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Surface waves and normal modes
Our treatment to this point has been limited to body waves, solutions to the seismic
wave equation that exist in whole spaces. However, when free surfaces exist in a
medium, other solutions are possible and are given the name surface waves. There
are two types of surface waves that propagate along Earth’s surface: Rayleigh waves
and Love waves. For laterally homogeneous models, Rayleigh waves are radially
polarized (P/ SV ) and exist at any free surface, whereas Love waves are transversely
polarized and require some velocity increase with depth (or a spherical geometry).
Surface waves are generally the strongest arrivals recorded at teleseismic distances
and they provide some of the best constraints on Earth’s shallow structure and
low-frequency source properties. They differ from body waves in many respects –
they travel more slowly, their amplitude decay with range is generally much less,
and their velocities are strongly frequency dependent. Surface waves from large
earthquakes are observable for many hours, during which time they circle the Earth
multiple times. Constructive interference among these orbiting surface waves, to-
gether with analogous reverberations of body waves, form the normal modes , or
free oscillations of the Earth. Surface waves and normal modes are generally ob-
served at periods longer than about 10 s, in contrast to the much shorter periods
seen in many body wave observations.
8.1 Love waves
Love waves are formed through the constructive interference of high-order SH
surface multiples (i.e., SSS , SSSS , SSSSS , etc.). Thus, it is possible to model Love
waves as a sum of body waves. To see this, consider monochromatic plane-wave
propagation for the case of a vertical velocity gradient in a laterally homogeneous
model, a situation we previously examined in Section 6.4. In this case, a plane
wave defined by ray parameter p will turn at the depth where β = 1 /p. Along the
215
216 8. S U R F A C E W A V E S A N D N O R M A L M O D E S
X(t)
surface the plane wave will propagate with horizontal slowness defined by p. If the
surface bouncepoints are separated by a distance X(t), then the travel time along the
surface between bouncepoints is given by pX(p). This follows from our definition
of a plane wave and does not depend upon the velocity model. In contrast, the travel
time along the ray paths is given by T(p) and is a function of the velocity–depth
profile. Because these travel times are not the same, destructive interference will
occur except at certain fixed frequencies. Along the surface, the phase (0 to 2π) of
a harmonic wave will be delayed by ωpX(p), where ω is the angular frequency of
the plane wave. The phase along the ray path is delayed by ωT(p) − π/2, where the
−π/2 comes from the WKBJ approximation for the phase advance at the plane-
wave turning point (see Section 6.4). The requirement for constructive interference
is thus
ωpX(p) = ωT(p) −
π
2
− n 2 π, (8.1)
where n is an integer. Rearranging, we obtain
ω =
n 2 π + π/ 2
T(p) − pX(p)
n 2 π + π/ 2
τ(p)
where the delay time τ(p) = T(p) − pX(p) (see Section 4.3.2). The wave travels
along the surface at velocity c = 1 /p; thus (8.2) defines the c(ω) function for the
Love waves, often termed the dispersion curve. The values of ω given for n = 0
are termed the fundamental modes; higher modes are defined by larger values of
n. The frequency dispersion in the Love waves results from the ray geometry and
does not require any frequency dependence in the body wave velocity β. Love
wave dispersion is much stronger than the small amount of dispersion in S-wave
velocities that results from intrinsic attenuation (see equation (6.91)).
The velocity defined by c = 1 /p is the velocity with which the peaks and troughs
at a given frequency move along the surface and is termed the phase velocity. When
the phase velocity varies as a function of frequency, as in (8.2), the wave is dispersed
and the group velocity (the velocity that energy propagates) will be different from
the phase velocity. In this example, the energy must move along the actual ray paths
218 8. S U R F A C E W A V E S A N D N O R M A L M O D E S
r
1
b
1
r
2
b
2
Figure 8.2 Love waves within a homogeneous layer can result from constructive interference between postcritically reflected body waves.
or
dω T − dk X + ω dX(dT/dX − k/ω) = 0. (8.5)
Since p = k/ω = dT/dX, the rightmost term drops out, and we have
dω
dk
X
T
= U, (8.6)
and we see that the group velocity is also given by dω/dk.
