Time-clustering analysis of volcanic occurrence sequences

Physics of the Earth and Planetary Interiors 131 (2002) 47–62

Time-clustering analysis of volcanic occurrence sequences Luciano Telesca a,∗ , Vincenzo Cuomo a , Vincenzo Lapenna a , Maria Macchiato b a

Istituto di Metodologie Avanzate di Analisi Ambientale, CNR, C. da S. Loja Z.I., 85050 Tito Scalo (PZ), Italy b Dipartimento di Scienze Fisiche, Università Federico II, Naples, Italy Received 19 December 2001; received in revised form 15 March 2002; accepted 25 March 2002

Abstract We investigate the time dynamics of worldwide volcanic activity in order to find the presence of time correlation structures, by applying the Fano factor (FF) method. The analysis, performed on 35 time-occurrence volcanic eruptive sequences with volcanic explosivity index (VEI ≥ 0) has revealed fractal behaviors in the most of the data sets considered, with fractal exponent ␣ ranging from ∼0.2 to 0.9. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Fano factor; Volcanic explosivity index; Fractal

1. Introduction The sequence of volcanic eruptions for a given volcano generally evolves with a complex temporal pattern. Recently, significant advances have been reached in understanding volcanic phenomena; in particular time series analysis appears the most suitable tool to extract information from the volcanic process (Marzocchi, 1996). The deep understanding of the time-correlation structures governing observational time-series coming from volcanic phenomena may provide useful information on the dynamical features of volcanic processes and on the involved geodynamical mechanisms. In particular, analyzing the temporal properties of volcanic eruptive sequences could have important implications in the improvement of probabilistic techniques for volcanic hazard analysis (Dubois and Cheminèe, 1988; Jaquet et al., 2000). The sequence of volcanic eruptive episodes for a given volcano is generally characterized by a complex ∗ Corresponding author. Tel.: +39-971-427206; fax: +39-971-427271. E-mail address: [email protected] (L. Telesca).

pattern in the time domain; this implies that the analysis of the volcanic process has to be performed by means of quantitative non-linear statistical methods. Dubois and Cheminèe (1993) analyzed the eruptive activity of Piton de la Fournaise volcano, applying the fractal Cantor dust method to the sequence of repose periods, finding two scaling regions in the distribution, with fractal dimension of 0.45 for timescales less than 10 months and 0.70 for timescales between 10 and 48 months. The different clustering behaviours of the eruptions for short and long timescales may correlate with feeding and draining of different magma chambers. The use of non-homogeneous Poisson processes is increasingly popular to model volcanic sequences, as it is a simple and versatile tool to assess the waxing or waning time-trends of a volcano and to assess volcanic hazard (Ho, 1991, 1995). Volcanic time-trend analysis has been performed in Ho (1996), providing quantitative comparisons among volcanoes and a methodology for volcanic model selection process. Bebbington and Lai (1996), analyzing data from two New Zealand volcanoes, Mt. Ruapehu and Mt. Ngauruhoe, fit the Poisson and the Weibull renewal

0031-9201/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 1 - 9 2 0 1 ( 0 2 ) 0 0 0 1 5 - 8

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models, concluding that simple Poisson process fits Ngauruhoue very well, while the strong correlation between successive repose lengths for Ruapehu reveals a more complex than a Poisson-like dynamics. Cox stochastic models have been proposed by Jaquet et al. (2000) to describe the statistical properties and to assess the probability of occurrence of volcanic eruptions; new methodology has been developed for the probabilistic modeling of volcanic hazards based on regional volcanic data that facilitates the production of probabilistic hazard maps for various volcanic scenarios (lava flows, tephra). Combined analyses of four properties exhibited by the time series of Vesuvius eruptions (power-law distribution of the after-eruptions; long period of quiescence preceding a large eruption; the self-affinity character and the constant average power) has been carried out in Palumbo (1999), with important implications for the eruption predictability. Like some random phenomena, such as noise and traffic in communication systems (Ryu and Meadows, 1994), biological ion-channel openings (Teich, 1989), trapping times in amorphous semiconductors (Lowen and Teich, 1993a,b), seismic events (Telesca et al., 1999, 2000a,b), volcanic eruptions occur at some random locations in time. A stochastic point process is a mathematical description which represents these events as random points on the time axis (Cox and Isham, 1980). Such a process may be called fractal if some relevant statistics display scaling, characterized by a power-law behavior, with related scaling coefficients, that indicate that the represented phenomenon contains clusters of points over a relatively large set of timescales (Lowen and Teich, 1995). In this paper, any volcanic sequence is assumed to be a realization of a point process, with events occurring at some random locations in time, and it is completely defined by the set of event times, or equivalently by the set of interevent-intervals. If the point process is Poissonian, the occurrence times are uncorrelated; the interevent-interval probability density function p(t) behaves as a decreasing exponential function p(t) = λ e−λt , for t ≥ 0,with λ the mean rate of the process. If the point process is characterized by fractal properties the interevent-interval probability density function p(t) generally decreases as a power-law function of the interevent time,

