Omori law
•  The modified Omori law
•  Omori law for foreshocks
•  Aftershocks of aftershocks
•  Physical aspects of temporal clustering
Omori law: the modified Omori law
Omori law (Omori, 1894):
C1
˙
N (t) =
,
t
the modified Omori law (Utsu, 1961):
C1
N˙ (t) =
,
p
€
(C 2 + t )
and its cumulative form (for p=1):
# t
&
N(t) = ∫ N˙ (t)dt = C1 ln% + 1( ,
€
$ C2 '
0
t
where t is time, N is earthquake count, C1, C2 and p are fitting
coefficients. The decay exponent, p, is commonly referred to as
the “p-value”.
€
Why study aftershocks?
Omori law: Aftershocks around the world
1995 Mw 6.9 Kobe, Japan
duration
background
Omori law: Aftershocks around the world
1979 Mw 6.6 Imperial Valley, CA
Omori law: Aftershocks around the world
1989 Mw 7.1 Loma Prieta, CA
Omori law: Aftershocks of small mainshocks
The traditional approach is to consider as mainshocks only
earthquakes that are large and infrequent. Recent studies show
that small-to-moderate earthquakes also enhance the seismicity
in their vicinity.
•  Aftershocks of aftershocks also decay according to the modified Omori
law.
Omori law: Aftershocks of small mainshocks
When analyzing spatio-temporal clustering with respect to small
earthquakes, it is useful to construct a composite catalog of
stacked aftershock sequences.
A recipe for analysing aftershocks of microearthquakes:
•  We consider each earthquake as a potential mainshock, and for
each such mainshock compute its rupture dimensions.
•  Calculate lag-times and distances between each potential
mainshock and all later earthquakes within the study area.
•  Stack mainshock-aftershock pairs with an inter-event distance
that is less than twice the mainshock radius to get a “composite
catalog”.
Omori law: Aftershocks of small mainshocks
•  Micro-earthquakes during “background activity” also trigger
aftershocks that decay according to the modified Omori law.
Omori law: Remote aftershocks
N˙ (Izmit + 10 days) − N˙ (Izmit - 100 days)
N˙ (1985 - 2002)
The Mw7.4 Izmit (Turkey):
€
Mw5.8
Two weeks later
Omori law: Remote aftershocks
N˙ (Izmit + 10 days) − N˙ (Izmit - 100 days)
N˙ (1985 - 2002)
cumulative Omori law
€
(See also Brodsky et al., 2000.)
•  The decay of remote aftershocks follows the modified Omori law!
Omori law: Remote aftershocks
The decay of M7.4 Izmit
aftershocks throughout Greece
is very similar to the decay of
M5.8 Athens aftershocks in
Athens area (just multiply the
vertical axis by 2).
Omori law: Remote aftershocks
N˙ (Landers + 10 days) − N˙ (Landers - 100 days)
N˙ (1985 - 2002)
days since mainshock
Omori law: Remote aftershocks
ΔCFF(t) = Δσ S (t) − µΔσ N (t) ,
•  The magnitude of static
stress changes decay as
disatnce-3.
•  The magnitude of the peak
dynamic stress changes
decay as distance-1.
•  At great distances from the
rupture, the peak dynamic
stresses are much larger than
the static stresss.
Figure from Kilb et al., 2000
Omori law: Remote aftershocks
No triggering
Stress
Instantaneous
triggering
Time
Time
Omori law: Remote aftershocks
Indeed, distant aftershocks are observed during the passage of
the seismic waves emitted from the mainshock rupture.
Izmit aftershocks in Greece.
Brodsky et al., 2000
Omori law: Remote aftershocks
Omori law: Remote aftershocks
A major aftershock (magnitude 7.1) can be seen at the closest
stations starting just after the 200 minutes mark. Note the relative
size of this aftershock, which would be considered as a major
earthquake under ordinary circumstances, compared to the
mainshock.
Omori law: Remote aftershocks
•  Dynamic stress changes trigger aftershocks that rupture during
the passage of the seismic waves.
•  But the vast majority aftershocks occur during the days, weeks
and months after the mainshock.
•  Dynamic stress changes cannot trigger “delayed aftershocks”,
i.e. those aftreshocks that rupture long after the passage of the
seismic waves emitted by the mainshock.
