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1.16 Deep Earth Structure – Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography
J-P. Montagner, Institut de Physique du Globe, Paris, France
ª 2007 Elsevier B.V. All rights reserved.

1.16.1

Introduction

559

1.16.2

Effects of Seismic Velocity and Anisotropy on Seismograms

560

1.16.2.1

First-Order Perturbation Theory

560

1.16.2.2

Effect of Anisotropic Heterogeneities on Normal Modes and Surface Waves

562

1.16.2.3

Comparison between Surface Wave Anisotropy and SKS Splitting Data

564

1.16.3

Upper Mantle Tomography of Seismic Velocity and Anisotropy

565

1.16.3.1

Forward Problem

565

1.16.3.1.1

Data space: d

565

1.16.3.1.2

Parameter space: p

568

1.16.3.2

Inverse Problem

568

1.16.3.3

Isotropic and Anisotropic Images of the Upper Mantle

571

1.16.4

Geodynamic Applications

573

1.16.4.1

Oceanic Plates

574

1.16.4.2

Continents

576

1.16.4.3

Velocity and Anisotropy in the Transition Zone

577

1.16.5

Numerical Modeling and Perspectives

579

References

584

1.16.1 Introduction
The first global isotropic tomographic models of the mantle were published in 1984 (Woodhouse and Dziewonski, 1984; Dziewonski, 1984). Since that time, many new tomographic models were published, and a large family of techniques was made available. This important progress was made possible by the extensive use of computers which can handle very large data sets and by the availability of good quality digital seismograms recorded by broadband seismic networks such as GEOSCOPE (Romanowicz et al., 1984), IRIS (Smith, 1986) and all networks coordinated by the Federation of Digital Seismograph Networks (FDSN); Romanowicz and Dziewonski, 1986). Thanks to the installation of modern digital networks, it is now possible to map the whole Earth from the surface down to its center by seismic tomography. However, most tomographic techniques only make use of travel times or phase information in seismograms and very few use the amplitude, even when seismic waveforms are used ( Woodhouse and Dziewonski, 1984; Li and Romanowicz, 1996). Global tomographic models have been improved over years

by an increase in the number of data and more importantly by using more general parametrizations, now including anisotropy (radial anisotropy in Nataf et al. (1986); general slight anisotropy in Montagner and Tanimoto (1990, 1991)) and to a lesser extent anelasticity (Tanimoto, 1990; Romanowicz, 1990). This chapter is focused on the imaging of largescale (>1000 km) lateral heterogeneities of velocity and anisotropy in the upper mantle (0–660 km depth) where the lateral resolution is the best, thanks to surface waves providing an almost uniform lateral and azimuthal coverage, particularly below oceanic areas. We will discuss how tomographic imaging completely renewed our vision of upper mantle dynamics. It makes it possible to relate surface geology and plate tectonics to underlying mantle convection, and to map at depth the origin of geological objects such as continents, mountain ranges, slabs, ridges, and plumes. The goal of this chapter is not to review all contributions to this topic, but to underline the main scientific issues, to present different approaches and to illustrate the different progress (partly subjectively) by some of our results or by other more recent models. This chapter aims to

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560 Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography

show why a major step, which takes a complete account of amplitude anomalies in the most general case and which will enable to map shorter scale heterogeneities, is now possible and presently ongoing.

1.16.2 Effects of Seismic Velocity and Anisotropy on Seismograms
For theoretical and practical reasons, the Earth was considered, for a long time, as composed of isotropic and laterally homogeneous layers. While an isotropic elastic medium can be described by two independent elastic parameters (! and ", the Lame´ parameters), the cubic symmetry requires three parameters, but the most commonly used anisotropic medium (transverse isotropy with vertical symmetry axis) necessitates five independent parameters (Love, 1927: Anderson, 1961) and the most general elastic medium requires 21 independent parameters. However, since the 1960s, it was recognized that most parts of the Earth are not only laterally heterogeneous but also anisotropic. Though the lateral heterogeneities of seismic velocities were used for a long time for geodynamical applications, the importance of anisotropy for understanding geodynamic processes has only been recognized recently.
Seismology is an observational field based on the exploitation of seismic recordings of the displacement (velocity or acceleration) of the Earth induced by earthquakes. Broadband three-component high dynamic seismometers have been installed in more than 500 stations around the world during the last 20 years (see Chapter 1.01). Thanks to progress in instrumentation and theoretical developments, it is now possible to observe and to take a simultaneous account of the effects of lateral heterogeneities of velocity and anisotropy on seismograms.

1.16.2.1 First-Order Perturbation Theory

The basic equation which governs the displacement u(r, t) is the elasto-dynamics equation:

&0

d2ui dt 2

¼

X 'ij ; j
j

þ Fli

þ FEi

½1Š

Fli et FEi represent, respectively, the whole ensemble of applied inertial and external forces (see Takeuchi and Saito (1972) or Woodhouse and Dahlen (1978)

for a complete description of all terms). Generally, by neglecting the advection term, this equation is written in a simple way:

ð&0qtt – H0Þuðr; t Þ ¼ FðrS; t Þ

½2Š

where H0 is an integrodifferential operator and F expresses all forces applied to the source volume in rS at time t (considered as external forces). F is assumed to be equal to 0 for t < 0. In the elastic case, there is a linear relationship between 'ij and the strain tensor kl : 'ij ¼ Ækl Àijkl kl (þ terms related to the initial stress). Àijkl is a fourth-order tensor, often written in its condensed form Cij as a 6 Â 6 matrix. By using the different symmetry conditions Àijkl ¼ Àjikl ¼ Àijlk ¼ Àklij , the tensor À is shown to have 21 independent elastic moduli in the most general anisotropic medium. In an isotropic medium, this number reduces to two, the Lame´ coefficients ! and ".
When solving for the free oscillations of the Earth, F ¼ 0. The solution u(r, t) of eqn [2] can be calculated for a spherically symmetric nonrotating reference
Earth model associated with the operator H0, according to the equation

&0qtt uðr; t Þ ¼ H0uðr; t Þ

½3Š

The solution of eqn [3] is beyond the scope of this
chapter and is described in Chapter 1.02. The eigenvalues of the operator H0 are equal to – &0n!2l , where n !l is the eigenfrequency characterized by two quantum numbers n and l, respectively termed radial and
angular orders. The corresponding eigenfunctions n uml (r, t) depend on three quantum numbers n, l, m, where m is the azimuthal order, with the property
that –l m l. Therefore, for a given eigenfrequency
n!l calculated in a spherically symmetric Earth model, 2l þ 1 eigenfunctions can be defined. The eigenfre-
quency n!l is said to be degenerate, with a degree of degeneracy 2l þ 1. There is a complete formal similar-
ity with the calculation of the energy levels of the atom
of hydrogen in quantum mechanics. The eigenfunctions numl (r, t) of the operator H0 are orthogonal and normalized.
The important point is that the basis of functions numl (r, t) is complete. This implies that any displacement at the surface of the Earth can be expressed as a
linear combination of these eigenfuctions:

X

uðr; t Þ ¼

nalmnuml ðr; t Þ

n; l; m

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Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography 561

Therefore, these eigenfunctions can be used to calculate the synthetic displacement at any point r, at time t, due to a force system F in the source volume. For a point force F at point rS, a step time function and its associated moment tensor M, which is a good starting model for earthquakes, the solution of eqn [2] is given by (Gilbert, 1971)

uðr;

tÞ

¼

X numl ðrÞ ð1 –

n; l; m
 e – n!l t =2Q

À M

cosn!l t Þ

n !2l

:

n

ml

Á
rs

½4Š

where  is the deformation tensor. Since eqn [4] is linear in M, it can be easily generalized to more complex spatial and temporal source functions, and can be rewritten as

uðr; t Þ ¼ Gðr; rS; t ; tSÞMðrS; tSÞ

where G(r, rS, t, tS) is the Green operator of the medium. Normal mode theory is routinely used to calculate synthetic seismograms at long periods (T ! 40 s.) and centroid moment tensor solutions (Dziewonski et al., 1981).
An example of real and synthetic seismograms is presented in Figure 1. However, there are still some discrepancies (usually frequency dependent) between the observed and synthetic seismograms. The simplest way to explain the observed phase shifts (time delays) is to remove the assumption that the Earth is spherically symmetric, that is, there are lateral heterogeneities between the source and the receiver. The next step is to characterize these lateral heterogeneities. Since the agreement between synthetic and observed seismograms is good at long periods (T ! 40 s), we can reasonably infer that the amplitude of heterogeneities is small (<10%). Behind

(a) Real seismogram

Synthetic seismogram

2500
(b) n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

3000

3500

4000

Time (s)

4500

5000
× 1/25 × 1/6.7 × 1/1.6 × 1/1.4 × 1/1.5 × 1/1.0 × 1/1.3

2500

3000

3500

4000

Time (s)

4500

5000

Figure 1 (a) Example of real and synthetic seismograms used for retrieving Rayleigh wave dispersion curve for the fundamental mode and overtones (Beucler et al., 2003). Behind body waves, the signal is composed of surface waves. (b) The complex phase before the high amplitude wave packet corresponding to the fundamental mode of Rayleigh wave (n¼0) can be synthetized by summing the first overtones.

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562 Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography

the surface wave train, a long coda is usually observed, interpreted as scattered waves. However, when filtering out periods shorter than 40 s, this coda vanishes, which means that the scattering effect is only large in the shallowest regions of the Earth (primarily the crust, and the upper lithosphere) but that it is probably negligible at larger depths. However, some groups are starting to use the information contained in these coda waves (Aki and Richards, 1980; Snieder et al., 2002), and even from seismic noise (Shapiro et al., 2005) for imaging the crust. For the sake of simplicity, our study is limited to long-period surface waves and it is hypothesized that the scale of lateral heterogeneities is large compared with the seismic wavelength. This point will be discussed in Section 1.16.3.1. A second hypothesis that must be discussed is the isotropic nature of the Earth materials. Actually, it is a poor assumption, because seismic anisotropy can be unequivocally observed at different scales. Finally, the influence of lateral variations in attenuation must also be taken into account and will be discussed elsewhere in this treatise.
1.16.2.2 Effect of Anisotropic Heterogeneities on Normal Modes and Surface Waves
Different geophysical fields are involved in the investigation of the manifestations of anisotropy of Earth materials: mineral physics and geology for the study of the microscopic scale, and seismology for scales larger than, typically, 1 km. The different observations related to anisotropy, at different scales are reviewed in Montagner (1998) and in Chapter 1.09.
Different kinds of observations have been used for investigating anisotropy in the upper mantle: the Rayleigh–Love wave discrepancy (Anderson, 1961), the azimuthal variation of phase velocities of surface waves (Forsyth, 1975) and the shear-wave splitting particularly for SKS waves (Vinnik et al., 1992). The lack of stations in oceanic areas explains why it is necessary to use surface waves to investigate upper mantle structure (isotropic or anisotropic) at the global or regional scales.
In the simplest case (fundamental modes, no coupling between branches of Rayleigh and Love waves), the frequency shift !/! (and the corresponding phase velocity perturbation V/V ), for a constant wavenumber k can be written by applying Rayleigh’s principle:

Z

! 