p(t) = kt−(1+α) , with α the so-called fractal exponent (Thurner et al., 1997). The exponent ␣ measures the strength of the clustering and represents the scaling coefficient of the decreasing power-law spectral density of the process S(f ) ∝ f −α (Lowen and Teich, 1993a,b). The power spectral density furnishes the information about how the power of the process is concentrated at various frequency bands (Papoulis, 1990), and it gives information about the nature of the temporal fluctuations of the process. If α ≈ 0 the temporal fluctuations of the process are purely random, typical of Poisson-like processes; if α = 0 the process displays clustering behavior, this indicating the presence of time correlation structures in the distribution of the occurrence times of the events. In this paper, we analyze the clustering properties of time-occurrence sequences of eruptions of 35 volcanoes sited in the world. Volcanic data were extracted from the National Geophysical Data Center (NGDC) Internet site (www.ngdc.noaa.gov); the index indicating the explosivity of the eruption is the volcanic explosivity index (VEI), an integer index ranging from 0 to 8, with 0 meaning non-explosive and 8 meaning destructive. We analyzed data sets with VEI ≥ 0. 2. Methods A sequence of volcanic eruptions can be mathematically expressed by a finite sum of Dirac’s ␦ functions centered on the occurrence times ti with amplitude Ai indicating the VEI of the i-th event: y(t) =

N


Ai δ(t − ti ).

(1)

i=1

Dividing the time axis into equally spaced contiguous counting windows of duration τ , we produce a sequence of counts (Nk (τ )), with Nk (τ ) denoting the number of eruptions in the k-th window:  tk
n Nk (τ ) = δ(t − tj ) dt. (2) tk−1 j =1

This sequence is a discrete-random process of non-negative integers. An important feature of this representation is that it preserves the correspondence between the discrete time axis of the counting process (Nk ) and the “real” time axis of the underlying point process, and the correlation in the process (Nk )

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refers to correlation in the underlying point process (Lowen and Teich, 1993a). With this representation the Fano factor (FF) statistics can be used. The FF (Thurner et al., 1997) is a measure of correlation over different timescales. It is defined as the variance of the number of events in a specified counting time τ divided by the mean number of events in that counting time; that is FF(τ ) =

Nk2 (τ ) − Nk (τ )2

Nk (τ )

(3)

where  denotes the expectation value. The FF of a fractal point process with 0 < α < 1 varies as a function of counting time τ as:  α τ FF(τ ) = 1 + (4) τ0 The monotonic power-law increase is representative of the presence of fluctuations on many timescales (Lowen and Teich, 1995); τ 0 is the fractal onset time, and marks the lower limit for significant scaling behaviour in the FF (Teich et al., 1996), so that for τ τ0 the clustering property becomes negligible. For Poisson processes the FF is always near unity for all counting times. The fractal exponent α ≈ 0 for Poissonian processes.

3. Data analysis and discussion We considered the events occurred in the temporal period ranging from 1800 to 2000, and we selected the data sets with a minimum of 25 events, thus analyzing 35 volcanic data sets. As with historical earthquakes, catalogue incompleteness may create serious problems. If not properly detected and treated, it may result in false pattern and lead to wrong interpretation. The whole point here is to account for as many events as possible by using the largest possible time interval in which the volcanic records are reliable (Marzocchi, 1996). Therefore, although the NGDC volcano data set contains volcanic eruptions from 79 to 2000, we selected only eruptions occurred after 1800. Fig. 1 shows the temporal distribution of the volcanic data sets. The y-axis shows the VEI variable, that is a quantitative measure of the intensity of the activity of a volcano (Newhall and Self, 1982). Such an index of activity is related to the volume (tephra) of the solid material of

