See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/282855459 [PRESENTATION] Extention of the Aki-Utsu b-value Estimator for Incomplete Catalogues Conference Paper · January 2012 CITATIONS READS 0 161 2 authors: Andrzej Kijko Ansie Smit Natural Hazard Centre, University of Pretoria University of Pretoria 172 PUBLICATIONS 2,941 CITATIONS 47 PUBLICATIONS 177 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Fluid-Induced Seismicity in the Central Basin Area: Ground Motion Prediction and the Development of an Early Warning System for Risk Reduction View project Reliability of earthquake hazard assessment in Iceland: improved models, uncertainties and sensitivities (SENSHAZ) View project All content following this page was uploaded by Ansie Smit on 15 October 2015. The user has requested enhancement of the downloaded file.
Knowledge exists to be imparted. (R.W. Emerson) Knowledge exists to be imparted. (R.W. Emerson) Extention of the Aki-Utsu -value Estimator for Incomplete Catalogues By Andrzej Kijko, Ansie Smit Email: andrzej.kijko@up.ac.za, ansie.smit@up.ac.za International Training Course on “Seismology, Seismic Data Analysis, Hazard Assessment and Risk Mitigation”. Session:
Contents 1. Introduction 2. Problem Statement 3. Previous Solutions to Problem 4. Solution Aki-Utsu b -Value Estimator 5. 6. Comparison of Methods 7. Area-Characteristic Seismic Activity Rate 8. Conclusion 9. References Kijko, Smit 2012
Introduction Gutenberg-Richer frequency magnitude relation a = level of seismicity b = ratio between weak and strong events Both parameters are used in a wide variety of seismological studies and especially in: •seismicity simulation ( Ogata and Zhuang 2006; Felzer, 2008 ) •earthquake prediction ( Kagan and Knopoff 1987; Geller 1997 ) •seismic hazard and risk assessment ( Cornell, 1968; Beauval &Scotti, 2004 ) Accurate assessment of parameters is therefore of critical importance. Kijko, Smit 2012
Introduction Classic Aki (1965) estimator where β�b ln10, = average magnitude and m min = level of completeness. Aki-estimator can only be applied to a complete seismic event catalogue (i.e. a catalogue that starts from a specified level of completeness m min ) Kijko, Smit 2012
Problem Statement Question: How to calculate a and b -value for incomplete seismic event catalogue? Incomplete catalogue A catalogue that can be divided into sub-catalogues, each with different levels of completeness. Kijko, Smit 2012
Problem Illustration for Incomplete Catalogue Kijko, Smit 2012
Previous Solutions to Problem Question: How to calculate a and b -value for incomplete seismic event catalogue? •Molchan et al (1970) •Rosenblueth (1986) •Rosenblueth and Ordaz (1987) •Kijko and Sellevoll (1989, 1992) •Weichert (1980) - most popular Kijko, Smit 2012
Solution (Forgotten/ Unknown) The joint likelihood function is defined as where L i is the i th likelihood function and β�b ln10 Kijko, Smit 2012
Solution (Forgotten/ Unknown) cont For each catalogue when assuming that the seismic events are i.i.d random variables, following the Gutenberg-Richter frequency-magnitude relation, and m is a continuous variable that is � m min . Any distribution model for magnitude can be applied to this procedure. Kijko, Smit 2012
Aki-Utsu b -value estimator The maximization of the likelihood function provides the Generalized Aki-Utsu estimator where The sample standard deviation: The confidence interval: Kijko, Smit 2012
Comparison of Methods Two complete catalogs simulation 1.4 Estimated b-value "true" b-value = 1 1.3 +/- Standard error Estimated b-value 1.2 1.1 1 0.9 0.8 0.7 100 150 200 250 300 350 400 450 500 Number of earthquakes Kijko, Smit 2012
Area-Characteristic Seismic Activity Rate Once the - value is known, the mean seismic activity rate is defined as follows: (Kijko & Sellevoll, 1989;1992). Assumption: The number of seismic events in unit time is a Poisson random variable. Kijko, Smit 2012
Conclusions •Derive a new maximum likelihood estimate of the Gutenberg-Richter b - value for multiple catalogues with different levels of completeness. •Easy to use. •Can estimate the margin of error with the confidence intervals •The slight overestimation , due to bias in catalogues with a small number of events ,is small and can be removed. •The formulism is not restricted to any model of earthquake magnitude distribution. Kijko, Smit 2012
References 1. Molchan , GM., V. L. Keilis-Borok, and V. Vilkovich (1970). Seismicity and principal seismic effects, Geophys. J. 21 , 323–335. 2. Rosenblueth , E. (1986). Use of statistical data in assessing local seismicity, Earthq. Eng. Struct. Dynam . 14 , 325–337. 3. Rosenblueth , E., and M. Ordaz (1987). Use of seismic data from similar regions, Earthq. Eng. Struct. Dyn . 15 , 619–634. 4. Kijko , A., and M. A. Sellevoll (1989). Estimation of earthquake hazard parameters from incomplete data files, Part I, Utilization of extreme and complete catalogues with different threshold magnitudes, Bull. Seismol. Soc. Am . 79 , 645–654. 5. Kijko , A., and M. A. Sellevoll (1992). Estimation of earthquake hazard parameters from incomplete data files, Part II, Incorporation of magnitude heterogeneity, Bull. Seismol. Soc. Am . 82 , 120–134.Weichert (1980) 6. Kijko , A., Smit, A. (2012) Extension of the b-value Estimator for Incomplete Catalogs . Bull. Seism. Soc. Am , Vol 102 , No 3, pp. 1283–1287. doi: 10.1785/0120110226. 7. Weichert , D.H., and Kijko, A., (1989), Estimation of earthquake recurrence parameters from incomplete and variably complete catalogue, Seism. Res. Lett., 60, p.28. 8. Ogata , Y., and J. Zhuang (2006). Space-time ETAS models and an improved extension, Tectonophysics 413 , no. 1-2, 13–23. 9. Felzer , K. R. (2008). Simulated aftershock sequences for aM 7.8 earthquake on the southern San Andreas fault, Seismol. Res. Lett . 80 , 21–25. 10. Kagan , Y. Y., and L. Knopoff (1987). Statistical short-term earthquake prediction, Science 236 , no. 4808, 1563– 1567. 11. Geller , R. J. (1997). Earthquake prediction: A critical review, Geophys. J. Int . 131 , 425–450. 12. Cornell , C. A. (1968). Engineering seismic risk analysis, Bull. Seism. Soc. Am . 58 , 1583–1606. 13. Beauval , C., and O. Scotti (2004). Quantifying sensitivities of PSHA for France to earthquake catalog uncertainties, truncation of groundmotion variability, and magnitude limits, Bull. Seismol. Soc. Am . 94 , 1579– 1594. 14. Weichert , D. H. (1980). Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes, Bull. Seismol. Soc. Am . 70 , 1337–1346.
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