Skip to main content
SpringerLink
Log in

On the possible correlation between the Gutenberg-Richter parameters of the frequency-magnitude relationship

  • ORIGINAL ARTICLE
  • Published:
Journal of Seismology Aims and scope Submit manuscript

Abstract

The study of the Gutenberg-Richter (GR) parameters a and b has been very important to describe and characterize the seismicity over the different seismic provinces around the world. As far as we know, the possible correlation between the GR parameters a and b has not received enough attention. Bayrak et al. reported the a and b values for 27 active seismic regions around the boundaries of the main tectonic plates of the world. From these data, we found that there exists a positive correlation between the a and b parameters (R = 0.85, R2 = 0.72). On the other hand, we made around 150 computer runs of a spring-block model proposed by Olami et al. (Phys Rev Lett 68(8):1244–1247, 1992). This model roughly emulates the interaction between two fault planes and it reaches a self-organized critical state. With these simulations, we also found that the a and b parameters are positively correlated. Motivated by these results, we propose an analytical demonstration that indeed a and b are positively correlated. In addition, we discuss on other possible applications of the spring-block model to actual seismicity and to frictional experiments made with sandpapers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Aki K (1981) A probabilistic synthesis of precursory phenomena. Earthq Predict: Int Rev Am Geophys Union 4:566–574

    Google Scholar 

  • Angulo-Brown F, Muñoz-Diosdado A (1999) Further seismic properties of a spring-block earthquake model. Geophys J Int 139(2):410–418

    Article  Google Scholar 

  • Angulo-Brown F, Ramírez-Guzmán AH, Yépez E, Rudolf-Navarro A (1998) Fractal geometry and seismicity in the mexican subduction zone. Geofis Int 37(1):29–33

    Google Scholar 

  • Bak P (1996) How nature works: the science of self-organized criticality. USA, New York

    Book  Google Scholar 

  • Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: an explanation of the 1/f noise. Phys Rev Lett 59(4):381–384

    Article  Google Scholar 

  • Bak P, Tang C, Wiesenfeld K (1988) Self-organized criticality. Phys Rev A 38(1):364–374

    Article  Google Scholar 

  • Barriere B, Turcotte DL (1994) Seismicity and self-organized criticality. Phys Rev E 49(2):1151–1160

    Article  Google Scholar 

  • Bayrak Y, Yılmaztürk A, Öztürk S (2002) Lateral variations of the modal (a/b) values for the different regions of the world. J Geodyn 34(5):653–666. https://doi.org/10.1016/S0264-3707(02)00037-6

    Article  Google Scholar 

  • Båth M, Duda SJ (1964) Earthquake volume, fault plane area, seismic energy, strain, deformation and related quantities. Ann Geophys 17(3):353–368

    Google Scholar 

  • Burridge R, Knopoff L (1967) Model and theoretical seismicity. Bull Seismol Soc Am 57(3):341–371

    Google Scholar 

  • Chen K, Bak P, Obukhov SP (1991) Self-organized criticality in a crack-propagation model of earthquakes. Phys Rev A 43(2):625–630

    Article  Google Scholar 

  • Christensen K, Olami Z (1992) Variation of the gutenberg-richter b values and nontrivial temporal correlations in a spring-block model for earthquakes. J Geophys Res Solid Earth 97(B6):8729–8735. https://doi.org/10.1029/92JB00427

    Article  Google Scholar 

  • Feder HJS, Feder J (1991) Self-organized criticality in a stick-slip process. Phys Rev Lett 66(20):2669–2672

    Article  Google Scholar 

  • Ferguson CD, Klein W, Rundle JB, Gould H, Tobochnik J (1998) Long-range earthquake fault models. Comput Phys 12(1):34–40

    Article  Google Scholar 

  • Frohlich C, Davis SD (1993) Teleseismic b values; or, much ado about 1.0. J Geophys Res Solid Earth 98(B1):631–644

    Article  Google Scholar 

  • Geller RJ, Jackson DD, Kagan YY, Mulargia F (1997) Earthquakes cannot be predicted. Science 275(5306):1616–1616

    Article  Google Scholar 

  • Gutenberg B, Richter CF (1944) Frequency of earthquakes in california. Bull Seismol Soc Am 34(4):185–188

    Google Scholar 

  • Gutenberg R, Richter C (1954) The seismicity of the earth 1904–1952

  • Gutenberg B, Richter CF (1956) Earthquake magnitude, intensity, energy, and acceleration: (second paper). Bull Seismol Soc Am 46(2):105–145

    Google Scholar 

  • Helmstetter A, Hergarten S, Sornette D (2004) Properties of foreshocks and aftershocks of the nonconservative self- organized critical olami-feder-christensen model. Physical Review E 70(4):046120. https://doi.org/10.1103/PhysRevE.70.046120

    Article  Google Scholar 

  • Hergarten S, Neugebauer HJ (2002) Foreshocks and aftershocks in the olami-feder-christensen model. Phys Rev Lett 88(23):238501. https://doi.org/10.1103/PhysRevLett.88.238501

    Article  Google Scholar 

  • Ito K, Matsuzaki M (1990) Earthquakes as self-organized critical phenomena. J Geophys Res Solid Earth 95(B5):6853–6860