¼ V 

¼

1

Ãij  Àijkl kl d Z

½5Š

!k

V k 2! &0numl Ãnuml d



where ij and Àijkl are, respectively, the deformation and the deviations of elastic tensor components from a spherically symmetric, nonrotating, elastic, isotropic (SNREI) model, and numl the eigenfunctions as defined in the previous section.
We only consider the propagation of surface waves in a plane-layered medium for a general slight elastic anisotropy, but it can be easily extended to the spherical Earth (Mochizuki, 1986; Tanimoto, 1986; Romanowicz and Snieder, 1988; Larson et al., 1998; Trampert and Woodhouse, 2003). Smith and Dahlen (1973, 1975) found that, to first order in anisotropy and at frequency !, the azimuthal variation of local phase velocity (Rayleigh or Love wave) can be expanded as a Fourier series of the azimuth É along the path and is of the form

V ð!; ; 0; ÉÞ – V0ð!Þ ¼ 0ð!; ; 0Þ þ 1ð!; ; 0Þcos 2É þ 2ð!; ; 0Þsin 2É þ 3ð!; ; 0Þcos 4É þ 4ð!; ; 0Þsin 4É ½6Š

where V0(!) is the reference velocity of the unperturbed medium, and É is the azimuth along the path with respect to the north direction. Montagner and Nataf (1986) present the expressions for the different azimuthal coefficients i (!, , 0) as depth integral functions dependent on 13 simple linear combinations of standard cartesian elastic coefficients Cij . Appendix 1 shows how to relate Àijkl to Cij and presents detailed calculation of azimuthal terms for Love waves in the geographical coordinate system:

 Constant term (0É-azimuthal term: 0)

A

¼

&VP2H

¼

3 8

ðC11

þ

C22Þ

þ

1 4

C12

þ

1 2

C66

C ¼ &VP2V ¼ C33

F

¼

1 2

ðC13

þ

C23Þ

L

¼

&VS2V

¼

1 2

ðC44

þ

C55Þ

N

¼

&VS2H

¼

1 8

ðC11

þ

C22Þ –

1 4 C12

þ

1 2 C66

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 2É-azimuthal term:

1 cos 2É

Bc

¼

1 2

ðC11

–

C22Þ

Gc

¼

1 2

ðC55

–

C44Þ

Hc

¼

1 2

ðC13

–

C23Þ

2 sin 2É Bs ¼ C16 þ C26 Gs ¼ C54 Hs ¼ C36

 4É-azimuthal term:

3 cos 4É

Ec

¼

1 8

ðC11

þ

C22Þ

þ

1 4 C12

–

1 2

C66

4 sin 4É

Es

¼

1 2

ðC16

– C26Þ

where indices 1 and 2 refer to horizontal coordinates (1: North; 2: East) and index 3 refers to vertical coordinate. & is the density, VPH, VPV are respectively horizontally and vertically ‘propagating’ P-wave velocities, VSH, VSV horizontal and vertical ‘polarized’ S-wave velocities. So, the different parameters present in the different azimuthal terms are simply related to elastic moduli Cij .
From a practical point of view, once source phase is removed and assuming that the scale of heterogeneities is larger than the wavelength, the total phase 0t (and the travel time) between the epicenter E and the receiver R is easily related to the measurement of phase velocity Vd(!), and therefore to the local phase velocity V (!, , 0, É):

0t

¼

!tE!R

¼

!Á Vdð!Þ

¼

ZR !
E V ð!;

ds ; 0;

ÉÞ

½7Š

Therefore, eqns [6] and [7] define the forward problem in the framework of first-order perturbation theory. We will see in the next section how to solve the inverse problem. This means that, ideally, surface waves in the plane case have the ability to provide information on 13 elastic parameters, which emphasizes the enormous potential of surface waves in terms of geodynamical and petrological implications. There are only 13 elastic moduli among 21, since propagation of surface waves is invariant against rotation by %, which corresponds to a monoclinic symmetry.
The 0-É term corresponds to the average over all azimuths and involves five independent parameters, A, C, F, L, N, which represent the equivalent transversely isotropic medium with a vertical symmetry axis (more simply named VTI or radial anisotropy). It must be noted that it is possible to retrieve the equivalent isotropic shear modulus from these five

parameters. By using a Voigt average, the shear modulus "iso is given by

"iso

¼

&VS2iso

¼

1 15

ðC11

þ C22

þ C33 – C12

– C13 – C23 þ 3C44 þ 3C55 þ 3C66Þ

According to the expressions of A, C, F, L, N in terms of

elastic moduli, "iso ¼ 115ðC þ A – 2F þ 6L þ 5N Þ So we can see that the equivalent isotropic velocity

depends not only on VSV and VSH, but also on P-wave velocity and anisotropy (0 ¼ C/A) and on

 ¼ F/(A – 2L). By rewriting this expression

"iso ¼ 115ðC þ ð1 – 2ÞA þ ð6 þ 4ÞL þ 5N Þ, neglecting anisotropy in P-wave (0 ¼ 1) and assuming  ¼ 1,

it is found that "iso ¼ &VS2iso % 23L þ 13N ¼ 23&VS2V þ 13&VS2H. Naturally, this choice is partly arbitrary, since usually, there is no S-wave anisotropy

without P-wave anisotropy. Another way might consist

in using correlations between anisotropic parameters

for petrological models as derived by Montagner and

Anderson (1989a).

The other azimuthal terms (2-É and 4-É) depend

on four groups of two parameters, B, G, H, E, respec-

tively describing the azimuthal variation of A, L, F, N.

These simple parameters make it possible to describe

in a simple way the two seismically observable effects

of anisotropy on surface waves, the ‘polarization’

anisotropy (Schlue and Knopoff, 1977) and the azi-

muthal anisotropy (Forsyth, 1975).

Another important point in these expressions is

that they provide the partial derivatives for the radial

and azimuthal anisotropy of surface waves. The cor-

responding kernels and their depth dependence are

plotted in Montagner and Nataf (1986) (Figures 14

and 15). These partial derivatives of the different

azimuthal terms with respect to the elastic para-

meters can be easily calculated by using a radial

anisotropic reference Earth model, such as PREM

(Dziewonski and Anderson, 1981). The partial deri-

vatives of the eigenperiod 0Tl with respect to parameter p, ( p/T) (qT/qp) can easily be converted

into phase velocity partial derivatives by using





p qV ¼ – V p qT

V qp T U T qp k

For example, the parameters Gc and Gs have the same kernel as parameter L (related to VSV) as shown by comparing the expressions of R1, R2, and R3 in eqn [29] of Appendix 1. For fundamental modes, the calculation of kernels shows that Love waves are almost insensitive to VSV (Figure 14) and Rayleigh

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564 Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography

waves to VSH. Rayleigh waves are the most sensitive to SV waves. However, as pointed out by Anderson and Dziewonski (1982), the influence of P-waves (through parameters A and C ) can be very large in an anisotropic medium. The influence of density is also very large for Love and Rayleigh waves but, as shown by Takeuchi and Saito (1972), it is largely decreased when seismic velocities are inverted for, instead of elastic moduli and density.

1.16.2.3 Comparison between Surface Wave Anisotropy and SKS Splitting Data

It can be noted that some of the linear combinations of elastic moduli Cij , derived from surface waves in the previous section, also come up when considering the propagation of body waves in symmetry planes for a weakly anisotropic medium (see, e.g., Crampin et al. (1984)), and their azimuthal dependence

&VP2 ¼ A þ Bccos 2É þ Bssin 2É þ Eccos 4É þ Essin 4É
&Vq2SH ¼ N – Eccos 4É – Essin 4É &Vq2SV ¼ L þ Gccos 2É þ Gssin 2É

where VqSH and VqSV correspond, respectively, to quasi-SH and quasi-SV waves.
A global investigation of anisotropy inferred from SKS body wave splitting measurements (delay times and directions of maximum velocities) has been undertaken by different authors (Vinnik et al., 1992; Silver, 1996; Savage, 1999). Unfortunately, most SKS measurements have been done in continental parts of the Earth, and very few in oceans. It turns out that a direct comparison of body wave and surface wave data sets is now possible (Montagner et al., 2000). If the anisotropic medium is assumed to be characterized by a horizontal symmetry axis with any orientation (this is a very strong assumption which can be alleviated as shown by Chevrot et al., 2004), and for a vertically propagating SKS wave, a synthetic data set of SKS delay times and azimuths can be calculated from the global distribution of anisotropy derived from surface waves, by using the following equations:

tSKS

¼

Z
0

h

rffiffiffi &
dz L

GcðzÞ LðzÞ

cosð2ÉðzÞÞ !

þ GsðzÞ sinð2ÉðzÞÞ

½8Š

LðzÞ

where tSKS is the integrated travel time for the depth range 0 to h for a propagation azimuth É,

where the anisotropic parameters Gc(z), Gs(z), and L(z) are the anisotropic parameters retrieved from

surface waves at different depths. It is remarkable to

realize that only the G -parameter (expressing the

SV wave azimuthal variation) is present in this

equation. From eqn [8], we can infer the maximum value of delay time tSmKaSx and the corresponding azimuth ÉSKS:

tSmKaSx

¼

s&ffiffiffiffiZffiffiffiffiffihffiffiffidffiffizffiffirffiffiffiffiffiffi&ffiffiffiffiffiffiGffiffifficffiðffiffizffiffiÞffiffi'ffiffiffiffi2ffiffiþffiffiffi&ffiffiffiffiZffiffiffiffiffihffiffidffiffiffizffiffirffiffiffiffiffiffi&ffiffiffiffiffiGffiffiffisffiffiðffiffizffiffiÞffiffi'ffiffiffi2ffiffi

0

L LðzÞ

0

L LðzÞ

½9Š

Zh

dzGsðzÞ=LðzÞ

tanð2ÉSKSÞ ¼

0

Z
0

h

dz

GcðzÞ LðzÞ

½10Š

However, eqn [8] is approximate and only valid when the wavelength is much larger than the thickness of layers. It is possible to make more precise calculations by using the technique derived for two layers by Silver and Savage (1994) or by using the general expressions given in Rumpker and Silver (1998), Montagner et al. (2000), and Chevrot et al. (2004).
With eqns [9] and [10], a synthetic map of the maximum value of delay time tSmKaSx can be obtained by using a 3-D anisotropic surface wave model. A detailed comparison between synthetic SKS derived from anisotropic upper mantle (AUM) model (Montagner and Tanimoto, 1991) and observed SKS (Silver, 1996) was presented in Montagner et al. (2000). Figure 2 shows such a map for the Earth

Synthetic SKS prem amax = 1.88 s

60° 30°

60° 30°

0°

0°

–30°

–30°

–60°

–60°

0.00 0.05 0.10 0.20 0.30 0.40 0.50 0.60 1.00 1.50 3.00
Figure 2 Map of synthetic SKS splitting delay time derived from the anisotropic surface wave model of Montagner (2002). The delay time is expressed in seconds; Amax ¼ 1.88 s.

