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near spherical features such as oolites will have a nearly ellipsoid shape after
deformation, but only a section of the ellipsoids will be seen in exposure. For
rocks with many such inclusions, a statistical analysis for the dominant orientation
and stretch can provide useful constraints on the strain ¬eld. In other cases where
there is a fairly uniform distribution of markers, such as pebbles, the pattern of
the markers after deformation can be used to examine the form of strain even
where the original shapes are not known. The distortion of the shapes of fossils
under deformation was recognised at an early date, indeed once the possibility of
changes of shape had been recognised the number of apparently distinct species of,
e.g., trilobites was substantially reduced. When comparison can be made with an
undeformed specimen, the determination of the strain ¬eld is fairly direct. Also the
mappings needed to restore two different fossils to a common shape can be used to
infer the relevant strain ¬elds. Elongated objects such as belemnites provide useful
markers for extension, since the segments of the original fossil break apart and the
intervening space is ¬lled with crystalline material, whose ¬bre directions align
with the separation direction of the fossil fragments. The change in the length of
the fossil and the ¬bre directions provide direct measures of the properties of the
strain ellipsoid.
Many of the features in rocks are associated with progressive deformation during
geological processes, occurring over a substantial time frame, so that the strain
10.2 Macroscopic structures 189

¬eld can change in time. The in¬‚uence of the strain rate can then be felt through
the rheology of the material, particularly the effective viscosity of different rock
components. The differences in resistance to deformation can have very signi¬cant
effects on the way that the different components of the rock behave under the same
deformation ¬eld.
In a shear zone a variety of mechanisms can operate to provide weakening
of the material. Geometric softening can arise from the development of
crystallographically preferred orientations for minerals associated with shear
deviations. Softening can also occur due to grain size reduction (e.g., by dynamic
recrystallisation); this allows a switch from dislocation creep to diffusion creep and
grain-boundary sliding creep with a consequent modi¬cation of the constitutive
equation. In the presence of reactive ¬‚uids, there can be chemical effects that
induce softening. Many chemical reactions produce a weaker mineral assemblage,
e.g., the breakdown of alkali feldspars to produce micas, or the production of
chlorite from amphiboles. There is a widespread belief that material softening leads
to strain localisation.
Even during plastic deformation, the permeability of rocks can be enhanced
during straining due to the presence of microcracks. This permeability facilitates
the localisation of ¬‚uid ¬‚ow and reaction in shear zones. The presence of ¬‚uid
commonly leads to reaction softening, but very intense hydrothermal alteration
can result in micas being replaced by assemblages rich in quartz and feldspar
with consequent reaction hardening. The presence of elevated pore ¬‚uid pressures
relative to hydrostatic pressure can lead to modi¬cation of the normal behaviour.
Thus, e.g., with high pore ¬‚uid pressures pf material may move from the ductile to
the brittle ¬eld even deep within the crust.

10.2.2 Folding and boudinage
Folding in rocks represents a common response to deformation and can be observed
over a very wide range of scales from the microscopic in ¬ne grained metamorphic
rocks to scales of tens to hundreds of kilometres (Figure 10.6). Frequently multiple
scales of folding can coexist representing different phases of deformation. The
diversity of observed fold features come from the interaction of a range of different
factors:
(a) the composition of the layers and the primary rheologies of each rock type,
(b) the way in which the rheological parameters change with pressure and
temperature during the period of fold formation,
(c) the development of oriented mineral grains during the deformation leading to
anisotropy in material properties,
(d) the mechanical properties of the interfaces between layers
(e) the thickness of the different rock layers, and the spacing between the most
competent layers,
(f) the nature of the boundary constraints on the rock units undergoing folding, e.g.,
an upper free boundary but a lower boundary de¬ned by sliding on a decollement
190 Geological Deformation




km
0 10 20 30




Figure 10.6. Large-scale and smaller-scale folds exposed by differential weathering of the
Proterozoic meta-sediments in the Davenport and Murchison Ranges, Northern Territory,
Australia.



surface (see Section 10.2.4), and
(g) the overall scale of the rock mass being folded.
One way in which fold structures can be developed that does not depend on
the actual rheology is by differential ¬‚ow with well-de¬ned ¬‚ow lines (passive
or shear folds “ Figure 10.7). The deformation of each layer is then the same;
this mechanism can be important in providing an ampli¬cation and distortion of
existing folds; in which case the folded layers provide geometric markers for the
imposed deformation.
Nevertheless, most of the observed deformation in rocks is related to the layering
and differential strength of a geological sequence. Under compression the stronger
layers tend to bend and then buckle, while weaker layers ¬‚ow (Figure 10.8), and
the nature of the folded structures will depend on the direction of the prevailing
stress relative to the layering.
When a compression is applied parallel to rock layers the initial effect is
homogeneous shortening along their length, but as the process continues there will
be a tendency for buckling with transverse de¬‚ection. The physical effects will vary
depending on the environment of the stronger layers. If they are widely spaced in
a softer matrix then the layers will buckle independently; whereas if they are close
10.2 Macroscopic structures 191




Figure 10.7. Development of folds by spatial varying but coherent ¬‚ow.


together the con¬guration is generally imposed by the most competent material and
the effect is that of a composite, thicker layer.
A simple model for the development of folds can be found for a linear viscous
rheology by examining the dependence on the viscosity ratio ζ between an isolated,
competent layer with viscosity ·1 and its less viscous surroundings (viscosity ·2).
Consider the initiation of a sinusoidal fold of long wavelength Lb relative to the
layer thickness d of the competent layer, in a two-dimensional situation with plane
strain and volume conservation. There will be two parts to the resistance to fold
formation. The ¬rst will come from the more competent layer through the force
required to stretch the outer arc of each fold and compress the inner arc; the force
can be represented as

2π2 d3 ‚
Fint = ·1 ln ex, (10.2.1)
3 L2 ‚t
b

where ex is the shortening strain along the orientation of the original layer; the
force will be least for long wavelengths Lb. The second resistive component comes
from the matrix through the transverse de¬‚ection,
Lb ‚
Fext = ·2 ln ex, (10.2.2)
π ‚t
and here the least energy would be associated with the shortest wavelength. The
balance of the two forces leads to a minimum when
4π2 d3 ‚
‚ 1‚
Fint + Fext = ’·1 ln ex + ·2 ln ex = 0, (10.2.3)
3 L3 ‚t
‚Lb π ‚t
b
192 Geological Deformation




Figure 10.8. In¬‚uence of compression on a multi-layered rock, showing the differential
effects of buckling depending on the resistance of the layers. [Courtesy of S. Cox.]


and thus when the wavelength Lb is given by

4π3 ·1 3
L3 = d. (10.2.4)
b
3 ·2

For such an isolated layer, the ratio of the wavelength of buckling Lb to the layer
thickness d thus depends on the relative strength of the layer to its surroundings ζ

Lb = 2πh[ζ/6]1/3, (10.2.5)

with a similar result for elastic buckling where ζ is the ratio of the elastic moduli.
Buckling of layers with a rheology governed by a power-law dependence of
strain rate on stress also displays a proportionality between wavelength and layer
thickness, but the dependence on relative strength can be reduced.
The effect of extension on a multi-layered material is somewhat different. The
stronger layers resist extension along their length and as a result tend to become
segmented (Figure 10.9). The material necks and may even separate to give a
cross-section reminiscent of sausages, from whence the name boudin from a French
sausage. There is a ¬‚ow of the softer material material into the necks and gaps that
imposes a distinct local fabric.
When both the stronger and weaker materials are in the ductile regime, the
boudins can form with wavelike regularity in a similar way to folds.
10.2 Macroscopic structures 193




Figure 10.9. In¬‚uence of extension on a multi-layered rock, showing the development of
boudins in the most resistant layers. [Courtesy of S. Cox.]


