Transcript presentation

Time-dependent b value for
aftershock sequences
Yuh-Ing Chen
Institute of Statistics
National 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)
k
 (t ) 
,
p
(t  c)
t  tL
Reasenberg-Jones(1989) model
 (t , M )   (t ) S ( M )
P = P{At least one aftershock with
magnitude between M 1 and M 2
occurs in (S, T)}

 1  exp  [S(M1 )-S(M 2 )]

T
S
 (t )dt

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 black 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 occurred
during [t+1, t+7) days
after the Chi-Chi
earthquake for 2≦t≦5
Days.
Evaluation of the PAH map - Phi coefficient
Alarming for M≧ M0 aftershocks in the grids with P ≧ P0
Number of
grids
Aftershock
Total

Alarming area
Total
Yes
No
Yes
a
b
a+b
No
c
d
c+d
a+c
b+d
a+b+c+d
a  b  c  d  (ad  bc)
(a  b)(c  d )(a  c)(b  d )
Claim, at significance level 0.01, that alarming area is correlated
to the location of M  M 0 aftershocks if   2.326 .
Phi coefficients for alarming M≧5.0 aftershocks
based on the RJ-PAH map
← 2.326
2. Distribution of aftershockmagnitude
The  value obtained from 250 aftershocks, sliding by 100,
post to the Chi-Chi event.
S(M |t)  exp  (t )  (M  M c ) for M  M c
RJ :
 t   
MRJL :  (t )   0  1 ln t
MRJQ :  (t )   0  1 ln t   2 (ln t ) 2
3. Modified Reasenberg-Jones model
 (t , M )   (t ) S(M | t)
  (t )  exp   (t )  ( M  M c )
P  P{At least one aftershock with
magnitude between M1 and M 2
occurs in (S, T)}

 1  exp    (t )S ( M 1 | t )  S ( M 2 | t )dt
T
S

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