地震活动的随机标度和非线性标度律
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摘要
把地震作为一个复杂系统，研究了地震活动的随机性质．当不考虑震级范围时，全球地震活动、人震的余震和区域震群均有以幂次律为特征的长尾现象．地震的强度由震级确定，具有一特定震级的地震可形成一个地震活动系列，很多这样的地震活动系列就形成具有各种震级的地震的集合．不同地震系列间的统计特征由随机标度来表征，随机标度表明了由地震震级分类的不同地震系列间统计时刻的标度关系．为了统一地方、区域和全球地震活动性的统计特性，引入了非线性标度率．
The randomness of earthquakes sources has been investigated in order to elucidatetheir universal nature as a complex system. The long-tail behavior characterized bythe power law has been pointed out in regional and world-wide seismicities,aftershocks of large earthquakes, and local earthquake swarms. Although earthquakesoccur more or less at random, these long-tail behaviors could not be explained by asimple stochastic process. Since the strength of each earthquake is classified by theearthquake magnitude, a series of an earthquake activity can be selected by a particular value of an earthquake magnitude. Taking another magnitued value, we can deriveanother series of the earthquake activity. Many of these series of the earthquake activity form a cluster of all the earthquakes with different magnitudes. Each series can beregarded as a point process with the same size or energy, because of the narrowrang of magnitude values. The stochastic property among the different seriesof earthquakes is characterized by a stochastic scaling. The stochastic scaling specifiesa scaling relation of statistical moments among different random processes (series ofearthquakes) classified by energy levels of events (earthquake magnitudes). In order tounify statistical properties of local, regional, and global seismicities, a nonlinearscaling law is a new concept to characterize the hierarchy of the complex earthquakeactivity in a general manner.
The randomness of earthquakes sources has been investigated in order to elucidatetheir universal nature as a complex system. The long-tail behavior characterized bythe power law has been pointed out in regional and world-wide seismicities,aftershocks of large earthquakes, and local earthquake swarms. Although earthquakesoccur more or less at random, these long-tail behaviors could not be explained by asimple stochastic process. Since the strength of each earthquake is classified by theearthquake magnitude, a series of an earthquake activity can be selected by a particular value of an earthquake magnitude. Taking another magnitued value, we can deriveanother series of the earthquake activity. Many of these series of the earthquake activity form a cluster of all the earthquakes with different magnitudes. Each series can beregarded as a point process with the same size or energy, because of the narrowrang of magnitude values. The stochastic property among the different seriesof earthquakes is characterized by a stochastic scaling. The stochastic scaling specifiesa scaling relation of statistical moments among different random processes (series ofearthquakes) classified by energy levels of events (earthquake magnitudes). In order tounify statistical properties of local, regional, and global seismicities, a nonlinearscaling law is a new concept to characterize the hierarchy of the complex earthquakeactivity in a general manner.
引文
[1] Koyama, J., General description of the complex faulting process and some empirical relations in seismology,J. Phys. Earth, 42, 103-148, 1994.
[2] Ogu, Y.. Abe, K., Some statishcal features of the long-term variation of the global and regional seismicactivity, Intem. Stat. Rev,59, 139-161, 1991.
[3] Mandelbrott, B. B., Wallis, J. R., Computer experiments with frartional Gaussian noises, Part 1, Averagesand variances, Water Resource Res., 5, 228-241, 1969.
[4] Koyama, J., Hara, H., Fractional Brownian motions described by scaled hangrvin equation, Chaos, Solitonsand Fractals, 3, 467-480, 1993.
[5] King, G., The accommodation of largr strains in the upper lithosphere of the earth and other solids byself-simialar fault systems: The grometrical origin of b-value, PAGEOPH, 121, 761-815, 1983.
[6] Hirata, T., Omori's power law aftershock sequences of microfracturing in rock fracturc experiment, JGeophys. Res.. 92, 6215- 6221, 1987.
