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We can write the equation of motion (11.3.20) in the operator form
H(u) + ПЃ0П‰2u = в€’f, (11.3.24)
11.3 Building the response of the Earth to a source 219

and then
cJH(ue) + ПЃ0П‰2 cJue = в€’f, (11.3.25)
J J
J J

and thus, taking the scalar product of (11.3.25) with [ue ]в€— ,
K

d3x cJH(ue) В· [ue ]в€— + П‰2cK = в€’ d3x f В· [ue ]в€— . (11.3.26)
J K K
Ve Ve
J

Each eigenvector ue satisп¬Ѓes the equation
I

H(ue) + ПЃ0П‰2ue = 0, (11.3.27)
I I

and hence

d3x H(ue) В· [ue ]в€— + П‰2ОґJK = 0. (11.3.28)
J K
Ve

With this substitution in (11.3.26) the coefп¬Ѓcients in the modal expansion can be
found as
1
d3x f В· [ue ]в€— .
cK = 2 (11.3.29)
K
П‰K в€’ П‰2 Ve
We have therefore a frequency-domain representation of the displacement induced
by the force system f:
ue
d3x f В· [ue ]в€— .
K
u(x, П‰) = (11.3.30)
K
2 в€’ П‰2
П‰K Ve
K

For a step function time history H(t) for f we can perform the inverse Fourier
transform and recover the time domain response, extracting the residue at each
pole П‰K using CauchyвЂ™s theorem, to produce
1 в€’ cos(П‰Kt)
ue d3x f В· [ue ]в€— .
u(x, t) = (11.3.31)
K K
П‰2 Ve
K
K

When we allow for the decay of the eigenmodes produced by slight anelasticity
в€’ cos(П‰Kt)eв€’iП‰K t/2QK
1
ue d3x f В· [ue ]в€— .
u(x, t) = (11.3.32)
K K
П‰2 Ve
K
K

The static response as t в†’ в€ћ is also given by a normal mode sum
1
ue d3x f В· [ue ]в€— .
u(x, t) = (11.3.33)
K K
П‰2 Ve
K
K

So far we have not speciп¬Ѓed the nature of the force system f, but if we specialise
to the point moment tensor formulation introduced in Section 11.2 above then

d3x f В· [ue ]в€— = [ee ]в€—: M(xs) = [ee ]в€— Mpq(xs) (11.3.34)
K pq
K K
Ve
220 Seismology and Earth Structure

where eK is the strain tensor associated with the Kth mode.
We can extend the treatment to a source that depends on time by introducing the
moment rate tensor
в€‚
M(t) = M(t). (11.3.35)
в€‚t
The moment rate tensor has the advantage that it differs from zero only during the
occurrence of an earthquake, e.g., if the moment tensor M behaves as a Heaviside
step function, the moment rate tensor M is a delta function Оґ(t).
We also introduce the modal time function

CK(t) = 1 в€’ cos(П‰Kt)eв€’iП‰K t/2QK H(t), (11.3.36)

and then the complete response can be written as a time convolution of the modal
response with the moment rate tensor,
в€ћ
1
CK(t в€’ t )ue [ee ]в€—: M(xs, t) .
u(x, t) = dt (11.3.37)
KK
2
П‰K
в€’в€ћ
K

11.3.2 Free oscillations of the Earth
We have seen that we can build the response of the Earth to a source through the
superposition of contributions from the normal modes of free oscillation of the
Earth. For all except the low frequency modes the inп¬‚uence of the rotation of
the Earth is small, and it is possible to analyse the modes in terms of two distinct
class of modes in a spherical harmonic representation: the toroidal modes with
purely torsional behaviour and the spheroidal modes which include the inп¬‚uence
of self-gravitation. A comprehensive treatment of the development of the low
frequency part of the seismic wave п¬Ѓeld in terms of these normal modes is presented
by Dahlen & Tromp (1998).
The presence of the п¬‚uid core means that for the toroidal modes the boundary
conditions are the vanishing of radial traction at both the surface and the
coreвЂ“mantle boundary. The spheroidal modes penetrate right to the centre of
the Earth and can have signiп¬Ѓcant energy in both the outer and inner core.
For each spherical harmonic, the displacement u and the radial traction tr are
expressed in terms of vector surface harmonics to yield quantities which describe
the displacement and radial traction п¬Ѓelds and which are linked through п¬Ѓrst-order
ordinary differential equations in radius. For spheroidal modes the inп¬‚uence of
self-gravitation is important for low frequencies. For each angular order l there will
be a set of modes described by a radial order n whose frequencies are determined
by satisfying the interior and exterior boundary conditions.
The calculation of the free oscillations of the Earth depends on the numerical
integration of the coupled sets of ordinary differential equations. These systems
11.3 Building the response of the Earth to a source 221

0 3 6
Y6 Y6 Y6

m
Figure 11.5. Behaviour of the spherical harmonics of degree 6. The real parts of Y6 are
plotted for m = 0, 3, 6 as deviations from a sphere of radius 5.

govern the radial behaviour of the normal mode eigenfunctions with an assumed
angular dependence through the spherical harmonics of order l,

2l + 1 (l в€’ m)! m
Yl (Оё, П†) = (в€’1)m
m
Pl (cos Оё)eimП†, (11.3.38)
4ПЂ (l + m)!

m
where Pl is an associated Legendre polynomial and в€’l в‰¤ m в‰¤ l. The behaviour
m
of Yl for l = 6 is illustrated in Figure 11.5.
For short wavelengths the standing waves of the spherical harmonic can be
decomposed into two travelling waves circling the Earth in opposite directions with
phase velocities

c = П‰re /(l + 1 ). (11.3.39)
2

For simplicity we will discuss the calculation of free oscillation periods for a
spherical non-rotating Earth model. With these assumptions the normal mode
periods are independent of the azimuthal order m, i.e., each period has (2l+1)-fold
degeneracy. This degeneracy is broken by the EarthвЂ™s rotation, the ellipticity of
the п¬Ѓgure of the Earth, and the presence of three-dimensional structure within the
Earth.
We project the displacement u and radial traction tr on a vector spherical
harmonic п¬Ѓeld