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Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography 565

centered in the Pacific, by using the anisotropic surface wave model of Montagner (2002) derived from the data of Montagner and Tanimoto (1991) and Ekstro¨m et al. (1997). First of all, the comparison shows that both data sets are compatible in magnitude but not necessarily in directions. Some contradictions between measurements derived from surface waves and from body waves have been noted. The agreement of directions is correct in tectonically active areas but not in old cratonic zones. The discrepancy in these areas results from the rapid lateral change of directions of anisotropy at a small scale. These changes stem from the complex history of these areas, which have been built by successive collages of continental pieces. It might also result from the hypothesis of horizontal symmetry axis, which was shown to be invalid in many areas (Plomerova et al., 1996). The positive consequence of this discrepancy is that a small-scale mapping of fossile anisotropy in such areas might provide clues for understanding the processes of growth of continents and mountain building opening a new field, the paleoseismology.
Unlike surface waves, SKS waves have a good lateral resolution, and are sensitive to the short wavelength anisotropy just below the stations. But their drawback is that they have a poor vertical resolution. On the other hand, global anisotropy tomography derived from surface waves only provides longwavelength anisotropy (poor lateral resolution) but enables the location at depth of anisotropy. The longwavelength anisotropy derived from surface waves will display the same direction as the short-wavelength anisotropy inferred from body waves only when large-scale vertical coherent processes are predominant. As demonstrated by Montagner et al. (2000), the best agreement between observed and synthetic SKS can be found when only layers in the uppermost 200 km of the mantle are taken into account. Moreover, tomographic models derived from surface waves lose resolution at depths greater than 200 km. In some continental areas, short-scale anisotropy, the result of a complex history, might be important and even might mask the large-scale anisotropy more related to present convective processes (see, e.g., Marone and Romanowicz, 2006 for North America). From a statistical point of view, good agreement is found between orientations of anisotropy and plate velocity motion for fast-moving plates. The differences between anisotropy and tectonic plate directions are related to more complex processes, as will be seen in Section 1.16.3.

1.16.3 Upper Mantle Tomography of Seismic Velocity and Anisotropy
We now show how to implement theory of Section 1.16.1 from a practical but general point of view, and how to design a tomographic technique in order to invert for the 13 different elastic parameters and density. A tomographic technique necessitates solving simultaneously a forward problem and an inverse problem. By using the results of the previous section, it successively considers how to set the forward problem, and how it is used to retrieve a set of parameters by inversion.

1.16.3.1 Forward Problem
First, it is necessary to define the data space d and the parameter space p. It is assumed that a functional g relating d and p can be found such that
d ¼ gðpÞ
where d is the set of data (which samples the data space), and p the set of parameters.

1.16.3.1.1 Data space: d

The basic data set is made of seismograms u(t). We

can try to directly match the waveform in the time

domain, or we can work in the Fourier domain, by

separating phase and amplitude on each component

ui (t):

Z1

ui ðt Þ ¼

Ai ð!Þeið!t – 0i Þd!

–1

The approach consisting in fitting seismic waveforms is quite general but, from a practical point of view, it does not necessarily correspond to the simplest choice. In a heterogeneous medium, the calculation of amplitude and phase effects makes it necessary to calculate the coupling between different multiplets (Li and Tanimoto, 1993; Li and Romanowicz, 1995; Marquering et al., 1996), which is very time consuming. When working in Fourier domain, different time windows can be considered and the phase of different seismic trains, body waves and surface waves can be separately matched (Nolet, 1990; Le´veˆque et al., 1991) under drastic simplifying assumptions. Figure 1 shows an example of observed and synthetic seismograms, the latter obtained by normal mode summation with the different higher modes. The fundamental wave train is well separated from other

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566 Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography

modes at large epicentral distances. The part of the seismogram corresponding to higher modes is more complex and shows overlap of these modes in the time domain. Therefore, from a practical point of view, the fitting of the fundamental mode wave train will not cause any problem and has been widely used in global mantle tomography. The use of higher mode wave trains and the separation of overtones is much more difficult. The first attempts were performed by Nolet (1975), Cara (1979), Okal and Jo (1985), and Dost (1990) by applying a spatial filtering method. Unfortunately, all these techniques can only be applied in areas where dense arrays of seismic stations are present, that is, in North America and Europe. By using a set of seismograms recorded at one station but corresponding to several earthquakes located in a small source area, Stutzmann and Montagner (1994) showed how to separate the different higher modes. A similar approach was also followed by Van Heijst and Woodhouse (1997). We only detail in this paper the technique which was designed for fitting the fundamental mode wave train and the reader is referred to Stutzmann and Montagner (1994), Van Heijst and Woodhouse (1997), and Beucler et al. (2003) for the description of the recovery of higher-mode dispersion properties and to Romanowicz (2002) for a general overview. Figure 3 presents an example of phase velocity

dispersion for different surface wave modes (funda-

mental and first higher modes (Beucler et al., 2003))

and how they compare with previous investigations

(Cara, 1979; Van Heijst and Woodhouse, 1997).

We take advantage of the fact that, according to

the Fermat’s principle, the phase velocity perturba-

tion is only dependent to second order on path

perturbations, whereas amplitude perturbations are

dependent, to first order, on these perturbations,

which implies that the eigenfunctions must be re-

calculated at each iteration. Therefore, the phase is

a more robust observable than the amplitude. The

amplitude A(!) depends in a complex manner on

seismic moment tensor, attenuation, scattering,

focusing effects, station calibration and near-receiver

structure whereas the contribution of lateral hetero-

geneities of seismic velocity and anisotropic

parameters to the phase 0(!) can be easily extracted.

The data set under investigation, is composed of

propagation times (or phase velocity measurements

for surface waves) along paths: d ¼ { Á/V(!)}.

On the other hand, the phase of a seismogram at

time t is decomposed, as where k is the wave vector, including several terms: 000

follows: 0 ¼ k?r 000 is the initial ¼ 00 þ 0S þ 0I;

þ 000, phase
0S is

the initial source phase, 00 is related to the number

of polar phase shifts, 0I is the instrumental phase.

0 can be measured on seismograms by Fourier

PREM This study 14 12 10 8

σ (km s–1)

σ (km s–1)

van Heijst et Woodhouse [1997] Cara [1979] 4.3 4.2 n = 0 4.1 4.0 3.9 3.8 25 50 75 100 125 150
Period (s)

6.8

6.4 6.0

n = 1

5.6

5.2

4.8

4.4

25 50

75 100 125 150 Period (s)

σ (km s–1)

6
4 50 100 150 200 250 300 Period (s)

σ (km s–1)

6.4 6.0 n = 2 5.6 5.2 4.8
30 40 50 60 70 Period (s)

Figure 3 Phase velocity of the fundamental mode and the first six higher modes of Rayleigh compared with PREM (right plot) and with results (center) obtained in previous studies along the same path between Vanuatu and California (SCZ geoscope station) (Beucler et al., 2003).

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transform. We usually assume that 0S is correctly given by the centroid moment tensor solution. For a path between epicenter E and receiver R with an epicentral distance Á, the phase 0 is given by

!Á

0 ¼ Vobs þ 00 þ 0S þ 0I

½11Š

In the general case, we want to relate the observed phase velocity Vobs(!) to the parameters of the Earth model p(r, , 0). Data and 3-D parameters can be related through integrals over the whole volume of the Earth. But for computing reasons, it is usual to use a multistep approach, where we first retrieve the local phase velocity V(!, , 0) including its azimuthal terms, and then perform the inversion at depth. These two steps can be reversed since the order of the integrations can be reversed. It is necessary to consider the nature of the perturbed medium. Following the approach of Snieder (1988), if the perturbed medium is at the same time smooth (long-wavelength heterogeneities) and weak (small amplitude of heterogeneities), the geometrical optics approximation (and ray theory) applies. This hypothesis is not necessarily met within the Earth where some geological objects (slabs, mantle plumes, etc.) have a length scale which can be close to the seismic wavelength. In the approximation of ray theory, the volume integral reduces to the curvilinear integral along the geometrical ray path. When ray theory is applicable, we have

0–

000

¼

!Á Vobsð!Þ

¼

ZR
E

! ds V ð!; ;

0Þ

½12Š

where the integral is evaluated along the ray path between the epicenter E and the receiver R. Following the results of the previous section, different approximations are implicitly made when using this expression of the phase:
 Large angular order l ) 1, but not too large (scattering problems). From a practical point of view, this means that measurements are performed in the period range 40 s < T < 200 s with seismic wavelengths between 200 and 1000 km.
 Geometrical optics approximation. If ! is the wavelength of the surface wave at period T, and ÃS the spatial wavelength of heterogeneity: ÃS ) ! ¼ VT ) ÃS & 2000 km. Epicentral distance Á must be larger than seismic wavelength.

 Slight anisotropy and heterogeneity. V/V ( 1. According to Smith and Dahlen (1973) for the plane case, the local phase velocity can be decomposed as a Fourier series of the azimuth É (eqn (6)): Each azimuthal term i(T, , 0) of eqn [6] can be related to the set of parameters pi(r, , 0) (density þ 13 elastic parameters), according to the expressions derived in Appendix 1:

Á

Á

–

VobsðT Þ X2
¼– j ¼0

V0ðT Þ X 14 Z R i¼1 E

ds V0

Z

a

" pi

0V

 qV piðr ; ;

qpi j

pi

0Þ

cosð2j ÉÞ

þ

pi qV V qpi

pi ðr ; ; 0Þ sinð2j ÉÞ

j

pi

½13Š

Equation [13] defines the forward problem in the framework of first-order perturbation theory, relating the data and the parameter spaces. This approach is usually named path average approximation (PAVA). Many terms in eqn [13] are equal to zero since all parameters are not present in each azimuthal term. A last important ingredient in the inverse problem formulation is the structure of the data space. It is expressed through its covariance function (continuous case) or covariance matrix (discrete case) of data Cd. When data di are independent, Cd is diagonal and its elements are the square of the errors on data 'di .

1.16.3.1.1.(i) Finite-frequency effects As mentioned previously, a strong hypothesis is that in the framework of geometrical optics approximation, only large-scale heterogeneities can be retrieved. But interesting geological objects such as slabs and plumes are smaller scale. To go beyond the ray theory, it is necessary to take account of the finitefrequency effect when scale length has the same order of magnitude as the seismic wavelength. It is possible to use the scattering theory based on the Born or Rytov approximations (see, e.g., Woodhouse and Girnius (1982) for normal mode approach, Snieder (1988) for surfaces waves, Yomogida (1992), and Dahlen et al. (2000) for body waves). Equation [13] shows that the sensitivity kernels are 1-D, meaning that only heterogeneities in the vertical plane containing the source and the receiver are taken into account, whereas, by using the scattering theory, it is possible to calculate 3-D kernels and consequently to take account of off-path

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heterogeneities. Equations [12] and [13] must be replaced by an integral over the volume :

Á

Á

X 14 Z Z Z

–

¼

VobsðT Þ V0ðT Þ i¼1



Â

Ki ðT ;

;

0Þ

piðr ; ; pi

0Þ d

½14Š

where K(T, , 0) is the scattering Fre´chet sensitivity kernel, which depends on wave type (Rayleigh or Love) and on the relative location of E and R (see, e.g., Romanowicz (2002) for a review). Very different strategies can be followed for calculating this triple integral, by separating the surface integral and the radial integral. For a point scatterer, the kernel displays a typical shape of banana-doughnut (Dahlen et al., 2000). Different approximations of K(T, , 0) have been proposed (Spetzler et al., 2002; Yoshizawa and Kennett, 2002; Ritzwoller et al., 2002), but Sieminski et al. (2004) claimed that ray theory surface wave tomography with a very dense path coverage can detect heterogeneities with length scales close and even smaller than the seismic wavelength. The discussion of the advantages and shortcomings of these different techniques is beyond the scope of this chapter, but some new tomographic models using 3-D sensitivity kernels are starting to be constructed (Zhou et al., 2004, 2006) for radially anisotropic media.

