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Physics of the Earth and Planetary Interiors 146 (2004) 113–124 Constraining large-scale mantle heterogeneity using mantle and inner-core sensitive normal modes Miaki Ishii a,,1 , Jeroen Tromp b a Department of Earth & Planetary Sciences, Harvard University, Cambridge, MA 02138 USA b Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125 USA Received 2 September 2002; received in revised form 5 February 2003; accepted 18 June 2003 Abstract Attempts to resolve density heterogeneity within the mantle using normal-mode data revealed an unexpected feature near the core-mantle boundary: regions of strong sheer velocity reduction, known as superplumes, are characterized by heavier than average material [Science 285 (1999) 1231]. Thus far, free-oscillation studies of mantle structure have relied upon modes which sample only the mantle (e.g., [J. Geophys. Res. 96 (1991) 551; Science 285 (1999) 1231; J. Geophys. Res. 104 (1999a) 993; Geophys. J. Int. 143 (2000b) 478; Geophys. J. Int. 150 (2002) 162]). In order to better constrain mantle heterogeneity, we add inner-core sensitive modes to our data set, and invert for mantle structure and inner-core anisotropy simultaneously. The additional modes do not alter the pattern of the density anomalies relative to model SPRD6 [Geophys. J. Int. 145 (2001) 77]. However, they help constrain the amplitude of lateral variations in density. In a previous study, obtaining a density model with reasonable amplitudes required supplemental gravity data, which are not included in the current study. Nonetheless, the new density model is generally weaker, especially at the bottom of the mantle. The root-mean square (RMS) amplitude of the model exhibits two maxima around 600 and 2300 km depth. Experiments indicate that these are robust components of the model and neither artifacts of the choice of radial basis functions nor the damping scheme. Comparisons of the density model with shear- and compressional-velocity models show negative or nearly zero correlation in the transition zone and near the base of the mantle, where the root-mean square density amplitude is high. Finally, the anomalous features near the core-mantle boundary are confirmed: strong anti-correlation between shear and bulk-sound velocities, and increased density at the locations of superplumes. © 2004 Elsevier B.V. All rights reserved. Keywords: Seismic tomography; Normal modes; Seismic velocities; Density 1. Introduction Body-wave travel times, the most extensively used data in seismic tomography, have revealed large Corresponding author. Tel.: +1-858-534-4643. E-mail address: [email protected] (M. Ishii). 1 Present address: Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California, La Jolla, CA 92093, USA. low-velocity structures near the core-mantle boundary (Dziewo´ nski, 1984). Although images of these veloc- ity anomalies have been refined using larger data sets and improved techniques (e.g., Inoue et al., 1990; Li and Romanowicz, 1996; Dziewo´ nski et al., 1997; van der Hilst et al., 1997; Masters et al., 2000c), mantle density anomalies have remained elusive. This is be- cause high-frequency body-wave data are insensitive to density variations. In contrast, the gravitational restoring force is important for low-frequency normal 0031-9201/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2003.06.012
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Physics of the Earth and Planetary Interiors 146 (2004) 113–124

Constraining large-scale mantle heterogeneity using mantleand inner-core sensitive normal modes

Miaki Ishii a,∗,1, Jeroen Trompba Department of Earth & Planetary Sciences, Harvard University, Cambridge, MA 02138 USA

b Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125 USA

Received 2 September 2002; received in revised form 5 February 2003; accepted 18 June 2003

