On secular spatial seismicity Yosihiko Ogata, ogata@ism.ac.jp

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 ogata@ism.ac.jp if you are interested in.

California ANSS data