Abstract
Assuming that a mainshock may be considered as a new phase, the natural time analysis of the seismicity reveals that the normalized power spectrum Π(φ) for small φ (φ → 0) or the quantity K1(= _χ2_–_χ_2) may be considered as an order parameter for seismicity. The probability distribution P(K1) of this order parameter is obtained from the calculation of the variance K1 when a time window of length l (= number of consecutive events) is sliding through an earthquake catalog. The K1 value at which this probability distribution P(K1) maximizes is designated by K1,p. By using P(K1), we find: first, studying the order parameter fluctuations relative to the standard deviation of its distribution, we observe that (a) the scaled distributions of different seismic areas (as well as that of the worldwide seismicity) fall on a universal curve and (b) this curve exhibits an “exponential tail” similar to that observed in certain non-equilibrium systems (e.g. 3D turbulent flow) as well as in several equilibrium critical phenomena, e.g., 2D Ising, 3D Ising, 2D XY. Second, the constant b in the Gutenberg–Richter (G-R) law for EQs, N(=M) = 10a-bM, is determined from the Maximum Entropy Principle which leads to b ≈ 1 in accordance with the b value obtained from real seismic data. Third, by analyzing either the original earthquake catalog or a shuffled one the following results are obtained for the Southern California Earthquake Catalog (SCEC) as well as for the Japanese Meteorological Agency Earthquake Catalog (Japan). Concerning the K1,p values, we find K1,p = 0.066 for the original data, while K1,p = 0.064 for the randomly shuffled data (with possible uncertainty of ±0.001). Both these K1,p values, the difference of which is shown to be associated with temporal correlations between the EQ magnitudes M, differ markedly from the value ? u = 1/12(≈ 0.083) of the “uniform” distribution, which is interpreted as reflecting that the process’s increments’ infinite variance contributes significantly to self-similarity. Fourth, upon employing multifractal cascades (generalized Cantor sets) in natural time an interconnection between K1,p and the parameter b of the G-R law is obtained which for b ≈ 1 leads to K1,p = 0.064 that coincides with the K1,p value obtained from the (randomly) shuffled earthquake data of Japan and SCEC. Fifth, by applying DFA to the earthquake magnitude time series of the SCEC and Japan data, we confirm that temporal correlations exist between EQ magnitudes. Sixth, focusing on the order parameter fluctuations of seismicity before and after mainshocks, we find the following. The P(K1) versus K1 plot before mainshocks exhibits a significant bimodal feature which is reminiscent of the bimodal feature observed in the pdf of the order parameter when approaching (from below) Tc in equilibrium critical phenomena. Finally, the G-R law or its generalization in the frame of the nonextensive statistical mechanics, if combined with natural time, which captures the temporal correlations between EQ magnitudes, can reproduce the features of real seismic data.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abe, S.: Essential discreteness in generalized thermostatistics with non-logarithmic entropy. EPL 90, 50004 (2010)
Abe, S.: Reply to the Comment by B. Andresen. EPL 92, 40006 (2010)
Abe, S., Bagci, G.B.: Necessity of q -expectation value in nonextensive statistical mechanics. Phys. Rev. E 71, 016139 (2005)
Abe, S., Okamoto, Y. (eds.): Non Extensive Statistical Mechanics and its Applications. Springer, Berlin (2001)
Abe, S., Suzuki, N.: Itineration of the internet over nonequilibrium stationary states in Tsallis statistics. Phys. Rev. E 67, 016106 (2003)
Abe, S., Suzuki, N.: Law for the distance between successive earthquakes. J. Geophys. Res. 108(B2), 2113 (2003)
Abe, S., Suzuki, N.: Aging and scaling of earthquake aftershocks. Physica A 332, 533–538 (2004)
Abe, S., Suzuki, N.: Scale-free statistics of time interval between successive earthquakes. Physica A 350, 588–596 (2005)
