Bayesian estimation of ETAS models using Rstan
(applied to seismic recurrence in Ecuador 2016)
PhD student Applied Mathematics Fausto Fabian Crespo
Fernández1 , PhD Statistics Carlos Jimenez Mosquera 1
1 San
Francisco de Quito University(USFQ)
StanCon2018
Introduction
There were four mega earthquakes in the Ecuadorian Coast in the
past century:
I
1906(magnitude 8.8)
I
1942(magnitude 7.8)
I
1958(magnitude 7.7)
I
1979(magnitude 8.2)
And then... April 16th, 2016 a quake of magnitude 7.8 according to
USGS (magnitude 7.4 according to Seismology Institute - EPN)
Methods
Omori law
The empirical law of Omori and the law of Omori-Utsu (also called
Modified Law of Omori) Utsu1995 describe the decreasing frequency
of aftershocks over time after an earthquake:
N(t) =
K
(t + c)
(1)
N(t) =
K
(t + c)p
(2)
Where N(t) is the occurrence rate of events, t is the time since the
earthquake and K , c, p are constants.
Methods
Gutenberg-Ritcher law
... relates the magnitude to the number of earthquakes with
magnitudes greater than M:
log10 N(≥ M) = a − bM
(3)
That is, the number of events of magnitude greater than a
threshold, decreases exponentially with the increase of that
threshold value by a power law.
So, the distribution of magnitudes by the Gutenberg-Richter law
s(m) = βe −βm
Gutenberg-Richter law
Gutenberg-Richter law: linear regression log (N) = a + bM and
then a = 5.064(sd = 0.039) and −0.694(sd = 0.007). From these
values, b is 0.694 and β = bln(10) = 1.598.
Methods
An earthquake T is represented by a tuple (xi , yi , zi , Mi )
Temporal ETAS model
The simplest ETAS model is the temporal model with constant
P
(p−1)c p−1
background seismicity: λ(t|Ht ) = µ + j:tj <t K(t−t
p
j +c)
where µ is the background intensity that is assumed to be constant
(measured in events / day) and
g (t) =
(p − 1)c p−1
(t + c)p
(4)
is the probability density function of the occurrence times of the
events triggered by previous earthquakes.
Methods
Temporal ETAS model
If the background intensity is not constant, but depends only on the
longitude and latitude x, y we have:
P
(p−1)c p−1
λ(t|Ht ) = µ(x, y ) + j:tj <t K(t−t
p
j +c)
where µ(x, y ) is now measured in events per day per unit of longitude
and per unit of latitude. In this case, it is generally assumed that
µ(x, y ) = µu(x, y ) where µ on the right side is a constant.
Methods
ETAS model with magnitudes
Considering the magnitudes of earthquakes we have the model
λ(t|Ht ) = µ +
α(Mj −M0 )
P
K (p−1)c p−1 Ae
j:tj <t
(t−tj +c)p
Methods
ETAS models stability
The event rate in the ETAS models may explode. Stability depends
on the branching ratio n =expected
number of descendants of a
R ∞ R Mmax
parent event. We have n= 0 dt M0 s(m)λ(t)dm
with s(m) = βe −βm the distribution of magnitudes by the GutenbergRichter law and λ(t) the branching term.
For the temporal ETAS model with magnitudes and p > 1 we have
Kc 1−p β 1−e −(β−α)(Mmax −M0 )
n = (p−1)(β−α)
1−e −β(Mmax −M0 )
Methods
For the temporal ETAS model with magnitudes and p > 1 we have
Kc 1−p β 1−e −(β−α)(Mmax −M0 )
n = (p−1)(β−α)
Assuming Mmax = ∞ the
1−e −β(Mmax −M0 )
previous formula can be reduced to: n =
Sornette2005 y Touati2011
and n is infinite if p < 1 or if α > β.
Kc 1−p β
(p−1)(β−α)
Methods
ETAS models stability
If each event induces another event: n = 1 then the process
propagates indefinitely. This justifies normalizing the functions
that
in the sum over the preceding events. For example
R ∞ appear
K
dt
= 1 implies that we need to add (p − 1)c p−1 to the
p
0 (t+c)
RM
constant K and similarly M0max βe −β(m) dm = 1 implies we need
to add
1/(exp(−βM0 ) − exp(−βMmax )) to the constant β .
