Stacey F.D., Davis P.M. Physics of the Earth

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resulting from the main shock (Stein, 1999). This

is given by Eq. (11.36):

  

 



þ S

; (15:4)

where 

is the limiting static friction at which

sliding begins, 

is the normal stress and S

the cohesion. However, this (variation as 1/r

)has

been observed not to fit the observed 1/r variation

in frequency of occurrence with distance r from a

main shock (Felzer and Brodsky, 2006). It appears

possible that aftershocks reflect the distribution

of sub-faults, referred to above, that are caused by

the dynamic stress of the main shock.

Geodetic measurements following large

earthquakes have shown that regional strain

continues in the same directions as in the main

events with characteristic time constants rang-

ing from months to years. Some post-earthquake

strain is expected from the aftershocks, but that

is not a complete explanation. The cumulative

aftershock moment is generally one to two

orders of magnitude less than the geodetically

observed moment released after the event. Two

models are under investigation. One involves

further aseismic slip of the rupture plane itself.

The other involves slip in the lower ductile crust

resulting from the increased stress external to

the fault plane. Recently inversions of post-seismic

slip from two large Californian earthquakes

(Landers, 1992 and Hector Mine, 1998) have

shown that aseismic slip both on the fault and

below it is required to satisfy the data. The after-

slip may continue for years. Thatcher (1983) esti-

mated a time constant of 30 years for the 1906

San Francisco earthquake. The Maxwell relaxa-

tion time (for the model in Fig. 10.4(a)) is given by



¼ =: (15:5)

If the lower crust is relaxing over 30 years then, for

 ¼ 5  10

Pa, the viscosity of the lower crust is

 ¼ 5  10

Pa s. A zone of steady deformation of

order 200 km wide in Southern California is seen

to be overlaid by patches of more concentrated

strain change. Jackson et al. (1997) concluded that

they can be attributed to earthquake after-slip of

historic earthquakes. This complicates any

attempt to use strain observations to infer strain

build-up for future earthquakes.

Static stresses fall off with distance too rap-

idly (as 1/r

) to cause triggering of remote earth-

quakes. Dynamic stresses of seismic waves

diminish less rapidly, particularly for surface

waves which are spreading over a surface and

not a volume so that wave energy falls off as 1/r

and amplitude and stress as 1=

ﬃﬃﬃ

. This is why, at

teleseismic distances, surface waves dwarf the

body waves, for which amplitude decreases as

1/r. Earthquakes may be triggered by surface

waves from distant events, but this is observed

to occur only under special conditions, found in

hydrothermal or volcanic zones that are charac-

terized by high pore pressure. An example of

triggering that appears to have been a prolonged

response to dynamic stressing is seen in a several-

year period of increased seismicity in the

magmatic zone of Long Valley, California, imme-

diately following the Landers 1992 earthquake.

Long Valley is several hundred kilometres from

Landers and static stress changes from the earth-

quake would have been negligible.

15.3 Fault friction and earthquake

nucleation: the quasi-static

regime

The classical description of faulting is that faults

break when the driving shear stress is greater

than the limiting static friction, and once slip

starts the friction drops to a lower value, the

so-called dynamic friction. When the reduction

in friction with slip is greater than the reduction

in the driving stress from the surrounding elastic

medium, the fault is unstable and accelerates,

causing an earthquake. This stick–slip model of

earthquakes was prompted by laboratory obser-

vations on friction between rock surfaces.There

are several possible reasons for the difference

between static and dynamic friction. The pres-

ence of fluids introduces several possibilities and

is probably important, but there are two mecha-

nisms that do not depend explicitly on fluids.

They both represent friction in terms of the

interactions between asperities on adjacent sur-

faces subjected to a normal stress. In the static

situation with prolonged contact, asperities

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become partly welded, and deform, increasing

the contact area and consequent friction. With

prolonged shear they may move sufficiently to

break contact so that new contacts are estab-

lished with smaller welded areas and this

reduces the friction, eventually precipitating an

earthquake. This is the essence of the rate–state

theory (Dieterich, 1979a, b, 1994). Dynamic fric-

tion, which takes over when fault motion is

established, involves different mechanisms.

