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

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common value, 2/3 (Table 14.1). Equation (14.44)

is empirical and the physical conditions that

determine M

are not fully understood, although

geometrical effects, such as the depth of the

seismogenic zone, appear to be controlling.

Subduction zones have higher values than

other regions, especially mid-ocean ridges.

The total moment of the Kagan distribution is

total

¼

M



expðM

þ=M

1

expðM

: ð14:45Þ

Noting that the gamma function is given by

GðzÞ¼

z1

t

dt, and that G(z þ1) ¼zG(z),

Eq. (14.45) integrates to

total

¼ Gð1  ÞM

1

: (14:46)

For  ¼2/3, G(1 ) ¼2.68, M

¼1.2 10

with  10

, the average annual global

moment release is M

0total

0.6 10

Nm, sub-

stantially less than the moments of the Chile

1960 or Sumatra 2004 earthquakes, but these

are rare events.

A particular interest in moments is the com-

parison of the total seismic moment along a

plate boundary with that expected from rela-

tive plate motion (Brune, 1968). Let M

be the

moment of the ith earthquake and assume that

a boundary of length L is slipping to a depth D.

Then the total seismic moment for all earth-

quakes on this area is LDu where u is the cumu-

lative slip from earthquakes. Seismic coupling,

C, is defined as the seismic slip rate u divided by

the tectonic rate u

u ¼

LD

total

LD

¼ Cu

; (14:47)

where D is the effective depth of the seismogenic

zone. u

is determined from plate reconstructions

or from geodetic measurements of current plate

Table 14.1 Analysis of the Harvard earthquake catalogue from Table 5 of Bird and Kagan (2004).

Events on a boundary may be shared between different zones, which accounts for the fractional

numbers. For annual average numbers divide by 25.9

Province

Events

1977–2002.9

(10

Nm) 

(corner

magnitude)

0total

(10

Nm/

year) C

Energy

(10

J/yr)

Cont. rift boundary 285.9 1.13 0.65 7.64 0.5 0.5 2.5

Cont. transform faults 198.5 3.50 0.65 8.01 1.2 0.7 6.0

Cont. convergent

boundary

259.4 3.50 0.62 8.46 3.3 0.9 16.5

Ocean ridge normal

faults

424.3 1.13 0.92 5.86 0.31 0.02 1.5

Ocean ridge other

mechanisms

77 1.17 0.82 7.39 0.06 0.05 0.3

Ocean transform

fault slow

398 2.00 0.64 8.14 2.1 0.9 10.5

Ocean transform

fault medium

406.9 2.00 0.65 6.55 0.30 0.1 1.5

Ocean transform

fault fast

376.6 2.00 0.73 6.63 0.29 0.1 1.5

Oceanic convergent

boundary

117.7 3.50 0.53 8.04 1.5 0.3 7.5

Subduction zones 2052.8 3.50 0.64 9.58 91.3 0.7 456

14.7 THE DISTRIBUTION OF EARTHQUAKE SIZES 217

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motions to ascertain whether all the motion is

being taken up by earthquakes or some is taken

up by aseismic creep, that is, C < 1. It is impor-

tant that the catalogue extends over sufficient

time to be representative of the earthquake

activity. While L is easy to measure, D is uncer-

tain. Because earthquakes with the largest mag-

nitudes have the greatest contribution to the slip

it is important to include their effect. However,

they are rare. This problem can be overcome by

fitting the distribution of moment from the cata-

logue to the Kagan distribution (Eq. (14.44)) to

find the corner moment and earthquake rate,

and then using Eq. (14.46) for total seismic

moment. From a global survey, Kagan (2002a, b)

found that on average the worldwide coupling

coefficient is C 0.5.