l
U(r)Pm(Оё, П†) + V(r)Bm(Оё, П†) + W(r)Cm(Оё, П†) , (11.3.40)
u= l l l
l m=в€’l

l
P(r)Pm(Оё, П†) + S(r)Bm(Оё, П†) + T (r)Cm(Оё, П†) ,
tr = (11.3.41)
l l l
l m=в€’l
222 Seismology and Earth Structure

where the orthogonal vector surface harmonics have the properties
Pm(Оё, П†) = Yl ^r,
m
e (11.3.42)
l
m m
1 в€‚Yl 1 в€‚Yl
m
Bl (Оё, П†) = ^Оё + ^П† ,
e e (11.3.43)
в€‚Оё sin Оё в€‚П†
ЕЃ
m m
1 в€‚Yl 1 в€‚Yl
m
Tl (Оё, П†) = в€’ ^Оё в€’ ^П† .
e e (11.3.44)
в€‚Оё sin Оё в€‚П†
ЕЃ
Here ^r, ^Оё, ^П† are the unit vectors in the r, Оё, П† directions of the spherical
eee
coordinates, and we have set ЕЃ = [l(l + 1)]1/2. The vector spherical functions
are orthogonal so that, e.g.,
ПЂ 2ПЂ
dП† [Rm]в€— Rm = Оґmm Оґll ,
sin ОёdОё l l
0 0
ПЂ 2ПЂ
dП† [Rm]в€— Sm = 0.
sin ОёdОё (11.3.45)
l l
0 0

With this representation the governing equations (11.3.20) separate into two
groups of coupled differential equations in terms of radius r. The toroidal
oscillations associated with the Cm(Оё, П†) separate from the spheroidal oscillations
l
that link the coefп¬Ѓcients of Pm(Оё, П†), Bm(Оё, П†).
l l
The п¬Ѓrst few overtone sequences of the normal modes link through the travelling
wave representations at short periods to Love waves for the toroidal oscillations,
and to Rayleigh waves for the spheroidal oscillations.

Toroidal oscillations
The toroidal oscillations involve no radial displacement and can be described
entirely through the coefп¬Ѓcients of the Cm(Оё, П†) vector harmonics, i.e., the
l
displacement coefп¬Ѓcient W, and traction coefп¬Ѓcient T . The toroidal modes do not
penetrate into the п¬‚uid core so disturbances are conп¬Ѓned to the mantle.
In the mantle, the quantities W and T are related by
rв€’1 Ојв€’1
d W W
= . (11.3.46)
в€’ПЃП‰2 + (l + 1)(l + 2)Ојrв€’2 в€’3rв€’1
T T
dr
The boundary conditions that have to be applied for these equations for a free
oscillation are
T (re ) = 0 at the surface r = re , and
T (rc ) = 0 at the coreвЂ“mantle boundary r = rc . (11.3.47)
The toroidal equations may thus be integrated from the coreвЂ“mantle boundary (r =
rc ) with a starting solution,
W(rc ) = 1, T (rc ) = 0, (11.3.48)
11.3 Building the response of the Earth to a source 223

and the secular equation for the torsional modes that determines the allowed
frequency П‰ to satisfy the boundary conditions is
FT (П‰, l) = T (re , П‰, l) = 0. (11.3.49)
For higher mode overtones of the torsional modes at large l, integration from the
coreвЂ“mantle boundary can lead to overп¬‚ow in the computations. This is because
the main variation in the eigenfunction is concentrated near the surface with only
an exponential tail decaying into the lower mantle. In this case it is worth starting
the integration from a higher level in the mantle. We assume that for a surface of
radius r1 the material is uniform, then a suitable starting solution is
W(r1) = 1,
T (r1) = (Ој/r1)[(l в€’ 1) в€’ zl(П‰r/ОІ)], (11.3.50)
where zl is deп¬Ѓned in terms of the spherical Bessel functions by
zl(x) = xjl+1(x)/jl(x), (11.3.51)
Takeuchi & Saito (1972) have given a recursion relation for zl
zlв€’1(x) = x2/[(2l + 1) в€’ zl(x)]. (11.3.52)
For a large wavenumber this recursion formula may be applied in order of
decreasing l from an initial value zl = x2/(2l + 3).
We work at п¬Ѓxed angular order l (i.e., п¬Ѓxed wavenumber) and integrate (11.3.46)
up from the initial values with a trial value of frequency П‰, and the surface value of
T (re ) is then the required secular function. If the trial frequency does not lead to the
vanishing of the secular function then the eigenfrequency estimate can be improved
by using variational results. For a trial frequency П‰t an improved estimate П‰n may
be found from
П‰2 = П‰2 + B/T , (11.3.53)
n t

where for this torsional mode case
re
dr r2ПЃ(r)W 2(r), B = r2W(re )T (re ),
T= (11.3.54)
e
0
with W and T calculated for the trial frequency П‰t. This scheme may be applied
iteratively until convergence to the eigenfrequency of the desired normal mode is
achieved. Once the eigenfrequency has been found the last set of values for W(r)
and T (r) deп¬Ѓne the corresponding eigenfunction, which may need to be normalised.
For a given angular order l the toroidal modes are designated nTl in order of
increasing frequency П‰n, n = 0, 1, 2, .... The index n is the radial order and
for the toroidal modes corresponds to the number of zero crossings in the radial
eigenfunctions. The (2l + 1)-fold degeneracy in the eigenfrequency for azimuthal
order m is broken in the presence of lateral heterogeneity, and can be accounted for
using perturbation theory.
224 Seismology and Earth Structure

CMB

0 T50 T34 2 T26
1
ICB

W

CMB

3 T21 T18 5 T12
4
ICB

Figure 11.6. Radial variation of the eigendisplacement W associated with toroidal modes
for model AK135, with frequencies close to 6 mHz. The upper row corresponds to modes
that are equivalent to turning S waves in the mantle and the lower row to ScS equivalent
modes.

The radial dependence of the eigenfunctions for the set of toroidal modes with
eigenfrequency close to 6 mHz is shown in Figure 11.6. The displacement W is
conп¬Ѓned to the mantle. The fundamental mode 0T50 is equivalent to a Love wave
with decay of the displacement W with depth. As the radial order n increases
the depth of penetration increases and the modes can be regarded as equivalent
to mantle S waves. For the higher overtones in the lower row of Figure 11.6
the displacement п¬Ѓlls the whole mantle; this behaviour represents ScS equivalent
modes.