Abstract

Attempts to resolve density heterogeneity within the mantle using normal-mode data revealed an unexpected feature nearthe core-mantle boundary: regions of strong sheer velocity reduction, known as superplumes, are characterized by heavierthan average material [Science 285 (1999) 1231]. Thus far, free-oscillation studies of mantle structure have relied upon modeswhich sample only the mantle (e.g., [J. Geophys. Res. 96 (1991) 551; Science 285 (1999) 1231; J. Geophys. Res. 104 (1999a)993; Geophys. J. Int. 143 (2000b) 478; Geophys. J. Int. 150 (2002) 162]). In order to better constrain mantle heterogeneity,we add inner-core sensitive modes to our data set, and invert for mantle structure and inner-core anisotropy simultaneously.The additional modes do not alter the pattern of the density anomalies relative to model SPRD6 [Geophys. J. Int. 145 (2001)77]. However, they help constrain the amplitude of lateral variations in density. In a previous study, obtaining a density modelwith reasonable amplitudes required supplemental gravity data, which are not included in the current study. Nonetheless, thenew density model is generally weaker, especially at the bottom of the mantle. The root-mean square (RMS) amplitude ofthe model exhibits two maxima around 600 and 2300 km depth. Experiments indicate that these are robust components ofthe model and neither artifacts of the choice of radial basis functions nor the damping scheme. Comparisons of the densitymodel with shear- and compressional-velocity models show negative or nearly zero correlation in the transition zone andnear the base of the mantle, where the root-mean square density amplitude is high. Finally, the anomalous features near thecore-mantle boundary are confirmed: strong anti-correlation between shear and bulk-sound velocities, and increased densityat the locations of superplumes.© 2004 Elsevier B.V. All rights reserved.

Keywords: Seismic tomography; Normal modes; Seismic velocities; Density

1. Introduction

Body-wave travel times, the most extensively useddata in seismic tomography, have revealed large

∗ Corresponding author. Tel.:+1-858-534-4643.E-mail address: [email protected] (M. Ishii).

1 Present address: Institute of Geophysics and Planetary Physics,Scripps Institution of Oceanography, University of California, LaJolla, CA 92093, USA.

low-velocity structures near the core-mantle boundary(Dziewonski, 1984). Although images of these veloc-ity anomalies have been refined using larger data setsand improved techniques (e.g.,Inoue et al., 1990; Liand Romanowicz, 1996; Dziewonski et al., 1997; vander Hilst et al., 1997; Masters et al., 2000c), mantledensity anomalies have remained elusive. This is be-cause high-frequency body-wave data are insensitiveto density variations. In contrast, the gravitationalrestoring force is important for low-frequency normal

0031-9201/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.pepi.2003.06.012

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modes, and hence they possess sensitivity to lateralvariations in density. Knowledge of the density distri-bution within the mantle would constitute a substantialstep toward understanding its dynamics and providesimportant constraints on separating effects due to tem-perature and chemistry(Forte and Mitrovica, 2001).

Although controversial(Resovsky and Ritzwoller,1999b; Masters et al., 2000b; Romanowicz, 2001), aprevious inversion of normal-mode data resulted indensity variations with an unexpected pattern nearthe core-mantle boundary(Ishii and Tromp, 1999).Underneath the Pacific and Africa, where seismicvelocities are anomalously low (features identifiedas superplumes), the density is found to be higherthan average. This observation is consistent with con-vection simulations where chemically distinct, andheavier, material accumulates under regions of activeupwellings with higher temperatures and lower shearvelocities (Christensen, 1984; Davies and Gurnis,1986; Hansen and Yuen, 1988; Tackley, 1998). Thedisadvantage of normal-mode tomography is that onlylarge-scale even-degree structure can be resolved.Nonetheless, the mantle is generally dominated bylarge-scale features, and there are structures with adominant even-degree component, such as the super-plumes. Therefore, a comparison between velocityand density heterogeneity should give us insight intothe petrology and dynamics of these structures.

Thus far, modeling of density variations within themantle has relied upon mantle sensitive modes(Ishiiand Tromp, 1999; Masters et al., 2000b; Kuo andRomanowicz, 2002). Despite their sensitivity to man-tle structure, inner-core sensitive modes have been ig-nored in studies of the mantle, because data from thesemodes exhibit a strong inner-core signature. To over-come this dilemma, we invert mantle and inner-coresensitive modes simultaneously for mantle hetero-geneity and inner-core anisotropy. The results for theinner core are discussed elsewhere(Ishii et al., 2002),and we focus on models of the mantle in this paper.

