Geophys. J. Int. (XXXX) XXX, XXX–XXX

Comparison of azimuthal seismic anisotropy from surface waves

and finite-strain from global mantle-circulation models

T.W. Becker1 , J.B. Kellogg2, G. Ekstrom¨ 2, and R.J. O’Connell2 1Cecil H. and Ida M. Green Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Drive, La Jolla CA 92093, USA 2Department of Earth and Planetary Sciences, Harvard University, 20 Oxford Street, Cambridge MA 02138, USA

Submitted to Geophysical Journal International on January 7, 2003

SUMMARY

We present global models of strain accumulation in mantle flow to compare the pre- dicted finite-strain ellipsoid (FSE) orientations with observed seismic anisotropy. The geographic focus is on oceanic and young continental regions where we expect our models to agree best with azimuthal anisotropy from surface waves. Finite-strain de- rived models and alignment with the largest FSE axes leads to better model fits than the hypothesis of alignment of fast propagation orientation with absolute plate mo- tions. Our modeling approach is simplified in that we are using a linear viscosity for flow and assume a simple relationship between strain and anisotropy. However, results are encouraging and suggest that similar models can be used to assess the validity of as- sumptions inherent in the modeling of mantle convection and lithospheric deformation. Our results substantiate the hypothesis that seismic anisotropy can be used as an indi- cator for mantle flow; circulation-derived models can contribute to the establishment of a quantitative functional relationship between the two. Key words: mantle convection – seismic anisotropy – finite strain – plate driving forces – intraplate deformation

1 INTRODUCTION & Tanimoto 1991) are reviewed by Montagner & Guillot (2000). Seismic wave propagation in the uppermost mantle is anisotropic, as has been demonstrated using a variety of The existence of anisotropy in upper-mantle rocks can methods and datasets (e.g. Hess 1964; Forsyth 1975; An- be associated with accumulated strain due to mantle con- derson & Dziewonski 1982; Vinnik et al. 1989; Montagner vection (e.g. McKenzie 1979), as reviewed by Montagner & Tanimoto 1991; Schulte-Pelkum et al. 2001). Vertically (1998). The use of seismic anisotropy as an indicator for polarized waves propagate more slowly than horizontally mantle and lithospheric flow is therefore an important av-

polarized waves in the upper ¡ 220 km of the mantle, imply- enue to pursue since other constraints for deep mantle flow ing widespread transverse isotropy with a vertical symme- are scarce. There are currently few quantitative models that try axis. Surface-wave studies have established lateral varia- connect anisotropy observations to the three-dimensional (3- tions in the pattern of this radial anisotropy (e.g. Ekstrom¨ D) geometry of mantle convection, and this paper describes & Dziewonski 1998). Inverting for azimuthal anisotropy our attempts to fill that gap using Rayleigh wave data and (where the fast propagation axis lies within the horizon- global circulation models. tal) is more difficult, partly because of severe trade-offs be- Surface waves studies can place constraints on varia- tween model parameters (e.g. Tanimoto & Anderson 1985; tions of anisotropy with depth, but the lateral data resolution Laske & Masters 1998). Ongoing efforts to map azimuthal is limited. Alternative datasets such as shear-wave splitting anisotropy in 3-D using surface waves (e.g. Montagner measurements have the potential to image smaller-scale lat- eral variations, though they lack depth resolution (e.g. Sil- ver 1996; Savage 1999). However, an initial comparison

between observations and synthetic splitting from surface- ¢ [email protected] wave models showed poor agreement between results from 2 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell both approaches (Montagner et al. 2000). Possible reasons anisotropy produced by these strains to the seismic data, us- for this finding include that the modeling of shear-wave split- ing a simplified version of Ribe’s (1992) model. We focus ting measurements needs to take variations of anisotropy on oceanic plates where the lithosphere should be less af- with depth into account (e.g. Schulte-Pelkum & Black- fected by inherited deformation (not included in our model) man 2002). The predicted variations of finite-strain with than in continental areas. The agreement between predicted depth from geodynamic models can be quite large (e.g. Hall maximum extensional strain and the fast axes of anisotropy et al. 2000; Becker 2002), implying that a careful treatment is found to be good in most regions which supports the in- of each set of SKS measurements might be required. We will ferred relationship between strain and fabric in the mantle. thus focus on surface-wave based observations of azimuthal We envision that future, improved global circulation models anisotropy for this paper and try to establish a general un- can be used as an independent argument for the validity and derstanding of strain accumulation in 3-D flow. In subse- the appropriate parameter range of fabric development mod- quent models, other observations of anisotropy should also els such as that of Kaminski & Ribe (2001), and intend to be taken into account. expand on our basic model in a next step.

1.1 Causes of seismic anisotropy Anisotropy in the deep lithosphere and upper mantle is most 2 AZIMUTHAL ANISOTROPY FROM RAYLEIGH likely predominantly caused by the alignment of intrinsi- WAVES cally anisotropic olivine crystals (lattice-preferred orienta- Tomographic models with 3-D variations of anisotropy tion, LPO) in mantle flow (e.g. Nicolas & Christensen 1987; based on surface waves exist (Montagner & Tanimoto 1991; Mainprice et al. 2000). Seismic anisotropy can therefore Montagner & Guillot 2000). However, since there are some be interpreted as a measure of flow or velocity-gradients in concerns about the resolving power of such models (e.g. the mantle and has thus received great attention as a possi- Laske & Masters 1998), we prefer to directly compare ble indicator for mantle convection (e.g. McKenzie 1979; our geodynamic models with a few azimuthally-anisotropic Ribe 1989; Chastel et al. 1993; Russo & Silver 1994; phase-velocity maps from inversions by Ekstrom¨ (2001). In Tommasi 1998; Buttles & Olson 1998; Hall et al. 2000; this way, we can avoid the complications that are involved in Blackman & Kendall 2002). There is observational (Ben Is- a 3-D inversion. mail & Mainprice 1998) and theoretical (Wenk et al. 1991; Phase velocity perturbations, δc, for weak anisotropy Ribe 1992) evidence that rock fabric and the fast shear-wave can be expressed as a series of isotropic, D0, and az- polarization axis will line up with the orientation of maxi- imuthally anisotropic terms with π-periodicity, D2φ, and mum extensional strain. (We shall distinguish between di- ¤ 4φ

£ π 2-periodicity, D , (Smith & Dahlen 1973):

rections, which refer to vectors with azimuths between 0

£

§ §

and 360 , and orientations, which refer to two-headed vec- dc 0 2φ § 2φ

¥ ¨ © ¨ © £ δc D DC cos 2φ DS sin 2φ tors with azimuths that are 180 -periodic. We will also use c ¦

4φ § 4φ

© ¨ © fabric loosely for the alignment of mineral assemblages in D cos ¨ 4φ D sin 4φ (1) mantle rocks such that there is a pronounced spatial cluster- C S ing of particular crystallographic axes around a specific ori- where φ denotes azimuth. D2φ and D4φ anisotropy imply entation, or in a well-defined plane.) More specifically, we that there are one and two fast propagation orientations in the expect that for a general strain state, the fast (a), intermediate horizontal plane, respectively. As discussed in the Appendix, (c), and slow (b) axes of olivine aggregates will align, to first fundamental mode Rayleigh waves are typically more sensi- order, with the longest, intermediate, and shortest axes of the tive to anisotropy variations with 2φ-dependence than Love finite strain ellipsoid (Ribe 1992). The degree to which the waves; we will thus focus on Rayleigh waves. fast axes cluster around the largest principal axis of the fi- The kernels for surface wave sensitivity to 2φ- nite strain ellipsoid (FSE), or, alternatively, the orientation anisotropy have a depth dependence for Rayleigh waves that

of the shear plane, will vary and may depend on the exact is similar to their sensitivity to variations in vSV (Montag- ¡ strain history, temperature, mineral assemblage of the rock, ner & Nataf 1986) with maximum sensitivity at ¡ 70 km,

and possibly other factors (e.g. Savage 1999; Tommasi 120 km, ¡ 200 km depth for periods of 50 s, 100 s, and 150 s, et al. 2000; Blackman et al. 2002; Kaminski & Ribe 2002). respectively (see Appendix). Figure 1 shows phase-velocity

The simple correlation between FSE and fabric may there- maps from inversions by Ekstrom¨ (2001) for those periods. fore not be universally valid. For large-strain experiments, Ekstrom¨ used a large dataset ( ¡ 120 000 measurements) and

the fast propagation orientations were found to rotate into the a surface-spline parameterization (1442 nodes, correspond- £ shear plane of the experiment (Zhang & Karato 1995), an ing to ¡ 5 spacing at the equator) to invert for lateral varia- effect that was caused partly by dynamic recrystallization of tions in the D terms of (1). To counter the trade-off between grains and that is accounted for in some of the newer theoret- isotropic and anisotropic structure, Ekstrom¨ (2001) damped ical models for fabric formation (e.g. Wenk & Tome 1999; 2φ and 4φ-terms 10 times more than the isotropic D0 param- Kaminski & Ribe 2001, 2002). Further complications for eters. The inversion method will be discussed in more detail LPO alignment could be induced by the presence of water in a forthcoming publication, and the resolving power for (Jung & Karato 2001). D2φ anisotropy will be addressed in sec. 5.1. Here, we shall Instead of trying to account for all of the proposed only briefly discuss some of the features of the anisotropy anisotropy formation mechanisms at once, we approach the maps. problem by using global flow models to predict the orien- The isotropic and 2φ-anisotropic shallow structure im- tation of finite strain in the mantle. We then compare the aged in Figure 1 appears to be dominated by plate-tectonic Seismic anisotropy and mantle flow 3

