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.]