1. Knowledge exists to be imparted. (R.W. Emerson)
FLAW IN THE BAYESIAN PROCEDURE FOR ESTIMATION OF THE MAXIMUM REGIONAL EARTHQUAKE MAGNITUDE By
Andrzej Kijko
2013 SSA Annual Meeting
Salt Lake City, Utah
17-19 April 2013.

2. Contents
Current Bayesian Procedure for mmax estimate (standard in nuclear industry !!!)
What is wrong with the procedure and why?
How to correct it? Illustration.
Conclusion and Remarks.
A Kijko

3. Estimation of Maximum Possible Magnitude
Two Approaches:
Based exclusively on past observations (catalogue).
2. Based on past observations and independent information (Bayesian approach).

4. Approach 1: Based on Seismic Event
Catalogue
Derivation: - Davies (1951)
- Quenouille (1956)
- Tate (1959)
- Cooke (1979)

5. Approach 1: Based on Seismic Event Catalogue
where Δ is defined as
(Cooke, 1979; Pisarenko et al., 1996)

6. Bayesian Procedure for mmax Estimation
Cornell (1994) …
combination of observations with
already existing knowledge!
A Kijko

7. Bayesian Procedure for mmax Estimation
Prior mmax distribution
for intraplate regions
Courtesy Mark Petersen,
USGS
Cratons
Margins
A Kijko

8. Petersen’s prior and Gaussian Prior
Bayesian Procedure for mmax Estimation
Petersen’s prior and Gaussian Prior
Gaussian Prior: Coppersmith (1994); Ordaz (2007)
A Kijko

9. Bayesian Procedure for mmax Estimation
Cornell (1994) …
Posterior probability
of mmax
given the sample
sample likelihood
function
prior probability
of mmax = C
A Kijko

10. Bayesian Procedure for mmax Estimation
Cornell (1994) …
pposterior (mmax) = c X L(x|mmax) X pprior(mmax)
A Kijko

11. Flaw in Current Procedure
For the sample likelihood function, the range of observations (magnitudes) depends on the unknown parameters.
This dependence violates the fundamental rules for the application of the maximum likelihood estimation procedure.
A Kijko

12. Flaw in Current Procedure
The currently used Bayesian procedure will by default underestimate / overestimate the value of mmax!!!
The currently used Bayesian procedure will locate mmax somewhere between the maximum observed magnitude and the true maximum.
A Kijko

13. Illustration 1
Prior distribution for Intraplate regions (Petersen et al, 2008)
A Kijko

14. Illustration 2
Gaussian Prior by Cornell (1994)
A Kijko

15. How to Correct Flaw in Current Procedure
Shift the Sample Likelihood Function from maximum
observed magnitude to maximum expected magnitude.
where
A Kijko

16. Illustration of Shift Correction
Correction: Shift of Sample Likelihood Function
A Kijko

17. Conclusion and Remarks
The current Bayesian procedure by default underestimates / overestimates mmax.
Underestimation / overestimation of mmax can reach a value of unit of magnitude.
A possible method to correct the flaw of the procedure is presented.
A Kijko

18. Details of the Approach
A Kijko

19. References
Cooke, P., Statistical Inference for bounds of random variables. Biometrika, 66,
Cornell, C. A. (1968). Engineering seismic risk analysis, Bull. Seism. Soc. Am. 58, 1583–1606.
Cornell CA. Statistical analysis of maximum magnitudes in the earthquakes of stable continental regions. In: J Schneider (ed). Seismic hazard methodology for the Central and Eastern United States. The earthquakes of stable continental regions. Vol. 1. Assessment of large earthquake potential. Palo Alto, CA, USA: EPRI;1994. NP-4726, pp 5-27.
Davis, R.C.,1951. On minimum variance in nonregular estimation, Ann. Math. Stat., 22,
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.
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).
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: /
Molchan, GM., V. L. Keilis-Borok, and V. Vilkovich (1970). Seismicity and principal seismic effects, Geophys. J. 21, 323–335.
Ordaz M, Aguilar A, Arboleda J ., Program for computing seismic hazard. CRISIS Ver Instituto de Ingenierı´a. UNAM, Mexico.
Pisarenko, V.F., Lyubushin A., Lysenko, V.B., Golubeva, T.V., Statistical estimation of seismic hazard parameters: maximum possible magnitude and related parameters. BSSA, 86, 3,
Quenouille , M.H., Notes on bias estimation. Biometrika. 43,
Rosenblueth, E. (1986). Use of statistical data in assessing local seismicity, Earthq. Eng. Struct. Dynam. 14, 325–337.
Rosenblueth, E., and M. Ordaz (1987). Use of seismic data from similar regions, Earthq. Eng. Struct. Dyn. 15, 619–634.
Weichert, D. H. (1980). Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes, Bull. Seismol. Soc. Am. 70, 1337–1346.
Tate, R.F., Unbiased estimation of location and scale parameters, Ann. Math. Statist. 30,

20. The End
Thank You 20