completeness of earthquake catalogues and statistical forecasting
TRANSCRIPT
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Completeness of Earthquake Catalogues and Statistical Forecasting
byAndrzej Kijko
Council for GeosciencePretoria, South Africa
International Conference on “Global Change”Islamabad, 13-17 November 2006
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OUTLINE
• Incompleteness of catalogues
• Uncertainties (location, magnitudes and origin times)
• Inadequate model of seismicity
• Seismogenic zones
• Maximum earthquake magnitude mmax
Surprise, Surprise!
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APPROACH 1: Parametric Procedure
(Cornell, 1968)
SITE SITE
PARAMETERS•activity rate λ•b-value•maximum magnitude mmax
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Advantages:
• Can account for seismic gaps, non-stationary seismicity, faults etc.
Disadvantages:
• Specification of seismogenic zones• Requires knowledge of seismic hazard
parameters (e.g. activity rate, b-value, mmax) for each zone
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APPROACH 2 Non-parametric “Historic” Procedure
(Veneziano, et al., 1984)
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Advantages
• No division into seismic zones needed• Seismic parameters no required
Disadvantages
• Unreliability at low probabilities, or at the areas with low seismicity
• The procedure does not take into account incompleteness and uncertainty of earthquake catalogues
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ALTERNATIVE Combination of both
The ‘Parametric - Historic’ procedure• Assessment of basic hazard parameters (e.g.
seismic activity rate, b-value, mmax) for the area in the vicinity of the particular site
• Assessment of the distribution function for amplitude of ground motion for a specified site.
• If the procedure is applied to all grid points a seismic hazard map can be obtained
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Epicentre500 km
PARAMETRIC-HISTORIC PROCEDURE
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INCOMPLETENESS AND UNCERTAINTIES OF SEISMIC CATALOGUES
Kijko, A. and Sellevoll, M.A. 1992. Estimation of earthquake hazard parameters from incomplete data files. Part II. Incorporation of magnitude heterogeneity. BSSA, 82(1),
120-134.
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MAGNITUDE DISTRIBUTION
Magnitude, m
log
n
log n = a - bm
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APPLICATION TO THE GUTENBERG-RICHTER MAGNITUDE DISTRIBUTION
.,)](exp[1)](exp[1
,1
0)(
max
maxmin
min,
minmax
min
mmformmmfor
mmfor
mmmmmFM
>≤≤
<
⎪⎪⎩
⎪⎪⎨
⎧
−−−−−−
=ββ
)10ln(bwhere =βHow???
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PGA DISTRIBUTION
• Please note:
2/ cβγ =
ln (PGA), x
lnn
ln n = α - γx
,lnln 4321 rcrcmcca ⋅+⋅+⋅+=2/ cβγ = where
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APPLICATIONSSeismic Hazard Map of Sub-Saharan Africa
10% Probability of exceedance of PGA in 50 years
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SOUTH AFRICASeismological Monitoring Network
$
$
$
$
$
$
$
$
$
$
$
$
$
$$
$
$
$
$
$
$
34° 34°
32° 32°
30° 30°
28° 28°
26° 26°
24° 24°
22° 22°
14°
14°
16°
16°
18°
18°
20°
20°
22°
22°
24°
24°
26°
26°
28°
28°
30°
30°
32°
32°
34°
34°
MSNA
MOPALEP
BFTSLRKSR
SWZNWL
POGPRYS
SEK
HVD KSD
GRM
SOE
PKA
CVNA
ELIM
CER
UPI
KOMG
Seismological station$
LEGEND
100 0 100 200 Kilometers
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Gold and Diamonds
Gold pouringKimberley Big Hole
Mine Shaft
Diamonds
Krugerrand
Star of Africa
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Magnitude = 5.2 at Welkom 1976
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Stilfontein, 9 March 2005, ML = 5.3
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Tulbagh, 29 September 1969, ML = 6.3
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Seismic Hazard Map of South Africa
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Maximum Possible Earthquake magnitude in Johannesburg/Pretoria?