2. Theory

In a seismic spectrum, peaks associated with thenormal modes of the Earth can be easily observed.Some of them split visibly due to three factors: ro-tation, excess ellipticity, and non-spherical material

properties (e.g., lateral heterogeneity and anisotropy).The first two effects can be determined accurately us-ing the precisely known rotation rate and ellipticity ofthe Earth(Woodhouse and Dahlen, 1978), facilitatingthe isolation of splitting due to internal structure. Thisprocess provides “splitting-function coefficients” forevery mode which combine to produce the “splittingfunction”

σ(r) =∑s=0

s∑t=−s

cstYst(r),

wherer denotes a point on the unit sphere,cst is thesplitting-function coefficient at spherical degrees andorder t, andYst represents fully-normalized sphericalharmonics(Edmonds, 1960). This splitting functionrepresents a radial average of the structure beneath thepoint r as uniquely sampled by a given mode. It isclosely related to surface-wave phase-velocity maps,with the important difference that the splitting functionof an isolated mode is limited to even degrees. This isbecause waves traveling in opposite directions destruc-tively interfere to cancel out the odd-degree signal.

Because a splitting function is a radial average ofthree-dimensional heterogeneity, its coefficients arerelated to internal propertiesδm and topographyδd by

cst =∫ a

0

∑m

δmKms dr +

∑d

δdstKds , (1)

where Ks denotes the degree-dependent sensitivitykernel of a given mode. The first term on the right-handside ofEq. (1) is the volumetric contribution with anintegration over radius from the center to the surface(a) of the Earth. Bothδm andKm

s are dependent on ra-dius and the summation overm represents a sum overmaterial properties, such as lateral variations in veloc-ity or anisotropy. The second term on the right-handside ofEq. (1) is a contribution from topographyδdon various boundaries represented by a sum overd,e.g., the core-mantle boundary and Moho. The sen-sitivity kernel for such undulations,Kd

s , tends to besmall, hence this term is often neglected. Because thetrade-off between density and topography is not sig-nificant (Ishii and Tromp, 2001), we also choose toignore this term in this study.

Previous inferences of density heterogeneity withinthe mantle are based upon mantle sensitive modes(Ishii and Tromp, 1999; Masters et al., 2000b; Kuo and

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M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124 115

Romanowicz, 2002), i.e., modes for which the sensi-tivity kernelsKm

s are zero in the inner core (Fig. 1). Inthis case, the volumetric term inEq. (1) is integratedover the mantle, with densityδρ, shear and compres-sional velocitiesδβ and δα, respectively, as materialpropertiesδm in Eq. (1):

cst =∫ a

b

(δβstKβs + δαstK

αs + δρstK

ρs ) dr,

where b is the radius at the core-mantle boundary.In contrast, inner-core sensitive modes have non-zerosensitivity in the inner core (Fig. 1), and they intro-duce an additional volumetric term associated withinner-core anisotropy,

cst =∫ a

b

(δβstKβs + δαstK

αs + δρstK

ρs ) dr

+∫ c

0

∑γ

δγKγst dr, (2)

wherec is the radius of the inner-core boundary, andδγ represents parameters which describe inner-coreanisotropy(Tromp, 1995). Eq. (2) defines the linearproblem

dm = Gmm, (3)

Fig. 1. Comparison of sensitivity kernels. Sensitivity kernels (K2) for isotropic variations in shear and compressional velocities and densityfor mantle and inner-core sensitive modes. The kernels are for modes1S4, a mantle sensitive mode (left), and6S3, an inner-core sensitivemode (right). Mode1S4 is insensitive to structure within the inner core, but is strongly sensitive to lateral variations in shear velocity(dashed curve) in the lower mantle, compressional velocity (dotted curve) near the surface, and density (solid curve) in the upper mantle.On the other hand,6S3 has a non-zero kernel within the inner core as well as significant sensitivity to heterogeneity within the mantle,especially to density.

wheredm is the data vector containing splitting-functioncoefficients from various modes,Gm a matrix con-taining sensitivity kernels, andm the model vectorwith mantle and inner-core components, i.e.,m =(δβk

st δαkst δρ

kst δγ)T for different values ofs and t (T

denotes the transpose). The additional superscriptk

is the index of the radial basis functions within themantle. In most of this paper, we use Chebyshevpolynomials(Su, 1992)up to order 13 (kmax = 13)as the radial basis function. Using both mantle andinner-core modes, we can simultaneously invert formantle and inner-core structure.

