Mechanics of Earthquakes

Mechanics of earthquakes

Friction of faults: often unstable & slip occurs rapidly as a rupture dynamically propagates over the fault surface

Sudden motions: generate seismic waves and this is the mechanism of the most common & important type of earthquake

Seismicity: short-timescale phenomenon of brittle tectonics.

Theoretical background

Dynamic energy balance: Earthquake may be considered to be a dynamically running shear crack

e s k fU W U U U U

W: work done by external forces;Ue: change in internal strain energy:Us: surface energy involved in creation of the crack.Uk: kinetic energyUf: work done against friction

Equilibrium energy balance

e s k fU W U U U U

Domains of integration for thedynamic energy balance.

12k i iV

U u u dVt

0

2sU dSt

0

ij i jS

W u n dS

0

f ij i jU u n dSt

0

12e ij ijV V

U dVt

Gc:The critical strain energy release rate

Griffith (1893-1963)the energy-balance approach

21

U

Radiated seismic energy ES

s k e f sE U U U U change in internal strain energy

Seismic energy

• Assume: during sliding the friction stress has some constant value determined by dynamic friction, f, and so define a dynamic stress drop d=(1-f)

Seismic efficiency

McGarr (1999) shows that

Simple slip weakening model for stress breakdown at the crack tip

102 ( )c y f d

21

2( )

cc

f

L

021

( )( )

y fc

f

L d

Stress history during rupture at points that exhibit velocity weakening

Stress history during rupture at points that exhibit velocity weakening

Earthquake phenomenology

Quantification of earthquakesMagnitude: logarithmic scale based on the

amplitude of a specified seismic wave measured at a particular frequency, suitably corrected for distance and instrument response.

Many types of magnitude (mL, mb, Ms, etc.) that are useful under different conditions.

Seismic moment

ui : mean slip vector averaged over the fault area A

Nj : unit normal : shear modulus.M0ij is a second rank tensor with a scalar

valueM0=uA

A scaling law forseismic spectra

The period (20 s) at which the surface wave magnitude Ms is traditionally measured.

As long as 1/f0 (f0: corner frequency) is shorter than 20 s, Ms ~log M0, so that a magnitude-moment relation may be defined empirically

Aki, 1967

A scaling law forseismic spectra

For earthquakes large enough that 1/f0 is longer than 20 s, Equation (4.26) no longer holds and Ms seriously underestimates moment.

At this point the Ms scale is said to saturate.

Aki, 1967

Gutenberg–Richter empirical relationship

02sE M

log 1.5 4.8s sE M

Earthquake stress drops are approximately constant and ~3 MPa

Diagram illustrating the definitions of small and large earthquakes, showing hypocenter (H), epicenter (E), moment centroid (MC), and the dimensions of rupture (a, L, and W).

Earthquake scaling relations

Average static stress drop

C: constant that depends on the geometry of the rupture

u: is mean slip a characteristic rupture dimension

uC

Relationship between M0 and source radius. Dashed lines are of constant stress drop. (From Hanks, 1977.)

These data illustrate scaling regime 1 in Table 4.1.

a3 dependence30

167

M a

Mean slip versus rupture length for large earthquakes

There is a broad cross-over between these two regimes, which are listed as regimes 2 and 3 in Table 4.1.

Frequency–size distribution for shallow

earthquakes

A change in slope from 1.0 to 1.5 is observed at M7.5

Change from small to large earthquakes for subductionzone events, which dominate this catalog.

0 0( ) , 3BN M aM b b

Gutenberg–Richter or Ishimoto–Aida relation

It is about 11⁄2 orders of magnitude larger than the extrapolation of the small earthquakes would indicate. The rolloff at M0

Cumulative size distribution of earthquakes

Single fault or plate boundary segment

A large region containing many faults or plateboundary segments

Observations of earthquakes

M > 4

CRF: Camp Rock faultEF: Emerson faultJVF: Johnson Valley fault NFFZ: North Frontal fault zone BMF: Burnt Mountain fault EPF: Eureka Peak faultHVF: Homestead Valley fault

1992/04/23

Why off-fault aftershocks?