2φ

(a) T 50 s (peak D -sensitivity: 70 km depth) 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

D0 [%] 60˚ 6

4 30˚ 2

0˚ 0

-2 -30˚

-4

-6 -60˚

inversion: D2φ: 2.5% D4φ: 1.8%

2φ

(b) T 100 s (peak D -sensitivity: 120 km depth) 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

D0 [%] 60˚ 4

30˚ 2

0˚ 0

-30˚ -2

-60˚ -4

inversion: D2φ: 2.9% D4φ: 2.0%

2φ

(c) T 150 s (peak D -sensitivity: 200 km depth) 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

D0 [%] 60˚

2 30˚

0˚ 0

-30˚

-2

-60˚

inversion: D2φ: 2.1% D4φ: 1.7%

Figure 1. Isotropic and anisotropic variations of phase velocity for Rayleigh waves at periods, T, of 50 s (a), 100 s (b), and 150 s (c) from Ekstrom¨ (2001). We show isotropic anomalies as background shading (D0 term in (1)), D2φ fast orientations as sticks, and D4φ terms as crosses with maximum amplitude as

2φ

  indicated in the legend. D sensitivity peaks at  70 km, 120 km, and 200 km depth for 50 s, 100 s, and 150 s, respectively (see Appendix). 4 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell

50 s recovery 50 s 100 s 150 s 2φ 0 0 2φ 2φ 4φ iso vs. D Dr D D vs.D D D seafloor_age2φ

craton continent

rum

|v avg r | |v h | avg

-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 r r r r

Figure 2. Correlation coefficient, r, for T  50 s recovery test and surface wave models of Figure 1 at periods of 50 s, 100 s, and 150 s. We show isotropic anomaly (D0, circles), 2φ-anisotropy (D2φ amplitude, filled triangles), and 4φ-anisotropy (D4φ amplitude, squares) correlations with seafloor-age, cratons, and continental regionalizations (from Nataf & Ricard 1996), uppermost mantle slabs (rum, average upper 400 km from Gudmundsson & Sambridge 1998),

avg avg

   average absolute radial (  vr ) and horizontal ( vh ) velocities from a flow calculation based on plate motions (upper 400 km average, no-net-rotation frame) and smean nt (sec. 3.1). Stars indicate r between the isotropic and D2φ-signal, and open triangles for the leftmost plot the correlations of surface

2φ 2φ 2φ 2φ 

wave inversion’s D amplitude-recovery, D log  D D , as discussed in sec. 5.1. All fields are expanded up to spherical harmonic degree 

r 10 recovered  model 

max  31 before calculating r; vertical dashed lines indicate the corresponding 99% significance level assuming bi-normal distributions. features such as the well-known seafloor spreading-pattern covery of azimuthal anisotropy is better in the oceanic plates close to the ridges (e.g. Forsyth 1975; Montagner & Tani- than in the continents, which leads to a positive and nega- 2φ moto 1991). There are, however, deviations from this sim- tive r of Dr with horizontal plate speeds and the continental 2φ ple signal, and the D pattern in the Western Pacific and function, respectively. south-western parts of the Nazca plate has no obvious rela- If anisotropy were purely due to a clearly defined fast tion to current plate motions. The patterns in anomaly ampli- propagation axis in the horizontal plane, we would expect tudes are quantified in Figure 2. We show the correlation of the D4φ signal to be much smaller than D2φ for fundamental 0 2φ 4φ D , D , and D -variations with tectonic regionalizations mode Rayleigh waves (see Montagner & Nataf 1986, and and plate-motion amplitudes. The isotropic signal shows a Appendix). The 4φ-signal in Figure 1 is indeed smaller than positive correlation with sea-floor age for T ¥ 50 s, since that from 2φ, but it is not negligible. Both 4φ and 2φ ampli-

avg  ridges are where slow wave-speeds are found; this signal tudes are positively correlated with plate boundaries (  vr is lost for deeper sensing phases. Especially for T ¥ 50 s, in Figure 2), especially with convergent margins where slabs both convergent and divergent plate boundaries are found are found (rum in Figure 2). This might indicate that the to be seismically slow, which is reflected in a negative cor- predominantly vertical orientation of strain in these regions relation coefficient, r, with the radial velocities from flow (sec. 3.4) causes the symmetry axis of anisotropy to have a

avg 0  models,  vr . D is also positively correlated with cratons large radial component. However, we will proceed to inter- for all periods, because the continental tectosphere is gen- pret only the D2φ signal for simplicity, and compare the ob- erally imaged as fast structure by tomography. There is a served fast axes with the horizontal projection of the longest 0 negative correlation of D with horizontal velocities; this axis of the flow-derived FSEs. Some of the largest deviations is partly the consequence of positive r with continental re- between our models and observations are in the regions of gions because oceanic plates move faster than continental large D4φ (sec. 5.2). This could be caused by either poor res- 2φ ones. The D -signal has a negative correlation with cratons olution of the data or a breakdown of our model assumptions for all T , and a positive correlation with horizontal and ra- about anisotropy, or both. dial velocities for T  100 s. The observed weaker regional anisotropy underneath cratons might be due to the higher viscosity of continental keels (and hence less rapid shear- ing) and/or frozen in fabric with rapidly varying orientations which is averaged out by the surface wave inversion. Posi- tive r for D2φ with plate speeds in turn may be caused by more rapid shearing underneath oceanic plates. However, 3 MODELING STRAIN AND LPO ANISOTROPY especially this last conclusion is weakened by the uneven raypath coverage which limits tomographic inversions. The We predict seismic anisotropy by calculating the finite strain ability of Ekstrom’¨ s (2001) dataset to recover synthetic D2φ- that a rock would accumulate during its advection through structure is reflected by the correlation of the D2φ-recovery mantle flow (e.g. McKenzie 1979). Given a velocity field 2φ that is known as a function of time and space, the passive- function, D , from sec. 5.1, shown in open triangles in Fig- r tracer method lends itself to the problem of determining ure 2. As a consequence of the event-receiver geometry, re- strain as outlined below. Seismic anisotropy and mantle flow 5

0 in radial flow at ¡ 300 km depth, while the low viscos- ity “notch” at 660 km that characterizes ηG leads to an 500 abrupt change between upper and lower mantle horizontal velocities. The rate of decrease of horizontal velocities with depth in the upper mantle is smaller for than for be- 1000 ηG ηF cause there is less drag exerted by the lower mantle. Since this implies slower shallow straining for the ηG models, we 1500 will consider ηG as an endmember case alongside the more depth [km] generic ηF. However, the physical mechanisms that might 2000 lead to an ηG-type viscosity profile are only poorly under- stood (e.g. Panasyuk & Hager 1998), and we consider ηF 2500 ηD (Hager & Clayton, 1989) to be closer to the current consensus on the mantle’s average ηF (Steinberger, 2000)

ηG (Mitrovica & Forte, 1997) viscosity. 0.01 0.1 1 10 100 Plate-motion related flow can provide a good approxi- viscosity η [1021 Pas] mation for large-scale, near-surface straining patterns. Such flow will, however, not be representative of the overall

Figure 3. Mantle viscosity profiles ηD, ηF, and ηG as obtained by previous straining and mixing properties of the mantle, and we ex- inversions (references given in the legend) and used for the flow modeling. pect vigorous thermal convection to exhibit stronger currents Dashed horizontal line indicates 660 km depth. at depth (e.g. van Keken & Zhong 1999). Hence, we also compared the flow characteristics of circulation models that include density in the mantle with a convection model by 3.1 Modeling flow Bunge & Grand (2000). We scale the present-day temper- We use the method of Hager & O’Connell (1981) to model ature field of Bunge & Grand’s (2000) 3-D spherical cal- circulation in the mantle. The Stokes equation for incom- culation to density and then recompute the corresponding pressible flow in the infinite Prandtl number regime, velocities using our method. Both the shape and amplitude

