We can write the equation of motion (11.3.20) in the operator form

H(u) + ρ0ω2u = ’f, (11.3.24)

11.3 Building the response of the Earth to a source 219

and then

cJH(ue) + ρ0ω2 cJue = ’f, (11.3.25)

J J

J J

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

K

d3x cJH(ue) · [ue ]— + ω2cK = ’ d3x f · [ue ]— . (11.3.26)

J K K

Ve Ve

J

Each eigenvector ue satis¬es the equation

I

H(ue) + ρ0ω2ue = 0, (11.3.27)

I I

and hence

d3x H(ue) · [ue ]— + ω2δJK = 0. (11.3.28)

J K

Ve

With this substitution in (11.3.26) the coef¬cients in the modal expansion can be

found as

1

d3x f · [ue ]— .

cK = 2 (11.3.29)

K

ωK ’ ω2 Ve

We have therefore a frequency-domain representation of the displacement induced

by the force system f:

ue

d3x f · [ue ]— .

K

u(x, ω) = (11.3.30)

K

2 ’ ω2

ωK 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 ωK using Cauchy™s theorem, to produce

1 ’ cos(ωKt)

ue d3x f · [ue ]— .

u(x, t) = (11.3.31)

K K

ω2 Ve

K

K

When we allow for the decay of the eigenmodes produced by slight anelasticity

’ cos(ωKt)e’iωK t/2QK

1

ue d3x f · [ue ]— .

u(x, t) = (11.3.32)

K K

ω2 Ve

K

K

The static response as t ’ ∞ is also given by a normal mode sum

1

ue d3x f · [ue ]— .

u(x, t) = (11.3.33)

K K

ω2 Ve

K

K

So far we have not speci¬ed 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 · [ue ]— = [ee ]—: M(xs) = [ee ]— 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

‚

M(t) = M(t). (11.3.35)

‚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 δ(t).

We also introduce the modal time function

CK(t) = 1 ’ cos(ωKt)e’iωK 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,

∞

1

CK(t ’ t )ue [ee ]—: M(xs, t) .

u(x, t) = dt (11.3.37)

KK

2

ωK

’∞

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 in¬‚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 in¬‚uence

of self-gravitation. A comprehensive treatment of the development of the low

frequency part of the seismic wave ¬eld in terms of these normal modes is presented

by Dahlen & Tromp (1998).

The presence of the ¬‚uid core means that for the toroidal modes the boundary

conditions are the vanishing of radial traction at both the surface and the

core“mantle boundary. The spheroidal modes penetrate right to the centre of

the Earth and can have signi¬cant 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 ¬elds and which are linked through ¬rst-order

ordinary differential equations in radius. For spheroidal modes the in¬‚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 ’ m)! m

Yl (θ, φ) = (’1)m

m

Pl (cos θ)eimφ, (11.3.38)

4π (l + m)!

m

where Pl is an associated Legendre polynomial and ’l ¤ m ¤ 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 = ωre /(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 Earth™s rotation, the ellipticity of

the ¬gure 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 ¬eld

l

U(r)Pm(θ, φ) + V(r)Bm(θ, φ) + W(r)Cm(θ, φ) , (11.3.40)

u= l l l

l m=’l

l

P(r)Pm(θ, φ) + S(r)Bm(θ, φ) + T (r)Cm(θ, φ) ,

tr = (11.3.41)

l l l

l m=’l

222 Seismology and Earth Structure

where the orthogonal vector surface harmonics have the properties

Pm(θ, φ) = Yl ^r,

m

e (11.3.42)

l

m m

1 ‚Yl 1 ‚Yl

m

Bl (θ, φ) = ^θ + ^φ ,

e e (11.3.43)

‚θ sin θ ‚φ

Ł

m m

1 ‚Yl 1 ‚Yl

m

Tl (θ, φ) = ’ ^θ ’ ^φ .

e e (11.3.44)

‚θ sin θ ‚φ

Ł

Here ^r, ^θ, ^φ are the unit vectors in the r, θ, φ directions of the spherical

eee

coordinates, and we have set Ł = [l(l + 1)]1/2. The vector spherical functions

are orthogonal so that, e.g.,

π 2π

dφ [Rm]— Rm = δmm δll ,

sin θdθ l l

0 0

π 2π

dφ [Rm]— Sm = 0.

sin θdθ (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(θ, φ) separate from the spheroidal oscillations

l

that link the coef¬cients of Pm(θ, φ), Bm(θ, φ).

l l

The ¬rst 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 coef¬cients of the Cm(θ, φ) vector harmonics, i.e., the

l

displacement coef¬cient W, and traction coef¬cient T . The toroidal modes do not

penetrate into the ¬‚uid core so disturbances are con¬ned to the mantle.