Equation (8.2) is not very accurate at small values of n since a high frequency
approximation was used to determine the phase shift at the turning point. However, it
does give some understanding of how Love waves are formed through the positive
interference of S surface multiples. More accurate Love wave calculations are
generally performed using homogeneous layer techniques. In these methods, the
plane wave response of a stack of layers is computed at a series of values of ray
parameter; the frequencies of the different Love wave branches are then identified
as the eigenvalues of the resulting set of equations.
8.1.1 Solution for a single layer
An exact equation may be derived for Love wave dispersion within a homogeneous
layer. Consider a surface layer overlying a higher velocity half-space (Fig. 8.2).
Equation (8.2) is still applicable, provided we replace the approximate π/2 phase
shift at the turning point with the phase shift, φ` S S´ , resulting from the SH reflection
off the bottom of the layer:
ω =
n 2 π − φ` S S´ τ(p)
From (4.33), we may express the delay time τ as
τ(p) = 2 h
1 /β^2 1 − p^2 , (8.8)
8.2 R A Y L E I G H W A V E S 219
where h is the layer thickness and β 1 is the shear velocity in the top layer. For
postcritical reflections, it can be shown from (6.46) and (6.59) that
φS` S´ = −2 tan
− 1
μ 2
p^2 − 1 /β 2 2
μ 1
1 /β 2 1 −^ p
2
⎦.^ (8.9)
Substituting (8.8) and (8.9) into (8.7), we have
2 hω
1 /β^2 1 − p^2 − n 2 π = 2 tan − 1
μ 2
p^2 − 1 /β 2 2
μ 1
1 /β 2 1 −^ p
2
or
tan
[
hω
1 /β 2 1 −^ p
2
]
μ 2
p^2 − 1 /β 2 2
μ 1
1 /β 2 1 −^ p
2
This defines the dispersion curves for Love wave propagation within the layer.
Note that the phase velocity, c = 1 /p, varies between β 1 and β 2 (c > β 2 is
not postcritical). For every value of c, there are multiple values of ω because of
the periodicity in the tangent function. The smallest of the ω values defines the
fundamental mode, the second smallest is the first higher mode, etc. There is no
analytical solution to (8.10) for c; the c(ω) values must be determined numerically
(see Exercise 8.1).
8.2 Rayleigh waves
For SH polarized waves, the reflection coefficient at the free surface is one, and the
interference between the downgoing SH waves and those turned back toward the
surface produces Love waves. The P/ SV system is more complicated because
the surface reflections involve both P and SV waves. In this case, the upgoing
and downgoing body waves do not sum constructively to produce surface waves.
However, a solution is possible for inhomogeneous waves trapped at the interface;
the resulting surface waves are termed Rayleigh waves. The displacements of Love
and Rayleigh waves are compared in Figure 8.3.
Let us begin by examining what occurs when P and SV waves interact with a
free surface. For a laterally homogeneous medium, the displacements for harmonic
8.2 R A Y L E I G H W A V E S 221
Now from (8.11), consider plane wave solutions for φ and 'y (the only part of '''
that produces SV motion for plane wave propagation in the x direction):
φ = Ae −iω(t−px−ηαz) , (8.13)
'y = Be −iω(t−px−ηβz) , (8.14)
where A and B are the amplitudes of the P and SV waves respectively, and the
vertical slownesses are given by
ηα = ( 1 /α 2 − p 2 ) 1 / 2 , (8.15)
ηβ = ( 1 /β
2 − p
2 )
1 / 2
. (8.16)
The ray parameter p is constant; both P and SV are assumed to have the same
horizontal slowness. Noting that ∂y and uy are zero for our P/ SV plane wave
geometry, the P-wave displacements are