10.2.3 Fractures and faulting
In the crust at shallow depth the rheology is brittle and as a result the response to
large deformation is for rocks to fail along narrow zones of frictional slip. At
greater depth the brittle to ductile transition re¬‚ects the change from localised
to distributed failure, with more plastic ¬‚ow. The interaction of plastic ¬‚ow
mechanisms and brittle cracking can occur in a number of ways. Cracks may
be nucleated at pile-ups of dislocations, from more rigid secondary phases or
boundaries. The nature of the growth of such cracks depends strongly on the
ambient conditions particularly with regard to mineral properties, stress conditions
including the pressure from pore ¬‚uids and temperature.
The consequence of brittle failure is that coherent bodies of rock slip past each
other to create a fault in which equivalent strata on the two sides of the slip surface
are displaced. Repeated slippage in the same place leads to a zone of localised
deformation with broken material between the fault surfaces.

Material failure
A convenient representation of the stress conditions associated with material failure
is provided by working with the normal stress σN = n · „ and the shear stress
σS = m · „ on a plane where, as in Section 3.3, the vector m is orthogonal to the
current normal vector n to the plane. We have already seen that we can represent
the stress state on a plane by the stress-circle construction as in Figure 3.9. We can
194 Geological Deformation

also represent the failure surface in the same domain, as a limit on acceptable shear
stresses (Figure 10.10).
In the presence of ¬‚uid pore pressure pf the entire effective stress state is shifted,
so that σN ’ σN = σN ’ pf. This shift is indicated by the movement between the
solid stress circle in Figure 10.10 and the chain-dashed variant in the presence of
¬‚uid pressure. The stress state for a normal at an angle θ to the σ1 principal stress
axis is still found at the point on the stress circle with inclination 2θ.
If the state of stress in the material plots on or above the fracture envelope the
material will fracture. The onset of fracture will occur when the stress circle
touches the fracture envelope. The ¬nal macroscopic fracture will generally be
oriented close to the plane that is a tangent to the fracture envelope (orientation
θ to the σ1 axis in Figure 10.10). However, the presence of ¬‚aws in the rock or




σS

ure
il
b fa
m
ulo
Co



pf

T0
’σN σ2 σ1 σN
’2θ
Tr
an
s iti
on
al
ten
sil
e




’σS


Figure 10.10. Three ¬elds of the Mohr fracture envelope in the normal and shear stress
domain. The tensile fracture ¬eld shows a ¬xed strength T0 . Coulomb failure shows a
linear increase of shear strength σS with normal stress σN associated with frictional sliding.
Transitional tensile behaviour occurs for normal stresses between the tensile and Coulomb
states. The stress circle is shifted to lower normal stresses by the presence of pore ¬‚uid
pressure pf . The onset of fracture occurs when the relevant stress circle touches the fracture
envelope, in this case for a stress state on the planes inclined at angle θ to the σ1 axis. For
high normal stresses the shear strength tends to drop below the straight line predicted by
the Coulomb relation.
10.2 Macroscopic structures 195

existing zones of weakness can modify the actual fracture development. There is
only one point at which the stress circle is a tangent to the fracture envelope and
only a single direction for potential failure.
For material under tension, or effective tensile stress, σN < 0, tensile fractures
can form perpendicular to the direction of maximum tensile stress. The tensile
strength T0 is independent of the other principal stresses and is typically in the
range ’5 to 20 MPa for dry rocks.
For intermediate con¬ning pressures σ1 ≥∼ |5T0|, and σ1 ’ σ3 ≥ 4T0, the
fracture strength increases approximately linearly with increasing normal stress.
This regime is characterised by a shear stress at failure on a fault plane
σS = c + μσN, (10.2.6)
where c is known as the cohesion and μ is the coef¬cient of internal friction.
This form was suggested by Coulomb in 1773 on the basis of an oversimpli¬ed
physical model, but similar behaviour can be derived for a model with small ¬‚aws
(Grif¬ths cracks) which can enlarge and propagate under the in¬‚uence of an applied
stress. The cracks will tend to close at high pressure and the equivalent form to
(10.2.6) is
σS = 2T0 + μσN, (10.2.7)
with c identi¬ed as 2T0 in terms of the tensile strength. As illustrated in Figure
10.10 the stress circle touches the fracture envelope twice so that there are two
possible directions for failure. For many rocks the coef¬cient μ is close to 0.8,
which also appears appropriate for rock friction. A linear relation of the form
(10.2.6) also applies approximately to many less consolidated materials such as
sand and concrete, but with a reduced value for μ.
At high con¬ning pressures the shear stress for fracture tends to drop below the
straight line predicted from (10.2.6) as ¬‚ow begins, and the strength increases more
slowly with pressure.
The transitional region between tensile failure and the Coulomb regime has
two potential directions of fracture since the stress circle will touch the fracture
envelope for both positive and negative shear stresses. This transitional region
is characterised by a rapid increase in the strength with increasing normal stress,
which for the Grif¬th crack model takes the form
σ2 = 4T0(σN ’ T0). (10.2.8)
S

The fracture orientation is typically oriented at about 30—¦ to the maximum
compression. Such transitional tensile behaviour is likely to be important in the
development of joints in rocks.

Faults
Faults represent discontinuities across which there is a relative displacement of
equivalent geological features. If the corresponding strata can be recognised on
196 Geological Deformation

the two sides of the fault then slip across the discontinuity can measured. By their
nature as zones of concentrated slip, faults are commonly zones of crushing, with
a weak in¬ll and often hot-water circulation. Fault zones are therefore rather easily
eroded and hence can be less visible than the surrounding rocks. As a result, many
faults are inferred from offsets in geological patterns across an unexposed zone that
is presumed to be the surface expression of the zone of slip. In some circumstances
faults can be directly observed in natural exposures, as, e.g., coastal cliffs (Figure
10.11) and canyons, roadcuts, quarries and mines.
The grinding of the two fault surfaces against each other during the process of
slip can produce polished and grooved surfaces called slickenslides. The scratches
and polish are produced by abrasion by the wall rock and the fault zone fragments
during the frictional sliding. The grooves record the orientation of the fault slip.
Multiple episodes of slip along a fault at low temperatures tend to produce a
weak in¬ll material, fault gouge, with frequently a signi¬cant clay component. For
major faults the gouge zone can be quite wide, e.g., on the North Anatolian fault
in Turkey fault gouge zones about 50 m wide are commercially quarried for sands
for building materials. The con¬guration within the gouge can be rather complex
without a simple relation between materials of different ages.
Slip on a fault zone at higher temperatures is accompanied by crystal plasticity




Figure 10.11. Faulting in Palaeozoic rocks exposed on the south coast of New South Wales,
Australia, which juxtaposes grey shales on the left against red sandstone on the right. The
near horizontal quartz veins imply that σ3 is close to vertical, and thus this is a reverse
fault.
10.2 Macroscopic structures 197




Dip-slip


Reverse

Strike-s
lip

Right-la
teral



Oblique


Right-la
teral
Dip-slip


Normal




Figure 10.12. Examples of faulting displacements in dip-slip, strike-slip and oblique faults.


rather than brittle failure. The plastic fault zones are generally composed of
mylonite, which is a ¬ne-grained laminate with common streaking of contrasting
materials. The mylonite is often very hard because of the ¬ne grains and
possible work-hardening of the crystals. Mylonites do not always retain the fabric
and texture of the faulting event since they can be subject to recrystallisation,
and possibly annealing, after deformation. Despite the recrystallisation,
the crystallographic preferred orientation is commonly preserved, as are the
mesoscopic fabrics that indicate shear strain.