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all sizes ejected by a volcano (Thorarinsson, 1974). In our case, VEI ranges from 0 to 6, indicating an ejection from less than 103 to 1010 m3 of tephra. The volcanic data sets are characterized by a moderate VEI value (1–3), with the exception of Bezymianny and Krakatau volcanoes, that, in the period investigated, were characterized by events of VEI = 5 and VEI = 6, respectively. Fig. 2 shows the results of the statistical analysis performed on the volcanic eruption sequences. The most of the volcanic data sets, that we have considered in our study, present temporal clustering, revealed by the power-law behavior of the FF versus the counting time τ . The FF curves increase monotonically with the counting time and is well fitted by an increasing power-law function of the counting time, for counting times greater than, approximately 103 h. This lower timescale could be put in relation with the mean interevent time t, that has the same order. The estimate of α has been performed calculating the slope of the line that fits, by the least square method, the linear range of the FF curve, plotted in log-log scales. In Table 1 for each volcano the geographical coordinates, the time span of the data set, the number of the eruptions, the mean interevent time and the α-value are indicated. The estimate of the fractal exponent α suggests a non-Poissonian modeling of the temporal distribution of the volcanic eruption occurrences for the most of the sequences analyzed. The fractal exponent α varies from almost 0.2 to 0.9, the numerical value quantifying the degree of the clusterization of the volcanic phenomenon. The mean value of the scaling exponents is α ≈ 0.5. Only for five volcanic sequences (Bezymianny, Merapi, Nyamuragira, Tarumai and Vesuvius) we do not recognize the presence of a timescale range, characterized by a scaling behavior of the underlying process, this implying an interpretation in terms of Poissonian-like dynamics of the temporal distribution of their eruptive activity. The Poisson feature of the temporal distribution of the eruptions of these volcanoes does not depend on the number of events, but it could possibly related to their particular volcanological specificity. Our analysis confirms the results obtained by Dubois and Cheminèe (1988) and Dubois and Cheminèe (1993) about the fractal investigation of the sequence of volcanic eruptions of the Piton de la Fournaise by the Cantor dust method; they found a

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Fig. 1. Temporal distribution of the volcanic sequences. The y-axis represents the VEI, that ranges from 0 (non-explosive) to 8 (destructive).

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Fig. 1. (Continued).

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Fig. 1. (Continued).

L. Telesca et al. / Physics of the Earth and Planetary Interiors 131 (2002) 47–62

Fig. 1. (Continued).

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Fig. 1. (Continued).

L. Telesca et al. / Physics of the Earth and Planetary Interiors 131 (2002) 47–62

Fig. 2. FF analysis of the volcanic data sets.

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Fig. 2. (Continued).

L. Telesca et al. / Physics of the Earth and Planetary Interiors 131 (2002) 47–62

Fig. 2. (Continued).

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Fig. 2. (Continued).

L. Telesca et al. / Physics of the Earth and Planetary Interiors 131 (2002) 47–62

Fig. 2. (Continued).

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Table 1 Characteristics of the volcanoes

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Volcano

Latitude

Longitude

Time span

N

t (days)

␣

Ambrym Asama Aso Bezymianny Cotopaxi Etna Fournaise Fuego Gamalama Izalco Karymsky Kilauea Kliuchevskoi Krakatau Lamongan Manam Marapi Mauna Loa Mayon Merapi Ngauruhoe Nyamuragira Oshima Pacaya Pavlof Raung Ruapehu Semeru Shin-Iwo-Jima Slamet Stromboli Tarumai Tengger Vesuvio White Island

−16.25 36.4 32.88 55.97 −0.677 37.734 −21.229 14.473 0.8 13.812 54.08 19.425 56.057 −6.102 −8.0 −4.1 −0.38 19.475 13.257 −7.542 −39.158 −1.38 34.73 14.381 55.42 −8.125 −39.28 −8.108 24.28 −7.242 38.789 42.68 −7.941 40.821 −37.52

168.12 138.53 131.1 160.6 −78.436 15.004 55.713 −90.88 127.325 −89.632 159.43 −155.292 160.638 105.423 113.341 145.061 100.471 −155.608 123.685 110.442 175.63 29.2 139.38 −90.601 −161.9 114.042 175.57 112.92 141.52 109.208 15.213 141.38 112.95 14.426 177.180

1894–1988 1803–1983 1804–1990 1955–2000 1800–1942 1802–2000 1800–2000 1855–1999 1811–1994 1840–1966 1854–2000 1823–2000 1813–1999 1883–1999 1806–1953 1877–1998 1845–1994 1832–1975 1800–2000 1820–1998 1839–1975 1894–2000 1803–1990 1846–2000 1844–1997 1817–1985 1861–1997 1818–1996 1904–1982 1825–1988 1822–1998 1867–1978 1804–1995 1804–1944 1826–2000

28 36 59 33 28 101 99 25 43 34 28 71 67 39 42 34 48 36 38 47 59 29 35 26 36 35 46 61 33 35 71 32 44 34 37

