final remark on power-law distributions - bgu · modi ed omori law [omori, 1894; utsu, 1961] omori...
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Final remark on power-law distributions
Many systems and phenomena are distributed according
to a power-law distribution. A power-law applies to a
system when large is rare and small is common.
The distribution of individual wealth is a good example
of this: there are a very few rich men and lots and lots
of poor folks. A familiar way to think about power laws
is the 80/20 rule: 80% of the wealth is controlled by
20% of the population (see:
http://management.about.com/cs/generalmanagement).
A few examples are shown below.
AOL Users:
Binned distribution of users to sites. (Taken from
http : //www.hpl.hp.com/research/idl/papers/ranking/ranking.html)
Forest Fires:
Noncumulative frequency-area distributions for actual forest fires and wildfires in
the United States and Australia: (A) 4284 fires on U.S. Fish and Wildlife Service
lands (1986-1995) (9), (B) 120 fires in the western United States (1150-1960)
(10), (C) 164 fires in Alaskan boreal forests (1990-1991) (11), and (D) 298 fires in
the ACT (1926-1991) (12). For each data set, the noncumulative number of fires
per year (dCF/dAF) with area (AF) is given as a function of AF (13). In each
case, a reasonably good correlation over many decades of AF is obtained by using
the power-law relation (Eq. 1) with = 1.31 to 1.49; is the slope of the best-fit line
in log-log space and is shown for each data set. (From Malamud and Turcotte,
Science, 281, p. 1840-1842, 1998)
Landslides Triggered by the Northridge Earthquake:
Noncumulative frequency-area distribution of 11000 landslides triggered by the
January 1994, Northridge, California earthquake. (From Guzzetti et al., EPSL, 195,
p. 169-183, 2002.)
Landslides Triggered by Snow Melt:
Noncumulative frequency-area distribution of landslides of Central Italy . (From
Guzzetti et al., EPSL, 195, p. 169-183, 2002.)
Loss:
Comparison of natural disaster fatalities in the United States. Cumulative
size-frequency distributions for annual earthquake, flood, hurricane, and tornado
fatalities. In addition to demonstrating linear scaling behavior over 2 to 3 orders
of magnitude in loss, these data group into two families. Hurricanes and
earthquakes are associated with relatively flat slopes (D= -0.4 to -0.6); while
floods and tornadoes have steeper slopes (D= -1.3 to -1.4). Open symbols were
not used to calculate slope of lines. (Taken from:
http : //coastal.er.usgs.gov/hurricanef orecast/hurrlosses.html)
Final remark regarding G-R relation andcharacteristic distribution
Extrapolation of the b−value inferred for small
earthquake may result in under-estimation of the actual
hazard, if earthquake size-distribution is characteristic
rather than power-law.
For a single fault, the size distribution is often
characteristic.
10-1
100100
101
102
103
104
105
# ev
ents
0 1 2 3 4 5 6 7 8Magnitude
A histogram of event magnitude for a small segment of the San-Andreas fault that
contains the Loma Prieta earthquake (your hw assigment from previous week).
For a population of faults, the distribution is consistent
with Gutenberg-Richter relation, i.e. power-law
(provided that the dataset is large).
10-1
100100
101
102
103
104
105
# ev
ents
0 1 2 3 4 5 6 7 8Magnitude
A histogram of event magnitude for Northern California (your hw assigment from
previous week).
Modified Omori Law [Omori, 1894; Utsu, 1961]
Omori studied the 1891 Nobi earthquake (Japan), and
noticed that aftershock decay rate is fittable with (see
next page):
N
N0
=1
(t0 + t)
where:
N is earthquake production rate
t is time since the mainshock time
N0 and t0 are empirical constant
Utsu modified Omori’s original formula as follows:
N
N0
=1
(t0 + t)p
where:
p is an empirical constant that is usually close to 1
Use of the modified Omori Law provides better fit to
the data.
Occurrence rate (top) and cumulative number (bottom) of felt earthquakes at
Gifu after the Nobi earthquake of 1891. (diagram taken from Utsu et al., J. Phys.
Earth., 43, 1-33, 1995.)