In order to better constrain inner-core anisotropy,we include PKP travel times in our data set. The re-lationship between the anisotropic parametersδγ andtravel times is also linear, i.e.,

dt = Gtm, (4)

where the data vectordt contains travel times andGt

describes their sensitivity to anisotropy in the innercore (seeIshii et al., 2002for a detailed discussion).CombiningEqs. (3) and (4), we have

d = Gm,

whered = (dTm dT

t )T, andG = (GTm GT

t )T. Since thisis generally an under-determined inverse problem,

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weighting and damping are introduced in the inver-sions, a detailed description of which can be foundin Ishii and Tromp (2001)and Ishii et al. (2002).Inner-core sensitive modes are strongly sensitive tothe compressional-velocity structure of the mantle,and we are forced to damp model parameters forcompressional velocity twice as hard as shear veloc-ity or density. Otherwise the compressional-velocitymodel in the mantle acquires a strong zonal patternwhich we know to be erroneous based upon othertomographic models.

3. Data

In the past several years, various groups have mea-sured mode splitting using a multitude of methods(e.g., Tromp and Zanzerkia, 1995; Resovsky andRitzwoller, 1998; Masters et al., 2000a). The currentlyavailable data set includes isolated spheroidal andtoroidal modes, and efforts have been made to deter-mine the splitting of coupled modes(Resovsky andRitzwoller, 1995), which provide valuable informa-tion about the odd-degree structure of the Earth’s inte-rior. The maximum angular degree,s, of the splittingfunction has been extended to degree 12(Ritzwollerand Resovsky, 1995), a three-fold improvementcompared to earlier studies (e.g.,Giardini et al.,1988). From this wealth of data we select splittingcoefficients of isolated modes at degrees 2, 4, and6. We considered higher-degree and coupled-modemeasurements to be insufficient at this time for de-riving reliable mantle models. These selection criteria

Table 1Mode and travel-time inversions

Model χ2/(N − M) χ2/(N − M7) VR (DF) VR (BC) VR (AB) VR (modes) VR (IC)

S 3.3 3.4 84.4 71.3 74.7 89.6 76.5SP 3.0 3.0 83.2 72.2 75.0 91.1 79.9SPR 2.7 2.7 84.4 70.5 74.2 92.0 82.6

Table summarizing the fit for different model parameterizations within the mantle. Inner-core anisotropy is assumed to be uniform.χ2-testsfor the overall fit are given byχ2/(N − M) and χ2/(N − M7), whereN denotes the number of data, andM and M7 are the number ofmodel parameters withkmax = 13 and 7, respectively. VR (DF) is the variance reduction for PKPDF data, VR (BC) the variance reductionfor PKPBC − PKPDF data, and VR (AB) the variance reduction for PKPAB − PKPDF data. VR (modes) is the fit to the entire normal-modedata set; VR (IC) the fit to inner-core sensitive modes. The “S” model involves an inversion for mantle and inner-core structure assumingthat compressional velocity and density are related to shear velocity through scaling. The scaling values areδ ln ρ = 0.2δ ln β andδ ln α = 0.55δ ln β, whereρ is the density,β the shear velocity, andα the compressional velocity. “SP” indicates that data are invertedfor shear- and compressional-velocity structure, but density is a scaled shear-velocity model. Finally, the “SPR” model has shear-velocity,compressional-velocity, and density variations.

reduce the number of modes in our data set to 123spheroidal and 42 toroidal modes, of which 55 areinner-core sensitive. For some modes, measurementsof splitting functions are available from independentresearch groups. Rather than excluding data, we keepmultiple measurements of splitting functions for thesame mode by different groups in our data set, muchthe same way as data for the same ray path are usedin body-wave tomography.