Methods
RT
P
For temporal models: logL(θ) = j log λ(tj |Ht ) − 0 max λ(t)dt
The closed forms for temporal and temporal with magnitudes
models: logL(µ, k, p, c) =
PN
PN
c p−1
i=1 log (λ(ti )) − µTmax − k
i=1 (1 − (Tmax −ti +c)p−1 )
P
logL(µ, k, p, c, A, α) = N
i=1 log (λ(ti )) − µTmax −
PN α(M −M0 )
c p−1
i
kA i=1 e
(1 − (Tmax −t
)
+c)p−1
i
Listing 1: Temporal Rstan model
24
with constant background seis25
26
micity
27
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5
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7
8
9
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28
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functions{
r e a l l o g l i k e l i h o o d ( i n t N,
30
r e a l mu ,
31
real k ,
32
real p ,
33
real c ,
v e c t o r t i m e s d i f f34
35
,
36
r e a l tmax ) {
37
r e a l s e i s m i c r a t e [N ] ;
38
r e a l i n t e g r a l o f r a t e [N ] ;
39
real integral mu ;
40
real log likelihood ;
41
s e i s m i c r a t e [ N ]=l o g (mu) ;
42
i n t e g r a l o f r a t e [ N ]=0 ;
43
f o r ( j i n 1 : ( N−1) ) {
44
v e c t o r [ N−j ] y ;
45
int start ;
46
i n t end ;
s t a r t =N∗( j −1)−( j ∗( j −1) ) /2 + 1 ; 47
48
end =j ∗N−( j ∗( j +1) ) / 2 ;
49
y=t i m e s d i f f [ s t a r t : end ] ;
y=( k ∗( p − 1 ) ∗ c ˆ ( p−1) ) ∗exp(−p∗
50
l o g ( y+c ) ) ;
s e i s m i c r a t e [ j ]=l o g (mu+sum ( y ) ) ; 51
i n t e g r a l o f r a t e [ j ]=( k ) ∗(1−c ˆ ( p
−1)/
( ( t i m e s d i f f [ j ]+ c ) ˆ ( p−1) ) ) ;
}
i n t e g r a l m u=mu∗tmax ;
l o g l i k e l i h o o d=sum ( s e i s m i c r a t e )−
i n t e g r a l m u −sum (
integral of rate ) ;
return ( l o g l i k e l i h o o d ) ;
}
}
data{
i n t <l o w e r=0> N ;
r e a l <l o w e r=0> m a x t i m e ;
v e c t o r [ N∗(N−1) / 2 ] t i m e s d i f f ;
}
parameters{
r e a l <l o w e r=0> mu ;
r e a l <l o w e r=0> k ;
r e a l <l o w e r=1.000005 > p ;
r e a l <l o w e r=0.00005 > c ;
}
model{
mu˜ e x p o n e n t i a l ( 2 . 8 ) ;
k˜ e x p o n e n t i a l ( 2 . 8 ) ;
p˜ e x p o n e n t i a l ( 0 . 3 ) ;
c˜ e x p o n e n t i a l ( 2 . 8 ) ;
i n c r e m e n t l o g p r o b ( l o g l i k e l i h o o d (N, mu
,k,
p , c , t i m e s d i f f , max time ) ) ;
}
Listing 2: Temporal with magni29
30
tudes Rstan model with constant
31
background seismicity
32
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27
28
33
functions{
34
r e a l l o g l i k e l i h o o d ( i n t N,
35
r e a l mu ,
36
real k ,
37
real p ,
38
real c ,
r e a l alpha ,
39
r e a l A,
40
vector t ,
41
v e c t o r m a g n i t u d e s42
,
43
v e c t o r t i m e s d i f f44
,
45
r e a l max time ,
46
r e a l magnitude0 ){
47
r e a l s e i s m i c r a t e [N ] ;
48
r e a l i n t e g r a l o f r a t e [N ] ;
49
real integral mu ;
50
real log likelihood ;
51
s e i s m i c r a t e [ N ]= l o g (mu) ;
52
i n t e g r a l o f r a t e [ N ]=0 ;
53
f o r ( j i n 1 : ( N−1) ) {
54
v e c t o r [ N−j ] y ;
55
int start ;
56
i n t end ;
57
s t a r t = N∗( j − 1 )−( j ∗( j −1) ) /2 +58
1;
59
end =j ∗N−( j ∗( j +1) ) / 2 ;
60
y=t i m e s d i f f [ s t a r t : end ] ;
61
y=( k∗A∗( p−1)∗c ˆ ( p−1) ) ∗exp ( a l p h a ∗
62
( m a g n i t u d e s [ ( j +1) :] −
63
magnitude0 ) )
. ∗ exp(−p∗ l o g ( y+c ) ) ;
s e i s m i c r a t e [ j ]=l o g (mu+sum ( y ) ) ;
i n t e g r a l o f r a t e [ j ]=( k∗A) ∗exp (
alpha∗
( m a g n i t u d e s [ j +1]−
m a g n i t u d e 0 ) ) ∗(1−c ˆ ( p−1)/
( ( t i m e s d i f f [ j ]+ c ) ˆ ( p−1) ) ) ;
}
i n t e g r a l m u =mu∗ m a x t i m e ;
l o g l i k e l i h o o d=sum ( s e i s m i c r a t e )−
i n t e g r a l m u −sum ( i n t e g r a l o f r a t e
);
return ( l o g l i k e l i h o o d ) ;
}
}
data{
