Source parameters I
• Magnitudes and moment
• Source time function
• Source spectra
• Stress drop
• Earthquake scaling and statistics
• Fault scaling and statistics
• Asperities and barriers
Richter in his living room
Source parameters I: Magnitudes and moment
Richter noticed that the vertical
offset between every two curves of
log(a)
maximum displacement 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(D)- log10 A0 (D),
where A and A0 are the largest
amplitudes of the event and the
reference event, respectively, and
 is the epicentral distance.
event1
event2
event3
distance
Source parameters I: Magnitudes and moment
The actual data Richter was using
is this:
Source parameters I: Magnitudes and moment
From a table of values of A0, an
approximate empirical formula has
been derived for the logarithm of
the reference event, and the local
magnitude is defined as:
M L = log10 A + 2.76log10 D - 2.48.
This definition is only valid for
Southern California and only when
the Wood-Anderson seismograph
is being used.
Source parameters I: Magnitudes and moment
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.
Nevertheless, Local Magnitudes are still reported sometimes because many
building have resonant frequencies near 1Hz, close to that of the WoodAnderson (0.8 Hz), so ML can serve as an indication of the structural damage
an earthquake can cause.
Source parameters I: Magnitudes and moment
Several other magnitude scales have been defined, but the most
commonly used are:
• Surface-wave magnitude (MS).
• Body-wave magnitude (mb).
• Moment magnitude (Mw).
Source parameters I: Magnitudes and moment
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:
MS or mb = log10 (A/T) + distance correction.
where the distance correction accounts for the decrease of A with distance due
to geometrical spreading and attenuation.
• mb is calculated from the early portion of the body wave train, usually the P
wave. Common US practice is to use the first 5 S of the record, and period
of less than 3 S, usually 1 S on instruments with peak response near 1 S.
• Ms is measured either using the largest amplitude (zero to peak) of the
surface waves, or using the amplitude of the Rayleigh waves at 20 S, which
often have the largest amplitude.
Magnitude scales are logarithmic, so an increase in one unit indicates a ten-fold
increase in the displacement.
Source parameters I: Magnitudes and moment
Main limitations:
• Totally empirical and as such have no physical meaning (note that the
expression inside the logarithm has physical units of mm/S).
• Magnitude estimates vary noticeably with azimuth due to amplitude radiation
pattern - although this difficulty is partially reduced by averaging.
• Body and surface wave magnitudes do not correctly reflect the size of large
earthquakes.
Source parameters I: Magnitudes and moment
The moment magnitude is a function of the seismic moment, M0, as follows:
2
M W = log10 (M 0 ) -10.7 ,
3
where M0 is in dyne-cm.
The seismic moment is a physical quantity (as opposed to earthquake
magnitude) that measures the strength of an earthquake. It is equal to:
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.)
Source parameters I: Magnitudes and moment
Note that an increase of one magnitude unit corresponds to a
101.5 ≈ 32 times increase in the amount of energy released:
the proportional difference in energy released = 10
where m1 and m2 are two moment magnitudes.
The pasta-quake analogy:
3
( m1-m2 )
2
Source parameters I: Magnitudes and moment
Note that:
• mb saturates above 6
• MS saturates above 8
Source parameters I: Magnitudes and moment
Just so that you don’t get the false
impression that earthquake slip
distribution is smooth and
everywhere close to the average slip
Source parameters I: Magnitudes and moment
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
Source parameters I: Magnitudes and moment
Richter
Magnitude
TNT for Seismic
Energy Yield
-1.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
6
30
320
1
4.6
29
73
1,000
5,100
32,000
80,000
1 million
5 million
32 million
7.5
8.0
8.5
9.0
10.0
12.0
160 million
1 billion
5 billion
32 billion
1 trillion
160 trillion
ounces
pounds
pounds
ton
tons
tons
tons
tons
tons
tons
tons
tons
tons
tons
tons
tons
tons
tons
tons
tons
Example
(approximate)
Breaking a rock on a lab table
Large Blast at a Construction Site
Large Quarry or Mine Blast
Small Nuclear Weapon
Average Tornado (total energy)
Little Skull Mtn., NV Quake, 1992
Double Spring Flat, NV Quake, 1994
Northridge, CA Quake, 1994
Hyogo-Ken Nanbu, Japan Quake, 1995;
Largest Thermonuclear Weapon
Landers, CA Quake, 1992
San Francisco, CA Quake, 1906
Anchorage, AK Quake, 1964
Chilean Quake, 1960
(San-Andreas type fault circling Earth)
(Fault Earth in half through center,
OR Earth's daily receipt of solar energy)
Source parameters I: Magnitudes and moment
Source parameters I: Source time function
Moment magnitude has become the standard measure of
earthquake size, but it requires more analysis and understanding
of the seismic signal.
Source parameters I: Source time function
Source parameters I: Source time function
Imagine the source time
function resulting from a
point source radiating an
impulse moving at velocity
VR along a line of length L
that is embedded within a
medium whose seismic
velocity is V.
The signal’s duration is:
æ L r ö r0
TR = ç + ÷ è VR V ø V
Source parameters I: Source time function
For points far from the fault, i.e.
r>>L:
r » r0 - L cosq.