of the variation of velocity amplitudes with depth are sim-

¥  ∇  τ ∇p ρg (2) ilar to what we obtain with our flow models using ngrand. with τ, p, ρ, and g denoting deviatoric stress, pressure, den- Since Bunge & Grand’s (2000) model is partly constructed sity, and gravitational acceleration, respectively, is solved such that the temperature heterogeneity matches that in- for a constitutive equation where the viscosity, η, varies ferred from tomography, we should indeed expect that the only radially. A spherical harmonic expansion of all physical profiles look similar. However, the similarity indicates that quantities is used, plate velocities are prescribed as surface our scaled density anomalies are of the right order, as ex- boundary conditions, the CMB is free-slip, and internal den- pected (Becker & O’Connell 2001a). sity variations are scaled from seismic tomography. We can Our flow calculation uses Newtonian rheology, as would then solve (2) for the instantaneous velocities in a realistic be expected for the diffusion-creep regime of the high- three-dimensional geometry. One of the major limitations of temperature deep mantle (e.g. Ranalli 1995). However, this approach is that η is not allowed to vary laterally. How- lattice-preferred orientation of olivine, and hence LPO- ever, we think that the method is accurate enough to give us related anisotropy, forms only under dislocation creep, a good first order estimate of large-scale flow in the mantle. which is characterized by a stress weakening power-law. We Our choice of input models (Table 1) and viscosity expect that Newtonian mantle flow will be similar to power- structures (Figure 3) is motivated by previous studies (e.g. law creep (e.g. Christensen 1984) and therefore think that Hager & Clayton 1989; Ricard & Vigny 1989; Mitrovica velocities in our models should resemble the actual large- & Forte 1997; Lithgow-Bertelloni & Silver 1998; Stein- scale circulation pattern in the mantle. A quantitative study berger 2000) and our own work on comparing mantle mod- of 3-D circulation with power-law rheology to validate this els (Becker & Boschi 2002) and inverting for plate veloci- assumption remains to be undertaken. ties (Becker & O’Connell 2001a). Mantle density structure is taken either directly from slab models, or from tomogra- 3.1.1 No-net-rotation phy where we scale velocity anomalies to density using a constant factor All of our flow calculations are performed in the no-net- S ¤ rotation (NNR) reference frame since our model cannot R ¥ d lnρ d lnv (3) ρ generate any net rotation of the lithosphere as it has no of 0.2 for S-wave models for all depths below 220 km, lateral viscosity contrasts (Ricard et al. 1991; O’Connell and zero else to account for the tectosphere where veloc- et al. 1991). A prescribed net rotation of the surface would ity anomalies may image compositional heterogeneity. For not produce any strain but simply lead to a rotation of the density models ngrand nt and smean nt, we also include whole mantle. Fast anisotropy orientations are, however, of- shallow structure above 220 km but only away from cratons ten compared with absolute plate motions in a hotspot refer- by masking out the regions given by 3SMAC (Nataf & Ri- ence frame (APM, e.g. Vinnik et al. 1992). In the simplest card 1996). hypothesis, the orientation of velocity vectors in the APM For all purely plate-motion related flow, the predicted frame at the surface is supposed to be indicative of a simple circulation is focused in the upper mantle since viscosity shear layer in an asthenospheric channel, causing straining increases in the lower mantle. The shallow asthenospheric between the plates and a stationary deep mantle. channel in ηF (Figure 3) results in a pronounced maximum Since lateral viscosity variations exist in the mantle, a 6 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell

Table 1. Mantle density models used for the flow calculations (see Becker & Boschi 2002, for details), z indicates depth. We use the tectonic regionalizations from 3SMAC by Nataf & Ricard (1996).

S name type Rρ density scaling for source tomography

lrr98d slab model, slablets sink at parameterized – Lithgow-Bertelloni & Richards (1998) speeds – stb00d slab model, includes advection in 3-D flow – Steinberger (2000)

ngrand S-wave tomography 0.2 for z  220 km S. Grand’s web site as of June 2001

0.0 for z  220 km ngrand nt ngrand with anomalies in cratonic regions 0.2 ngrand from 3SMAC removed

smean mean S-wave model based on published 0.2 for z  220 km Becker & Boschi (2002)

models 0.0 for z  220 km smean nt smean with anomalies in cratonic regions 0.2 smean from 3SMAC removed net rotation may be generated by convection. However, as this approximate method, we obtain satisfactory results for shown by Steinberger & O’Connell (1998), the net rotation backward advection when compared with a passive tracer component in hotspot reference frames may be biased ow- method in terms of overall structure. However, tracers are, ing to the relative motion of hotspots. We will compare our as expected, better at preserving sharp contrasts, because the NNR-frame derived strains with the quasi-null hypothesis of field method suffers from numerical diffusion. a correlation between anisotropy and surface-velocity ori- While active tracer methods (e.g. Schott et al. 2000) entations below. A better fit is obtained for strain-derived would be better suited for problems in which the detailed anisotropy, leading us to question the hypothesis of align- distribution of density anomalies matters, we shall not be ment with surface velocities alone, both for the APM and concerned with any improvement of the backward advection NNR reference frames. scheme at this point. The results from inversions that pursue a formal search for optimal mantle-flow histories such that the current density field emerges (Bunge et al. 1998, 2001) 3.1.2 Reconstructing past mantle circulation should eventually be used for strain modeling. However, if fabric does not influence the pattern of convection signifi- We show in sec. 4.2 that strain accumulation at most depths cantly, the method we present next is completely general; it is sufficiently fast that, under the assumption of ongoing re- can also be applied to a velocity field that has been generated working of fabric, the last tens of Myr are likely to domi- with more sophisticated methods than we employ here. nate present-day strain and anisotropy. However, we include results where our velocity fields are not steady-state but change with time according to plate-motion reconstructions and backward-advected density fields. We use reconstruc- tions from Gordon & Jurdy (1986) and Lithgow-Bertelloni et al. (1993) within the original time-periods without in- terpolating between plate configurations during any given 3.2 The tracer method stage. To avoid discontinuities in the velocity field, the tran- We use a fourth-order Runge-Kutta scheme with adaptive sition at the end of each tectonic period is smoothed over stepsize control (e.g. Press et al. 1993, p. 710) to numer- 1% of the respective stage length; velocities change to that ically integrate the tracer paths through the flow field. All ¦ 2 7 of the next stage according to a cos -tapered interpolation. fractional errors are required to be smaller than 10  for all The width of this smoothing interval was found to have little unknowns, including the finite strain matrix as outlined be- effect on the predicted long-term strain accumulation. low. For this procedure, we need to determine the velocity While we can infer the current density anomalies in the at arbitrary locations within the mantle. Velocities and their mantle from seismic tomography, estimates of the past dis- first spatial derivatives are thus interpolated with cubic poly- tributions of buoyancy sources are more uncertain. To model nomials (e.g. Fornberg 1996, p. 168) from a grid expansion

circulation patterns for past convection, we also advect den- of the global flow fields. Mantle velocities are expanded on

£ £ sity anomalies backward in time using the field method of 1 1 grids with typical radial spacing of ¡ 100 km; they

Steinberger (2000). We neglect diffusion, heat production, are based on flow calculations with maximum spherical har-

¥  ¥ and phase changes and assume adiabatic conditions. These monic degree  max 63 for plate motions and max 31 for simplifications make the problem more tractable, but advec- density fields (tomographic models are typically limited to tion can be numerically unstable under certain conditions long wavelengths, cf. Becker & Boschi 2002). To suppress 2 (e.g. Press et al. (1993), p. 834ff). To damp some short- ringing introduced by truncation at finite  , we use a cos ta- wavelength structure which is artificially introduced into the per for the plate motions. We conducted several tests of our density spectrum at shallow depths, we taper the density tracer advection scheme, and found that we could accurately

2 ρ

  time-derivative using a cos filter for  0 75 max. With follow closed streamlines for several overturns. Seismic anisotropy and mantle flow 7

2 3.3 Finite strain the si after sorting accordingly. Numerically, L is faster to calculate since it does not involve finding the inverse of F. The strain accumulation from an initial x to a final posi- tion r can be estimated by following an initial infinitesimal displacement vector dx to its final state dr (e.g. Dahlen & Tromp 1998, p. 26ff). We define the deformation-rate tensor G based on the velocity v as

T 3.3.1 Anisotropy based on the FSE

©  G ¥!¨ ∇rv (4) T We introduce natural strains as a convenient measure of Here, ∇r is the gradient with respect to r and indicates the stretching transpose. For finite strains, we are interested in the defor-

mation tensor F, λ1 λ2

¥ * ¥ *

T ζ ln and ξ ln (11)

¥"¨ © + F ∇xr (5) λ2 + λ3 where ∇x is the gradient with respect to the tracer x, because following Ribe (1992) who shows that the orientations of F transforms dx into dr as fast shear wave propagation rapidly align with maximum

1 stretching eigenvectors in numerical deformation experi-



 ¥  

dr ¥ F dx or dx F dr (6) $

ments, regardless of the initial conditions. After ζ and ξ ¡

1  The latter form with the inverse of F, F , (which exists for 0  3, there are essentially no further fluctuations in fast ori- realistic flow) allows us to solve for the deformation that cor- entations. Using a logarithmic measure of strain is also ap- responds to the reverse path from r to x. To obtain F numer- propriate based on the other result of Ribe (1992) that the

ically, we make use of the relation between G and F: amplitude of seismic anisotropy grows rapidly with small  linear strain and levels off at larger values, above ζ ¡ 0 7.