    Article  Google Scholar 

  • Kanamori H, Anderson DL (1975) Theoretical basis of some empirical relations in seismology. Bull Seismol Soc Am 65(5):1073–1095

    Google Scholar 

  • Kawamura H, Yamamoto T, Kotani T, Yoshino H (2010) Asperity characteristics of the olami-feder-christensen model of earthquakes. Physical Review E 81(3):031119. https://doi.org/10.1103/PhysRevE.81.031119

    Article  Google Scholar 

  • Kotani T, Yoshino H, Kawamura H (2008) Periodicity and criticality in the olami-feder-christensen model of earthquakes. Phys Rev E 77(1):010102. https://doi.org/10.1103/PhysRevE.77.010102

    Article  Google Scholar 

  • Legrand D (2002) Fractal dimensions of small, intermediate, and large earthquakes. Bull Seismol Soc Am 92(8):3318–3320

    Article  Google Scholar 

  • Mendenhall W, Sincich TL (2003) A second course in statistics: regression analysis. Prentice Hall, Upper Saddle River

    Google Scholar 

  • Muñoz-Diosdado A, Rudolf-Navarro AH, Angulo-Brown F (2012) Simulation and properties of a non-homogeneous spring-block earthquake model with asperities. Acta Geophys 60(3):740–757

    Article  Google Scholar 

  • Olami Z, Feder HJS, Christensen K (1992) Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys Rev Lett 68(8):1244–1247

    Article  Google Scholar 

  • Pacheco JF, Scholz CH, Sykes LR (1992) Changes in frequency–size relationship from small to large earthquakes. Nature 355(6355):71–73

    Article  Google Scholar 

  • Richter CF (1958) Elementary seismology. WH Freeman and Company, USA

    Google Scholar 

  • Sornette A, Sornette D (1989) Self-organized criticality and earthquakes. EPL (Europhys Lett) 9(3):197–202

    Article  Google Scholar 

  • Turcotte DL (1989) A fractal approach to probabilistic seismic hazard assessment. Tectonophysics 167(2-4):171–177

    Article  Google Scholar 

  • Turcotte DL (1997) Fractals and chaos in geology and geophysics. Cambridge University Press, New York

    Book  Google Scholar 

  • Utsu T, Seki A (1955) A relation between the area of aftershock region and the energy of main shock. J Seism Soc Jpn 7:233–240

    Google Scholar 

  • Vargas CA, Basurto E, Guzman-Vargas L, Angulo-Brown F (2008) Sliding size distribution in a simple spring-block system with asperities. Phys A: Stat Mech Appl 387(13):3137–3144

    Article  Google Scholar 

Download references

Funding

FAB and AMD thank the partial support from COFAA-IPN and EDI-IPN; AGS thanks ABACUS-CINVESTAV and JPO thanks support from CONACYT-México.

Author information

Authors and Affiliations

  1. Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, UP Zacatenco, C.P. 07738, Mexico City, Mexico

    J. Perez-Oregon, A. H. Rudolf-Navarro & F. Angulo-Brown

  2. Unidad Profesional Interdisciplinaria de Biotecnología, Instituto Politécnico Nacional, Mexico, Mexico

    A. Muñoz-Diosdado

  3. Computational Genomics, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA

    A. Guzmán-Sáenz

Authors
  1. J. Perez-Oregon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Muñoz-Diosdado
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. H. Rudolf-Navarro
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Guzmán-Sáenz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. F. Angulo-Brown
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Perez-Oregon.

Appendix

Appendix

Let us consider the following modified Utsu law:

$$ log~ S = 1.02M - 4.01. $$
(A.1)

The Gutenberg-Richter law establishes that

$$ log\ \dot{N}=a-bM. $$
(A.2)

and Aki’s relation is

$$ \dot{N} = CS^{-b}. $$
(A.3)

Taking logarithm in both sides of Eq. A.3, it follows that

$$ log\ \dot{N} = log\ C- b\ log\ S. $$
(A.4)

Substituting Eq. A.1 in the right side of Eqs. A.4 and A.2 in the left side of Eq. A.4, we have

$$a - bM = log\ C - b (1.02M - 4.01) , $$
$$\Rightarrow \quad a = log\ C - 1.02bM + bM + 4.01b. $$

Therefore,

$$ a = (4.01 - 0.02M)b + log\ C. $$
(A.5)

On the other hand, consider now another modified Utsu relation; that is,

$$ log\ S = 1.21M - 5.05. $$
(A.6)

Following the same steps to obtain Eq. A.5, we arrive at

$$ a = (5.05 - 0.21M)b + log\ C . $$
(A.7)

As we can see in Eqs. A.5 and A.7, the positive correlation between the GR parameters a and b remains for both modified Utsu relations. Even in the extreme case of M = 10, the coefficient of b in Eq. A.5 is 3.90 and that for Eq. A.7 is 2.95; that is, in both cases, this coefficient is positive, such as that we obtain for the Bayrak et al. (2002) data.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Perez-Oregon, J., Muñoz-Diosdado, A., Rudolf-Navarro, A.H. et al. On the possible correlation between the Gutenberg-Richter parameters of the frequency-magnitude relationship. J Seismol 22, 1025–1035 (2018). https://doi.org/10.1007/s10950-018-9757-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10950-018-9757-6

Keywords

Search

Navigation