i n t <l o w e r=0> N ;
v e c t o r [N] times ;
r e a l <l o w e r=0> m a x t i m e ;
v e c t o r [N] magnitudes ;
v e c t o r [ N∗(N−1) / 2 ] t i m e s d i f f ;
r e a l <l o w e r=0> t h r e s h o l d m a g n i t u d e ;
}
parameters{
r e a l <l o w e r=0> mu ;
r e a l <l o w e r=0> k ;
r e a l <l o w e r=1.000005 > p ;
r e a l <l o w e r=0.00005 > c ;
r e a l <l o w e r=0> a l p h a ;
r e a l <l o w e r=0> A ;
}
model{
mu˜ e x p o n e n t i a l ( 2 . 8 ) ;
k˜ e x p o n e n t i a l ( 2 . 8 ) ;
p˜ e x p o n e n t i a l ( 0 . 3 ) ;
c˜ e x p o n e n t i a l ( 2 . 8 ) ;
alpha ˜ exponential (2.8) ;
64
65
66
A˜ e x p o n e n t i a l ( 2 . 8 ) ;
i n c r e m e n t l o g p r o b ( l o g l i k e l i h o o d (N, 67
mu , k , p , c
68
, alpha , A, times , magnitudes , t i m e s d i f f
,
max time , t h r e s h o l d m a g n i t u d e ) ) ;
}
Methods
Isotropic spatio temporal ETAS model
In 1998 Ogata proposed a modified version of the spatio-temporal
ETAS model: P
λ(t, x, y ) = µ + j:tj <t g (t − tj , x − xj , y − yj , Mj |Ht )
where
α(M−M )
x 2 +y 2
g (t, x, y , M) Ke(t+c)p 0 { e γ(M−M
+ d}−q and the background
0)
intensity µ is constant. We can normalize:
g (t, x, y , M) =
K (p−1)c p−1 (q−1)d q−1 αe (α−γ)(M−M0 )
x 2 +y 2
{ e γ(M−M
π(t+c)p
0)
+ d}−q
Methods
Anisotropic spatio temporal ETAS model
λ(t, x, y ) = µ +
P
j:tj <t
where g (t, x, y , M) =
g (t − tj , x − xj , y − yj , Mj |Ht )
pSj p T
Ke α(M−M0
(t+c)p { e α(Mj −M0 )
+ d}−q
and p = (x − xj , y − yj ) is a row vector, xj , yj are the coordinates
of the earthquake j preceding the earthquake with epicenter x, y
(both in the same cluster) and Sj (j = 1, 2, ...) are positive definite
symmetric matrices representing the normalized covariance matrix
of the earthquake cluster obtained by applying MBC or Magnitude
Based Cluster algorithm
Methods
Magnitude Based Cluster MBC
This method is based on selecting the greatest magnitude earthquake(with magnitude Mj ) between those that are not in any cluster yet (if there are two with equal magnitude the oldest one is
chosen) and then the earthquakes of the cluster associated with the
previous selected earthquake, are those with latitude and longitude
± 3.33 ∗ 100.5Mj −2 km (Utsu spatial distance) from the latitude
and longitude of the selected earthquake and with a time difference
therefrom (towards the future) of
max(100, 100.5Mj −1 ) days Ogata1998. Then the process is repeated
with earthquakes that are not yet in any cluster until all earthquakes
belong to a cluster.
Bivariate normal distribution
For the anisotropic
normal bivariate models
model,
2 four x1
σ̃
0
Ogata1998: N(
,
)
y
0 σ̃ 2
2
1
x̄
σ̂
0
N(
,
)
ȳ
0 σ̂ 2
x1
σ˜1 2
ρ̃σ˜1 σ˜2
N(
,
)
y
ρ̃σ˜ σ˜
σ˜2 2
1 12 2
x̄
σˆ1
ρ̂σˆ1 σˆ2
N(
,
)
ȳ
ρ̂σˆ1 σˆ2
σˆ2 2
x
σ˜1 2
ρ̃σ˜1 σ˜2
N( 1 ,
)
y
ρ̃σ˜ σ˜
σ˜2 2
1 12 2
x̄
σˆ1
ρ̂σˆ1 σˆ2
N(
,
)
ȳ
ρ̂σˆ1 σˆ2
σˆ2 2
where (x1 , y1 ) is the position of the cluster’s main earthquake,
(x̄, ȳ ) and centroid of the cluster.
(Ogata 1998)
Bivariate normal distribution
P
P
And σ̃ 2 = [ j (xj − x1 )2 + j (yj − y1 )2 ]/(2n)
P
P
σ̂ 2 = [ j (xj − x̄)2 + j (yj − ȳ )2 ]/(2n)
P
P
σ˜1 2 =P[ j (xj − x1 )2 ]/n σ˜2 2 = [ j (yj − y1 )2 ]/n
ρ̃ = [ j (xj − x1 )(yj − y1 )]/(nσ˜1 σ˜2 )
P
σ˜1 2 = [ j (xj − x1 )2 ]/n
P
P
σ˜2 2 = [ j (yj − y1 )2 ]/n ρ̃ = [ j (xj − x1 )(yj − y1 )]/(nσ˜1 σ˜2 )
P
σˆ1 2 = [ j (xj − x̄)2 ]/n
P
σˆ2 2 =P[ j (yj − ȳ )2 ]/n
ρ̂ = [ j (xj − x̄)(yj − ȳ )]/(nσˆ1 σˆ2 )
(Ogata 1998)
Methods
Bivariate normal
We select the model with the lowest AIC = −nln(det(S))+2k where
S is the variance covariance matrix for each of the four models and
k is the corresponding number of parameters. Ogata1998.
