Distribution of volcanic earthquake recurrence intervals

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi:10.1029/2008jb005942, 2009 Distribution of volcanic earthquake recurrence intervals M. Bottiglieri, 1 C. Godano, 1 and L. D Auria 2 Received 21 July 2008; revised 17 February 2009; accepted 12 May 2009; published 15 October 2009. [1] We analyze the distribution of volcanic earthquake recurrence intervals in the Vesuvio, Campi Flegrei, and Hawaii regions and compare it with tectonic recurrence rates in California. We find that the distribution behavior is similar for volcanic and tectonic seismic events. In both cases, the recurrence interval distributions collapse onto the same master curve if time is rescaled by the average occurrence rate. This implies that both phenomena have the same temporal organization, and it is possible to adopt for volcanic areas that the same occurrence models used for tectonic regions. Citation: Bottiglieri, M., C. Godano, and L. D Auria (2009), Distribution of volcanic earthquake recurrence intervals, J. Geophys. Res., 114,, doi:10.1029/2008jb005942. 1. Introduction [2] During the last fifty years many statistical models for seismic hazard evaluation have been proposed. They share some common features: they consider earthquake occurrence as due to a complex nonlinear dynamics based on some empirical scaling laws, namely the Gutenberg-Richter (GR) distribution [Gutenberg and Richter, 1944] and the Omori law [Omori, 1894]. The first states that the number of earthquakes with magnitude larger than M is N / 10 bm, where b is an empirical constant. The Omori law states that the number of aftershocks decays in time as n(t) (t + c) p, where t is the time elapsed from the main shock, p and c are empirical parameters characterizing respectively the aftershock number decay and the time after which the decay is observed. [3] Recent interest has focused on the investigation of time intervals Dt between successive events. Some authors propose that the functional behavior of this distribution is universal [Bak et al., 2002; Corral, 2003, 2004; Davidsen and Goltz, 2004], i.e., that it does not depend on the region characteristics. Specifically, they state that the average rate of earthquake occurrence m represents a non universal quantity defining the characteristic scale of the time intervals. Therefore when the recurrence intervals are rescaled by the average rate, the distributions collapse onto a unique universal master curve. More precisely the transformation DðDtÞ ¼ mf ðmdtþ ð1þ makes D(Dt) independent of the tectonic environment or any other local property. Corral [2003, 2004] applies relation (1) to periods of stationary earthquake occurrence (i.e., when m constant in time). 1 Department of Environmental Sciences and CNISM, Second University of Naples, Caserta, Italy. 2 Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Napoli Osservatorio Vesuviano, Napoli, Italy. Copyright 2009 by the American Geophysical Union. 0148-0227/09/2008JB005942$09.00 [4] Corral [2006] shows that the scaling function D(Dt) can be represented by a generalized Gamma function fðþ¼ x Cjdj agðg=dþ ðx=aþg 1 exp ðx=aþ d ð2þ where x = mdt, a is a dimensionless parameter and C is a normalization factor. If g and d are positive, the former controls the behavior at small Dt and d the behavior at large Dt. Corral s fitted parameter values are g = 0.67 ± 0.05, d = 1.05 ± 0.05 and a = 1.64 ± 0.15. This result has been generalized to non stationary periods by Shcherbakov et al. [2005], who show that the recurrence interval distribution of the major Californian sequences (Landers, Northridge and Hector Mine) fulfils the scaling relation (1). They found g = 0.2, in very good agreement with the derivation of the recurrence interval distribution for seismic sequences obtained by Utsu [2002], D(Dt) / (Dt) q with q =2 p 1. [5] For sake of completeness we discuss some criticisms to the hypothesis of universality. As an example, Lindman et al. [2005] generate synthetic data using a nonhomogeneous Poisson process derived from Omori s law and are able to reproduce some of the Bak et al. [2002] results. They suggest that this indicates a trivial origin for the unifying scaling law. Conversely, Corral and Christensen [2006] show that the Bak et al. [2002] approach is not trivial, because it provides a way to measure important properties of seismic catalogues. Moreover, Molchan [2005] shows that, if at least two regions in the data sets are independent, and if a scaling relation must hold, this scaling function could only be exponential. Finally, Saichev and Sornette [2006, 2007] extend Molchan s arguments and show that the approximate data collapse of the waiting times could be explained by the Epidemic Type Aftershock Sequences (ETAS) model [Ogata, 1988]. [6] Here we investigate the recurrence interval distribution for some volcanic areas of the world. We show that the behavior of the recurrence interval distribution can be considered universal even if we consider the whole catalogue. Specifically, we show that the distribution can be fitted by a Gamma function with same exponent, identifying 1of6

