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On secular spatial seismicity Yosihiko Ogata, [email protected]
The Institute of Statistical Mathematics; and Earthquake Research Institute, University of Tokyo
Reference Ogata, Y. (2017). Statistics of Earthquake Activity: Models and Methods for Earthquake Predictability Studies, Annu. Rev. Earth Planet. Sci. 45, no. 1, doi: 10.1146/annurev-earth-063016-015918
JMA catalog 1994 – 2011.3.10, M ≥ 4
LONGITUDE
LATI
TUDE
TIME (days)
1( , ) ( , ) jj j j j
j
x xQ x y x x y y S
y y−
− = − − −
where Iso-contour of λ(t, x, y| Ht)
lati
tude
longitude
{ }( , , , );t j j j j jH t x y M t t= <where
{ }Pr [ ) [ ) [ ) |( , , | )t
t
anevent in t t x x y y Ht x y H
t x yλ
+ ∆ × + ∆ × + ∆≈
∆ ∆ ∆
Space-Time ETAS model
{ : }
( , )( , , | ) ( , )
( ) jj
qj j j
t Mpj t t j
Q x x y yKt x y H x y dt t c eαθλ µ ν
−
<
− − = ⋅ + + − +
∑
01
log ( ) log ( , , | ) ( , , | )i
N T
i i i t tAi
L t x y H t x y H dxdydtθ θθ λ λ=
= −∑ ∫ ∫∫ˆˆ ˆ ˆˆ ˆ ˆ ˆ( , , , , , , )K c p d qθ µ α= Maximum likelihood estimate (MLE)
( , , , , , , )K c p d qθ µ α=parameters:
( , ){ ; }
( , , | ) ( , )0 ( , )
( ) j jj
t p x yj t t j
t x y H x yj jK x y
t t cλ µ
<= +
− +∑ ( , ) ( )
1( , ) ( , )j j
t qj j j jx y M Mj c
jx x y y S x x y y
de
α
−
−
−− − − −+
µ (x,y) K0 (x,y)
α (x,y) p (x,y)
Mogi (1967, BERI)
Piecewise linear function
4/14 00:00 4/14 22:26
4/15 01:03 4/16 13:25
event/day/100 km^2
event/day/100 km^2
Rates of M≧4 event during the 2016 Kumamoto sequence
( , ){ ; }
( , , | ) ( , )0 ( , )
( ) j jj
t p x yj t t j
t x y H x yj jK x y
t t cλ µ
<= +
− +∑ ( , ) ( )
1( , ) ( , )j j
t qj j j jx y M Mj c
jx x y y S x x y y
de
α
−
−
−− − − −+
M≧4.0
M6.5
M6.4
M7.3
Harvard global data
:
Background seismicity
Mw≧5.4
{ : }
( , ) ( , )( , , | ) ( , )
( ) jj
qj j j
t Mpj t t j
K x y Q x yt x y H x y d
t t c eαλ µ−
<
= + + − +
∑
even
ts d
eg^2
day
Estimated from M≥5.0 for 1926-1995
= earthquakes of M>= 6.7 during 1996 – 2011 Mar
( , ){ ; }
0( , , | )
( , )( , )
( ) j jj
t p x yj t t j
j jt x y H
K x yx y
t t cλ µ
<= +
− +∑ ( , ) ( )
1( , ) ( , )j j
t qj j j jx y M Mj c
jx x y y S x x y y
de
α
−
−
−− − − −+
Background seismicity rates of the HIST-ETAS model
1885~1925 4.0M ≥ 1926~1995 6.0M ≥ 1885~1925
4.0M ≥ 1926~2010 6.0M ≥ 1885~1925
4.0M ≥ 1926~2012 6.0M ≥
1885~1925 4.0M ≥ 1926~1995 6.0M ≥ 1885~1925
4.0M ≥ 1926~2010 6.0M ≥ 1885~1925
4.0M ≥ 1926~2012 6.0M ≥
+
Estimation period1926 ~1995; M≧4.0
M7.2 M7.2
M6.8 M6.7
M6.7
M7.2
M7.3
M7.0 M6.9
M7.0
Predictive period 2000 ~2016; M≧6.7
10-6
10-7
10-8
M≧
4 ev
ents
/ km
^2 /
day
{ ; }( , , | ) ( , )
( , )
( )jt p
j t t j
t x y H x yK x y
t t cλ µ
<= +
− +∑ ( )
( , ) ( , ) t qj j j j jM Mj c
x x y y S x x y yd
eα
−
−
− − − −+
Background seismicity rates of the HIST-ETAS model
Among the characteristic parameters of the HIST-ETAS models as mathematically defined below, we are primary interested in intensity of the background seismicity (BS) that subtracted the triggering effect of the Hierarchical Space-Time ETAS (HIST-ETAS) model. The BS solution values vary regionally in the range of a few orders within a seismogenic zone, but are confirmed to always take the same pattern, independent of observed periods. In particular, in inland Japan, the spatial intensity is highly correlated with the spatial variation of shear stress accumulation rates or volumetric stressing rates calculated from the geodetic GPS data. Thus, we think the estimated background activity to be secular spontaneous seismicity. Hence, this is quite useful for the secular prediction of large earthquake locations, in conjunction with Gutenberg-Richter distribution where the b-value is also location dependent.