Asperities brush past one another rapidly with-

out establishing welded contacts and, in doing

so, generate high-frequency elastic waves, mak-

ing the asperity interactions less effective. With

rapid motion fluid lubrication probably also

reduces the friction.

We need a dynamic model of earthquakes into

which the stick–slip (rate–state) mechanism can

be introduced. A convenient analogue is the

spring–block slider system (Fig. 15.2). Progressive

extension of the spring simulates the tectonic

motion that drives the fault movement. The

block in the figure moves intermittently, repre-

senting earthquakes. The size of the block corres-

ponds to the size of the earthquakes with a 10 km

block representing a magnitude 6 event. Let y be

the position of the block, and y

the position of

the driving plate. The force acting on the block,

kðy

 yÞ, where k is the spring constant, is

opposed by the viscous force 

y and the friction

F. The equation of motion is

y ¼ kðy

 yÞ

y  F: (15:6)

This is the equation for decaying oscillatory

motion. There is a transitional phase between

static and dynamic friction in the development

of instability during which we ignore viscosity

and for which there are no inertial or dynamic

effects. This is the quasi-static regime. Now we

examine the friction term, F, in terms of rate–

state theory. Once slipping begins, the change

from static to dynamic friction does not take

place instantaneously. It requires the surfaces

to move a critical distance, D

, to displace inter-

acting asperities with respect to each other suffi-

ciently to erase any memory of the previous

regime. D

is taken as the average size of the

asperities. For this transitional, quasi-static

regime the friction depends on the time asper-

ities are in contact with one another, and there-

fore on velocity. Since plastic failure has less

time to develop than in the static case, we

have a quasi-static friction that decreases with

velocity as



¼ 

 a lnð

Þ; for

y >

; (15:7)

where a is a constant, and 

is the friction when

the velocity equals the reference velocity

. For

the static case, the friction, 

, arising from the

plastic failure (or chemical changes) of indivi-

dual contacts increases logarithmically with time.



¼ 

þ a lnðt=D

Þ: (15:8)

Rabinowicz (1965) unified the concepts of tran-

sient and static friction by suggesting that the

static friction of two surfaces that have been held

together for time t should equal the friction

when they are moving at a constant velocity,

y ¼ D

=t. Then, for those asperities that are in

02y

0F 2y

FI G U R E 15.2 An earthquake is modelled as two blocks

in frictional contact that slide relative to one another. The

lower figure shows the spring–block analogue used to

investigate the effects of quasi-static and dynamic

friction. After a displacement y

the force exerted by the

spring equals the dynamic friction F.

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contact in the quasi-static case, plastic flow occurs

for the same interval of time, t.Equation(15.7)

becomes 

¼ 

þ a lnðt=D

Þ¼

,thatis

equivalent to the static case, Eq. (15.8), a conclu-

sion that has been confirmed experimentally

(Scholz, 1990).

These results can be summarized by the

rate–state friction law,



¼ 

þ A lnð

ÞþB lnðÞ (15:9)

where A and B are constants (of order 0.01 and

0.013). The second term, the A-term, represents

effects that increase friction with velocity in a

manner similar to viscosity. The third term,

the B-term, describes the chemical adhesion

between the asperities that increases with con-

tact time so that y is the effective time of contact.

Dieterich (1994) shows that the state variable is

given by

d ¼







dy: (15:10)

For steady-state creep at an arbitrary creep

rate

y,d=dy ¼ 0 and then Eq. (15.10) gives

 ¼ D

y so that the friction coefficient is



¼ 

þ B lnðD

ÞþðA  BÞlnð

Þ; for

y >

(15:11)

Comparing Eq. (15.7) with Eq. (15.11), we see that

a ¼ A  B and 

¼ 

þ B lnðD

Þ.IfB > A then an

increase in

y causes a decrease in friction. This is

an unstable situation, called velocity weakening,

which is a necessary preliminary to an earth-

quake. If B < A this is a stable condition called

velocity strengthening. In earthquake regions

the boundary between zones of velocity weaken-

ing and velocity strengthening is thought to

occur at the base of the seismogenic zone mark-

ing a change in material properties, controlled

by temperature.