Then the selected
matrix isnormalized:
σ
/σ
−ρ
2
1
(√ 1 2 )
(1−ρ )
−ρ
σ1 /σ2
Methods
Hypocentral ETAS model
Guo2015 introduces a modification of the spatio temporal ETAS
model that includes
P the depths of earthquakes
λ(t, x, y ) = µ + j:tj <t g (t − tj , x − xj , y − yj , Mj |Ht )
where g (t, x, y , M) =
z
0
h(z, z ) =
0
pSj p T
Ke α(M−M0
(t+c)p { e α(Mj −M0 )
+ d}−q h(z − zi , zi ) and
0
z )
( Zz )η Z (1− Zz )η(1− Z
0
0
ZB(η zZ +1,η(1− zZ )+1)
with Z theRthickness of the seismogenic layer and
1
B(p, q) = 0 t p−1 (1 − t)q−1 is the Beta function.
Methods
For spatio P
temporal models
RT R R
logL(θ) = j log λ(tj , xj , yj |Ht ) − 0 max
S λ(t, x, y )dxdydt
For the hypocentral ETAS model, the logarithm of the likelihood
is
RT R R RZ
P
logL(θ) = j log λ(tj , xj , yj |Ht ) − 0 max
S 0 λ(t, x, y )dzdxdydt
In spatio temporals models, the log likelihood does not have a closed
form.
Methods
We approximate the log likelihood for spatio temporals models
using polar coordinates and S the minimum covering circle
Minimum Covering Circle Welzl 1991
a
b
Methods(Cont)
Approximation
logL(µ,
P k, p, c, d, α, γ)
≈ N
log (λ(ti )) − µTmax πr 2 −
Pi=1
c p−1
α(Mi −M0 ) (1 −
)(1 −
kα N
i=1 e
(Tmax −t +c)p−1
i
d q−1
r2
( α(M i−M ) +d)q−1
0
i
e
)
where r is the radius of the minimum covering circle, ri is the
greatest distance between earthquake i and the previous
earthquakes according to the metric defined in the cluster.
Methods
For the anisotropic model, the Euclidean distance is changed by
the metric defined by the standardized variance covariance matrix
Sj where now ri is calculated as the maximum distance (according
to the previous metric) between each earthquake and the previous
earthquakes in the cluster.
Methods
Pre-processing
We sorted the data in reverse chronological time and calculated the
time, latitude, and longitude differences for each earthquake j for
which we know the initial and final positions where the differences of
its time, latitude, and longitude with respect to the previous earthquakes are:
start = N ∗ (j − 1) − (j ∗ (j − 1))/2 + 1
end = j ∗ N − (j ∗ (j + 1))/2
Listing 3: Anisotropic spatio26
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temporal Rstan model with con28
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stant background seismicity 30
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r e a l s e i s m i c r a t e [N ] ;
r e a l i n t e g r a l o f r a t e [N ] ;
real integral mu ;
real log likelihood ;
s e i s m i c r a t e [ N ]=l o g (mu) ;
31
i n t e g r a l o f r a t e [ N ]=0 ;
functions{
32
f o r ( j i n 1 : ( N−1) ) {
r e a l l o g l i k e l i h o o d ( i n t N,
33
v e c t o r [ N−j ] y ;
r e a l mu ,
34
v e c t o r [ N−j ] z ;
real k ,
35
v e c t o r [ N−1] x ;
real p ,
36
r e a l temp ;
real c ,
37
r e a l temp1 ;
real q ,
38
r e a l temp2 ;
real d ,
39
int start ;
r e a l alpha ,
40
i n t end ;
r e a l gamma ,
41
s t a r t =N∗( j −1)−( j ∗( j −1) ) /2 + 1 ;
vector t ,
42
end =j ∗N−( j ∗( j +1) ) / 2 ;
v e c t o r m a g n i t u d e s43
y=t i m e s d i f f [ s t a r t : end ] ;
,
44
z=exp(−q∗ l o g ( q u a d r a t i c f a c t o r A n i
vector times diff
[ s t a r t : end ] . /
,
45
( exp ( gamma∗( m a g n i t u d e s [ ( j +1) :] −
vector latitudes ,
m a g n i t u d e 0 ) ) )+d ) ) ;
v e c t o r l o n g i t u d e s46
y=( k∗ a l p h a ∗( p−1)∗c ˆ ( p−1)∗( q−1)∗d
,
ˆ ( q−1)∗(1/ p i ( ) ) )
vector
47
∗exp ( ( a l p h a−gamma ) ∗( m a g n i t u d e s
quadratic factorAni
[ ( j +1) :] −
,
48
magnitude0 ) )
vector
49
. ∗ exp(−p∗ l o g ( y+c ) ) ;
q u a d r a t i c f 50
actorIso
y=y . ∗ z ;
,
51
