1. The “EPRI” Bayesian M max Approach for Stable Continental Regions (SCR) (Johnston et al. 1994) Robert Youngs AMEC Geomatrix USGS Workshop on Maximum Magnitude Estimation September 8, 2008 Figure A6–1

2. Statistical Estimation of m u (M max ) Assumption - earthquake size distribution in a source zone conforms to a truncated exponential distribution between m 0 and m u Likelihood of m u given observation of N earthquakes between m 0 and maximum observed, m max-obs Figure A6–2

3. Plots of Likelihood Function for m max-obs = 6 Figure A6–3

4. Results of Applying Likelihood Function m max-obs is the most likely value of m u Relative likelihood of values larger than m max-obs is a strong function of sample size and the difference m max-obs – m 0 Likelihood function integrates to infinity and cannot be used to define a distribution for m u 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 m max-obs for domains from catalog of SCR earthquakes Figure A6–5

6. Bias Adjustment (1 of 2) “bias correction” from m max-obs to m u based on distribution for m max-obs given m u For a given value of m u and N estimate the median value of m max-obs, Use to adjust from m max-obs to m u Figure A6–6

7. Bias Adjustment (2 of 2) Example: m max-obs = 5.7 N(m ≤ 4.5) = 10 m u = 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 m max-obs for extended and non extended crust Use average sample size to adjust to m u Figure A6–9

10. EPRI (1994) Regression Prior Regress m max-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 r 2 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 m u Requires an assessment of a prior distribution for m u Johnston et al. (1994) developed two types: –crustal type category: extended or non-extended –a regression model (low r 2 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