At high velocities, still within the quasi-static

regime, the first term in Eq. (15.10) becomes

small. If it is neglected we have  ¼ expðy=D

and Eq. (15.9) becomes



¼ 

þ A lnð

ÞBy=D

: (15:12)

Using 

(Eq. (15.12)) to substitute for F by F ¼ S

in Eq. (15.6), where S is area and  is normal

stress, the equation of motion for the block

becomes

y ¼ kðy

 yÞ

S  AS lnð

ÞþBSy=D

(15:13)

Since we are considering the quasi-static regime,

we assume the inertial term is zero (m

y ¼ 0) and

Eq. (15.13) reduces to

0 ¼

AS

ðy

 yÞþ



þ lnð

Þ

(15:14)

which is a differential equation for y with the form

0 ¼ lnð

ÞCy þ E; (15:15)

where C ¼



AS



and E ¼





AS



are

constants. Equation (15.15) is solved by rewriting

it as

¼ expðCy  EÞ¼expðEÞexpðCyÞ; then

multiplying by expðCyÞdt to get two integrals,

expðCyÞdy ¼

expðEÞdt, which integrates to

ð1=CÞexpðCyÞ¼

expðEÞt þ const. The con-

stant is found by setting y ¼ 0att ¼ 0; which

gives the solution

y ¼



lnðexp E 

CtÞ: (15:16)

Differentiating Eq. (15.16),

y ¼

exp E 

: (15:17)

The time to instability for C > 0 is obtained by

setting the denominator in Eq. (15.17) to zero,

failure

exp





AS



/ exp 



A



;

(15:18)

where t is the shear stress. The essential conclu-

sion from this equation is that the time to failure

depends exponentially on the ratio of shear to

normal stresses. Alternatively, if C 0 this model

implies stability, which means failure at nega-

tive time, that is in that case the model is recov-

ering from a hypothetical instability in past

time.

We now use the rate–state theory to model

the behaviour of a fault in the ductile region

below the seismic zone. This is the velocity

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strengthening condition, with (B < A) in Eq. (15.11)

(Marone et al., 1991; Hearn, 2003). Consider the

blocks in the aseismic zone which are stressed by

earthquake displacements of the overlying

blocks in the seismogenic zone above them.

They can be modelled by a similar spring–block

system to that represented in Fig. 15.2. After the

upper elastic blocks have slipped in an earth-

quake they exert a force kðy

 yÞ on the lower

blocks that then slip aseismically along the

boundary separating them, with friction given

by Eq. (15.11), that is



¼ 

þðA  BÞlnð

Þ; (15:19)

where 

is the friction coefficient at the critical

velocity

, marking the boundary between sta-

ble and unstable states, (y

 y) is the position of

a lower block relative to the one above, and k is

the spring constant for the force exerted by an

upper block on the lower one. As the lower block

moves, the stress decreases. The equation of

motion is given by the quasi-static version of

Eq. (15.14), which becomes

0 ¼

ðA  BÞS

ðy

 yÞ



ðA  BÞ

 ln

y þ ln

(15:20)

Let the initial velocity, immediately after the

earthquake, be V

. Then Eq. (15.20) integrates to

y ¼

A  B



t þ 1



: (15:21)

Astheseblocksmovetorelievetheirown

stresses they apply stress to the region above

and this is observed as post-seismic strain.

Hearn (2003) concluded that Eq. (15.21) gives a

satisfactory fit to geodetic observatio ns o f post-

seismic displacements following several large

earthquakes (Landers, Izmit and Hector Mine).

This model appears relevant to the effect of the

layer immediately below the seismogenic zone,

but less so for deeper material which, being

hotter, is probably better represented as a vis-

cous medium, as in Eq . (15.5). Long-term relax-

ation after large events probably includes

asthenospheric flow and accounts for l arge-

scale, long-term clustering of earthquakes

(Section 15.7).

