Time-dependent b value for aftershock sequences

Yuh-Ing Chen

Institute of StatisticsNational Central University

Chungli, Taiwan

Statsei4, Japan, 2006/1/10

Outline

1. Reasenberg-Jones (RJ) model

2. Distribution of aftershock-magnitude

3. Modified Reasenberg-Jones (MRJ) model

4. Comparison of RJ and MRJ models

1. Reasenberg and Jones model

Gutenberg-Richter (1944) law

log N(M)= a- bM, for M ≧Mc,

where N(M) is the number of earthquakes with magnitude M or larger and Mc is the cut-off magnitude.

The distribution of M is left-truncated exponential with survival distribution

S(M)=exp {-β(M -Mc)} , M ≧Mc

Modified Omori’s Law (Utsu, 1961)

( ) , ( ) Lp

kt t t

t c

)()(),( MStMt

P = P{At least one aftershock with magnitude between and occurs in (S, T)}

1 21 exp [S( )-S( )] ( )T

SM M t dt

Reasenberg-Jones(1989) model

2M1M

The blue dots represent the

aftershocks with magnitude

M 2.0 occured in 7 days

after the 1999/9/21 Chi-Chi,

Taiwan, earthquake. The red

star locates the epicenter of

the Chi-Chi earthquake.

The blac

k line corresponds to

the Che-Long-Pu fault.

The rectangular indicates

the region under study.

Determination of Mc －Wiemer (2000)

b-value and p-value

Probabilitic aftershock hazard (PAH) map－ Wiemer (2000)

The probability that at least one M≧5.0 aftershocks occurredduring [t+1, t+7) days after the Chi-Chi earthquake for 2 t 5 ≦ ≦Days.

Number of grids

Alarming area

TotalYes No

After-shock

Yes a b a+b

No c d c+d

Total a+c b+d a+b+c+d

( )

( )( )( )( )

a b c d ad bc

a b c d a c b d

Evaluation of the PAH map - Phi coefficient

0

Claim, at significance level 0.01, that alarming area is correlated

to the location of aftershocks if 2.326 .M M

Alarming for M ≧ M0 aftershocks in the grids with P ≧ P0

Phi coefficients for alarming M 5.0 ≧aftershocks based on the RJ-PAH map

← 2.326

The value obtained from 250 aftershocks, sliding by 100,

post to the Chi-Chi event.

2. Distribution of aftershock- magnitude

exp ( ) ( ) for c cS(M | t) t M M M M

210 )(lnln)( ttt 2 :MRJQ

tt ln)( :MRJL 10

: RJ t

3. Modified Reasenberg-Jones model

( , ) ( )

( ) exp ( ) ( )c

t M t S(M | t)

t t M M

dttMStMSt

P

T

S

)|()|()(exp1

T)} (S,in occurs

M and Mbetween magnitude

withaftershock oneleast P{At

21

21

Sequential prediction

The observed and predicted cumulative number of aftershocks next day based on the fitted models

obtained from the available aftershocks post to the Chi-Chi earthquake.

Model: MRJL

Model: MRJQ

4. Comparison of RJ and MRJ models