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

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

obsuNu

obsu

u

mmmmb

mmmL

max0

max

for ))(10ln(exp1

for 0][

Figure A6–2

Plots of Likelihood Function for mmax-obs = 6

0

0.5

1

1.5

2

2.5

3

3.5

4 5 6 7 8 9

Magnitude

Lik

eli

ho

od

m0 = 4, N = 1

m0 = 5, N = 1

m0 = 4, N = 10

m0 = 5, N = 10

Figure A6–3

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

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

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

uobs

N

uobs

obs mmmmmb

mmbmF

max0

0

0maxmax for

))(10ln(exp(1

))(10ln(exp(1][

obsm maxˆ

obsu mm maxˆ

Figure A6–6

Bias Adjustment (2 of 2)Example:

mmax-obs = 5.7

N(m ≤ 4.5) = 10

mu = 6.3 produces = 5.7

4.5

5

5.5

6

6.5

7

7.5

8

4.5 5 5.5 6 6.5 7 7.5 8

mu

N = 1

N = 3

N = 10

N = 30

N = 100

N = 1000

Med

ian m

max-

obs

obsm maxˆ

Figure A6–7

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

EPRI (1994) Category Priors• Compute statistics of mmax-obs for extended

and non extended crust

• Use average sample size to adjust to mu

5.03.6crust extended-nonfor

84.04.6crust extendedfor

max

max

obs

obs

mu

mu

m

m

5.02.6crust extended-nonfor

84.004.6crust extendedfor

max

max

max

max

obs

obs

mobs

mobs

m

m

Figure A6–9

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

Example Application Using Category Prior

Extended crustMu = 6.4Mu = 0.84

5 events recorded between M 4.5 and M 5

0

0.001

0.002

0.003

0.004

0.005

4 5 6 7 8 9

Magnitude

Pri

or

Pro

ba

bili

ty

0

5

10

4 5 6 7 8 9

Magnitude

Lik

elih

oo

d

0

0.002

0.004

0.006

0.008

4 5 6 7 8 9

Magnitude

Po

ste

rio

r P

rob

ab

ility

0

0.05

0.1

0.15

0.2

0.25

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

Magnitude

Pro

ba

bil

ity

Figure A6–11

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