Aftershock Relaxation for Japanese and Sumatra Earthquakes
description
Transcript of Aftershock Relaxation for Japanese and Sumatra Earthquakes
Aftershock Relaxation for Japanese and Sumatra
Earthquakes
Kazu Z. Nanjo1, B. Enescu2, R. Shcherbakov3, D.L. Turcotte3,
T. Iwata1, & Y. Ogata1
1, ISM, Tokyo, Japan2, Kyoto Univ., Kyoto, Japan
3, UC Davis, CA, USA
Objective: Analyze the decay of the aftershock activity for Japanese and Sumatra earthquakes, using catalogs maintained by Japan Meteorological Agency and Advanced National Seismic System.
Approach: Generalized Omori’s law proposed by Shcherbakov et al. (2004, 2005).
The Gutenberg-Richter (GR) law (Gutenberg and Richter, 1954) N: # of earthq. with mag. ≥ mA and b: constants
The modified Bath’s law (Shcherbakov and Turcotte, 2004)
Δm* = mms - m*
m*: mag. of the inferred largest aftershock (m* = A/b) or mag. at the intercept between an extrapolation of the applicable GR law and N=1mms: main shock mag.
bmAN 10log
The GR law can be rewritten for aftershocks as
The modified Omori’s law (Utsu, 1962)
dN/dt: rate of occurrence of aftershocks with mag. ≥ mt: time since the main shock c and τ: characteristic timesp: exponent
mmmbN ms *log10
pctdtdN
/11
Requirement among the parameters
Assume: p is a constant independent of m and mms (Utsu, 19
62) b, mms, and Δm* are known parameters
Three possible hypotheses:1. c is a constant c = c0 andτis dependent on m2. τis a constant τ= τ0 and c is dependent on m3. c and τ are dependent on m (Shcherbakov et al., 2
004, 2005)
mmmb mspc *101
Hypoth. I, c = c0
Hypoth. II, τ = τ0
Hypoth. III, c and τ are dependent of m
c(m*): the characteristic time; β: a constantHypoth. III Hypoth. I if c(m*) = c0 and β = b Hypoth. II if c(m*) = τ0(p-1) and β = bp
pctmdtdN
0/11
mmmb ms
pcm
*0 10
1
pmctdtdN
/11
0
mmmb mspmc *0 101
pmctmdtdN
11
mmmpb
ms
mcmc
*
110*
mmmbpb
ms
pmcm
*110
1*
The list of main shocks
Spatial distribution and GR law for Kobe
Mag. ≥ 2t (days) < 1000
(
Δm*=1.1
m*=A/b=6.2
mms=7.3
t (days) < 1000A=4.85, b=0.78
L (km) = 0.02 X 100.5m_ms [Kagan, 2002]
Aftershock relaxation for Kobe and small aftershocks in the early periods
0.1 ≤ t < 1.00.01 ≤ t < 0.1
t (days) < 1000
How to find the best hypothesisTo find the optimal fitting of the prediction to the data f
or individual hypotheses
Point process modeling with max. likelihood (e.g., Ogata, 1983).
AIC (Akaike, 1974) to find the best hypothesis.
AIC = -2(max. log-likelihood) + 2(# of parameters)
# of parameters1.Hypoth. I: two (c0 and p)2.Hypoth. II: two (τ 0 and p)3.Hypoth. III: three (c(m*), β, and p)
Test of the generalized Omori’s law for KobeHypoth. I, c = c0 Hypoth. II, τ=τ 0
Hypoth. III,c and τare dependent on m
AIC=-3376.95 AIC=-3405.00
AIC=-3403.00
Aftershocks of Sumatra earthq.
A=8.88 b=1.20
mms=9.0
t (days) < 251
(m*=A/b=7.4
Δm*=1.6
Mag. ≥ 4.5t (days) < 251
Test of the generalized Omori’s law for Sumatra
Hypoth. I Hypoth. II
AIC=-925.42 AIC=-936.76
AIC=-934.76
Hypoth. III
Summary of the results
Test of the generalized Omori’s law for Tottori
Hypoth. I Hypoth. II
AIC=-6630.54 AIC=-6654.70
AIC=-6658.58
Hypoth. III
Establishment of the GR law (1)
Hypoth. II
pmctdtdN
/11
0
mmmb mspmc *0 101
Kobe earthq.
mms=7.3, b=0.78, Δm*=1.1p=1.16, τ0=0.000508 (days)
At time t = 0, dN/dt = 1/τ0
c values for different m
0.01 ≤ t < 0.1, b=0.79
t (days) < 1000, b=0.78
0.1 ≤ t < 1.0, b=0.72
Kobe
Establishment of the GR law (2)
0.01 ≤ t < 0.1, b=0.79
t (days) < 1000, b=0.78
0.1 ≤ t < 1.0, b=0.72
0.1 ≤ t < 1.0, b=1.27
0.01 ≤ t < 0.1, b=1.44
t (days) < 251, b=1.20
10 ≤ t < 100, b=1.37
1.0 ≤ t < 10, b=1.14
Kobe
Sumatra
Establishment of the GR law (2)
ConclusionThe generalized Omori’s law proposes:
Hypoth. I: τ scales with a lower cutoff mag. m and c is a constant.
Hypoth. II: c scales with m and τ is a constant.Hypoth. III: Both c and τ scale with m.
6 main shocks in Japan and Sumatra.Earthq. catalogs of JMA and ANSS. AIC and maximum likelihood to find the best hypoth. The hypoth. II is best applicable to the entire sequenc
e for different cutoff mag. from a state defined immediately after the main shock.
The c value is the characteristic time associated with the establishment of the GR law.
Test of the generalized Omori’s law for Niigata
Hypoth. I Hypoth. II
AIC=-7151.79 AIC=-7169.11
AIC=-7167.16
Hypoth. III
Summary of parameter values
m* = A/bΔm* = mms- masmax
Δm* = mms- m*
Hypothesis I, c = c0
Hypothesis II, τ = τ0
Hypothesis III, c and τ are dependent of m (Shcherbakov et al., 2004, 2005)
c(m*): the characteristic time; β: a constant
pctmdtdN
0/11
mmmb ms
pcm
*0 10
1
pmctdtdN
/11
0
mmmb mspmc *0 101
pmctmdtdN
11
mmm
pb
ms
mcmc
*
110*
mmmbpb
ms
pmcm
*110
1*
OutlineIntroduction of the generalized Omori’s law6 main shocks considered in this study
5 Japanese earthquakes1 Sumatra earthquake
Application of the law to these earthquakesMethods to find optimal fitting to the observed aftershock decayExamples of the applicationSummary of the application
DiscussionConclusion