The Harvard moment tensor catalogue

(Dziewonski et al., 1981) contains seismologically

determined moment tensors for earthquakes

worldwide since 1977, and is thought to be com-

plete above magnitude 5.6 (threshold moment

¼3 10

Nm). Let N

be the number of events

per unit time above the threshold. Then from

Eq. (14.44),

 ¼ N



: (14:48)

Substituting Eq. (14.48) in Eq. (14.46), the total

moment becomes

0total

¼N



Gð1  ÞM

1

: (14:49)

and Eq (14.47) gives

C ¼

LDu



Gð1  ÞM

1

: (14:50)

Bird and Kagan (2004) have used this model to

examine parameters in Eq. (14.47) and coupling

as functions of tectonic province. They divided

the Harvard moment tensor catalogue into

different regions, such as subduction zones,

ocean spreading centres and continental conver-

gence zones, as in Table 14.1, and determined N

, M

, and M

0total

from the moment distributions

with an estimation of C by Eq. (14.50). The con-

tinental collision zones are particularly well

observed, with the Alpine–Himalayan chain as

the major component. They estimated a total

boundary length of 12 516 km and average slip

of 5.26 cm y

1

, taking into account dip angles

and slip on variously oriented faults. If the depth

of the seismogenic zone is taken as 20 km, the tec-

tonic moment rate is LDu

¼3.65 10

Nm y

1

assuming  ¼27.7 GPa. The seismic moment

rate was obtained by fitting Eq. (14.49) to the

moment tensor data for the years 1977–

2002.9 giving M

¼5.82 10

Nm,  ¼0.62,

¼8.6, M

¼3.5 10

Nm, N

¼10.01/ y. With

these parameters the annual total seismic

moment, for the Alpine–Himalayan chain, is

0total

¼3.29 10

Nm y

1

. In this case the ratio

of the seismic and tectonic rates (3.29/3.65)

gives a coupling coefficient of 0.9. Of the other

provinces examined, ocean spreading ridges had

the lowest corner magnitudes and coupling

¼5.86, C ¼0.02), which might be expected

because they are hot, favouring aseismic creep.

By contrast, subduction zones are well coupled

¼9.6, C ¼0.7).

We use these ideas to calculate the global

energy release in the various tectonic zones

(Table 14.1). Using Eqs. (14.36) and (14.38),

E ¼ 5:0  10

5

J: (14:51)

Then from Eq. (14.49) we obtain

total

¼ 5  10

5



Gð1  ÞM

1

(14:52)

Summing t he energies in Table 14.1, the aver-

age total energy release is 5.04 10

1

.We

assign an average ‘efficiency’ of 6% to the con-

version of strain energy to seismic waves in

Chapter 15, making t he annual dissipation

of strain energy about 8.4 10

1

2.7 10

W. This is 3.5% of the total conv ective

power of the mantle (Section 22.5). The fraction

of mantle volume that is subject to earthquakes

is much smaller than 3%, so there is a strong

concentration of dissipation in the subduction

zones. These numbers confirm that earth-

quakes are hiccups at sticking points in mantle

convection. The average annual energy release

corresponds to that of a single magnitude 8.6

event and is exceeded by each of the largest

shocks (Chile, 1960; Alaska, 1964; Sumatra,

2004).

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14.8 Tsunamis

There are occasional reports of earthquakes hav-

ing been felt at sea, but the secondary effect of

wave generation by submarine earthquakes is

much more significant. The waves sweep across

open ocean at high speeds and cause severe dam-

age and loss of life on coastal areas many thou-

sands of kilometres from the earthquakes that

generate them. These waves were once referred

to as ‘tidal waves’ but are not related to tides and

the Japanese word ‘tsunami’ (literal translation:

harbour wave) is now used universally. Much of

the perimeter of the Pacific Ocean is seismically

active and it has received most attention.

This was deemed to have paid off after the suc-

cessful prediction of the damaging tsunami from

the May 1960 Chilean earthquake. However,

all oceans are at risk, as the devastating

Indian Ocean tsunami triggered by the 2004

Sumatra–Andaman earthquake, reminded us.

The principles of tsunami propagation, as

reviewed by Stevenson (2005), are reasonably

well understood, being an application of classi-

cal hydrodynamics, although with complica-

tions in detail arising from complexity of sea

floor topography.

