### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Detrended Fluctuation Analysis
- 3. Data Analysis and Discussion
- 4. Conclusions
- Acknowledgments
- References

[1] The detrended fluctuation analysis (DFA) is a powerful method for capturing scaling behavior in nonstationary time series. Using an appropriate instability index, it is possible to identify and quantify deviations from uniform power-law scaling, which suggest the presence of changing dynamics in the system under study. In this context, the scaling behavior of the 1981–2007 seismicity in Umbria-Marche (central Italy), which is one of the most seismically active areas in Italy, was investigated. Significant deviations from uniform power-law scaling in the seismic temporal fluctuations were revealed mostly linked with the occurrence of rather large earthquakes or seismic clusters.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Detrended Fluctuation Analysis
- 3. Data Analysis and Discussion
- 4. Conclusions
- Acknowledgments
- References

[2] Geophysical processes are featured by complex behavior that often is characterized by self-similarity, which suggests that their dynamics may be interpreted as due to many components interacting over a wide range of time or space scales [*Ashkenazy et al.*, 2003]. Earthquakes are a typical example of complex phenomena, where power-law statistics characterize the distribution of spatial, temporal and energy parameters. The existence of power-laws describing the main characteristics of a seismic process qualifies earthquakes as a good example of fractal systems.

[3] In particular, tectonic processes are generally considered to display fractal properties in time. The interevent intervals (times between two successive seismic events) follow a Poissonian distribution for completely random seismic sequences, while they are generally power-law distributed for time-clusterized sequences [*Telesca et al.*, 2001]. But the probability density function of the interevent intervals is only one window into a process, because it yields only first-order information and it reveals none about the correlation properties [*Turcott et al.*, 1994]. Therefore, time-fractal second-order methods are necessary to investigate the temporal fluctuations of seismic sequences more deeply. The use of statistics like the Allan Factor [*Allan*, 1966], the Fano Factor [*Lowen and Teich*, 1995], and the Detrended Fluctuation Analysis (DFA) [*Peng et al.*, 1995] has provided additional insight into the time dynamics of seismicity. All these measures lead to the determination of a scaling exponent, estimated by a linear fitting procedure performed on the power-law statistics plotted in log-log scales. Such scaling exponent informs about the dynamical properties of an earthquake sequence, in terms of its correlation structures and its memory phenomena and several studies have been performed aiming at the: (1) discrimination between Poissonian and clusterized sequences, (2) spatial variability of time-clustering behavior, and (3) magnitude-variability of the property of time-clusterization [*Telesca et al.*, 2003]. Furthermore, the numerical value of the scaling exponent allows us to determine quantitatively the strength of the temporal fluctuations of seismicity.

[4] In the present study, the scaling analysis was performed on the series of earthquakes that occurred in Umbria-Marche, central Italy. This area was struck by a strong earthquake occurred on September 26, 1997, with a duration magnitude of 5.8 (this event will be indicated as EQ0 hereafter). After that, the Umbria-Marche region, central Italy, was deeply studied and many investigations have been performed regarding its geodynamical features [*Boncio and Lavecchia*, 2000; *Calamita et al.*, 2000], active tectonics [*Galadini et al.*, 1999; *Cinti et al.*, 2000], spatio-temporal seismic distribution [*Ripepe et al.*, 2000; *Di Giovambattista and Tyupkin*, 2000], and induced geophysical effects [*Esposito et al.*, 2000]. This area is characterized by a well-documented historical and instrumental seismicity, mainly confined within the upper part of the crust (<16 km) [*Lavecchia et al.*, 1994].

### 2. Detrended Fluctuation Analysis

- Top of page
- Abstract
- 1. Introduction
- 2. Detrended Fluctuation Analysis
- 3. Data Analysis and Discussion
- 4. Conclusions
- Acknowledgments
- References

[5] A seismic sequence can be represented by a temporal point process that describes events that occur at some random locations in time [*Cox and Isham*, 1980], and it can be expressed by a finite sum of Dirac's delta functions centered on the occurrence times t_{i}, with amplitude A_{i} proportional to the magnitude of the earthquake:

(N + 1) represents the number of events recorded. This process can be described by the set of the interevent times, which is the series of the time intervals between two successive earthquakes.

[6] The DFA was proposed by *Peng et al.* [1995], and it 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 behavior 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 behavior of data in the presence of possible trends without knowing their origin and shape.