Types of faults
The main classes of faults are illustrated in Figure 10.12. In a normal fault
the hanging wall drops relative to the foot wall and such faults are common in
extensional environments. The dip of a normal fault is commonly quite steep,
usually in the range 55—¦ to 70—¦ , and can sometimes approach near vertical in the
near-surface. Frequently normal faults occur in conjugate sets with parallel strikes,
but opposite dip; although the slip displacements are frequently signi¬cantly
different between the two sets. Good examples of normal faults are provided by
the bounding faults of the East African rift, and other graben structures. Collapse
structures such those above an emptied magma chamber are marked by a ring of
normal faults. Deep seated normal faults associated with crustal extension tend to
show lower dips at depth as they pass into the zone of plastic ¬‚ow in the lower crust
(listric faults).
198 Geological Deformation

Thrust faults are dip-slip faults in which the hanging wall has moved up relative
to the foot wall. The dip of a thrust fault is usually less than 30—¦ during active slip
and commonly thrusts dip at 10—¦ to 20—¦ at their time of formation. The structural
settings for thrust faults occur in (i) convergent plate boundaries in both continental
and oceanic settings and (ii) secondary faulting developed in response to folding,
¬‚exure or intrusions. The great Mw 9.3 2004 Sumatra“Andaman earthquake was
a very major thrust event associated with Indo“Australian plate subduction under
Asia. The thrust initiated towards the northern end of Sumatra and in a series of
episodes lasting nearly 5 minutes propagated more than 1300 km to the north along
the upper edge of the gently dipping plate. The dip along strike was somewhat
variable but generally between 10—¦ and 15—¦ . The slip pattern shows distinct
reductions at some points leading to lower energy release in these areas (see Figure
11.26), that are associated with changes in the strike of the subduction zone and
changes in the physical properties of the slab.
Reverse faults are thrusts with dips greater than 45—¦ and can arise from the
reactivation of former listric normal faults in periods of subsequent compression,
such as, e.g., in the Zagros mountains of Iran associated with the convergence of
Africa and Asia.
Strike-slip faults have nearly horizontal slip parallel to the strike of vertical or
very steeply dipping faults. Such prominent strike-strip faults such as the San
Andreas fault in California or the North Anatolian fault in Turkey are often regarded
as vertical faults, but can have some variations in dip. For example the segment of
the San Andreas fault near Santa Cruz has an inclination to the vertical as indicated
by earthquake locations; this dip is needed to compensate for the bend in the fault
line. Other prominent continental strike-slip systems are the Dead Sea system and
the Denali fault in Alaska with lengths of hundreds to thousands of kilometres.
A common feature of strike-slip faults is that the fault surface is not a continuous
plane, but is instead composed of en-echelon segments that accommodate slip
by compressive or extensional deformation, producing local uplift or depressions
(˜pull-apart™ basins). Figure 10.13(a) shows how the offset will require the
movement on the strike-slip fault to lead to uplift. The effect is illustrated in
Figure 10.13(b) by a photograph taken immediately after the 1891 Neo Valley
earthquake in Central Japan that produced a 16 m vertical scarp in a strike-slip
event. The horizontal displacement away from the uplift is clearly indicated by the
deformation of the pre-1891 ¬elds, which are still outlined by tea-bushes (Figure
10.13b). A large-scale example of a similar type is the zone of thrusting and folding
in the Transverse Ranges of southern California associated with an en-echelon step
to the left of the San Andreas fault (viewed from the north or south).
Many faults have some oblique component superimposed on the main motion
that can be deduced from the displacement of the strata on the two sides of the
fault. Thus, the thrust associated with the Mw 7.6 1999 Chi-Chi earthquake in
10.2 Macroscopic structures 199


(a)
Uplift




Figure 10.13. En-echelon segments of strike-slip faults have the effect of producing local
compression in the overlap region between the segments: (a) illustration of the development
of an uplift block, (b) the 16 m fault scarp in Central Japan formed during the 1891 Neo
Valley strike-slip earthquake [1891 photograph courtesy of the Seismological Society of
Japan], (c) the strike-slip component illustrated by the offset of tea-bushes marking the
edge of ¬elds before the earthquake [1991 photograph, B. Kennett].


Taiwan has signi¬cant oblique motion, and the dominantly strike-slip Alpine fault
in New Zealand has associated vertical uplift.
A simple mechanical theory of faulting was developed by Anderson (1951)
on the basis of the assumption that shallow-level faults are Coulomb fractures
(Figure 10.10) in an isotropic medium. The line of the intersection of the two
fracture orientations is parallel to the intermediate principal stress σ2. The direction
of maximum compression, σ1 bisects the acute angle between the two fracture
directions and the axis of least principal compression σ3 will bisect the obtuse
angle. The material will shorten parallel to σ1 and expand parallel to σ3 as a result
of slip along the fractures. Thus, given the orientation of the principal stresses and
the slope of the fracture envelope μ, the orientation and sense of the slip in the
Coulomb fractures can be determined.
The Earth™s surface is a plane without shear stress, as a ¬‚uid“solid boundary,
and as a result one of the principal stress directions must be perpendicular to the
surface. For shallow faulting it can reasonably be assumed that this principal stress
direction is maintained and the other two principal stress directions will then be
horizontal. Different con¬gurations will be achieved depending on which of σ1, σ2
or σ3 is vertical (Figure 10.14). Normal faults will be associated with σ1 vertical
with fractures dipping more than 45—¦ and slip will be downdip. When σ3 is vertical
the fractures will be inclined at less than 45—¦ and slip will be updip in thrust faults.
200 Geological Deformation

(a) Normal (b) Thrust (c) Strike-slip
σ2
σ1 σ3




σ3 σ1 σ1


σ3
σ2
σ2




Figure 10.14. Relation of principal stress directions and fault planes (indicated by shading)
predicted from Coulomb failure: (a) Normal faulting, (b) Thrust faulting, (c) Strike-slip
faulting. The sense of slip is indicated in each case.


