Xinlei Sun - Tomographic Inversion for Three-dimensional Anisotrophy of Earth's Inner Core

Physics of the Earth and Planetary Interiors 167 (2008) 5370

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Physics of the Earth and Planetary Interiors

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Tomographic inversion for three-dimensional anisotropy of Earths inner core

Xinlei Sun , Xiaodong SongGeology Department, University of Illinois at Urbana-Champaign, United States

a r t i c l e i n f o

Article history:Received 29 August 2007Received in revised form 23 December 2007Accepted 11 February 2008

Keywords:Three-dimensional inner core anisotropyInner inner coreThree-dimensional ray tracing

a b s t r a c t

Seismological studies generally suggest that the Earths inner core is anisotropic and the anisotropicstructure changes signicantly both laterally and with depth. Previous body-wave studies of the innercore have relied on ray tracing or waveform modeling using onepresent non-linear tomographic inversions of the inner core anisoray tracing, spline parameterization, and a large collection of PKPpseudo-bending ray tracing (PBR)method in spherical coordinates fo

contis minerogegraphodelreatengitu

B), whabochan

about 150km) at the radius of about 600km, slightly less than half of the inner core radius, forming adistinct inner inner core (IIC). The velocity in the IIC hasmaximums at equatorial and polar directions andminimum at an angle of about 40 from the equatorial plane. The velocity in the outer inner core (OIC),however, changes little for ray directions 040 from the equatorial plane. (3) Despite large variation ofthe anisotropy, the isotropic velocity (Voigt average) throughout the inner core is nearly uniform. The

1. Introduction

The Earths inner core is impolution of the Earth and the geodymagnetic eld. It is dominated baxis aligned nearly NS (e.g., More1986; Creager, 1992; Song and HShearer, 1994). In last decade or s

Corresponding author at: Geology DepChampaign, 245 NHB, 1301 West Green S

E-mail address: [email protected] (X. Sun

0031-9201/$ see front matter 2008 Eldoi:10.1016/j.pepi.2008.02.011results suggest that the OIC is likely composed of the same type of iron crystals with uniform chemistry,but the IIC may be composed of a different type of crystal alignment, a different iron phase, or a differentchemical composition. Our tests onmodel parameterization,mantle correction, and linear and non-linearinversion suggest themain features of our model are very robust. However, ne scale structures are likelyto differ, particularly in the major transition zones, e.g., in the topmost QWH (isotropy to anisotropy),between OIC and IIC (change in the form of anisotropy), and between QEH and QWH in OIC (differencein anisotropy strength). Searches for possible waveform complications from these boundaries need to beaware of the directional dependence and geographical variation to be successful.

2008 Elsevier B.V. All rights reserved.

rtant in understanding the evo-namo responsible for the Earthsy cylindrical anisotropy with fastlli et al., 1986; Woodhouse et al.,elmberger, 1993a; Tromp, 1993;o, increased studies have shown

artment, University of Illinois at Urbana-treet, Urbana, IL 61801, United States.).

many interesting features of the inner core and complexities of theinner core anisotropy. The topmost of the inner core (top 100km orso) is nearly isotropic with anisotropy less than 1% (Shearer, 1994;Song and Helmberger, 1995a). Between the upper isotropic and thelower anisotropic regions, a transition zonewas suggested, and thedepth of this transition may be quite different in the two hemi-spheres (Song and Helmberger, 1998). In the deeper part, stronganisotropy (2.53% on average) seems to persist to the center of theinner core (Vinnik et al., 1994; Song, 1996; Sun and Song, 2002).The inner core attenuation also appears to change with depth(Song and Helmberger, 1995a; Cormier et al., 1998; Garcia, 2002a;Yu and Wen, 2006; Cao et al., 2007). The lateral variation of theinner core structure is just as pronounced. An important observa-

sevier B.V. All rights reserved.core (PKP(DF) phase). The method iteratively perturbs each disof the ray through 3D earth structure so that its travel time iof the inner core is approximated to the rst order as 3D hetgiven ray. The data are corrected using a scaled mantle tomomodel obtained has the following major features. (1) The mvariation. The isotropic velocity in the topmost inner core is g(40160E) than in quasi-western hemisphere (QWH) (other loto the depth of 600700km below the inner core boundary (ICat much shallower depth (about 100200km below the ICB) tothroughout the rest of the inner core. (2) The anisotropy form-dimensional (1D) models. Here wetropy using three-dimensional (3D)differential travel times. We adapt ar seismic rays that traverse the innernuity point and continuous segmentimum. The 3D anisotropic structureneous (but isotropic) structure for aic model. The inner core anisotropyhas strong hemispherical and depthr in quasi-eastern hemisphere (QEH)des). The anisotropy is weak in QEHile in QWH, the anisotropy increasesut 34%, then remains at about 24%ges abruptly (over a depth range of

