Presentazione di PowerPoint

1
R. Console, M. Murru, F. Catalli Applying the
Rate-and-State friction law to an epidemic model
of earthquake clustering
4th International Workshop on Statistical
Seismology (Statsei4) in memory of Tokuji
Utsu Graduate University for Advanced Studies,
Shonan Village Campus, Kanagawa Prefecture, Japan
9 - 13 January, 2006
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Outline of the talk
An example of stochastic model the classic
epidemic model for earthquake clustering
A modified stochastic model incorporating the
rate-and-state constitutive law
Towards the full application of the
rate-and-state constitutive law to the epidemic
model
Problems still open
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Definition of the occurrence density and of the
likelihood in the case of a continuous
distribution
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Time dependent model (epidemic model) - I

• The magnitude distribution is the same for all
  earthquakes
  (Gutenberg-Richter law)

• The occurrence rate density is the superposition
  of a time
  independent (poissonian) component and the
  activity triggered by previous earthquakes

• The occurrence rate of triggered events depends
  exponentially on the magnitude of every
  preceeding event

• The spatial distribution of triggered events is
  described by an isotropic function around the
  epicenter of every previous event

• The temporal behaviour of triggered events is
  described by the Omori law starting from the
  occurrence time of every previous event

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Time dependent model (epidemic model) - II

• Every event is potentially triggered by all the
  previous events and every event can trigger
  subsequent events according to their relative
  time-space distance

• A definition of the words foreshock, mainshock
  and aftershock is not necessary

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• Time dependent model
• (epidemic model) III

(Ogata, 1998)
Occurrence rate density
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(No Transcript)
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Observed seismicity and its continuous
representation (function m0 (x, y), events per
year in 1000 km2) (Central Italy, January 1981
December 1996, M ? 2.0)
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Time dependent distribution of the epicenters and
magnitude for triggered seismicity
where
and
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Seismic sequence of Umbria-Marche
(1997) Comparison between the number of observed
events (a) and the number of expected events (b)
for time windows of 12 hours
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Epidemic model Umbria-Marche, 1 September 1997,
0000
Occurrence rate density (events per day in 100
km2, Ml ? 2.0)
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Epidemic model Umbria-marche, 26 September 1997,
0033 (before the event near Colfiorito,
Ml5.6)
Occurrence rate density (events per day in 100
km2, Ml ? 2.0)
1
6
0
.
0
0
1
5
0
.
0
0
(a)
1
4
0
.
0
0
1E001
1
3
0
.
0
0
1
1
2
0
.
0
0
0.1
)
m
0.01
k
1
1
0
.
0
0
(

Y
0.001
1
0
0
.
0
0
0.0001
9
0
.
0
0
1E-005
8
0
.
0
0
1E-006
7
0
.
0
0
6
0
.
0
0
-60.00
-40.00
-20.00
0.00
20.00
40.00
X (km)
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Epidemic model Umbria-Marche, 26 September 1997,
940 (before the second event near Colfiorito,
Ml5.8)
Occurrence rate density (events per day in 100
km2, Ml ? 2.0)
1
6
0
.
0
0
(b)
1
4
0
.
0
0
10
1
0.1
1
2
0
.
0
0
)
m
0.01
k
(

Y
0.001
1
0
0
.
0
0
0.0001
1E-005
8
0
.
0
0
1E-006
6
0
.
0
0
-60.00
-40.00
-20.00
0.00
20.00
40.00
X (km)
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Epidemic model Umbria-Marche, 14 October 1997
(before the event near Sellano, Ml5.5)
Occurrence rate density (events per day in 100
km2, Ml ? 2.0)
1
6
0
.
0
0
(c)
1
4
0
.
0
0
10
1
0.1
1
2
0
.
0
0
)
m
0.01
k
(

Y
0.001
1
0
0
.
0
0
0.0001
1E-005
8
0
.
0
0
1E-006
6
0
.
0
0
-60.00
-40.00
-20.00
0.00
20.00
40.00
X (km)
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Epidemic model Umbria-Marche, 3 April 1998
(before the event near Nocera Umbra, Ml5.0)
Occurrence rate density (events per day in 100
km2, Ml ? 2.0)
1
6
0
.
0
0
(d)
1
4
0
.
0
0
10
1
1
2
0
.
0
0
0.1
)
m
k
0.01
(

Y
0.001
1
0
0
.
0
0
0.0001
1E-005
8
0
.
0
0
1E-006
6
0
.
0
0
-60.00
-40.00
-20.00
0.00
20.00
40.00
X (km)
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Modified time dependent model (with physical
constraints)

