1 – stress contributions 2 – probabilistic approach 3 – deformation transients small...

1 – Stress contributions1 – Stress contributions

2 – Probabilistic approach2 – Probabilistic approach

3 – Deformation transients3 – Deformation transients

Small earthquakes contribute as much as large earthquakes do to stress changes

Extract the « influence » of small earthquakes directly from seismicity data

Evaluate how anomalous a given seismicity pattern can be

Meier et al (JGR 2014)


Estimation of the contribution in static stress transfer from small (unresolved) sources

with -1 ≤ ≤ 1, C constant

r = hypocentral distances btw earthquakes

Mo = seismic moment

r

Source MSource Moo

ReceiverReceiver

3L+rCM=τ o


Holtsmark (1919), Chandrasekhar (1943)

Gravitational field at any given location, caused by a distribution of stars (with random masses)

V

g(r1,M

1)

RrπR

πr=rfr

3

2

3/4

4

R

} f g (g )=K g−5 /2

For one star :


2)( GMrrgx

X 1≡X 2≡X 3 ...

X= X 1+ X 2+ X 3+ ...+ X N≡ c X 1

c=N 1 /2For normal (Gaussian) distributions :

There exist other stable distributions, known as Lévy distributions, for which

They have power-law decays in

c=N 1 / α

f X (x ) = K x−1−α

Stability of random distributions for addition



V

g(r1,M

1)

R

For all stars :

V = g1 + g

2 + ... f V (V )=K ' V −5 /2

Stable for additionStable for addition




V

g(r1,M

1)

R

}For all stars :


2)( GMrrgx

RrπR

πr=rfr

3

2

3/4

4 f V (V )=K ' V −5 /2


Mo = seismic moment

r

Source MSource Moo

ReceiverReceiver


Estimation of the contribution in static stress transfer from small (unresolved) sources

with -1 ≤ ≤ 1, C constant 3L+rCM=τ o


Mo = seismic moment

r

Source MSource Moo

ReceiverReceiver

3L+rCM=τ o

1Dr Kr=rf

} 3/1 Doτ τMK=τf


For one source:

Kagan (Nonl. Proc. Geophys., 1994)



Mo = seismic moment

r

Source MSource Moo

ReceiverReceiver

} 3/1/3 Do

Dmτ, τMmNK=τf


For all N(m) sources of magnitude m:1 – Stress contributions1 – Stress contributions

3L+rCM=τ o

1Dr Kr=rf

N

Marsan (GJI 2004)

}Stable for additionStable for addition

For all N(m) sources of magnitude m:

Gutenberg-Richter : N (m )=N 010−b m

Kanamori : M o=10

1.5m+ 9

NB : far field approximation


3/1/3 Do

Dmτ, τMmNK=τf

3L+rCM=τ o

1Dr Kr=rf

Compute stress on a regular grid

Largest earthquakes

dominate

3/1/3 Do

Dmτ, τMmNK=τf

D=3

10(1.5−b)m



M

7

6

5

4

3

Compute stress on hypocenters

All earthquakes contribute

101.5 (1−b )m

D=2


3/1/3 Do

Dmτ, τMmNK=τf


FM from Yang et al. (BSSA 2012)

Hypocenters from Hauksson et al. (BSSA 2012)


Stress change at the hypocenters of future earthquakes :

ALL magnitude bands contribute equally

Uncertainty on stress change grows as cut-off magnitude decreases


pi , j

probability that #i triggered #j

j<k

jk,

ji,ji, λ+λ

λ==p

0

Contribution from #i

Sum of all contributions

pi , j

i

j

Probabilistic approachProbabilistic approach


1... ,1,2,1,0 jjjjj pppp

lk,j,i λ=t,xλContribution from #i

Distance from #i to x

Time t-ti

Magnitude mi

Marsan and Lengliné (2008)

Inversion from data by Expectation – Maximization


Shearer et al. (2005) catalogue

N>70,000 earthquakes m ≥ 2

No decoupling between space and time

Correction for lack of detection following large shocks

Distances from fault to target hypocenter

target

mai

n fa

ult r


7M

76 <M

65 <M

54 <M

43 <M

32 <M

Mainshock magnitude



B

C

A

A

B

C

A B COROROROR

????????????

C is an indirect aftershock of A


0 – 5 km0 – 5 km

5 – 20 km5 – 20 km

> 20 km> 20 km

Probability of being a direct aftershock of a M>7 earthquake

1%1%

1 w

ee

k1

we

ek

3 m

on

ths

3 m

on

ths

60%60%


Probability of being a direct aftershock of a 3<M<4 earthquake

0 – 1 km0 – 1 km

1 – 5 km1 – 5 km

5 – 20 km5 – 20 km


Marsan and Lengliné (JGR 2010)

Our method: Direct ASDirect AS

Modified from Felzer & Brodsky (2006):

Background removedBackground removedL

ine

ar

de

nsi

ty (

1/k

m/d

ay)

Lin

ea

r d

en

sity

(1

/km

/da

y)

Distance (km)

0 < t < 15'

0.5 < t < 1 day

r -1.76 +- 0.35r -1.76 +- 0.35

3 ≤ mMS

< 4

mAS

≥ 2


M = 3 earthquake (L = 400 m, u = 1 cm)

1 km

2 km

km

km

CFF (bars)

Q: can static stress triggering explain this distribution?Q: can static stress triggering explain this distribution?


a

AσΔCFFatt

a t

t

Aσ

ΔCFF++tR,Rμ=n 1eeln //

21

Rate-and-state friction

Dieterich (JGR 1994)

= # of direct aftershocks in time and distance n [0, t ] [R1,R

2]


a

AσΔCFFatt

a t

t

Aσ

ΔCFF++tR,Rμ=n 1eeln //

21

Rate-and-state friction

Dieterich (JGR 1994)

= # of direct aftershocks in time and distance n [0, t ] [R1,R

2]


Background rate – density at R1 < r < R2

∝ # of background earthquakes at these distances

Background earthquakes

Mainshock 3 ≤ m < 4

No clustering

Clustering

Euclidean volume (km3)

~ r 1.65


r -2.4

r -2.2

t ~ 1 hourt ~ 1 hour

NO

T R

ES

OL

VE

DN

OT

RE

SO

LV

ED


Observations: ~ r -1.76 ± 0.35

Static stress model: ~ r -2.30 ± 0.27

Mainshock m = 3Aftershocks up to ~ 1 hour, at 1 < r < 30 km

Mainshock m = 3Aftershocks up to ~ 1 hour, at 1 < r < 30 km

Given the uncertainties, we cannot reject triggering by static stress

Given the uncertainties, we cannot reject triggering by static stress


3 – Deformation transient3 – Deformation transient

Marsan et al. (GRL 2014)

With precursory acceleration

Without precursory acceleration

3 – Transient deformation3 – Transient deformation3 – Deformation transient3 – Deformation transient

With precursory acceleration

Without precursory acceleration


Extra aftershocks from extra foreshocks

ETAS simulations


3 – Transient deformation3 – Transient deformation3 – Deformation transient3 – Deformation transient!!! POSTER by Thomas Reverso

Small earthquakes contribute significantly to the stress budget

They add great spatial variability to stress changes caused by large sources

They cannot be accounted for deterministically (lack of information, number)

Probabilistic approach that can be parameterized given the observed seismicity

Observation of anomalous activity reveals slow deformation transients

Conclusions

1 – stress contributions 2 – probabilistic approach 3 – deformation transients small...

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