For the intermediate case, σ2 vertical, the fractures will be vertical and the slip
horizontal. With the failure criterion (10.2.7) and the typical value of μ of 0.85,
the predicted angle of the failure plane to the σ3 principal axis will be about 25
degrees, in general agreement with the patterns in natural fault systems.
This simple theory captures much of the behaviour of near-surface faulting, but
fails to predict, e.g., high-angle reverse faults. Thrust faults often dip at smaller
angles than predicted (less than 20—¦ ) because their properties are dominated by
bedding slip. Rocks are anisotropic and when the maximum principal stress is close
to foliation, the anisotropy can have profound effects on the orientation of faulting.
The stress ¬eld is also somewhat variable in space, and models that incorporate
the stress ¬eld due to tectonic forces indicate modi¬cation of the Coulomb fracture
directions in ways that are closer to ¬eld observations. The stress state considered
in the Anderson theory is de¬nitely a considerable oversimpli¬cation, because the
presence of faults changes the stress state in their vicinity that can give rise to
secondary faulting.
The process of slip in an earthquake has to be accommodated by the surrounding
material, and complex residual stress and strain patterns can be left around the
edge of the region in which there has been major slip. The discomfort is relieved
by subsequent smaller earthquakes and so the initial aftershock pattern tends to
map out the edges of the slipped zone or regions within that have proved resistive
to slip.
The nature of the termination of fault segments depends on the level of
displacements relative to the fault length. If displacement is modest, the space
and continuity issues associated with fault termination can be taken up by gradual
reduction in displacement towards the end of the fault with distributed elastic strain
(Figure 10.15). Blind thrusts as in Figure 10.15a have proved to be important in
recent earthquakes in California, such as the 1994 Northridge event in the San
Fernando valley, north of Los Angeles, where signi¬cant surface disruption and
10.2 Macroscopic structures 201


(a) (b)




Figure 10.15. Accommodation of thrust faulting in three dimensions: (a) blind thrust with
surface uplift, (b) termination at the surface. Note the similarities to the structures in screw
and edge dislocations.

very strong ground acceleration occurred without faulting actually reaching the
surface.
Where displacement is larger relative to the fault dimensions, accommodation
can still be by elastic strains provided that they are distributed through a large
volume of rock with, e.g., a set of splay faults branching from near the end of
the main fault. Such splays may occur both vertically and horizontally. A very ¬ne
example is provided by the set of dominantly strike-slip faults branching from the
northern end of the Alpine fault in New Zealand (Figure 10.16).
A somewhat different scenario applies at the depth for major strike-slip faults
such as the San Andreas fault. The lower part of the fault lies within the ductile
regime so that part of the motion has to be taken up by plastic ¬‚ow that will become
more widespread, but less intense, as the depth increases.

Motion on a fault
The observed displacements on faults are typically built up from many episodes of
slip. Although there are some examples of slow progressive slip, the most common




Figure 10.16. The con¬guration of
faults at the northern end of South
Island, New Zealand, showing the
well-developed set of splay faults
branching from the northern end of
the Alpine Fault. [Active faults
from the QMAP: Geological Map
of New Zealand and Active Faults
ult databases courtesy of GNS Science,
Fa
ne New Zealand.]
lpi
A
202 Geological Deformation

mode of failure is in the concentrated mode of an earthquake. Since even the largest
earthquakes, Mw 9, do not produce much more than 12 m slip at a time, a very large
number of seismic events and a considerable time period is required to achieve the
displacements of hundreds of kilometres seen on major strike-slip systems. On the
thrusts at the top of the shallow part of the subducting slab the faulting process is
part of the subduction system, and recurrence times are of the order of hundreds of
years for the largest events, since a signi¬cant strain must be imposed.
Early progress in understanding the way in which ruptures can propagate came
from models of the development of cracks. However, one of the dif¬culties in
understanding the faulting process is the complex nature of the fault zone. The
surfaces that move are not completely planar and en-echelon features can occur at
many scales. Further, the fault may contain gouge material and be subjected to ¬‚uid
pressures. The speci¬c conditions have therefore a strong in¬‚uence on the apparent
friction between the two sides of the fault, and it is therefore very dif¬cult to
specify a constitutive equation for the gouge material. Such information is needed
for dynamically consistent simulations of fault motion that take into consideration
the interaction of the fault with its environment. Considerable development has
been made for such computational models that are needed for understanding strong
ground motion in the immediate vicinity of the fault.

N N
(a) Normal (c) Strike-Slip
N
(b) Thrust




E E
E




Figure 10.17. P wave radiation patterns, lower hemisphere equal-area projection, for the
three basic fault types for a north“south strike with compression indicated by pluses and
dilatation by minuses, the size of the symbol is scaled by relative amplitude: (a) normal
fault with a dip of 60—¦ , (b) thrust fault with a dip of 30—¦ , and (c) strike-slip fault. Note the
ambiguity between the fault plane and the perpendicular auxiliary plane, and the relatively
broad zone of diminished amplitude around the nodal planes.


For most purposes a rather simpler formulation of a slip distribution across a
fault surface is adequate to explain seismological observations (see Section 11.2).
Small events can be adequately described by a ˜point source™ with an associated
moment that allows for the actual size of the slip area. The equivalent force
system to localised slip on a plane is a double-couple, i.e., a matched pair of
couples with no net moment (Figure 11.4). The resulting radiation for P waves
is a four-lobed pattern with alternating zones of compression and dilatation (Figure
10.2 Macroscopic structures 203

distance (km)
-10 10
0 -10 0 10
X
0.0
-1.0
10 1.0
-2.0
distance (km)




0.0
0
1.0
-2.0
-10
0.0
fault
strike
A B
X™
X X™
1.0
-2.0
2.0 fault
10
depth (km)




plane
2
0 1
-2 -1
stress change (bars)
20

cross-section
C

Figure 10.18. Map and cross-section views of the change in Coulomb failure stress
generated by a right-lateral slip event on a vertical fault. The fault is 10 km long, extending
from 5“15 km in depth. (A) Map-view calculation of the stress change that is resolved
on optimally oriented faults in the wall-rock around the mainshock. In all areas other
than very close to the mainshock fault, optimally oriented faults have the same orientation
and slip rake as the mainshock fault. The results show that two clusters of positive stress
changes are generated around both tips of the mainshock fault. (B) Map-view calculation
of the stress change that is resolved on thrusts, which generate spatially restricted single
clusters of positive stress change around the tips of the fault. (C) Cross-section view of the
strike-slip fault along line X“X™ shown in A. This ¬gure shows a long-section projection
of the rupture plane and associated stress changes with depth. Positive stress changes form
a doughnut-like distribution around the rupture plane in three dimensions; such positive
Coulomb failure stress changes indicate where the crust is brought closer to failure after
slip on the mainshock fault. [Courtesy of S. Micklethwaite.]



10.17), with the nodal planes accompanied by zero radiation along the fault plane
and perpendicular to it (see Figure 5.5). The patterns of observed ground motion
at distant stations can be projected back to the immediate neighbourhood of the
source and used to constrain the fault mechanism of the event.
Figure 10.18 shows the stress distribution associated with right-lateral slip on a
vertical fault, calculated in three dimensions as a dislocation in an elastic, isotropic,
half-space with a free surface. The imposed far-¬eld stresses have Andersonian
orientations relative to the strike-slip fault, that is the maximum principal stress
is horizontal with an azimuth of 30—¦ . The fault extends from 5“15 km in depth
and the magnitude of slip is 0.5 m, which would be approximately equivalent to a
204 Geological Deformation