Table 1. Time Periods and Areas Covered by the Four Earthquake Catalogues Used in This Study Catalogue Period Latitude ( N) Longitude Hawaii 1975 2008 16.9 23.0 154.7 162.0 W Vesuvio 1972 2007 40.8 40.9 14.4 14.5 E Campi Flegrei 1982 1984 40.8 40.9 14.0 14.2 E California 1981 2005 32.0 37.0 122.0 114.0 W universal behavior in the temporal organization of earthquake occurrence. This result seems to be linked to the self similar nature of earthquake occurrence [Vere-Jones, 2005; Lippiello et al., 2007, 2008]. 2. Data Analysis [7] Our data set is composed of earthquake catalogues for the Vesuvio volcano, Campi Flegrei volcanic area, and the Hawaiian islands. The first two catalogues can be obtained courtesy of the Istituto Nazionale Geofisica e Vulcanologia (INGV)-Osservatorio Vesuviano, whereas the other is available online at the Advanced National Seismic System (ANSS) site (http://earthquake.usgs.gov). The periods and the areas covered by each catalogue are reported in Table 1. We analyze these catalogues and establish their completeness thresholds on the basis of the scaling region of the magnitude distribution (Figure 1). In Table 2 we report the magnitude completeness thresholds M c, the fitted b values, the total number of earthquakes and the number of events with magnitude M M c for each catalogue. [8] The distributions of recurrence intervals for the different volcanoes are shown in Figure 2. When we apply the transformation (1) the individual curves collapse onto the same master curve (Figure 3). Here the average occurrence rate m has been evaluated as 1/Dt, which is the best estimate of m for non stationary periods. In order to better demonstrate the universal behavior of the recurrence interval distributions, we show for comparison the distribution of the Californian recurrence intervals (Southern California Earthquake Center), which again collapse onto the other curves (Figures 2 and 3). Note that in the Californian case the completeness of the catalogue has been set at M c = 2.5. The best fit of the distributions, obtained by a maximum likelihood method, is a Gamma function with d equal to 1 and the parameters g = 0.30 ± 0.05 and a = 3.0 ± 0.2 (Figure 3). [9] Notice that the Gamma distribution is intrinsically universal under the transformation (1), since it becomes independent of m, the only local parameter. Therefore the rescaling of Dt by the average rate transforms relation (2) into a universal distribution, provided that the g value does not change. In this sense universality assumes a similar meaning as in statistical mechanics. More precisely, different phenomena belong to the same class of universality if the complete set of exponents for different characteristic properties is the same. In our case this requirement implies the same power law behavior for recurrence interval, spatial separation and energy distributions. Unfortunately, the events in the Campi Flegrei and Vesuvian catalogues are not localized. Moreover, the seismic Figure 1. Distributions of earthquake magnitudes for the three volcanic earthquake catalogues analyzed in this study. Each distribution follows the Gutenberg-Richter law. Arrows indicate the completeness magnitude threshold. 2of6

Table 2. The Catalogue b Value, Completeness Cutoff M c, Number of Events N, and Number of Events N c With Magnitude M M c Catalogue b M c N N c Hawaii 1.01 4.5 5,500 3,200 Vesuvio 1.49 1.0 10,700 4,000 Campi Flegrei 0.95 1.5 19,000 15,000 California 0.95 2.5 430,000 31,500 moments are not available and therefore it is not possible to make the nonlinear magnitude-energy conversion. Hence we cannot perform a complete comparison of the exponent set. [10] According to Utsu s relation q =2 p 1, our result g = 0.3 implies p 0.8. We evaluate D(Dt) for the whole catalogue, therefore the derived p value represents the average over the different Omori sequences dominating the catalogue seismicity. This is evident for tectonic catalogues, whereas it is less clear in volcanic areas [Lemarchand and Grasso, 2007]. It is well known that the number of events, occurring in a given sequence, scales exponentially with the main shock magnitude [Helmstetter, 2003]. In volcanic areas the low main