s e i s m i c r a t e [ j ]=l o g (mu+sum ( y ) ) ;
r e a l tmax ,
52
temp=exp ( a l p h a ∗( m a g n i t u d e s [ j +1]−
r e a l mag nitu de0 ,
magnitude0 ) ) ;
real lat min ,
53
temp1=exp ( gamma∗( m a g n i t u d e s [ j
r e a l lat max ,
+1]−m a g n i t u d e 0 ) ) ;
r e a l long min ,
54
temp2=max ( q u a d r a t i c f a c t o r A n i [
r e a l long max ,
s t a r t : end ] ) ;
real radius
55
i n t e g r a l o f r a t e [ j ]=k∗ a l p h a ∗temp
){
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
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75
76
77
78
79
80
81
∗(1−c ˆ ( p−1)/
82
( ( t i m e s d i f f [ j ]+ c ) ˆ ( p−1) ) ) ∗(1−d83
ˆ ( q−1)/
84
( ( temp2 / ( temp1 )+d ) ˆ ( q−1) ) ) ;
85
}
86
i n t e g r a l m u=mu∗tmax∗ p i ( ) ∗ r a d i u s ˆ 2 87
;
l o g l i k e l i h o o d=sum ( s e i s m i c r a t e )−88
i n t e g r a l m u −sum ( i n t e g r a l o f r a t e89
);
90
return ( l o g l i k e l i h o o d ) ;
91
92
}
93
}
94
data{
95
i n t <l o w e r=0> N ;
96
v e c t o r [N] times ;
97
r e a l <l o w e r=0> m a x t i m e ;
98
v e c t o r [N] magnitudes ;
99
v e c t o r [ N∗(N−1) / 2 ]
100
quadratic factorAni ;
101
v e c t o r [ N∗(N−1) / 2 ]
quadratic factorIso ;
v e c t o r [ N∗(N−1) / 2 ] t i m e s d i f f ;
102
r e a l <l o w e r=0> t h r e s h o l d m a g n i t u d e ;
v e c t o r [N] l a t i t u d e s ;
103
v e c t o r [N] l o n g i t u d e s ;
real lat min ;
104
r e a l lat max ;
r e a l long min ;
105
r e a l long max ;
r e a l <l o w e r=0> r a d i u s ;
106
}
parameters{
r e a l <l o w e r=0> mu ;
r e a l <l o w e r=0> k ;
r e a l <l o w e r=1.000005 > p ;
r e a l <l o w e r=0> c ;
r e a l <l o w e r=0> d ;
r e a l <l o w e r=1.00005 > q ;
r e a l <l o w e r=0> a l p h a ;
r e a l <l o w e r=0> gamma ;
}
model{
mu˜ e x p o n e n t i a l ( 2 . 8 ) ;
k˜ e x p o n e n t i a l ( 2 . 8 ) ;
p˜ e x p o n e n t i a l ( 2 . 8 ) ;
c˜ e x p o n e n t i a l ( 2 . 8 ) ;
d˜ e x p o n e n t i a l ( 2 . 8 ) ;
q˜ e x p o n e n t i a l ( 2 . 8 ) ;
gamma˜ e x p o n e n t i a l ( 2 . 8 ) ;
a l p h a−gamma˜ e x p o n e n t i a l ( 5 ) ;
i n c r e m e n t l o g p r o b ( l o g l i k e l i h o o d (N,
mu , k , p , c , q , d ,
a l p h a , gamma , t i m e s , m a g n i t u d e s ,
times diff ,
latitudes , longitudes ,
quadratic factorAni ,
q u a d r a t i c f a c t o r I s o , max time ,
threshold magnitude ,
l a t m i n , lat max , long min , long max ,
radius ) ) ;
}
Listing 4:
Anisotropic spa23
24
tio temporal hypocentral Rstan
25
26
model with constant background
27
28
seismicity
29
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
30
functions{
31
r e a l l o g l i k e l i h o o d ( i n t N,
32
r e a l mu ,
33
real k ,
34
real p ,
35
real c ,
36
real q ,
37
real d ,
38
r e a l alpha ,
39
r e a l gamma ,
40
r e a l eta ,
41
vector t ,
42
v e c t o r m a g n i t u d e s43
,
44
v e c t o r t i m e s d i f f45
,
46
vector
47
d e p t h s d i f f 48
,
v e c t o r l a t i t u d e s 49
,
v e c t o r l o n g i t u d e s50
,
51
vector depths ,
vector
52
quadratic factorAni
,
53
vector
54
quadratic factorIso
,
55
r e a l tmax ,
56
r e a l mag nitu de0 ,
real
real
real
real
real
real
){
lat min ,
lat max ,
long min ,
long max ,
radius ,
layer depth
r e a l s e i s m i c r a t e [N ] ;
r e a l i n t e g r a l o f r a t e [N ] ;
real integral mu ;
real log likelihood ;
s e i s m i c r a t e [ N ]=l o g (mu) ;
i n t e g r a l o f r a t e [ N ]=0 ;
f o r ( j i n 1 : ( N−1) ) {
v e c t o r [ N−j ] y ;
v e c t o r [ N−j ] z ;
v e c t o r [ N−j ] x ;
v e c t o r [ N−j ] w ;
r e a l temp ;
r e a l temp1 ;
r e a l temp2 ;
r e a l temp3 ;
int start ;
i n t end ;
