Annals 47, 1, 2004, 01/07def 11 ANNALS OF GEOPHYSICS, VOL. 47, N. 1, February 2004 Key words self-potential signals – multifractal for- malism 1. Introduction The dynamics underlying tectonic processes could be directly revealed by the investigation of the temporal fluctuations of self-potential sig- nals, which may be useful to monitor and un- derstand many seemingly complex phenomena linked to seismic activity (Johnston, 1997; Park, 1997). Self-potential field variability may be in- duced by stress and fluid flow field variability (Scholz, 1990). Therefore, the analysis of these induced fluctuations could yield information on the geophysical mechanisms governing normal as well as intense seismic activity. In this con- text, this work investigated the dynamic proper- ties of geoelectrical signals, as they can be de- tected from observational time series. Self-potential signals are the result of the in- teraction among very heterogeneous and not well known mechanisms, which can be influ- enced by the particular structure of the moni- tored zone (Patella et al., 1997). This means that local features can be mixed with the gener- al thereby increasing the difficulty of rightly characterizing and interpreting the signal time variations. In addition, as occurs for many envi- ronmental signals, observational data are made even more erratic by the presence of anthropic phenomena: electrical signals coming from an- thropic sources may be added, e.g., to the natu- ral signal, hindering its dynamical characteriza- tion (Cuomo et al., 1997; Pham et al., 1998). In a previous paper, Cuomo et al. (1998) ana- lyzed the geoelectrical daily means to give infor- mation on the statistical features of the geoelec- trical background noise and the inner dynamics of geophysical processes producing the electrical phenomena observed on Earth surface in seismic areas. They discussed the statistical analysis of dynamic systems based on the estimation of their A preliminary study of the site-dependence of the multifractal features of geoelectric measurements Luciano Telesca (1), Gerardo Colangelo (1), Vincenzo Lapenna (1) and Maria Macchiato (2) (1) Istituto di Metodologie per l’Analisi Ambientale (IMAA), CNR, Tito Scalo (PZ), Italy (2) Dipartimento di Scienze Fisiche, Università degli Studi di Napoli «Federico II», Italy Abstract Multifractal analysis was performed to characterize the fluctuations in dynamics of the hourly time variability of self-potential signals measured from January 2001 to September 2002 by three stations installed in the Basil- icata region (Southern Italy). Two stations (Giuliano and Tito) are located in a seismic area, and one (Laterza) in an aseismic area. Multifractal formalism leads to the identification of a set of parameters derived from the shape of the multifractal spectrum (the maximum α0, the asymmetry B and the width W ) and measuring the «complexity» of the signals. Furthermore, the multifractal parameters seem to discriminate self-potential signals measured in seismic areas from those recorded in aseismic areas. Mailing address: Dr. Luciano Telesca, Istituto di Meto- dologie per l’Analisi Ambientale (IMAA), CNR Area della Ricerca di Potenza, Contrada S. Loja, 85050 Tito Scalo (PZ), Italy; e-mail: ltelesca@imaa.cnr.it 12 Luciano Telesca, Gerardo Colangelo, Vincenzo Lapenna and Maria Macchiato degree of predictability, distinguishing random- ness from chaos and providing a parsimonious representation in terms of autoregressive models of observations, by means of the only information coming from the time series itself. In the study of seemingly complex phenome- na, like those generating self-potential signals, methodologies able to capture the dynamic pecu- liarities in observational time series are particular- ly useful tools to obtain information on the fea- tures and causes of signal time variability. In par- ticular, fractal techniques, developed to extract qualitative and quantitative information from time series, have been applied recently to the study of a large variety of irregular, erratic signals and have proved very useful to reveal deep dynamic features. Cuomo et al. (2001) detected scaling be- haviour in the power spectra of geoelectrical time series, revealing the antipersistent