Earthquakes are not the only cause of tsuna-

mis. The 1883 eruptions of Krakatoa volcano, in

the Sunda Strait between Java and Sumatra, pro-

duced several tsunamis, the final one being

locally very big and destroying hundreds of

coastal villages in western Java and southern

Sumatra. Slumping of marine sediments also

generates tsunamis and, in at least some cases,

the slumping is triggered by earthquakes,

leading to confusion over the mechanism of

tsunami generation. A recent example that

drew att ention to the problem is the tsunami

that struck the Aitape area of New Guinea in

July 1988, following a magnitude 7.0 earth-

quake north of the island. The wave height,

peaking close to 15 m, appeared too great to be

explained by the earthquake it self and

is attributed to a sediment slump triggered by

the earthquake. It appears possible that slump-

ing contributes to most major tsunamis, even if

they are clearly identified with earthquakes.

Geological evidence of very big prehistoric

tsunamis links them to sediment slumping,

but with no indication whether they were

earthquake-triggered. There is an area of mas-

sive slumps between Norway and Iceland, each

involving more than 1000 km

of sediment

that slipped hundreds of kilometres into the

deep Atlantic from the continental slope off

Norway. Best documented is an isotopically

dated event about 7000 years ago, coincident

with widespread tsunami deposits along the

Norweg ian an d Scottish coasts. The Hawaiian

islands also deposit thick sediments on steep

offshore slopes and are therefore a likely source

of slump-generated tsunamis. A major slump

south of the island of Lanai about 100 000 years

ago is postulated to have caused a tsunami still

evident as marine deposits on the east coast of

Australia.

Earthquakes such as Chile, 1960, Alaska,

1964 and Sumatra, 2004, that are major tsunami

generators, cause movements of the sea floor

that are hundreds of kilometres in extent and

so generate waves hundreds of kilometres in

wavelength. They are shallow-water waves,

meaning that water depth is much less than the

wavelength The general equation for the speed

of a wave of wavelength l ¼2p/k in water of

arbitrary depth h is

v ¼

tanhðkhÞ

1=2

; (14:53)

where g is gravity (see, for example, Proudman,

1953). The definition of a shallow-water wave is

(kh) 1, in which case Eq. (14.53) reduces to

v ¼

ﬃﬃﬃﬃﬃ

: (14:54)

In water 5 km deep the speed is 220 m s

1

(800 km/hour). Although this is very fast by the

standards of ocean waves, it is twenty times

slower than Rayleigh waves and forty times

slower than P-waves, making possible the tsu-

nami warning system that operates in the

Pacific. The waves are also slower than the

usual speed of fault propagation (by a factor

10), so that, to a good approximation, tsunami

generation by an earthquake is synchronous

over the fault area. By Eq. (14.54) the wave

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speed is depth dependent but not frequency

dependent.

An understanding of tsunamis was one of the

targets of the International Geophysical Year

(IGY), a period of focussed international collabo-

ration in 1957–8. The rarity of major tsunamis

had restricted progress and a new series of sensi-

tive recorders was developed, with hydraulic

tuning that made them sensitive to waves with

periods between 3 minutes and 2 hours while

almost ignoring both tides and normal wind-

driven waves. They were deployed around the

Pacific with the expectation that numerous

small tsunamis would be seen, yielding much

more data in a short time than would otherwise

be possible. But the IGY period was virtually

tsunami-free. Small tsunamis are not as frequent

as had been supposed. To see a reason for this we

compare the energy of a tsunami with that of the

earthquake that generates it.

First, we note that earthquakes on faults that

are not very large (in both dimensions), com-

pared with the depth of water below which

they occur, cannot be significant tsunami gener-

ators because the integral of sea floor displace-

ments is zero and adjacent movements of

opposite signs cancel within the water column.