∂

 F ¥ G F (7) When we average fast strain orientations with depth for the

∂t comparison with seismic anisotropy, we weight by ζ to in- ¤ where ∂ ∂t denotes the time derivative. Our algorithm cal- corporate these findings. This is a simplification since some culates G at each timestep to integrate (7) (starting from of Ribe’s (1992) experiments indicate a more rapid satura-

F # I at x, where I denotes the identity matrix) with the same tion of anisotropy amplitude at large strains. However, dif- Runge-Kutta algorithm that is used to integrate the tracer po- ferences between logarithmic and linear averaging of strain sition from x to r. To ensure that volume is conserved, we set orientations are usually not large. We therefore defer a more the trace of G to zero by subtracting any small non-zero di- detailed treatment of the anisotropy amplitudes to future vergence of v that might result from having to interpolate work when we can incorporate fabric development more re- v. We tested our procedure of estimating F against analyti- alistically, e.g. using Kaminski & Ribe’s (2001) approach. cal solutions for simple and pure shear (McKenzie & Jack- We formulate the following ad hoc rules to determine son 1983). the finite strain from circulation models. The deformation matrix can be polar-decomposed into an orthogonal rotation Q and a symmetric stretching matrix (i) Follow a tracer that starts at r backward in time for a

in the rotated reference frame, the left-stretch matrix L, as constant interval τ to location x while keeping track of the ¡ deformation F , (τ 5 Myr). Then, calculate the strain that

T 1

 ¥"¨  ©  F ¥ L Q with L F F 2 (8) would have accumulated if the tracer were to move from an

1

,  © unstrained state at x to r, given by ¨ F . Such an approach For the comparison with seismic anisotropy, we are only in- would be appropriate if fabric formation were only time- terested in L which transforms an unstrained sphere at r into dependent; the τ - 0 result is related to the instantaneous an ellipsoid that characterizes the deformation that material strain rates. accumulated on its path from x to r. The eigenvalues of L,

(ii) Alternatively, define a threshold strain ζc above which

$

λ1 $ λ2 λ3 (9) any initial fabric gets erased, as would be expected from the  results of Ribe (1992) (ζc ¡ 0 5). In this case, we only have

measure the length and the eigenvectors the orientation of 1

 ,.© to advect backward until ζ or ξ, as based on ¨ F , reaches

the axes of that finite strain ellipsoid at r after the material © ζ ; we match ζ ¨ to ζ within 2%. The tracer trajectory will has undergone rotations. c r c correspond to different time intervals depending on the ini- Our approach is similar to that of McKenzie (1979), as tial position of the tracer (sec. 4.2). applied to subduction models by Hall et al. (2000). However, those workers solve (7) by central differences while we use We stop backward advection in both schemes if tracers orig- Runge-Kutta integration. Moreover, Hall et al. calculate the 1 inate below 410 km depth, where we expect that the phase FSE based on B  where transition from olivine to the β-phase (e.g. Agee 1998) will T

1 1 ' 1

 

 erase all previous fabric. For most of the models, we will

  B ¥&% F F (10) consider the flow field as steady-state but not advect back in Hall et al. define a stretching ratio, si, from the deformed time for more than 43 Ma, the age of the bend in the Hawaii- to the undeformed state in the direction of the i-th eigen- ¤)( Emperor seamount chain that marks a major reorganization

1 ¥ vector of B  with eigenvalue γi as si 1 γi. Since B is of plate motions (e.g. Gordon & Jurdy 1986). Changes in equivalent to L2, and B as well as L are symmetric, both ap- plate motions affect only the very shallowest strains for con- proaches yield the same results when we identify the λi with tinuous strain accumulation (sec. 4.2). 8 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell

3.4 Examples of finite strain accumulation porates only plate-related flow using ηF. For simplicity, plate motions are assumed to be constant in time for 43 Myr. We We examine strain accumulation by following individual focus on the horizontal projection of the largest stretching tracers close to plate boundaries in order to develop an un- direction of L, showing the orientation of the FSE axis as derstanding of the global, convection-related strain field. Our sticks whose lengths scale with ζ. The background shad-

examples are similar, and should be compared to, previous  ing in Figure 6 shows ∆L ¥ L 1 as an indication of e.g. rr rr work ( McKenzie 1979; Ribe 1989; Hall et al. 2000) stretching in the radial, r, direction. Strain was calculated

but they are unique in that they are fully 3-D and based on £

for ¡ 10 000 approximately evenly distributed tracers ( 2 estimates of mantle circulation that include realistic plate ge- spacing at equator) for each layer, which were placed from¦ ometries. 50 km through 400 km depth at 50 km intervals to sample the For simplicity, the flow field used for our divergent mar- upper mantle above the 410-km phase transition. The hori- gin example for the East Pacific Rise (EPR, Figure 4) in- zontal projections of the largest axes of the FSE are depth av- cludes only plate-motion related flow, calculated by pre- eraged after weighting them with the ζ-scaled strain at each scribing NUVEL1 (DeMets et al. 1990) NNR veloci- location (sec. 3.3), while the radial, ∆Lrr, part is obtained ties at the surface using viscosity profile ηF. The largest from a simple depth average. The horizontal strain orienta- stretching axes align roughly perpendicular to the ridge and tions will be interpreted as a measure of the depth-averaged mostly in the horizontal plane. The ridge-related strain pat- azimuthal anisotropy as imaged by Rayleigh waves. e.g. tern has been observed globally for surface waves ( The finite-strain field of Figure 6a is similar to instanta- Forsyth 1975; Nishimura & Forsyth 1989; Montagner & neous strain-rates in terms of the orientations of largest ex- SKS Tanimoto 1991) and splitting (Wolfe & Solomon 1998) tension, which are dominated by the shearing of the upper and can be readily interpreted in terms of a general plate tec- mantle due to plate motions. However, orientations are not e.g. tonic framework ( Montagner 1994). identical to those expected to be produced directly from the For the strain evolution example at convergent margins surface velocities because of 3-D flow effects. The depth- (Figure 5), we include ngrand nt density in the flow calcula- averaged strain for τ ¥ 10 Myr is furthermore dominated by tion. The strain development can be divided into two stages: radial extension (positive ∆L ) at both ridges and trenches, first, we see compression in the plane of the slab as material rr unlike for instantaneous strain (or small τ) where ridges are enters the mantle. The largest stretching axes are nearly ra- under average radial compression. This effect is due to the dial, and the horizontal part of the FSE shows trench-parallel radial stretching of material that rises underneath the ridges elongation. In the second stage, for greater depths and partic- before being pulled apart sideways and radially compressed ularly owing to the inclusion of slab pull forces, deep stretch- at the surface. This effect has been invoked in qualitative ing becomes more important and leads to deformations such models of radial anisotropy (e.g. Karato 1998) and may ex-

that the largest stretching axes rotate mostly perpendicular ¡ plain the fast vSV (vSV $ vSH at 200 km depths) anomaly in to the trench and become more horizontal (Figure 5c). surface wave models for the EPR (Boschi & Ekstrom¨ 2002). Where the largest eigenvector points in a nearly ra- Not surprisingly, strain accumulation for constant τ models dial direction, the elongated horizontal part of the FSE ap- is strongest underneath the fast-moving oceanic plates.

proximately follows the trench geometry and shows vary-  Figure 6b shows results for constant strain, ζc ¥ 0 5, as- ing degrees of trench-parallel alignment. In regions where suming that this is a relevant reworking strain after which measurements for SKS waves, traveling nearly radially, all previous fabric is erased (Ribe 1992). As a conse- yield zero or small anisotropy but where more horizontally quence, most horizontal strains are of comparable strength,

propagating local S phases show splitting (Fouch & Fis- with some exceptions, such as the Antarctic plate around £ cher 1998), this small horizontal deformation component 30 £ W/60 S. There, shearing is sufficiently slow that it may be important. This is particularly the case if crystallo- would take more than our cutoff value of 43 Ma to accu- graphic fast axes are distributed in a ring rather than tightly mulate the ζ strain. clustered around the largest FSE eigendirection, in which c case the resolved anisotropy may be trench parallel. How- ever, Hall et al. (2000) show that synthetic splitting is mostly 4.2 Mantle density driven flow

trench-perpendicular in the back-arc region for simple flow  geometries, if fast wave propagation is assumed to be always Figure 7 shows ζc ¥ 0 5 depth-averaged strain for flow that oriented with the largest FSE axis. Escape flow around a slab includes the effect of plate-related motion plus internal den- that impedes large-scale currents in the mantle is therefore sities as derived from tomography model smean (Table 1). usually invoked as an explanation for trench-parallel split- With the caveat that we are interpreting the instantaneous ting (e.g. Russo & Silver 1994; Buttles & Olson 1998). flow that should be characteristic of mantle convection at the Yet, at least for some regions, 3-D circulation that is not present day as steady state for several tens of Myr, we find due to slab impediment might be invoked alternatively (Hall that the inclusion of mantle density leads to a concentration et al. 2000; Becker 2002). of radial flow and deformation underneath South America and parts of East Asia. These features are related to sub- duction where the circum-Pacific downwellings exert strong forces on the overlying plates (cf. Steinberger et al. 2001; 4 GLOBAL FINITE-STRAIN MAPS Becker & O’Connell 2001b). For ηF (Figure 7a), smaller- scale structure owing to density anomalies is most clearly 4.1 Plate-scale circulation visible within continental plates, which were characterized

Figure 6a shows the global, τ ¥ 10 Myr time interval, depth- by large-scale trends for plate motions alone. Other patterns averaged strain field for a circulation calculation that incor- include a west-east extensional orientation in East Africa, Seismic anisotropy and mantle flow 9