character of the self-potential fluctuations. Telesca et al. (2001) proposed a new approach to investigate correla- tions between geoelectrical signals and earth- quakes, analyzing the time variations of the frac- tal parameters, characterizing their dynamics. Balasco et al. (2002) found that self-potential measurements seem to be featured by long-range correlations with scaling exponents that indicate that the underlying geophysical process is charac- terized by stabilizing mechanisms. In all the previous works, monofractal analy- ses were performed leading to the estimation of only one scaling exponent. Monofractals are ho- mogeneous objects, in the sense that they have the same scaling properties characterized by a single singularity exponent (Stanley et al., 1999; and references therein). The need for more than one scaling exponent to describe the scaling properties of the process uniquely indicates that the process is not a monofractal but could be a multifractal. A multifractal object requires many indices to characterize its scaling properties. Mul- tifractals can be decomposed into many – possi- bly infinitely many – sub-sets characterized by different scaling exponents. Thus multifractals are intrinsically more complex and inhomoge- neous than monofractals (Stanley et al., 1999), and characterize systems featured by very irregu- lar dynamics, with sudden and intense bursts of high frequency fluctuations (Davis et al., 1994). The most adequate manner to investigate multi- fractals is to analyze their fractality or singularity spectra. The singularity spectrum quantifies the fractal dimension of the sub-set characterized by a particular exponent, that is gives information on the relative dominance of various fractal expo- nents present in the process. In particular, the maximum of the spectra furnishes the dominant fractal exponent and the width of the spectrum denotes the range of the fractal exponents. There- fore, the multifractal formalism appears to fur- nish more deep information on the complexity of a time series. In the present work, we investigate the tem- poral fluctuations of self-potential data, meas- ured in Southern Italy from January 2001 to September 2002, using multifractal formalism, to disclose typical dynamic features. 2. Data Our data consist in eight geoelectrical time series recorded at three monitoring stations: Giu- liano (Giul1 and Giul2), Tito (Tito1, Tito2, Tito3 and Tito4) and Laterza (Lat1 and Lat2). The first two stations are located in seismic sites, the third in an aseismic one. Figure 1 shows the locations of the monitoring stations. As far as the technical features of the experimental equipment are con- cerned, we refer the reader to Cuomo et al. (1997) and for the results of mono- and multi- parametric preliminary statistical analysis of the monitored variables to Di Bello et al. (1994). Fig- ure 2 shows the time variations of the self-poten- tial signals measured. 3. Methods The concept of multifractal object was de- veloped by Mandelbrot (1974) to investigate several features in the intermittency of turbu- lence (Meneveau and Sreenivasan, 1991). Many authors have applied multifractality to several fields of scientific research. Multifractal formalism is based on the defi- nition of the so-called partition function Z (q, ε) i 1= , .Z q ( ) i q N boxes =f n f f !_ ^i h8 B (3.1) 13 A preliminary study of the site-dependence of the multifractal features of geoelectric measurements The quantity µi (ε) is a measure and it depends on ε, the size or scale of the boxes used to cov- er the sample. The boxes are labelled by the in- dex i and Nboxes (ε) indicates the number of box- es of size ε needed to cover the sample. The ex- ponent q is a real parameter, giving the order of the moment of the measure. The choice of the functional form of the measure µi (ε) is arbi- trary, provided that the most restrictive condi- tion µi (ε) ≥ 0 is satisfied. The parameter q can be considered a power- ful microscope, able to enhance the smallest differences of two very similar maps (Diego et al., 1999). Furthermore, q represents a selective parameter: high values of q enhance boxes with relatively high values for µi (ε); while low val- ues of q favour boxes with relatively low values of µi (ε). The box size ε can be considered a fil- ter, so that large values of the size are equiva- lent to apply a large scale filter to the map. Changing the size ε, one explores the sample at different scales. Therefore, the partition func- tion Z (q, ε) furnishes information at different scales and moments. The generalized dimensions are defined by the following equation: Fig. 1. Location of the geoelectrical monitoring stations Giuliano, Laterza and Tito in Southern Italy. 