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Definition of mmax
• The maximum regional magnitude, mmax, is the upper limit of magnitude for a given region
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Δ+= obsmm maxmaxˆ
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The Generic Formula for Estimation of the Maximum Regional Magnitude, mmax
[ ]∫+=max
min
d)(ˆ maxmax
m
m
nM
obs mmFmm
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THREE REAL LIFE CASES
1. Earthquake Magnitudes are Distributed according to the Gutenberg-Richter relation
2. Earthquake Magnitude Distribution deviates largely from the Gutenberg-Richter relation
3. No specific model for The Earthquake Magnitude Distribution is assumed
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CASE 1: Earthquake Magnitudes are Distributed according to the Gutenberg-Richter relation
Magnitude, m
log
n
log n = a - bm
.,)](exp[1)](exp[1
,1
0)(
max
maxmin
min,
minmax
min
mmformmmfor
mmfor
mmmmmFM
>≤≤
<
⎪⎪⎩
⎪⎪⎨
⎧
−−−−−−
=ββ
where β = b ln 10
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Case 1
( ) ( )( ) ( )nm
nnEnEmm obs −+
−−
+= expexp
ˆ min2
1121maxmax β
)(1 •E
)](exp[ minmax12 mmnn obs −−= β
where
And denotes an exponential integral function
)](exp[1{ minmax1 mm
nn obs −−−=
β
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Case 2: The Earthquake
Magnitude Distribution
deviates largely from the
Gutenberg-Richter relation
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Case 2: The Earthquake Magnitude Distribution deviates largely from the
Gutenberg-Richter relationEarthquake magnitude distribution follow the Gutenberg-
Richter relation with some uncertainty in the value
β
( )⎪⎩
⎪⎨
⎧
>≤≤
<−+−=
max
maxmin
min
min) },)]/([1{1
0
mmformmmfor
mmformmppCmF q
M β
β
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Case 2
][ ),/1(),/1()]1/(exp[ˆ/1
maxmax δδβ
δ qrqrnrmm qqqq
obs −Γ−−Γ−
+=
βσ)(•Γ
β
where
,minmax mmp
pr−+
=,βδ nC= ,11
qrC
−=β
,)/( 2βσβ=p 2)/( βσβ=q
is the incomplete delta functionand is the known standard deviation of
This formula can be used where there exists temporal trends, cycles, short-term oscillations and pure random fluctuations
in the seismic process
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What does one do when the emperical distribution of earthquake magnitude is of the following form…
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Case 3: No specific model for The Earthquake Magnitude Distribution is
assumed• A non-parametric estimator of an unknown
PDFfor sample data mi , i=1,…n,:
• Where h is a smoothing factor and is a kernel function
∑=
⎟⎠⎞
⎜⎝⎛ −
=n
i
iM h
mmKnh
mf1
1)(ˆ
)(•K
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Case 3: Continued…
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Case 3: Continued…
• From the functional form of the kernel and from the fact that the data comes from a finite interval, one can derive the estimator of the CDF of earthquake magnitude:
• Where denotes the standard Gaussian CDF
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
>
≤≤
<
⎟⎠⎞
⎜⎝⎛ −
−⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ −
−⎟⎠⎞
⎜⎝⎛ −
=
∑
∑
=
=
max
maxmin
min,
1
minmax
1
min
,
,1
,0
)(ˆ
mmfor
mmmfor
mmfor
hmm
hmm
hmm
hmm
mF n
i
ii
n
i
ii
M
φφ
φφ
)(•φ
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Application 1: Southern California
CASE Assumptions mmax ± SD
1
2
3
Gutenberg-Richter 8.32 ±0.43
Gutenberg-Richter+ Uncertainty in b-value
8.31 ±0.42
No model for distribution is assumed (Non-parametric procedure)
8.34 ±0.45
Field et al. 1999 7.99
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Fitting of Observations using the Non-Parametric Procedure
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Application 2: New Madrid Zone
5 5.5 6 6.5 7 7.5 8 8.5 910
1
102
103
104
105
106
Magnitude [mb]
Mea
n R
etur
n P
erio
d [Y
EAR
S]
New Madrid Seismic Zone
Prehistoric events excludedPrehistoric events included
mmax = 7.9
mmax = 8.7
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CONCLUSIONS
• It is possible to develop a lot of useful tools which is capable of taking even the most diverse behaviour of seismic activity into account
• A lot needs to be done to improve it.
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THE END
THANK YOU