Spheroidal oscillations
The spheroidal oscillations of the Earth give rise to a perturbation in the
gravitational potential; we therefore need to consider coupled elasticвЂ“gravitational
disturbances and satisfy the elastic equations of motion and PoissonвЂ™s equation
11.3 Building the response of the Earth to a source 225

simultaneously. Unlike the toroidal modes, the spheroidal mode involves the whole
Earth with motion in both the solid and liquid parts of the core and the mantle.
The displacement and traction п¬Ѓelds link the Pm(Оё, П†) and Bm(Оё, П†) vector
l l
harmonics through the quantities U, V, P, S. We express the perturbation in
gravitational potential as
m
П•(x, П‰) = в€’О¦(r)Yl (Оё, П†), (11.3.55)
l+1
dОЁ
ОЁ= + 4ПЂGПЃU + О¦. (11.3.56)
r
dr
This scalar ОЁ has the desirable properties of being continuous, zero at the EarthвЂ™s
surface r = re and identically zero for r > re .
The displacement coefп¬Ѓcient U, V, traction coefп¬Ѓcients P, S and the gravitational
terms О¦, ОЁ satisfy the coupled set of ordinary differential equations with respect to
О· 1
dU
= (ЕЃV в€’ 2U) + P,
r О» + 2Ој
dr
1 1
dV
= в€’ (V в€’ ЕЃU) + S,
r Ој
dr
П‡ ПЃg ПЃg П‡
dP
= в€’ПЃП‰2 + 4 2 в€’ U+ЕЃ в€’2 2 V
r r r r
dr
2(О· в€’ 1) (l + 1)ПЃ
ЕЃ
+ P+ Sв€’ О¦ + ПЃОЁ,
r r r
ЕЃ2 в€’ 2Ој
ПЃg П‡ О· 3 ПЃ
dS 2
=ЕЃ в€’ 2 2 U в€’ПЃП‰ + V в€’ ЕЃ P в€’ S + ЕЃ ОЁ,
r2
r r r r r
dr
l+1
dО¦
= в€’4ПЂGПЃU в€’ ОЁ + ОЁ,
r
dr
4ПЂGПЃ lв€’1
dОЁ
= [ЕЃV в€’ (l + 1)U] + ОЁ; (11.3.57)
r r
dr
with
О» Ој(3О» + 2Ој) 4Ој(О» + Ој)
О·= , П‡= , ОЅ= . (11.3.58)
О» + 2Ој О» + 2Ој О» + 2Ој
In this form the set of elements of the coupled differential equations reduce
asymptotically to the forms for plane stratiп¬Ѓcation for large П‰ and l.
The boundary conditions that have to be satisп¬Ѓed at the EarthвЂ™s surface are that
the surface traction and gravitational scalar ОЁ should vanish, i.e.,
P(re ) = S(re ) = 0, ОЁ(re ) = 0. (11.3.59)
The set of six coupled equations (11.3.58) will have six independent solutions, but
the solutions of interest are the three that are regular at the origin. We therefore
226 Seismology and Earth Structure

start the integrations for these three solutions in the solid inner core and integrate
up to the EarthвЂ™s surface. The secular equation for the spheroidal oscillations is
then given by the vanishing of the determinant
P1(re ) P2(re ) P3(re )
FS(П‰, l) = S1(re ) S2(re ) S3(re ) = 0. (11.3.60)
ОЁ1(re ) ОЁ2(re ) ОЁ3(re )
We model the outer core of the Earth as a liquid with vanishing rigidity and, in
consequence, the system of differential equations is simpliп¬Ѓed since no tangential
stress can be supported. In the equation system (11.3.57) О· = 1, П‡ = 0, ОЅ = 0 and
ЕЃ
V= [P + ПЃ(gU в€’ ОЁ)] , (11.3.61)
ПЃП‰2r
so that in the п¬‚uid
2 1
dU ЕЃ
= в€’ U + V + P,
r r О»
dr
ПЃg ПЃg (l + 1)ПЃ
dP
= в€’ ПЃП‰2 + 4 U+ЕЃ V в€’ О¦ + ПЃОЁ,
r r r
dr
l+1
dО¦
= в€’4ПЂGПЃU в€’ ОЁ + ОЁ,
r
dr
4ПЂGПЃ lв€’1
dОЁ
= [ЕЃV в€’ (l + 1)U] + ОЁ; (11.3.62)
r r
dr
V becomes indeterminate at zero frequency (П‰ = 0).
We can summarise the set of displacement, traction and gravitational variables
through the column vector
b = [U, V, P, S, О¦, ОЁ]T . (11.3.63)
The solution of (11.3.57) in the inner core can be written as a superposition of the
three regular solutions
bI = CI bI + CI bI + CI bI , i = 1, 2, ..., 6; (11.3.64)
i 1 1i 2 2i 3 3i

whereas in the liquid outer core we have
bK = CI bK + CI bK + CI bK , i = 1, 3, 5, 6; (11.3.65)
i 1 1i 2 2i 3 3i

as we are now only dealing with four coupled equations. In the solid mantle we
have again
bM = CI bM + CI bM + CI bM, i = 1, 2, ..., 6; (11.3.66)
i 1 1i 2 2i 3 3i

When we integrate the coupled equations outward from the inner core we must
be able to connect the solutions in the different regions. We can, e.g., use the
requirement that at the inner core boundary r = ri the tangential stress SI must
11.3 Building the response of the Earth to a source 227

vanish and so one of the constants of integration can be eliminated in the inner
core,

SI I SI I
CI = в€’ 1 C1 в€’ 2 C2. (11.3.67)
3
SI SI
3 3

The continuity relations at the inner core interface are

SI I SI I
bK (ri ) bI в€’ bK (ri ) bI в€’
1 2
= b, = b, i = 1, 3, 5, 6, (11.3.68)
1i 1i
SI 3i 2i 2i
SI 3i
3 3

with

CK = CI , CK = CI . (11.3.69)
1 1 2 2

Once we reach the core mantle boundary we may continue two solutions directly
into the mantle, i.e., at the interface r = rc ,

bK (rc ) = bM(rc ), bK (rc ) = bM(rc ), i = 1, 3, 5, 6 (11.3.70)
1i 1i 2i 2i

supplemented by the requirements

V1 (rc ) = SM(rc ) = V2 (rc ) = SM(rc ) = 0, CM = CK, CM = CK. (11.3.71)
M M
1 2 1 1 2 2