If we consider an earthquake with mean dis-

placement b across a fault of length L and width

W in a medium of rigidity  then its moment is

¼bLW (Eq. 14.6). We can relate it to earth-

quake energy, E

, by combining Eqs. (14.36) and

(14.38), assuming M

¼M

¼ M

 10

4:3

: (14:55)

This effectively assumes that seismic strain is

independent of magnitude, which is found to

be a satisfactory approximation in Section 15.6,

at least for the large shocks that are relevant to

tsunamis. Thus, seismic energy is proportional

to the product of dimensions, bLW. But, with the

assumption of constant strain, b is proportional

to the smaller fault dimension, W, so that we

have

/ LW

: (14:56)

The sea floor displacement is some fraction, f,of

the fault displacement, b, depending on the

inclination of the fault, and, since the fault

movement is rapid compared with ocean wave

propagation, the movement of the water column

is almost synchronous everywhere and follows

the sea floor motion. On average it is balanced,

that is the total of movements up and down is

zero, and the average gravitational energy

imparted to the water column is proportional

to (fb)

. This becomes the tsunami energy, for

which the total is, with substitution of the pro-

portionality of b to W,

/ gðfbÞ

LW / gf

: (14:57)

From Eqs. (14.56) and (14.57) we see that, for

faults that are equidimensional (L W), E

is pro-

portional to W

but E

is proportional to W

. This

appears to be appropriate for small to moderate

shocks and demonstrates that tsunami energy is

more dependent on size than is the earthquake

energy itself. We can see why small tsunamis are

rarer than might have been expected from the

distribution of earthquake magnitudes. For the

largest shocks (magnitudes 8 to 9þ), fault dimen-

sion ratios (L/W ) are observed to increase with

magnitude; in the approximation that W is con-

stant for this size range and only L increases with

magnitude, the two energies are proportional to

one another. The rare very large shocks are dis-

proportionately effective tsunami generators.

An IGY tsunami recorder was maintained

on Norfolk Island, 1200 km east of mainland

Australia, for several years after the IGY period

and was operating at the time of the 1960 Chile

earthquake. For much of the resulting record the

instrument was driven off scale (drawing square

waves), but satisfactorily recorded the first hour

(1.5 cycles) of the arriving wave (Fig. 14.17(a)).

The interest in this record arises from the fact

that it was obtained at a site that is about as near

as one can get to an open ocean situation: an

instrument mounted on an exposed jetty on an

island roughly 7 km square (small compared

with the tsunami wavelength), standing in deep

water and remote from other land. But the ocean

floor between Chile and Norfolk Island is

far from featureless and the depth dependent

wave speed means that the arriving wave had

been refracted, diffracted and reflected in

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complicated ways, just as seismic waves are

affected by irregularities in the mantle. The

lowest frequencies (longest wavelengths) are

least affected by the irregularity and so travel

faster than the higher frequencies, as in the ‘ran-

dom media’ problem in seismology, discussed

in Chapter 16. As Fig. 14.17(a) shows, and

Fig. 14.17(b) confirms with observations on an

estuary near Hilo, Hawaii, the first arriving wave

had a distinctly longer period than the following

waves, which is what is normally observed, but

also that it was relatively small. This observation

does not invalidate Eq. (14.54), but it draws

attention to the fact that the equation assumes

a uniform ocean floor and that for the real ocean

there is some dispersion. It means that inferen-

ces about source characteristics from wave fea-

tures must be treated with caution. The ‘frosted

glass effect’ is difficult to allow for.

The problem of tsunami amplitude is of par-

ticular concern. If a wave approaches a shore line

from a perpendicular direction up a gently slop-

ing sea floor, then the speed decreases by

Eq. (14.54), but the wave energy is conserved

and the amplitude increases. For a wave of ampli-

tude a and wavelength l, the energy in one

19.00 hr

20.00

hr 21.00 hr

22.00

Local time

FI G U R E 14.17(a) Norfolk Island record of the tsunami from the Chile 1960 earthquake.

FI G U R E 14.17(b) Observations by Eaton et al. (1961) of the water level at the Wailuku River bridge, Hilo, Hawaii,

during arrival of the tsunami from Chile in May, 1960.

