Transcript Document

Earthquake scaling and statistics
• The scaling of slip with length
• Stress drop
• Seismic moment
• Earthquake magnitude
• Magnitude statistics
• Fault statistics
The scaling of fault length and slip
Normalized slip profiles of normal faults
of different length.
From Dawers et al., 1993
The scaling of fault length and slip
Displacement versus fault length
What emerges from this
data set is a linear scaling
between displacement and
fault length.
Figure from: Schlische et al, 1996
The seismic moment
The seismic moment is a physical quantity (as opposed to
earthquake magnitude) that measures the strength of an
earthquake. It is equal to:
moment G A D ,
where:
G is the shear modulus
A = LxW is the rupture area
D is the average co-seismic slip

(It may be calculated from the
amplitude spectra of the seismic
waves.)
The scaling of seismic moment with rupture length
• What emerges from this is that
co-seismic stress drop is
constant over a wide range of
sizes.
• The constancy of the stress
drop implies linear scaling
between co-seismic slip and
rupture length.
slope=3
Figure from: Schlische et al, 1996
Earthquake magnitude
Richter noticed that the
vertical offset between every
two curves is independent of
the distance. Thus, one can
measure the magnitude of a
given event with respect to
the magnitude of a reference
event as:
M L  log10 A()  log10 A0 (),
log(a)
event1
event2
event3
where A0 is the amplitude of
the reference event and  is
the epicentral distance.
distance
Earthquake magnitude
Richter arbitrarily chose a magnitude 0 event to be an earthquake
that would show a maximum combined horizontal displacement of
1 micrometer on a seismogram recorded using a Wood-Anderson
torsion seismometer 100 km from the earthquake epicenter.
Problems with Richter’s magnitude scale:
• The Wood-Anderson seismograph is no longer in use and
cannot record magnitudes greater than 6.8.
• Local scale for South California, and therefore difficult to
compare with other regions.
Earthquake magnitude
Several magnitude scales have been defined, but the most
commonly used are:
• Local magnitude (ML), commonly referred to as "Richter
magnitude".
• Surface-wave magnitude (MS).
• Body-wave magnitude (mb).
• Moment magnitude (Mw).
Earthquake magnitude
• Both surface-wave and body-waves magnitudes are a function of
the ratio between the displacement amplitude, A, and the
dominant period, T, and are given by:
M S or mb  log10 (A /T)  distance correction .
• The moment magnitude is a function of the seismic moment, M0,
asfollows:
2
MW  log10 (M0 ) 10.7 .
3
where M0 is in dyne-cm.

Earthquake magnitude
The diagrams to the right
show slip distribution inferred
for several well studied
quakes. It is interesting to
compare the rupture area of
a magnitude 7.3 (top) with
that of a magnitude 5.6
(smallest one near the
bottom).
Earthquake magnitude
Magnitude classification (from the USGS):
0.0-3.0 :
3.0-3.9 :
4.0-4.9 :
5.0-5.9 :
6.0-6.9 :
7.0-7.9 :
8.0 and greater :
micro
minor
light
moderate
strong
major
great
Intensity scale
The intensity scale, often referred to as the Mercalli scale,
quantifies the effects of an earthquake on the Earth’s surface,
humans, objects of nature, and man-made structures on a scale
of 1 through 12. (from Wikipedia)
I
V
VIII
XII
shaking is felt by a few people
shaking is felt by almost everyone
cause great damage to poorly built structures
total destruction
The Gutenberg-Richter statistics
Fortunately, there are many more small quakes than large ones.
The figure below shows the frequency of earthquakes as a
function of their magnitude for a world-wide catalog during the
year of 1995.
This distribution may be fitted
with:
log N( M)  a  bM ,

Figure from simscience.org
where n is the number of
earthquakes whose magnitude
is greater than M. This result
is known as the GutenbergRichter relation.
The Gutenberg-Richter statistics
• While the a-value is a measure of earthquake productivity, the bvalue is indicative of the ratio between large and small quakes.
Both a and b are, therefore, important parameters in hazard
analysis. Usually b is close to a unity.
• Note that the G-R relation describes a power-law distribution.
1. logN( MW )  a  bMW .
Recall that:
2
2. MW  log10 M 0 10.7 .
3
Replacing 1 in 2 gives:
3a. logN( MW )  a blogM 0 ,
which is equivalent to:
 b 
3b. N( MW )  aM 0 .
The Gutenberg-Richter distribution versus characteristic
distribution
G-R distribution
characteristic distribution
Two end-member models can explain the G-R statistics:
• Each fault exhibits its own G-R distribution of earthquakes.
• There is a power-law distribution of fault lengths, with each fault
exhibiting a characteristic distribution.
Fault distribution and earthquake statistics
Cumulative length distribution of subfaults of the San Andreas
fault.
Scholz, 1998
Fault distribution and earthquake statistics
Loma Prieta
Fault distribution and earthquake statistics
In conclusion:
• For a statistically meaningful population of faults, the distribution
is often consistent with the G-R relation.
• For a single fault, on the other hand, the size distribution is often
characteristic.
• Note that the extrapolation of the b-value inferred for small
earthquakes may result in under-estimation of the actual hazard, if
earthquake size-distribution is characteristic rather than powerlaw.
Question: what gives rise to the drop-off in the small magnitude
with respect to the G-R distribution?
The controls on rupture final dimensions
Seismological observations show that:
1. Co-seismic slip is very heterogeneous.
2. Slip duration (rise time) at any given point is much shorter than
the total rupture duration
Example from the 2004 Northern Sumatra giant earthquake
Preliminary result by Yagi.
Uploaded from: www.ineter.gob.ni/geofisica/tsunami/com/20041226-indonesia/rupture.htm
The controls on rupture final dimensions
• Barriers are areas of little slip in a single earthquake (Das and
Aki, 1977).
• Asperities are areas of large slip during a single earthquake
(Kanamori and Stewart, 1978).
The origin and behavior with time of barriers and asperities:
1. Fault geometry - fixed in time and space?
2. Stress heterogeneities - variable in time and space?
3. Both?
The controls on rupture final dimensions
According to the
barrier model (Aki,
1984) maximum slip
scales with barrier
interval.
If this was true, fault
maps could be used to
predict maximum
earthquake magnitude in
a given region.
The controls on rupture final dimensions
But quite often barriers fail to stop the rupture…
The 1992 Mw7.3 Landers (CA):
The 2002 Mw7.9 Denali (Alaska):
Figure from: pubs.usgs.gov
Figure from: www.cisn.org
The controls on rupture final dimensions
While in the barrier model ruptures stop on barriers and the bigger
the rupture gets the bigger the barrier that is needed in order for it
to stop, according to the asperity model (Kanamori and Steawart,
1978) earthquakes nucleate on asperities and big ruptures are
those that nucleate on strong big asperities.
That many ruptures nucleate far from areas of maximum slip is
somewhat inconsistent with the asperity model.
The controls on rupture final dimensions
In the context of rate-state friction:
• Asperities are areas of a-b<0.
• Barriers are areas of a-b>0.
Further reading:
• Scholz, C. H., The mechanics of earthquakes and faulting, NewYork: Cambridge Univ. Press., 439 p., 1990.
• Aki, K., Asperities, barriers and characteristics of earthquakes, J.
Geophys. Res., 89, 5867-5872, 1994.