54 X. Sun, X. Song / Physics of the Earth and Planetary Interiors 167 (2008) 5370

tion is the hemispherical variation in seismic anisotropy (Tanakaand Hamaguchi, 1997; Creager, 1999), in isotropic velocity of thetopmost inner core (Tanaka and Hamaguchi, 1997; Niu and Wen,2001; Cao and Romanowicz, 2004), and in attenuation (Cao andRomanowicz, 2004; Yu and Wen, 2006). Lateral variation appearssignicant even at the ne scale of a few km to cause strong scat-tering (Vidale and Earle, 2000; Cormier and Li, 2002; Koper et al.,2004). More recently, Ishii and Dziewonski (2002) showed thatthere may be a distinct inner most inner core (IMIC), which hasminimum velocity at about 45 from Earths rotation axis and aradius of about 300km. Evidence for a possible anisotropy changewas subsequently reported (Beghein and Trampert, 2003; Sun andSong, 2004; Cormier and Stroujkova, 2005; Calvet et al., 2006; Caoand Romanowicz, 2007) at perhaps a larger radius of 400500km.A notable exception to the above studies is the work by Ishii etal. (2002a,b), which favors a model of constant anisotropy for thewhole inner core.

The lateral variability of the inner core has provided a markerfor the detection of the inner core rotation (e.g., Song and Richards,1996; Creager, 1997; Vidale et al., 2000; Laske and Masters, 2003;Zhang et al., 2005; Sun et al., 2006; Song and Poupinet, 2007). Therecent study by Zhang et al. (2005) provided strong support for aninner core motion, based on direct comparison of seismic wavesfrom earthquake doublets (identical in location and source mech-anisms, producing identical wave shapes). Using one of the bestdoublets, Wen (2006) and Cao et al. (2007) reported evidence forsignicant topography of the inner core boundary (ICB) or localizedrapid changes of the ICB.With a larger set of 14waveform doublets,Song and Dai (submitted for publication) conrmed the observa-tion and suggested ICB topography of a few kilometers on local toregional scales. Regions with large topography (beneath Africa andCentral America) coincide with large anomalies at the core-mantle

Fig. 1. PKP ray paths and travel-time curtravel timesbetweenPKP(DF) andother thfrom 130 to 180 are used in this study.

boundary (CMB),whichmay suggest strong thermal coupling of themantle and the core.

Seismic tomographyof the3Danisotropyof thewhole inner corehas not been carried out, although there have been some attemptsto map both the depth and lateral variation of the anisotropy of thewhole inner core. Using normal-mode and differential PKP travel-time data, Romanowicz et al. (1996) and Durek and Romanowicz(1999) inverted for models of axisymmetric inner core anisotropythat include latitudinal anddepthvariations.A fewstudies exploredbody-wavePKP travel times, includingarrival times fromthe ISC (SuandDziewonski, 1995), hand-measured differential times (Creager,1999), or the combination of both (Garcia and Souriau, 2000).Mod-els are parameterized approximately. At the radial direction, theinner core is divided into three or four layers, or, the data at dif-ferent distances are divided into groups to represent roughly theanisotropy at different depth. At the longitudinal direction, the bot-toming point longitude of a ray or the longitude of a cylindricalstacking (Su and Dziewonski, 1995) is used to represent the lateralsample of the ray.

In this study, we attempt to mthe inner core using seismic toma large collection of differential trother threebranches (AB, BCandCa 3D ray tracing method for theof apparent large variation of themay depart signicantly from thoparameterize our models using cuand linear interpolation in the lsion is done iteratively by retracinmedia. This paper discusses the deimportant features of our nal mpaper, which is used to map the t(Sun and Song, submitted for pub

2. Data

We dene the ray angle () ainner core with the equatorial plaEarths rotation would be = 90with < 50 as equatorial pathspaths. We also dene quasi-easply eastern hemisphere as longitinner core as quasi-western hemern hemisphere. The division is ghemispherical variation of the seiduction above.

