Journal of Seismology

, Volume 21, Issue 1, pp 127–135 | Cite as

Gutenberg-Richter b-value maximum likelihood estimation and sample size

  • F. A. Nava
  • V. H. Márquez-Ramírez
  • F. R. Zúñiga
  • L. Ávila-Barrientos
  • C. B. Quinteros


The Aki-Utsu maximum likelihood method is widely used for estimation of the Gutenberg-Richter b-value, but not all authors are conscious of the method’s limitations and implicit requirements. The Aki/Utsu method requires a representative estimate of the population mean magnitude; a requirement seldom satisfied in b-value studies, particularly in those that use data from small geographic and/or time windows, such as b-mapping and b-vs-time studies. Monte Carlo simulation methods are used to determine how large a sample is necessary to achieve representativity, particularly for rounded magnitudes. The size of a representative sample weakly depends on the actual b-value. It is shown that, for commonly used precisions, small samples give meaningless estimations of b. Our results give estimates on the probabilities of getting correct estimates of b for a given desired precision for samples of different sizes. We submit that all published studies reporting b-value estimations should include information about the size of the samples used.


Gutenberg-Richter b-value Maximum likelihood estimation Aki/Utsu method 



We sincerely thank two anonymouis reviewers for useful comments and suggestions. Thanks to Dr. J.L. Brioso for his guidance and patience. Many thanks to José Mojarro for technical support.