• The magnitude distribution is the same for all
  the earthquakes
  (Gutenberg-Richter law)

• The occurrence rate density is the
  superposition of a time
  independent (poissonian) component
  and that of the triggered seismic activity

• The occurrence rate of the triggered events
  depends exponentially on the magnitude of every
  preceeding event

• The spatial distribution of triggered events is
  described by an isotropic function around the
  epicenter of every previous event

• The temporal behaviour of the triggered events
  is derived from the rate-and-state constitutive
  law

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Rate-and-state model - I (Dieterich, 1994)
where
R is the occurrence rate density of the induced
events R0 is the background rate density Dt is
the Coulomb stress change A, s and ta are
constitutive parameters
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Rate-and-state model - II (Dieterich, 1994)
is the stress rate in the area
Where
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Rate-and-state model - III (Dieterich, 1994)
Time dependence of the rate of triggered
events for different values of the stress change
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Rate-and-state model - IV (Console et al., 2004)
The total number of triggered events is
proportional to the stress change Dt
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Rate-and-state model - V (Console et al., 2004)
Dependence of
on R0
where
b is the parameter of the Gutenberg-Richter
law
is the seismic moment of an earthquake of
magnitude m0
is the stress drop, assumed constant
m0 and mmax are the minimum and maximum
magnitude, respectively
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The rate-and-state model applied to a catalog of
earthquakes
where
Dt0 is the stress change at the center of the
fault
and d0 is the radius of a fault of magnitude m0
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d0 is related to the smallest seismic moment
and the smallest magnitude m0 through
and
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Modified epidemic model Umbria-Marche, 1
September 1997, 0000
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Modified epidemic model Umbria-marche, 26
September 1997, 0000 (before the event near
Colfiorito, Ml5.6)
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Modified epidemic model Umbria-marche, 26
September 1997, 940 (before the secondth event
near Colfiorito, Ml 5.8)
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Modified epidemic model Umbria-Marche, 14 October
1997 ( before the event near Sellano, Ml 5.5)
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Full application of the stress transfer and
rate-state model

• The Coulomb stress change is computed by the
  information on the source mechanism of any
  triggering event (stress drop fixed at 2.5 MPa)
• Only one free parameter (As) is necessary in the
  model
• The expected seismicity rate is compared with the
  real observations of seismic activity

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Stress transfer and rate-state model Umbria-Marche
, 26 September 1997, 0000 (before the event
near Colfiorito, Ml5.6)
Occurrence rate density (events per day in 1000
km2, Ml ? 2.0)
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Stress transfer and rate-state model
Umbria-Marche, 26 September 1997, 940 (before
the second event near Colfiorito, Ml5.8)
Occurrence rate density (events per day in 1000
km2, Ml ? 2.0)
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Stress transfer and rate-state model
Umbria-marche, 14 October 1997 (before the
event near Sellano, Ml5.5)
Occurrence rate density (events per day in 1000
km2, Ml ? 2.0)
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Stress transfer and rate-state model Umbria-marche
, 3 April 1998 (before the event near Nocera
Umbra, Ml5.0)
Occurrence rate density (events per day in 1000
km2, Ml ? 2.0)
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Stress transfer and rate-state model (short term)
Stress transfer and rate-state model (long term)
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Discussion - I
It is possible to observe how the Coulomb stress
change affects the spatial and temporal
distribution of the seismicity. However, a
complete match of the modeled earthquake
distribution with the observations is not
achievable.
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Discussion - II

• Possible causes of mismatch
• Uncertainties in the source parameters
• Stress drop assumed constant
• Lack of details on the slip distribution
• Uncertainties in the background seismicity
• Uncertainties in hypocentral locations
• Depth assumed constant
• Ignoring previous stress hetereogeneties
• Focal mechanism assumed constant
• Ignoring the real shape of seismogenic
  structures
• Ignoring the triggering potential of moderate
  and minor seismicity.

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Conclusions  
Stochastic modeling allows the computation of the
expected earthquake rate density on a continuous
space-time volume, suitable for the validation of
a model with respect to others and for real time
forecasts. The significant steps made during the
last decades in the physical modeling of
earthquake clustering provide a tool for the
refinement of these stochastic models. Jointly
with the improvement of the seismological
observations, these steps appear as a progress
towards the possible practical application for
earthquake forecast.
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Thank you!