u
P x =^ ∂xφ^ =^ pAiωe
−iω(t−px−ηαz) , (8.17)
u
P z =^ ∂zφ^ =^ ηαAiωe
−iω(t−px−ηαz) , (8.18)
and the SV -wave displacements are
u
S x =^ −∂z'y^ =^ −ηβBiωe
−iω(t−px−ηβz) , (8.19)
u
S z =^ ∂x'y^ =^ pBiωe
−iω(t−px−ηβz)
. (8.20)
Now consider the boundary conditions at a free surface z = 0. Both the normal and
shear tractions must vanish: τxz = τzz = 0. From (3.13), we have
τxz = μ(∂zux + ∂xuz), (8.21)
τzz = λ(∂xux + ∂zuz) + 2 μ∂zuz. (8.22)
Substituting (8.17)–(8.20) into (8.21) and (8.22), we obtain
τ P xz =^ −A(^2 μpηα)ω
2 e −iω(t−px−ηαz) , (8.23)
τ P zz =^ −A
[
(λ + 2 μ)η 2 α +^ λp
2
]
ω 2 e −iω(t−px−ηαz) , (8.24)
τ S xz =^ −Bμ
p 2 − η 2 β
ω 2 e −iω(t−px−ηβz) , (8.25)
τ S zz =^ −B(^2 μηβp)ω
2 e −iω(t−px−ηβz)
. (8.26)
222 8. S U R F A C E W A V E S A N D N O R M A L M O D E S
At the free surface, we require
τxz = τ P xz +^ τ
S xz =^0 ,^ (8.27)
τzz = τ P zz +^ τ
S zz =^0.^ (8.28)
Substituting (8.23)–(8.26) into (8.27) and (8.28) at z = 0, and canceling the com-
mon terms, we obtain
A( 2 pηα) + B
p 2 − η 2 β
A
[
(λ + 2 μ)η^2 α + λp^2
]
The equations for τzz can be written in terms of the P and S velocities by substituting
λ + 2 μ = ρα 2 , μ = ρβ 2 , and λ = ρ(α 2 − 2 β 2 ) to give
A[2pηα] + B
[
p 2 − η 2 β
]
A
[
α 2
η 2 α +^ p
2
− 2 β 2 p 2
]
This coupled set of equations describes the free surface boundary condition for P-
and SV -waves with horizontal slowness p. Recall that the vertical slownesses are
given by ηα = ( 1 /α 2 − p 2 ) 1 / 2 and ηβ = ( 1 /β 2 − p 2 ) 1 / 2
. When p < 1 /α, there are
two real solutions, a positive value of ηα for downgoing P waves and a negative
value for upgoing P waves (assuming the z axis points downward). Similarly, when
p < 1 /β, then ηβ is real and there exist both downgoing and upgoing SV waves.
By defining different amplitude coefficients for the downgoing and upgoing waves,
one could use (8.31) and (8.32) to solve for the P/SV reflection coefficients at the
free surface.
However, our interest is in the case where p > β − 1
α − 1 and both ηα and ηβ
are imaginary. From (8.11), if we factor out the depth dependence, we obtain
u = A e iωηz e −iω(t−px) , (8.33)
and we see that imaginary values of η will lead to real values in the exponent. In this
case we have the evanescent waves discussed in Chapter 6, for which amplitude
grows or decays exponentially as a function of depth. The sign of η is chosen to
give the solution that decays away from z = 0. For single imaginary values of ηα
and ηβ, the linear system of equations for A and B given in (8.31) and (8.32) has a
non-trivial solution only when the determinant vanishes, that is, when
p
2 − η
2 β
) [
α
2 (η
2 α +^ p
2 ) − 2 β
2 p
2
]
− 4 β
2 p
2 ηαηβ = 0. (8.34)
224 8. S U R F A C E W A V E S A N D N O R M A L M O D E S
and horizontal components are out of phase by π/2; the resulting elliptical motion
changes from retrograde at the surface to prograde at depth, passing through a node
at which there is no horizontal motion. For Rayleigh waves propagating along the
surface of a uniform half-space there is no velocity dispersion (since there is no
scale length in the model). However, in the Earth velocity dispersion results from
the vertical velocity gradients in the crust and upper mantle; longer period waves
travel faster since they sense the faster material at greater depths. As in the case of
Love waves, Rayleigh wave dispersion curves for vertically stratified media may
be computed using propagator matrix methods.