1.16.3.1.2 Parameter space: p
It is quite important to consider the structure of the parameter space in detail. First of all, it is necessary to define which parameters are required to explain our data set, how many physical parameters can be effectively inverted for, in the framework of the theory that is considered. For example, if the Earth is assumed to be elastic, laterally heterogeneous but isotropic, only three independent physical parameters, VP, VS, and density & (or the elastic moduli !, ", and &) can be inverted for, from surface waves. In a transversely isotropic medium with a vertical symmetry axis (Anderson, 1961; Takeuchi and Saito, 1972), the number of independent physical parameters is now six (five elastic parameters þ density). In the most general case of a weak anisotropy, 14 physical parameters (13 combinations of elastic moduli þ density) can actually be inverted for, using surface waves. Therefore, the number of ‘physical’ parameters pi depends on the underlying theory which is used for explaining the data set.

Once the number of physical independent parameters is defined, we must define how many spatial (or geographical) parameters are required to describe the 3-D distributions pi (r, , 0). This is a difficult problem because the number of spatial parameters that can be reliably retrieved from the data set is not necessarily sufficient to provide a correct description of pi (r, , 0), that is, of the real Earth. The correct description of pi (r, , 0) depends on its spectral content: for example, if pi (r, , 0) is characterized by very large wavelengths, only a small number of spatial parameters is necessary, but if pi (r, , 0) presents very small scale features, the number of spatial parameters will be very large. In any case, it is necessary to assess the range of possible variations for pi (r, , 0) in order to provide some bounds on the parameter space. This is done through a covariance function of parameters in the continuous case (or a covariance matrix for the discrete case) Cpipj (r, r9) at two different points r, r9. These a priori constraints can be provided by other fields in geosciences, geology, mineralogy, numerical modeling, etc.
Consequently, a tomographic technique must not be restricted to the inversion of parameters p ¼ {pi (r, , 0)} that are searched for, but must include the calculation of the final covariance function (or matrix) of parameters Cp . This means that the retrieval of parameters is contingent to the resolution and the errors of the final parameters and is largely dependent on the resolving power of data (Backus and Gilbert, 1967, 1968, 1970). Finally, the functional g which expresses the theory relating the data space to the parameter space is also subject to uncertainty. In order to be completely consistent, it is necessary to define the domain of validity of the theory and to assess the error 'T associated with the theory. Tarantola and Valette (1982) showed that the error 'T is simply added to the error on data 'd .
1.16.3.2 Inverse Problem
So far, we did not make assumption on the functional g relating data and parameters. But in the framework of first-order perturbation theory, the forward problem is usually linearized and eqn [13] can be simply written in the linear case:
d ¼ Gp
where G is now a matrix (or a linear operator) composed of Fre´chet derivatives of d with respect to p,

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which has the dimensions nd  np (number of data  number of parameters). This matrix usually is not square and many different techniques in the past have been used for inverting G. In any case, the inverse problem will consist in finding an inverse for the functional g, which we will write g˜ – 1, notwithstanding the way it is obtained, such that

p ¼ g˜ – 1ðdÞ

To solve the inverse problem, different algorithms can be used. The least-squares solution is usually solved by minimizing a cost function J. Making the data space and the parameter space symmetric, Tarantola and Valette (1982) define the cost function J as

J ¼ ðd – GpÞtCd– 1ðd – GpÞ þ ðp – p0ÞtCp– 1ðp – p0Þ

The first term corresponds to classical least-squares with no damping, whereas the second term corresponds to norm damping, which imposes smoothness upon the parameter space. Different choices were proposed for this second term. For example, Montagner (1986b) uses a Gaussian covariance function characterized by a correlation length and an a priori error 'p on parameters, whereas Su et al. (1984) prefer to minimize the roughness of the model. Other choices consist in taking a constant value such that ptCp– 1p ¼ !2ptp (Yoshizawa and Kennett, 2004). Or the covariance operator can be replaced by a Laplacian operator (see, e.g., Zhou et al., 2006):

Z ptCp– 1p ¼ "

Z

Z

jr2

 p j2

d

1=2 

p

A discussion about damping can be found in Trampert and Snieder (1996), who prefer Laplacian over model damping to reduce the spectral leakage.
As an example, by using the expression of J, a quite general and widely used algorithm has been derived by Tarantola and Valette (1982):

p

–

p0

¼

 GtCd– 1G

þ

Cp–0 1 – 1GtCd– 1ðd

–

gðpÞ

þ

Gðp

–

p0ÞÞ

¼ Cp0 GtðCd þ GCp0 GtÞ – 1ðd – gðpÞ

þ Gðp – p0ÞÞ

½15Š

where Cd is the covariance matrix of data, Cp0 the covariance function of parameters p, and G is the Frechet derivative of the operator g at point p(r). This algorithm can be made more explicit by writing it in its integral form:

XXZ

pðrÞ ¼ p0ðrÞ þ

dr9Cp0ðr; r9Þ

ijV

 Gi ðr9ÞðS – 1Þij Fj

½16Š

with Z
Sij ¼ Cdij þ dr1dr2Gi ðr1ÞCp0 ðr1; r2ÞGj ðr2Þ VZ
Fj ¼ dj – g j ðpÞ þ dr0Gj ðr0Þðpðr0Þ – p0ðr0ÞÞ
V
This algorithm can be iterated and is suited for solving slightly nonlinear problems. Different strategies can be followed to invert for the 3-D models p(r), because the size of the inverse problem is usually enormous in practical applications and a compromise must be found between resolution and accuracy (and also computing time). For the example of mantle tomography, a minimum parameter space will be composed of 13 (þdensity) physical parameters multiplied by 30 layers (if the mantle is divided into 30 independent layers. If geographical distributions of parameters are searched for up to degree 40 (lateral resolution around 1000 km), this implies a number of $700 000 independent parameters. Such a problem is still very difficult to handle from a computational point of view. A simple approach for solving this problem consists in dividing the inversion procedure into two steps. The first step consists in regionalizing phase (or group) velocity data in order to retrieve the different azimuthal terms, and the second step is the inversion at depth. It was implemented by Montagner (1986a, 1986b) and a very similar technique is presented by Barmin et al. (2001). In case of a large data set, Montagner and Tanimoto (1990) showed how to handle the inverse problem by making a series expansion of the inverse of matrix S. It was recently optimized from a computational point of view by Debayle and Sambridge (2004) and Beucler and Montagner (2006). One advantage of this technique is that it can be applied indifferently to regional studies or global studies. In case of imperfect spatial coverage of the area under investigation, it does not display ringing phenomena commonly observed when a spherical harmonics expansion is used (Tanimoto, 1986).
From a practical point of view, the choice of the model parameterization is also very important and different possibilities can be considered:

 Discrete basis of functions. A simple choice consists in dividing the Earth into 3-D blocks with a surface block size different from the radial one.

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The size of block depends on the lateral resolution

expected from the path coverage. A variant of this

parametrization is the use of a set of spherical

triangular grid points (see, e.g., Zhou et al.

(2006)). The block decomposition is valid as well

for global investigations as for regional studies.

Usually, the Earth surface parametrization is

different for the radial one. For global study,

the natural basis is composed of the spherical

harmonics piðr ; ; 0Þ

¼

fPorlmax
l ¼0

Pthel
m¼

–

horizontal l almðr ÞYlmð;

variations 0Þ. Other

choices are possible, such as spherical splines

(Wang and Dahlen, 1995). When data coverage is

very uneven, other strategies are proposed using

irregular cells or adaptative meshes (see, e.g.,

Zhang and Thurber (2005)).

 Continuous function p(r). In this case, the function

is directly inverted for. Since the number of para-

meters is then infinite, it is necessary to regularize

the solution by defining a covariance function of

parameters Cp0(r, r9). For the horizontal variations, a Von Mises distribution (Montagner, 1986b) can be

used for initial parameters P0(r):

Cp0 ðr;

r9Þ

¼

'pðrÞ'pðr9Þ exp

cos

Árr9 L2cor

–

1

%

'p ðrÞ'p ðr9Þ

exp

– Á2rr9 2L2cor

where Lcor is the correlation length, which defines the smoothness of the final model. This kind of distribution is well suited for studies on a sphere and is asymptotically equivalent to a Gaussian distribution when Lcor ( a (a radius of the Earth). When distributions of different azimuthal terms are searched for, it is possible to define cross-correlated covariance functions of parameters Cpipj (r, r9), but since the different terms of the Fourier expansion in azimuth correspond to orthogonal functions, the cross-correlated terms off the diagonal can be taken equal to zero.

It is interesting to note that in eqn [16] the Frechet derivatives G along the path are multiplied by the Gaussian covariance operator Cp0. It means that the technique, which can be named Gaussian tomography, is equivalent to use fat rays: when the correlation length is wider than the Fresnel zone, ray theory applies and, consequently, the finite-frequency effects can be neglected. As discussed by Ritzwoller et al. (2002) and Sieminski et al. (2004), there might be some slight differences in amplitude between Gaussian tomography and diffraction tomography (taking account of finite-frequency effects),

but not in the location of heterogeneities provided

that the spatial path coverage is sufficiently dense.

The radial parametrization must be related to the

resolving capability of the data at depth, according

to the frequency range under consideration. For the

radial variations, polynomial expansions can be used

(see, e.g., Dziewonski and Woodhouse (1987) for

Tchebyshev polynomials, or Boschi and Ekstro¨m

(2002) for radial cubic splines). Since the number of

physical parameters is very large for the inversion at

depth, physical parameters are usually correlated.

The different terms of the covariance function Cp between parameters p1 and p2 at radii ri and rj can be defined as follows:

"

#

Cp1; p2 ðri ; rj Þ ¼ 'p1 ðri Þ'p2 ðrj Þp1; p2 exp

– ðri – rj Þ2 2Lri Lrj

Where p1,p2 is the correlation between physical parameters p1 and p2 inferred for instance from different petrological models (Montagner and Anderson, 1989a) such as pyrolite (Ringwood, 1975) and piclogite (Anderson and Bass, 1984; Bass and Anderson, 1986). Lri ; Lrj are the radial correlation lengths, which are used to smooth the inverse model.
The a posteriori covariance function is given by

Cp

¼ ¼

Cp0 G

T–CCdp–01GGTþðCCd p–þ0 1G–C1p0

G

t

Þ

–

1

GCp0

½17Š

The resolution R of parameters can be calculated as well. It corresponds to the impulsive response of the system: p ¼ g˜ À1 d ¼ g˜ À1 gp9 ¼ Rp9. If the inverse problem is perfectly solved, R is the identity function or matrix. However, the following expression of resolution is only valid in the linear case (Montagner and Jobert, 1981):

R ¼ Cp0 Gt ðCd þ GCp0 Gt Þ – 1G ¼ ðGt Cd G þ Cp0 Þ – 1Gt Cd– 1G ½18Š
It is interesting to note that the local resolution of parameters is imposed by both the correlation length and the path coverage, unlike the Backus-Gilbert (1967, 1968) approach, which primarily depends on the path coverage. The effect of a damping factor in the algorithm to smooth the solution is equivalent to the introduction of a simple covariance function on parameters weighted by the errors on data (Ho-Liu et al., 1989). When the correlation length is chosen very small, the algorithms of Backus-Gilbert (1968, 1970) and Tarantola and Valette (1982) are equivalent.