magnitude 6 earthquake. The coseismic static stress changes are illustrated for a
depth of 14.5 km, but the same distribution would be obtained at any depth along
the fault plane between 5 and 15 km. Positive Coulomb failure stress changes have
been shown to have a good correlation with the distribution of aftershocks and
the triggering of subsequent earthquakes (see, e.g., King et al., 1994). The stress
transfer model results indicate that, after a right-lateral strike-slip event, strike-slip
related aftershocks may be triggered both along-strike and in lateral distributions
away from the tips of the mainshock rupture (Figure 10.18A). In contrast, any
thrust-related aftershocks will be spatially distributed in tightly constrained clusters
(Figure 10.18B).
The results of this simulation replicate the distributions of aftershock reported
from many strike-slip earthquakes, such as the 1992 Mw 7.3 Landers, California
event (Liu et al., 2003). Most of the aftershocks, triggered by these earthquakes
are likely to have nucleated on pre-existing small-displacement faults surrounding
the mainshock. A striking feature of the 1992 Mw 7.3 Landers example is
that thrust aftershocks occurred within the aftershock sequence, despite being
triggered by a strike-slip mainshock. Indeed, the full gamut of focal mechanisms
(strike-slip, thrust and normal) has been observed in aftershock sequences from
recent earthquakes (Beroza and Zoback, 1993), which presents a challenge to
conventional structural geology and the construction of deformation histories.
Cox & Ruming (2004) and Micklethwaite & Cox (2004) have identi¬ed a
relationship between the distribution of aftershocks and fault-related mineral
deposits by applying the stress transfer modelling technique to fossil fault systems.
Aftershocks can occur over months to decades after a large earthquake, and can
also display a large range of orientations and kinematics. Where ¬‚uids access
these aftershock networks, aftershock clusters associated with strike-slip ruptures
will produce near vertical pipe-like ¬‚ow. Permeability is potentially enhanced in
aftershock domains for months to years after a mainshock rupture, long after the
mainshock fault plane would have lost permeability due to hydrothermal sealing.
For a mineral deposit to form, such a process would have to have operated multiple
times, with repeated mainshock arrest at certain geometrical barriers along a fault
system.


10.2.4 Development of thrust complexes
Some of the most important examples of thrust faults occur in the complex belts
in continents along convergent plate boundaries. In many environments there is
evidence for displacement of material over long distances as in the nappe structures
of the Alps. The deformed, and frequently folded, structure is detached from its
base along a decollement surface with a narrow zone of intense shear.
Individual thrust faults within the compressional complexes tend to cut upwards
through the stratigraphic layering in a sequence of ramps that impose fault-bend
folding as displacement continues to occur along the decollement surface. The
10.2 Macroscopic structures 205


W E




Figure 10.19. Con¬guration of the thrust fault associated with the 1999 Chi-Chi earthquake
in Taiwan. The black arrows indicate projected GPS vectors that are consistent with lower
dip thrusting at depth. [Courtesy of John Suppe.]


ramps may cut through earlier parts of the thrust sheet to give a complicated
sequence of fault splays.
A good example of such thrust ramping associated with recent faulting is
provided by the 1999 Chi-Chi earthquake in Taiwan (Yue, Suppe & Hong, 2005)
as illustrated in Figure 10.19. The earthquake produced several metres of surface
displacement along an 85 km surface break. Seismic re¬‚ection control is available
on the structures away from the mountain belt, and the Chelungpu Thrust on
which the main ground displacement occurred shows a distinct set of ramps. The
earthquake in fact initiated on a deeper fault below the Chelungpu thrust, at about
10 km depth approximately vertically below the TSK-1 borehole, but the main
motion in the event was taken up on the thrust with a very gentle dip at depth.
A number of shallow seismic lines and drill holes con¬rm that the near-surface
thrust is parallel to bedding in the upper 500“2000 m, and there therefore have to
be distinct fault-bend folds to accommodate the motion along the ramped thrust.
These fault-bend folds are commonly associated with localised kink bands with
a rapid change in direction re¬‚ecting the changes in dip on the underlying thrust.
Shallow fault-bend folds produced some distinct surface scarps of a few metres
height displaced about 3“4 km to the east of the main fault trace.
In a few cases a major thrust can be followed from the surface to depth with
the aid of deep seismic re¬‚ection pro¬ling. A good example is the Red Bank zone
in central Australia (Figure 10.20), a mylonite zone that separates two Proterozoic
terrains. The thrust shows clearly in the surface exposure through the juxtaposition
of rocks with very different colour and texture. This distinctive surface feature can
be correlated with a major structure penetrating through the whole crust revealed
by deep seismic re¬‚ections (Goleby et al., 1989). The quality of re¬‚ection data
in this area is very high and relatively high-frequency energy is returned from
depths below 40 km (more than 14 s two-way time). The re¬‚ection section
(Figure 10.20) shows a distinct change in deep re¬‚ection character across the
thrust structure associated with the Red Bank zone (RDZ). A splay thrust links
to the Ormiston Nappe and Thrust Zone (ONTZ) to the south. To the north the
206 Geological Deformation




Figure 10.20. Deep seismic re¬‚ection pro¬ling across the region around the Red Bank
thrust complex in Central Australia showing the major displacement of the crust-mantle
boundary associated with the thrust and a sequence of related features to the north [courtesy
of B. Goleby].

pattern of re¬‚ections can be interpreted in terms of a stack of secondary thrust
slices, indicating thick-skinned tectonics. The crust“mantle boundary appears to
be displaced upwards on the north side of the thrust zone, bringing lower crustal
material to the surface. The Red Bank zone is accompanied by a large gravity
signature that can be well matched by the upward displacement of the crust“mantle
boundary. The last deformation in this region is associated with the intraplate
Alice Springs orogeny (∼300 Ma) and the structure has remained out of isostatic
equilibrium since that time.
11
Seismology and Earth Structure




The theory of linearised elasticity introduced in Chapter 5 provides the basis for
understanding the behaviour of seismic waves except in the immediate vicinity
of an earthquake fault or explosion. Once the incremental strains associated with
wave disturbances drop below 10’5 , the linearised treatment provides an effective
description of the situation. The dominant variation in the properties of the Earth is
with radius and the treatment of wave propagation in a spherically strati¬ed Earth
gives a useful reference. The complexities of the pre-stressed, three-dimensional
Earth can then generally be addressed by a perturbation treatment about the
reference state. Although most minerals are anisotropic, the incremental properties
of the mineral aggregates in the Earth are close to isotropic. The complexities of
anisotropic propagation are therefore needed only for limited regions, mostly in
boundary or transition zones.


11.1 Seismic waves
In Chapter 5 we demonstrated the properties of plane elastic waves in a uniform
medium, and illustrated the radiation from simple sources. For application to the
Earth we need to take account of the variations in elastic parameters with depth, the
presence of material interfaces, and energy loss through intrinsic anelasticity and
scattering.
The waves exploited in seismology span a considerable range of frequencies.
The longest period free oscillations of the Earth have a period close to an hour,
whereas frequencies above 100 Hz can be employed in seismic exploration for
shallow structure. The underlying principles are the same, but the styles of analysis
vary signi¬cantly between re¬‚ection seismic techniques employed for exploration
and studies of the deep Earth.
In exploration work the source is under control and closely spaced seismic
recorders can be deployed to achieve multiple spatial coverage of the relevant
part of the wave ¬eld. The weak signals returned from depth can then be tracked
by making use of the redundancy in the recording through stacking to suppress
incoherent noise but enhance coherent arrivals.

207
208 Seismology and Earth Structure

For most deep Earth studies we are dependent on where earthquakes occur and
have to make use of the global network of high quality seismic stations. The
coverage and signal quality is best for continental regions. Major efforts are being
made to install relatively dense networks of recording stations in some localities,
e.g., the USarray, so that signal coherence can be exploited in appropriate frequency
bands.


11.1.1 Re¬‚ection and refraction
In a spherical Earth model the in¬‚uence of the wavespeed variation with depth
is supplemented by the spherical geometry. When the wavespeed depends only
on radius r the path of a ray through the structure is determined by a continuous
version of Snell™s law. The ray parameter „˜ is constant along the path,

„˜ = r sin i/v(r), (11.1.1)

where i is the inclination to the radial vector and v(r) is the current wavespeed
(either P or S).
10

0 20




6200




6000




5800




Figure 11.1. Wavefront pattern for P waves in the upper mantle of model AK135 with
increasing velocity with depth and velocity discontinuities. The wavefronts of refracted
and re¬‚ected waves are indicated at 10 s intervals.