8.3 Dispersion
When different frequency components travel at different phase velocities, pulse
shapes will not stay the same as they travel but will become dispersed as the
frequencies separate. This leads to interference effects that cancel the wave energy
except at particular times defined by the group velocity of the wave. This may
be illustrated by considering the sum of two harmonic waves of slightly different
frequency and wavenumber:
u(x, t) = cos(ω 1 t − k 1 x) + cos(ω 2 t − k 2 x). (8.37)
Relative to an average frequency ω and wavenumber k, we have
ω 1 = ω − δω, k 1 = k − δk, (8.38)
ω 2 = ω + δω, k 2 = k + δk. (8.39)
Substituting into (8.37), we obtain
u(x, t) = cos(ωt − δωt − kx + δkx) + cos(ωt + δωt − kx − δkx)
= cos [(ωt − kx) − (δωt − δkx)] + cos [(ωt − kx) + (δωt − δkx)]
= 2 cos(ωt − kx) cos(δkx − δωt), (8.40)
where we have used the identity 2 cos A cos B = cos(A + B) + cos(A − B).
The resulting waveform consists of a signal with the average frequency ω whose
amplitude is modulated by a longer period wave of frequency δω (Fig. 8.5).
In acoustics, this phenomenon is termed beating and may be observed when two
musical notes are slightly out of tune. The short-period wave travels at velocity ω/k
and the longer period envelope travels at velocity δω/δk. The former is the phase
velocity c; the latter is the group velocity U. In the limit as δω and δk approach
8.3 D I S P E R S I O N 225
Time
Distance
Group velocity
Phase velocity
Figure 8.5 The sum of two waves of slightly different frequencies results in a modulated wave.The group velocity is the velocity of the wave packets; the phase velocity is the velocity of the individual peaks.
zero, we thus have
U =
dω
dk
which agrees with our previous result in (8.6). Using the various relationships
between the harmonic wave parameters (see Table 3.1), the group velocity may be
alternatively expressed as
U =
dω
dk
= c + k
dc
dk
= c
1 − k
dc
dω
For Earth, the phase velocity c of both Love and Rayleigh waves generally in-
creases with period; thus dc/dω is negative and from (8.42) it follows that the
group velocity is less than the phase velocity (U < c). Figure 8.6 plots Love
and Rayleigh dispersion curves computed from the PREM model. A minimum or
maximum point on the group velocity dispersion curve will result in energy from
a range of periods arriving at nearly the same time. This is termed an Airy phase
and occurs in Earth for Rayleigh waves at periods of about 50 and 240 s.
8.4 G L O B A L S U R F A C E W A V E S 227
R 1
R 2
R 3
Quake
Receiver
Figure 8.7 Ray paths for the first three Rayleigh wave arrivals.
(^0 30 60 90 120 )
Time (minutes)
Vertical
Radial
Transverse
R
R P
SV
SH
G
G
Figure 8.8 The vertical, radial, and transverse components of motion for a March 11, 1989,
earthquake at 230 km depth in the Tonga trench recorded at IRIS/IDA station NNA in Peru. P, SV,
and Rayleigh waves are most visible on the vertical and radial components; SH and Love waves
appear on the transverse component.
228 8. S U R F A C E W A V E S A N D N O R M A L M O D E S
At long periods, Rayleigh waves are sufficiently coherent that it is possible to
stack records from many different events to produce a global picture (Fig. 8.9) of
the seismic wavefield that images the surface wave arrivals (Shearer, 1994). This
vertical-component image illustrates many of the concepts that we have developed
in this chapter. The dispersion of the Rayleigh waves is clearly apparent, particularly
in the later part of the image. Very long period (≥ 300 s) waves travel the fastest,
arriving before the pronounced shorter-period banding in the Airy phase. The high
amplitude of the Airy phase results from a local minimum in the group velocity
dispersion curve near 240 s. The difference between phase and group velocity can
be seen clearly in the image of the Airy phase. The lines of constant phase, defined
by the peaks and troughs in the seismograms, are not parallel to the overall direction
of energy transport. Rather, they cut across at a slightly more horizontal orientation,
since the phase velocity is higher than the group velocity.
