the “epri” bayesian m max approach for stable continental regions (scr) (johnston et al. 1994)...
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![Page 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](https://reader036.vdocuments.net/reader036/viewer/2022082517/56649eb55503460f94bbe484/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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