Derivation of the Gutenberg-Richter empirical formula from the solution of the generalized logistic equation

Affiliation(s)

Aims College, Greeley, CO, USA and Computer Center FEBRAS, Russia.

All-Russia Gramberg Research Institute for Geology and Mineral Resources of the Ocean (VNIIOKEANGEOLOGIA), St. Petersburg, Russia.

Aims College, Greeley, CO, USA and Computer Center FEBRAS, Russia.

All-Russia Gramberg Research Institute for Geology and Mineral Resources of the Ocean (VNIIOKEANGEOLOGIA), St. Petersburg, Russia.

ABSTRACT

We have written a new equation to study the statistics of earthquake distributions. We call this equation “the generalized logistic equation”. The Gutenberg-Richter frequency-magnitude formula was derived from the solution of the generalized logistic equation as an asymptotic case for the approximation of large magnitudes. To illustrate how the solution of the generalized logistic equation works, it was used to approximate the observed cumulative distribution of earthquakes in four different geological provinces: the Central Atlantic (40N - 25N, 5W - 35W), Canary Islands, Magellan Mountains (20N - 9S, 148E - 170E), and the Sea of Japan. This approximation showed an excellent correlation between the theoretical curves and observed data for earthquakes of magnitudes 1 < m < 9.

We have written a new equation to study the statistics of earthquake distributions. We call this equation “the generalized logistic equation”. The Gutenberg-Richter frequency-magnitude formula was derived from the solution of the generalized logistic equation as an asymptotic case for the approximation of large magnitudes. To illustrate how the solution of the generalized logistic equation works, it was used to approximate the observed cumulative distribution of earthquakes in four different geological provinces: the Central Atlantic (40N - 25N, 5W - 35W), Canary Islands, Magellan Mountains (20N - 9S, 148E - 170E), and the Sea of Japan. This approximation showed an excellent correlation between the theoretical curves and observed data for earthquakes of magnitudes 1 < m < 9.

Cite this paper

Maslov, L. and Anokhin, V. (2012) Derivation of the Gutenberg-Richter empirical formula from the solution of the generalized logistic equation.*Natural Science*, **4**, 648-651. doi: 10.4236/ns.2012.428085.

Maslov, L. and Anokhin, V. (2012) Derivation of the Gutenberg-Richter empirical formula from the solution of the generalized logistic equation.

References

[1] Ishimoto, M. and Iida, K. (1939) Observations sur les seismes enregistres parle microsismographe construit dernierement (1). Bulletin of the Earthquake Research Institute, 17, 443-478.

[2] Gutenberg, B. and Richter, C.F. (1942) Earthquake magnitude, intensity, energy and acceleration. Bulletin of the Seismological Society of America, 32, 163-191.

[3] Main, I. (1996) Statistical physics, seismogenesis, and seismic hazard. Reviews of Geophysics, 34, 433-462. doi:10.1029/96RG02808

[4] Bundle, J.B., Turcott, D.L., Shcherbakov, R., Klein, W., Sammis, C. (2003) Statistical physics approach to understanding the multiscale dynamics of earthquake fault system. Reviews of Geophysics, 41, 1-30.

[5] Ayele, A. and Kulhanek, O. (1997) Spatial and temporal variation of seismicity in the horn of Africa from 1960 to 1993. Geophysical Journal International, 130, 805-810. doi:10.1111/j.1365-246X.1997.tb01875.x

[6] Gerstenberger, M., Wiemer, S. and Gardini, D. (2001) A systematic test of the hypothesis that the b-value varies with depth in California. Geophysical Research Letters, 28, 57-60. doi:10.1029/2000GL012026

[7] Gibowicz, S.J. and Lasocki, S. (2001) Seismicity induced by mining: Ten years later. In: Dmowska, R. and Saltzman, B., Eds., Advances in Geophysics, Academic Press, Academic Press, 39-181.

[8] Smyth, C. and Mori, J. (2009) Temporal variations of the Gutenberg-Richter distribution prior to the Kobe earthquake. Annals of Disaster Prevention Research Institute, 52B, 255-261.

[1] Ishimoto, M. and Iida, K. (1939) Observations sur les seismes enregistres parle microsismographe construit dernierement (1). Bulletin of the Earthquake Research Institute, 17, 443-478.

[2] Gutenberg, B. and Richter, C.F. (1942) Earthquake magnitude, intensity, energy and acceleration. Bulletin of the Seismological Society of America, 32, 163-191.

[3] Main, I. (1996) Statistical physics, seismogenesis, and seismic hazard. Reviews of Geophysics, 34, 433-462. doi:10.1029/96RG02808

[4] Bundle, J.B., Turcott, D.L., Shcherbakov, R., Klein, W., Sammis, C. (2003) Statistical physics approach to understanding the multiscale dynamics of earthquake fault system. Reviews of Geophysics, 41, 1-30.

[5] Ayele, A. and Kulhanek, O. (1997) Spatial and temporal variation of seismicity in the horn of Africa from 1960 to 1993. Geophysical Journal International, 130, 805-810. doi:10.1111/j.1365-246X.1997.tb01875.x

[6] Gerstenberger, M., Wiemer, S. and Gardini, D. (2001) A systematic test of the hypothesis that the b-value varies with depth in California. Geophysical Research Letters, 28, 57-60. doi:10.1029/2000GL012026

[7] Gibowicz, S.J. and Lasocki, S. (2001) Seismicity induced by mining: Ten years later. In: Dmowska, R. and Saltzman, B., Eds., Advances in Geophysics, Academic Press, Academic Press, 39-181.

[8] Smyth, C. and Mori, J. (2009) Temporal variations of the Gutenberg-Richter distribution prior to the Kobe earthquake. Annals of Disaster Prevention Research Institute, 52B, 255-261.