The third set of mantleвЂ“core boundary values can be determined by recalling that
the tangential displacement V M need not be continuous across the interface. We
may thus take
M
UM = P3 = SM = О¦M = ОЁM = 0.
M
V3 = 1, (11.3.72)
3 3 3 3

The constants of integration CM, CM, CM are to be determined by satisfying the
1 2 3
free surface boundary conditions for an eigenfrequency.
The starting conditions at the centre of the Earth are based either on a locally
uniform medium or on power series. For a given trial frequency, the spheroidal
equations have to be integrated to the surface using the continuation scheme we
have just described. The determinantal secular function (11.3.60) has then to be
constructed from the surface values of Pi, Si, ОЁi from the separate integrations.
In this direct approach very high order accuracy is needed in the integration
scheme to overcome the вЂњstiffвЂќ nature of the differential equation system. In
consequence modern programs favour working directly with the minors of the
secular determinant, with a much more complex set of differential equations but
Iterative improvement of the estimated frequency П‰t can be achieved with the
variational result for an improved frequency П‰n

П‰2 = П‰2 + B/T , (11.3.73)
n t
228 Seismology and Earth Structure

with
re
dr r2ПЃ(r)[U2(r) + V 2(r)],
T=
0
2
+ V(re )S(re ) + (4ПЂG)в€’1О¦(re )ОЁ(re )].
B= re [U(re )P(re ) (11.3.74)
The particular set of functions b to be used in these equations must represent a good
approximation to the eigenfunction at the true eigenfrequency. The combination
of the three independent solutions must therefore be made as if П‰t were the
eigenfrequency.
Spheroidal modes nSl have a similar notation to that used for the toroidal modes.
The index n again indicates the frequency order at п¬Ѓxed l, but unlike the toroidal
case does not correspond directly to the number of nodal surfaces in radius.
The eigenfunctions for the spheroidal modes are illustrated in Figure 11.7, once
again for eigenfrequencies close to 6 mHz. The range of behaviour is much more
complex than for the toroidal modes of similar frequency shown in Figure 11.6,
because the spheroidal modes include P wave behaviour and are no longer conп¬Ѓned
to the mantle. Whereas U, О¦ are continuous, the eigendisplacement V displays
jumps at the п¬‚uidвЂ“solid interfaces at the top and bottom of the inner core. The
progression of the higher overtones with eigenfunctions extending through more
of the Earth is interrupted by two modes that correspond to trapped Stoneley
waves concentrated at these п¬‚uidвЂ“solid interfaces. The mode 3S28 is trapped at the
coreвЂ“mantle boundary, and mode 7S15 at the inner-core boundary. The variations in
the gravitational scalar are concentrated in the core and have very little expression
at the surface.
The fundamental mode 0S54 is equivalent to a Rayleigh wave. The modes 1S34
and 4S24 correspond to S waves turning in the mantle, and 5S19, 6S17 are equivalent
to ScS waves reп¬‚ected from the coreвЂ“mantle boundary which is no longer a specular
reп¬‚ector since transmitted P waves can make their way into the п¬‚uid core. The
bottom row in Figure 11.7 shows modes with much of their energy in the inner
core.

If we consider spheroidal free oscillations for angular order zero, the deformation
is purely in the radial direction so we only need to consider the solution of two
coupled equations
в€’2О·rв€’1 (О» + 2Ој)в€’1
d U U
= . (11.3.75)
в€’ПЃП‰2 + 4(П‡rв€’2 в€’ ПЃgrв€’1) (2О· в€’ 2)rв€’1
P P
dr
where О·, П‡ are the composite moduli (11.3.58). The gravitational potential in this
case can be found from
dО¦
= в€’4ПЂGПЃU. (11.3.76)
dr
The surface boundary condition is as before P(re ) = 0.
11.3 Building the response of the Earth to a source 229

CMB

0 S54 1 S34 3 S28 4 S24
ICB
U
V
О¦

CMB

5 S19 6 S17 7 S15 8 S13
ICB

CMB

10 S9 14 S5 15 S3
ICB

Figure 11.7. Radial variation of the displacements U, V and gravitational term О¦ associ-
ated with spheroidal modes for model AK135, with frequencies close to 6 mHz. A more
complex behaviour is displayed than for the toroidal modes of similar frequency shown
in Figure 11.6. The low radial order modes are largely conп¬Ѓned to the mantle, but the
spheroidal modes can penetrate into the core. There are now modes that are equivalent to
Stoneley waves concentrated at the coreвЂ“mantle boundary 3 S28 and the boundary between
the inner and outer cores 7 S15 . Some modes such as 14 S5 have almost all their energy
concentrated in the inner core.
230 Seismology and Earth Structure

Since we have only two equations the treatment of these radial nS0 modes
parallels closely that for the torsional modes. For a starting solution we may use a
U = r, P = 3О» + 2Ој, (11.3.77)
and start the integration at some small radius r1. The same coupled equations
apply in the liquid and solid parts of the Earth so that there is no problem with the
continuity of solutions. The secular function is simply the surface value of P. As
before the estimate of the eigenfrequency can be improved by using the variational
technique; for a trial frequency П‰t an improved estimate is given by
П‰2 = П‰2 + B/T , (11.3.78)
n t

re
dr r2ПЃ(r)U2(r), B = r2[U(re )P(re ) + (4ПЂGre )в€’1О¦2(re )], (11.3.79)
T= e
0
with U, P and О¦ calculated for the trial frequency П‰t.

The RayleighвЂ“Ritz method
For an eigenfrequency of an Earth model we have equality of the total kinetic and
potential energies
re re
2 2
dr r2V(r) = 0,
П‰ dr r T(r) в€’ (11.3.80)
0 0

where П‰2T is the kinetic energy density per unit volume at radius r and V the
potential energy density. This expression is stationary around the eigenfrequency
and so approximated eigenfunctions and eigenfrequencies can be found by a
RayleighвЂ“Ritz procedure. A comprehensive treatment of this approach is provided
by Dahlen & Tromp (1998).
If the radial part of an eigenfunction ue(r) is represented as a linear combination
of N test functions О¶i(r) which each satisfy the boundary conditions then
N
u(r) в‰€ biО¶i(r). (11.3.81)
i=1

The energy balance equation results in a matrix eigenvalue problem of the form
(П‰2T в€’ V)b = 0. (11.3.82)
Each eigenvector b represents a projection of the modal eigenfunction u(r) into the
space spanned by the test functions О¶i and the corresponding value of П‰2 will be
an upper bound to the eigenfrequency associated with u. Since T and V depend on
the angular order l of a normal mode, we have a different set of matrix equations
for each angular order. The successive eigenvectors b obtained for a given angular
order represent higher radial order (overtones). With a suitable set of test functions
11.4 Probing the Earth 231

such as piecewise cubic Hermitian splines, the model-dependent parts of T and V
can be evaluated by GaussвЂ“Legendre integration with not less than six grid points
per wavelength. The computation time per eigenfrequency is independent of the
node, and no advantage can be taken of the properties of the mode. However,
difп¬Ѓculties such as problems with nearly coincident eigenfrequencies are avoided.