When v(r)/r grows with depth, the angle i must also increase with depth
for a ray descending into the Earth, and when „˜ = r/v(r) the ray will travel
perpendicular to the radius vector, i = π/2. Subsequently i continues to grow,
but the ray climbs back to the surface. The effect can be seen in the pattern of
wavefronts at 10 s intervals shown for upper mantle propagation in Figure 11.1;
the wavefronts turn and sweep back to the surface with a shape determined by the
variation of P wavespeed with depth.
At an interface, there is partition of energy into re¬‚ected and transmitted waves,
with the possibility of conversion between P waves and vertically polarised SV
waves. All the re¬‚ected and transmitted waves at the interface with radius ri share
11.1 Seismic waves 209

the same ray parameter „˜ as the incident wave; so that the inclinations to the radial
vector of waves with wavespeeds v1, v2 are determined by Snell™s law,
„˜ = ri sin i1/v1 = ri sin i2/v2. (11.1.2)
Horizontally polarised SH waves do not undergo conversion at an interface that is
a surface of constant radius. The amplitudes of the re¬‚ected and transmitted waves
are determined by the conditions of continuity of displacement and traction for
plane wave components with the slowness „˜; this leads to a system of four coupled
equations for P-SV waves and two coupled equations for SH waves. The re¬‚ection
and transmission coef¬cients depend on the contrasts in elastic wavespeeds and
density as well as the angle of incidence (represented through the slowness „˜).
The wavefronts of re¬‚ected P waves from the upper mantle discontinuities
are displayed in Figure 11.1 together with the refracted phases. At moderate
inclinations the re¬‚ected wavefronts link to the incident refracted waves at the
interface. For wide angle re¬‚ections the sharp junction of the re¬‚ected and
refracted wavefronts separates from the interface and moves progressively towards
the surface.
A detailed presentation of the properties of refracted and re¬‚ected waves, and
the treatment of the interface conditions can be found in, e.g., Kennett (2001).


11.1.2 Attenuation effects

The passage of a seismic wave through the Earth is accompanied by energy loss,
due to conversion of energy to heat (intrinsic anelasticity) or from the cumulative
effect of interaction with small-scale heterogeneity (scattering attenuation). We can
use the theory of linear viscoelasticity introduced in Chapter 5 to include the effects
of intrinsic anelasticity through the use of complex elastic moduli in the frequency
domain.
One useful model for intrinsic attenuation is the standard linear solid, as in
(5.7.14)“(5.7.16), but now with a spectrum of relaxation times to match the
observations of little frequency dependence of Q over the seismic frequency band.
The form suggested by Anderson & Minster (1979) for the distribution of relaxation
times „ is
±„±’1
, „L < „ < „H
D(„) = ± (11.1.3)
„H ’ „±
L

with D(„) = 0 outside the band between the lower and upper cut-off times. The
exponent ± characterises the weak frequency dependence.
An alternative form based on a Burgers model has been used by Faul & Jackson
(2005) to interpret the results of laboratory experiments on the wavespeed and
attenuation of synthetic olivine aggregates, without melt, as a function of both
temperature and grain size (Figure 11.2). The in¬‚uence of temperature becomes
210 Seismology and Earth Structure

appreciable well before the solidus is reached with a very signi¬cant increase in
dissipation Q’1 as temperature increases at ¬xed grain size. The experiments are
carried out in torsion over the seismic frequency band from 0.001 - 1.0 Hz and the
use of synthetic aggregates provides control on grain size.
The Burgers material includes elastic and viscoelastic effects with the further
possibility of long term viscosity (5.6.5). For a crystalline material, the viscous
relaxation time will depend on the temperature and grain size sensitivity of the
viscosity and can be approximated as
„M = Adm exp[EM/RT ], (11.1.4)
where A is a constant, R is the gas constant and T is the absolute temperature. The
behaviour is controlled by an activation energy EM and the dependence on the grain
size d carries exponent m. The anelastic relaxation times are expected to have a
similar dependence on temperature and size.
The various relaxation times can then be written in a way that emphasises
deviations from a reference state with temperature TR and grain size dR
m —
pVi
d Ei 1 1 1 1
„i = „iR ’ ’ ,
exp exp (11.1.5)
dR R T TR R T TR

where „iR is the relaxation time in the reference state, p is the pressure, and Vi is
an activation volume.
With a distribution of anelastic relaxation times, the full representation of the
creep function for the Burgers solid, including pressure dependence, then takes the
form
„H
ψU(p)t
d„ D(„)[1’e’t/„]+
ψ(t, d, T, p) = ψU(p)+δψU(T )+”ψ ,(11.1.6)
„M
„L

where ψU is the reciprocal of the relevant instantaneous modulus and δψU(T ) is
an adjustment for the effect of temperature. ”ψ describes the extent of viscoelastic
relaxation with a relaxation time distribution and the Maxwell relaxation time for
the in¬‚uence of viscosity is „M = ·ψU, where · is the Newtonian viscosity.
The single expression (11.1.6) is able to provide a satisfactory representation
of the variation of both shear modulus and dissipation for the synthetic olivines
as a function of period (1“1020 s), temperature (1000“1300—¦ C) and grain size
(3-165 μm). The experiments do not cover the full range of the absorption band;
the temperature correction δψU(T ) compensates for the cumulative effect of the
relaxation times shorter than 1 s, and can be approximated as
’mJ
‚ ln ψU d
δψU(T ) = ψU(p) (T ’ TR). (11.1.7)
‚T dR
R
For a harmonic oscillation with frequency ω, the complex compliance (the
reciprocal of the shear modulus)
ψ— (ω, d, t, p) = ψ1(ω, d, t, p) + iψ2(ω, d, t, p), (11.1.8)
11.1 Seismic waves 211

Shear Modulus [Gpa]




(a) (b)
3 μm 23 μm
log10 (Dissipation 1/Q)




(c) (d)

log10 (Period [s]) log10 (Period [s])


Figure 11.2. Shear modulus and strain energy dissipation as a function of temperature (in
—¦
C) and oscillation period for two samples of synthetic olivine aggregate with grain sizes of
3 μm and 23 μm. The dissipation shows a mild period dependence with constant exponent
±, except for the highest temperatures and longest periods for the ¬ne-grained sample when
the transition to viscous behaviour is evident. The dots represent the experimental data and
the lines the ¬t to the modulus from (11.1.10) and the attenuation from (11.1.11). [Courtesy
of U. Faul & I. Jackson.]

with real and imaginary parts
’mJ
‚ ln ψU d
ψ1(ω, d, t, p) = ψU(p) 1 + (T ’ TR)
‚T dR
R
„H
„±’1
±”ψ
+± ,
d„
„H ’ „± 1 + ω2„2
„L
L
„H
„±’1
ω± ”ψ 1
ψ2(ω, d, t, p) = ψU(p) + ,
d„ (11.1.9)
„± ’ „± 1 + ω2„2 ω„M
„L
H L

The frequency-dependent shear modulus is then to be found from
m(ω) = [ψ2(ω) + ψ2(ω)]’1/2, (11.1.10)
1 2

and the energy dissipation
Q’1(ω) = ψ2(ω)/ψ1(ω). (11.1.11)
212 Seismology and Earth Structure

The extended Burgers model provides a good representation of the experimental
results as a function of temperature, frequency and grain size as illustrated in Figure
11.2 for synthetic olivine aggregates with mean grain sizes 3 μm and 23 μm. With
suitable choices for the grain size and temperature distributions with depth, it also
proves possible to achieve a good match to shear wavespeed and attenuation pro¬les
derived directly from seismological models.
This parametric model for intrinsic attenuation based on detailed experimental
observations captures many signi¬cant aspects of attenuation within the Earth.
The temperature derivatives of both wavespeed and attenuation vary rapidly as
temperature is increased towards the solidus, but strong change is not by itself an
indicator of partial melt.