The data set for this study differs from that of ourprevious study(Ishii and Tromp, 2001)in that it incor-porates measurements from the study ofMasters et al.(2000b), which includes surface-wave equivalent aswell as other mantle sensitive modes, and inner-coresensitive modes fromHe and Tromp (1996), Resovskyand Ritzwoller (1998), and Durek and Romanowicz(1999). Before inversion, these data are corrected forcrustal structure using model Crust5.1(Mooney et al.,1998). Because the data set includes splitting coeffi-cients of isolated modes up to and including sphericalharmonic degree 6, corresponding mantle models willconsist only of the even degrees 2, 4, and 6.

The travel-time data included in this study only con-strain inner-core anisotropy, hence we omit a discus-sion of this data set. A detailed description of the dataprocessing applied to the body-wave data can be foundin Ishii et al. (2002).

4. Results

Statistical results of inversions with different pa-rameterizations within the mantle are presented in

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M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124 117

Fig. 2. Models of shear velocity, compressional velocity, density, and bulk-sound velocity within the mantle. Three-dimensional modelsof the mantle plotted in map view at six discrete depths. Blue indicates regions where the relative perturbation is higher than average andred indicates values that are lower than average. The scale is fixed at a saturation level of±1% for all maps. These maps are plottedusing even spherical harmonic degrees 2, 4, and 6, hence some of the common features seen in tomographic maps such as the contrastbetween oceans and continents at shallow depths is not obvious. (Shear Velocity) Comparison of the shear-velocity model derived frommantle and inner-core sensitive modes (left) and SKS12WM13 (right). (Compressional Velocity) Comparison of the compressional-velocitymodel derived from mantle and inner-core sensitive modes (left) and P16B30 (right). (Density) Comparison of density models based uponmantle and inner-core sensitive modes (left) and SPRD6(Ishii and Tromp, 2001)which is constrained only by mantle sensitive modes(right). (Bulk Sound Velocity) Comparison of bulk-sound velocity models derived using shear- and compressional-velocity models fromthis study (left) and from SPRD6 (right).

Table 1. An increase in the number of parametersfrom a shear-velocity only (S) to a shear-velocity,compressional-velocity, and density (SPR) inversionis supported by a systematic decrease inχ2/(N −M),whereN is the number of data andM is the numberof model parameters. In a previous study, this de-crease inχ2/(N −M) is obtained when the maximumradial parameterization (kmax) is 7 (Ishii and Tromp,2001), but the improved modal database supports adecrease inχ2/(N − M) even whenkmax = 13. Thefit to the body-wave data does not change more thana couple of percent, suggesting that additional pa-rameters in the mantle do not trade-off significantlywith inner-core anisotropy. On the other hand, thefit to inner-core sensitive modes improves with anincreased number of mantle parameters, confirmingthat mantle structure makes a substantial contributionto the splitting coefficients of these modes.

In Fig. 2, velocity and density models of the man-tle are shown at discrete depths. These models are

obtained using the following set of starting mod-els: shear-velocity model SKS12WM13(Dziewonskiet al., 1997), compressional-velocity model P16B30(Bolton, 1996), and a zero density model. Corre-sponding statistical results appear inTable 1 underthe model name SPR.Fig. 2 also includes plots ofbulk-sound velocity obtained using the shear- andcompressional-velocity models from the inversion andreference model PREM(Dziewonski and Anderson,1981). Unlike compressional velocity, bulk-soundvelocity has no dependence on shear velocity, whichtends to mask anomalies associated with incompress-ibility in compressional-velocity maps. The modelsobtained from our inversions are not readily compa-rable to existing mantle models because they consistonly of the even degrees 2, 4, and 6. For example,the ocean-continent distribution observed in all to-mographic models near the surface is difficult to seewith only even degrees. We therefore compare ourmodels with degrees 2, 4, and 6 of SKS12WM13,