s t a r t =N∗( j −1)−( j ∗( j −1) ) /2 + 1 ;
end =j ∗N−( j ∗( j +1) ) / 2 ;
y=t i m e s d i f f [ s t a r t : end ] ;
z=exp(−q∗ l o g ( q u a d r a t i c f a c t o r A n i
[ s t a r t : end ]
. / ( exp ( gamma∗( m a g n i t u d e s [ ( j +1)
:] −
m a g n i t u d e 0 ) ) )+d ) ) ;
y=( k∗ a l p h a ∗( p−1)∗c ˆ ( p−1)∗( q−1)∗d
ˆ ( q−1)∗
( 1 / p i ( ) ) ) ∗exp ( ( a l p h a−gamma ) ∗
( m a g n i t u d e s [ ( j +1) :] − m a g n i t u d e 0
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
))
82
. ∗ exp(−p∗ l o g ( y+c ) ) ;
83
x=y . ∗ z ;
84
z=( 1 / l a y e r d e p t h ) ∗
85
86
d e p t h s d i f f [ s t a r t : end ] ;
87
y=( 1 / l a y e r d e p t h ) ∗
d e p t h s [ ( j +1) : N ] ;
88
f o r ( i i n 1 : ( N−j ) ) {
89
w [ i ]=( 1 / ( l a y e r d e p t h ∗ exp (
lbeta ( eta ∗
90
( y [ i ] ) + 1 , eta − eta ∗ ( y [
i ] )+ 1 ) ) ) ) ∗
91
( z [ i ] ˆ ( e t a ∗ ( y [ i ] ) ) ) ∗ 92
((1 − z [ i ] ) ˆ
93
( e t a−e t a ∗ ( y [ i ] ) ) ) ;
94
95
}
s e i s m i c r a t e [ j ]=l o g (mu+sum ( x . ∗ 96
w) ) ;
97
temp=exp ( a l p h a ∗( m a g n i t u d e s [ j +1]−
98
magnitude0 ) ) ;
99
temp1=exp ( gamma∗( m a g n i t u d e s [ j 100
+1]−m a g n i t u d e 0 ) ) ;
101
temp2=max ( q u a d r a t i c f a c t o r A n i [ 102
s t a r t : end ] ) ;
103
i n t e g r a l o f r a t e [ j ]=k∗ a l p h a ∗temp
104
∗(1−c ˆ ( p−1)
105
/ ( ( t i m e s d i f f [ j ]+ c ) ˆ ( p−1) ) ) ∗(1−
106
d ˆ ( q−1)
107
/ ( ( temp2 / ( temp1 )+d ) ˆ ( q−1) ) ) ; 108
}
109
i n t e g r a l m u=mu∗tmax∗ p i ( ) ∗
110
layer depth∗
111
( r a d i u s ˆ2) ;
112
l o g l i k e l i h o o d=sum ( s e i s m i c r a t e )−
113
i n t e g r a l m u −sum ( i n t e g r a l o f r a t114
e
);
115
return ( l o g l i k e l i h o o d ) ;
116
}
}
data{
i n t <l o w e r=0> N ;
v e c t o r [N] times ;
r e a l <l o w e r=0> m a x t i m e ;
v e c t o r [N] magnitudes ;
v e c t o r [ N∗(N−1) / 2 ]
quadratic factorAni ;
v e c t o r [ N∗(N−1) / 2 ]
quadratic factorIso ;
v e c t o r [ N∗(N−1) / 2 ] t i m e s d i f f ;
v e c t o r [ N∗(N−1) / 2 ] d e p t h s d i f f ;
r e a l <l o w e r=0> t h r e s h o l d m a g n i t u d e ;
v e c t o r [N] l a t i t u d e s ;
v e c t o r [N] l o n g i t u d e s ;
v e c t o r [N] depths ;
real lat min ;
r e a l lat max ;
r e a l long min ;
r e a l long max ;
r e a l <l o w e r=0> r a d i u s ;
r e a l <l o w e r=0> l a y e r d e p t h ;
}
parameters{
r e a l <l o w e r=0> mu ;
r e a l <l o w e r=0> k ;
r e a l <l o w e r=1.000005 > p ;
r e a l <l o w e r=0> c ;
r e a l <l o w e r=0> d ;
r e a l <l o w e r=1.00005 > q ;
r e a l <l o w e r=0> a l p h a ;
r e a l <l o w e r=0> gamma ;
r e a l <l o w e r=0> e t a ;
}
model{
mu˜ e x p o n e n t i a l ( 2 ) ;
117
118
119
120
121
122
123
124
125
k˜ e x p o n e n t i a l ( 2 ) ;
126
p˜ e x p o n e n t i a l ( 2 ) ;
c˜ e x p o n e n t i a l (2) ;
127
d˜ e x p o n e n t i a l ( 2 ) ;
128
q˜ e x p o n e n t i a l ( 2 ) ;
129
eta ˜ exponential (2) ;
130
gamma˜ e x p o n e n t i a l ( 2 ) ;
a l p h a−gamma˜ e x p o n e n t i a l ( 5 ) ;
131
i n c r e m e n t l o g p r o b ( l o g l i k e l i h o o d (N,132
mu , k , p , c , q , d ,
a l p h a , gamma , e t a , t i m e s , m a g n i t u d e s ,
times diff ,
depths diff , latitudes , longitudes ,
depths , q u a d r a t i c f a c t o r A n i ,
q u a d r a t i c f a c t o r I s o , max time ,
threshold magnitude , lat min ,
lat max , long min ,
lo n g m a x , r a d i u s , l a y e r d e p t h ) ) ;
}
Listing 5: Anisotropic Rstan
27
28
model with variable background
29
30
seismicity
31
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
32
functions{
33
r e a l l o g l i k e l i h o o d ( i n t N,
