Source parameters II
• Stress drop determination
• Energy balance
• Seismic energy and seismic efficiency
• The heat flow paradox
• Apparent stress drop
Source parameters II: use of empirical Green function for sourcetime function retrieval
• Recall that:
seismogram = s Ä q Ä i
• A way to retrieve s is by
deconvolving an empirical q Ä i .
• This is done by use of a seismic
recording from a small earthquake
located near a larger event of
interest.
• If the the source depth and focal
mechanism of the two events are
identical, the earth response to each
station will be identical.
Note the directivity effect!
Source parameters II: stress drop
M 0 = GuA = h1GuL2
Dt = Ge = h2G
u
L
u 3
L = hDt L3
L
hM
hM 0
Dt = 3 0 =
3
L
(VRTR )
M 0 = h1G
 Seismic moment, stress drop and rupture dimensions are
related.
Source parameters II: stress drop
How can the stress drop be estimated?
Rupture dimensions may sometimes be estimated from:
•
the aftershock distribution
•
geodetic observations.
•
the corner frequency of the amplitude spectra.
Source parameters II: stress drop
Use of aftershock distribution for stress drop determination
Figures from Lay and Wallace
Source parameters II: stress drop
Use of corner frequency for stress drop determination
Even in cases where the spectra is
of excellent quality, the precision of
the corner frequency is modest.
Source parameters II: stress drop
The other problem is that the use
of amplitude spectra for stress
drop determination is extremely
model-dependent.
Note that small differences in time
function duration correspond to
large differences in stress drop,
even for assumed rupture velocity
and fault geometry.
Source parameters II: stress drop
How uncertain is the stress drop?
The uncertainty is related to the uncertainties of each parameter in this
expression:
cM 0
cM 0
Dt = 3 =
3
L
(VRTR )
A common approach for the standard deviation is to use the propagation of error
relation:
2
2
2
2
s
2
Dt
æ ¶Dt ö
æ ¶Dt ö
æ
ö
æ ¶Dt ö
2
2
2 ¶Dt
=s ç
÷ + s VR ç
÷ + s TR ç
÷
÷ + s M0 ç
è ¶c ø
è ¶M 0 ø
è ¶VR ø
è ¶TR ø
2
c
It seems that the uncertainty of a stress drop is often a factor of 2-3.
Source parameters II: stress drop
Despite this uncertainty, the result
that earthquake stress drops are
typically 10-100 bars over a very
wide range of seismic moments is
convincing.
Source parameters II: stress drop
The near constancy of the stress drop
implies that the ratio of average slip to
fault length, i.e. the strain release is
constant.
Taking a stress drop of 50 bars, shear
modulus of 50x1011 dyn/cm2 yields:
u Dt
e= =
=10-4
L G
Source parameters II: seismic energy
• The physical size of an earthquake is often described in terms of its seismic
moment.
• An alternative measure of earthquake size is the energy release.
• To calculate the energy release, we consider the kinetic and potential
energies of a material particle during the passage of the seismic waves.
•
Consider a monochromatic source of seismic energy. In that case, the
ground displacement at the station is given by:
æ 2p t ö
x = Acos ç
÷
è T ø
where A is the amplitude of a wave whose period is T.
Source parameters II: seismic energy
• The ground velocity is then:
2p A æ 2 p t ö
v=sin ç
÷
è T ø
T
• The kinetic energy is just:
1
EK = r v 2
2
• The average of this over one cycle gives the kinetic energy density:
1r
v=
2T
T
òv
2
dt =
0
2
r æ 2p A ö
2 æ 2p t ö
sin
ç
÷ ò
ç
÷dt =
è T ø
2T è T ø 0
A2
rp 2
T
2
T
Source parameters II: seismic energy
• And since the kinetic and the potential energies are equal, we can write:
æ Aö
ES = EK + EP = 2 rp 2 ç ÷
èT ø
2
• To integrate over a spherical wavefront, the particle motion should be
corrected for amplitude attenuation due to geometrical spreading:
æ Aö
ES = F ( r, r, c) ç ÷
èT ø
2
• This can be recast as:
æ Aö
log ES = log F ( r, r, c) + 2 log ç ÷
èT ø
• Thus, if F is know, the seismic energy may be related to magnitude.
Source parameters II: seismic energy
• Gutenberg-Richter obtained the following empirical relation between the
seismic energy and the surface wave magnitude:
log ES =11.8+1.5M S
• This result highlights the tremendous range of earthquake size!
Source parameters II: a long-standing question
• Are faults weaker or stronger than the surrounding crust?
• Do earthquakes
release most, or just a
small fraction of the
strain energy that is
stored in the crust?
Source parameters II: Griffith criteria
The static frictionless case:
U =UM +US
• UM is the mechanical energy.
• US is the surface energy.
dU dc < 0
crack extends if:
crack at equilibrium if dU dc = 0
dU dc > 0
crack heals if:
Source parameters II: dynamic shear crack
Dynamic shear crack:
U =UM +US +UK +UF
Here, in addition to UM and US:
• UK is the kinetic energy.
• UF is the work done against friction.
During an earthquake, the partition of energy (after less before) is
as follows:
ES = DUK = -DUM - DUS - DUF ,
where ES is the radiated seismic energy.
Source parameters II: dynamic shear crack
ES = -DUM - DUS - DUF .
Question: what are the signs of UM, US and UF?