Harmonic tremor (low-frequency micro-

earthquakes) originating at the base of the seis-

mogenic layers of the Cascades subduction zone

(Dragert et al., 2001, Rogers and Dragert, 2003)

and the San Andreas fault south of Parkfield,

California (Nadeau and Dolenc, 2005) is expli-

cable as behaviour of material just marginally

in the velocity weakening state. This means a

stick–slip type process moderated by viscosity

of fluids in the fault plane.

The critical distance D

for establishing a new

asperity regime as applied in Eq. (15.8) is not

identifiable in earthquake studies. Values of

microns to tens of microns are found in labora-

tory experiments, but seismic observations have

too low a resolution for its detection. The asper-

ities in the laboratory samples depend on grain

size and on the preparation of slipping surfaces.

Natural failure surfaces are much rougher, so

that D

is expected to be larger.

The maximum displacement of an equi-

dimensional crack of radius l, responding to a

shear stress , is obtained from Eq. (15.3), with

D ¼, x ¼0, c ¼l,as

¼ l=: (15:22)

We can relate this to the seismic moment and

hence micro-earthquake magnitude that such a

displacement would represent. For a stress drop

D in medium of rigidity , the moment corres-

ponding to D

¼ D

 ¼ l

; (15:23)

so that l ¼ M

=ðÞ

1=3

. This imposes a minimum

size on the magnitudes of earthquakes that

can be triggered by the rate–state process. The

Gutenberg–Richter number–magnitude relation-

ship (Eq. (14.39)), is linear down to at least

¼ 0 (Section 14.7), corresponding to M

1:3  10

Nm. For D ¼5 MPa, l ¼6.3 m and there-

fore with ambient shear stress  ¼30 MPa and

¼ 6:3 mm. The Gutenberg–Richter relation-

ship may extend down further but at such

small magnitudes earthquake catalogues are

incomplete. If, instead, D

is as small a 100 mm,

then the cut-off magnitude is M

¼3:6. Lack of

evidence for accelerating strain before earth-

quakes (Johnston and Linde, 2002; Johnston

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et al., 2006) indicates that the sizes of the nucleat-

ing patches are very small.

15.4 The dynamic regime

Once nucleation is complete and the blocks

either side of a fault start to slide, the system

passes from a quasi-static to a dynamic regime,

and inertial effects in Eq. (15.13) become important.

The rate–state mechanism no longer applies and

we consider a simplified version of the spring–

block model. We neglect viscosity and consider

the spring constant, inertia and friction. When

the tension in the stretching spring reaches the

limiting static frictional force on the block,

¼ ky

, and it starts to move, the frictional

force becomes the dynamic friction F

¼ ky

and the block starts to accelerate at a rate

ðF

 F

Þ=m ¼ kðy

 y

Þ=m. At arbitrary y,

y ¼ kðy  y

Þ: (15:24)

This is the equation of simple harmonic motion

about y

, noting that

y is positive when y < y

When the block reaches y

the net force on it

reverses sign and it completes a half cycle about

, to reach an extreme position of 2y

. At this

point it becomes stationary, static friction takes

over and there is no further motion even if there

is overshoot of the equilibrium position. The

solution to Eq. (15.24) is

y ¼ y

ð1  cosð!tÞÞ; (15:25)

where ! ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

k=m

. The difference between static

and dynamic stress determines how far and how

fast a block travels. The rise time of the pulse is

the duration of the motion  ¼ p=!. The maxi-

mum velocity, V ¼ !y

, occurs at y ¼ y

, and the

maximum acceleration, a ¼ !

, occurs at y ¼ 0.

We put in some numbers for illustration, trans-

lating the block model parameters to seismic

parameters. Let the limiting static stress be



¼ ky

¼ 30 MPa and the dynamic stress



¼ kðy

 y

Þ=L

¼ 25 MPa. The final stress is



¼ kðy

 2y

Þ=L

¼ 20 MPa, giving a stress drop

 ¼ð

 

Þ¼2ð

 

Þ¼2ky

¼ 10 MPa.