The major P and SV body-wave phases can also be seen in this image, in the
triangular shaped region before the first Rayleigh wave (R1). Additional body-wave
arrivals are visible between R1 and the second Rayleigh wave (R2). These include
some P phases, but most prominent are the high-order S surface multiples and the
families of S-to-P converted phases that they spawn upon each surface reflection.
These can be traced to beyond 720◦^ and are the major source of seismic energy
between the Rayleigh wave arrivals. In the surface wave literature, these arrivals are
termed overtone packets and are sometimes referred to as X phases (e.g., Tanimoto,
1987).
8.5 Observing surface waves
Surface waves are generally the strongest arrivals at teleseismic distances and con-
tain a great deal of information about crust and upper mantle structure as well as
the seismic source. Much of the power of surface wave observations comes from
the fact that velocity can be measured at a number of different frequencies from a
single seismogram, providing direct constraints on the velocity versus depth pro-
file everywhere along the source–receiver path. In contrast, the corresponding body
wave observations provide only a single travel time per phase, and recovering the
complete velocity structure requires stations at a wide range of source–receiver
distances.
A major goal in most surface wave studies is to determine the group or phase
velocity at a number of periods. This can be done in several ways. If the location
and origin time of the source are known, then the group velocity may be estimated
from a surface wave record at a single station by measuring the travel time to the
station for energy at a particular frequency. This can be done by applying narrow
230 8. S U R F A C E W A V E S A N D N O R M A L M O D E S
passband filters to the record to isolate the wave packet for a target frequency,
or, more crudely, by measuring the time between successive peaks in a single
dispersed seismogram. The same approach can be used to determine the group
velocity between two stations along a great circle ray path through the source by
measuring the difference in the arrival times at the stations. In this case (the two-
station method ), precise details of the source are not required, provided the location
is approximately correct.
Many modern surface wave analyses measure the phase velocity rather than the
group velocity. This is done by computing the Fourier spectrum of the record to
determine the phase of each frequency component. If the phase is known at the
source (this requires the focal mechanism or moment tensor for the event), then
phase velocity measurements are possible from a single receiver; alternatively the
two-station method can be used to determine the phase velocity between a pair of
receivers. The tricky part of phase velocity measurements is that the observed phase
φ at a particular frequency varies only between 0 and 2π and there will typically be
many cycles between observation points, so that the total phase shift - is actually
2 πn + φ, where n is an integer.
For example, consider measuring the Rayleigh wave phase velocity in Fig. 8.
at a period of 240 s (close to the dominant period of the high-amplitude Airy phase)
using stations at 90 ◦ and 120 ◦
. Phase measurements at these ranges alone do not tell
us how many cycles, n, occurred between the stations; the phase velocity cannot be
determined without independent knowledge of n. At long periods this is not a signif-
icant problem since n can be accurately estimated from standard one-dimensional
Earth models. However, at shorter periods it becomes increasingly difficult to cal-
culate n with confidence, since lateral velocity variations in the upper mantle cause
n to vary with position as well as range. In this case, a useful approach is to measure
the phase velocity at the longest periods first, and then gradually move to shorter
periods, keeping track of the total accumulated phase shift -. This will work pro-
vided the phase velocity dispersion curve is a smooth and continuous function of
frequency.
Comprehensive studies of surface wave phase velocities, using a global distri-
bution of sources and receivers, can be used to invert for maps of phase velocity
for both Love and Rayleigh waves. This is done separately for each period using
techniques analogous to the body-wave velocity inversion problem discussed in
Chapter 5. The structure seen in these maps is related to Earth’s lateral velocity
variations; the depth dependence in this heterogeneity is constrained by the results
at different periods. Inverting surface-wave phase velocity observations is currently
one of the best ways to resolve three-dimensional velocity variations in the upper
few hundred kilometers of the mantle.
Thus far we have considered the propagation of body and surface waves largely
as if the Earth were of infinite extent. However, the Earth is a finite body in which
8.6 N O R M A L M O D E S 231
50 s
150 s
Raw –2.5 0 2.5% Crustal corrected
Figure 8.10 Rayleigh wave phase velocity at (top) 50 s and (bottom) 150 s period.The right
panels have been corrected for crustal thickness variations using the model CRUST 2.0 (Laske
et al ., http://mahi.ucsd.edu/Gabi/rem.dir/crust/crust2.html).Velocity perturbations are contoured
at 2.5% intervals, with black indicating regions that are 2.5% faster than average, and white
indicating velocities over 2.5% slower than average. Maps produced by Guy Masters (personal
communication) using measurements from Ekström et al. (1997).
