The Centenary of the Omori Formula for a Decay Law of Aftershock Activity

Presenter§1~§7 Shota Sakurai (Hatano Lab.M2)§8~§15 Takashi Okuda (Ide Lab.M2)

Theory of Earthquake Occurrence Seminar

Tokuji Utsu, Yosihiko Ogata, and Ritsuko S. Matsuʼura

地震発⽣論セミナー 2017年4⽉17⽇

Outline

地震発⽣論セミナー 2017年4⽉17⽇

Introduction : Review Omori Law to cerebrate the centenary of the original one.

Purpose : To know about the study of aftershocks

Method : Considering interaction between aftershocks, foreshocks

Conclusion : the ETAS model seems to be useful.

Omori Law

n(t) = K(t+ c)�1

l Represent the decay of aftershock activity with time

l Omori found it from Nobi EQ(1891)

t

Occurrence rate through 1891-1899 of felt EQ at Gifu after the Nobi EQ of 1891

the frequency of felt aftershocks per unit time

※K, c; constantelapsed time from mainshock

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Modified Omori Law

http://www.zisin.jp/modules/pico/index.php?content_id=514

T.Utsu(1928-2004)

The decay of aftershock activity of several EQs is somewhat faster than that expected from original Omori formula

地震発⽣論セミナー 2017年4⽉17⽇

Cumulative number of aftershocks

Nobi EQ(1891)Hokkaido-NanseiOki EQ(1993)

Modified Omori Law

地震発⽣論セミナー 2017年4⽉17⽇

N(t) =

Z t

0n(s)ds = K[c1�p � (t+ c)1�p]/(p� 1)

n(t) = K(t+ c)�p

Modified Omori law

( p = 0.9-1.8, frequently:1.1-1.4)

p and c value may relate to somewhat value of mainshock, largest aftershock, b-value of GR Law?

地震発⽣論セミナー 2017年4⽉17⽇

Author Earthquake p valueAdams and Le Fort (1963) Westport (1962) 0.9 ± 0.1

Papazachos et al (1967) Greece (1894?) 1.13 ‒ 2.5

Papazachos et al (1975a) Greece (1894?) 0.83 ‒1.86

Lukk (1968) Hindu Kush intermediate depth (1965) 1.4Page (1968) great Alaska (1964) 1.14

±0.06Pyall and Savage (1969) Nevada (1968) 0.79

Iio (1986a) Nevada (1968) 1.12Usami Kyoto(827),Meio(1498),Ansei(1854),Edo(1855) 0.8 -1.7

l Some researchers tried to find p value relation to aftershocks, but little has been known

p value estimates from various sequences

p value dependence on the lowest limit of magnitudel Some researchers tried to find p value relation to aftershocks, but little has

been knownEstimates of the parameters of the modified Omori formula fore the aftershocks of the 1993 Hokkaido-Nansei-Oki EQ

→Mz vs p : seems to have no relation地震発⽣論セミナー 2017年4⽉17⽇

Lowest limit of magnitude

Omori Law on complex casesl Anomalous cases are found

Fukuoka double EQ(1899)San Francisco EQ(1906)

l One or more large aftershocks accompanied by many secondary aftershocks

n(t) = K(t+ c)�p +H(t� T2)K2(t� T2 + c2)�p2 +H(t� T3)K3(t� T3 + c3)

�p3

main shock secondary aftershock of large aftershocks occurring at time T2 and T3

地震発⽣論セミナー 2017年4⽉17⽇

Aftershock activitydecrease→constant→increase

l In relation to swarm, there is the case where aftershock activity doesnʼt obey eq.(9)

(9)

Estimation of Parametersl K valueAssuming a non-stationary Poisson process for N aftershocks occurring at time ti (Ts≦ti≦Te) with intensity λ(t) (=n(t))

L =

"NY

i=1

�(ti)

#exp

"�Z Te

Ts

�(t)dt

#

@(lnL)/@c = 0

@(lnL)/@K = 0

@(lnL)/@p = 0

K = N(p� 1)/[(Ts + c)�p+1 � (Te + c)�p+1]

地震発⽣論セミナー 2017年4⽉17⽇

Defining likelihood function

NX

i=1

ln(ti + c)� N

p� 1�N

ln(Ts + c)(Ts + c)�p+1 � ln(Te + c)(Te + c)�p+1

(Ts + c)�p+1 � (Te + c)�p+1= 0

pNX

i=1

1

ti + c� N(p� 1)(Ts + c)�p � (Te + c)�p

(Ts + c)�p+1 � (Te + c)�p+1= 0

l c valueØ represents the complex feature of rupture process of the main

shock?[Yamakawa(1968)]

l p valueØ relation to heat flow(p high → heat flow high)[Kisslinger and

Jones(1991)]

Ø correlation with the degree of heterogeneity of the fault zone of main shock[Mikumo and Miyatake(1979)](but data is too few to discuss the relation to p)

Ø Regional variation of the p value uSuperposed sequencesuETAS model

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Estimation of Parameters