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