Radar interferogram of the coseismicdisplacements of the 1992 Landers

earthquake

Gilles Peltzer, JPL

差分合成孔徑雷達干涉觀測(D-InSAR: Differential Interferometric SAR)

(http://www.futura-sciences.com)

(Erwan , 2004)

Image of an Earthquake1992 Landers M=7.3 earthquakeat Mojave Desert of California

Massonnet et al., 1993, Nature

Repeat pass inteferometry

Flight direction

Looking direction

Radar image: Foreshortening (前坡縮短) and layover effect (疊置效應)

Courtesy of Pei-Ling Wang

SRTM: How it was done?

http://www2.jpl.nasa.gov/srtm/

SRTM was a fixed-baseline (固定基線) interferometry mission. Reflected radar signals collected at 2 antennas, providing 2

sets of radar signals separated by a distance.

Calculate altitude

( ) cosz y h

2 2 2

2 2

( ) 2 cos(90 )2 sin( )

B BB B

2

2 2( )2( ) cos

2 sin( )

Bz y h

B

is the fractional phase (value 0-2 radians), λ is wavelength

(1)

(2)

(3)

(4)

B: baseline

Θ: looking angle

Height: 800 km

Velocity: 7 km/s

The geometry of InSAR

D-InSAR: How it works?

1 12 2R

2 22 2R

1 2 1 22 42 2

4 sin( )

R R

B

R1

Hsat

Hp

SAR interferometry during Chi-Chi earthquake

A

A’

A

A’

990401_000316 990715_000106

2.8cm

0

Profile AA’ of interferograms A and B in SRD. Notice that the profile is slightly increasing when passing through Dadushan.

A

A’

What is PS-InSAR (持久性散射體合成孔徑雷達干涉) ?

(Hooper et al., 2004) (Funning et al., 2006)

PS: Persistent scatterers (持久性散射體)

Rupture model of the 1992 Landers earthquake

N S

3.6 km/s

MW 7.0

Loma Prieta EQ: oblique slip

1989/10/18

Aftershocks tended not to occur in the areas of high slip but were concentrated in areas of low slip or high slip gradients.

NW SE

Thick curves indicate the slip distribution (contour interval 0.4 m) for an inversion of waveform and geodetic data

Northridge earthquakes

Aftershocks and faulting in the Borah Peak, Idaho, earthquake of 1983

Aftershocks & faulting in the Borah Peak, Idaho, earthquake of 1983

Schematic diagram illustrating the various types of earthquake sequences

mainshock (MS) with foreshocks and aftershocks

mainshock–aftershocksequence

swarm

Omori law

• Named for his observation of it following the 1891 Nobi earthquake

• n(t): number of aftershocks in an interval at time t after the mainshock

• K and p are constants and c is a positive number near zero

( )( ) p

Kn tc t

Aftershocks and slip in the 1984 Morgan Hill (California) earthquake, astrike-slip earthquake on the Hayward fault

Compound earthquakes on noncontiguous faults have occurred repeatedly in the

South Iceland seismic zone

Cluster of earthquakesEvents a, b, c, d, and e (M6.6, 6.4, 6.8, 5.8, 5.5) occurred from July to September: rupture on the Rainbow Mountain fault (RMF).

Fairview Peak eq. (f, M7.1) occurred at 1107 on December 16 and ruptured bilaterally along the Fairview fault (FF).

The Dixie Valley eq. (g, M6.8) occurred at 1111 on December 16 and ruptured the Dixie Valley fault (DVF).

Rupturing on the North Anatolian fault in a sequence of six large earthquakes that

progressed from east to west over the period 1939–67

Coulomb Failure Most successes in modeling earthquake interactions

are based on the widely used Coulomb failure criterion (Jaeger and Cook, 1969; Scholz, 1990).