14 Luciano Telesca, Gerardo Colangelo, Vincenzo Lapenna and Maria Macchiato -30 -20 -10 0 10 20 30 40 Giul1 t (hour) -30 -20 -10 0 10 20 Lat2 S P ( m V ) t (hour) -50 -40 -30 -20 -10 0 10 Lat1 t (hour) -70 -60 -50 -40 -30 -20 -10 0 10 Tito1 S P ( m V ) t (hour) -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 Tito3 S P ( m V ) t (hour) -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 S P ( m V ) S P ( m V ) S P ( m V ) -30 -20 -10 0 10 20 30 40 Giul2 t (hour) S P ( m V ) -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100 110 Tito2 t (hour) S P ( m V ) 0.0 5.0x10 3 1.0x10 4 1.5x10 4 0.0 5.0x10 3 1.0x10 4 1.5x10 4 0.0 5.0x10 3 1.0x10 4 1.5x10 4 0.0 5.0x10 3 1.0x10 4 1.5x10 4 t (hour) 0.0 5.0x10 3 1.0x10 4 1.5x10 4 0.0 5.0x10 3 1.0x10 4 1.5x10 4 0.0 5.0x10 3 1.0x10 4 1.5x10 4 0.0 5.0x10 3 1.0x10 4 1.5x10 4 Fig. 2. Hourly variability of the 8 geoelectrical signals recorded at stations Giuliano, Laterza and Tito. 15 A preliminary study of the site-dependence of the multifractal features of geoelectric measurements 0 , .lim ln ln D q q Z q 1 1 = - f f "f _ _ i i (3.2) D (0) is the capacity dimension; D (1) is the in- formation dimension, and D (2) is the correla- tion dimension. An object is called monofrac- tal if D (q) is constant for all values of q, oth- erwise is called multifractal. In most practical applications the limit in eq. (3.2) cannot be cal- culated, because we do not have information at small scales, or because below a minimum physical length no scaling can exist at all (Diego et al., 1999). Generally, a scaling re- gion is found, where a power-law can be fitted to the partition function, which in that scaling range behaves as . ( )q ,Z q "f fx_ i (3.3) The slope τ (q) is related to the generalized di- mension by the following equation: .q q D q1= -x _ _ _i i i (3.4) A usual measure in characterizing multifractals is given by the singularity spectrum or Legen- dre spectrum f (α) that is defined as follows. If for a certain box j the measure scales as j j"n f fa^ h (3.5) the exponent α, which depends on the box j, is called Hölder exponent. If all boxes have the same scaling with the same exponent α, the sample is monofractal. The multifractal is given if different boxes scale with different exponents α, corresponding to different strength of the measure. Denoting as Sα the subset formed by the boxes with the same value of α, and indi- cating as Nα (ε) the cardinality of Sα, for a mul- tifractal the following relation holds: . ( )a N f"f f-a^ h (3.6) By means of the Legendre transform the quan- tities α and f (α) can be related to q and τ (q) q dq d q =a x _ _ i i (3.7) ( ) ( ).f q q q= -a a x^ h (3.8) The curve f (a) is a single-humped function for a multifractal, while it reduces to a point for a monofractal. In order to quantitatively recognize possible differences in Legendre spectra stemming from different signals, it is possible to fit the spectra to a quadratic function around the position of their maxima at α 0 by a least square method (Shimizu et al., 2002) .f A B C0 2 0= - + - +a a a a a^ _ _h i i (3.9) Parameter B measures the asymmetry of the curve, which is zero for symmetric shapes, pos- itive or negative for left-skewed or right-skewed shapes respectively. Another parameter is the width of the spec- trum, that estimates the range of α where f (α) > > 0, obtained extrapolating the fitted curve to zero; thus the width is defined as W max min= -a a (3.10) where f (α max) = f (α min) = 0. These three parameters serve to describe the «complexity» of the signal. If α 0 is low, the signal is correlated and the underlying process «loses fine structure», becoming more regular in appear- ance (Shimizu et al., 2002). The width W meas- ures the length of the range of fractal exponents in the signal. Therefore, the wider the range, the «richer» the signal in structure. The asymmetry parameter B informs us about the dominance of low or high fractal exponents with respect to the other. A right-skewed spectrum denotes relatively strongly weighted high fractal exponents, corre- sponding to fine structures, and low ones (more smooth-looking) for left-skewed spectra. 