11.4 Probing the Earth
In this section we look at various aspects of the Earth as seen by seismic waves.
We start by considering relatively high-frequency wave propagation for which the
pattern of arrivals can be understood through ray tracing and wavefronts. We
introduce the terminology for seismic phases and show the way in which P and S
waves interact with the structure in the EarthвЂ™s mantle and core. Because the outer
core behaves as a п¬‚uid only P waves propagate through the core; the P wavespeed
at the top of the core is lower than the P wavespeed at the base of the mantle, but
higher than the corresponding S wavespeed. This leads to very different behaviour
for the transmitted P waves in the core, depending on whether they are derived from
P or S waves in the mantle. Horizontally polarised SH waves are totally reп¬‚ected
from the coreвЂ“mantle boundary and so are conп¬Ѓned to the mantle.
The different classes of seismic propagation are also useful in understanding the
way in which the array of frequencies of the normal modes of the Earth relate to
structure. Toroidal modes correspond to SH waves and so have displacements only
in the mantle. In contrast, the spheroidal modes penetrate the whole Earth and the
pattern of modes is inп¬‚uenced by the structure of the core and the mantle.
The two complementary viewpoints on seismic wave propagation provide
convenient ways to look at the higher and lower frequency characteristics of
the seismic wave п¬Ѓeld. A spherical Earth model provides a good basis for
understanding the timing and general characteristics of seismic arrivals. However,
the presence of three-dimensional structure in the Earth leads to variations in the
properties of seismic phases that depend on the particular path between source and
receiver, rather than just their angular separation. This information can be exploited
in the techniques of seismic tomography to provide images of internal structure
exploiting both ray-based and normal-mode-based methods.

11.4.1 Seismic phases
The body wave portion of a seismogram is marked by the arrival of distinct bursts of
energy that can be associated with different classes of propagation path through the
Earth. This is illustrated in Figure 11.8, for three-component seismograms at 100в—¦
from an intermediate event in Vanuatu that displays a rich set of seismic phases that
are marked with their phase code. The individual seismic arrivals sample different
parts of the Earth in their passage between the source and the receiver and their
properties are dictated by the structure they encounter.
232 Seismology and Earth Structure

pPdiff
Pdiff PP SS
SP
Z sPdiff
SKKSAC
SKSAC PS
R

T

Sdiff

1200. 1800. 2400
Time [s]
KURK 100.49Лљ

Figure 11.8. Three-component seismogram from an intermediate depth earthquake in Van-
uatu recorded in Kazakhstan at approximately 100в—¦ from the source, showing a rich set of
seismic phase arrivals.

The phase code describing each propagation path is built up by combining the
different elements of the path through the major zones in the structure of the Earth
introduced in Chapter 1:
вЂў a leg in the mantle is denoted by P or S depending on wave type;
вЂў a compressional leg in the outer core is labelled as K;
вЂў a compressional leg in the inner core is indicated with I, the corresponding shear
wave leg would be denoted J;
вЂў waves leaving upward from the source are indicated by lowercase letters, thus
pP represents a surface reп¬‚ection for P near the source, and sP a reп¬‚ection with
conversion above the source; and
вЂў reп¬‚ected waves from major interfaces are indicated by lowercase letters: m for
the MohoroviЛ‡ iВґ discontinuity (Moho), c at the coreвЂ“mantle boundary and i at
cc
the boundary between the inner and outer cores.
A P wave which returns to the surface after propagation through the mantle and
is then reп¬‚ected at the surface to produce a further mantle leg will be represented
by PP. PKIKP is a P wave which has travelled through the mantle and both the
inner and outer cores, whilst PKiKP is reп¬‚ected back from the surface of the inner
core. Similarly an S wave that is reп¬‚ected at the coreвЂ“mantle boundary is indicated
by ScS, and if conversion occurs on reп¬‚ection we have ScP.
The times of passage of the different classes of seismic waves through the Earth
are sensitive to different aspects of Earth structure. For example, the time of arrival
of the PcP phase, which is reп¬‚ected from the coreвЂ“mantle boundary, is a strong
function of the radius of the core. The primary control on the wavespeed structure
in the AK135 model illustrated in Figure 1.6 comes from the times of P, S and the
major core phases, but in all the observed times of 19 different phases were used to
assess the properties of different models (Kennett et al., 1995).
PKJKP would correspond to a wave that traversed the inner core as a shear wave,
11.4 Probing the Earth 233

90

60 120

P
30 150

PKP

PKIKP

0 180

Figure 11.9. Ray paths for major P phases. Wavefronts are indicated by ticks on the ray
paths at 1 minute intervals.

a number of claims have been made for the observation of this phase but none have
been conп¬Ѓrmed. However, a very careful analysis by Duess et al. (2000) suggests
that the arrival SKJKP has been detected using recordings from many stations for a
very deep event in the Flores Sea, Indonesia.
The вЂ˜depth phasesвЂ™ (pP, sP and pS, sS) can be very distinct for deep events. The
time difference between the arrival of such phases reп¬‚ected at the surface near the
source and the main phases P, S provides a useful estimate of the source depth.
The way in which seismic energy travels through the Earth to emerge at the
surface can be illustrated by the nature of the ray paths and wavefronts associated
with different classes of arrivals. Figures 11.9вЂ“11.12 show the behaviour for major
P and S phases for surface source in the model AK135 of Kennett et al. (1995).
P wave legs are shown in black and S wave legs are indicated by using grey tone.
The time progression of the waves through the Earth is shown by marking the
wavefronts with ticks along the ray paths at one minute intervals; the spreading
and concentration of the ray paths provides a simple visual indication of the local
amplitude associated with each phase.
Kennett (2002) provides an extended treatment of the nature of seismic body
wave propagation through the Earth accompanied by illustrations of seismograms
for the major phases. Shearer (1999) shows stacks of seismograms recorded around
the globe that reveal the complexity of the range of propagation paths through
the Earth, including a variety of minor phases that are rarely recognised on single
seismograms.
The direct P waves refracted back from the velocity gradients in the mantle
extend to about 100в—¦ away from the source, but then reach the coreвЂ“mantle
boundary at grazing incidence. P waves with steeper take-off angles at the source
234 Seismology and Earth Structure