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P16B30, the density model of SPRD6(Ishii andTromp, 2001), and a bulk-sound velocity model basedupon the shear- and compressional-velocity mod-els of SPRD6. In general, these models are similarto SPRD6, with average correlation coefficients be-tween SPRD6 and the new set of models of 0.9, 0.8,and 0.7 for shear velocity, compressional velocity,and density, respectively. Even though the pattern ofthe density model obtained in this study is similarto that of SPRD6, there are two regions within themiddle mantle, at around 1000 and 1800 km depth,where the correlation is relatively low (0.5 and 0.6,respectively). These are noticable inFig. 2 as slightdifferences in the density distribution pattern in themaps at depths of 1300 and 1800 km. Furthermore,strong anti-correlation of bulk-sound velocity withshear velocity at the base of the mantle is observed, aswell as the regional anti-correlation of shear-velocityand density underneath the central Pacific and Africa.

In a previous study based upon mantle-sensitivemodes, the amplitude of density was highly uncer-tain and the addition of gravity data was required to

Fig. 3. Comparison of the RMS amplitude of the density modelsfrom SPRD6 (Ishii and Tromp, 2001; dashed curve), and this study(solid curve).

Fig. 4. Correlation coefficients between models of shear and com-pressional velocities (solid), shear velocity and density (dotted),and compressional velocity and density (dashed) as a function ofdepth. For the number of free parameters in these models, thecorrelation is statistically significant at the 90% confidence levelif the correlation coefficient is greater than 0.25.

constrain it to a reasonable value(Ishii and Tromp,2001). Here, we do not use gravity data, yet the ampli-tude of the density model is comparable to or smallerthan that of the compressional-velocity model. In par-ticular, unlike the density model of SPRD6, the newmodel has smaller root-mean square (RMS) amplitudenear the core-mantle boundary. As a consequence, theRMS profile as a function of depth contains two peaks:one within the transition zone and another at around2300 km depth (Fig. 3). As we demonstrate in the nextsection, these characteristics are independent of theradial basis functions used in the inversion.

As shown inFig. 4, the correlation between veloc-ities and density is rather poor. Furthermore, sometrends observed previously in SPRD6 are strength-ened, such as the decorrelation of density and veloc-ities in the transition zone. In fact, the correlationcoefficients in these regions suggest that the modelsmay be significantly anti-correlated. In the bulk lower

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M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124 119

Fig. 5. Map views of laterally varying ratios of density-to-shear-velocity (left; R-to-S), density-to-compressional-velocity (center; R-to-P),and Poisson’s ratio (right) at six discrete depths. Blue indicates regions where the relative perturbation is higher than average and redindicates values that are lower than average. Perturbations from the reference ratio as in PREM(Dziewonski and Anderson, 1981)areshown at 0.05, 0.02, and 0.05% levels for the three ratios, respectively.

mantle, the correlations between density and seismicvelocities initially decrease monotonically but flat-ten or increase at the core-mantle boundary. Simpleradially-dependent scaling relations between densityand velocity have been calculated both theoreticallyand experimentally (e.g.,Anderson et al., 1968;Anderson, 1987; Karato, 1993) and have been usedextensively in geodynamic modeling (e.g.,Hager andClayton, 1989; Forte et al., 1994), but their meaningis ambiguous when the models are not highly corre-lated(Ishii and Tromp, 2001). Therefore, we calculatelaterally varying ratios of density-to-shear-velocity,density-to-compressional-velocity, and Poisson’s ratio(Fig. 5). Note that Poisson’s ratio is another represen-tation of the ratio between shear and compressionalvelocities. The ratios are usually dominated by thepattern of models with stronger lateral variations,

although effects due to density variations can be dis-cerned in the density-to-compressional-velocity ratio.Anomalies associated with superplumes clearly standout in these plots.