In the mantle, the quantities W and T are related by

r’1 μ’1

d W W

= . (11.3.46)

’ρω2 + (l + 1)(l + 2)μr’2 ’3r’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 core“mantle boundary r = rc . (11.3.47)

The toroidal equations may thus be integrated from the core“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 ω to satisfy the boundary conditions is

FT (ω, l) = T (re , ω, l) = 0. (11.3.49)

For higher mode overtones of the torsional modes at large l, integration from the

core“mantle boundary can lead to over¬‚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) = (μ/r1)[(l ’ 1) ’ zl(ωr/β)], (11.3.50)

where zl is de¬ned 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

zl’1(x) = x2/[(2l + 1) ’ 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 ¬xed angular order l (i.e., ¬xed wavenumber) and integrate (11.3.46)

up from the initial values with a trial value of frequency ω, 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 ωt an improved estimate ωn may

be found from

ω2 = ω2 + B/T , (11.3.53)

n t

where for this torsional mode case

re

dr r2ρ(r)W 2(r), B = r2W(re )T (re ),

T= (11.3.54)

e

0

with W and T calculated for the trial frequency ωt. 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) de¬ne 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 ωn, 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

con¬ned 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 ¬lls 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 elastic“gravitational

disturbances and satisfy the elastic equations of motion and Poisson™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 ¬elds link the Pm(θ, φ) and Bm(θ, φ) vector

l l

harmonics through the quantities U, V, P, S. We express the perturbation in

gravitational potential as

m

•(x, ω) = ’¦(r)Yl (θ, φ), (11.3.55)

and introduce the additional variable

l+1

dΨ

Ψ= + 4πGρU + ¦. (11.3.56)

r

dr

This scalar Ψ has the desirable properties of being continuous, zero at the Earth™s

surface r = re and identically zero for r > re .

The displacement coef¬cient U, V, traction coef¬cients P, S and the gravitational

terms ¦, Ψ satisfy the coupled set of ordinary differential equations with respect to

radius r:

· 1

dU

= (ŁV ’ 2U) + P,

r » + 2μ

dr

1 1

dV

= ’ (V ’ ŁU) + S,

r μ

dr

χ ρg ρg χ

dP

= ’ρω2 + 4 2 ’ U+Ł ’2 2 V

r r r r

dr

2(· ’ 1) (l + 1)ρ

Ł

+ P+ S’ ¦ + ρΨ,

r r r

Ł2 ’ 2μ

ρg χ · 3 ρ

dS 2

=Ł ’ 2 2 U ’ρω + V ’ Ł P ’ S + Ł Ψ,

r2

r r r r r

dr

l+1

d¦

= ’4πGρU ’ Ψ + Ψ,

r

dr

4πGρ l’1

dΨ

= [ŁV ’ (l + 1)U] + Ψ; (11.3.57)

r r

dr

with

» μ(3» + 2μ) 4μ(» + μ)

·= , χ= , ν= . (11.3.58)

» + 2μ » + 2μ » + 2μ

In this form the set of elements of the coupled differential equations reduce

asymptotically to the forms for plane strati¬cation for large ω and l.

The boundary conditions that have to be satis¬ed at the Earth™s surface are that

the surface traction and gravitational scalar Ψ should vanish, i.e.,

P(re ) = S(re ) = 0, Ψ(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 Earth™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(ω, l) = S1(re ) S2(re ) S3(re ) = 0. (11.3.60)

Ψ1(re ) Ψ2(re ) Ψ3(re )

We model the outer core of the Earth as a liquid with vanishing rigidity and, in

consequence, the system of differential equations is simpli¬ed since no tangential

stress can be supported. In the equation system (11.3.57) · = 1, χ = 0, ν = 0 and

Ł

V= [P + ρ(gU ’ Ψ)] , (11.3.61)

ρω2r

so that in the ¬‚uid

2 1

dU Ł

= ’ U + V + P,

r r »

dr

ρg ρg (l + 1)ρ

dP

= ’ ρω2 + 4 U+Ł V ’ ¦ + ρΨ,

r r r

dr

l+1

d¦

= ’4πGρU ’ Ψ + Ψ,

r

dr

4πGρ l’1

dΨ

= [ŁV ’ (l + 1)U] + Ψ; (11.3.62)

r r

dr

V becomes indeterminate at zero frequency (ω = 0).