34
r e a l mu ,
35
real k ,
36
real p ,
37
real c ,
38
real q ,
39
real d ,
40
r e a l alpha ,
41
r e a l gamma ,
42
vector t ,
43
v ec t or magnitudes ,
44
vector times diff ,
45
vector latitudes ,
46
vector longitudes ,
vector
47
background seismic
,
48
vector
q u a d r a t i c f a c t o r A49n i
,
vector
50
quadratic factorIso
,
51
r e a l tmax ,
52
r e a l mag nitu de0 ,
53
real lat min ,
r e a l lat max ,
r e a l long min ,
54
r e a l long max ,
real radius
55
){
r e a l s e i s m i c r a t e [N ] ;
r e a l i n t e g r a l o f r a t e [N ] ;
real integral mu ;
real log likelihood ;
s e i s m i c r a t e [ N ]=l o g (mu) ;
i n t e g r a l o f r a t e [ N ]=0 ;
f o r ( j i n 1 : ( N−1) ) {
v e c t o r [ N−j ] y ;
v e c t o r [ N−j ] z ;
v e c t o r [ N−1] x ;
r e a l temp ;
r e a l temp1 ;
r e a l temp2 ;
int start ;
i n t end ;
s t a r t =N∗( j −1)−( j ∗( j −1) ) /2 + 1 ;
end =j ∗N−( j ∗( j +1) ) / 2 ;
y=t i m e s d i f f [ s t a r t : end ] ;
z=
exp(−q∗ l o g ( q u a d r a t i c f a c t o r A n i
[ s t a r t : end ] . /
( exp ( gamma∗( m a g n i t u d e s [ ( j +1)
rates
:] − m a g n i t u d e 0 ) ) )+d ) ;
y=( k∗ a l p h a ∗( p−1)∗c ˆ ( p−1)∗( q−1)∗d
ˆ ( q−1)∗
( 1 / p i ( ) ) ) ∗exp ( ( a l p h a−gamma ) ∗(
m a g n i t u d e s [ ( j +1) : ]
−m a g n i t u d e 0 ) ) . ∗ exp(−p∗ l o g ( y+c )
);
y=y . ∗ z ;
s e i s m i c r a t e [ j ]=
l o g (mu∗
background seismic rates [
j +1]+sum ( y ) ) ;
temp=exp ( a l p h a ∗( m a g n i t u d e s [ j +1]−
magnitude0 ) ) ;
temp1=exp ( gamma∗( m a g n i t u d e s [ j
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
+1]−m a g n i t u d e 0 ) ) ;
83
temp2=max ( q u a d r a t i c f a c t o r A n i [ 84
85
s t a r t : end ] ) ;
i n t e g r a l o f r a t e [ j ]=k∗ a l p h a ∗temp86
87
∗(1−c ˆ ( p−1)/
88
( ( t i m e s d i f f [ j ]+ c ) ˆ ( p−1) ) ) ∗
(1−d ˆ ( q−1) / ( ( temp2 / ( temp1 )+d ) ˆ (89
90
q−1) ) ) ;
91
}
92
i n t e g r a l m u=mu∗sum (
b a c k g r o u n d s e i s m i c r a t e s ) ∗ 93
94
tmax∗ p i ( ) ∗ r a d i u s ˆ 2 ;
l o g l i k e l i h o o d=sum ( s e i s m i c r a t e )−95
i n t e g r a l m u −sum ( i n t e g r a l o f r a t e96
97
);
return ( l o g l i k e l i h o o d ) ;
98
99
}
100
}
101
data{
102
i n t <l o w e r=0> N ;
103
v e c t o r [N] times ;
104
r e a l <l o w e r=0> m a x t i m e ;
v e c t o r [N] magnitudes ;
105
v e c t o r [ N∗(N−1) / 2 ]
quadratic factorAni ;
106
v e c t o r [ N∗(N−1) / 2 ]
quadratic factorIso ;
107
v e c t o r [ N∗(N−1) / 2 ] t i m e s d i f f ;
r e a l <l o w e r=0> t h r e s h o l d m a g n i t u d e ; 108
v e c t o r [N] l a t i t u d e s ;
v e c t o r [N] l o n g i t u d e s ;
109
v e c t o r [ N ] b a c k g r o u n d s e i s m i c r a t e s ;110
real lat min ;
r e a l lat max ;
111
r e a l long min ;
r e a l long max ;
r e a l <l o w e r=0> r a d i u s ;
}
parameters{
r e a l <l o w e r=0> mu ;
r e a l <l o w e r=0> k ;
r e a l <l o w e r=1.000005 > p ;
r e a l <l o w e r=0> c ;
r e a l <l o w e r=0> d ;
r e a l <l o w e r=1.00005 > q ;
r e a l <l o w e r=0> a l p h a ;
r e a l <l o w e r=0> gamma ;
}
model{
mu˜ e x p o n e n t i a l ( 2 . 8 ) ;
k˜ e x p o n e n t i a l ( 2 . 8 ) ;
p˜ e x p o n e n t i a l ( 2 . 8 ) ;
c˜ e x p o n e n t i a l ( 2 . 8 ) ;
d˜ e x p o n e n t i a l ( 2 . 8 ) ;
q˜ e x p o n e n t i a l ( 2 . 8 ) ;
gamma˜ e x p o n e n t i a l ( 2 . 8 ) ;
a l p h a − gamma ˜ e x p o n e n t i a l ( 5 ) ;
i n c r e m e n t l o g p r o b ( l o g l i k e l i h o o d (N,
mu , k , p , c , q , d ,
a l p h a , gamma , t i m e s , m a g n i t u d e s ,
times diff ,
latitudes , longitudes ,
background seismic rates ,