Andresen, B.: Comment on “Essential discreteness in generalized thermostatistics with nonlogarithmic entropy” by Abe Sumiyoshi EPL 92, 40005 (2010)
Ausloos, M., Lambiotte, R.: Brownian particle having a fluctuating mass. Phys. Rev. E 73, 011105 (2006)
Bak, P., Christensen, K., Danon, L., Scanlon, T.: Unified scaling law for earthquakes. Phys. Rev. Lett. 88, 178501 (2002)
Balankin, A.S.: Dynamic scaling approach to study time series fluctuations. Phys. Rev. E 76, 056120 (2007)
Balankin, A.S., Morales Matamoros, D., Pati˜no Ortiz, J., Pati˜no Ortiz, M., Pineda Le´on, E., Samayoa Ocha, D.: Scaling dynamics of seismic activity fluctuations. EPL 85, 39001 (2009)
Barnett, R.M., et al.: Review of particle physics. Phys. Rev. D 54, 1–708 (1996)
B˚ath, M.: Lateral inhomogeneities of the upper mantle. Tectonophysics 2, 483–514 (1965)
Beck, C., Schl¨ogl, F.: Thermodynamics of Chaotic Systems: An Introduction. Cambridge University Press, Cambridge, U.K. (1993)
Bramwell, S.T., Christensen, K., Fortin, J.Y., Holdsworth, P.C.W., Jensen, H.J., Lise, S., L´opez, J.M., Nicodemi, M., Pinton, J.F., Sellitto, M.: Universal fluctuations in correlated systems. Phys. Rev. Lett. 84, 3744–3747 (2000)
Bramwell, S.T., Christensen, K., Fortin, J.Y., Holdsworth, P.C.W., Jensen, H.J., Lise, S., L´opez, J.M., Nicodemi, M., Pinton, J.F., Sellitto, M.: Bramwell et al. reply. Phys. Rev. Lett. 87, 188902 (2001)
Bramwell, S.T., Christensen, K., Fortin, J.Y., Holdsworth, P.C.W., Jensen, H.J., Lise, S., L´opez, J.M., Nicodemi, M., Pinton, J.F., Sellitto, M.: Bramwell et al. reply. Phys. Rev. Lett. 89, 208902 (2002)
Bramwell, S.T., Fortin, J.Y., Holdsworth, P.C.W., Peysson, S., Pinton, J.F., Portelli, B., Sellitto, M.: Magnetic fluctuations in the classical XY model: The origin of an exponential tail in a complex system. Phys. Rev. E 63, 041106 (2001)
Bramwell, S.T., Holdsworth, P.C.W., Pinton, J.F.: Universality of rare fluctuations in turbulence and critical phenomena. Nature 396, 552–554 (1998)
Buchel, A., Sethna, J.P.: Statistical mechanics of cracks: Fluctuations, breakdown, and asymptotics of elastic theory. Phys. Rev. E 55, 7669–7690 (1997)
Carlson, J.M., Langer, J.S., Shaw, B.E.: Dynamics of earthquake faults. Rev. Mod. Phys. 66, 657–670 (1994)
Chen, H., Sun, X., Chen, H.,Wu1, Z.,Wang, B.: Some problems in multifractal spectrum computation using a statistical method. New J. Phys. 6, 84 (2004)
Clusel, M., Fortin, J.Y., Holdsworth, P.C.W.: Criterion for universality-class-independent critical fluctuations: Example of the two-dimensional Ising model. Phys. Rev. E 70, 046112 (2004)
Darooneh, A.H., Mehri, A.: A nonextensive modification of the Gutenberg-Richter law: q-stretched exponential form. Physica A 389, 509–526 (2010)
Davy, P., Sornette, A., Sornette, D.: Some consequences of a proposed fractal nature of continental faulting. Nature 348, 56–59 (1990)
Eichner, J.F., Kantelhardt, J.W., Bunde, A., Havlin, S.: Statistics of return intervals in long-term correlated records. Phys. Rev. E 75, 011128 (2007)
Fawcett, T.: An introduction to ROC analysis. Pattern Recogn. Lett. 27, 861–874 (2006)
Garbaczewski, R.: Differential entropy and time. Entropy 7, 253–299 (2005)
Garber, A., Hallerberg, S., Kantz, H.: Predicting extreme avalanches in self-organized critical sandpiles. Phys. Rev. E 80, 026124 (2009)
Gluzman, S., Sornette, D.: Self-consistent theory of rupture by progressive diffuse damage. Phys. Rev. E 63, 066129 (2001)
Gutenberg, B., Richter, C.F.: Seismicity of the Earth and Associated Phenomena. Princeton Univ. Press, Princeton, New York (1954)
Hanks, T.C., Kanamori, H.: Moment magnitude scale. J. Geophys. Res. 84(B5), 2348–2350 (1979)
Holliday, J.R., Rundle, J.B., Turcotte, D.L., Klein,W., Tiampo, K.F., Donnellan, A.: Space-time clustering and correlations of major earthquakes. Phys. Rev. Lett. 97, 238501 (2006)
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)
Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, New York (2003)
Kagan, Y.Y.: Short-term properties of earthquake catalogs and models of earthquake source. Bull. Seismol. Soc. Am. 94, 1207–1228 (2004)