[7] Kagan, Y. Y., Knopoff, L., Spatial distribution of earthquaks: the two-point correlation funtion, Geophys.J Roy. astr. Soc., 62, 303-320, 1980.
[8] Rundle, J. B., Derivation of the complete Gutenberg-Richter maghtude-frequency relation using the principle of scale invdriance, J Geophys. Res., N, 12337-12342, 1989.
[9] Shaw, B. E., Genertheed Omori law for afershocks and foreshocks from a simple dynamics.Geophys.Res.Lett., 20, 907- 910, 1993.
[10] Gutenberg, B., Richter, C. F., Seismicity of the Earth, Princeton Univ., 1-310, 1954.
[11] Bak, P., Tang, C., Earthquaks as self-organized critical phenomenon, J Geophys.Res.,94, 635-637,1989.
[12] Carson, J. M., Time intervals between characteristic earthquaks and correlations with smaller events: Ananalysis based on a mecbocal model of a fault, J Geophys. Res., 96, 4255-4267. 1991.
[13] Koyama, J., Hara, H., Scaled Langevin equation to desribe 1/fx spectrum,Phys.Rev.Rev A, 46, 1844-1849,1992.
[14] Hara, H., Koyama, J., Activation of complex system and its autocorrelation (in Japanese), StatisticalMathematics, 40, 217-226, 1992.
[15] Chen, Y. T., Knopoff, L., Simulation of earthquak sequences, Geophys.J Roy.astr.Soc.,91, 693-709,1987.
[2] Ogu, Y.. Abe, K., Some statishcal features of the long-term variation of the global and regional seismicactivity, Intem. Stat. Rev,59, 139-161, 1991.
[3] Mandelbrott, B. B., Wallis, J. R., Computer experiments with frartional Gaussian noises, Part 1, Averagesand variances, Water Resource Res., 5, 228-241, 1969.
[4] Koyama, J., Hara, H., Fractional Brownian motions described by scaled hangrvin equation, Chaos, Solitonsand Fractals, 3, 467-480, 1993.
[5] King, G., The accommodation of largr strains in the upper lithosphere of the earth and other solids byself-simialar fault systems: The grometrical origin of b-value, PAGEOPH, 121, 761-815, 1983.
[6] Hirata, T., Omori's power law aftershock sequences of microfracturing in rock fracturc experiment, JGeophys. Res.. 92, 6215- 6221, 1987.
[7] Kagan, Y. Y., Knopoff, L., Spatial distribution of earthquaks: the two-point correlation funtion, Geophys.J Roy. astr. Soc., 62, 303-320, 1980.
[8] Rundle, J. B., Derivation of the complete Gutenberg-Richter maghtude-frequency relation using the principle of scale invdriance, J Geophys. Res., N, 12337-12342, 1989.
[9] Shaw, B. E., Genertheed Omori law for afershocks and foreshocks from a simple dynamics.Geophys.Res.Lett., 20, 907- 910, 1993.
[10] Gutenberg, B., Richter, C. F., Seismicity of the Earth, Princeton Univ., 1-310, 1954.
[11] Bak, P., Tang, C., Earthquaks as self-organized critical phenomenon, J Geophys.Res.,94, 635-637,1989.
[12] Carson, J. M., Time intervals between characteristic earthquaks and correlations with smaller events: Ananalysis based on a mecbocal model of a fault, J Geophys. Res., 96, 4255-4267. 1991.
[13] Koyama, J., Hara, H., Scaled Langevin equation to desribe 1/fx spectrum,Phys.Rev.Rev A, 46, 1844-1849,1992.
[14] Hara, H., Koyama, J., Activation of complex system and its autocorrelation (in Japanese), StatisticalMathematics, 40, 217-226, 1992.
[15] Chen, Y. T., Knopoff, L., Simulation of earthquak sequences, Geophys.J Roy.astr.Soc.,91, 693-709,1987.
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