shock magnitude generates a small increase in the occurrence rate which cannot be measured due to the rate fluctuations. Therefore sequences cannot be easily identified. Nevertheless, the existence of Omori sequences can be demonstrated by implementing an averaging procedure. For each catalogue we define a main event as an event with magnitude M M m. An Omori sequence starts at the occurrence time of a given main event and ends when the next main event occurs. In Figure 4 we show the Omori law fits for the different volcanic catalogues. Here we set M m = 6.0 for Hawaii, M m = 3.0 for Campi Flegrei and M m = 2.5 for Vesuvio. We find that the typical behavior of the rate is characterized by a plateau for t t M < c followed by a power law decay. The existence of a second plateau at greater times is the signature of a sequence ending when the stationary rate is reached again. The obtained p values are all compatible with the estimated g value, i.e., p = 0.8 ± 0.2 for Vesuvio and Campi Flegrei, p = 0.7 ± 0.1 for Hawaii. The Utsu relationship is confirmed also by Shcherbakov et al. [2005a, 2005b], who obtained g 0.2 and p 1.2 for three analyzed sequences (Landers, Northridge and Hector Mine). [11] The difference between our g value and that of Corral [2006] can be explained by observing that Corral [2006] considers only stationary periods and evaluates the average rate as N/T, where N is the number of events occurred in the time interval T. In contrast, we consider the whole catalogues and evaluate the average rate as 1/Dt. [12] Finally, we investigate how the choice of magnitude lower cutoff M* influences the behavior of the distribution. We considered all catalogues, setting the magnitude cutoff M* =M c ± 0.5. In all cases, the curve collapse is very good (Figures 5 and 6) indicating that M* does not influence significantly the rescaling property (1). Notice that in Figure 6 the collapse is worse than in the other two cases (Figures 3 and 5) because the increased magnitude threshold implies fewer events and poorer statistics. For the same reason we Figure 2. Distributions of earthquake recurrence intervals for the three volcanic and one tectonic catalogue analyzed in this study. A different value of the completeness magnitude threshold is adopted for each catalogue. 3of6

Figure 3. Distributions of earthquake recurrence intervals rescaled by the average occurrence rate m. The renormalized Probability Density Function (PDF) f(mdt) is equivalent to mm 1 D(Dt) as stated in relation (1). All curves collapse onto the same master curve. The solid line represents the Gamma function fit. Figure 4. The Omori law for the sequences identified in each of the catalogue analyzed in this study. The number of aftershocks as a function of the time elapsed from the main event occurrence has been normalized to 1 in order to have a better comparison of the three curves. 4of6

Figure 5. Rescaled recurrence interval distributions with a magnitude cutoff M* =M c 0.5. The renormalized probability density function (PDF) f(mdt) is equivalent to m 1 D(Dt) as stated in relation (1). All curves collapse onto the same master curve. The solid line represents the Gamma function fit. Figure 6. Rescaled recurrence interval distributions with a magnitude cutoff M* =M c + 0.5. The renormalized probability density function (PDF) f(mdt) is equivalent to m 1 D(Dt) as stated in relation (1). All curves collapse onto the same master curve. The solid line represents the Gamma function fit. 5of6

cannot analyze higher cutoff values. For lower cutoff values the statistics cannot be further improved, since there are very few events with M* <M M c 0.5 and, therefore they do not significantly affect D(Dt). 3. Conclusions [13] We have shown that recurrence interval distributions are similar for volcanic and tectonic earthquakes. In both cases we observe a collapse onto a unique master curve when the rescaling equation (1) is applied. The Gamma distribution obtained for D(Dt) is compatible with the previous results of Utsu [2002] linking the distribution exponent to the Omori p exponent. Therefore we conclude that volcanic and the tectonic events have the same temporal organization. This implies that, although the seismic stress acts at very different scales (a scale of kilometers for volcanic events and thousand kilometers for tectonic events) and the sources are very different (magma motion and tectonic plate displacement, respectively), the mechanism of stress redistribution in the earth crust appears to be the same. This is also consistent with the Diodati et al. [1991] results, which indicate a Gamma Distribution for the recurrence times between acoustic emissions