•  It is, therefore, unclear what gives rise to delayed aftershocks in
regions that are located very far from the mainshock.
Omori law: Aftershocks of aftershocks and the origin of remote
aftershocks
The mainshock index quantifies the degree to which the triggering
effect of a given aftershock is locally more important than the
mainshock. The mainshock index of event i is defined as:
N (Δt i < t ≤ 2Δt i ,r < 2Ri )
λi =
.
N (0 < t ≤ Δt i ,r < 2Ri )
•  t is time measured from the mainshock time
•  Δt is the€lag time between the mainshock and aftershock I
•  r is inter-event distance
•  R is the rupture radius
Omori law: Aftershocks of aftershocks and the origin of remote
aftershocks
Mainshock index
Omori law: Aftershocks of aftershocks and the origin of remote
aftershocks
•  λi>1 is indicative of seismicity
rate increase in the vicinity of
the aftershock in question,
suggesting that the triggering
effect of that aftershock in that
region is stronger than the
triggering effect of the
mainshock and the previous
aftershocks.
λ in north1
Omori law: Aftershocks of aftershocks and the origin of remote
aftershocks
Comparison with a mainshock index of a sequence decaying
locally according to the Omori law:
2Δt i
∫
λOmori
=
i
Δt i
Δt i
∫
0
C1
dt
p
(C2 + t i )
C1
dt
p
(C2 + t i )
,
which has the properties:
€
For Δt → 0 , λOmori
→1
i
and
For Δt → ∞ , λOmori
→0 .
i
Omori law: Aftershocks of aftershocks and the origin of remote
aftershocks
•  In conclusion, most (if
not all) Landers remote
aftershocks were not
directly triggered by
landers, but are
aftershocks of previous
aftershocks.
Comparison with theoreticalA.λZiv
Figure 6.
Percentage of k ! kth (p ! 1) as a function of the threshold magnitude for earthquakes that
occurred during the 100 days after the Landers earthquake within regions North1 (solid) and North2
(dashed). Earthquakes that occurred during the first
24 hr were excluded from this analysis.
Omori law: Aftershocks of aftershocks and the origin of remote
aftershocks
Hector Mine aftershocks
Omori law: Aftershocks of aftershocks and the origin of remote
aftershocks
Note that:
On the Roleof
of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies
•  The sequence consists
several sub-sequences,
Hector Mine aftershocks
and the onset of activity
migrated southward.
•  Many of the quakes that
occurred between 33N
and 33.5N are aftershocks
of a M4.3 that ruptured 10
minutes after the
mainshock.
•  M4.37 that occurred 2.4
days after the mainshocks
Figure 8. Time-space diagram for the Hector Mine aftershocks in area South. The
triggered a burst of
size of the circles is proportional to the earthquake magnitude. The vertical dashed
lines indicate the timing of the three largest earthquakes.
seismicity near latitude
33N.
Omori law: Foreshocks
•  The increase in
foreshock rate too
follows an Omori law,
with t being the time to
the mainshock.
From Jones and Molnar, 1979
Omori law: Physical aspects
Implications of static-kinetic friction on earthquake timing:
The “clock advance” does NOT depend on the time of the stress
application.
Omori law: Physical aspects
Implications of rate-and-state friction on earthquake timing:
The “clock advance”
depends on the time of the
stress application.
Omori law: Physical aspects
Implications of the friction law on temporal clustering:
Summary:
•  Not only aftershocks of large quakes, but also aftershocks of
aftershocks decay according to the modified Omori law.
•  Micro-earthquakes during “background activity” also trigger
aftershocks that decay according to the modified Omori law.
•  The decay of remote aftershocks follows the modified Omori law.
•  Most (if not all) Landers remote aftershocks were not directly
triggered by the Landers earthquake, but are aftershocks of
previous aftershocks.
•  The increase in foreshock rate too follows an Omori law, with t
being the time to the mainshock.
•  Stress perturbation applied on a population of faults governed by
static-kinetic friction cannot give rise to seismicity rate change.
Further reading:
•  Scholz, C. H., The mechanics of earthquakes and faulting, NewYork: Cambridge Univ. Press., 439 p., 1990.
•  Ziv, A., On the Role of Multiple Interactions in Remote Aftershock
Triggering: The Landers and the Hector Mine Case Studies, Bull.
Seismol. Soc. Am., 96(1), 80-89, 2006.