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