For shear modulus ,stiffnessisgivenby

k ¼ L,whereL is the block dimension. From

Eq. (15.25) the equilibrium position is y

¼ L

2k ¼ L=2 ¼ 1:5m; the maximum velocity

is V ¼ =

ﬃﬃﬃﬃﬃﬃﬃﬃ

=

¼ "V

,where" is maximum

strain and V

is the shear wave velocity. These

relations are list ed in Table 15.1, which gives

Table 15.1 Spring–block analogue for a 10 km 10 km earthquake (M 6)

Block dimension L 10 km

Shear modulus  3:3 10

Density  310

Shear wave velocity V

ﬃﬃﬃﬃﬃﬃﬃﬃ

=

3.3 km s

1

Stiffness k ¼ L 3:310

1

Limiting static friction 

30 Mpa

Dynamic friction 

25 Mpa

Stress drop  ¼ð

 

Þ 10 Mpa

Maximum strain " ¼ = 310

4

Equilibrium position y

¼ "L=2 1.5 m

Maximum displacement 2y

Rise time  ¼ p=! 9.5 s

Maximum velocity V ¼ !y

¼ "V

=2 0.5 m s

1

Maximum acceleration a ¼ !

¼ "V

=2L 0.16 m s

2

Radiation efficiency (Eq. (15.52)) 

2.5%

Seismic efficiency (Eq. (15.51))





40ð

þ 

 =2Þ

0.5%

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values in the range of those observed for earth-

quakes of this magnitude. This is of course a

simplified model in which the block is moving

over an inertially fixed plane. The simple fric-

tional model predicts that earthquakes should

be periodic and not irregular as is observed.

This is a shortcoming of the assumptions of the

simple frictional model and is the essential rea-

son for considering more complicated frictional

theories such as rate–state.

15.5 Omori’s aftershock law

Omori (1894) found that the rate of occurrence of

aftershocks, R(t), as a function of time, t, after a

mainshock decays as inverse time (Fig. 15.3),

RðtÞ¼A=t: (15:26)

This law was generalized by Utsu (1961) to

RðtÞ¼A

ðt þ cÞ

; (15:27)

with p 1, and c a constant, avoiding the singu-

larity at t ¼0. A law of this type, where t is the

time to the mainshock, has also been used to

represent foreshocks for those events for which

they are clearly identified.

Such power-law clustering has been

described by a number of physical mechanisms

that have in common the observation that fail-

ure is not an instantaneous process, but develops

on a time scale that is non-linearly dependent on

stress. They include fault healing, static friction

increasing with time (the rate–state theory) and

stress corrosion, and it is possible that several

operate simultaneously. The principle can be

FI G U R E 15.3 The decay of aftershock activity after the 1891 Nobi, Japan, magnitude 8 earthquake, fitted to

Eq. (15.27). n(t) is the number of shocks per day, identified as aftershocks, which are seen to have persisted for

80 years. Figure from Utsu (2002).

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illustrated by examining equations that have

been used to describe the stress corrosion mech-

anism. Material subjected to high stress, espe-

cially at the tips of cracks, is vulnerable to

corrosion by fluids because the atomic bonds

are weakened by the stress. Two alternative

expressions have been used to describe the

time to failure of an aftershock fault in terms

of stress corrosion theory:

failure

¼ t



n

expðH=RTÞ (15:28)

and

failure

¼ t

exp½ð þ HÞ=RT; (15:29)

where t

and t

are constants,  is stress, H is

an activation enthalpy and n and  are coeffi-

cients that are fitted to observations (Atkinson,

1982; Meredith and Atkinson, 1983). As we saw

in Section 15.3, the rate–state model gives a

stress dependence with the form of Eq. (15.29),

where temperature is assumed constant, and we

examine the application of this equation.

Equation (15.29) can be written

t ¼ t

expðlÞ: (15:30)

The number of patches with shear stress

between  and  þd, SðÞd, is equated to the

number with failure times between t and t þdt,

RðtÞdt

RðtÞdt ¼SðÞd: (15:31)

Substituting from Eq. (15.30) we obtain

RðtÞ¼SðÞ

d

SðÞ

expðlÞ; (15:32)

RðtÞ¼

SðÞ

;



; (15:33)

where 

is an upper bound, the threshold stress

for plastic failure, caused by slip weakening.