11.2 Seismic sources
In the application of linearised elasticity to the representation of seismic waves we
have made an explicit assumption about the relationship of the incremental stress
σ and the displacement ¬eld u. At any point where this constitutive relationship
is inadequate we can rectify the situation by introducing an additional stress tensor
component or an equivalent force system.
The main places where the true stress differs from that implied by the linearised
relations will be in the immediate vicinity of the generation of seismic waves where
strains exceed 10’5 . For an explosive source this zone will approximate a sphere
around the point of initiation. Whereas for an earthquake, the difference “ between
the actual and linearised estimate of the stress tensor will be concentrated in a
narrow zone surrounding the portion of a fault that has slipped, including any gouge
zone.
The stress difference “ij, which is often referred to as the ˜stress glut™, can be
viewed as a local density of source excitation with an equivalent body force system
e = ’∇ · “. This stress glut is an example of a moment tensor density mij. For an
internal source there can be no net change in linear and angular momentum of the
whole Earth, so that mij is symmetric, mij = mji.
We can examine the radiation from a dislocation source, representing the action
on an earthquake, by exploiting the representation for displacement introduced
in Section 5.3.2. We consider a region V containing some form of dynamic
discontinuity across a fault surface Σ (Figure 11.3).
We apply the integral representation (5.3.18) for the displacement ¬eld u to
a surface S consisting of the two surfaces Σ+, Σ’ lying on either side of the
fault surface Σ and joined at the termination of the dislocation (Figure 11.3). We
work in the frequency domain so that the convolution integrals in time in (5.3.18)
reduce to products of Fourier components, and we can include anelastic effects.
When the Green™s tensor corresponds to the con¬guration in V before the action
of the discontinuity, Gk(x, ξ, ω) and its associated traction Hk(x, ξ, ω) will be
continuous across Σ. The two surface integrals over Σ+, Σ’ can therefore be
11.2 Seismic sources 213




n
Σ
Σ+
Σ’


V

‚V

Figure 11.3. Surface of discontinuity in displacement and tractions Σ representing a source
of seismic energy.


combined into a single integral

d2ξ {Gkq(x, ξ, ω)[tq(ξ, ω)]+ ’ [uq(ξ, ω)]+Hkq(x, ξ, ω)},
uk(x, ω) = ’ ’ ’
Σ
(11.2.1)

where [tq(ξ, ω)]+, [uq(ξ, ω)]+ represent the jumps in traction and displacement
’ ’
’ to Σ+. The normal n to Σ is taken to be directed from Σ’
in passing from Σ
to Σ+. The equivalent source distribution will lie along the fault surface Σ. A
general dislocation with both displacement and traction jumps is allowed to provide
a ¬‚exible parameterisation of possible sources associated with a fault surface. The
traction associated with the Green™s tensor is given by

Hkq(x, ξ, ω) = np(ξ)cpqrs(ξ, ω)‚rGks(x, ξ, ω). (11.2.2)

If the surface Σ intersects the surface or internal surfaces of material discontinuity
additional surface tractions are introduced.
In (11.2.1) we are only concerned with the values of Gkq and hkq on the fault
surface Σ, so we introduce the Dirac delta function δΣ(ξ, ·) and its derivative
localised on the surface Σ. The expression for the radiated seismic displacement
(11.2.1) can then be cast as a surface integral over Σ of the form

d3· Gkq(x, ·, ω) [tq(ξ, ω)]+δΣ(ξ, ·)
uk(x, ω) = ’ (11.2.3)

V
+[us(ξ, ω)]+nr(ξ)cpqrs(ξ, ω)‚pδΣ(ξ, ·) ,


where ’‚pδ extracts the derivative of the function it acts upon and we have
exploited the symmetry of the moduli crspq = cpqrs.
214 Seismology and Earth Structure

The traction jump can be represented directly in terms of a set of force elements
distributed along Σ weighted by the size of the discontinuity,
= ’nr(ξ)[„qr(ξ, ω)]+δΣ(ξ, ·).
q(·, ω) (11.2.4)


The displacement jump [u] leads to force doublets along Σ which are best
represented in terms of a moment tensor density mpq,
mpq(·, ω) = nr(ξ)cpqrs(ξ, ω)[us(ξ, ω)]+δΣ(ξ, ·). (11.2.5)


The seismic radiation predicted by (11.2.1) is determined solely by the
displacement and traction jumps imposed across the surface Σ. The properties
of the material surrounding the fault appear only indirectly through the nature of
the Green™s tensor Gkq. Frequently, some assumed model of the slip behaviour
on the fault is used to specify [u]+. A full solution for [u]+ requires a much more
’ ’
complex calculation in which the propagating fault interacts with its surroundings;
such calculations are just becoming feasible for simple situations.
Earthquake models normally prescribe only tangential displacement jumps and
then nr[u]+ = 0. A normal displacement will occur for an opening crack that

occurs in, e.g., rock bursts in mines.
In general we can represent the effect of a distributed dislocation source (as in a
major earthquake) through an integral over the source volume

d3· ‚pGkq(x, ·, ω)mpq(·, ω).
uk(x, ω) = (11.2.6)
V
Frequently, we are interested in the best equivalent point source for which
uk(x, ω) ≈ ‚pGkq(x, xs , ω)Mpq(ω), (11.2.7)
where the point moment tensor

d3· mpq(·, ω).
Mpq(ω) = (11.2.8)
V
The moment tensor Mpq is the most commonly used description of a seismic
source, and as we see from (11.2.7) represents the integral of the moment tensor
density across the source region. The centroid of the disturbance may therefore be
displaced from the point of initiation of the event. We note that the moment tensor
acts on ‚pGkq or, equivalently, the strain associated with the Green™s tensor G.
The moment tensor M, represents the weights to be applied to the set of 9
force doublets produced by displacements along the coordinate axes, as illustrated
in Figure 11.4. The diagonal elements of M correspond to dipoles and the
off-diagonal elements to pure couples. For an internal source M is required to
be symmetric, e.g., M12 = M21 and so each couple is accompanied by another
that neutralises the induced moment, giving an effective double-couple source.
If we represent an earthquake by displacement [u] in the slip direction ν , over
an area A, on a planar fault surface with normal n, the point moment tensor is
Mij(ω) = A[u(ω)] (κ ’ 2 μ)nkνk + μ(niνj + njνi) ,
¯ (11.2.9)
3
11.2 Seismic sources 215

3
M 11 M 12 M 13
2



1


M 21 M 22 M 23




M 31 M 32 M 33




Figure 11.4. The representation of the elements of the moment tensor as weights for a set
of dipoles and couples.