We can summarise the set of displacement, traction and gravitational variables

through the column vector

b = [U, V, P, S, ¦, Ψ]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 = ’ 1 C1 ’ 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 ’ bK (ri ) bI ’

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 mantle“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 = ¦M = ΨM = 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, Ψi from the separate integrations.

In this direct approach very high order accuracy is needed in the integration

scheme to overcome the “stiff” 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

the advantage of numerical stability.

Iterative improvement of the estimated frequency ωt can be achieved with the

variational result for an improved frequency ωn

ω2 = ω2 + B/T , (11.3.73)

n t

228 Seismology and Earth Structure

with

re

dr r2ρ(r)[U2(r) + V 2(r)],

T=

0

2

+ V(re )S(re ) + (4πG)’1¦(re )Ψ(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 ωt 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 ¬xed 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 con¬ned

to the mantle. Whereas U, ¦ are continuous, the eigendisplacement V displays

jumps at the ¬‚uid“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 ¬‚uid“solid interfaces. The mode 3S28 is trapped at the

core“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 re¬‚ected from the core“mantle boundary which is no longer a specular

re¬‚ector since transmitted P waves can make their way into the ¬‚uid core. The

bottom row in Figure 11.7 shows modes with much of their energy in the inner

core.

Radial oscillations

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

’2·r’1 (» + 2μ)’1

d U U

= . (11.3.75)

’ρω2 + 4(χr’2 ’ ρgr’1) (2· ’ 2)r’1

P P

dr

where ·, χ are the composite moduli (11.3.58). The gravitational potential in this

case can be found from

d¦

= ’4πGρU. (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

¦

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 ¦ 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 con¬ned 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 core“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

power series about the origin

U = r, P = 3» + 2μ, (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 ωt an improved estimate is given by

ω2 = ω2 + B/T , (11.3.78)

n t

where for the radial modes

re

dr r2ρ(r)U2(r), B = r2[U(re )P(re ) + (4πGre )’1¦2(re )], (11.3.79)

T= e

0

with U, P and ¦ calculated for the trial frequency ωt.

The Rayleigh“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,

ω dr r T(r) ’ (11.3.80)

0 0

where ω2T 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

Rayleigh“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 ζi(r) which each satisfy the boundary conditions then

N

u(r) ≈ biζi(r). (11.3.81)

i=1

The energy balance equation results in a matrix eigenvalue problem of the form

(ω2T ’ 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 ζi and the corresponding value of ω2 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 Gauss“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,

dif¬culties 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 Earth™s mantle and core. Because the outer

core behaves as a ¬‚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 re¬‚ected

from the core“mantle boundary and so are con¬ned 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 in¬‚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 ¬eld. 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 100—¦

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 100.49˚

Figure 11.8. Three-component seismogram from an intermediate depth earthquake in Van-

uatu recorded in Kazakhstan at approximately 100—¦ 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:

• a leg in the mantle is denoted by P or S depending on wave type;

• a compressional leg in the outer core is labelled as K;

• a compressional leg in the inner core is indicated with I, the corresponding shear

wave leg would be denoted J;

• waves leaving upward from the source are indicated by lowercase letters, thus

pP represents a surface re¬‚ection for P near the source, and sP a re¬‚ection with

conversion above the source; and

• re¬‚ected waves from major interfaces are indicated by lowercase letters: m for

the Mohoroviˇ i´ discontinuity (Moho), c at the core“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 re¬‚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 re¬‚ected back from the surface of the inner

core. Similarly an S wave that is re¬‚ected at the core“mantle boundary is indicated

by ScS, and if conversion occurs on re¬‚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 re¬‚ected from the core“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 con¬rmed. 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 ˜depth phases™ (pP, sP and pS, sS) can be very distinct for deep events. The

time difference between the arrival of such phases re¬‚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 11.9“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 100—¦ away from the source, but then reach the core“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 re¬‚ected from the core“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 ¬‚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 Snell™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 145—¦ from the source. The apparent ˜shadow zone™

with no P arrivals from 100 to 145—¦ led to the discovery of the core by Oldham, at

the end of the 19th century. Re¬‚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 ¬ll in the gap between

direct P out to 100—¦ and the PKP caustic. The identi¬cation of these arrivals in

the ˜shadow zone™ 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 suf¬cient to produce turning points that combine to

produce the PKP caustic that emerges at the surface near 144—¦ .