5. Discussion

The resolution of these mantle models can be ad-dressed most conveniently by looking at the resolutionmatrix of the inversion (seeIshii and Tromp, 2001, forthe definition of the resolution matrix and for variousresolution tests). If the model parameters are perfectlyresolved, i.e., if no damping is used in the inversion,the resolution matrix becomes the identity matrix,therefore deviations of the resolution matrix fromthe identity matrix reflect the effects of damping on

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120 M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124

Fig. 6. Resolution matrix of the SPR inversion when Chebyshev polynomials are used as the radial basis functions. The elements arearranged by model, i.e., shear (S) velocity, compressional (P) velocity, or density, and in increasing Chebyshev polynomial order (left toright). Parts of the matrix corresponding to model parameters at degree 2 order 1 (top), degree 4 order 1 (middle), and degree 6 order 1(bottom).

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M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124 121

model parameters. InFig. 6components of the resolu-tion matrix are shown. Because neither the sensitivitykernels nor the applied damping depend upon angularorder t, the resolution for different values oft withinthe same angular degree is nearly identical. The ef-fect of stronger damping for higher-order Chebyshevpolynomials is evident, because higher-order modelcoefficients are poorly resolved (diagonal elementswith high values ofk are close to zero). In addition,stronger damping on higher spherical harmonic de-grees and the compressional-velocity models manifestthemselves as poorer resolution of the related param-eters.Fig. 6 also shows that there does not appearto be substantial leakage from model to model (indi-cated by smaller off-diagonal elements in comparisonto diagonal components).

To address the effects of off-diagonal terms associ-ated with density model parameters and to understandthe full effects of damping and the starting model, weperform a resolution test. In this test, splitting-functioncoefficients of all modes used in the inversion are cal-culated using shear- and compressional-wave modelsSKS12WM13 and P16B30, respectively, a zero modelfor density, and an inner-core model described inIshiiet al. (2002). These synthetic data (without noise) areinverted with the same damping, weighting, and start-ing models as in inversions based upon the real data.This test focuses on the leakage of power from veloc-ity heterogeneity to density, because any density vari-ations observed in the retrieved model are due to theinversion process. The RMS amplitudes and correla-tion between the two density models obtained fromthe real and synthetic data are compared inFig. 7.The influence of the velocity models on the density issmall, and the density distribution obtained from thisresolution test is considerably different from that pre-sented inFig. 2. Both the RMS amplitude and corre-lation indicate that the depths affected most stronglyby velocity structure are between 2000 and 2500 km.The weak trade-off between the velocity models anddensity is in contrast with a similar analysis byKuoand Romanowicz (2002), who used less than 20% ofthe modes considered in our study.

We assume in our analysis that the mantle exhibitsonly isotropic variations. In the bulk mantle, this as-sumption is reasonable given that most modes havekernels with broad sensitivity in radius. However, thisis not a valid assumption in the upper mantle, where

Fig. 7. Resolution test for density model. Comparison of densitymodels obtained from the inversion of data and from a resolutiontest where shear and compressional velocity models are used asthe input models. The RMS amplitudes of the density model (solidcurve) and that obtained from the resolution test (dashed curve)are shown in the left panel, and the correlation between the twomodels is shown in the right panel.

there is evidence of significant anisotropic variations(e.g.,Ekström and Dziewonski, 1998). To investigatethe effects of anisotropy near the Earth’s surface, weperform an inversion without the splitting-function co-efficients of surface-wave equivalent modes. Two al-terations to the models are observed when we use thissubset of the data. The first is a change in pattern.When correlation coefficients are calculated betweenmodels with and without theMasters et al. (2000b)data, they are smaller near the surface, as expected.The minimum correlation coefficient is observed forthe density models and occurs around 150 km depth.However, it is still large at 0.83, so the effect of pat-tern modification due to the omission of the surfacewaves is not very significant, especially in the lowermantle. The second effect is related to the amplitudeof the models. Because we have excluded a signifi-cant number of data, the constraints on amplitude areweaker and hence the resulting mantle models haveamplitudes that are closer to those ofIshii and Tromp(2001).