  1. Aki K (1965) Maximum likelihood estimate of b in the formula log(N) = a - bM and its confidence limits. Bull Earthq Res Inst Tokio Univ 43:237–239Google Scholar
  2. Aki K (1981) A probabilistic synthesis of precursory phenomena In: Simpson DW Richards PG (eds) Earthquake prediction: an international review (vol. 4). American Geophysical Union, Washington, p 566–574Google Scholar
  3. Bender B (1983) Maximum likelihood estimation of b values for magnitude grouped data. Bull Seism Soc Am 73:831–851Google Scholar
  4. Bhattacharya P, Majumdar R, Kayal J (2002) Fractal dimension and b-value mapping in northeast India. Curr Sci 82:1486–1491Google Scholar
  5. Enescu B, Ito K (2001) Some premonitory phenomena of the 1995 Hyogo-Ken Nanbu (Kobe) earthquake: seismicity, b-value and fractal dimension. Tectonophysics 338:297–314CrossRefGoogle Scholar
  6. Epstein B, Lomnitz C (1966) A model for the occurrence of large earthquakes. Nature 211:954–956CrossRefGoogle Scholar
  7. Ghosh A, Newman A, Amanda M, Thomas A, Farmer G (2008) Interface locking along the subduction megathrust from b-value mapping near Nicoya Peninsula, Costa Rica. Geophys Res Lett 35:L01301. doi: 10.1029/2007GL031617 Google Scholar
  8. Gutenberg B, Richter C (1944) Frequency of earthquakes in California. Bull Seismol Soc Am 34:185–188Google Scholar
  9. Ishimoto M, Iida K (1939) Observations sur les seisms enregistrés par le microseismograph construit dernièrement (I). Bull Earthquake Res Inst Univ of Tokyo 17:443–478Google Scholar
  10. Kamer Y, Hiemer S (2015) Data-driven spatial b-value estimation with applications to California seismicity: to b or not to b. doi:  10.1002/2014JB011510.
  11. Khan P (2005) Mapping of b-value beneath the Shillong plateau. Gondwana Res 8:271–276. doi: 10.1016/S1342-937X(05)71126-6 CrossRefGoogle Scholar
  12. Kijko A (1988) Maximum likelihood estimation of Gutenberg-Richter b parameter for uncertain magnitude values. PAGEOPH 127:573–579CrossRefGoogle Scholar
  13. Kijko A, Selevoll M (1989) Estimation of earthquake hazard parameters from incomplete data files. Part I. Utilization of extreme and complete catalogs with different threshold magnitudes. Bull Seism Soc Am 79:645–654Google Scholar
  14. Kijko A, Selevoll M (1992) Estimation of earthquake hazard parameters from incomplete data files. Part II. Incorporation of magnitude heterogeneity. Bull Seism Soc Am 82:120–134Google Scholar
  15. Lomnitz C (1966) Statistical prediction of earthquakes. Rev Geophys 4:377–393CrossRefGoogle Scholar
  16. Lomnitz C (1974) Global tectonics and earthquake risk. Elsevier Sc. Pub. Co., CH.Google Scholar
  17. Márquez-Ramírez V. (2012) Análisis multifractal de la distribución espacial de sismicidad y su posible aplicación premonitora. Exploración de un posible mecanismo para la fractalidad mediante modelado semiestocástico. PhD Thesis, Programa de Posgrado en Ciencias de la Tierra, Centro de Investigación Científica y de Educación Superior de Ensenada, Baja California, México.Google Scholar
  18. Márquez-Ramírez V, Nava F, Zúñiga F (2015) Correcting the Gutenberg-Richter b-value for effects of rounding and noise. Earthq Sci 28:129–134. doi: 10.1007/s11589-015-0116-1 CrossRefGoogle Scholar
  19. Montuori C, Falcone G, Murru M, Thurber C, Reyners M, Eberhart-Phillips D (2010) Crustal heterogeneity highlighted by spatial b-value map in the Wellington region of New Zealand. Geophys J Int 183:451–460. doi: 10.1111/j.1365-246X.2010.04750.x CrossRefGoogle Scholar
  20. Richter C (1958) Elementary seismology. W H Freeman and Co, USAGoogle Scholar
  21. Scholz C (1968) The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes. Bull Seismol Soc Am 58:399–415Google Scholar
  22. Shaw E, Carlson J, Langer J (1992) Patterns of seismic activity preceding large earthquakes. J Geophys Res 97(B1):479–488CrossRefGoogle Scholar
  23. Shi Y, Bolt B (1982) The standard error of the magnitude-frequency b value. Bull Seismol Soc Am 72:1677–1687Google Scholar
  24. Singh C, Singh S (2015) Imaging b-value variation beneath the Pamir-Hindu Kush region. Bull Seismol Soc Am 105:808–815. doi: 10.1785/0120140112 CrossRefGoogle Scholar
  25. Singh C, Singh A, Chadha R (2009) Fractal and b-value mapping in Eastern Himalaya and Southern Tibet. Bull Seismol Soc Am 99:3529–3533. doi: 10.1785/0120090041 CrossRefGoogle Scholar
  26. Tinti S, Mulargia F (1987) Confidence intervals of b-values for grouped magnitudes. Bull Seismol Soc Am 77:2125–2134Google Scholar
  27. Utsu T (1965) A method for determining the value of b in a formula 329 log n = a - bM showing the magnitude-frequency relation for 330 earthquakes. Geophys Bull Hokkaido Univ 13:99–103Google Scholar
  28. Wiemer S, Wyss M (1997) Mapping the frequency-magnitude distribution in asperities: an improved technique to calculate recurrence times? J Geophys Res 102(B7):15115–15128CrossRefGoogle Scholar
  29. Wiemer S, Wyss M (2000) Minimum magnitude of completeness in earthquake catalogs: examples from Alaska, the Western United States, and Japan. Bull Seismol Soc Am 90:859–869CrossRefGoogle Scholar
  30. Wiemer S, Wyss M (2002) Mapping spatial variability of the frequency-magnitude distribution of earthquakes. Adv Geophys 45:259–302CrossRefGoogle Scholar
  31. Wyss M, Wiemer S (2000) Change in the probability for earthquakes in southern California due to the Landers magnitude 7.3 earthquake. Science 290:1334CrossRefGoogle Scholar
  32. Zúñiga R, Wyss M (2001) Most-and least-Likely locations of large to great earthquakes along the Pacific coast of Mexico estimated from local recurrence times based on b-values. Bull Seismol Soc Am 91:1717–1728CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • F. A. Nava
    • 1
  • V. H. Márquez-Ramírez
    • 2
    • 3
  • F. R. Zúñiga
    • 2
  • L. Ávila-Barrientos
    • 1
  • C. B. Quinteros
    • 1
  1. 1.SismologíaCentro de investigación Cientifica y de Educación Superior de EnsenadaEnsenadaMexico
  2. 2.Centro de GeocienciasUniversidad Nacional Autónoma de MéxicoJuriquillaMexico
  3. 3.SisVoc, CUCUniversidad de GuadalajaraPuerto VallartaMexico

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