This bound disallows infinite stress, requiring

Sð !1Þ¼0 in Eq. (15.33). Equation (15.33)

gives the 1/t variation of Omori’s law, provided

that SðÞ is a weak function of  compared with

the exponential, which makes R(t) a strong func-

tion of t. Imposition of the stress threshold can

account for the round-off in the Omori law at

short times. Shaw (1993) presented a similar

analysis using Eq. (15.28).

15.6 Stress drop and radiated

energy

One of the fundamental problems of earthquake

dynamics is the fraction of the released elastic

energy that goes into seismic waves, and how

much is dissipated in the fault as heat and the

breakage of new surfaces. Consider a fault of

area S across which slip b occurs during an earth-

quake. If the stress across S is initially 

and

decreases during the earthquake to 

, then the

total energy release by the earthquake is

total

ð

þ 

ÞSb: (15:34)

We distinguish two components of E

total

,the

energy radiated as elastic waves, E

,andthebal-

ance that is dissipated in the fault zone. The break-

age of rocks in the vicinity of the fault and the

production of powdered fault gouge is generally

recognized, and conventional frictional heating,

even to the point of forming a thin layer of melt,

has been inferred from exhumed faults. However,

seismic wave observations cannot distinguish

heating from rock breakage. We represent the

total dissipation in the fault zone as D.Then

total

¼ E

þ D: (15:35)

The essential difficulty in apportioning the

energy partitioning is that seismic observations

do not give values of 

or 

but, as we show

below, give their difference  ¼ 

 

, which

is referred to as the stress drop. E

is found as

elastic wave energy from seismograms. There is

no direct observation of E

, and E

total

can be esti-

mated, where observed displacements can be

fitted to dislocation models, only with assump-

tions about 

and 

There are two distinct quantities, referred to

as efficiencies, that are ratios of energies, illus-

trated in Fig 15.4. The seismic efficiency, 

,is

the ratio of radiated to total energy



total

: (15:36)

This can be determined if 

and 

are estimated

independently of seismic wave observations. We

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consider this further below to compare E

total

with the tectonic energy budget. We now con-

sider the second efficiency, 

, termed radiation

efficiency, that can be estimated from seismic

waves without additional information. We iden-

tify with the stress drop a notional energy, E



that can be estimated from the moment and

stress drop if a characteristic dimension of an

event is determined from either its spectrum,

geodetically observed displacements, or, for

large events, the distribution of aftershocks.

Then the radiation efficiency is





: (15:37)

Values of about 50% are typical, but have a wide

range (Kanamori and Brodsky, 2004).

Recalling that seismic moment (Eq. (14.6)) is

¼ Sb: (15:38)

the elastic energy, E



, identified with a stress

drop  is



2

: (15:39)

is estimated from the zero-frequency asymp-

tote of the spectrum of seismic waves

(Section 14.5);  can be calculated from the

moment if we can estimate a characteristic

fault dimension, l, since for an equi-dimensional

fault

 ¼ M

: (15:40)

To obtain l we use the corner frequency, !

(Section 14.3), for which there is a direct relation-

ship with the duration of the rupture process, so

that we can write

¼ V

=l; (15:41)

where V

is the speed of propagation of rupture,

and the numerical constant   3:5 is somewhat

model-dependent. Observations give

 0:9V

; (15:42)

that is, a rupture spreads at about 90% of the

speed of shear waves in the rupturing medium.

Equations (15.41) and (15.42) can be used to find l

from measurements of moment and corner

(a)

(b)

(c)

(σ

)/2

Stress

(σ

)/2

Stress

total

Stress

Δσ

Displacement

FI G U R E 15.4 Energy terms used in defining seismic

and radiation efficiencies. Stress during an earthquake

drops from 

to 

, taken here as linear with

displacement, but it can have a more complicated

variation. (a) Total energy. (b) Radiated energy, E

, and

energy dissipated as frictional heat and rock breakage,

D. (c) Stress drop energy, E



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frequency. There are numerous determinations

of stress drop, and for earthquakes of magni-

tudes exceeding 3 or 4 the universal conclusion

is that  is centred on the range 1 to 10 MPa

(10 to 100 bar), as in Fig. 15.5(a). Thus, assuming

the stress drop to be approximately a universal

constant (at least for earthquakes with M

> 3,

= ¼ "  10

4

), the moment (or magnitude) is

determined by the area of the fault that breaks,

 S

3=2

: (15:43)

The observation that M

/ S

3=2

for moderate to

large events (Henry and Das, 2001; Kanamori and

Brodsky, 2004) has provided the strongest evi-

dence that stress drops have a limited range.