90

60 120

S
30 150

SKS

SKIKS

0 180

Figure 11.10. Ray paths for major S phases for the AK135 model of seismic wavespeeds.

are either reп¬‚ected from the coreвЂ“mantle boundary (PcP) or refracted into the core
(PKP). The refracted PKP waves have a relatively complex pattern of propagation
in the core (see Figure 11.14). The absence of shear strength in the п¬‚uid outer
core means that the P wavespeed at the top of the core is markedly less than at
the base of the mantle. The result of this wavespeed reduction is that P waves
entering the core have a much steeper inclination than in the mantle as a result
of SnellвЂ™s law. The refraction into the lower velocity medium combined with the
wavespeed gradients in the outer core produce a pronounced PKP caustic with a
concentration of arrivals near 145в—¦ from the source. The apparent вЂ˜shadow zoneвЂ™
with no P arrivals from 100 to 145в—¦ led to the discovery of the core by Oldham, at
the end of the 19th century. Reп¬‚ection from the boundary between the inner and
outer cores (PKiKP) and refraction through the outermost part of the inner core
(PKIKP) produce small-amplitude arrivals which help to п¬Ѓll in the gap between
direct P out to 100в—¦ and the PKP caustic. The identiп¬Ѓcation of these arrivals in
the вЂ˜shadow zoneвЂ™ led to the discovery of the inner core by Inge Lehmann. The
reduction in P wavespeed is so large that there are no turning points for P waves
in the upper part of the core. Eventually the effect of sphericity and the increase
in wavespeed with depth is sufп¬Ѓcient to produce turning points that combine to
produce the PKP caustic that emerges at the surface near 144в—¦ .
For S waves (Figure 11.10) the pattern of propagation in the mantle is similar
to that for P. However, since the P wavespeed in the outer core is slightly higher
than the S wavespeed at the base of the core, it is possible for an SKS wave, with
a P wave leg in the core, to travel faster in the core and overtake the direct S wave
travelling just in the mantle. For distances beyond 82в—¦ , the SKS phase arrives before
S. Unlike the PKP waves, the SKS family of waves (including the set of underside
11.4 Probing the Earth 235

90

60 120

PcP

30 150

PKiKP

0 180

Figure 11.11. Ray paths for the reп¬‚ected phases PcP and PKiKP.

90

60 120

SKP
ScP

30 150

SKIKP

0 180

Figure 11.12. Ray paths for the converted phases ScP and SKiKP.

reп¬‚ections at the coreвЂ“mantle boundary as in SKKS) sample the whole of the core
The P reп¬‚ections from the major internal boundaries of the Earth can often be
seen as distinct phases. In Figure 11.11 we show the ray paths for PcP reп¬‚ected
from the coreвЂ“mantle boundary and PKiKP reп¬‚ected from the inner-core boundary.
Such reп¬‚ections can sometimes be seen relatively close to the source, but their time
trajectories cut across a number of other phases and so they can often be obscured
by other energy. The P wave refracted in the mantle and the PcP wave reп¬‚ected
from the coreвЂ“mantle boundary have very similar ray paths for near grazing
236 Seismology and Earth Structure

0 0

330 30 330 30

300 60 300 60

PP SS
270 90

PKKP SKKS
240 120 240 120

210 150 210 150

180 180

Figure 11.13. Ray paths for multiply reп¬‚ected phases (a) PP, PKKP, (b) SS, SKKS.

incidence, and so their travel time curves converge close to 90в—¦ epicentral distance.
A similar effect occurs for S and ScS. Sometimes, for horizontally polarised shear
waves, a minor arrival can be seen between S and ScS as the branches converge,
which provides evidence for a discontinuity in shear wavespeed at the top of the
D layer (see, e.g., Young & Lay, 1987).
The strong contrast in physical properties at the coreвЂ“mantle boundary has the
effect of inducing conversion between S and P waves. An incident S wave at the
coreвЂ“mantle boundary can give rise to the converted reп¬‚ection ScP which can be
quite prominent out to 60в—¦ from the source. In addition conversion on transmission
into the core generates SKP as illustrated in Figure 11.12.
The EarthвЂ™s surface and the underside of the coreвЂ“mantle boundary can give rise
to multiply reп¬‚ected phases (Figure 11.13) with very clear internal caustics. The
surface multiples can be tracked for P and S to great distances and can often be
recognised as distinct phases for the third or higher multiples (PPP, SSS, etc.). The
internal core reп¬‚ections also carry energy to substantial distance, and retain the
character of the original wave system. Thus PKKP only has strong sensitivity to P
wave structure in the upper part of the core near the bounce point at the coreвЂ“mantle
boundary, whereas SKKS samples the P wavespeed distribution through the whole
core. The higher-order multiples SKKKS etc. provide the closest probing of the
structure in the outermost parts of the core.
Multiply reп¬‚ected waves from the coreвЂ“mantle boundary are of particular
signiп¬Ѓcance for horizontally polarised shear waves (SH), since the solidвЂ“п¬‚uid
interface is close to a perfect reп¬‚ector. With efп¬Ѓcient reп¬‚ection at the EarthвЂ™s
surface, a long train of multiple (ScS)H can be established that carry with them
11.4 Probing the Earth 237

information on the internal structure of the Earth, in terms of the inп¬‚uence of
internal discontinuities.

