V, Macchiato M, Telesca L. 1=f b fluctuations and self-similarity in earthquake dynamics: observational evidences in southern Italy. Phys Earth Planet Int 1998;106:115–27. Lowen SB, Teich MC. Estimating the dimension of a fractal point process. In: 64/SPIE chaos in biology and medicine, vol. 2036; 1993. Lowen SB, Teich MC. Estimation and simulation of fractal stochastic point processes. Fractals 1995;3:183–210. Mandelbrot BB. In: The fractal geometry of Nature. San Francisco: Freeman Press; 1982. Morelli A, Ekstr€ om G, Olivieri M. Source properties of the 1997–98 central Italy earthquake sequence from inversion of longperiod and broad-band seismograms. J Seismol 2000;4:365–75. Patacca E, Sartori R, Scandone P. Tyrrhenian basin and Apennine arcs: kinematics relations since late Tortonian times. Mem Soc Geol It 1992;3:1–8. Peng C-K, Havlin S, Stanley HE, Goldberger AL. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 1995;5(1):82–7. Pergalani F, Romeo R, Luzi L, Petrini V, Pugliese A, San o T. Seismic microzoning of the area struck by Umbria-Marche (central Italy) Ms 5.9 earthquake of 26 September 1997. Soil Dyn Earthq Eng 1999;18:279–96. Quattrocchi F, Pik R, Pizzino L, Guerra M, Scarlato P, Angelone M, et al. Geochemical changes at the Bagni di Triponzo thermal spring during the Umbria-Marche 1997–1998 seismic sequence. J Seismol 2000;4:567–87. Ripepe M, Piccinini D, Chiaraluce L. Foreshock sequence of the September 26th, 1997 Umbria-Marche earthquakes. J Seismol 2000;4:387–99. Shi Y, Bolt BA. The standard error of the magnitude-frequency b value. Bull Seism Soc Am 1982;72:1677–87. Sornette A, Sornette D. Self-organizing criticality and earthquakes. Europhys Lett 1989;9(2):197. Sornette A, Dubois J, Cheminee JL, Sornette D. Are sequence of volcanic eruptions deterministrically chaotic? J Geophys Res 1991;96(B7):11931–45. Teich MC, Heneghan C, Lowen SB, Turcott RG. Estimating the fractal exponent of point processes in bilogical systems using wavelet- and Fourier-transform methods. In: Aldroubi A, Unser M, editors. Wavelets in medicine and biology. Boca Raton, FL: CRC Press; 1996. p. 383–412. Telesca L, Cuomo V, Lapenna V, Macchiato M. Analysis of the time-scaling behaviour in the sequence of the aftershocks of the Bovec (Slovenia) April 12, 1998 earthquake. Phys Earth Planet Int 2000;120:315–26. Telesca L, Cuomo V, Lapenna V, Macchiato M. Identifying space-time clustering properties of the 1983–1997 Irpinia-Basilicata (southern Italy) seismicity. Tectonophysics 2001;330:93–102. Telesca L, Lapenna V, Vallianatos F. Monofractal and multifractal approaches in investigatine scaling properties in temporal patterns of the 1983–2000 seismicity in the western Corinth graben, Greece. Phys Earth Planet Int 2002;131:63–79. Thurner S, Lowen SB, Feurstein MC, Heneghan C, Feichtinger HG, Teich MC. Analysis, synthesis, and estimation of fractal-rate stochastic point processes. Fractals 1997;5:565–96. Turcotte DL. Fractals and chaos in geology and geophysics. Cambridge University Press; 1995. Vittori E, Maschio L, Ferreli L, Michetti AM, Serva L. Carta e base di dati delle faglie capaci per lÕItalia centro-meridionale: presentazione e stato di avanzamento del progetto ITHACA. Il Quatern 1997;10(2):305–21. Vittori E, Deiana G, Esposito E, Ferreli L, Marchegiani L, Mastrolorenzo G, et al. Ground effects and surface faulting in the September–October 1997 Umbria-Marche (central Italy) seismic sequence. J Geodyn 2000;29:535–64.