Another term, in common usage, that can be

estimated from observations is apparent stress,



, defined as



E

; (15:44)

which is generally found to be lower than stress

drop, as can be seen by comparing Figs. 15.5(a)

and (b). Combining Eqs. (15.37), (15.39) and (15.42)

we see that the radiation efficiency is



2



: (15:45)

Values of 

range from a few per cent to 100%

(Fig. 15.6), with the higher values corresponding

to higher rupture velocities. For small earth-

quakes, constant 

¼ 0:66 appears to be a better

approximation than constant stress drop (Ide

et al., 2003).

Seismic radiation has near-field terms, which

decay rapidly with distance from an exciting

earthquake and include static terms. They are

important to the strong ground motion near to

an earthquake, but for the present discussion we

are more interested in the far-field radiated P or S

energy from a dislocation (Aki and Richards,

2002), which depends on the acceleration of the

moment as

15pV

; (15:46)

10pV

: (15:47)

Since V

 1, the radiated energy is dominated

by the S-waves. A very slow earthquake with

 0

generates negligible radiatedenergy(e.g.,Fig.15.6).

Using the block model from Section 15.4, we

can evaluate Eq. (15.47) using the relations in

(b)

(a)

Seismic Energy (J)

Source Dimension (m)

Constant stress drop (MPa)

Constant Apparent Stress (MPa)

Cajon Pass Borehole

Various Combined Studies

100

0.1

0.01

Seismic Moment (Nm)

TERRAscope (Kanamori et al. 1993)

Corrected for Q

= Q

= 1000

FI G U R E 15.5 (a) Seismic moment versus source

dimension from various studies compiled by

Abercrombie and Rice (2005), including small events

detected by a low-noise seismic station at the base of

a borehole in Southern California (Cajon Pass).

Stress drops calculated from M

¼ l

(Eq. (15.40)),

plotted as dashed lines, are centred on the range 1–10

MPa. (b) Radiated energy versus moment estimates by

Abercrombie and Rice (2005). Corresponding apparent

stresses calculated from E

(Eq. (15.44)) are plotted

as dashed lines. Disregarding the smallest shocks, the

data are centred on the range 0.1–1 MPa. Stress drop

is about three times the apparent stress. Thus, by

Eq. (15.45), 

¼ 2

= and radiation efficiency is 0.6.

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Table 15.1. An earthquake is approximated by

two blocks moving relative to one another

(Fig. 15.2), with total moment given by

ðtÞ¼

ð1  cos !tÞ; for 0

; (15:48)

where, taking into account the blocks on either

side of the fault, M

¼ 4y

. Using expressions

from Table 15.1, Eq. (15.47) becomes

4  10pV

cos

ð!tÞdt ¼

80V

: (15:49)

The total change in elastic energy is

total



þ 

2

; (15:50)

so that seismic efficiency, 

¼ E

total



80V

2

ð

þ 

l

80V

2

ð

þ 



ð

þ 

(15:51)

For the example chosen in Table 15.1, 

¼ 0:5%.

Because we cannot normally measure 

, seismic

efficiencies are uncertain. In cases where mea-

surements have been made, efficiencies less than

6% have been inferred (McGarr, 1999). Radiation

efficiency, Eq. (15.45), is obtained by replacing

ð

þ 

Þ in Eq. 15.51) by  (Eq. (15.39)):



: (15:52)

FI G U R E 15.6 Radiation efficiency and rupture velocity. While for small earthquakes radiation efficiency is more or

less constant, 0.6, large earthquakes show the range given here. Symbols I, II, III refer to different crack models

to describe earthquakes. (After Venkataraman and Kanamori, 2004.) Note: c

 V

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