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wavelength is proportional to a

l and since

l /V /h

1/2

, a increases with decreasing water

depth h as h

1/4

. On this basis a 1 m wave in

4 km deep ocean becomes a 4 m wave in 15 m

of water, by which point non-linearity is signifi-

cant and the simple rule breaks down. Other

important factors that affect wave amplitude

are refraction towards shallow water, which

selects certain sections of coastline for enhanced

waves according to the sea floor topography, and

local resonances (seiches) in bays, estuaries and

harbours. A naturally defended coastline is one

with sharp submarine contours offshore (per-

pendicular to the wave direction). The amplitude

calculation above assumes a gently sloping sea

floor, but if there is a sharp change of depth,

relative to the wavelength, then partial reflec-

tion occurs. The water motion extends through

the full depth of the sea and such depth changes

can be modelled as impedance mismatches. A

reef, with a steep outer edge distant from the

shore, is ideal, blocking motion in much of the

water column and reflecting the corresponding

fraction of the wave energy.

The floor of the Indian Ocean is subducting

beneath Sumatra and the Andaman Islands to

the north of it, and the 2004 earthquake was a

sharp increment in this process. The subducting

(western) plate jerked downwards and the over-

riding (eastern) plate upwards in the manner

illustrated by Fig. 10.14. For waves propagating

in the two opposite directions, the first motions,

illustrated by a set of five tide records repro-

duced by Lay et al. (2005), were the reverse of

what this simple picture would suggest.

Similarly, the Norfolk Island record of the Chile

tsunami (Fig. 14.17(a)) shows an initial rise in

level, although the island is on the subducting

side of the fault. In both cases the fault dips were

shallow and the simple view of up or down

motion on opposite sides is not valid. All of the

major tsunamigenic subduction zone earth-

quakes have focal mechanisms similar to the

model of the Alaska earthquake in Fig. 14.7,

with shallow dipping fault planes. It is necessary

to recognize that the stress release is always of

double couple form (Fig. 14.9(b)). As Fig. 14.7

shows, the initial sea floor motion is downwards

at the land edge, causing an initial water

withdrawal at the nearby shore and initial inun-

dation at remote sites. We need to recognize also

that there is probably always some sediment

slumping. In the Sumatra case this is consistent

with evidence for a very slow component of the

tsunami excitation (Lay et al., 2005). The speed of

slumping is much slower than the wave speed,

so this component behaves quite differently

from earthquake excitation and would not have

contributed to the first motion of the wave.

14.9 Microseisms

A limit to the useful sensitivity of seismic record-

ing is imposed by background vibration, to

which the term microseisms is applied. It is

most serious at periods between 5 s and 12 s.

The word does not imply that microseisms are

caused by numerous small earthquakes. There

are several causes, some of which are local,

such as the shaking of trees by wind (forests are

seismically noisy). Of greater interest is the gen-

eration of microseisms by storm waves at sea.

Storm-generated microseisms are observed at

great distances. They are principally Rayleigh

waves (vertically polarized surface waves – see

Section 15.3).

Ocean waves are caused by friction between

the wind and sea surface. The amplitudes and

wavelengths grow with wind velocity and its

duration up to a limit. This limit occurs when

the phase velocity of the waves is approximately

equal to the wind speed, a condition that gives

rise to what is called a fully developed sea.

For deep water waves (kh 1, tanh (kh) !1),

Eq. (14.53) becomes V ¼(g/k)

1/2

, so that the

period is

T ¼ 2pV=g: (14:58)

For 30 to 40 knot winds (15.4 to 20.6 m s

1

), typ-

ical of storms, Eq. (14.58) gives periods of 10 to

13 seconds and wavelengths of 150 to 270 m.

