The

1. The �EPRI� Bayesian Mmax Approach for Stable Continental Regions (SCR)(Johnston et al. 1994)Robert YoungsAMEC Geomatrix

USGS Workshop on Maximum Magnitude Estimation September 8, 2008 Figure A6�1

2. Statistical Estimation of mu (Mmax)

Assumption - earthquake size distribution in a source zone conforms to a truncated exponential distribution between m0 and mu Likelihood of mu given observation of N earthquakes between m0 and maximum observed, mmax-obs Figure A6�2

3. Plots of Likelihood Function for mmax-obs = 6

Figure A6�3

4. Results of Applying Likelihood Function

mmax-obs is the most likely value of mu Relative likelihood of values larger than mmax-obs is a strong function of sample size and the difference mmax-obs � m0 Likelihood function integrates to infinity and cannot be used to define a distribution for mu Hence the need to combine likelihood with a prior to produce a posterior distribution Figure A6�4

5. Approach for EPRI (1994) SCR Priors

Divided SCR into domains based on: Crustal type (extended or non-extended) Geologic age Stress regime Stress angle with structure Assessed mmax-obs for domains from catalog of SCR earthquakes Figure A6�5

6. Bias Adjustment (1 of 2)

�bias correction� from mmax-obs to mu based on distribution for mmax-obs given mu For a given value of mu and N estimate the median value of mmax-obs , Use to adjust from mmax-obs to mu Figure A6�6

7. Bias Adjustment (2 of 2)

Example: mmax-obs = 5.7 N(m = 4.5) = 10 mu = 6.3 produces = 5.7 Figure A6�7

8. Domain �Pooling�

Obtaining usable estimates of bias adjustment necessitated pooling �like� domains (trading space for time) �Super Domains� created by combining domains with the same characteristics Extended crust - 73 domains become 55 super domains, average N = 30 Non-extended crust � 89 domains become 15 super domains, average N = 120 Figure A6�8

9. EPRI (1994) Category Priors

Compute statistics of mmax-obs for extended and non extended crust Use average sample size to adjust to mu Figure A6�9

10. EPRI (1994) Regression Prior

Regress mmax-obs against domain characterization variables Default region is non-extended Cenozoic crust �Dummy� variables indicating other crustal types, ages, stress conditions, and a continuous variable for ln( activity rate ) indicate departure from default. Model has low r2 of 0.29 � not very effective in explaining variations Figure A6�10

11. Example Application Using Category Prior

Figure A6�11

12. Summary

Bayesian approach provides a means of using observed earthquakes to assess distribution for mu Requires an assessment of a prior distribution for mu Johnston et al. (1994) developed two types: crustal type category: extended or non-extended a regression model (low r2 and high correlation between predictor variables) Bayesian approach is not limited to the Johnston et al. (1994) priors, any other prior may be used Figure A6�12