Failure initiates & spreads on the fault plane when the Coulomb Failure Function exceeds a specific value

)( PCFF

Shear stress on the failure plane (left: +)

Normal stress on the failure plane (extension:+)Coefficient

of friction

Pore fluid pressure

Coulomb Failure If the failure plane is orientated at β to the σ1 axis we

can express, under plane-stress conditions, the normal stress applied to the plane in terms of the principle stresses

Mechanics of earthquake interactions

• if the CFS>0, slip potential is enhanced and if CFS

Coulomb stress change for right-lateral faults

Coulomb stress change induced by the 1979 Homestead valley earthquake

White line is the mainshock rupture and white symbols are aftershocks.

Changes of CFS of as little as 1 bar separate regions of enhanced and decreased seismicity.

King & Cocco, 2000

King & Cocco, 2000

Coulomb stress produced by the Landers earthquake & the earthquakes that just preceeded it

Case Study: North Anatolian fault Westward-migrating earthquakes ruptured 725 km of the

North Anatolian fault during 1939-1944

How seismic stress transfer alters the probability of subsequent earthquakes?

What conditions promote progressive failure?

Stein, et al., 1997

Fault Model

Stein, et al., 1997

Stein, et al., 1997

Summary

Stress shadows produced by large earthquakes produce seismic quiescences which last until they are overcome by tectonic loading

It appears that static stress changes as low as 0.1 bar can trigger seismicity. This is just a small fraction of earthquake stress drops

These very low triggering levels require that the triggered events be very close to their rupture point

The commonness of this phenomenon suggests that there are some fault segments everywhere that are very near their critical point, a conclusion that is also reached from observations of reservoir induced seismicity

Is there a threshold for the triggering stress? Results to date have been negative (Ziv & Rubin, 2000)

Mechanisms for the time delaySimple Coulomb friction offers no explanation for the time delays that are observed, which range from a few tens of seconds to decades.

In the case of time delays of the order of years to decades, one may appeal to the simple case of Coulomb loading from prior earthquakes being augmented by continued tectonic loading

The more interesting cases are those with time delays short enough that tectonic loading may be ignored

Seismic waves generated by earthquakes do not trigger widespread seismicity even though the dynamic Coulomb stress loads in seismic waves are often several times higher than the static ones that later did trigger earthquakes

These observations suggest that earthquakes are relatively insensitive to transient loading (e.g., earth tides?)

Mechanisms for the time delay Rate–state-friction laws: The direct effect states that a fault subjected to

a suddenly imposed shear will be strengthened, and that such strengthening will be erased only by finite slip.

The time required for this subsequent weakening is determined by the evolution law and depends on the magnitude and duration of the load, the position of the fault within its seismic cycle, as well as the friction parameters (Gomberg et al., 1998)

‘clock advance’ of the seismic cycle: typically to produce the same clock advance as a static shear stress increase, a 10 s transient (similar to a seismic wave packet) would have to be 1000 times larger and a 104 s transient (comparable to an earth tide) would have to be 10–100 times larger

Directivity effect (the amplification of seismic wave radiation in the direction of rupture propagation): triggering by transients played an important role, where the transients were of order a hundred times larger than the static stress changes

Mechanisms for the time delay

Overall time decay is faster than the Omori law for short times (< 200 days) and slower than Omori for long times.

rate–state-friction effect the decay should be Omori

Time evolution of off-fault aftershocks of the Landers earthquake

Decay of the frequency of events with time

Mechanisms for the time delay

The fast decay at short times is dominated by events for which CFS is primarily due to and the slow time decay for long times is dominated by events for which n is the main contributor

Unclamping case: probability of triggering increases with time, hence the net decay rate becomes slower than Omori.

Even among the dominated events, those which are clamped have faster decay rates than those that are unclamped, because for the clamping cases, the probability decreases with time

Compoundearthquake sequence

12 h later

November, 1987 near Superstition Hills, CA

Stress interactionis almost pure unclamping: the reduction of normal stress was about 30 bars