Kanamori, H.: Quantification of earthquakes. Nature 271, 411–414 (1978)
Keilis-Borok, V.I., Kossobokov, V.G.: Premonitory activation of earthquake flow: algorithm M8. Phys. Earth Planet. Inter. 61, 73–83 (1990)
Keilis-Borok, V.I., Rotwain, I.M.: Diagnosis of time of increased probability of strong earthquakes in different regions of the world: algorithm CN. Phys. Earth Planet. Inter. 61, 57–72 (1990)
Kun, F., Herrmann, H.J.: Transition from damage to fragmentation in collision of solids. Phys. Rev. E 59, 2623–2632 (1999)
Lennartz, S., Bunde, A., Turcotte, D.L.: Missing data in aftershock sequences: Explaining the deviations from scaling laws. Phys. Rev. E 78, 041115 (2008) Uncorrected Proof
288
Lennartz, S., Livina, V.N., Bunde, A., Havlin, S.: Long-term memory in earthquakes and the distribution of interoccurrence times. EPL 81, 69001 (2008)
Lippiello, E., de Arcangelis, L., Godano, C.: Influence of time and space correlations on earthquake magnitude. Phys. Rev. Lett. 100, 038501 (2008)
Lippiello, E., Godano, C., de Arcangelis, L.: Dynamical scaling in branching models for seismicity. Phys. Rev. Lett. 98, 098501 (2007)
Meneveau, C., Sreenivasan, K.R.: Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424–1427 (1987)
Miguel, M.C., Zapperi, S.: Fluctuations in plasticity at the microscale. Science 312, 1151–1152 (2006)
O’Neil, J., Meneveau, C.: Spatial correlations in turbulence: Predictions from the multifractal formalism and comparison with experiments. Phys. Fluids A 5, 158–172 (1993)
Papanastassiou, D., Latoussakis, J., Stavrakakis, G.: A revised catalogue of earthquakes in the broader area of Greece for the period 1950–2000. Bulletin of the Geological Society of Greece 34, 1563–1566 (2001)
Pastor-Satorras, R.: Multifractal properties of power-law time sequences: Application to rice piles. Phys. Rev. E 56, 5284–5294 (1997)
Roumelioti, Z., Kiratzi, A., Theodoulidis, N., Papaioannou, C.: S-wave spectral analysis of the 1995 Kozani-Grevena (NW Greece) aftershock sequence. Journal of Seismology 6, 219–236 (2002)
Rundle, J.B., Turcotte, D.L., Shcherbakov, R., Klein, W., Sammis, C.: Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems. Rev. Geophys. 41, 1019 (2003)
Saichev, A., Sornette, D.: Power law distributions of seismic rates. Tectonophysics 431, 7–13 (2007)
Sarlis, N.V., Skordas, E.S., Varotsos, P.A.: See (the freely available) EPAPS Document No. EPLEEE8- 80–014908 originally from N.V. Sarlis, E.S. Skordas and P.A. Varotsos, Phys. Rev. E 80, 022102 (2009). For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.
Sarlis, N.V., Skordas, E.S., Varotsos, P.A.: Multiplicative cascades and seismicity in natural time. Phys. Rev. E 80, 022102 (2009)
Sarlis, N.V., Skordas, E.S., Varotsos, P.A.: Nonextensivity and natural time: The case of seismicity. Phys. Rev. E 82, 021110 (2010)
Sarlis, N.V., Skordas, E.S., Varotsos, P.A.: Order parameter fluctuations of seismicity in natural time before and after mainshocks. EPL 91, 59001 (2010)
Schultka, N., Manousakis, E.: Finite-size scaling in two-dimensional superfluids. Phys. Rev. B 49, 12,071–12,077 (1994)
Sethna, J.P.: Order parameters, broken symmetry, and topology. In: L. Nagel, D. Stein (eds.) 1991 Lectures in Complex Systems, Santa Fe Institute Studies in the Sciences of Complexity, Proc. Vol. XV. Addison-Wesley, New York (1992)
Shcherbakov, R., Turcotte, D.L., Rundle, J.B.: A generalized Omori’s law for earthquake aftershock decay. Geophys. Res. Lett. 31, L11613 (2004)
Shore, J.E., Johnson, R.W.: Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory IT-26, 26–37 (1980)
Shore, J.E., Johnson, R.W.: Properties of cross-entropy minimization. IEEE Trans. Inf. Theory IT-27, 472–482 (1981)
Shore, J.E., Johnson, R.W.: Comments on and correction to ‘axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy’ (Jan 80 26–37) (corresp.). IEEE Trans. Inf. Theory IT-29, 942–943 (1983)
Silva, R., Franc¸a, G.S., Vilar, C.S., Alcaniz, J.S.: Nonextensive models for earthquakes. Phys. Rev. E 73, 026102 (2006)
Sornette, D.: Critical Phenomena in Natural Science, 2nd edn. Springer, Berlin (2004)
