1. Deprem Tehlike Analizine Giriş:
Türkiye’den Örnekler
Introduction to Seismic Hazard
Analysis: Examples from
Turkey
Ali Osman Öncel, Turkey
Knowledge exists to be imparted.
(R.W. Emerson(
2. Deprem Tehlike Analizi
• Erzincan ve Çevresi
• Kuzey Anadolu Fay Zonu
• Artçı Şokların Etkisi
• Tehlike Haritaları
• Mmax Estimation
Ali Osman Öncel, Turkey
20. Deprem Tehlike Analizi
• Erzincan ve Çevresi
• Kuzey Anadolu Fay Zonu
• Artçı Şokların Etkisi
• Tehlike Haritaları
• Mmax Estimation
Ali Osman Öncel, Turkey
38. Deprem Tehlike Analizi
• Erzincan ve Çevresi
• Kuzey Anadolu Fay Zonu
• Artçı Şokların Etkisi
• Tehlike Haritaları
• Mmax Estimation
Ali Osman Öncel, Turkey
49. A. Kijko
Flaw in the EPRI Procedure
for maximum earthquake
magnitude estimation and
its correction
ESC 2010
6-10 September 2010 Montpeller, France
Andrzej Kijko, South Africa
Knowledge exists to be imparted.
(R.W. Emerson(
50. Andrzej Kijko, South Africa
Contents
1. EPRI Bayesian Procedure for mmax
estimate
2. What is wrong with the procedure and
why?
3. How to cure it? Illustration
4. Conclusion and Remarks
51. Andrzej Kijko, South Africa
EPRI Procedure for mmax estimation
(Cornell, 1994(
Splendid idea …
- combination of
observations with already
existing knowledge!
52. EPRI Procedure for mmax Estimation
(Cornell, 1994)
Andrzej Kijko, South Africa
Prior mmax distribution
for intraplate regions
Courtesy Mark
Petersen,
USGS
Cratons Margins
53. EPRI Procedure for mmax Estimation
(Cornell, 1994)
Gaussian prior mmax distribution
(e.g. M Ordaz, 2007)
Andrzej Kijko, South Africa
54. EPRI Procedure for mmax Estimation
(Cornell, 1994)
Petersen's prior & Gaussian prior
Andrzej Kijko, South Africa
5.5 6 6.5 7 7.5 8 8.5
0
0.5
1
1.5
2
2.5
M agnitude m
m a x
PriorPDF
P rior D istributions of m
m a x
G aussian prior (m ean m
max
= 6.92 S D = 0.5)
P rior for intraplete regions by M .P etersen (U S G S )
M ean of prior m
max
55. EPRI Procedure for mmax estimation, (Cornel
Andrzej Kijko, South Africa
⋅=
maxmax
max
mof
yprobabilitprior
mgiven
likelihoodsample
const
samplethegiven
mof
yprobabilitPosterior
56. Andrzej Kijko, South Africa
)()|()( maxmaxmax mpmLkmp priorposterior ⋅⋅= x
5.5 6 6.5 7 7.5 8 8.5 9
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
E xam ple of sam ple likelihod functions
M agnitude
ln(likelihoodfunction)
S am ple likelihood function
"true" m
m ax
= 6.92
m
max
obs = 5.89
EPRI Procedure for mmax estimation,
(Cornell, 1994)
5.5 6 6.5 7 7.5 8 8.5
0
0.5
1
1.5
2
2.5
M agnitude m m a x
PriorPDF
P rior D istributions of m
m a x
G aussian prior (m ean m
m ax
= 6.92 S D = 0.5)
P rior for intraplete regions by M .P etersen (U S G S )
M ean of prior m
m ax
57. Flow in EPRI Procedure
Andrzej Kijko, South Africa
• For the sample likelihood function,
the range of observations
(magnitudes) depends on the
unknown parameters.
• This dependence violates the
fundamental rules of application of
maximum likelihood estimation
procedure.
58. • EPRI Bayesian procedure by
default will underestimate value
of mmax !
• EPRI Bayesian procedure will
locate mmax somewhere between
maximum observed magnitude
and “true” mmax
Andrzej Kijko, South Africa
Flow in EPRI Procedure
59. Confirmation 1: Prior Distribution for
Intraplate Regions
(by M. Petersen, USGS)
Andrzej Kijko, South Africa
100 200 300 400 500 600 700 800 900 1000
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
E stim ated m
m a x
w ith prior of m
m ax
for intraplate regions
A ctivity rate Lam bda * Tim e span of catalogue [Y]
mmax
m
max
estim ated
m
max
observed
"true" m
max
= 6.92
60. Andrzej Kijko, South Africa
Confirmation 2: Gaussian Prior (by Cornell, 1
100 200 300 400 500 600 700 800 900 1000
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
E stim ated m
m a x
w ith G aussian P rior
A ctivity rate Lam bda * Tim e span of catalogue [Y]
mmax
m
m ax
estim ated
m
m ax
observed
"true" m
max
= 6.92
61. How to Correct the Flaw
in the EPRI Procedure?
Andrzej Kijko, South Africa
• Eliminate effect
• Eliminate cause
62. Approach #1: Eliminate Effect
Andrzej Kijko, South Africa
Shift the Likelihood Function from
maximum observed magnitude to
maximum expected mmax
Δmmˆ obs
maxmax +=
[ ]∫=∆
max
min
d)(
m
m
n
M mmF
63. Approach #1: Eliminate Effect
Correction by Shift of Sample Likelihood
Function
Approach #1: Correction
by shift of Sample
Likelihood Function
0 100 200 300 400 500 600 700 800 900 1000
6.5
6.6
6.7
6.8
6.9
7
7.1
E ffect of shift of sam ple likelihood function
N um ber of events
mmax
C urrent E P R I P rocedure
A fter correction by shift of S am ple Likelihood F unction
"true" m
max
= 6.92
Andrzej Kijko, South Africa
64. Our Problem: For the sample likelihood function,
the range of observations (magnitudes)
depends on the unknown parameters
Approach #2: Eliminate Cause
Correction by Account of Magnitude
Uncertainty
4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
10
-6
10
-4
10
-2
10
0
M ag n itu d e
G R
G R -apparent
mmax
Andrzej Kijko, South Africa
65. 0 100 200 300 400 500 600 700 800 900 1000
6.5
6.6
6.7
6.8
6.9
7
7.1
E ffect of account of m agnitude uncertainty
N um ber of events
mmax
C urrent E P R I P rocedure
A fter correction by account of m agnitude uncertainty
"true" m
max
= 6.92
Andrzej Kijko, South Africa
Approach #2: Eliminate Cause
Correction by Account of Magnitude
Uncertainty
66. Comparison of Two Correction
Procedures
0 100 200 300 400 500 600 700 800 900 1000
6.5
6.6
6.7
6.8
6.9
7
7.1
C om parison of m
m a x
estim ation procedures
N um ber of events
mmax
C urrent E P R I P rocedure
A fter correction by account of m agnitude uncertainty
A fter correction by shift of S am ple Likelihood F unction
"true" m
m ax
= 6.92
Andrzej Kijko, South Africa
67. Andrzej Kijko, South Africa
Conclusions and Remarks
•Current EPRI Bayesian procedure by
default underestimates value of mmax and
locates mmax somewhere between maximum
observed magnitude and “true” mmax.
•Underestimation of mmax can reach value
of ½ a unit of magnitude.
•Two ways to correct the flaw of the
procedure are presented.
