Post on 08-Feb-2016
description
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