in terms of the bulk modulus κ and shear modulus μ. This is equivalent to a
double-couple mechanism with axes along the orthogonal directions n, ν. The
symmetry of n, ν in (11.2.9) means that the radiation patterns carry no distinction
between the fault plane and the orthogonal auxiliary plane. For P wave radiation
both will be nodal planes; hence additional evidence is needed to resolve the
ambiguity.
The trace of the moment tensor for this displacement source vanishes, tr M =
i Mii = 0, since there is no volumetric component. Estimates of earthquake
mechanisms from waveform ¬tting often suggest the presence of a minor isotropic
component. Such effects can arise from a non-planar rupture surface, directivity
in rupture inadequately captured in a point estimate, and the in¬‚uence of
three-dimensional structure in the Earth.
The scalar moment spectrum
Mo = M0(ω) = A[u(ω)],
¯ (11.2.10)
characterises the behaviour of the event. The extrapolation to zero frequency
M0(0) provides a convenient measure of the size of the event, and forms the basis
of the earthquake magnitude scale Mw introduced by Kanamori:
Mw = (log Mo/1.5) ’ 10.7, (11.2.11)
when Mo is measured in dyn cm [10’7 N m]; the constant is introduced to tie to
216 Seismology and Earth Structure

earlier scales. The moment magnitude Mw has the advantage that it avoids the
problems of saturation encountered in more traditional measures such as mb, using
body waves, and Ms from surface waves.


11.3 Building the response of the Earth to a source
As an approximation to the whole Earth we will consider a spherical, non-rotating
model incorporating the possible effects of self-gravitation associated with slower
deformation. This will provide the framework for building representations of the
behaviour to which further complexities such as rotation and three-dimensional
structure can be added.
We will work in spherical polar coordinates (r, θ, φ) so that
x1 = r sin θ cos φ, x2 = r sin θ sin φ, x3 = r cos θ. (11.3.1)
In the undisturbed state before the source is activated
‚0 ‚
σij = ρ0 •0, (11.3.2)
‚xi ‚xi
where the gravitational potential •0 satis¬es Poisson™s equation
∇2•0(r) = 4πG•0(r). (11.3.3)
We will assume that the initial state is in hydrostatic equilibrium and then as in
Section 6.1.1 we have
‚p0 ‚g0 2 ‚•0
’ = ρ0g0, + g0 = 4πG•0(r), where g0(r) = , (11.3.4)
‚r ‚r r ‚r
and the acceleration due to gravity is g0 = ’g0^r, where ^r is the unit vector in the
e e
radial direction. The equations (11.3.4) are subject to the boundary conditions that
•0, g0, p0 are continuous across all surfaces of discontinuity in physical properties
[•0]+ = 0, [g0]+ = 0, [p0]+ = 0, (11.3.5)
’ ’ ’

and p0 must vanish at the free surface. The gravitational acceleration and potential
must diminish to zero at in¬nity and so in terms of the density ρ0:
r
4πG
ds s2ρ0(s),
g0(r) = 2
r 0
∞ re
GM
•0(r) = ’ ds g0(s) = ’ ’ ds g0s, (11.3.6)
re
r r
re
p0(r) = ds ρ0(r)g0(r), (11.3.7)
r
where re is the radius of the Earth and M is the total mass
re
ds s2ρ0(s).
M = 4π (11.3.8)
0
11.3 Building the response of the Earth to a source 217

With a time-dependent displacement u(x, t) induced by a force system f, the
linearised equation of motion takes the form
‚2u
ρ0 2 = f + ∇ · σ + ∇(u · ∇p0) ’ ρ0∇•1 ’ ρ1∇•0; (11.3.9)
‚t
the second term on the right hand side of (11.3.9) is the familiar gradient of the
incremental stress σ associated with u, and the remaining contributions come from
the perturbations in the density distribution induced by u. From the conservation
of mass
ρ1 = ’∇(ρ0u), (11.3.10)
so the incremental gravitational potential satis¬es the Poisson equation
∇2•1 = 4πGρ1 = ’4πG ∇(ρ0u). (11.3.11)
With the aid of (11.3.3) and (11.3.11) we can write the momentum equation
(11.3.9) in component form
‚2ui ‚σij ‚uk ‚•0 ‚•1 ‚ ‚•0
ρ0 2 = fi + + ρ0 ’ ’ uk
‚t ‚xj ‚xk ‚xi ‚xi ‚xi ‚xk
‚ρ0 ‚•0 ‚ρ0 ‚•0
+ ’ uk. (11.3.12)
‚xk ‚xi ‚xi ‚xk
In the hydrostatic state both ρ0 and •0 depend only on r and so the ¬nal term in
(11.3.12) vanishes.
The system of partial differential equations for the displacement u and
incremental gravitational potential •1 (which depends on u) thus takes the form
‚2ui ‚ ‚uk ‚uk ‚•0 ‚•1 ‚ ‚•0
ρ0 2 = fi + cijkl + ρ0 ’ ’ uk ,
‚t ‚xj ‚xk ‚xk ‚xi ‚xi ‚xi ‚xk

∇2•1 = 4πG (ρ0uk), (11.3.13)
‚xk
where we have used the generalised Hooke™s law (5.2.7) to express the incremental
stress in terms of the displacement gradient.
The boundary conditions on the incremental gravitational potential •1 are that it
vanishes at in¬nity and the continuity conditions
‚•1
+ +
•1 = 0, + 4πGρ0ur = 0 (11.3.14)
‚r
’ ’

apply at all surfaces of material discontinuity, including the free surface. Here
ur = u · ^r is the radial component of displacement.
e
The boundary conditions on the displacement depend on the nature of the
discontinuity surface: at the free surface r = re the traction must vanish
‚uk
σijnj = 0, i.e., cijkl nj = 0; (11.3.15)
‚xl
218 Seismology and Earth Structure

at ¬‚uid“solid boundaries, we require continuous traction
[σijnj]+ = 0, (11.3.16)


the traction must be normal to the boundary
(δik ’ nink)σkjnj = 0; (11.3.17)
and the normal displacement must be continuous
[uknk]+ = 0; (11.3.18)


whereas at solid“solid (welded) boundaries both traction and displacement must be
continuous
[σijnj]+ = 0, [ui]+ = 0. (11.3.19)
’ ’

The displacement must also be ¬nite at the centre of the Earth (r = 0).
In the frequency domain, (11.3.13) becomes
‚ ‚uk ‚uk ‚•0 ‚•1 ‚ ‚•0
ρ0ω2ui+ cijkl + ρ0 ’ ’ uk = ’fi,
‚xj ‚xk ‚xk ‚xi ‚xi ‚xi ‚xk
(11.3.20)
There is a set of frequencies ωI for which a displacement solution ue exists that
I
satis¬es (11.3.20) and the full set of boundary conditions in the absence of external
forcing (i.e., f = 0). These normal modes with index I form a basis from which we
can build the response to excitation.


11.3.1 Displacements as a normal mode sum
The eigendisplacements corresponding to different modes of oscillation are
orthogonal and can be normalised so that

d3x ρ0(r)ue(x) · [ue(x)]— = δIJ, (11.3.21)
I J
Ve

where the integral includes the entire volume of the Earth and the star denotes a
complex conjugate. We expand the displacement ¬eld in the presence of a source
as a sum of the normal modes

cIue(x),
u(x, ω) = (11.3.22)
I
I=0

and from the orthonormality of the eigenfunctions

d3x ρ0(r)u(x) · [ue(x)]— .
cI = (11.3.23)
I
Ve

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