1263.111000 1875.943000 1169.000000 506.718800 1921.556000 721.070000 742.734700 2188.500000 1597.548000 1403.697000 1967.630000 923.885700 1031.545000 1112.763000 1308.927000 1338.151000 1155.298000 1492.714000 1967.757000 1415.065000 856.655200 1378.393000 2009.118000 2251.120000 1594.971000 1811.647000 1109.356000 1080.467000 890.687500 1748.735000 917.714300 1310.903000 1618.163000 1549.151000 1757.917000

0.56 ± 0.04 0.29 ± 0.03 0.84 ±.01 – 0.28 ± 0.02 0.55 ± 0.03 0.61 ± 0.02 0.23 ± 0.08 0.53 ± 0.03 0.32 ± 0.05 0.14 ± 0.03 0.96 ± 0.04 0.81 ± 0.03 0.39 ± 0.03 0.53 ± 0.04 0.26 ± 0.05 0.30 ± 0.03 0.60 ± 0.05 0.43 ± 0.05 – 0.38 ± 0.06 – 0.49 ± 0.02 0.71 ± 0.05 0.38 ± 0.04 0.40 ± 0.06 0.49 ± 0.02 0.88 ± 0.03 0.69 ± 0.03 0.89 ± 0.08 0.70 ± 0.04 – 0.4 ± 0.1 – 0.47 ± 0.03

Number of events N, latitude, longitude, time span, mean interevent time, α-value. The α-value is not indicated for volcanoes with Poissonian-like dynamics.

clustering behavior in the volcanic inter-occurrence time series, revealing two distinct scaling regions with different fractal dimensions, probably correlated with feeding and draining of different magma chambers. In our analysis the counting process associated with the Piton de la Fournaise volcanic sequence has shown a clustering behavior, starting from timescale, approximately near to 103 h, with scaling exponent α∼0.6, this evidencing the presence of correlation structures in the temporal distribution of the data. Ho (1996) performed the time-trend analysis of the

repose time series of eruptive sequences of three volcanoes of New Zealand (White Island, Ngauruhoe and Ruapehu); he found in all the three cases an increasing trend during the observation period, fitting the eruption sequences by a Weibull (power law) process. Bebbington and Lai (1996) found a strong correlation between successive repose lengths for Ruapehu, but a Poissonian-like process seems to fit the Ngauruhoe volcanic eruptions. In our analysis, the volcanic sequences previously analyzed (White Island, Ngauruhoe and Ruapehu) show clustering non-Poissonian

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behavior with fractal exponent α∼0.47, 0.49 and 0.38, respectively; this seems to confirm Ho’s analysis and, concerning Ruapehu, Bebbington and Lai’s analysis. Marzocchi (1996) has found that a Poisson process can fit the eruptive sequences of two Italian volcanoes, Etna and Vesuvius. In our study, Vesuvius seems to be characterized by purely random temporal fluctuations, showing, approximately an absence of scaling behavior and indicating a Poissonian-like dynamics; while Etna displays a very clear scaling behavior with fractal exponent α∼0.55, informing on a time-correlated mechanism governing its dynamics. The most volcanic data sets analyzed in the present paper show power-law behaviour in the time-occurrence structure of the eruptions. The observed temporal fractal distribution of eruptions argues for the self-organized criticality (SOC) interpretation of the volcanic dynamical system, as pointed out in Grasso and Bachèlery (1995). SOC is a property of a class of phenomena occurring in continuously driven out of equilibrium systems made of many interactive components, characterized by: (i) highly non-linearity, (ii) slow driving force from the critical state, and (iii) scale invariance. Applying to active volcanoes, the first feature corresponds to the rupture threshold for the rapid nucleation and growth of fluid transfer associated with an eruption; the second reflects the slow long term magma supply; the third aspect, that we investigated in the present paper, refers to mechanisms that couple the fluid flow to one single observable, including scaling organizations of different sources of volcanic seismic signals (Chouet and Shaw, 1991; Shaw and Chouet, 1991) and eruptive activity (Sornette et al., 1991).

4. Conclusions The occurrence-time sequences of volcanic eruptions in the world in the period 1800–2000 have been studied in order to evidence the presence of fractal behavior. Volcanic time-occurrence sequences characterized by VEI ≥ 0 have been investigated by means of the FF in the range of timescales stretching from 10 to T/10 days, where T indicates the period of the sequence. For the most of volcanoes considered, the FF curve increases monotonically with the counting time

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and is well fitted by an increasing power-law function of the counting time, for counting times greater than, approximately 103 h. This monotonic, power-law increase indicates the presence of fluctuations on many timescales and therefore of fractal clustering. Our results are in good agreement with the most of the works dealing with the analysis of the temporal properties of time-occurrence sequences of volcanic eruptions.

Acknowledgements We acknowledge the NGDC for rendering available volcanic data. The authors are grateful to two anonymous referees, whose suggestions improved the present paper.

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