A1 x [km] B1 -1500 -1000 -500 0 500 1000 1500 (a) 0˚ (b) 0

100 -5˚ A1 B1

z [km] 200 time [Myr] -10˚ 0 10 20 300 A2 A2 x [km] B2 -15˚ -1500 -1000 -500 0 500 1000 1500 0 B2

-20˚ 100

z [km] 200 -25˚ A3 time [Myr] 0 10 20 300

-30˚ B3 A3 x [km] B3 -1500 -1000 -500 0 500 1000 1500 0 230˚ 235˚ 240˚ 245˚ 250˚ 255˚ 260˚ 265˚ depth [km] 100 0 50 100 150 200 250 300 350 400

z [km] 200 time [Myr] 0 10 20 300

(c) A2 x [km] B2 -1500 -1000 -500 0 500 1000 1500 0

100

z [km] 200 time [Myr] 0 10 20 30 300

Figure 4. (a) Projection of the largest principal axes of strain ellipsoids (sticks) and horizontal cut through FSEs (ellipses), shown centered at tracer positions in purely plate-motion driven flow (ηF) around the East Pacific Rise. Tracer positions are shown at 5 Myr intervals, starting at 300 km depth, with depth grayscale-coded. (b) Projection of FSEs for profiles 1, 2, and 3 as indicated by dashed lines in (a), ellipse shading corresponds to time. (c) Time evolution of strain for an expanded set of tracers along profile 2 (surface projection not shown in (a)) with 2.5 Myr time intervals.

related to an upwelling that correlates with the rift-zone tec- ers, where shearing is stronger, strain is accumulated more  tonics. The predicted strain for ηG (Figure 7b) is, expectedly, rapidly such that ζ ¥ 0 5 is reached by most tracers before more complex. Since the low viscosity notch of ηG partly 20 Myr and at smaller horizontal advection distances than decouples the upper and lower mantle in terms of shearing, at shallow depth. Most regions of slow strain accumulation

upwellings such as the one in the southwestern Pacific (the are underneath continents where minima in the amplitudes  superswell region) are able to cause a stronger anisotropy of surface velocities are found. In general, however, ζ ¡ 0 5

signal than for the ηF model. strains are accumulated in a few Myr and small advection

¨ ©/© In contrast to the plate-motion only models, finite strain distances ¨ O 500 km for all but the shallowest depths. from density-driven flow is quite sensitive to the input mod- The depth average of Figure 7 hides variations of the els and viscosity structures in terms of the local orienta- orientation of the largest strain with depth. The rotation of tions of the largest stretching axes. This is to be expected, the horizontal projection of the largest FSE axis between given that tracer advection will amplify small differences be- 400 km depth and the surface can be in excess of 180 £ tween tomographic models. Figure 7 is therefore only an il- (Becker 2002). Such complexity in strain is strong in, but lustration of the large-scale features predicted by the lowest- not limited to, regions of predominantly radial flow (cf. smean common-denominator tomography model ; individual Figure 5). This finding may complicate the interpretation e.g. ngrand high resolution models ( ) lead to more irregular of anisotropy measurements, especially shear-wave splitting strain predictions, but not to better model fits (sec. 5.2). (Saltzer et al. 2000; Schulte-Pelkum & Blackman 2002).

Figures 8 shows histograms for the time required to However, comparisons of our global horizontal projections  achieve ζc ¥ 0 5 (“age”) for different depths based on the of the largest principal axes of the FSE with observed split- model shown in Figure 7a. We find that many of the shallow ting orientations indicate agreement between strain and split- strain markers (within the high-viscosity lithospheric layer) ting in (sparsely sampled) oceanic and in young continental have our age-limit of tc ¥ 43 Ma, implying that strain accu- regions, e.g. the western US (Becker 2002). In older conti- mulation is slow within the “plates” that move coherently nental regions (e.g. the eastern US and eastern South Amer- without large interior velocity gradients. For deeper lay- ica) and shear-dominated regions (e.g. New Zealand and NE 10 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell

A1 x [km] B1 -500 0 500 (a) (b) 0 -10˚

B1 100

z [km] time [Myr] -15˚ 200 0 10 20

A1 A2 x [km] B2 -500 0 500 0 -20˚ B2

100 z [km] -25˚ time [Myr] A2 200 0 10 20

A3 x [km] B3 B3 -500 0 500 -30˚ 0

A3 100

-35˚ z [km] 200 time [Myr] 285˚ 290˚ 295˚ 0 10 20 depth [km] 0 50 100 150 200 250 300 350 400

(c) A2 x [km] B2 -1000 -500 0 500 1000 0 100 200 300 400 z [km] time [Myr] 500 0 10 20 30 600

Figure 5. (a) Tracer paths in a smean nt density-included flow field (ηF) at 5 Myr intervals for part of the South American subduction zone. Ellipses are drawn at tracer positions as in Figure 4, moving from west to east and starting at 50 km depth in the Nazca plate. Note the variations in predicted slab dip along strike as indicated by the final depth of the tracer locations. (b) Projection of FSEs on profiles 1,2, and 3 as shown in (a). (c) Finite strain accumulation along different streamlines close to profile 2 (surface trace not shown in (a)) with 2.5 Myr intervals.

Tibet) agreement is poor. It is difficult to interpret these find- since the smaller-scale density-related currents change over ings given the uneven distribution of the SKS data that sam- time. If we include non-tectosphere density from smean for

ples mostly continents where we expect that our method will z 0 220 km, details in the fast orientations change slightly, lead to less reliable estimates of anisotropy. and more pronounced straining occurs underneath Eastern Africa. Since evolving plate boundaries mainly affect shallow 4.3 Changes in plate motions structure, we can expect that SKS-based shear-wave split- ting observations will be only slightly modified by the un- The global models described in the previous section were all certainties in such reconstructions and the overall effect of based on steady-state flow with a cut-off age, tc, of 43 Ma. non steady-state flow. For short-period surface waves with We conducted additional experiments where we allowed strong sensitivity to shallow structure, such as 50-s Rayleigh for evolving plate boundaries and backward-advected den- waves (see Appendix), the effect may be larger. We there- sity anomalies as described in sec. 3.1.2. Given our find- fore analyze whether strain predictions that include evolving

ing of rapid shearing for large depths from above, results plate boundaries lead to a better fit to the surface-wave data $ for z ¡ 150 km were, as expected, similar to the steady- than steady-state models (sec. 5.2). state models; depth averages with uniform weight for each layer as in Figure 7a are, thus, not strongly modified. Fig- ure 9 explores some of the variations in predicted strains for

shallow depth, at z ¥ 100 km. Plate boundaries that change 5 GLOBAL COMPARISON WITH SURFACE with time will introduce more complexity at small scales. We WAVES find that the large-scale effects of evolving plates are mod- 5.1 Resolution of Rayleigh wave 2φ-anisotropy est, however, even when we go back beyond the bend in the inversions Hawaii-Emperor seamount chain. The shallow strain pattern underneath regions that move coherently for steady plate Inversions of phase velocities for azimuthal anisotropy are motions (Figure 9a) is weakened or disrupted (Figure 9b) complicated by trade-offs between isotropic and anisotropic Seismic anisotropy and mantle flow 11

(a) τ 1 10 Myr 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

60˚

30˚

0˚

-30˚

-60˚

ζ = 0.93

∆Lrr

-0.5 0.0 0.5 2 (b) ζc 1 0 5 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

60˚

30˚

0˚

-30˚

-60˚

ζ = 0.46

∆Lrr

-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3

   3 Figure 6. Depth-averaged (upper mantle, 50 km  z 400 km) finite strain for τ 10 Myr (a) and ζc 0 5 (b) strain accumulation. We use plate-motion related

flow only and viscosity profile ηF. Thick sticks are plotted centered at every  5th tracer location and indicate the orientation of the horizontal projection of the  largest axis of the FSE, scaled with the logarithmic strain ζ (see legend). Background shading denotes ∆Lrr Lrr 4 1, and thin sticks denote the orientation of surface plate motions from NUVEL1-NNR. Colorscale for ∆Lrr is clipped at 70% of the maximum absolute value.

structure (e.g. Laske & Masters 1998) and the uneven ray- 2φ-anisotropy was inferred from the steady-state circulation  path coverage might map itself into apparent anisotropic model using smean, ηF, and ζc ¥ 0 5 (cf. Figure 7a). The structure (e.g. Tanimoto & Anderson 1985). To ad- horizontal projections of the largest FSE axes were scaled dress these issues, we performed recovery tests, shown for with ζ to roughly account for the strength of inferred seis-

Rayleigh waves at T ¥ 50 s in Figure 10. The same inversion mic anisotropy; we additionally weighted each layer ac- procedure that was used to obtain the phase-velocity maps cording to the 50-s Rayleigh wave sensitivity-kernel of Fig- shown in Figure 1 was employed to test our ability to recover ure A1. Maximum predicted strains were scaled to a 1.5% a synthetic input model using the available data coverage. 2φ-anisotropy amplitude (resulting in 0.6% RMS variation) To make structure in the input model as realistic as possible, and the isotropic and D4φ variations were set to zero. Ran- 12 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell