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By considering the a posteriori covariance function and the resolution, it is possible to assess the reliability of the hypotheses made about the independence of parameters. For example, Tanimoto and Anderson (1985) and Montagner and Jobert (1988) showed that there is a tradeoff between azimuthal terms and constant term in case of a poor azimuthal coverage. For the inversion at depth, Nataf et al. (1986) also display the tradeoff between physical parameters VPH, VSV, $, 0, and  when only Rayleigh and Love wave 0-É terms are used in the inversion process.
Though 13 elastic parameters (þ density) are necessary to explain surface wave data (Rayleigh and Love waves), only four parameters are well resolved for small anisotropy (Montagner and Jobert, 1988): the azimuthally averaged S-wave velocity VS, the radial anisotropy expressed through the $ parameter ($ ¼ (VSH/VSV)2), where VSH (resp. VSV) is the velocity of S-wave propagating horizontally with horizontal transverse polarization (resp. with vertical polarization), and the G (Gc, Gs) parameters expressing the horizontal azimuthal variation of VSV. $ was introduced in the reference Earth model PREM (Dziewonski and Anderson, 1981) down to 220 km in order to explain a large data set of free oscillation eigenfrequencies and body wave travel times. The other elastic parameters can be derived by using constraints from petrology in order to reduce the parameter space (Montagner and Anderson, 1989a). This approach was followed by Montagner and Anderson (1989b) to derive an average reference earth model, and by Montagner and Tanimoto (1991) for the first global 3-D anisotropic model of the upper mantle.
1.16.3.3 Isotropic and Anisotropic Images of the Upper Mantle
The complete anisotropic tomographic procedure has been implemented for making different regional and global studies. Many global isotropic tomographic models of the upper mantle were published since Wooodhouse and Dziewonski (1984) and the recent results have been reviewed by Romanowicz (2003). Many models inverting only for radial anisotropy but neglecting azimuthal anisotropy have also been published (Nataf et al., 1984, 1986; Ekstro¨m and Dziewonski, 1998; Shapiro and Ritzwoller, 2002, Gung et al., 2003; Panning and Romanowicz, 2004; Zhou et al., 2006) The complete anisotropic tomographic technique (including azimuthal anisotropy) has been applied for investigating the upper mantle

structure either at a regional scale of the Indian Ocean (Montagner, 1986a; Montagner and Jobert, 1988; Debayle and Le´veˆque, 1997), the Atlantic Ocean (Mocquet and Romanowicz, 1989; Silveira et al., 1998), Africa (Hadiouche et al., 1989; Debayle et al., 2001; Sebai et al., 2005; Sicilia et al., 2005), Pacific Ocean (Nishimura and Forsyth, 1989; Montagner, 2002; Ritzwoller et al., 2004), Antarctica (Roult et al., 1994), Australia (Debayle and Kennett, 2000; Simons et al., 2002), and Central Asia (Griot et al., 1998a, 1998b); Villasen˜or et al., 2001) or at a global scale (Montagner and Tanimoto, 1990, 1991; Montagner, 2002; Debayle et al., 2004). The reader is also referred to a quantitative comparison of tomographic and geodynamic models by Becker and Boschi (2002).
An important issue when constructing tomographic models is the correction for crustal structure, where sedimentary thickness and Moho depth variations are so strong that they affect dispersion of surface waves at least up to 100 s: it has been shown (Montagner and Jobert, 1988) that standard perturbation theory is inadequate to correct for crustal correction and more rigorous approaches have been proposed (Li and Romanowicz, 1996; Boschi and Ekstro¨m, 2002; Zhou et al., 2005) using the updated crustal models 3SMAC (Nataf and Ricard, 1996; Ricard et al., 1996) or CRUST2.0 (Mooney et al., 1998; Laske et al., 2001).
As an example of the results obtained after the first step of the tomographic procedure, Figure 4 shows different maps of 2-É azimuthal anisotropy for Rayleigh waves at 100 s period for the first three modes, n ¼ 0,1,2, superimposed on the isotropic part (0-É term) of phase velocity (Beucler and Montagner, 2006). From petrological and mineralogical considerations, Montagner and Nataf (1988) and Montagner and Anderson (1989a, 1989b) showed that the predominant terms of phase velocity azimuthal expansion are the 0-É and 2-É for Rayleigh waves, and 0-É and 4-É for Love waves. However, Trampert and Woodhouse (2003) carefully addressed the requirement of azimuthal anisotropy, and demonstrated that Rayleigh wave data need both 2-É and 4-É terms, which is also confirmed by Beucler and Montagner (2006). It was shown that for the same variance reduction, a global parametrization of anisotropy including azimuthal anisotropy requires fewer parameters than an isotropic parametrization. This apparent paradox can be explained by the fact that the increase of physical parameters is largely compensated by the smaller number of geographical

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n = 2 60° 30° 0°
–30° –60°

T = 100 s

60° 30° 0° –30°
–60°

n = 1 60° 30° 0°
–30° –60°
n = 0 60° 30° 0°
–30° –60°

60° 30° 0° –30°
–60°
60° 30° 0° –30°
–60°

–10.0 –5.0 –4.0 –3.0 –2.0 –1.0 –0.5 0.5 1.0 2.0 3.0 4.0 5.0 10.0
Figure 4 Rayleigh wave phase velocity maps at period T ¼ 100 s for the first three modes (n ¼ 0, 1, 2). Modified from Beucler E and Montagner J-P (2006) Computation of large anisotropic seismic heterogeneities. Geophysical Journal International 165: 447–468.

parameters, that is, larger-scale heterogeneities. Other tests have questioned whether phase data are sensitive enough to detect azimuthal anisotropy (Larson et al., 1998; Laske and Masters 1998) and the use of additional polarization data has been proposed.
Most tomographic models agree that down to $250–300 km, the deep structure is closely related to plate tectonics and continental distribution. Figure 5 presents two horizontal cross-sections from the most recent model of Debayle et al. (2005), which illustrates and confirms the robust features of the upper mantle models published so far since Montagner and Tanimoto (1991). In the upper mantle depth range around 100 km, all plate boundaries are slow: ridges and back-arc areas are slow, shields are fast, and seismic velocity in oceanic areas is increasing with the age of the seafloor. Except at few places, it is found that radial anisotropy expressed through the $ parameter ð$ ¼ ðVS2H – VS2VÞ=VS2VÞ is positive, as large as 10% in some oceanic areas and decreases with depth.

(a)
100 km

(b)
200 km

60° 120° 180° 240° 300° 0°

–10 –5

5 10

δ V s(%) V sref = 4.41 km s–1

60° 120° 180° 240° 300° 0°

–10 –5

5 10

δ V s(%) V sref = 4.44 km s–1

Figure 5 (a, b) Two cross-sections at 100 km (top) and 200 km (bottom) depths of the global tomographic model of Debayle et al. (2005). Directions of azimuthal anisotropy are superimposed on S-wave velocity heterogeneities. The length of bars is proportional to its amplitude (<2%).

The amplitude of SV-wave azimuthal anisotropy (G parameter) presents an average value of $2% below oceanic areas (Figure 5). Montagner (1994, 2002) noted a good correlation between seismic azimuthal anisotropy and plate velocity directions (primarily for fast moving plates) given by Minster and Jordan (1978) or DeMets et al. (1990). However, the azimuth of G parameter can vary significantly as a function of depth. For instance, at shallow depths (down to 60 km), the maximum velocity can be parallel to mountain belts or plate boundaries (Vinnik et al., 1992; Silver, 1996; Babuska et al., 1998), but orthogonal to them at large depth. This means that, at a given place, the orientation of fast axis is a function of depth, which explains why the interpretation of SKS splitting with a simple model is often difficult.
As depth increases, the amplitude of heterogeneities rapidly decreases, some trends tend to vanish, and some distinctive features come up: most fast

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ridges are still slow but slow ridges are hardly visible and back-arc regions are no longer systematically slow below 200 km. Large portions of fast ridges are offset with respect to their surface signatures. Below 300 km of depth (not shown here), high-velocity body below the western and the eastern Pacific rim is the most striking feature, which can be related to subducting slabs.
A visual and quantitative comparison of existing models can be found in the reference Earth model (REM) web site.
1.16.4 Geodynamic Applications
The most popular application of large-scale tomographic models is the understanding of mantle convection. Seismic velocity anomalies can be converted, under some assumptions, into temperature anomalies, density anomalies, and also into chemical or mineralogical heterogeneities. The application of seismic anisotropy to geodynamics in the upper mantle is straightforward if we assume that, due to the lattice-preferred orientation (LPO) of anisotropic crystals such as olivine (Christensen and Lundquist, 1982; Nicolas et al., 1973), the fast-polarization axis of mineralogical assemblages is in the flow plane parallel to the direction of flow. Figure 6 shows what is expected for the observable parameters VS, $, G, G in the case of a simple convective cell with LPO. Radial anisotropy $ expresses the vertical ($ < 1) or horizontal character ($ > 1) of convective flow, and the azimuthal anisotropy G can be related to the horizontal flow direction. Conversely, the three maps of VS, $, G, can be interpreted in terms of convective flow. These three pieces of information are necessary to correctly interpret the data. For example, upwellings or downwellings are both characterized by a weak or negative $ parameter, but a correlative positive or negative VS discriminates between these possibilities. By simultaneously inverting at depth for the different azimuthal terms of Rayleigh and Love waves, it is therefore possible to separate the lateral variations in temperature from those induced by the orientation of minerals. Such an interpretation might, however, be erroneous in water-rich mantle regions where LPO of minerals such as olivine is not simply related to the strain field (e.g., Jung and Karato, 2001). We will only present some examples of interesting applications of anisotropy in large-scale geodynamics and tectonics.

120°
60° 30°
dVs 0° –30°
60° –60° 30°
δξ 0°
–30° 60° –60°
30° G 0°
–30° –60°

180°
120° 120°

240°
180° 180°

60° 30°
0° –30° –60°
30°
0° –30°
60° –60° 240° 30°
0° –30° –60°
240°

120° 180° 240°

depth

–200

120° 180° 240°
–7.00 –3.00 –2.00 –1.00 –0.50 –0.25 0.25 0.50 1.00 2.00 3.00 7.00
Figure 6 Horizontal cross sections of the seismic observable parameters Vs, $, G, ÉG associated with a simple convecting cell in the upper mantle, assuming LPO of anisotropic minerals such as olivine. A vertical flow is characterized by a negative $ radial anisotropy (ratio between VSH and VSV, and a small azimuthal anisotropy (G % 0). An upwelling (resp. downwelling) is characterized by a large positive (resp. negative) temperature anomaly inducing Vs < 0 (resp. Vs > 0). A predominant large-scale horizontal flow will be translated into a significant amplitude of the G azimuthal anisotropy and its orientation will reflect the direction of flow (with a 180 ambiguity). Modified from Montagner J-P (2002) Upper mantle low anisotropy channels below the Pacific plate. Earth and Planetary Science Letters 202: 205–227.
Seismic anisotropy in the mantle primarily reflects the strain field prevailing in the past (frozen-in anisotropy) for shallow layers or present convective processes in deeper layers. Therefore, it makes it possible to map convection in the mantle. It must be noted that, when only the radial anisotropy is retrieved, its interpretation is nonunique. A fine layering of the mantle can also generate such a kind of anisotropy, and neglecting the azimuthal anisotropy can bias the amplitude of radial anisotropy and its interpretation.