4. Results and discussion We performed multifractal analysis, calcu- lating the Legendre spectra by means of the software FRACLAB, developed at INRIA and available at the Internet site http://www.rocq.in- ria.fr. Since the data present gaps, we consid- ered for each signal the longest segments with- 16 Luciano Telesca, Gerardo Colangelo, Vincenzo Lapenna and Maria Macchiato 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 a Giul2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 a Lat2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Tito2 a 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 a Tito4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 Giul1 a 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 a Lat1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 a Tito1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.2 0.4 0.6 0.8 1.0 a Tito3 f( a ) f( a ) f( a ) f( a ) f( a ) f( a ) f( a ) f( a ) Fig. 3. Legendre spectra of the signals measured in Southern Italy. For each signal, we selected the 3-4 longest segments without gaps, whose order of magnitude of length is 103. All the spectra show a single-humped shape, typical of multifractal signals. 17 A preliminary study of the site-dependence of the multifractal features of geoelectric measurements out data missing. The order of the magnitude of the length of each segment is about 103, thus yielding reliable estimates of the singularity spectrum and multifractal parameters. Figure 3 shows the Legendre spectra for the selected segments for each signal. All the spectra present the typical single-humped shape, which charac- terizes multifractal signals. Figure 4 shows the average spectra. The differences among the spectra are very clear. Thus, after the determi- nation of the maxima α0, we fitted each average spectrum by eq. (3.9), thus estimating the asymmetry B and the width W. The results are summarized in fig. 5a-c. The time series Lat1 and Lat2 measured in station Laterza located in an aseismic area present the lowest value for the maximum α 0 (fig. 5a), indicating a more regu- lar process governing the time variability of such signals that seem to be characterized by more «coarse» structures. The smooth-looking behaviour displayed by Lat1 and Lat2 is also recognized in fig. 5b, where the asymmetry B assumes the highest positive value for such sig- nals. Furthermore, the lowest values for the width W indicate that a narrower range of frac- tal exponents characterizes Lat1 and Lat2 sig- nals. The signals measured by station Tito (Tito1, Tito2, Tito3 and Tito4) and Giuliano (Giul1 and Giul2) located in a seismic area ap- pear more complex because they present large maximum α 0 values, large width W values and relatively low value for the asymmetry B. In fig. 6 the geoelectric signals are represented as points in the space (α 0, B, W) of the multifrac- tal parameters: the signals Lat1 and Lat2, meas- ured in aseismic sites, are well discriminated from the other signals (Tito and Giuliano), recorded in seismic areas. The complex geophysical phenomenon un- derlying the geoelectrical variations is influ- enced by the geological and tectonic environ- ment outlined in the previous sections. The use of multifractal methods to investigate the tem- poral fluctuations of geoelectrical signals can 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Giul1 Giul2 Lat1 Lat2 Tito1 Tito2 Tito3 Tito4f( α ) α Fig. 4. Average Legendre spectra, calculated, for each signal, averaging the segment multifractal spectra. 