Estimation of aftershock duration ta, defined as the time until the seismicity rate
returns to the background rate before the 1995 Kobe earthquake. (diagram taken
from Toda et al., JGR, 103(10), 24,543–24,565, 1998)
Remote Aftershocks
Izmit aftershock sequence on continental Greece
19˚
19˚
20˚
20˚
21˚
21˚
22˚
22˚
23˚
23˚
24˚
24˚
25˚
25˚
26˚
26˚
35˚ 35˚
36˚ 36˚
37˚ 37˚
38˚ 38˚
39˚ 39˚
40˚ 40˚
41˚ 41˚
42˚ 42˚
0 100 200
km
1 to 55 to 25> 25
M=5.87 Sept. 1999
A map of earthquake rate change following the Izmit earthquake, calculated for
non-overlapping spatial windows with dimensions of 0.5 × 0.5 degrees. A star
indicating the epicentral location of the magnitude 5.8, of 7 September 1999,
Athens earthquake. In the following page, aftershock activities are examined in
two regions; the first is the sum of all the dotted rectangles and the second is the
solid box surrounding Athens.
10
20
30
40
50
60
70
even
t co
unt
10 20 30 40days after Izmit
(A) Izmit triggered areas
20
40
60
80
100
120
140
even
t co
unt
10 20 30 40days after Athens
(B) Athens region
Plots of cumulative earthquak counts as a function of time for (a) Izmit
aftershocks, and (b) Athens aftershocks. The cut-off magnitude is 3. Dashed lines
are best fits to Omori’s formula.
Landers and Hector Mine - ”Twin earthquakes”
238˚ 240˚ 242˚ 244˚ 246˚
32˚ 32˚
34˚ 34˚
36˚ 36˚
38˚ 38˚
40˚ 40˚0 100 200
km
Mw=7.3
1 to 5
5 to 10
> 10
(A)
North1
LVC
North2
IV
E1
E2
E3
238˚ 240˚ 242˚ 244˚ 246˚
32˚ 32˚
34˚ 34˚
36˚ 36˚
38˚ 38˚
40˚ 40˚0 100 200
km
Mw=7.1
1 to 5
5 to 10
> 10
(B)
North1
LVC
North2
IV
E1
E2
E3
A map of earthquake rate change following (a) the Landers earthquake, and (b)
the Hector Mine earthquake. Stars are indicating the epicentral locations of the
Landers and the Hector Mine earthquakes. For clarity purposes, results in an area
surrounding the mainshock hypocenters that is indicated by a dashed square are
not shown.
Observed seismicity rate increase following large
earthquakes in sites that are located several source
lengths away from the mainshock centroid poses a
major problem. This is because the static stress change
induced by a mainshock in that region seems to be
insignificant. This led to the idea that dynamic stresses,
since they decay slower with distance than static
stresses, are the cause for remote triggering [e.g.,
Anderson et al., 1994; Gomberg and Bodin, 1994].
Cartoon time-histories of Coulomb function. [Kilb et al., Nature, 408(30), 571–574,
2000.]
Foreshocks
Foreshocks are more difficult to study than aftershocks.
This is because large earthquakes have many
aftershocks, but only a few (if any) foreshocks. In order
to study foreshock properties, it is useful to stack many
foreshock sequences.
Foreshock-mainshock pairs in the Harvard CMT catalog (1977–1994), shown as a
function of interevent distance (kilometers) and interevent time (days) [Reasenberg,
JGR, 104, 4755–4768, 1999.]
The ongoing improvement in the recording completeness
of seismicity led to an improved view of foreshock
activity, and to the recognition that foreshocks rate may
also be fitted with an Omori law [e.g., Papazachos,
1975; Kagan and Knopoff, 1978; Jones and Molnar,
1979; Shaw, 1993, Helmstetter and Sornette, 2003].
Foreshock activity (a-c) as a function of time before the mainshocks of different
magnitudes and (d) for all mainshocks from all three data sets. Also shown are
the exponential (solid line) and power-law (dotted line) equations that best fit
these data. [Jones and Molnar, JGR, 84, 3596–3608, 1979.]