Consider a water wave propagating across

deep ocean. The particle motion is circular, in

the sense of a wheel rotating backwards relative

to the forward motion of the wave, giving the

surface a trochoidal form, with crests sharper

than troughs. But the vertical motion of the

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water is sinusoidal and so there is a sinusoidal

oscillation in pressure below the surface. The

amplitude of the pressure oscillation decreases

exponentially with depth on a scale deter-

mined by the wavelength, because at increas-

ing depth there is an increasingly effective

cancellation of contributions by points on the

wave that are out of phase. Thus, there is no

pressure pulsation on the ocean floor and no

microseism generation by propagating waves

with wavelengths that are short compared

with the depth.

When two similar waves travelling in oppo-

site directions interfere, a standing wave is set up

and in this situation a pressure oscillation may

be transmitted to indefinitely great depths. A

simple version of the theory is given by

Longuet-Higgins and Ursell (1948). Consider the

sequence of surface profiles in Fig. 14.18. The

total gravitational potential energy of the water

is greater at t ¼T/4 and 3T/4 than at the instants

when the sea surface is flat. There is therefore a

synchronous pressure oscillation of period T/2

over the whole of the area over which the stand-

ing wave is coherent. If the dimensions of the

area are large compared with the depth of water,

then the pressure oscillation is transmitted to

the sea floor. The effect is a second order one.

The frequency of the pressure cycle is twice the

frequency of the surface wave and its amplitude

is proportional to the square of the wave ampli-

tude. To see why this is so, note that the mass of

water that is moved (represented by the cross-

section of the shaded areas in Fig. 14.18) and the

average height of the movement are each pro-

portional to wave amplitude and that the product

of these gives the integrated mass-times-vertical

acceleration of the water column and therefore

the variation in bottom pressure.

Microseisms may be caused also by storm

waves breaking on shorelines. Ocean waves are

refracted so that they are more or less parallel to

a coastline when they break. This gives them

greater coherence and more effective micro-

seism generation than the same energy distributed

irregularly. In this case there is no frequency

doubling. Storm microseisms originating at

shorelines may be distinguished from deep-

ocean microseisms by their dominant periods,

10 to 12 s instead of 5 to 6 s. Although the coastal

mechanism does not require the special condi-

tion of crossing wave-trains and is therefore a

more common occurrence, it is the deep-ocean

effect that produces the largest microseisms.

However, even vigorous storms are not necessa-

rily strong microseism generators. Large waves

produced by steady monsoonal winds are not

effective. Microseism generation requires

cyclonic disturbances that move about and so

produce crossing wave-trains.

t = 0

t = T / 4

t = T / 2

t =

3T / 4

t = T

FI G U R E 14.18 A series of sea surface profiles at

intervals of a quarter of the period, T, of a standing wave.

Relative to the flat surface at t ¼0, T/2 and T, water is

removed from troughs (lightly shaded) and added to the

crests (heavily shaded) twice in each wave period.

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Earthquake dynamics

15.1 Preamble

After more than a century of scientific observa-

tion, earthquakes have given a detailed picture

of the structure of the Earth, but only a rudimen-

tary idea of the physical processes in the fault

zones where they occur. We know that earth-

quakes are local hiccups in convectively driven

tectonic motion and the pattern of their occur-

rences is a central component of the theory of

plate tectonics. Elasticity theory explains the

geometry of stress release during earthquakes

but not the mechanism that triggers them.

Although this depends on the physical proper-

ties of materials in the fault zones, we are still far

from a level of understanding that might lead to

reliable prediction. It is possible that the detailed

information that would be required is beyond

reach. Nevertheless prediction has been the ulti-

mate target of much seismological research. This

chapter is concerned with the information and

ideas that the work has produced, with a brief

discussion of the possibility of prediction itself

in the final section.

A basic problem is that earthquakes are very

diverse phenomena. The range of sizes is

extremely wide, probably bounded at the upper

end of the scale by magnitudes not much greater

than that of the Chile 1960 event (Section 14.7)

and with no observed lower bound. The range of

energies exceeds 10

:1. The range of speeds of

fault movement during earthquakes is also very

wide, bounded at the upper end by the acceler-

ations that can be produced by stresses of order

10 MPa if fault friction suddenly vanishes. Most

well-studied earthquake ruptures have speeds of

2 to 3 km/s, near to the S-wave speed. There are

also ‘slowquakes’ that may occur over hours to

months and can be regarded as creep events.