18 Luciano Telesca, Gerardo Colangelo, Vincenzo Lapenna and Maria Macchiato lead to a better understanding of such complex- ity. The development of the Southern Apen- nines fold-and-thrust belt has been attributed to the subduction of the Ionian lithosphere be- neath the Adriatic plate (Doglioni et al., 1999; Bonini et al., 2000). This part of the Apennines comprises a stack of east-verging tectonic units representing an accretionary wedge composed of Mesozoic-Cenozoic sediments originally de- posited in different paleogeographic domains, both basinal (Lagonegro) and shallow-water which were thrust onto the Apulian foreland to the east (Mostardini and Merlini, 1986; Pesca- tore, 1988; Schiattarella 1998) (fig. 7). The monitoring network area extends along the Southern Apennine Chain where only Lat- erza station is out of an active seismic zone. In fact Laterza station is located in Puglia region, this area is part of the Apulian foreland which represents the Plio-Pleistocene foreland of the Southern Apennines orogenic system («Avam- paese Apulo» of Selli, 1962). The Apenninic foreland basins not yet incorporated into the orogenic wedge is characterized by a 6 km thick Mesozoic carbonate succession (Apulia carbonate platform – D’Argenio, 1974; Richet- ti, 1980). The uplift of the Apulian foreland (Middle-Late Pleistocene) has induced a low to Giul1 Giul2 Lat1 Lat2 Tito1 Tito2 Tito3 Tito4 0.50 0.55 0.60 0.65 0.70 0.75 m a xi m u m a 0 station Giul1 Giul2 Lat1 Lat2 Tito1 Tito2 Tito3 Tito4 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 a sy m m e tr y B station Giul1 Giul2 Lat1 Lat2 Tito1 Tito2 Tito3 Tito4 0.9 1.0 1.1 1.2 1.3 1.4 w id th W station Fig. 5a-c. Variation with the station of the multi- fractal parameters: a) maximum α 0, b) asymmetry B and (c) width W. b c -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.9 1.0 1.1 1.2 1.3 1.4 0.55 0.60 0.65 0.70 Lat2Lat1 Tito4 Giul1 Tito3 Tito2 Tito1 Giul2 m a xi m u m a 0 wid th W asymmetry B Fig. 6. 3D plot of the multifractal parameters: the signals Lat1 and Lat2, recorded in an aseismic area, are well discriminated from the signals (Tito and Giul) measured in seismic areas. a 19 A preliminary study of the site-dependence of the multifractal features of geoelectric measurements moderate-energy seismicity documented by pa- leoliquefaction features (Moretti, 2000) and historical and instrumental recorded seismic events (Pieri et al., 1997): it is considered a poorly tectonized area in the Apenninic fore- land (Doglioni et al., 1995). On the other hand, Giuliano and Tito sta- tions are located in the Campania-Basilicata area, one of the most active seismic zones of the Southern Apennines. Large destructive earth- quakes occurred both in historical and recent times in this region, which was struck in 1980 by the strongest event (ms = 6.9) of the past cen- tury in the Southern Apennines (Improta et al., 2003). The highest number of disastrous events with I ≥ X MCS took place in the Irpinia area in historical times. Such events (1964, 1930) ap- pear to be concentrated in the same zone; only one strong event, the December 16, 1857 earth- quake affected the southern sector of this area (Agri Valley – Basilicata) and 1550 and 1561 earthquakes have been located in the west of the area, in the Vallo di Diano (Campania) (Alessio et al., 1995). Ten year after the devasting Ir- pinia earthquake, a moderate event on May 1990 (ML = 5.2) and May 1991 (mb = 5.1) oc- curred approximately 40 km east of the southern end of the 1980 aftershock zone, causing dam- age in the nearby town of Potenza (Ekström, 1994). The normal faults affecting the internal side of the fold-and-thrust belt are commonly related to this extensional regime, which fol- lowed the progressive migration of the compres- sive thrust-front toward the Apulian foreland (Bonini et al., 2000). 5. Conclusions The determination of the multifractal pa- rameters of geoelectrical time series recorded in Southern Italy was performed by means of the calculation of the Legendre spectrum. We de- rived three parameters, the maximum α 0 of the spectrum, the asymmetry B and the width W of the curve. The time series Lat1 and Lat2, recorded in an aseismic site, are characterized by low maximum α 0, high positive asymmetry B and low width W. These features qualify these time series as less complex than the oth- ers (Tito1-4 and Giul1-2), which were meas- ured in seismic sites. Therefore, this set of mul- tifractal parameters seem to well discriminate signals measured in seismic from signal record- ed in aseismic sites. 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