Transcript The “EPRI” Bayesian M Approach for Stable Continental Regions (SCR)

Figure A6–1
The “EPRI” Bayesian Mmax
Approach for Stable Continental
Regions (SCR)
(Johnston et al. 1994)
Robert Youngs
AMEC Geomatrix
USGS Workshop on Maximum
Magnitude Estimation
September 8, 2008
Figure A6–2
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
L[mu ] 
0
1  exp b ln( 10)(m
u

 m0 )
N
for mu  mmax obs
for mu  mmax obs
Figure A6–3
Plots of Likelihood Function for
mmax-obs = 6
3.5
m0 = 4, N = 1
m0 = 5, N = 1
3
m0 = 4, N = 10
m0 = 5, N = 10
Likelihood
2.5
2
1.5
1
0.5
0
4
5
6
7
Magnitude
8
9
Figure A6–4
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 A6–5
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 A6–6
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 , m
ˆ max obs
1  exp( b ln( 10)( mmax obs  m0 ) 
F [mmax obs ]  

u
1

exp(

b
ln(
10
)(
m

m
)
0


N
for m0  mmax obs  m u
u
m
 mˆ max obs to adjust from mmax-obs to mu
• Use
Figure A6–7
Bias Adjustment (2 of 2)
Example:
8
mmax-obs = 5.7
N(m ≤ 4.5) = 10
mu = 6.3 produces
mˆ max obs = 5.7
N=1
N=3
N = 10
7.5
N = 30
N = 100
N = 1000
Median m max-obs
7
6.5
6
5.5
5
4.5
4.5
5
5.5
6
6.5
mu
7
7.5
8
Domain “Pooling”
Figure A6–8
• 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 A6–9
EPRI (1994) Category Priors
• Compute statistics of mmax-obs for extended
and non extended crust
for extended crust
for non - extended crust
m max obs  6.04  mmax o b s  0.84
m
m max obs  6.2
max o b s
 0.5
• Use average sample size to adjust to mu
for extended crust
for non - extended crust
mu  6.4  mmax obs  0.84
mu  6.3
m
max obs
 0.5
Figure A6–10
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 A6–11
Example Application Using
Category Prior
Prior Probability
0.005
0.004
0.003
0.002
0.001
0
4
5
6
7
8
9
8
9
Likelihood
Magnitude
Extended crust
Mu = 6.4
Mu = 0.84
5 events recorded
between M 4.5 and M 5
10
5
0
4
5
6
7
Magnitude
0.25
P roba bility
Posterior Probability
0.2
0.008
0.006
0.004
0.15
0.1
0.05
0.002
0
0
4
5
6
7
Magnitude
8
9
4.5
5.0
5.5
6.0
6.5
7.0
M a gn itud e
7.5
8.0
8.5
9.0
Summary
Figure A6–12
• 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