(a) ηF 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

60˚

30˚

0˚

-30˚

-60˚

ζ = 0.46

∆Lrr -0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3

(b) ηG 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

60˚

30˚

0˚

-30˚

-60˚

ζ = 0.47

∆Lrr

-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3 3 Figure 7. Depth-averaged finite-strain for ζc  0 5 strain accumulation in plate-motion and smean-driven flow for viscosity profiles ηF (a, cf. Figure 6b) and ηG (b). Thin sticks in background are NNR plate velocities as in Figure 6. dom noise was added to mimic the observational uncertain- multiplying ∆α with the input model amplitudes to give less

ties. The resulting variance reduction of the inversion was weight to small-signal regions; ∆α7 is shown as background 0 only 14% for D , implying that little of the spurious signal shading in Figure 10 and normalized such that weighted and was fit. Figure 10 shows that the orientations of azimuthal original maximum deviations are identical. anisotropy of the input model are generally well recovered. Exceptions are found in the Middle East, in the Aleutians, The amplitude recovery of the test inversion is not as the Cocos-Nazca plate area, and along the northern mid- good as the azimuthal one. As shown in Figure 2, recovery

Atlantic ridge system. To quantify these azimuthal devia-

5 £ £65 is worse on average in continental regions than in oceanic tions, we calculate the angular misfit ∆α (0 ∆α 90 ), ones; there, regions of poor recovery are found in the south- shown in a histogram in Figure 11a. We also compute ∆α7 by west Indian Ocean, the northern Atlantic, the Scotia plate region, and the north-west Pacific. The average of the log10 Seismic anisotropy and mantle flow 13

30 (a) steady-state up to 43 Ma 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

20 z = 75 km 60˚ [%] 0

30˚ 10 N/N

0˚

0 -30˚ 30

-60˚ 20 z = 175 km

[%] ζ = 0.50 0 ∆Lrr -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 10

N/N (b) plate-motion history up to 60 Ma 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

0 60˚ 30

30˚ 20 z = 275 km

[%] 0˚ 0

10 -30˚ N/N

0 -60˚ 0 10 20 30 40 age [Ma] ζ = 0.50 ∆Lrr

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

3  3 Figure 8. Histograms of tracer ages, i.e. duration for ζc  0 5-strain accu- Figure 9. Largest FSE axes for ζc 0 5 at 100 km depth, ηF, and plate- mulation, for plate motion and smean related flow and ηF (Figure 7a) for motion and smean related flow. Plot (a) is from steady-state flow without

tracers that end up at 75 km, 175 km, and 275 km depth. N N denotes rela- evolving plates, allowing for tracer backtracking until the cut-off time t 

 0 c tive frequency, and tracer locations are chosen such that they evenly sample 43 Ma (Figure 7a), (b) is for evolving plates and backward-advected smean the surface. Dotted histograms are limited to tracers that end up in oceanic densities until tc  60 Ma. plate regions of 3SMAC (Nataf & Ricard 1996). Plate velocities are as- sumed to be steady-state and the cut-off age is 43 Ma, hence the spike in

0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚ 3 the top panel corresponding to tracers that have not reached ζ  0 5 after 43 Ma. The age distributions correspond to a range of horizontal distances 60˚ between initial and final tracer positions of up to  2000 km for 75 km

depth, and the distribution maxima at  6 Ma for depths of 175 km and

275 km correspond to  250 km horizontal distance. Vertical distance trav- 30˚ eled is 89 50 km.

0˚

2φ

of the recovered 2φ signal over the input 2φ amplitude, Dr , -30˚  is  0 062, corresponding to a mean amplitude recovery of

86.7% (Figure 11b). Regions in which anisotropy strength -60˚ is overpredicted are found at some trenches (e.g. Central Chile), where the inversion gives a trench-perpendicular sig- inversion: D2φ: 1.5% D4φ: 0.8% input model: D2φ: 1.5% nal while the input model has small amplitude with FSEs ∆α [deg]

primarily radial. The area-weighted mean orientational mis- 0 10 20 30 40 50 60 70

£ £

;  fit, : ∆α , is 18.8 , which is reduced to 13 4 if we weight the misfit by the input-model 2φ amplitudes. Restricting our- Figure 10. Input (open bars) and recovered (black sticks) azimuthal anisotropy for 50 s Rayleigh waves. The inverted structure has some artifi- selves further to oceanic lithosphere (as given by 3SMAC; 4φ

£ cial D -anomalies (thin crosses), zero in the input model which is based on

: ;< ¥ 

Nataf & Ricard, 1996), ∆α oc = 11 8 . If we weight the 3 ζc  0 5, plate motions, and smean (cf. Figure 7a). All anisotropy shown on global misfit by input and output model amplitudes (as- the same, linear scale; maximum amplitudes are indicated below the map.

suming small output amplitudes indicate poor resolution), ?

£ Background shading is ∆α, the orientational deviation ∆α scaled by the lo-

2φ

;>¥  : ∆α 9 9 . These values can guide us in judging misfits cal D amplitude of the input model, normalized such that the maximum between observations and model predictions. of ∆α? and ∆α are identical. To evaluate the trade-offs that occur in the presence of coherent isotropic structure, we have analyzed a second res- olution test in which the random noise in the input D0 sig- nal was replaced by slowness anomalies derived from the 14 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell

(a) 20

20 [%] 0 [%] 0 10 10 N/N N/N

0 0 10 20 30 40 50 60 70 80 90 ∆α [deg] 0 0 10 20 30 40 50 60 70 80 90 (b) 30 ∆α [deg]

Figure 13. Histogram of orientational misfit, ∆α, between T  50 s 3 Rayleigh wave 2 anisotropy and model prediction for  0 5 strain, , 20 φ ζc ηF

evolving plates (tc  60 Ma), and smean nt advected density as in Fig-

[%] ure 12a. Solid bars: all data; open bars: data restricted to regions where 0

2φ 3 D of the inversion is @ 0 25 times its maximum; and open bars with dot- N/N 10 ted lines: further restricted to oceanic lithosphere. Dashed horizontal line indicates the expected random distribution of ∆α.

0 -1 0 1 scaled horizontal projection of the largest axes of the FSE.

2φ 2φ  log10(D out/D in) The calculation uses ζc ¥ 0 5-strain and includes the effects of evolving plate motions, smean nt buoyancy, and changes

Figure 11. (a) Orientational misfit histogram ∆α (gray bars) for equal-area in plate-configurations for tc ¥ 60 Ma. We find that much of spatial sampling of the recovery test of Figure 10, N N denotes relative the measured signal in the oceans can be explained by LPO

 0 frequency. Open bars show misfit when restricted to regions with input D2φ- orientations and finite strain as predicted from our global cir-

amplitudes @ 25% of the maximum input. Dashed horizontal line denotes culation model. We show a histogram of the angular misfit

the expected random distribution of misfit. (b) Histogram of amplitude re- for T ¥ 50 s in Figure 13; the largest misfits are typically 2φ 2φ 2φ covery, Dr , as expressed by the decadic logarithm of inversion D over found in regions of small D -amplitudes or in the conti- input D2φ-amplitudes (gray bars, open bars restricted to strong-amplitude nents, where anisotropy may be related to past deformation input as in (a)). episodes.

The laterally averaged misfits are compared for several ¥ T ¥ T crustal model CRUST5.1 (Mooney et al. 1998). The recov- different models in Figure 14a and b for 50 s and ered anisotropic signal is similar to that of the first experi- 150 s, respectively. We tried weighting the misfit in a num-

£ ber of ways but think that, in general, accounting for both the

;>¥  ment, with mean azimuthal misfit : ∆α 13 7 (weighted inversion’s and the model’s D2φ-amplitudes is most appro- D2φ by input ) and mean amplitude recovery of 87.1% for priate in order to avoid having poorly sampled regions bias

D2φ. If the 4φ-terms are suppressed by damping them much ; : ∆α . For consistency, we have also weighted the surface more strongly than D2φ, the result is, again, similar to that plate-velocity derived misfits (APM and NNR) by the model shown in Figure 10, but the azimuthal misfit and amplitude

£ (i.e. plate-velocity) amplitudes. However, such velocity-

;A¥  recovery are slightly improved to : 12 5 (weighted ∆α weighted misfits are biased toward the oceanic plates, which

by input D2φ) and 90.3%, respectively. Assuming that the  move faster than continental ones, relative to ζc ¥ 0 5-strain T ¥ 50 s results are representative of deeper-sensing waves, models, which show a more uniform anisotropy-amplitude the resolution tests are encouraging since they imply that Ek- globally (e.g. Figure 7). strom’¨ s (2001) inversions for azimuthal anisotropy are ro- 2φ We find that most circulation-based strain models out- bust. The number of parameters of an inversion for D is perform the hypotheses of alignment with NNR or APM larger than that of an isotropic inversion and the correspond- plate motions. This distinction in model quality supports ing increase in degrees of freedom cannot be justified based our modeling approach and indicates that further study of on the improvement in variance reduction alone (Laske & finite strain models could lead to a better understanding of Masters 1998). However, the pattern that such an anisotropic lithospheric deformation and mantle flow. The average fit inversion predicts is likely to be a real feature of the Earth to Ekstrom’¨ s (2001) anisotropy maps is better for shorter

and not an artifact. ¥ (T ¥ 50 s in Figure 14a) than for longer periods (T 150 s in Figure 14b). This could be due to differences in resolution of the surface waves, or the dominance of the plate-related 5.2 Global azimuthal-anisotropy model fit

strains at shallow depths, presumably the best-constrained ;