[7] The methodology operates on the series of the interevent times *τ*_{i}, where i = 1, 2, …, N and N is the length of the series. With we indicate the mean interevent time. The series is first integrated

with k = 1. N.Next, the integrated time series is divided into boxes of equal length, n. In each box a least-squares polynomial y_{n}(k) of degree p is fit to the data, representing the trend or order p in that box. Next we detrend the integrated time series y(k) by subtracting the local trend y_{n}(k) in each box. The root-mean-square fluctuation of this integrated and detrended time series is calculated by

Repeating this calculation over all box sizes, we obtain a relationship between F(n), that represents the average fluctuation as a function of box size, and the box size n. If F(n) behaves as a power-law function of n, data present scaling:

[8] Under these conditions the fluctuations can be described by the scaling exponent d, representing the slope of the line fitting logF(n) to log n. The estimate of the d-value, performed by a least square method, furnishes information about the type of correlations in the interevent series; furthermore, it quantifies the growth of the root mean square fluctuations F(n). For an uncorrelated seismic sequence, d = 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 n. d > 0.5 indicates the presence of persistent long-range correlations, meaning that a large (compared to the average) interevent interval is more likely to be followed by large one and vice versa. d < 0.5 indicates the presence of antipersistent long-range correlations, meaning that a large (compared to the average) interevent interval is more likely to be followed by small one and vice versa. d = 1 indicates flicker-noise dynamics, typical of systems in a self-organized critical state. d = 1.5 characterizes processes with Brownian-like dynamics.

### 3. Data Analysis and Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Detrended Fluctuation Analysis
- 3. Data Analysis and Discussion
- 4. Conclusions
- Acknowledgments
- References

[9] We analyzed the seismic interevent time series of earthquakes occurred in Umbria-Marche area, central Italy, from 1981 to 2007. We considered the earthquakes whose epicenters were located within a 40 km-radius circular area centered on the epicenter of the strongest event of the series (EQ0). The choice of such selection was given by the great seismological importance given by this event, and by the high activity characterizing this seismic area, featured by several rather large events during the investigation period. The seismicity data were obtained merging the Catalogo della Sismicita' Italiana (CSI) 1.0 [*Castello et al.*, 2006] from 1981 to 2002 and available at http://legacy.ingv.it/CSI/catalogo/doc11/SUM-FILES11.htm with the data from 2003 to 2007 taken from the instrumental bulletin of the Istituto Nazionale di Geofisica e Vulcanologia (INGV) available from site http://www.ingv.it/∼roma/reti/rms/bollettino. The CSI catalogue represents an extension of the previous Catalogo strumentale dei terremoti italiani dal 1981 al 1996 Version 1.1 [*CSTI Working Group*, 2001], where all the earthquakes have been relocated with uniform methods and the magnitudes revalued from original amplitudes and coda-durations, according to *Gasperini* [2002], by the calibration with real and synthetic Wood-Anderson magnitudes.

[10] In order to select the order of detrending polynomial, we applied the DFA to the whole seismic interevent series with different order p from 1 to 4 (we indicate as DFAp the DFA applied with a fitting polynomial of order p). Figure 1 shows the results: the DFA1 curve, even if it shows approximately the same linear fit as the other DFA curves, presents a relatively high fluctuation around this fit. The fluctuations shown by the other DFA curves are smaller. Furthermore the good similarity between DFA3 and DFA4 curves indicates that the 4-th order of DFA is sufficient to remove all the trends up to 4-th order in the profile curve or up to the 3-th order in the original curve. To examine transient phenomena, the seismic interevent time series was analyzed in overlapping windows of fixed number of events. Therefore the DFA4 was performed in a sliding window of 300 events, in order to have a sufficient amount of data. The shift between two successive windows was set to one event, in order to have a good smoothing among the calculated values. We applied the Gutenberg-Richter analysis to estimate the completeness of the seismic series in each time window, and we obtained the value of 2.1, which represents the threshold magnitude for the analysis. Furthermore, after calculating the fluctuation curve log[F(n)]∼log(n), in each window we calculated the scaling instability index *β*, as follows: the local scaling exponent *χ*(n) can be defined as the local derivative (slope) of the fluctuation function log[F(n)]∼log(n),

