Earthquake source mechanics
Lecture 6
Seismic moment
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Earthquake magnitude
Richter magnitude scale
M = log A(∆) - log A0(∆)
where A is max trace
amplitude at distance ∆
and A0 is at 100 km
Surface wave magnitude MS
MS = log A + α log ∆ + β
where A is max amp of 20s period surface waves
Magnitude and energy
log Es = 11.8 + 1.5 Ms (ergs)
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Seismic moment
ß
ß
ß
ß
Seismic intensity measures
relative strength of shaking
locally
Instrumental earthquake
magnitude provides measure
of size on basis of wave
motion
Peak values used in
magnitude determination do
not reveal overall power of
the source
Seismic Moment: measure
of quake rupture size related
to leverage of forces
(couples) across area of fault
slip
Moment = FL
F
L
F
Applying couple
to fault
- two equal & opposite forces
= force couple
- size of couple = moment
- numerical value = product of
value of one force times
distance between
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Seismic Moment II
ß
Stress &
strain
accumulation
ß
F
Applying couple
to fault
ß
Can be applied to
seismogenic faults
Elastic rebound along a
rupturing fault can be
considered in terms of
resulting from force couples
along and across it
Seismic moment can be
determined from
a
F
Fault rupture
and rebound
a
fault slip dimensions
measured in field or from
aftershock distributions
analysis of seismic wave
properties (frequency
spectrum analysis)
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Seismic moment
Seismic moment = Area of fault plane x stress drop of earthquake x
coseismic slip
[NB: Area x stress = force
force x distance = moment]]
ß provides estimate of overall size of the seismic source
Units: units of moment = newton-metres = Nm =
= joule = J = unit of energy
So seismic moment is also a measure of the energy of the earthquake
Empirical relation between moment and magnitude is
log10 M0 = c M + d
or Mw = 2/3 log10 M0 – 6
(Moment-magnitude scale (Kanamori)
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Seismic moment
ß
ß
ß
ß
A few great fracturing events
totally dominate the
earthquake seismic moment
released (moments more
realistic than comparing
magnitudes)
The moment release, even
at the major plate
boundaries is distributed
very unevenly
About ¼ of the seismic
moment released between
1904-86 was by the great
Chilean earthquake of 1960
that ruptured 100km of the
subduction zone interface at
the Peru-Chile trench
San Francisco 1906 doesn’t
even get a mention!
Seismic moment does not saturate
e.g. Alaskan earthquake: Ms=8.4;
Mw=9.2
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Seismic moment tensor
The nine different force
couples that make up the
components of the
moment tensor:
⎛ M 11
⎜
M jk = ⎜ M 21
⎜M
⎝ 31
M 12
M 22
M 32
M 13 ⎞
⎟
M 23 ⎟
M 33 ⎟⎠
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Seismic moment tensor
Example of right lateral movement on a strike-slip fault in the x1
direction corresponds to:
⎛ 0
⎜
M = ⎜M0
⎜ 0
⎝
M0
0
0
0⎞
⎟
0⎟
0 ⎟⎠
Note Mij = Mji
where Mo is defined as the scalar seismic moment:
M0 = µ A s
where µ is the rigidity modulus, A is the area of the fault and s
is the slip or displacement of the fault
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Scalar seismic moment
Define the scalar seismic moment (Aki)
M0 = µ A s
µ - rigidity
A - fault area
s - slip (vector)
l
s
µ
w
Units: force x length (Nm or J)
Aspect ratio:
usually l/w ∼ 2,
except for long strike-slip
faults
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Far-field radiation pattern for double couple source
P-waves
S-waves
The orientation of the small
arrows shows the direction of
first motion; their length is
proportional to the wave
amplitude. The P-wave first
motions are outward in the
compressional quadrant and
inward in the dilatational
quadrant, with nodal lines in
between. S-wave first motions
are generally away from the
pressure axis and toward the
tension axis; there are 6 nodal
points and no nodal lines in S
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Example of focal spheres
Example of focal spheres
and their corresponding
fault geometries. The lower
half of the focal sphere is
plotted to the left, with
compressional quadrants
shaded. The block
diagrams show th two fault
geometries (the primary
and auxillary fault planes)
that could have produced
the observed radiation
pattern.
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Far-field pulse shapes
The near-field displacements
caused by an earthquake will
be permanent. M(t) would look
like this
In the far field there is a
displacement pulse. dM/dt is
proportional to the far-field
dynamic response, such as
observed with P-wave arrivals.
Note most seismometers
measure velocity or
acceleration rather than
displacement.
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Seismic moment from seismograms
Define seismic moment
M0 ∝ area under pulse
Rupture length ∝ duration of pulse : l ∝ τ
Time domain - instrument corrected pulse
Amplitude
dM/dt
30
Pulse duration
τ
0
Time
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Seismic moment from seismograms
Define seismic moment
M0 ∝ A0
Rupture length ∝ 1/ frequency: l ∝ 1 / f0
Amplitude
Frequency domain
Frequency
f0
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Earthquake characteristics
M0 = µ A s ∼ µ 2 l s
by dimensional analysis slip is:
s = 2 M0 / µ l2
stress drop with earthquake:
∆σ = 2 M0 / 2π w2 l
Note can get ∆σ from seismogram as rupture length,
l ∝ τ , pulse duration and M0 is proportional to
amplitude
Stress drops in range 1-30 MPa (Kanamori)
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Stress drop in earthquakes
ß Stress drop is
typically a small
fraction of total
stresses in double
couple earthquakes
a Varies according
to crustal
properties, fault
maturity
ß Single-force
earthquakes
(landslides) have
much larger stress
drops
Fault length versus moment for large
earthquakes (Scholz et al. 1986): larger stress
drop produces more seismic moment for a
given rupture area
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Tectonics
Infer seismic and tectonic slip rate
slip rate = (sum of M0) / (µ l.w T)
M0 - summing and earthquake catalogue
l.w - overall tectonic setting
T - duration of earthquake catalogue
Find active slip rate (SAF) is 3 cm/yr
Aseismic (creep) rate is 3 cm/yr
Compare with satellite data, GPS
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Seismic moment - from fault area
Scalar seismic moment (Aki)
M0 = µ A s
l
s
µ
w
Get area of fault plane from
aftershocks
Measure slip in the field
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Regional tectonics from seismotectonics
ß Fault plane solutions
a Type of faulting
a Slip direction
a Stress field orientation
ß Quantitative seismotectonics
a Seismic moment
a Slip
a Stress drop
a Seismic and tectonic slip rate
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Moment tensor inversions
1. NEIC fast moment tensors - from teleseismic P waveforms
http://gldss7.cr.usgs.gov/neis/FM/fast_moment.html
2. Harvard CMT solutions - Centroid momen-tensor (from long
perdiod body waves)
http://www.seismology.harvard.edu/projects/CMT/
3. EMSC rapid source parameter determination - EuropeanMediterranean Seismological Centre - uses P & S waves – results in
24 hours
http://www.emsc-csem.org/
4. NEIC broadband depths and fault-plane solutions
http://neic.usgs.gov/neis/nrg/bb_processing.html
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