Sornette, D., Davy, P.: Fault growth model and the universal fault length distribution. Geophys. Res. Lett. 18, 1079–1082 (1991)
Sotolongo-Costa, O., Posadas, A.: Fragment-asperity interaction model for earthquakes. Phys. Rev. Lett. 92, 048501 (2004)
See the document SCSN/README.old included in SCSN catalogs.tar.gz available at http://www.data.scec.org/ftp/catalogs/SCSN/
Uncorrected
Telesca, L.: Nonextensive analysis of seismic sequences. Physica A 389, 1911–1914 (2010)
Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)
Tsallis, C.: Introduction to Nonextensive Statistical Mechanics. Springer, Berlin (2009)
Tsallis, C., Mendes, R.S., Plastino, A.R.: The role of constraints within generalized nonextensive statistics. Physica A 261, 534–554 (1998)
Turcotte, D.L.: Fractals and Chaos in Geology and Geophysics, 2nd edn. Cambridge University Press, Cambridge (1997)
Turcotte, D.L., Malamud, B.D., Guzzetti, F., Reichenbach, P.: Self-organization, the cascade model, and natural hazards. Proc. Natl. Acad. Sci. USA 99, 2530–2537 (2002)
Utsu, T.: A statistical study of the occurrence of aftershocks. Geophys. Mag. 30, 521 (1961)
Utsu, T.: Seismology. Kyoritsu (in Japanese), Tokyo (2001)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Spatio-temporal complexity aspects on the interrelation between seismic electric signals and seismicity. Practica of Athens Academy 76, 294–321 (2001)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Seismic Electric Signals and seismicity: On a tentative interrelation between their spectral content. Acta Geophys. Pol. 50, 337–354 (2002)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Lazaridou, M.S.: Entropy in natural time domain. Phys. Rev. E 70, 011106 (2004)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Tanaka, H.K., Lazaridou, M.S.: Attempt to distinguish long-range temporal correlations from the statistics of the increments by natural time analysis. Phys. Rev. E 74, 021123 (2006)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Tanaka, H.K., Lazaridou, M.S.: Entropy of seismic electric signals: Analysis in the natural time under time reversal. Phys. Rev. E 73, 031114 (2006)
Varotsos, P.A., Sarlis, N.V., Tanaka, H.K., Skordas, E.S.: See (the freely available) EPAPS Document No. E-PLEEE8-72-058510 originally from P.A. Varotsos, N.V. Sarlis, H.K. Tanaka and E.S. Skordas, Phys. Rev. E 72, 041103 (2005). For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.
Varotsos, P.A., Sarlis, N.V., Tanaka, H.K., Skordas, E.S.: Similarity of fluctuations in correlated systems: The case of seismicity. Phys. Rev. E 72, 041103 (2005)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Tanaka, H.K.: A plausible explanation of the b-value in the Gutenberg-Richter law from first principles. Proc. Japan Acad., Ser. B 80, 429–434 (2004)
Vilar, C.S., Franc¸a, G.S., Silva, R., Alcaniz, J.S.: Nonextensivity in geological faults? Physica A 377, 285–290 (2007)
Watkins, N.W., Chapman, S.C., Rowlands, G.: Comment on “universal fluctuations in correlated systems”. Phys. Rev. Lett. 89, 208901 (2002)
Woodard, R., Newman, D.E., S´anchez, R., Carreras, B.A.: Persistent dynamic correlations in selforganized critical systems away from their critical point. Physica A 373, 215–230 (2007)
Zapperi, S., Ray, P., Stanley, H.E., Vespignani, A.: First-order transition in the breakdown of disordered media. Phys. Rev. Lett. 78, 1408–1411 (1997)
Zheng, B.: Generic features of fluctuations in critical systems. Phys. Rev. E 67, 026114 (2003)
Zheng, B., Trimper, S.: Comment on “universal fluctuations in correlated systems”. Phys. Rev. Lett. 87, 188901 (2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Varotsos, P.A., Sarlis, N.V., Skordas, E.S. (2011). Natural Time Analysis of Seismicity. In: Natural Time Analysis: The New View of Time. Springer Praxis Books(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16449-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-16449-1_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16448-4
Online ISBN: 978-3-642-16449-1
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)