[11] If *χ*(n) is constant for different timescales n, the scaling is stable. Substantial variation of *χ*(n) with n indicates that the scaling behavior is unstable and deviates from the uniform power-law [*Viswanathan et al.*, 1997]. To quantitatively measure the deviation from stable power-law scaling, the scaling instability index *β* can be defined as the standard deviation of the local scaling exponents *χ*(n). High values of the index *β* indicate large deviations from uniform power-law. Using this definition of scaling instability index, each calculated *β* value was associated with the time of occurrence of the last event in the window. Figure 2 shows the time variation of *β*. Several features are revealed by this analysis: (1) the variation of the index *β* is characterized by a spiky behavior, where the highest values denote increase of instability; (2)eight single events or clusters of events with magnitude M ≥ 3.5 (which can be considered relatively high magnitude) were identified in the seismicity of the area, and are indicated in Figure 2 with numbers from 1 to 8; (3) almost always in correspondence with those eight events or clusters the instability index *β* shows an increasing spiky behavior indicated by letters a to j; (4) it seems quite clear a “precursory” pattern in the *β* variability before the occurrence of the events 1 (spike a), 3 (spike d), the cluster 4 (spike e) and the cluster 7 (spike h); (5) the spike b and b1 are clearly correlated with the cluster 2, in relationship with the seismic crisis, which struck the area in 1997 due to the occurrence of EQ0; (6) the spike c seems not to be connected with any rather strong event; (7) the spike g seems to be quite co-seismic with the event 6, even if a clear increasing behavior of *β* starts before the occurrence of such event; (8) the cluster 8 seem to be characterized by co-seismic quite irregular increase in the instability index *β*; and (9) in order to evaluate the significance of the estimates, we used a surrogate method [*Theiler et al.*, 1992]. For each window we generated one hundred shuffled series of the series, randomly permutating the interevent time series, thus having the same probability density function of the interevent time. Our aim is to test whether the calculated values of the instability index are significantly different from those displayed by the surrogates. Being *β* the instability index of the original series, let *μ*_{S} and *σ*_{S} indicate the mean and the standard deviation of the instability index values calculated for the shuffled sequences. We define the significance of our measure by the difference between the original and the mean surrogate value, divided by the standard deviation of the surrogate values [*Theiler et al.*, 1992]:

If the distribution of the statistic is Gaussian (and numerical experiments indicate that this is often a reasonable approximation), then the p-value is calculated by means of the formula p = erfc(*σ*/√2) [*Theiler et al.*, 1992]; this is the probability of observing a significance *σ* or larger if the null hypothesis is true, that is if the values are obtained by chance. Figure 3 shows the time variation of *β* after removing all the values with p > 0.05 (values obtained by chance). It is very clear the strong similarity with the curve shown in Figure 2 and, in particular, the identified peaks connected with earthquakes are still present, indicating the significance of our results.

[12] Considering the seismic settings of aftershocks in Italy [*Lolli and Gasperini*, 2003], it is unlikely that the spikes in the *β* variation (with the exception of those indicated by letters b, b1) can be considered as aftershock-related effects. In fact, what is in general evidenced by these results is that most of the rather large events are correlated (and, in certain cases, preceded) by an increase of the instability index, which indicates that the time-scaling behavior of the seismicity, given by the power-law shape of the fluctuation function F(n), becomes unstable.

[13] The approaching or the occurrence of an intense seismic event contributes to break the stability of the scaling behavior of the background seismicity, mainly given by the smaller events. The rupture of the scaling stability indicates that the earthquake distribution is likely to be “disturbed” by the occurrence of a large event. Such result seems to confirm what has been obtained in a rock fracture experiment with driven control [*Kuksenko et al.*, 2007]. In fact, it was found that at the initial stage, the events occur predominantly in certain “weak points” whose distribution is mainly determined by the structural heterogeneity and mechanical prehistory of samples. The system is dynamically connected through elastic interactions and oscillation processes and its scaling behavior shows its critical state that could be maintained as long as the structural rearrangements of the given scale level would enable to compensate the energy release. The appearance of larger-scale events (primary damages of upper hierarchic level) reduces the contribution of smaller-scale events to the formation of the fault geometry; in this period of hierarchic transition the scaling properties of the system become disturbed due to low occurrence of large events that determine the further evolution of the system. This behavioral transition was also found by *Tyupkin and Di Giovambattista* [2005], analyzing the time evolution of correlation length of events of acoustic emissions recorded in laboratory experiments on rock destruction and correlation length of seismicity of intermediate magnitude in the area of preparation of strong earthquakes on Kamchatka and Italy. They found that the decrease of correlation length preceding its precursory growth before the occurrence of large events, if the area in which seismic activity is correlated grows with time, can be interpreted as the beginning of formation of a potential source of earthquake.