[PRESENTATION] Extention of the Aki-Utsu b-value Estimator for - - PDF document

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[PRESENTATION] Extention of the Aki-Utsu b-value Estimator for Incomplete Catalogues

Conference Paper · January 2012

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2 authors: 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 Andrzej Kijko Natural Hazard Centre, University of Pretoria

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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:

Knowledge exists to be imparted. Knowledge exists to be imparted. (R.W. Emerson) (R.W. Emerson)

Contents

Kijko, Smit 2012

1. Introduction 2. Problem Statement 3. Previous Solutions to Problem 4. Solution 5. Aki-Utsu b-Value Estimator 6. Comparison of Methods 7. Area-Characteristic Seismic Activity Rate 8. Conclusion 9. References

Introduction

Kijko, Smit 2012

Gutenberg-Richer frequency magnitude relation 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.

a = level of seismicity b = ratio between weak and strong events

Introduction

Kijko, Smit 2012

Classic Aki (1965) estimator where βb ln10, = average magnitude and mmin = 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 mmin )

Problem Statement

Kijko, Smit 2012

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.

Problem Illustration for Incomplete Catalogue

Kijko, Smit 2012

Previous Solutions to Problem

Kijko, Smit 2012

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

Solution (Forgotten/ Unknown)

where Li is the i th likelihood function and βb ln10

Kijko, Smit 2012

The joint likelihood function is defined as

Solution (Forgotten/ Unknown) cont

Kijko, Smit 2012

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 mmin. Any distribution model for magnitude can be applied to this procedure.

Aki-Utsu b-value estimator

where

Kijko, Smit 2012

The maximization of the likelihood function provides the Generalized Aki-Utsu estimator The sample standard deviation: The confidence interval:

Comparison of Methods

100 150 200 250 300 350 400 450 500 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Two complete catalogs simulation Number of earthquakes Estimated b-value

Estimated b-value "true" b-value = 1 +/- Standard error

Kijko, Smit 2012

Area-Characteristic Seismic Activity Rate

Kijko, Smit 2012

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.

Conclusions

Kijko, Smit 2012

• 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
• f events ,is small and can be removed.
• The formulism is not restricted to any model of earthquake magnitude

distribution.

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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