Figure 12 compares maps for Rayleigh wave 2φ anisotropy large-scale features. We also observe a wider range in : ∆α ¥ from Ekstrom’¨ s (2001) inversions (Figure 1) with predic- for different types of models at T ¥ 50 s than at T 150 s. tions from our preferred geodynamic model, obtained by This is expected, given that strains accumulate more rapidly depth averaging (using the appropriate kernels) of the ζ- at depth, so that it is, for instance, less important whether Seismic anisotropy and mantle flow 15

2φ C (a) T B 50 s (peak D -sensitivity: 70 km depth) 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

n ∆αinv [deg] 60˚ 70

60 30˚ 50

0˚ 40

30 -30˚ 20

10 -60˚ 0

inversion: D2φ: 2.5% model: D2φ: 2.2%

2φ C (b) T B 150 s (peak D -sensitivity: 200 km depth) 0˚ 60˚ 120˚ 180˚ 240˚ 300˚ 360˚

n ∆αinv [deg] 60˚ 70

60 30˚ 50

0˚ 40

30 -30˚ 20

10 -60˚ 0

inversion: D2φ: 2.1% model: D2φ: 1.6%

2φ  Figure 12. Comparison of Rayleigh wave 2φ-azimuthal anisotropy for T  50 s (a) and T 150 s (b), shown as black sticks (maximum D values shown to

4φ

3  scale below map) and model anisotropy for ζc  0 5, evolving plates (tc 60 Ma), ηF, and smean nt advected density, shown as open bars. We omit D from the inversions and scale the predicted anisotropy such that the RMS D2φ are identical to those from the inversions. Background shading indicates azimuthal misfit scaled by the inverted D2φ amplitude such that maximum ∆α remains constant.

16 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell D (a) T D 50 s (b) T 150 s

tc = 60 Ma smean tc = 60 Ma smean smean_nt smean_nt

tc = 60 Ma smean_nt tc = 60 Ma smean_nt

lrr98d ηG tc = 120 Ma tc = 60 Ma stb00d lrr98d

smean area smean area

ηG tc = 120 Ma tc = 60 Ma 〈∆α〉recovery base model 〈∆α〉recovery

ηG ocean tc = 60 Ma ocean

2φ 2φ tc = 60 Ma stb00d D inv tc = 120 Ma D inv

2φ 2φ 2φ 2φ ngrand_nt D inv D mod stb00d D inv D mod

2φ 2φ 2φ 2φ ηG smean D inv D mod oc. ηG D inv D mod oc.

tc = 60 Ma ngrand_nt tc = 60 Ma stb00d

tc = 60 Ma nuvel.nnr

tc = 120 Ma nuvel

base model ngrand_nt

ηG tc = 60 Ma smean tc = 60 Ma ngrand_nt

nuvel.nnr ηG smean

nuvel ηG tc = 60 Ma smean

ηG tc = 60 Ma ngrand_nt ηG tc = 60 Ma ngrand_nt

25 30 35 40 45 50 30 35 40 45 50

〈∆α〉 [deg] 〈∆α〉 [deg] 

Figure 14. Mean orientational misfit between 2φ anisotropy of Rayleigh waves at T  50 s (a, cf. Figures 12a and 13) and T 150 s (b) for a selection of 3 circulation-derived, ζc  0 5 finite-strain models, and surface plate-velocity orientations. The y-axis indicates different flow models; all models based on plate

motions and viscosity profile ηF unless indicated otherwise. Types of density models abbreviated as in Table 1. If no time interval is specified, velocities are  steady-state with a cut-off in backward advection time, tc, of 43 Ma; tc  60 Ma or tc 120 Ma implies backward advection using evolving plate-boundaries

and possibly density sources until tc. nuvel and nuvel.nnr refer to the azimuthal misfit as calculated from surface plate-velocity orientations for hotspot (HS2, F APM) or NNR reference frames, respectively. The misfits, E ∆α , are obtained by area-weighted averaging over a grid interpolation of predicted and observed

anisotropy (filled stars). We additionally weight models by the surface-wave recovery function ∆α? as shown in the background of Figure 12 (open stars, mostly hidden by filled ones), restrict misfit to oceanic regions (open diamonds), weight by the inversion’s D2φ-amplitude (filled circles), the inversion’s and the

model’s D2φ-amplitudes (filled boxes), and the latter quantity furthermore restricted to oceanic plates (open boxes). (For APM and NNR velocities, weighting 3

by the model amplitudes implies a strong bias toward the oceanic plates relative to ζc  0 5-models.) If we randomize the 2φ orientations of our models, we

3 H E FI89 H H

find a standard deviation of G 0 4 indicating that ∆α 43 is significantly different from the random mean of 45 at the 5σ-level. £

flow is treated as steady-state or with evolving plate bound- random alignment and ¡ 35 for alignment with plate ve-

aries. For ηF, models that include buoyancy-driven flow lead locities. Taking the resolution of the surface wave inversion

£

: ; ¡ to ∆α improvements of 6 compared to those with plate as imaged by the ∆α7 function in Figure 10 into account im- motions only. Comparing different density models, smean proves the misfit, but not significantly. Regional variations typically leads to better results than higher resolution to- in misfit might therefore be due to poor surface-wave reso- mography (ngrand) or models that are based on slabs only lution but, globally, ∆α is independent of the surface-wave (stb00d or lrr98d). Including shallow density variations un- resolution pattern. derneath younger plate regions (non-tectosphere models, “nt”), slightly improves the model fit. There are a number of second-order observations that have guided us in the choice of the preferred model shown Globally weighted results typically show the same de- in Figure 12. τ-limited (time limited) strain-accumulation

pendence on model type as those that include only the models lead to results that are similar to those of ζ-models

: ;< oceanic lithosphere, ∆α oc = ; the latter misfits are, however, (strain limited) without weighting. Results from τ calcu-

£ 2φ

: ;

generally smaller than the global estimates by ¡ 4 . The best lations with D model-amplitude weighted ∆α are al-

£ £

: ;J ¥ : ;< =

models yield ∆α oc = 24 for T 50 s and ∆α oc 28 ways, as expected, better than ζ-models because of the ad-

£

¦ ¦ ¥ for T ¥ 100 s and T 150 s, to be compared with 45 for ditional bias that is introduced toward the oceans (where Seismic anisotropy and mantle flow 17 all models are more similar to the inversions). We there- ACKNOWLEDGMENTS fore limit consideration to ζc-models, for which we find that density-included flow calculations lead to better results for All figures were produced with the GMT software by Wes-

£ sel & Smith (1991). Bernhard Steinberger provided the orig-

: ; η than for η ( ¡ 2 difference in average ∆α ). For shal- F G inal version of the flow code used for the circulation cal- low (T ¥ 50 s) structure where strain accumulation is rep-

culation, which would not have been possible without nu- ; resentative of a longer timespan, : ∆α values are slightly improved (not shown) when we include a “freezing” mech- merous authors’ willingness to share inversion and model anism for strains at shallow depths where the temperature results. Discussions with Karen Fischer and Paul Silver fur- in the lithosphere might be too low for continuous fabric thered our understanding of the implications of shear-wave reworking (Becker 2002) but this is, again, partly due to splitting measurements. the resulting bias toward the oceanic regions. We cannot find any significant global improvement compared to contin- uous strain-accumulation models. Comparing models with