There is probably no lower bound short of the

speed of plate motion, centimetres per year.

Seismological observations have emphasized

the more rapid events because they radiate elas-

tic waves. The energy release is converted to

seismic waves with dissipation by local friction

and breakage of bonds. The diversity and wide

range of phenomena require an explanation in

terms of material properties and fault behaviour.

A phenomenological theory, known as the rate–

state theory, is the most successful of the current

alternatives.

For most earthquakes more energy is

released in the fault zones themselves than is

radiated and for ‘slowquakes’ virtually no energy

at all is radiated. This implies strong heating of

fault zones, so that the absence of geothermal

anomalies along fault zones requires an explan-

ation. Removal of heat by flow of ground water

can be invoked, but invites doubt about the val-

idity of heat flow measurements.

In pondering the problem of earthquake trig-

gering, basic questions are: What distinguishes

large from small events? Are they self-similar

with only a scale difference and if so what stops

them from growing further? Or can we suppose

that large events are just superpositions of

numerous small ones – ‘instantaneous after-

shocks’? The really big events, Sumatra, 2004

being an impressive example, start at a particular

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point and spread, often in one direction. From a

study of this event by Ishii et al. (2005) the dura-

tion of fault movement was about ten minutes,

which, for the 1600 km fault, implies an average

break speed of 2.8 km s

1

. A break in one section

transfers stress, which breaks the next, and so

on. In this situation fault instability may not be a

useful concept. An individual section of fault

may be nowhere near to breaking on its own

but respond to triggering by a neighbour or

neighbours. The concept of vulnerability

appears to be more relevant. A large earthquake

requires a large fault area that is vulnerable to

triggering by any small patch within it. This puts

the prediction question into context. Detailed

prediction (time, place, magnitude) would

require, as a starting point, identification and

monitoring of unstable patches that may be

very small individually, but capable of triggering

neighbours, and this appears to be an impossible

task. The best that can be expected is recognition

of vulnerability, that is, evidence that any earth-

quake that occurs is likely to grow into a big one.

The initiation process itself is almost certainly a

small-scale one.

On a longer time scale, aftershocks give an

indication of the rate of adjustment of stresses in

seismogenic zones. Large shocks are followed by

smaller ones that outline the fault zone; typically

the largest aftershock is one unit on the magni-

tude scale smaller than the main shock. But fore-

shocks are also known and a crucial question is:

Can they be recognized as such before the main

shock has occurred? A particularly striking exam-

ple was the occurrence of two M 8 events

shortly before the M ¼9.5 Chile 1960 earth-

quake, which appears to have been preceded

also by a very large, deep ‘slowquake’. Repeated

patterns of seismicity have often been sought,

using a frequency of occurrence vs magnitude

relationship discussed in Section 14.7, but no

clear conclusion has emerged.

15.2 Stress fields of earthquakes

When an earthquake occurs the crack that forms

releases stress in its immediate vicinity by rela-

tive motion between the two sides of the fault

plane, but increases stress in nearby regions. The

slip process starts from a nucleation patch and

grows at the rupture velocity, generally observed

to be slightly less than the S-wave velocity, that

is, about 3 km/s. For small events the slipping

zone can be modelled as a growing crack that

spreads until it is stopped by a barrier or reaches

a region of low stress. Until the stopping point is

reached, the whole fault is assumed to be slip-

ping, but that may be only a conclusion based on

inadequate observation of very small events.

Large earthquakes can be regarded as multiple

events of this kind. For very long faults such as

the San Andreas, the slipping zone grows to a

given size, e.g., a roughly rectangular shaped

patch extending over the depth of the seismo-

genic zone, and travels as a slip-patch disturb-

ance from one end of the fault to the other,

or perhaps in both directions, from the initia-

tion. The slipping zone may be made up of a

heterogeneous distribution of sub-faults or

asperities that trigger each other as the stresses

of their seismic waves travel to neighbouring

regions.