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The uppermost mantle down to 410 km is the depth range where the existence of seismic anisotropy is now widely recognized and well documented. Azimuthal variations have been found for body waves and surface waves in different areas of the world. During the last years, the shear wave splitting, primarily for SKS waves has extensively been used to study continental deformation, but very few studies using body waves are devoted to oceanic areas. Conversely, global anisotropic upper mantle models have been primarily derived during the last 10 years from surface waves, which are sensitive to structure below oceanic areas in the absence of ocean bottom stations and consequently of dense body wave data. The intercomparison of anisotropic body wave and surface wave data is still in its infancy. However, as shown by Montagner et al. (2000), Vinnik et al. (2003), and Simons et al. (2002), such a comparison is providing encouraging results.
1.16.4.1 Oceanic Plates
Oceans are the areas where plate tectonics applies almost perfectly and this is particularly the case in the largest one, the Pacific plate. Figure 7 presents three vertical cross-sections at two different latitudes, displaying VSV velocity anomalies (Figure 7(a)) and the two kinds of anisotropy, which can be retrieved by simultaneous inversion of Rayleigh and Love waves constant 0-É and azimuthal terms of eqn (1) from the model of Montagner (2002). In Figure 7(b), the equivalent radial anisotropy of the medium, for S-wave expressed through the $ parameter, is displayed. The maps of Figure 7(c) are the distributions of the G parameter related to the azimuthal variation of SV-wave velocity. The maximum amplitude of G is $5% and rapidly decreases as depth increases. The distributions of velocity and anisotropy are completely different for these different crosssections. The thickening of lithosphere with the age of the seafloor is well observed on VSV velocity maps, but lithosphere is much thicker in the northern cross-section. When compared with the cooling half-space model, bathymetry, heat flux and lithospheric thickness flatten with age (see Ritzwoller et al. (2004) for recent results). This flattening is explained by basal reheating, especially in the Central Pacific and the birth of small-scale convection below the lithosphere (Davaille and Jaupart, 1994; Solomatov and Moresi, 2000).

Radial cross-sections (Figure 7(b)) show that the $ ¼ $ – $PREM parameter is usually negative and small, where flow is primarily radial (mid-ocean ridges and subduction zones). For the East-Pacific Rise, Gu et al. (2005) found that a negative radial anisotropy is observed at least down to 300 km. Between plate boundaries, oceans display very large areas with a large positive radial anisotropy such as in the Pacific Ocean (Ekstro¨m and Dziewonski, 1998), characteristic of an overall horizontal flow field. This very large anisotropy in the asthenosphere might be the indication of a strong deformation field at the base of the lithosphere (Gung et al., 2003), corresponding to the upper boundary layer of the convecting mantle (Anderson and Regan, 1983; Montagner, 1998).
Since convective flow below oceans is dominated by large-scale plate motions, the long-wavelength anisotropy found in oceanic lithospheric plates and in the underlying asthenosphere should be similar to the high-resolution anisotropy measured from body waves. Incidentally, one of the first evidences of azimuthal anisotropy was found in the Pacific Ocean by Hess (1964) for Pn-waves. So far, there are very few measurements of anisotropy by SKS splitting in the oceans. Due to the lack of seismic stations on the sea floor (with the exception of H2O half-way between Hawaii and California), the only measurements available for SKS were performed in stations located on ocean islands (Ansel and Nataf, 1989; Kuo and Forsyth, 1992; Russo and Okal, 1999; Wolfe and Silver, 1998), which are by nature anomalous objects, such as volcanic hotspots, where the strain field is perturbed by the upwelling material and not necessarily representative of the main mantle flow field. SKS splitting was measured during the temporary MELT experiment on the East-Pacific Rise (Wolfe and Solomon, 1998) but the orientation of the splitting is in disagreement with the petrological predictions of Blackman et al. (1996). Walker et al. (2001) presented a first measurement of SKS splitting at H2O, but it is in disagreement with independent SKS splitting measurements at the same station by Vinnik et al. (2003) and with surface wave anisotropy (Montagner, 2002).
The large-scale azimuthal anisotropy within and below lithosphere in the depth range 100–300 km is closely related to plate motions (Montagner, 1994; Ekstro¨m, 2000) and modeled in this framework (Tommasi et al., 1996). Fast-moving oceanic plates are zones where the comparison between directions of plate velocities (Minster and Jordan, 1978) or

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(a) 120° 150° 180° 210° 240° 270° 300°

120° 150° 180° 210° 240° 270° 300°

60°

60° 60°

60°

30°

30° 30°

30°

0°

0°

0°

0°

–30° 120°

150°

180°

210°

240°

270°

–30° –30°

300°

120°

150°

180°

210°

240°

270°

6350

6350

6300

6300

6250

6250

6200

6200

6150

6150

6100

6100

6050

6050

6000

6000

120 150 180 210 240 270 300

120 150 180 210 240 270

Vs

(b)
6350

6350

6300

6300

6250

6250

6200

6200

6150

6150

6100

6100

6050

6050

6000 120
δξ
(c)
6350

6000

150 180 210 240 270 300

120 150 180 210 240 270

–3.00 –2.00 –1.00 –0.50 –0.25 –0.10 0.10 0.25 0.50 1.00 2.00 3.00 6350

6300

6300

6250

6250

6200

6200

6150

6150

6100

6100

6050

6050

6000

6000

120 150 180 210 240 270 300
G

120 150 180 210 240 270

–30° 300°
300
300
300

0.0 0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.5 2.0 3.0 %
Figure 7 Vertical cross-sections of the distribution of Vs, $ À $0, G in the Pacific plate at À20 south and 20 north between radii 6000 km (370 km depth) and 6350 km (20 km depth) (after Montagner, 2002). The color scales in percents are the same for Vs and $. The vertical scale is exaggerated to make the figures more readible. (a) Vs; (b) $ ¼ $ À $0: deviation of $ with respect to a reference model ($0 PREM model); (c) G: amplitude of azimuthal anisotropy parameters.

NUVEL-1 (DeMets et al., 1990) and directions of G parameter is the most successful (Figure 8). Conversely, such a comparison is more difficult and controversial below plates bearing a large proportion of continents, such as the European–Asian plate,

characterized by a very small absolute motion in the hotspot reference frame and probably a large influence of inherited anisotropy.
The map with the G parameter at 100 km (Figure 5) as well as the cross-sections of Figure 7(c)

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–90 –75 –60 –45 –30 –15 0 15 30 45 60 75 90

(a)

1.0

1.0

Depth (km)

4.5

0.8

0.8

100

4.4

4.4

200

0.6

0.6

0

50

100

150

Lithospheric age (Ma)

0.4

0.4

0.2

0.2

0.0

0.0

–90 –75 –60 –45 –30 –15 0 15 30 45 60 75 90

Figure 8 Histogram of the difference between plate velocities directions and synthetic SKS anisotropy azimuths in the Pacific plate. It is calculated by summing the contributions of each grid point (5Â5) weighted by the latitude and the amplitude of anisotropy. Modified from Montagner J-P (2002) Upper mantle low anisotropy channels below the Pacific plate. Earth and Planetary Science Letters 202: 205–227.

show that the azimuthal anisotropy is very large along spreading ridges with a large asymmetry for the EastPacific Rise. The direction of anisotropy is in very good agreement with plate motion, which is also found in all other available models (Ekstro¨m, 2000; Smith et al., 2004; Debayle et al., 2005). The anisotropy is also large in the middle of the Pacific plate, but a line of very small azimuthal anisotropy almost parallel to the East-Pacific Rise is observed there (see also Figure 2 for synthetic SKS). This linear area of small anisotropy was named low anisotropy channel (LAC) by Montagner (2002). When calculating the variation of the amplitude of azimuthal anisotropy as a function of depth, a minimum comes up between 40 and 60 Ma age of the seafloor (Figure 9(a)). The LAC is presumably related either to cracking within the Pacific plate and/or to secondary convection within and below the rigid lithosphere, predicted by numerical and analog experiments and also translated in the VS velocity structure (Ritzwoller et al., 2004; Figure 9(b)). These new features provide strong constraints on the decoupling between the plate and asthenosphere. The existence and location of these LACs might be related to the current active volcanoes and hotspots (possibly plumes) in Central Pacific. LACs, which are dividing the Pacific plate into smaller units, might indicate a

4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75
Shear velocity (km s–1) (b)

1.85 1.6

Amplitude

1.2

0.8

0.4

0.0 0

50

100

150

200

Age (Ma)

Figure 9 Variations of (a) average S-wave velocity and (b) azimuthal anisotropy (through the delay time of synthetic SKS splitting) plotted versus the age of lithosphericsea floor. In both cases the structure below the plate for age between 60 and 100 Ma looks anomalous. (a) Modified from Ritzwoller MH, Shapiro NM, and Zhong S-J (2004) Cooling history of the Pacific lithosphere. Earth and Planetary Science Letters 226: 69–84. (b) After Montagner J-P (2002) upper mantle low anisotropy channels below the Pacific plate. Earth and Planetary Science Letters 202: 205–227.

future reorganization of plates with ridge migrations in the Pacific Ocean. They call for more thorough numerical modeling.
1.16.4.2 Continents
Differences in the thickness of high-velocity layer underlying continents as imaged by seismic tomography have fuelled a long debate on the origin of continental roots (Jordan, 1975, 1978). Some global tomographic models provide a continental thickness of $200–250 km in agreement with heat-flow analysis or electrical conductivity, but others suggest thicker zones up to 400 km.
Seismic anisotropy can provide fundamental information on the structure of continents, their root, and the geodynamic processes involved in mountain building and collision between continents (Vinnik et al., 1992; Silver, 1996) such as in Central Asia (Griot et al., 1998a, 1998b). Radial anisotropy $ is

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usually very heterogeneous below continents in the first 150–200 km of depth with positive or negative areas according to geology. But it seems to display a systematic tendency of being positive at larger depth (down to 300 km), whereas it is very large in the oceanic lithosphere in the depth range 50–200 km and decreases rapidly at larger depths (Montagner, 1994). Conversely, radial anisotropy displays a maximum (though smaller than in oceanic lithosphere) below very old continents (such as Siberian and Canadian Shield) in the depth range 200–400 km (Montagner and Tanimoto, 1991). Seismic anisotropy below continents, sometimes confined to the upper 220 km (Gaherty and Jordan, 1995) can still be significant below. A more quantitative comparison of radial anisotropy between different continental provinces is presented in Babuska et al. (1998), and demonstrates systematic differences according to the tectonic context. The existence of positive large-scale radial anisotropy below continents at depth might be a good indicator of the continental root which was largely debated since the presentation of the model of tectosphere by Jordan (1975, 1978, 1981). If this maximum of anisotropy is assumed to be related to an intense strain field in this depth range, it might be characteristic of the boundary between continental lithosphere and ‘normal’ upper mantle material. Gung et al. (2003) showed that it is possible to reconcile different isotropic tomographic models by taking into account seismic anisotropy. They find that significant radial anisotropy (with VSH > VSV) under most cratons in the depth range 250–400 km, similar to that found at shallower depths (80–250 km) below oceanic basins. Such a result is also in agreement for the Australian continent (Debayle and Kennett, 2000; Simons et al., 2002). So, all results seem to show that the root of continents as defined by radial anisotropy is located between 200 and 300 km. However, this result is not correlated with a maximum in azimuthal anisotropy in this depth range (Debayle and Kennett, 2005): the fast-moving Australian plate seems to be the only continental region with a sufficiently large deformation at its base to be transformed into azimuthal anisotropy. They propose that, for continents other than Australia, weak influence of basal drag on the lithosphere may explain why azimuthal anisotropy is observed only in a layer located in the uppermost 100 km of the mantle. This layer shows a complex organization of azimuthal anisotropy suggesting a frozen-in origin of deformation, compatible with SKS splitting.

Frozen anisotropy

H
Weak anisotropy
L
Anisotropy SH > SV

G
Anisotropy SH > SV

100 km 200 km

300 km

Mechanical lithosphere

Asthenosphere stear cene

Figure 10 Scheme illustrating the difference in the location of maximum anisotropy between oceans and continents. Modified from Gung Y, Panning M, and Romanowicz B (2003) Global anisotropy and the thickness of continents. Nature 422: 707–711.