steady-state velocities or evolving plates in Figure 14 we do, REFERENCES £ however, see a small ( ¡ 2 ) decrease in global misfit when Agee, C. B., 1998. Phase transformations and seismic the last 60 Ma of plate reconfiguration and density advection structure in the upper mantle and transition zone, in are taken into account. Ultrahigh-Pressure Mineralogy. Physics and Chemistry of In summary, we find that the strain produced by the Earth’s Deep Interior, edited by R. J. Hemley, vol. 37 global mantle circulation is a valid explanation for the az- of Reviews in Mineralogy, pp. 165–203, Mineralogical imuthal anisotropy mapped by surface waves, and that such Society of America, Washington DC. circulation-produced strain explains the pattern of azimuthal Anderson, D. L. & Dziewonski, A. M., 1982. Upper mantle anisotropy better than does the pattern of absolute plate mo- anisotropy: evidence from free oscillations, Geophys. J. R. tions. Further exploration of regional and global model per- astr. Soc., 69, 383–404. formance can guide us in our understanding of anisotropy in Becker, T. W., 2002. Lithosphere–Mantle Interactions, the upper mantle. While some of the regions where our mod- Ph.D. thesis, Harvard University, Cambridge MA. els disagree with anisotropy observations might be due to Becker, T. W. & Boschi, L., 2002. A comparison of tomo- limitations in our method (such as the assumptions about the graphic and geodynamic mantle models, Geochemistry, relationship between strain and imaged anisotropy, sec. 2), Geophysics, Geosystems, 3(2001GC000168). others might indicate real effects such as intraplate deforma- Becker, T. W. & O’Connell, R. J., 2001. Predicting plate tion (e.g. in the Australian plate) for which we have not ac- velocities with geodynamic models, Geochemistry, Geo- counted so far. Short-period surface-waves with good shal- physics, Geosystems, 2(2001GC000171). low sensitivity may be able to provide us with constraints on Becker, T. W. & O’Connell, R. J., 2001. Lithospheric the strain history for times longer than a few tens of Myr and stresses caused by mantle convection: The role of plate will allow us to evaluate the degree of mantle-lithosphere rheology (abstract), EOS Trans. AGU, 82(47), T12C– coupling in different tectonic settings. 0921. Ben Ismail, W. & Mainprice, D., 1998. An olivine fabric database; an overview of upper mantle fabrics and seismic anisotropy, Tectonophysics, 296, 145–157. Blackman, D. K. & Kendall, J.-M., 2002. Seismic anisotropy of the upper mantle: 2. Predictions for current plate boundary flow models, Geochemistry, Geophysics, Geosystems, 3(2001GC000247). Blackman, D. K., Wenk, H.-R., & Kendall, J.-M., 2002. 6 CONCLUSIONS Seismic anisotropy of the upper mantle: 1. Factors that af- Observations of seismic anisotropy in the upper mantle can fect mineral texture and effective elastic properties, Geo- be explained by global circulation models under the assump- chemistry, Geophysics, Geosystems, 3(2001GC000248). tion that fast orientations are aligned with the largest axis of Boschi, L. & Ekstrom,¨ G., 2002. New images of the Earth’s the finite strain ellipsoid, as suggested by the theory of Ribe upper mantle from measurements of surface-wave phase (1992). Our models are very simplified in that we are using velocity anomalies, J. Geophys. Res., in press. a linear viscosity for the flow calculations and assume that Bunge, H.-P. & Grand, S. P., 2000. Mesozoic plate-motion anisotropy obeys a straightforward relationship with finite- history below the northeast Pacific Ocean from seismic strain. However, it is encouraging that mantle-flow derived images of the subducted Farallon slab, Nature, 405, 337– models lead to smaller misfits with azimuthal anisotropy 340. from surface waves than models based on alignment with Bunge, H.-P., Richards, M. A., Lithgow-Bertelloni, C., surface velocities. A coupled model that uses the strain his- Baumgardner, J. R., Grand, S. P., & Romanowicz, B. A., tory predicted from a, possibly more sophisticated, circula- 1998. Time scales and heterogeneous structure in geody- tion model as input for fabric development algorithms such namic earth models, Science, 280, 91–95. as that of Kaminski & Ribe (2001) should be attempted next. Bunge, H.-P., Hagelberg, C., & Travis, B., 2001. Mantle In this way, we should be able to both evaluate the validity of circulation models with variational data assimilation: In- our model assumptions and the generality of theories about ferring past mantle flow and structure from plate motion the origin of seismic anisotropy and tectonic deformation in histories and seismic tomography (abstract), EOS Trans. the upper mantle. AGU, 82(47), NG51C–07. 18 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell

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Lattice preferred orienta- Schott, B., Yuen, D. A., & Schmeling, H., 2000. The signif- tion of olivine aggregates deformed in simple shear, Na- icance of shear heating in continental delamination, Phys. ture, 375, 774–777. Earth Planet. Inter., 118, 273–290. Schulte-Pelkum, V. & Blackman, D. K., 2002. A synthesis of seismic P and S anisotropy, Geophys. J. Int., submit- ted. APPENDIX A: DEPTH SENSITIVITY OF SURFACE Schulte-Pelkum, V., Masters, G., & Shearer, P. M., 2001. WAVES TO AZIMUTHAL ANISOTROPY Upper mantle anistropy from long-period P polarization, J. Geophys. Res., 106, 21917–21934. Our treatment closely follows that of Montagner & Nataf Silver, P. G., 1996. Seismic anisotropy beneath the con- (1986) and Montagner & Tanimoto (1991). The eigenfunc-

tinents: probing the depths of geology, Ann. Rev. Earth tions of high-  normal modes can be used to construct 20 T.W. Becker, J.B. Kellogg, G. Ekstrom,¨ and R.J. O’Connell

Frechet´ derivatives (or sensitivity kernels), Km, that charac- Table A1. Ratios of elastic coefficients for 2φ contributions as in (A3) terize the effect of small perturbations in a parameter m at a for two different rock samples from the indicated sources. cos and sin de-

certain depth r on changing the phase velocity c of a surface note the C T S constants. Since the partitioning of these ratios depends on the

2 V 2 wave: horizontal coordinate frame, we also give the amplitude U cos sin norm,

∂c normalized by the maximum ratio in each row.

©K¥ ¥  k ¨ c r mK m (A1)

m m ∂m

W W BC W S HC S GC S Here, m stands for an elastic modulus or density ρ (e.g. type A F L Dahlen & Tromp 1998, p. 335ff). For a transversely

isotropic Earth model such as PREM, the elasticity tensor Peselnick & Nicolas (1978)

3 3

is reduced from its most general form with 21 independent cos 0 3 034 0 003 0 028

3 3 3 4 components to five constants, A, C, F, L, and N in the no- sin 4 0 002 0 006 0 008

2 V 2

3 3 3 tation of Love (1927). We write the perturbations in c in a U cos sin norm 1 0 0 176 0 853

transversely isotropic medium as Nunuvak (Ji et al. 1994)

3 3

cos 0 3 075 0 014 0 054

3 3 3

4 4

R sin 4 0 009 0 004 0 009

§ §

dC § dA dL

2 V 2

3 3 3

dc kC kA kL (A2) U cos sin norm 1 0 0 197 0 724 ¦ML

0 N C A L §

dN § dF dρ  kN kF k O 0 N F ρ ρ For wave propagation in an elastic medium that has the most general form of anisotropy, Smith & Dahlen (1973) showed 100 that the azimuthal dependence of Love and Rayleigh wave speed anomalies can be expanded as given in (1). Montagner 200 & Nataf (1986) demonstrated that the sensitivity kernels for the D-terms can be retrieved from the kernels already con- structed for transversely isotropic models as in (A2). Mon- 300 tagner & Nataf also found that the 4φ terms should gener- ally be small for predominantly horizontal alignment of fast propagation axes based on measurements of elasticity ten- depth [km] 400 sors from the field. Rayleigh waves are more sensitive to the 2φ component than Love waves, and only the 2φ term is readily interpreted in terms of fast horizontal propagation 500 axes in a strained medium with lattice-preferred orientation. Following our expectations, the observed 4φ signal is smaller than the 2φ signal in most regions, with aforementioned ex- 600 50 s (0S202) ception in regions of radial flow (sec. 2). 100 s (0S97) We shall therefore focus on D2φ for Rayleigh waves. 150 s (0S62) Montagner & Nataf (1986) find that sensitivity kernels for 700

0 0.2 0.4 0.6 0.8 1

© ¨ © the cos ¨ 2φ and sin 2φ terms can be written as

normalized sensitivity [A.U.]

R R R §

Rayleigh BC S § HC S GC S ¥

k kA kF kL  (A3) 2φ

QSR P Q cos P 2φ sin 2φ A F L Figure A1. D sensitivity kernels for Rayleigh waves with periods of 50 s,

100 s, and 150 s for anisotropic PREM, normalized to their respective max-

R R The six fractions with BC R S, HC S, and GC S are simple func- imum value. Kernels are shown for the amplitude of all 2φ terms and are

tions of Ci j (Montagner & Nataf 1986, eq. 5) and determine calculated from (A3) and the data of Peselnick & Nicolas (1978) in Ta- R the relative weights of the kA R F L kernels. While Montagner ble A1. & Nataf’s (1986) analysis of the effect of anisotropy was re- stricted to a flat Earth, Romanowicz & Snieder (1988) were

able to recover Montagner & Nataf’s approximation asymp- 2φ anisotropy, shown in Figure A1 for surface waves with

¥ ¥ periods of T ¥ 50 s, T 100 s, and T 150 s. totically for large  in a more complete treatment. Follow- ing Montagner & Tanimoto (1991), we can therefore ob- Phase-velocity sensitivity kernels were estimated based tain the depth sensitivity of Rayleigh waves to azimuthal on normal-mode eigenfunctions computed for radially anisotropy with (A3) if we use elastic moduli for some rep- anisotropic PREM with the MINEOS program by G. Masters resentative anisotropic material whose fast propagation axis (e.g. Dahlen & Tromp 1998, p. 335). For the elastic param- is assumed to be in the horizontal plane. Table A1 lists the eters under consideration, the sensitivity of Rayleigh waves relevant parameters for an oceanic sample from Peselnick & to azimuthal anisotropy has a depth dependence that is very Nicolas (1978) (as used by Montagner & Nataf 1986) and similar to their kVSV kernels. We use the 2φ kernels to average for Nunuvak from Ji et al.’s (1994) data (as used by Sav- the horizontal projections of the inferred fast-propagation

axes from strain at each depth when we compare our model- R age 1999). The BC R S and GC S ratios are consistently larger ing results with surface-wave based anisotropy.

than the HC R S ratio. Since the kL kernel is additionally some- ¤

what larger than kA and kF , the GC R S LkL term is mostly re- sponsible for the depth sensitivity of Rayleigh waves to the