Branches of core phases
The propagation of P waves into the EarthвЂ™s core in the PKP wave group includes
refraction back from the velocity gradients in the outer core and reп¬‚ection (PKiKP)
and refraction (PKIKP) from the inner core. The components of the wave group
are also commonly designated by a notation based on the different branches of the
travel-time curve (Figure 11.14).
The patterns of the travel times and the associated ray paths can be followed
in Figure 11.14, where the critical slowness points corresponding to the transition
between branches are clearly labelled. The details of the positions of these critical
slownesses vary slightly between different models for the EarthвЂ™s core, but the
general pattern is maintained.
The PKPAB branch corresponds to the waves entering the core at the shallowest
angles and PKPBC to waves refracted back from the lower part of the outer core.
The CD branch corresponds to wide-angle reп¬‚ection from the inner boundary
(the extension of the PKiKP phase). The PKPDF phase, which is equivalent to

0
180 F
330 30
A
PKIKP

160 300 60
C

PKP
B
Delta [deg]

140 270 90

D
120
240 120
D

PKiKP B
210 150
C
19 20 21 22 A
F
Time [min]

Figure 11.14. Rays and travel times for PKP for the AK135 model, wavefronts are indi-
cated by tick marks at 60 s intervals. The critical points for the various PKP branches are
indicated on both the travel-time curve and the ray pattern. The dashed segment indicates
the locus of precursors to PKIKP from scattering at the coreвЂ“mantle boundary.
238 Seismology and Earth Structure

PKIKP is refracted through the inner core. Rays penetrating into the upper part
are strongly bent and emerge near 110в—¦ at the D point, steeper entry leads to more
direct propagation through the inner core with the F point at 180в—¦ corresponding to
transmission without deп¬‚ection.
The concentration of rays near the PKP caustic at B is reп¬‚ected in localised
large-amplitude arrivals for PKP near 145в—¦ . The observations of the BC branch
tend to extend beyond the ray-theoretical predictions. For epicentral distances
beyond the C point, there is the possibility of diffraction around the inner
core. At the B caustic the real branch does not just stop and there will be
scattering in the mantle from PKP produces short-period arrivals as precursors to
PKIKP that can be seen because they arrive in a quiet portion of the seismic record.
The envelope of possible precursors is indicated in Figure 11.14 by a dashed line.

0
180 F
330 30

SKIKS
160

300 60
C
140
A
Delta [deg]

120 270 90
SKSAC D

100
D

240 120
80
S SKiKS
ScS A C
210 150
20 21 22 23 24 25 26 27
F
Time [min]

Figure 11.15. Rays and travel times for SKS for the AK135 model. S legs are plotted in
grey and wavefronts are indicated by tick marks at 60 s intervals. The critical points for the
various SKS branches are indicated on both the travel-time curves and the ray pattern.

The pattern of branches for the SKS phase is somewhat different because the P
wavespeed at the top of the core is higher than the S at the base of the mantle.
The AC branch extends from S incident on the core-mantle boundary that can just
propagate as P at the top of the outer core and emerge at A вЂ“ 63в—¦ , through to grazing
incidence at the inner core boundary at C. Diffracted waves around the inner core
can extend the branch beyond the formal C point. The DF branch (SKIKS) again
corresponds to refracted waves in the inner core. The post-critical reп¬‚ections from
11.4 Probing the Earth 239

the inner-core boundary (SKiKS) form the CD branch and connect directly into the
pre-critical reп¬‚ection at shorter distances than the D point at 104в—¦ (Figure 11.15).
When the refraction just begins at A, the S wave path to the same epicentral
distance is shorter and SKS is about 75 s behind S. However, as the proportion
of faster P wave path in the core increases, the discrepancies in S and SKS travel
time are reduced. Eventually, the travel time of SKS becomes less than that for
S at the same epicentral distance. Beyond 83в—¦ SKS becomes the onset of the
shear wave group and vertically polarised S, Sdiff have to be sought in the SKS
coda. The transversely polarised S wave is very distinct and small precursory SKS
contributions (as in Figure 11.8) can arise from either anisotropy or heterogeneity
in passage through the mantle.

11.4.2 Normal mode frequencies
The frequencies of the normal modes in the frequency/angular order domain are
displayed in п¬Ѓgure 11.16 for the toroidal modes and in п¬Ѓgure 11.17 for the more
complex spheroidal mode pattern.
For the toroidal modes the patterns of the modal frequencies is relatively simple.
Radial lines from the origin represent lines of constant phase velocity al/П‰ and can
be used to examine the physical character of the modes. Modes lying to the left of
the ScS line in п¬Ѓgure 11.16 have eigenfunctions that sample the whole mantle and
can be identiп¬Ѓed with multiple reп¬‚ections between the coreвЂ“mantle boundary and
the surface. The modes to the right of this line have their energy concentrated above
the coreвЂ“mantle boundary and correspond to multiple free-surface reп¬‚ections of S
body waves. As the fundamental mode is approached the crowding of the multiple
reп¬‚ection processes fuses into a representation of Love waves. The progression
of the behaviour of the eigendisplacement W of toroidal modes for frequencies
near 6 mHz has been illustrated in Figure 11.6. The modes equivalent to ScS have
displacement throughout the mantle, whereas the modes corresponding to multiple
S decay below the tuning depth for a body wave of the same angular slowness.
The modal pattern for spheroidal modes is much more complicated but can be
understood in terms of the major features of the structure of the Earth. The critical
sets of phase velocities are those corresponding to the existence ranges for different
types of propagation processes as indicated in Figure 11.17. As in the toroidal
case the ScS line separates processes which involve multiple S reп¬‚ections in the
mantle from phase velocities for which an S wave can be reп¬‚ected back from the
coreвЂ“mantle boundary. For the vertically polarised S waves in the spheroidal case
there is also the possibility of transmission into the core as a P wave for phases
of the type SKS. Modes with this type of character lie to the left of the SKS line in
п¬Ѓgure 11.17. At even higher phase velocities P wave propagation in the outer core is
possible, as indicated by the PKP phase velocity line, and п¬Ѓnally P waves penetrate
into the inner core. The presence of a range of different propagation patterns for
240 Seismology and Earth Structure

ScS
16

14

12

10
Frequency [mHz]

8

6

4

2

0
0 10 20 30 40 50 60 70 80 90 100 110 120
Angular order

Figure 11.16. Frequencies of toroidal modes as a function of angular order l for model
PREM.