Thus, the time pulse becomes:
æ 1 cosq ö L æ V
ö
TR = L ç ÷ = ç - cosq ÷
V ø V è VR
è VR
ø
It follows that the maximum
duration occurs at 180 degrees
of the rupture propagation
direction, whereas the minimum
duration occurs at the rupture
direction.
Source parameters I: Source time function
The Haskell fault model:
But slip at a given point does not
occur instantaneously.
The slip history is often modeled as
a ramp function.
The source time function depends
on the time derivative of the slip
history.
A ramp time history has a derivative
of a boxcar.
When convolved with the boxcar
time function due to rupture
propagation, a trapezoidal time
function results, whose width is
equal to the rupture and rise times.
Source parameters I: Source time function
The duration of the radiated
pulse varies with azimuth.
The area of the pulse is the
same at all azimuths, and
therefore the amplitude of the
source time function is
inversely proportional to the
duration. This effect is known
as the directivity effect.
In some cases the directivity
effect helps to identify rupture
plane, since no similar effect is
associated with the auxiliary
plane.
æ 1 cosq ö L æ V
ö
TR = L ç ÷ = ç - cosq ÷
V ø V è VR
è VR
ø
Source parameters I: Source time function
The ratio between the time pulse (the width of the boxcar) and the
predominant period of the seismic wave is:
TR L V L
»
=
T lV l
If this ratio is small, the source is “seen” as a point source,
whereas if this ratio is large, there are finite source effects.
Source parameters I: Source spectra
How would this source time function look in the frequency
domain?
According to the Haskell fault model the source time function
results from the convolution of two boxcar time functions.
The transform of a boxcar of height 1/T and length T is:
F (w ) =
1 iwt
1 iwT /2 -iwT /2 sin (wT / 2)
e
dt
=
(e - e ) = wT / 2 = sinc (wT / 2)
òT
Ti
w
-T /2
+T /2
Source parameters I: Source spectra
Thus, the spectral amplitude of the source signal is the product of
the seismic moment of two sinc terms:
A (w ) = M 0
sin (wTR / 2) sin (wTD / 2)
wTR / 2
wTD / 2
where TR and TD are the rupture and rise times, respectively.
Often, amplitude spectra are plotted using a logarithm scale.
Taking the logarithm of the above gives:
log A (w ) = log M 0 + logéësinc (wTR / 2)ùû + logéësinc (wTD / 2)ùû
Source parameters I: Source spectra
log A (w ) = log M 0 + logéësinc (wTR / 2)ùû + logéësinc (wTD / 2)ùû
It is useful to approximate sinc(x) as
1 for x<1 and as 1/x for x>1.
This approach yields:
Note the two corner frequencies
dividing the into 3 parts.
Source parameters I: Source spectra
While in principle, by
studying the spectra of real
earthquakes, we can
recover M0, TR and TD, in
practice things are more
complicated than that.
Source parameters I: Source spectra
As rupture lengths increase,
the seismic moments, rise
times and rupture durations
increase. Thus, the corner
frequencies move to the left,
i.e. to lower frequencies.
Source parameters I: Source spectra
• Ms is measured at periods
of 20S, and it saturates
beyond M0>10^25 dyncm.
• mb is measured at periods
of 1S, and it saturates
beyond M0>10^22 dyncm.
• Similar saturation effects
occur for other magnitude
scales which are
measured at specific
frequencies.
Source parameters I: Source spectra
Again, this is how the moment magnitude is defined:
2
M W = log10 (M 0 ) -10.7 ,
3
where M0 is in dyne-cm.
In contrast to body and surface wave magnitudes, the moment magnitude does
not saturate.
Source parameters I: Stress drop
M 0 = GuA = hGuL2
u
Dt = Ge » G
L
u
M 0 = hG L3 » hDt L3
L
One approach to find L is to
analyze the amplitude of the
spectrum, identify the corner
frequency, and use that to
assess the rupture duration.
The latter divided by an
average rupture speed of 0.70.8 of the shear wave speed
gives L.
Source parameters I: Stress drop
• 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: Hanks, 1977
Source parameters I: The scaling of fault length and slip
Normalized displacement
Normalized slip profiles of normal faults
of different length.
From Dawers et al., 1993
Source parameters I: 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
Source parameters I: 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:
logN(> 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.
Source parameters I: 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. log N(> MW ) = a - bMW .
Recall that :
2
2. MW = log10 M 0 -10.7 .
3
Replacing 1 in 2 gives :
3a. log N(> MW ) = a¢ - b¢ log M 0 ,
which is equivalent to :
- b¢
¢
3b. N(> MW ) = a M 0 .
Source parameters I: 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.
Source parameters I: Fault distribution and earthquake statistics
Cumulative length distribution of subfaults of the San Andreas
fault.
Scholz, 1998
Source parameters I: Fault distribution and earthquake statistics
Loma Prieta
Source parameters I: 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?
Source parameters I: 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
Source parameters I: 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?
Source parameters I: 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.
Source parameters I: 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
Source parameters I: 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.
Further reading:
• Scholz, C. H., The mechanics of earthquakes and faulting, New-York: Cambridge Univ. Press.,
439 p., 1990.
• Aki, K., Asperities, barriers and characteristics of earthquakes, J. Geophys. Res., 89, 5867-5872,
1994.