The difference in radial and azimuthal anisotropies between oceans and continents might reflect a difference of coupling between lithosphere and asthenosphere, through the basal drag. The coupling might be weak below most continental roots, in contrast with the Pacific plate, where the coupling (reflected by plate direction) is the first-order effect in the uppermost 200 km for young ages, before thermal instabilities take place at the base of the lithosphere, as evidenced by the existence of lowanisotropy channels. These results on the difference between oceanic and continental anisotropies are illustrated in Figure 10.

1.16.4.3 Velocity and Anisotropy in the Transition Zone
The transition zone plays a key role in mantle dynamics, particularly the 660 km discontinuity, which might inhibit the passage of matter between the upper and the lower mantle. Its seismic investigation is made difficult on the global scale by the poor sensitivity of fundamental surface waves in this depth range and by the fact that teleseismic body waves recorded at continental stations from earthquakes primarily occurring along plate boundaries have their turning point below the transition zone. For body waves, many different techniques using SS precursors (Shearer, 1991) or P-to-S converted waves (Chevrot et al., 1999) were used at global scale to investigate the thickness of the transition zone. In spite of some initial controversies, a recent model by Lawrence and Shearer (2006) provides a coherent large-scale image of the transition zone thickness.
Whatever the type of data (normal mode, higher modes of surface waves, or body waves), an important feature of the transition zone is that, contrarily to the

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rest of the upper mantle, the upper transition zone is characterized by a large degree-2 pattern (Masters et al., 1982), and to a less extent, a strong degree-6. The degree-2 pattern (as well as degree-6) can be explained by the predominance of a simple largescale flow pattern characterized by two upwellings in central Pacific Ocean and Eastern Africa and two downwellings in the Western and Eastern Pacific Ocean (Montagner and Romanowicz, 1993), proposed initially in the lower mantle (Busse, 1983). This scheme was corroborated by the existence, in the upper transition zone, of a slight but significant degree-4 radial anisotropy displayed by Montagner and Tanimoto (1991) and Roult et al. (1990) in agreement with the prediction of this model. Therefore, the observations of the geographical distributions of degrees 2, 4, 6 in the transition zone are coherent and spatially dependent. Montagner (1994) compared these different degrees to the corresponding degrees of the hotspot and slab distribution. In this simple framework, the distribution of plumes (degree 2þ6) are merely a consequence of the large-scale simple flow in the transition zone. The degree 6 of velocity in the transition zone is well correlated with the distribution of hotspots and might indicate that many mantle plumes might originate in the transition zone. Ritsema and van Heijst (2004) observe lowerthan-average shear velocity at eight hotspots in this depth range (Figure 11). These results suggest that there are different families of plumes, some of them originating in the transition zone.
As for anisotropy in the transition zone, Montagner and Kennett (1996), by using eigenfrequency data, display some evidence of radial anisotropy in the

Shear-velocity variation from 1-D

–2%

+2%

Depth = 575 km

Figure 11 Shear-velocity variation and hotspot distribution. Modified from Ritsema J and van Heijst HJ (2004) Global transition zone tomography. Journal of Geophysical Research 109: B02302 doi:10.1029/ 2003JB002610.

upper (410–660 km) and lower (660–900 km) transition zones. Gung et al. (2003) also display a slight maximum of the degree-0 $ in the transition zone. The existence of anisotropy close to the 660 km discontinuity was also found by Vinnik and Montagner (1996) below Germany, and by Vinnik et al. (1998) in Central Africa. By studying P-to-S converted waves at the GRF network and at GEOSCOPE station BNG in Central Africa, they observed that part of the initial P-wave is converted into SH wave. This signal can be observed on the transverse component of seismograms. The amplitude of this SH wave cannot be explained by a dipping 660 km discontinuity and it constitutes a good evidence for the existence of anisotropy just above this discontinuity. However, there is some evidence of lateral variation of anisotropy in the transition zone as found by the investigation of several subduction zones (Fischer and Yang, 1994; Fischer and Wiens, 1996). Fouch and Fischer (1996) present a synthesis of these different studies and show that some subduction zones such as Sakhalin Islands require deep anisotropy in the transition zone, whereas others, such as Tonga, do not need any anisotropy. They conclude that their data might be reconciled by considering the upper transition zone (410–520 km) intermittently anisotropic, and the rest of the transition zone might be isotropic.
Anisotropy in the transition zone was also advocated by two independent studies, using different data sets. The observations of Wookey et al. (2002), though controversial, present evidence of very large S-wave splitting (up to 7 s) in the vicinity of the 660 km discontinuity between Tonga–Kermadec subduction zone and Australia. On a global scale, Trampert and van Heijst (2002) show a long-wavelength azimuthal anisotropic structure in the transition zone. The root-mean-square amplitude of lateral variations of G is $1%. Beghein and Trampert (2003), using probability density functions and separating $, 0, and  anisotropies, suggest a chemical component to explain these different parameters. The interpretation of these new tentative results is not obvious and new data are necessary to close the debate on the nature of velocity and anisotropy heterogeneities in the transition zone. The transition zone might be a mid-mantle boundary layer, and a detailed and reliable tomographic model of S-wave velocity and anisotropy in the transition zone will provide fundamental insights into the dynamic of the whole mantle.

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1.16.5 Numerical Modeling and Perspectives
In the previous sections, we have highlighted the presence of lateral heterogeneities in seismic velocity and anisotropy in different parts of the Earth’s upper mantle. However, anisotropy is not present in all depth ranges nor at all scales. There is some consensus for the presence of radial anisotropy in many parts of the upper mantle in order to simultaneously explain Love-wave and Rayleigh-wave dispersion, and even in the lower mantle (Panning and Romanowicz , 2004). The existence of azimuthal anisotropy is more controversial, though, from petrological reasons, it turns out that radial anisotropy and azimuthal anisotropy are intimately related and should be searched for simultaneosuly. Additional data, such as polarization data, might help to provide additional constraints on both kinds of anisotropy (Yu and Park, 1993; Pettersen and Maupin, 2002). But it requires the development of improved theoretical and numerical methods in order to work on the amplitude of seismograms.
Thanks to the access to very powerful computers, we are at the beginning stage of a new era for seismology. The twentieth century was dominated by the use of ray theory, and later on the normal mode theory. Since it is now feasible to numerically compute synthetic seismograms in complex 3-D structures in global spherical geometry (Komatitsch and Vilotte, 1998; Komatitsch and Tromp, 1999; Capdeville et al., 2003), it is possible to model the complex interaction between seismic waves and 3-D heterogeneity, particularly in anisotropic, anelastic media. Some new and sophisticated tomographic methods are presently developed (Montelli et al., 2004; Capdeville et al., 2005; Tromp et al., 2005; Zhou et al., 2006) that should provide access to the complexity of the Earth mantle by the mapping of short-scale heterogeneities such as mantle plumes, in anisotropic and anelastic media.
A second important challenge is the complete understanding of the origin of anisotropy from the mineral scale up to global scale in the different layers of the Earth. In the upper mantle, seismic anisotropy is due to LPO of anisotropic minerals such as olivine at large scales, requiring several strong conditions, starting with the presence of anisotropic crystals up to the existence of an efficient large scale present or past strain field. In order to fill the gap between grain scale modeling (McKenzie, 1979; Ribe, 1989; Kaminski and Ribe, 2001) and large-scale anisotropy measurements in a convective system (Tommasi et al.,

2000), there is now a real need to make more quantitative comparisons between seismic anisotropy and numerical modeling. Gaboret et al. (2003) and Becker et al. (2003) calculated the convective circulation in the mantle by converting perturbations of S-wave velocity into density perturbations. Figure 12 shows two crosssections through the Pacific hemisphere and the associated flow lines (Gaboret et al., 2003) derived from the tomographic model of Ekstro¨m and Dziewonski (1998). This kind of modeling makes it possible to calculate the strain tensor and to test different hypotheses for the prevailing mechanisms of alignment, by comparison with seismic data.
The upper mantle is the best known of the deep layers of the earth, where there is now good agreement between many isotropic global tomographic models. But the account of seismic anisotropy is mandatory to avoid biased isotropic heterogeneities. The main application of anisotropy is the mapping of mantle convection and its boundary layers (Karato, 1998; Montagner, 1998). The finding of anisotropy in the transition zone (if confirmed) will provide strong constraints on the flow circulation and the exchange of matter between the upper and the lower mantle. Pursuing the first pionneering efforts, the systematic modeling of the complete seismic waveform in 3-D heterogeneous, anisotropic and anelastic media associated with new techniques of numerical modeling of seismograms will probably enhance our vision of the whole mantle.
In parallel to these theoretical and numerical challenges, there is a crucial need for instrumental developments since there are still many areas at the surface of Earth devoid of broadband seismic stations. These regions are primarily located in Southern Hemisphere and more particularly in oceanic areas where no islands are present. Therefore, an international effort is ongoing, coordinated through International Ocean Network (ION) in order to promote the installation of geophysical ocean bottom observatories in order to fill the enormous gaps in the station coverage (for a description of ION).
Appendix 1: Effect of Anisotropy on Surface Waves in the Plane-Layered Medium
The half-space is assumed to be homogeneous and may be described by its density &(z) and its fourthorder elastic tensor À(z) with 21 independent elastic coefficients. All these parameters are so far supposed independent of x and y coordinates (in Figure 13, z is

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A

A′

A

A′

B

B′

B

B′

5 cm yr –1

d Vs/Vs (%)

–1.5 –1.2 –0.9 –0.6 –0.3 –0.0 0.3 0.6 0.9 1.2 1.5

Figure 12 Mantle heterogeneities and convective flow below the Pacific Ocean. Modified from Gaboret C, Forte A, and Montagner J-P (2003) The unique dynamics of the Pacific Hemisphere mantle and its signature on seismic anisotropy. Earth and Planetary Science Letters 208: 219–223.

x (North)

Ψ

k

y (East)

z
Figure 13 Definition of the Cartesian coordinate system (x, y, z) used in the calculations; É is the azimuth of the wavevector with respect to north.

the vertical component). This condition will be released in the next section. The unperturbed medium is assumed isotropic with an elastic tensor À0(z). In that medium, the two cases of Love and Rayleigh wave dispersion can be successively considered.
The unperturbed Love wave displacement is of the form:

0

1

ÀW ðzÞsinÉ

uðr; t Þ ¼ BB@ W ðzÞcosÉ CCAexpði½kðx cosÉ þ y sinÉÞ – !t ŠÞ

0

½19Š

where W(z) is the scalar depth eigenfunction for Love waves, k is the horizontal wave number, and É is the azimuth of the wave number k measured

clockwise from the north. The unperturbed Rayleigh wave displacement is
of the form

0

1

V ðzÞcosÉ

uðr; t Þ ¼ BB@ V ðzÞsinÉ CCAexpði½kðx cos É þ y sin ÉÞ – !t ŠÞ

iU ðzÞ

½20Š

where V(z) and U(z) are the scalar depth eigenfunctions for Rayleigh waves. The associated strain tensor (r, t) is defined by

ij ðr; t Þ ¼ 1=2ðui; j þ uj ; i Þ

½21Š

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where ,j denotes the differentiation with respect to the jth coordinate. The medium is perturbed from À0(z) to À0(z) þ (z), where (z) is small compared to À0(z) but quite general in the sense that there is no assumption on the kind of anisotropy. This means that in this approximation we can still consider quasiLove modes and quasi-Rayleigh modes (Crampin, 1984). From Rayleigh’s principle, the first-order perturbation V (k) in phase velocity dispersion is (Smith and Dahlen, 1973, 1975):

Z1

V

ðkÞ

¼

V 2!2

Z

0 1

ijkl ij Ãkl &0 uk ukà dz

dz

½22Š

0

where ui and ij are, respectively, the displacement and the strain for the unperturbed half-space, and the asterisk denotes complex conjugation. Now because of the symmetry of the tensors (z) and , we use the simplified index notation cij and i for the elements ijkl and ij , but the number, nij, of coefficients ijkl for each cij must be taken into account. The simplified index notation for the elastic tensor ijkl is defined in a coordinate system (x1, x2, x3) by

8

>>>>><

if if

i¼j k¼l

) )

p¼i q¼k

ijkl

!

cpq

>>>>>:

if if

i 6¼ j k 6¼ l

) )

p ¼ 9–i–j q ¼ 9–k–l

½23Š

This kind of transformation enables us to relate the fourth-order tensor  (3Â3Â3Â3) to a matrix c (6Â6). The same simplified index notation can be applied to the components of the strain tensor "ij, transforming the second-order tensor  (3Â3) into a vector with six components. However, it is necessary to be careful, because to a given cpq correspond several ijkl, and ijkl must be replaced by npqcpq, where npq is the number of ijkl giving the same cpq. Therefore, eqn [22] expressing Rayleigh’s principle can be rewritten as

Z 1X

V

ðkÞ

¼

V 2!2

0Z 1

ij nij cij i Ãj dz

&0 uk ukà dz

½24Š

0

We only detail the calculations for Love waves.