In idealized models the final slip distribu-

tion is one that reduces the stress everywhere

on a fault to a value near the dynamic friction

limit, though some overshoot may occur

before friction brings the fault to a stop. Such

a slip distribution can be calculated analyti-

cally for elliptical cracks, a circular one being

mathematically simplest. The results have

several features in common. The slip distribu-

tion is elliptical. The stress drop is uniform

along the fault but stress has singularities that

appear at the boundary, and diminishes as

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

distance

from it. The distribution of the dis-

location b(x

), that is required to give constant

stress drop is found by solving the integral

equation

ðxÞ¼constant ¼

ðð

bðx

ÞG



ðx; x

ÞdS

(15:1)

where G



(x,x

) is the Green’s function for stress at

x due to a displacement at x

and may be obtained

by differentiating Eq. (14.8) and using Eqs. (11.4)

and (11.5) to convert displacements to stress

components.

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The infinitely long fault can be treated as a

special case of an elliptical fault. If displacement

occurs in the direction of the long axis, in the

sense of a screw dislocation (Fig. 14.4), this is

referred to as an antiplane crack, for which

stresses, ðxÞ, and the slip distribution, b(x ), are

(Knopoff, 1958)

ðxÞ¼initial stress þ D  1 



1=2

1

; x

> c;

ðxÞ¼

; x

 c;

(15:2)

bðxÞ¼





ðc

 x

: (15:3)

Figure 15.1(a) plots the stress and displacement

for this case. The singularity at each edge of the

crack arises because the gradient of displace-

ment becomes infinite. The solution is made

more realistic by assuming that the material

in the vicinity of the crack edge fails plastically,

and that the singularity is rounded off in what is

referred to as a slip-weakening zone (Fig. 15.1(b)).

After faulting, stresses are decreased along the

fault but are increased beyond the edges for dis-

tances comparable to the fault dimensions. A

feature of this model that is of interest is that

the condition for constant stress drop is elliptical

displacement (Eq. (15.3)) with the maximum

value equal to the strain times the fault dimen-

sion. This general behaviour is common to all

models.

While observed displacements are approxi-

mately elliptical they are much rougher t han

the theoretical ones of Fig. 15.1. The fault

plane may contain over- and u ndershoots of

stress release, resulting in a patchwork of stress

concentrations of all shapes and sizes, both

within and external to the fault. Even the con-

cept of an individual fault plane is a simplifi-

cation. A more realistic model describes t he

faulted region as a zone of sub -faults with a

power-law distribution of sizes. Each sub-fault

is surrounded by its own stress concentration

and aftershocks are delayed response to this

stress distribution. Laboratory observations,

which indicate that the time to rupture is expo-

nentially dependent on stress (Section 15.3), can

explain the long sequences of aftershocks that

may continue for years. The sub-fault model

explains why aftershocks are concentrated in

the fault zone of an earthquake, where reduced

stress might be expected. Sub-faults with their

associated stress concentrations leave behind

very rough stress fields. An explanation of the

aftershock delays is explored in Sections 15.3

and 15.5.

The distribution of aftershocks in the region

of an earthquake has usually been fitted to the

pattern of the change in Coulombic stress

(a) Shear crack

(b) Shear crack with slip-weakening zone

displacement

–c +c

initial stress

final stress

initial stress

final stress

Δσ

slip

weakening

zone

slip

weakening

zone

FI G U R E 15.1(a) A shear crack, c  x  c, with relative

displacement of elliptical form, as shown, causes a

uniform stress drop on the crack, but singular stress

outside, infinite on the boundary. (b) A slip-weakening

zone is added to make the model in (a) more realistic,

with displacement graded smoothly to zero and all

stresses remaining finite.

226 EARTHQUAKE DYNAMICS