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 82—¦ , 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 re¬‚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.

re¬‚ections at the core“mantle boundary as in SKKS) sample the whole of the core

(see also Figure 11.13).

The P re¬‚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 re¬‚ected

from the core“mantle boundary and PKiKP re¬‚ected from the inner-core boundary.

Such re¬‚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 re¬‚ected

from the core“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 re¬‚ected phases (a) PP, PKKP, (b) SS, SKKS.

incidence, and so their travel time curves converge close to 90—¦ 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 core“mantle boundary has the

effect of inducing conversion between S and P waves. An incident S wave at the

core“mantle boundary can give rise to the converted re¬‚ection ScP which can be

quite prominent out to 60—¦ from the source. In addition conversion on transmission

into the core generates SKP as illustrated in Figure 11.12.

The Earth™s surface and the underside of the core“mantle boundary can give rise

to multiply re¬‚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 re¬‚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 core“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 re¬‚ected waves from the core“mantle boundary are of particular

signi¬cance for horizontally polarised shear waves (SH), since the solid“¬‚uid

interface is close to a perfect re¬‚ector. With ef¬cient re¬‚ection at the Earth™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 in¬‚uence of

internal discontinuities.

Branches of core phases

The propagation of P waves into the Earth™s core in the PKP wave group includes

refraction back from the velocity gradients in the outer core and re¬‚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 Earth™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 re¬‚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 core“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 110—¦ at the D point, steeper entry leads to more

direct propagation through the inner core with the F point at 180—¦ corresponding to

transmission without de¬‚ection.

The concentration of rays near the PKP caustic at B is re¬‚ected in localised

large-amplitude arrivals for PKP near 145—¦ . 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

frequency-dependent decay into the shadow side of the caustic. In addition

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 “ 63—¦ , 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 re¬‚ections from

11.4 Probing the Earth 239

the inner-core boundary (SKiKS) form the CD branch and connect directly into the

pre-critical re¬‚ection at shorter distances than the D point at 104—¦ (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 83—¦ 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 ¬gure 11.16 for the toroidal modes and in ¬gure 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/ω and can

be used to examine the physical character of the modes. Modes lying to the left of

the ScS line in ¬gure 11.16 have eigenfunctions that sample the whole mantle and

can be identi¬ed with multiple re¬‚ections between the core“mantle boundary and

the surface. The modes to the right of this line have their energy concentrated above

the core“mantle boundary and correspond to multiple free-surface re¬‚ections of S

body waves. As the fundamental mode is approached the crowding of the multiple

re¬‚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 re¬‚ections in the

mantle from phase velocities for which an S wave can be re¬‚ected back from the

core“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

¬gure 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 ¬nally 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 ¬rst 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 Earth™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 Earth™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 modi¬ed slightly by the changed

boundary conditions at the core“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 core“mantle boundary or inner core“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 ¬gure 11.17. The set of open triangles cutting

across the mantle S branches arises from the presence of a Stoneley wave trapped

at the ¬‚uid“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 ¬‚uid outer core. The upper row of modes in Figure 11.7

represent propagation largely con¬ned 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 Peru“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 signi¬cant interference from other events is desirable for high

precision estimates of the normal mode frequencies. The great Sumatra“Andaman

earthquake (Mw 9.3), in late 2004, generated large amounts of low-frequency

energy and signi¬cantly 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 identi¬ed 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

˜stacking™ methods are quite powerful, particularly with the enhanced coverage of

the globe with high-¬delity instruments in recent years since they depend on a good

knowledge of the source parameters including the moment tensor. A stack aimed

at a speci¬c mode may well contain contributions from overtones of the same type,

and a further step known as ˜stripping™ 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 signi¬cant earthquakes. Such

observations have been made in many different parts of the world and suggest that

there is continual low-level excitation of the Earth™s normal modes, possibly due to

coupling between the solid Earth and the atmosphere.