from volcanic rocks. Moreover, our result has important implications for the evaluation of the occurrence probability of volcanic earthquakes since it is possible to adopt the same occurrence models as for tectonic earthquakes. Notation n number of aftershocks (dimensionless) t time (sec) c empirical constant (sec) p exponent of the Omori Law (dimensionless) D probability density function (PDF) (dimensionless) Dt recurrence time interval between successive events (sec) m occurrence rate of events (sec 1 ) C normalizing factor of the Gamma distribution (dimensionless) a empirical constant of the Gamma distribution (dimensionless) g Gamma distribution exponent controlling the behavior at small Dt (dimensionless) d Gamma distribution stretching exponent controlling the behavior at large Dt (dimensionless) q exponent of the Dt distribution within the Omori sequences (dimensionless) M earthquake magnitude (dimensionless) M c threshold of magnitude completeness (dimensionless) main event magnitude (dimensionless) M m M* cutoff magnitude (dimensionless) t M occurrence time of the main event (sec) References Bak, P., K. Christensen, L. Danon, and T. Scanlon (2002), Unified scaling law for earthquakes, Phys. Rev. Lett., 88, 178501. Corral, A. (2003), Local distributions and rate fluctuations in a unified scaling law for earthquakes, Phys. Rev. E, 68, 035102. Corral, A. (2004), Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes, Phys. Rev. Lett., 92, 108501. Corral, A. (2006), Modelling critical and catastrophic phenomena in Geoscience: A statistical physics approach, in Lecture Notes in Physics, vol. 705, edited by P. Bhattacharyya and B. K. Chakrabarti, pp. 191 221, Springer, Berlin. Corral, A., and K. Christensen (2006), Comment on Earthquakes descaled: On waiting time distributions and scaling laws, Phys. Rev. Lett., 96, 109801. Davidsen, J., and C. Goltz (2004), Are seismic waiting time distributions universal?, Geophys. Res. Lett., 31, L21612, doi:10.1029/ 2004GL020892. Diodati, P., F. Marchesoni, and S. Piazza (1991), Acoustic emission from volcanic rocks: An example of self-organized criticality, Phys. Rev. Lett., 67(17), 2239 2243. Gutenberg, B., and C. F. Richter (1944), Frequency of earthquakes in California, Bull. Seismol. Soc. Am., 34, 185 188. Helmstetter, A. (2003), Is earthquakes triggering driven by small earthqukes?, Phys. Res. Lett., 91, 058501. Lemarchand, N., and J. Grasso (2007), Interaction between earthquakes and volcano activity, Geophys. Res. Lett., 34, L24303, doi:10.1029/ 2007GL031438. Lindman, M., K. Jonsdottir, R. Roberts, B. Lund, and R. Bdvarsson (2005), Earthquakes descaled: On waiting time distributions and scaling law, Phys. Rev. Lett., 94, 108501. Lippiello, E., C. Godano, and L. de Arcangelis (2007), Dynamical scaling in branching models for seismicity, Phys. Rev. Lett., 98, 98,501 98,504. Lippiello, E., L. de Arcangelis, and C. Godano (2008), Influence of time and space correlations on earthquake magnitude, Phys. Rev. Lett., 100, 038501 038504. Molchan, G. (2005), Interevent time distribution in seismicity: A theoretical approach, Pure Appl. Geophys., 162, 1135 1150, doi:10.1007/s00024-0004-2664-5. Ogata, Y. (1988), Statistical models for earthquakes occurrence and residual analysis for point processes, J. Am. Stat. Assoc., 83, 9 27. Omori, F. (1894), On the after-shocks of earthquakes, J. Coll. Sci., Imp. Univ. Tokyo, 7, 111 200. Saichev, A., and D. Sornette (2006), Universal distribution of interearthquake times explained, Phys. Rev. Lett., 97, 079501. Saichev, A., and D. Sornette (2007), Theory of earthquake recurrence times, J. Geophys. Res., 112, B04313, doi:10.1029/2006jb004536. Shcherbakov, R., G. Yacovlev, D. L. Turcotte, and J. B. Rundle (2005a), Model for distribution of aftershocks interoccurrence times, Phys. Rev. Lett., 95, 218501. Shcherbakov, R., D. L. Turcotte, and J. B. Rundle (2005b), Aftershock statistics, Pure Appl. Geophys., 162, 1051 1076, doi:10.1007/s00024-004-2661-8. Utsu, T. (2002), Statistical features of seismicity, in International Handbook of Earthquake and Engineering Seismology, Int l Assoc. Seismol. and Phys. Earth s Interior, Committee on Education., edited by W. K. Lee et al., Part A, pp. 719 732, Academic Press, Amsterdam. Vere-Jones, D. (2005), A class of self-similar random measure, Adv. Appl. Prob., 37, 908 914. M. Bottiglieri and C. Godano, Department of Environmental Sciences, Second University of Naples, Via Vivaldi 43, I 81100 Caserta, Italy. (milena.bottiglieri@unina2.it; cataldo.godano@unina2.it) L. D Auria, Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Napoli Osservatorio Vesuviano, Via Diocleziano, 328 80124 Napoli, Italy. 6of6