Love Waves
By using previous expressions for u(r, t), [19], and ij (r, t), [21], the various expressions of strain are
1 ¼ – i cos É sin ÉkW
2 ¼ i cos É sin ÉkW
3 ¼ 0 ½25Š
4 ¼ 1=2 cos ÉW 9
5 ¼ – 1=2 sin ÉW 9 6 ¼ 1=2 ðcos2 É – sin2 ÉÞkW
where W9 ¼ dW/dr. In Table 1, the different terms nij cij i Ãj are given. We note that when cij iÃj is a purely imaginary complex, its contribution to V (k, É) is null. When all the contributions are summed, the different terms cosk É sinl É are such that k þ l is

Table 1 Calculation of the various cijij for Love waves, with the simplified index notation ( ¼ cos É;  ¼ sin É)

n

ij

cijij

1

11

c112 2 k2 W2

1

22

c222 2 k2 W2

1

33

0

2

12

Àc122 2 k2 W2

2

13

0

2

23

0

2

24

0

4

14

c14

ð

–

i2

Þ

kWW9 2

4

15

c15

ði2Þ

kWW9 2

4

16

c16

ð

–

Þð2

–

2

Þ

k2W2 2

4

24

c24

ð

–

i2

Þ

kWW 2

9

4

25

c25

ð

–

i2

Þ

kWW 2

9

4

26

c26

ðÞð2

–

2

Þ

k2W 2

2

4

34

0

4

35

0

4

36

0

4

44

c44

2

W92 4

8

45

W92

c45ð – Þ 4

8

46

c46

ð

–

iÞð2

–

2

Þ

kWW 2

9

4

55

c55 2

W92 4

8

56

c56

ðiÞð2

–

2

Þ

kWW 2

9

4

66

c66

ð2

–

2

Þ

k

2W2 4

Treatise on Geophysics, vol. 1, pp. 559-589

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582 Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography

even, which is not surprising in the light of the reciprocity principle. Therefore, each term can be developed as a Fourier series in É with only even terms. Finally, it is found that

VL

ðk;

ÉÞ

¼

V 2!2L0

Z
0

1

& dz k2W 2
!

!

1 8

ðc11

þ

c22

–

2c12

þ

4c66

þW 92

1 2

ðc44

þ c55Þ !

þ cos2ÉW 92

Â

1 2

ðc44

–

c55Þ

– sin2ÉW 92c45 – cos4Ék2W 2 !

Â

1 8

ðc11

þ

c22

– 2c12 !'

–

4c66

Þ

þ sin4 É k2W 2

Â

1 2

ðc26

–

c16Þ

½26Š

where

Z1

L0ðkÞ ¼ &W 2dz

L1ðkÞ L2ðkÞ L3ðkÞ L4ðkÞ L5ðkÞ

¼ ¼ ¼ ¼ ¼

0
1 L0 1 L0 1 L0 1 L0 1 L0

Z Z0 Z0 Z0 Z0
0

1 W

2N

þ

W 92 k2

1

 W

92



– Gc k2 dz

1

 W

92



– Gs k2 dz

1
– Ec:W 2dz

1
– Es:W 2dz

 L dz

In the particular case of a transversely isotropic

medium with a vertical symmetry axis (also named

radial anisotropic medium), we have: c11 ¼ c22 ¼ A, c33 ¼ C, c12 ¼ (A – 2N), c13 ¼ c23 ¼ F, c44 ¼ c55 ¼ L, c66 ¼ N, and c14 ¼ c24 ¼ c15 ¼ c25 ¼ c16 ¼ c26 ¼ 0. The local azimuthal terms vanish and eqn [26]

reduces to

VLðk;

ÉÞ

¼

1 2VLL0

Z 1& W 2N
0

þ

W 92 k2

' L dz

½27Š

Therefore, the same expressions as in Takeuchi and Saito (1972, p. 268) are found in the case of radial anisotropy. The 0-É term of eqn [26] corresponds to the averaging over azimuth É, which provides the equivalent transversely isotropic model with vertical symmetry axis by setting

N

¼

1 8

ðc11

þ

c22Þ –

1 4 c12

þ

1 2 c66

L

¼

1 2

ðc44

þ

c55Þ

If we call Cij the elastic coefficients of the total elastic tensor, we can set

N

¼

&VS2H

¼

1 8

ðC11

þ

C22Þ –

1 4 C12

þ

1 2 C66

L

¼

&VS2V

¼

1 2

ðC44

þ

C55Þ

According to eqn [26], the first-order perturbation in Love wave phase velocity VL(k, É) can then be expressed as

1 VLðk; ÉÞ ¼ 2V0L ðkÞ ½L1ðkÞ þ L2ðkÞcos 2É þ L3ðkÞsin 2É

þ L4ðkÞcos 4É þ L5ðkÞsin 4É

½28Š

Rayleigh Waves
The same procedure holds for the local Rayleigh wave phase velocity perturbation VR, starting from the displacement given previously (Montagner and Nataf, 1986):

VRðk;

ÉÞ

¼

1 2V0R

ðkÞ

½R1

ðkÞ

þ

R2ðkÞcos 2É

þ R3ðkÞsin 2É þ R4ðkÞcos 4É

þ R5ðkÞsin 4É

½29Š

where

Z1

R0ðkÞ ¼ &ðU 2 þ V 2Þdz

R1ðkÞ

¼

0Z 1

1

R00 þ V9

h V

2

A

þ

U 92 k2

2 i

– U L dz

C

þ

2U 9V k

:F

k

R2ðkÞ R3ðkÞ R4ðkÞ R5ðkÞ

¼ ¼ ¼ ¼

1 R0
1 R0 1 R0 1 R0

Z Z0 Z0 Z0
0

1 1 1 1

h V 2Bc þ h V 2Bs þ EcV 2dz EsV 2dz

2U 9V k
2U 9V k

Hc Hs

þ þ

 V9 k
 V9 k

– –

2 i U Gc dz
2 i U Gs dz

The 13 depth-dependent parameters A, C, F, L, N, Bc, Bs, Hc, Hs, Gc, Gs, Ec, and Es are linear combinations of the elastic coefficients Cij and are explicitly given as follows:

 Constant term (0É-azimuthal term: independent of azimuth)

Treatise on Geophysics, vol. 1, pp. 559-589

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Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography 583

A

¼

&VP2H

¼

3 8

ðC11

þ

C22Þ

þ

1 4

C12

þ

1 2

C66

C ¼ &VP2V ¼ C33

F

¼

1 2

ðC13

þ

C23Þ

L

¼

&VS2V

¼

1 2

ðC44

þ

C55Þ

N

¼

&VS2H

¼

1 8

ðC11

þ

C22Þ –

1 4 C12

þ

1 2 C66

 2É-azimuthal term

cos 2É

Bc

¼

1 2

ðC11

– C22Þ

Gc

¼

1 2

ðC55

–

C44Þ

1 Hc ¼ 2 ðC13 – C23Þ

sin 2É Bs ¼ C16 þ C26 Gs ¼ C54 Hs ¼ C36

 4É-azimuthal term

cos 4É

Ec

¼

1 8

ðC11

þ

C22Þ –

1 4

C12

–

1 2

C66

sin 4É

Es

¼

1 2

ðC16

– C26Þ

where indices 1 and 2 refer to horizontal coordinates (1: north; 2: east) and index 3 refers to vertical coordinate. & is the density, VPH, VPV are, respectively, the horizontal and vertical propagating P-wave velocities, and VSH, VSV the horizontal and vertical polarized Swave velocities. So, the different parameters present in the different azimuthal terms are simply related to elastic moduli Cij. We must bear in mind that A, C, L, N anisotropic parameters can be retrieved from measurements of the P- and S-wave velocities propagating perpendicular or parallel to the axis of symmetry.

The corresponding kernels are plotted in Figure 14 for long waves and Figure 15 for Rayleigh waves.

0T40 (200 s) 0.

0T120 (73 s) 0.

[ ] L ∂T
T ∂L
–0.138 0.

–0.112 0.

[ ] N ∂T
T ∂N
–1.779 2.025

–3.820 3.737

[ ] ρ ∂T
T ∂ρ

0. 0 100 200 300 400 500

0. Depth (km) 0 100 200 300 400 500

Figure 14 Partial derivatives for Love waves of the period of fundamental normal modes 0T40 (left) and 0T120 (right) with respect to the elastic coefficients of a transversely isotropic Earth, L, N, and density &, as a function of depth in the upper mantle (from Montagner and Nataf, 1986). The partial derivatives with respect to A, C, F are null for these modes. The plots are normalized to their maximum amplitudes, given for a Áh ¼ 1000 km thick perturbed layer. The combinations of elastic coefficients that have the same partial derivative as L are ÀGc, ÀGs for the azimuthal terms 2-É, and as N are Ec, Es for the azimuthal term 4-É. Note that the amplitude of the L-partial is very small for the fundamental modes, which is not the case for higher modes.

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584 Upper Mantle Structure: Global Isotropic and Anisotropic Elastic Tomography

0S40 (212 s) 0.

0S120 (82 s) 0.

A ∂T T ∂A
–1.607 0.

–3.498 0.

C ∂T T ∂C
–0.555 0.412

–0.854 1.044

F ∂T T ∂F
–0.070 0.

–0.134 0.

L ∂T T ∂L
–0.627 2.075

–2.034 4.335

ρ ∂T T ∂ρ

0. 0 100 200 300 400 500

0. 0 100 200 300 400 500
Depth (km)

Figure 15 Kernels for Rayleigh waves. Same conventions are applicable as for Figure 13 in the same depth range. The partial derivative with respect to N has not been plotted since its amplitude is very small for fundamental modes. Note that three partials contribute to the 2-É-azimuthal terms, A-partial for Bc, Bs, F-partial for Hc, Hs, and the largest one L-partial for Gc, Gs.

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Relevant Websites
http://seismo.berkeley.edu – Berkeley Seismological Lab.
http://matri.ucsd.edu – Whole Eart’h Geophysics at IGPP.

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