The choice of Chebyshev polynomials as the radialbasis function is somewhat arbitrary, although their

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Fig. 8. Comparison of density models based upon various radial basis functions. Density models using layers (left), Chebyshev polynomials(center), and cubic B-splines (right) at six discrete depths. Blue indicates regions where the relative perturbation is higher than averageand red indicates values that are lower than average. The scale is fixed at a saturation level of±0.5% for all maps.

gradual variation is consistent with the smoothly vary-ing sensitivity kernels. The disadvantage of global ba-sis functions, such as Chebyshev polynomials, is thatthe termination of polynomials at some order (in ourcase,kmax = 13) can lead to structure due to “ringing”near the end points (i.e., near the surface and thecore-mantle boundary). In what follows, we presentresults for density models using two local basis func-tions: layers and cubic B-splines(de Boor, 1978). Inthese inversions, the number of unknowns in the radialdirection is kept constant (i.e.,kmax = 13) and bothlayers and B-spline knots are spaced evenly through-out the mantle. It should be remembered that dampingin the radial direction has an entirely different meaningfor local and global basis functions. Therefore, damp-ing parameters in the radial direction are chosen so thatthe traces of the resolution matrices (i.e., the number

of resolved model parameters) are similar, while keep-ing damping in the lateral directions the same. Thefits to data with local basis functions are then similarto that obtained with Chebyshev polynomials. In gen-eral, the observed patterns in the density distributionare compatible between models with different radialbases, including features near the core-mantle bound-ary (Fig. 8). The models are least correlated near thesurface and the core-mantle boundary, as expected,but the correlation coefficients remain well above the95% significance level. Furthermore, the amplitudesof these models are relatively consistent with one an-other (Fig. 9). The two peaks in RMS amplitude in thetransition zone and around 2300 km depth are presentregardless of the choice of basis function or dampingscheme, implying that they are robust features and notthe results of “ringing”.

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M. Ishii, J. Tromp / Physics of the Earth and Planetary Interiors 146 (2004) 113–124 123

Fig. 9. Comparison of the RMS amplitude of density based uponvarious radial basis functions. Plot of the RMS amplitudes ofdensity models based upon Chebyshev polynomials (grey curve),cubic B-splines (solid black curve), and layers (dashed blackcurve).

6. Conclusions

We present three-dimensional models of shear ve-locity, compressional velocity, and density withinthe mantle based upon a mode data set that in-cludes inner-core sensitive free oscillations. Thelow-frequency inner-core modes possess substantialsensitivity to the mantle, and we simultaneously invertfor mantle heterogeneity and inner-core anisotropy.The mantle models obtained from this inversion donot trade-off significantly with inner-core anisotropy,and the improved database supports inversions forshear-velocity, compressional-velocity, and den-sity structure with a maximum radial expansion oforder 13. The models obtained from our inversionsexhibit a similar distribution of positive and negativeanomalies compared to our previous study, includingthe regional anti-correlation of velocities and den-sity near the core-mantle boundary. Analysis of the

resolution matrix indicates that the model parame-ters are well-resolved, with limited leakage betweenmodels.

Unlike our previous inversions, where the ampli-tude of the density model needed to be constrainedby gravity data, the amplitude of the new densitymodel is stable using only the modal data set. Thedensity RMS amplitude is generally smaller than thatof the velocity models, and exhibits two peaks around600 and 2300 km depth. These depths coincide withpoor or negative correlations between density andvelocities. Experiments with local radial basis func-tions demonstrate that the two maxima in the densityRMS amplitude are robust, and that the pattern of thedensity distribution does not depend on the choiceof the basis or damping scheme. The presence ofheavier material at the locations of superplumes sug-gests that these large-scale anomalies differ from theambient mantle not only in temperature but also inchemistry.

Acknowledgements

M.I. was supported by a Julie Payette ResearchScholarship from the Natural Sciences and Engi-neering Research Council of Canada. ContributionNumber 8887, Caltech Division of Geological andPlanetary Sciences.

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