quadratic factorAni ,
quadratic factorIso ,
max time , t h r e s h o l d m a g n i t u d e ,
l a t m i n , lat max , long min , long max ,
radius ) ) ;
}
Methods
We can estimate the probability that a given event is spontaneous
or is triggered by others
Kagan1980Zhuang2002. The contribution of the spontaneous
seismicity rate to the occurrence of an event i can be taken as the
probability that the event i is spontaneousZhuang2008:
µ(xi ,yi )
φ(i) = λ(t
i ,xi ,yi )
Similarly, the probability that the event j is produced by the event
κ(M)g (tj −ti )f (xj −xi ,yj −yi ,mi )
i is ρij =
λ(ti ,xi ,yi )
We can also obtain the expected
number of direct aftershocks
P
from the earthquake i as j ρij Zhuang2008
Methods
Inter-event times
The probability density function of the recurrence times (time
between two successive events) τ
H(τ ) ≈ λf (λτ )
where the function f (x) has been found practically the same in
different regions and λ is the average rate of events observed in
the analyzed region. Saichev2007
The scaling factor of times between earthquakes is taken as the
inverse of its mean.
The form of the function f (x) which is demonstrated in
Saichev2007 is
f (x) = (nθ x −1−θ + [1 − n + nθ x −θ ]2 ) ∗
ϕ(x, )
Results
Anisotropic Distribution
The covariance matrix for the cluster associated with the April
16th big earthquake (804 earthquakes in the cluster) is
1.469 −0.696
−0.696 0.709
and corresponds to the fourth bivariate model fitted.
Results
Kernel bivariate density
Kernel bivariate density using the R package kde.
Priors
Weakly informative priors
For the ETAS parameters(must be positive) we used exponential
priors and for p and q we use a minimum value close to 1:1.000005.
The radius of the minimum covering circle using Welzl algorithm
was, r = 1.70479.
Results
Figure: Chains for anisotropic
ETAS model with constant
background seismicity
Figure: A posteriori parameter
distributions for anisotropic ETAS
model with constant background
seismicity
Results
Figure: Correlation of parameter
values in the chains for anisotropic
ETAS model with constant
background seismicity
Figure: Correlation of parameter
values in the chains for anisotropic
ETAS model with constant
background seismicity
Results
Residuals
Figure: Acumulated number of
events for temporal Etas model
with constant background
seismicity
Figure: Residuals for Etas
temporal temporal Etas model
with constant background
seismicity
Figure: Accumulated number of
events for temporal ETAS model
with variable background
seismicity
Figure: Residuals for temporal
ETAS model with variable
background seismicity
Figure: Accumulated number of
events for temporal ETAS model
with constant background
seismicity vs variable background
seismicity
Figure: Residuals for temporal
ETAS model with constant
background seismicity vs variable
background seismicity
Figure: Accumulated number of
events for temporal ETAS with
magnitudes model and constant
background seismicity
Figure: Residuals for temporal
ETAS with magnitudes model and
constant background seismicity
Figure: Accumulated number of
events for spatio temporal
anisotropic ETAS model and
variable background seismicity
Figure: Residuals for spatio
temporal anisotropic ETAS model
and variable background seismicity
Results
Parent earthquakes
95% credibility intervals for the index of the most probable parent
earthquake with 1000 draws from the posterior distributions.
The green line is the big earthquake on April 16 and blue points are
the medians of the credibility intervals using the anisotropic spatio
temporal model with variable background seismicity.
Thanks