DU M < 0
DU S > 0
DU F >0
Let us now write expressions for UM , US and UF .
Source parameters II: elastic strain energy
To get a physical sense of what UM is, it is useful to consider the
spring-slider analog.
The reduction in the elastic strain
energy stored in the spring during
a slip episode is just the area
under the force versus slip curve.
For the spring-slider system, UE is equal to:
F1 + F2
DU M = Du.
2
Source parameters II: elastic strain energy
Similarly, for a crack embedded within an elastic medium, UM is
equal to:
DU M = -
s1 + s 2
2
DuA.
where 1 and 2 are initial and final stresses, respectively, and the
minus sign indicates a decrease in elastic strain energy.
Source parameters II: frictional dissipation and surface energy
The frictional dissipation:
where F is the friction, V is sliding speed, and t is time. Frictional
work is converted mainly to heat.
The surface energy:
where  is the energy per unit area required to break the atomic
bonds, and A is the rapture dimensions. Experimental studies
show that  is very small, and thus its contribution to the energy
budget may be neglected (but not everyone agrees with this
argument).
Source parameters II: the simplest model
Consider the simplest model, in
which the friction drops
instantaneously from 1 to 2.
In such case: F=2, and we
get:
Source parameters II: seismic efficiency
We define seismic efficiency, , as the ratio between the seismic
energy and the negative of the elastic strain energy change, often
referred to as the faulting energy.
ES
h=
,
-DU M
which leads to:
with  being the static stress drop. While the stress drop may be
determined from seismic data, absolute stresses may not.
Source parameters II: seismic efficiency
The static stress drop is equal to:
where G is the shear modulus, C is a geometrical constant, and
the tilded L is the rupture characteristic length.
The characteristic rupture
length scale is different for
small and large earthquakes.
For small earthquakes, L˜ = r and C = 7p 16. Combining this with the
expression for seismic moment we get:
Both M and r may be inferred from seismic data.
Source parameters II: seismic efficiency
Stress drops vary between 0.1
and 10 MPa over a range of
seismic moments between 1018
and 1027 dyn cm.
Figure from: Hanks, 1977
Source parameters II: seismic efficiency
constraints on absolute stresses: In a hydrostatic state of stress,
the friction stress increases with depth according to:
s F (z) = m(rc - rw )gz,
where  is the coefficient of friction, g is the acceleration of gravity,
Friction
measured
at maximum
and c and w are the
densities
of crustal
rocks and water,
stress
respectively.
Laboratory experiments show:
m » 0.6.
Byerlee, 1978
Source parameters II: seismic efficiency
Using:
, the coefficient of friction = 0.6
c, rock density = 2600 Kg m-3
w, water density = 1000 Kg m-3
g, the acceleration of gravity = 9.8 m s-2
D, the depth of the seismogenic zone, say 12x103 m
We get an average friction of:
m(rc - rw )gD
2
= 56MPa,
and the inferred seismic efficiency is:
Source parameters II: seismic efficiency
So, the radiated energy makes only a small fraction of the energy
that is available for faulting.
Based on this conclusion a strong heat-flow anomaly is expected
at the surface right above seismic faults.
Source parameters II: the heat flow paradox
At least in the case of the San-Andreas fault in California, the
expected heat anomaly is not observed.
Heat flow as a function of
distance from the San
Andreas fault in the Mojave
segment. Theoretical
anomaly is for a slip velocity
of 25 mm/yr and average
friction of 50 MPa
(Lachenbruch and Sass,
1988).
The disagreement between the expected and observed heat-flow
profiles is often referred to as the HEAT FLOW PARADOX.
Source parameters II: the heat flow paradox
A section parallel to the SAF plane:
Figure from: Scholz, 1990
Source parameters II:
The basis for the heat flow paradox is a
simplified model, according to which the slip
is occurring along a single plane. In practice,
however, the deformation is distributed
among several faults.
Source parameters II:
Other conceptual models:
constant friction
slip weakening
quasi-static
• The simple model.
• The slip-weakening model. Significant amount of energy is
dissipated in the process of fracturing the contact surface. In the
literature this energy is interchangeably referred to as the breakup energy, fracture energy or surface energy.
• A silent (or slow) earthquake - no energy is radiated.
Source parameters II:
In reality, things are probably more complex than that.
We now know that the distribution of slip and stresses is highly
heterogeneous, and that the source time function is complex.
Source parameters II: radiated energy versus seismic moment
and the apparent stress drop
Radiated energy and seismic moment of a large number of
earthquakes have been independently estimated. It is interesting
to examine the radiated energy and seismic moment ratio.
Figure from: Kanamori, Annu. Rev. Earth Planet. Sci., 1994
Source parameters II: radiated energy versus seismic moment
and the apparent stress drop
Remarkably, the ratio of
radiated energy to seismic
moment is fairly constant
over a wide range of
earthquake magnitudes.
Figure from: Figure from Kanamori and
Brodsky, Rep. Prog. Phys., 2004
Source parameters II: radiated energy versus seismic moment
and the apparent stress drop
What is the physical interpretation of the ratio ES to M0? Recall
that the seismic moment is:
and the radiated energy for constant friction (i.e., F = 2):
Thus, ES/M0 multiplied by the shear modulus, G, is simply:
E S s1 - s 2
G
=
.
M0
2
This is often referred to as the 'apparent stress drop’(apparent,
because it is based on a highly simplified model)