# Does the Scaling of Strain Energy

Τεχνίτη Νοημοσύνη και Ρομποτική

1 Δεκ 2013 (πριν από 4 χρόνια και 5 μήνες)

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Does the Scaling of Strain Energy
Release with Event Size Control the
Temporal Evolution of Seismicity?

Steven C. Jaum
é

Department of Geology And

Environmental Geosciences

College of Charleston

Charleston, South Carolina, USA

Overview of Statistical Stress/Strain Release Model

(Vere
-
Jones, 1978; Jaume
´
&Bebbington, 2000)

1.
Stress/strain history:

X(t) = X(0) +

t
-

(t)
t = time;

= stressing rate;

(t) = stress release of
earthquake at time t

2.
Earthquake magnitude distribution:

Prob (

> y) = (1 + y)
-

e
-
y/U

= linear slope at small magnitudes; U = magnitude
threshold for rolloff

3.
Rate of earthquakes:

(t) = exp[

+

X(t)]

and

are constants

Overview of Statistical Stress/Strain Release Model

(continued)

4.
Location of magnitude rolloff:

U = A + Bexp[X(t)]
A and B are constants

5.
Stress/strain release as a function of magnitude:

a.

i

= 10
0.75Mi + 2.4

“Benioff strain” stress drops
b.

i

= 10
1.5Mi + 9.0

Seismic moment stress drops

6.
Simulations:

A series of simulations were run where
either the event rate (

), magnitude distribution (U) or
both were a function of regional stress (X). All
resulting catalogs were then tested to see if they
contained accelerating moment release sequences.

Example of an accelerating
moment release sequence from
a stress release model
simulation. Solid circles are
simulated earthquakes and the
line is a fit to the equation
e
(t) = A

B(t
f

t)
m

where
e

is
the cumulative Benioff strain, t
is time, and t
f

is the time of the
mainshock (Bufe & Varnes,
1993).

Magnitude
-
frequency
distributions resulting
from stress release model
simulations where both the
rate and size of
earthquakes are a function
of regional stress. Note
that the relative scaling of
event size vs. number is
different even though
starting distribution is the
same.

Summary of Stress Release Models and
Accelerating Moment Release (AMR)

Model Type

% with AMR

“Benioff strain” stress drops

0
-
30%

Seismic moment stress drops

78
-
100%

Randomized catalogs

38
-
69%

Tentative Conclusion:

Scaling of stress/strain
release with magnitude influences the
dynamics of earthquake systems.

Comparison of stress
history (red line) to
earthquake history (vertical
black lines) for a model
where both event rate and
size are controlled by X(t)
and stress drop scaling
follows 5a. Compare with
cellular automaton model
which shows SOC behavior
(next slide).

Comparison of stress
history (red line) to
earthquake history
(vertical black lines) in
a cellular automaton
model that shows SOC
behavior.

Comparison of stress
history (red line) to
earthquake history (vertical
black lines) for a model
where both event rate and
size are controlled by X(t)
and stress drop scaling
follows 5b. Compare with
cellular automaton model
which shows Intermittent
Critical behavior (next
slide).

Comparison of stress
history (red line) to
earthquake history
(vertical black lines)in
a cellular automaton
model which shows
intermittent critical
behavior.

How Does Strain Energy Release Scale
With Earthquake Magnitude?

From Benioff (1951):

Strain energy release (E
st
)
scales as the square root of radiated seismic
energy(
E
s
)
-

E
st

E
s

0.5

Common Present Assumption:

Strain energy release
(E
st
) scales with seismic moment release (M
0
)
-

E
st

M
0

Strain energy (potential energy stored in elastically
deformed crust), W =
½

V

ij

e
ij

dV

ij

= stress tensor

e
ij

= strain tensor

Estimating the Scaling of Strain Energy
Release Scale With Earthquake Magnitude

Calculated W for a pre
-
strained (i.e., locked fault model)
volume (W
int
) and then for the same volume after an
earthquake event (W
fin
). The strain energy release by
the event (E
st
) should then be:

E
st

= W
int

W
fin

In practice the strain energy release by a particular event
depends upon the assumed loading model.

Sketch of Strike
-
Slip Fault Model

Model strike
-
slip fault buried in an elastic halfspace. Strain
energy is calculated (Okada, 1992) at the center of 1 km
3

cubes to a depth of 20 km; horizontal grid is scaled to be 5
times rupture length.

Both the “virtual slip”
loading

model and the
earthquake slip models
consist of 10 over
-
lapping dislocations
that taper to zero at the
edges. For M
w

< 6.5
earthquakes the slip
model tapers in all
directions; for M
w

6.5
earthquakes they mimic
the loading in depth
and taper in the
horizontal dimension.

Strain energy released as
a function of M
w

in the
strike
-
slip faulting model
described above. The
scaling of seismic
moment with magnitude
is shown for comparison.

Release of strain energy
stored in shear component
parallel to assumed fault
motion in model. Note
change in scale from
previous figure. Again,
seismic moment scaling
shown for comparison.

Energy dissipation as a
function of magnitude
from a pair of stress
-
release models. In those
models where energy
release is scaled as
seismic moment large
events dominate the
energy dissipation,
whereas small events
dominate when energy
release scales as Benoiff
strain.

Energy dissipation versus
event size (Energy
“Release”) in two cellular
automaton models.
Although seismicity
evolves very differently in
the two models, in both
cases most of the energy
dissipation occurs in the
largest events.

Conclusions?

Results from modified stress
-
release models
suggest that scaling of strain energy release
with magnitude helps control the temporal
evolution of seismicity.

An elastic dislocation model of a strike
-
slip
fault suggests that strain energy release may
scale as Log
10

E
st

2.1
-
2.2 M
w
.

However
…a check back with cellular
automaton models suggest that strain energy
scaling may not be the only or even most
important factor.