PKIKP PKP SKS ScS
16

14

12

10
Frequency [mHz]

8

6

4

2

0
0 10 20 30 40 50 60 70 80 90 100 110 120
Angular order

Figure 11.17. Frequencies of spheroidal modes as a function of angular order l for model
PREM.

modes with high phase velocity leads to the complex observed dispersion patterns
in Figure 11.17.
In both Figures 11.16 and 11.17 an attempt has been made to indicate the
physical character of the propagation processes associated with each part of the
11.4 Probing the Earth 241

mode branches, by distinguishing the individual modes in terms of their associated
group velocity. Those modes with low group velocity are indicated by solid
triangles, and as the group velocity increases the number of sides of the polygon
is increased and for the spheroidal modes with higher group velocities we move to
open symbols. This representation enhances the visibility of the mantle S and ScS
equivalent modes for the spheroidal modes.
At larger angular orders, the fundamental mode branch and part of the п¬Ѓrst radial
overtone branch are indicated by solid triangles; these parts are direct analogues
of Love waves for toroidal modes and Rayleigh waves for spheroidal modes. The
next prominent pattern (denoted by the solid diamonds) represents the trapping
of S wave energy in the mantle and can be seen clearly in both the toroidal and
spheroidal modes. For spheroidal modes some modulation of these branches is
introduced by the transfer of energy into SKS type propagation.
Two separate classes of contribution can be recognised arising from the structure
of the EarthвЂ™s core (Figure 11.17). Mode segments with shallower slopes are
associated with propagation in the outer core [PKP, PKKP, SKS, SKKS etc.]. The
segments with steeper slopes involve energy concentrated in the inner core and
so are rarely observable at the EarthвЂ™s surface. The modes can be separated into
different groups associated with the dominant character of the physical processes
as discussed in some detail in Chapter 8 of Dahlen & Tromp (1998). The
different classes of reverberation within the Earth are characterised by a spacing in
frequency between the different branches for the appropriate modes that is inversely
proportional to the size of the zone.
The ScS equivalent spheroidal modes for smaller angular orders have a similar
pattern to those for the toroidal modes, but modiп¬Ѓed slightly by the changed
boundary conditions at the coreвЂ“mantle boundary. The tightest mode spacing
comes from PKIKP equivalent modes that sample the whole Earth. Reverberations
dominantly in the outer core and inner core provide other sequences that contribute
to the complex patterns of spheroidal modes for smaller slownesses.
In addition to the main types of propagation phenomena it is also possible to get
energy trapped at the coreвЂ“mantle boundary or inner coreвЂ“outer core boundary with
exponential decay of amplitude away from the boundary. Such modes form a line
of constant phase velocity cutting across the major branches and can be recognised
from their differing properties in п¬Ѓgure 11.17. The set of open triangles cutting
across the mantle S branches arises from the presence of a Stoneley wave trapped
at the п¬‚uidвЂ“solid boundary at the top of the core. There is an equivalent set of
modes for the inner core boundary which lies just to the left of the SKS propagation
line.
The illustrations of the eigenfunctions for the spheroidal modes near 6 mHz in
Figure 11.7 show the concentration of displacement in the Stoneley modes at the
top and bottom of the п¬‚uid outer core. The upper row of modes in Figure 11.7
represent propagation largely conп¬Ѓned to the mantle. The middle row include ScS
242 Seismology and Earth Structure

equivalent modes and progressively deeper penetration into the outer core as the
radial order increases. The bottom row in Figure 11.7 shows the behaviour of
modes where displacement extends through the whole Earth, even though for an
inner core mode such as 14S5 comparatively little displacement is actually at the
surface.

Observations of modal eigenfrequencies
A single strain record for the 1952 Kamchatka earthquake recorded in Pasadena
suggested that the fundamental mode of the Earth had been observed. This result
stimulated work on the calculation of the frequencies of the normal modes of
the Earth, so that by the time of the great Chilean earthquake in May 1960
instrumental and theoretical seismology had converged to identify a wide range
of normal modes. Subsequent studies used the 1964 Alaska event and a large
deep earthquake near the PeruвЂ“Chile border in 1965 to develop an extensive mode
catalogue. Many additional data of very high quality were collected from the very
deep magnitude 8 Bolivian event in 1994. A long observation period (several
days) without signiп¬Ѓcant interference from other events is desirable for high
precision estimates of the normal mode frequencies. The great SumatraвЂ“Andaman
earthquake (Mw 9.3), in late 2004, generated large amounts of low-frequency
energy and signiп¬Ѓcantly improved measurements of the lower-frequency spheroidal
modes. Indeed the fundamental radial mode 0S0 with a period close to 20 minutes,
and little attenuation, could still be detected on sensitive instruments some months
after the event.
Heavy dots in Figure 11.18 show those spheroidal modes for which high
precision frequency estimates have been determined. The limited coverage of
seismic stations around the globe means that such frequency observations require
the combination of all available records for several days after a very large
earthquake.
The spectra of seismic records following a great earthquake show a sequence of
frequency peaks that can be enhanced by simple summation of the spectra from
many stations. These peaks are normally the fundamental modes (n = 0) and
the angular order can be identiп¬Ѓed by comparison with theoretical calculations.
More sophisticated methods can be used to extract the spectral peak associated
with a particular target mode; in particular the set of available records can be
combined in such a way as to enhance the target mode and reduce noise. Such
вЂ˜stackingвЂ™ methods are quite powerful, particularly with the enhanced coverage of
the globe with high-п¬Ѓdelity instruments in recent years since they depend on a good
knowledge of the source parameters including the moment tensor. A stack aimed
at a speciп¬Ѓc mode may well contain contributions from overtones of the same type,
and a further step known as вЂ˜strippingвЂ™ can help to separate the different modes. For
low-frequency modes, biases associated with mode coupling due to the rotation of
the Earth need to be removed. Masters & Widmer (1995) present a comprehensive
catalogue of modal frequencies and related information.
11.4 Probing the Earth 243

5

4

3
Frequency [mHz]

2

1

0
0 4 8 12 16 20
Angular order l

Figure 11.18. Low frequency spheroidal modes for which high precision frequency infor-
mation is available are indicated by solid symbols, superimposed on the theoretical values
for the PREM model.

Results from stations in low-noise environments, such as Antarctica, indicate the
presence of normal mode peaks in the absence of signiп¬Ѓcant earthquakes. Such
observations have been made in many different parts of the world and suggest that
there is continual low-level excitation of the EarthвЂ™s normal modes, possibly due to
coupling between the solid Earth and the atmosphere.
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