Nishimura (2017, Newton) 2005ー2009 Max. Shear Strain Rate
#(event of M≧4.0) / km^2 / day Estimation period 1926 ~1995
JMA 1926 ~ 1995 M≧5.0
M≧
4 ev
ents
/deg
^2/d
ay
2009 Nov 1 2012 May 1
µ (x,y) Background intensity
2000/01/01 2011/03/10
M 4.0≥
2011/03/01 2015/06/02
M 4.0≥
10
3.
1 3
1 10
1
2
1
2
1
( , , | )( , , | )
t
t
t x y Ht x y H
λλ
t1 = 2009.11.01
t2 = 2012.05.01
2011/3 /1-2015/6/2
2000/1/1 – 2011/3/10
M 4.0≥
M 4.0≥
t2= 2012.05.01
4.0M ≥
t1 = 2009.11.01
4.0M ≥
( , ){ ; }
( , )( , )
(( , |
), )
jp x y
j t t jt
K x yx y
t tt x y H
cλ µ
<= +
− +∑
( , )
( , ) ( )
( , ) ( , )α
−
−
− − − −+
t q x yj j j j jx y M Mj c
x x y y S x x y yd
e
Estimated period 1926 ~1995; M≧4.0 (1885~1925 ; M≧6.0)
○: Historical M≧6.8 ; 599 – 1884 by Utsu
Area Distortion Rates (Sagiya et al., 2000) Background Seismicity Rates
α-value
M >= 4 1932 - 2011
K0-value
M >= 4 1932- 2011
µ-value
M >= 4 1932- 2011
p-value
M >= 4 1932- 2011
q-value
( , ){ ; }
( , , | ) ( , )0 ( , )
( ) j jj
t p x yj t t j
t x y H x yj jK x y
t t cλ µ
<= +
− +∑ ( , )
( , ) ( )
( , ) ( , ) j j
j j
t q x yj j j j jx y M Mj c
x x y y S x x y yd
eα
−
−
− − − −+
M≧
5 event / year / (0.1deg x 0.1deg)
1984-2005 M>=3
1984-2005 M>=3
b-value µ-value (event / day / deg^2)
2011 ( , )
5.0 200610 ( , )b x y M
ETASPIXEL
dM dt dxdy x yµ∞ −∫ ∫ ∫∫
( , )b x y
( , )ETAS x yµ = 2006 – 2011, M>=5 earthquakes
( , ){ ; }
( , )( , , ) ( , )
( )λ µ
<= +
− +×∑
jp x y
j t t j
K x yt x y x y
t t c
( , )
( , ) ( )
( , ) ( , )α
−
−
− − − −+
t q x yj j j j jx y M Mj c
x x y y S x x y yd
e
Background rates of HIST-ETAS model Gutenberg-Richter Law ×
p-values
Computer programs with English manuals are available for the Delaunay-based nonstationary Poisson process, location-dependent b-values, Hierarchical Space-Time ETAS (HIST-ETAS) models. Contact the author or email [email protected] if you are interested in.
California ANSS data