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
January, 2006

2. 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

3. Definition of the occurrence density and of the likelihood in the case of a continuous distribution

4. 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

5. 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

6. Occurrence rate density
Time dependent model
(epidemic model) – III
(Ogata, 1998)
Occurrence rate density

7. l(x,y,t,m) = fr · l0 (x,y,m) +  li (x,y,t,m)
Time dependent model
(epidemic model) - IV
l(x,y,t,m) = fr · l0 (x,y,m) +  li (x,y,t,m)
ti<t
Time independent distribution of the epicenters and magnitude for the spontaneous seismicity
l0 ( x, y, m ) = m0 ( x, y ) 10 - b ( m-m0 )
where m0 is the completeness magnitude of the catalog

8. 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)

9. Time dependent model (epidemic model) - V
l(x,y,t,m) = fr · l0 (x,y,m) +  li (x,y,t,m)
ti<t
Time dependent distribution of the epicenters and
magnitude for triggered seismicity
where
and

10. 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

11. Epidemic model Umbria-Marche, 1 September 1997, 00:00 -60 -40 -20 20
Occurrence rate density (events per day in 100 km2, Ml  2.0)
-60
-40
-20 20 40
X (km) 6 8 1 2 4 Y ( k m )
1E-006
1E-005
0.0001
0.001
0.01
0.1 10

12. Epidemic model Umbria-marche, 26 September 1997, 00:33
(before the event near Colfiorito, Ml=5.6)
Occurrence rate density (events per day in 100 km2, Ml  2.0) 1 6 . 1 5 .
(a) 1 4 .
1E+001 1 3 . 1 1 2 .
0.1 ) m k
0.01 ( 1 1 . Y
0.001 1 .
0.0001 9 .
1E-005 8 .
1E-006 7 . 6 .
-60.00
-40.00
-20.00
0.00
20.00
40.00
X (km)

13. Epidemic model Umbria-Marche, 26 September 1997, 9:40
(before the second event near Colfiorito, Ml=5.8)
Occurrence rate density (events per day in 100 km2, Ml  2.0) 1 6 .
(b) 1 4 . 10 1
0.1 1 2 . ) m
0.01 k ( Y
0.001 1 .
0.0001
1E-005 8 .
1E-006 6 .
-60.00
-40.00
-20.00
0.00
20.00
40.00
X (km)

14. Epidemic model Umbria-Marche, 14 October 1997
(before the event near Sellano, Ml=5.5)
Occurrence rate density (events per day in 100 km2, Ml  2.0) 1 6 .
(c) 1 4 . 10 1 1 2 .
0.1 ) m
0.01 k ( Y
0.001 1 .
0.0001
1E-005 8 .
1E-006 6 .
-60.00
-40.00
-20.00
0.00
20.00
40.00
X (km)

15. Epidemic model Umbria-Marche, 3 April 1998
(before the event near Nocera Umbra, Ml=5.0)
Occurrence rate density (events per day in 100 km2, Ml  2.0) 1 6 .
(d) 1 4 . 10 1 1 2 .
0.1 ) m k (
0.01 Y
0.001 1 .
0.0001
1E-005 8 .
1E-006 6 .
-60.00
-40.00
-20.00
0.00
20.00
40.00
X (km)

16. 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

17. 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

18. Rate-and-state model - II
(Dieterich, 1994)
Where
is the stress rate in the area

19. Rate-and-state model - III
(Dieterich, 1994)
Time dependence of the rate of triggered events
for different values of the stress change

20. Rate-and-state model - IV
(Console et al., 2004)
The total number of triggered events is proportional to the stress change Dt

21. 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

22. 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

23. d0 is related to the smallest seismic moment
d0 is related to the smallest seismic moment and the smallest magnitude m0 through:
and

24. Modified epidemic model Umbria-Marche, 1 September 1997, 00:00

25. Modified epidemic model
Umbria-marche, 26 September 1997, 00:00
(before the event near Colfiorito, Ml=5.6)

26. Modified epidemic model
Umbria-marche, 26 September 1997, 9:40
(before the secondth event near Colfiorito, Ml 5.8)

27. Modified epidemic model
Umbria-Marche, 14 October 1997
( before the event near Sellano, Ml 5.5)

28. 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

29. Stress transfer and rate-state model
Umbria-Marche, 26 September 1997, 00:00
(before the event near Colfiorito, Ml=5.6)
Occurrence rate density (events per day in 1000 km2, Ml  2.0)

30. Stress transfer and rate-state model
Umbria-Marche, 26 September 1997, 9:40
(before the second event near Colfiorito, Ml=5.8)
Occurrence rate density (events per day in 1000 km2, Ml  2.0)

31. Stress transfer and rate-state model
Umbria-marche, 14 October 1997
(before the event near Sellano, Ml=5.5)
Occurrence rate density (events per day in 1000 km2, Ml  2.0)

32. Stress transfer and rate-state model
Umbria-marche, 3 April 1998
(before the event near Nocera Umbra, Ml=5.0)
Occurrence rate density (events per day in 1000 km2, Ml  2.0)

33. Stress transfer
and rate-state model
(short term)
Stress transfer
and rate-state model
(long term)

34. 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.

35. 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.

36. 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.

37. Thank you!