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Detecting spatial variations of
earthquake clustering
parameters via maximum
weighted likelihood estimates
Jiancang Zhuang
[email protected]
Inst. Statist. Math.
Motivations
1. Seismicity clusters differ from place to place. It is
important to quantify such difference, for a
better understanding of the connection between
seismicity and tectonic environment and for
better probability forecasts of earthquakes for
different types of seismicity.
2. Fitting the space-time ETAS model for a small
region with a few events makes the estimation
and forecasts unstable.
3. Realization of weighted likelihood models is
simple and straightforward.
Kernel Estimates
Location-dependent
Density /rate
estimation
Location-dependent
model parameters of
point process
Space-Time Epidemic Type Aftershock Sequence
(ETAS) model
 Seismicity rate = "background" + “Triggered seismicity":


 (t , x, y, m)  s (m)   ( x, y )    (mi ) g (t  ti ) f ( x  xi , y  yi )
i:ti t


 Time distribution:
the Omori-Utsu law
p
g (t ) 
p 1  t 
1   , t  0
c  c
 Spatial location distribution of children:
q
q 1 
x y 
1   ( m  m )  , q  1
f ( x, y; m) 
 ( m  mc ) 
c
De
 De

2
2
productivity: mean number of children
 (m)  Ae ( mm ) , m  mc
c
Likelihood and weighted likelihood
• Likelihood function
log L 

( ti , xi , yi , mi )[0,T ] AM
log  (ti , xi , yi , mi )  
T
0
   (t, x, y, m)dtdxdydm
A M
• Weighted likelihood: For each point (𝜉, 𝜁), take the kernel
function ℎ(⋅,⋅), the weighted likelihood is (Zhuang, 2006)
log WL( ,  )   h( xi   ,  i  y ) log  (ti , xi , yi , m j )
i
T
   h( x   , y   ) (t , x, y, m)dtdxdydm
0 S
• Maximum weighted likelihood estimate (MWLE)
ˆ( ,  )  arg max logWL( ,  )
Comparing to Ogata's
Bayesian method with a smoothness prior
Ogata’s HIST ETAS model
Weighted likelihood
Parameters
estimation
The parameters for each location
The parameters for each location
must be estimated simultaneously. can be estimated independently.
Programming
for parallel
computation
Not so easy.
Easy.
Triggering
parameters
Source location dependent.
Target location dependent.
Selection of
Smoothness
Objective selection of smoothness
Subjectively selection of kernel
functions


 (t , x, y, m)  s (m)   ( x, y )    (mi ) g (t  ti ) f ( x  xi , y  yi )
i:ti t


p
p 1  t 
g (t ) 
1   , t  0
c  c
target
dependent
 (m)  Ae ( mm ) , m  mc
c
q
q 1 
x y 
1   ( m  m )  , q  1
f ( x, y; m) 
 ( m  mc ) 
c
De
 De

2
2
source
dependent
• Parameters should be all dual-way dependent on both the source
and the target locations.
• To reduce the complexity of model implementation, since the
events in an earthquake sequence are quite close in space, such
differences could be neglected when investigating seismicity in a
much larger scale.
Dataset 1 -- Japan
•
•
•
•
•
•
JMA (Japan Meteorological Agency) catalog
Longitude: 121 ◦ ∼ 155 ◦ E,
Latitude: 21 ◦ ∼ 48 ◦ N,
Depth: 0 ∼ 100 km,
Time: 1 Jan 1965 to 31 Dec 2009
Magnitude: MJ ≥4.0
• 19,019 events
Background rate: event/(day∙deg2)
A
𝛼
𝑞
𝛾
𝜶 values and tectonics
𝛼
𝑝
Volcanic line
Dataset 2 - Italy
•
•
•
•
•
•
ITAG catalog (INGV)
Longitude: 8 ◦ ∼ 18 ◦ E,
Latitude: 35 ◦ ∼ 48 ◦ N,
Depth: 0 ∼ 70 km,
Time: 17 Apr 2005 to 28 May 2014
Magnitude: M ≥ 2.6
• 5,627 events
Tectonic settings and seismicity
(Billi et al, 2007, Geosphere)
A
𝒒
𝜶
𝜸
Clustering
Parameters
𝒑
I
II III IV
A
𝛼
Subregions of clustering features
H
I
H
𝑝
L
𝑞
H H
𝛾
H H
H: high value
L: low value
II
III
L
L
IV
Conclusions
1. Results from fitting both the Japan and Italy regions
show how the clustering characteristics of
seismicity vary in space and their connections to
the tectonic environments.
2. The weighted likelihood estimator can be used to
obtain stable estimates of spatial changes of the
ETAS parameters.
3. With complicated seismicity in both regions, the
ETAS model with weighted likelihood estimates is
potentially powerful for improving earthquake
forecast from using a constant ETAS model.