Figure 8.10 plots maps of Rayleigh wave phase velocity at 50 and 150 s period.
Notice the ocean-continent signal is enhanced after corrections are applied for varia-
tions in crustal thickness. In general, the thicker crust beneath continents compared
to the oceans causes slower surface-wave velocities, but this is counteracted by
generally faster upper-mantle velocities beneath continents, which are especially
strong in shield regions. When corrections for variations in crustal properties are
applied (to obtain what the velocity would be for a globally uniform crust), the
fast continental roots become even more prominent, particularly at shorter periods,
which are more sensitive to shallow structure. Global mantle tomography mod-
els rely heavily on surface-wave analyses to constrain upper-mantle heterogeneity.
Notice the similarity between the 50 s crustal-corrected phase velocity map and the
velocity structure at 150 km depth in Figure 1.7.
8.6 Normal modes
8.6 N O R M A L M O D E S 233
which are analogous to P/ SV and Rayleigh wave motion, and toroidal modes , which
are analogous to SH and Love wave motion. The Earth’s departures from spherical
symmetry mean that this separation is not complete, but it is a very good first-order
approximation. Toroidal modes involve no radial motion and are sensitive only
to the shear velocity, whereas spheroidal modes have both radial and horizontal
motion and are sensitive to both compressional and shear velocities. Spheroidal
mode observations at long periods are also sensitive to gravity and provide the best
direct seismic constraints on Earth’s density structure.
The lateral variations in normal mode eigenfunctions are best described in terms
of spherical harmonics , which provide an orthogonal set of basis functions on a
spherical surface (spherical harmonics are useful in many areas of geophysics for
representing functions on the surface of a sphere, and descriptions are available in
many of the standard texts; see Aki and Richards (1980, 2002) or Lay and Wallace
(1995) for treatments focusing on seismology). A common normalization for the
spherical harmonics is
Y
m l (θ,^ φ)^ =^ (−^1 )
m
[
2 l + 1
4 π
(l − m)!
(l + m)!
] 1 / 2
P
m l (cos^ θ)e
imφ (8.43)
where θ and φ are spherical polar coordinates (θ is the polar angle) and Pm l is the
associated Legendre function. The spherical harmonic function is written as Y m l ,
where l is termed the angular order number and m is the azimuthal order number.
The index l is sometimes also termed the spherical harmonic degree and is zero
or a positive integer up to any value. The angular order number, m, may take on
2 l + 1 integer values between ±l. The order numbers determine the number of
lines of zero crossings that are present in the function. The total number of zero
crossings is given by l; the number of zero crossings through the pole is given by
|m|. Figure 8.12 plots examples of Y m l for some of the lower harmonic degrees. Note that the harmonics are defined relative to a particular coordinate system
and depend upon the location of the poles. If the coordinate system is rotated, any
spherical harmonic function in the old coordinate system may be expressed as a
sum of spherical harmonics of the same l but differing m in the new coordinate
system. A rotation of coordinates does not affect the angular order number but will
change the relative weights of the azimuthal order numbers. For example, a rotation
of 90◦^ can change Y 0 1 to Y 1 1
Expansions of global observations in terms of spherical harmonics are common
in geophysics. Examples include Earth’s surface geoid and seismic velocity per-
turbations at a particular depth. For Earth’s normal modes, we are interested in
displacement, which is a vector quantity and most conveniently expressed in terms
234 8. S U R F A C E W A V E S A N D N O R M A L M O D E S
l = 0 l = 1 l = 2 l = 3
m = –
m = –
m = –
m = 0
m = 1
m = 2
m = 3
Figure 8.12 Spherical harmonic functions Ym l up to degree l = 3. Positive values are shown as
white, negative as black, with near-zero values as gray.There are 2l + 1 values of m at each
degree. Note that the negative m harmonics are rotated versions of the positive m harmonics.