Thedataweused are PKPdifferand CDDF. Because of the care referential time measurements mannot driven by the goal of obtaininments. Rather, we have focused s

ray dare miffer

nett et, 197the bSongt of Pmay1D rinnerdy. Ter yeong,ves for a 1D reference model. Differentialreebranches (AB, BCandCD)of PKPwaves

spatial coverage at differenting polar path data, whichanisotropy. Residuals of the drelative toAK135model (Kention (Dziewonski and Gilberthat the velocity gradient at(Souriau andPoupinet, 1991;1999), as in AK135, than tha1981), although the gradientet al., 2005). The choice of aaverage radial model of theanisotropy pattern of this stuthatwehave accumulated ov1995a; Song, 1996; Sun and Sap the 3D anisotropic structure ofography techniques. Our data areavel times between PKP(DF) andD)of PKPwaves (Fig. 1).Weadaptanisotropic inner core. Becauseanisotropy, the actual ray pathsse for a 1D reference model. Webic splines in the radial directionongitudinal direction. Our inver-g the rays through 3D anisotropictails of the inversion. Someof theodel are discussed in a separateexturing of iron in the inner corelication).

s the angle of the DF ray in thene; the angle of the ray with the (in degree). We refer pathsand those with > 50 as polartern hemisphere (QEH) or sim-udes 40160 and the rest of theisphere (QWH) or simply west-uided by previous studies of thesmic properties cited in the intro-

ential travel timesABDF, BCDF,quired to obtain high-quality dif-ually, our data selection effort is

g the largest amount of measure-trategically on obtaining uniformirections, particularly on obtain-ost sensitive to the inner core

ential travel times are calculatedtal., 1995)withellipticity correc-6). Some studies have suggestedottom of the outer core is lowerandHelmberger, 1995b; Creager,REM (Dziewonski and Anderson,also vary with hemispheres (Yueference model would affect thecore, but does not affect the 3Dhese data include measurementsars (Song andHelmberger, 1993a,2002). Ourmore recentmeasure-

X. Sun, X. Song / Physics of the Earth and Planetary Interiors 167 (2008) 5370 55

ments are obtained using waveforms from the Data ManagementCenter of Incorporated Research Institutions for Seismology (IRIS)up to 2006 and from the new China Seismological Network (CSN),a Chinese national backbone network installed in the recent years.The differential travel times of PKP aremeasured using cross corre-lation. Only data with cross correlation coefcient greater than 0.5are kept. Other data include those from previous studies (Poupinetet al., 1993; Vinnik et al., 1994; Tanaka and Hamaguchi, 1997;Creager, 1999; Niu and Wen, 2001). We nally have a total of 4439differential travel-timemeasurements from 2065 earthquakes and582 stations. The total numbers of equatorial and polar paths are3448 and 991, respectively.

Fig. 2 shows all the PKP differential time residuals we used.These raw data show some important features, which serve asan excellent guide for the verication and interpretation of theinversion model. Most of these features have been identied fromregional studies asmentioned in the introduction. (1) Largepositiveresiduals are populated in polar regions (large or small values),indicating a dominant cylindrical anisotropy with the fast axis

along NS direction. (2) At small distances (


Fig. 3. ResidualsofABDFandBCDFdiffesphere (QEH) and quasi-western hemisphsolid diamonds are polar data, and the greferent behaviors of the polar and equatotwo hemispheres, suggesting the anisotro

larger than the equatorial ones acating weak anisotropy in the upwestern hemisphere, even at theare clearly larger thanequatorial ostarts at shallow depth.

PKP differential travel time hearthquake origin time errors andquake location errors and strongmantle because the ray paths ofclose down to mid-mantle. In addanomalies of PKP differential trapolar directions are mostly fromof mantle heterogeneity, althougequatorial directions (Sun and So

However, using the differentbacks. First, differential ABDF titimes, can be affected by lowermHelmberger, 1997; Breger et al., 2far away fromeachother at the costrongheterogeneities havebeenof the distribution of earthquakeis limited, especially for polar dirpling the same regionmay dominfrom SSI earthquakes to stationsmake the inversion unstable or bdata set is relatively small.

Resolution of seismic tomograsteadily over the years (see recenRecent direct comparison of PKtions from S. Grands S tomograpshows good correlation betweentain regions (Sun et al., 2007; Zha subset of the observed PKP ditions for the most recent MIT P t(submitted for publication). The m

P differential time residuals and mantle modele most recent P-wave tomographic model fromon). The data that are usedhere to comparewithDFdifferential times fromequatorial paths (raythan 170 (a total of 2134 measurements). Then this study. Polar paths and antipodal data arem inner core anisotropy. (a) Observed residualsof AB or BC rays at the core-mantle boundary.

rallymean slowand fast anomalies, respectively.ed residuals for the MIT model. Symbols are inesiduals vs. predictions. The observed residualswell with cross-correlation coefcient of 0.65.rential travel times inquasi-easternhemi-ere (QWH) as a function of distance. They circles are equatorial data. Note the dif-rial paths as a function of distance in thepy changes with depth and laterally.

t distances less than 160, indi-per part of the inner core. In thedistance of 145, the polar datanes, suggesting stronganisotropy

as its advantages: it eliminatesreduces the inuences of earth-

heterogeneity in crust and upperdifferent PKP branches are veryition, it has been shown that thevel times at larger distances forthe inner core anisotropy insteadh this is not true for some of theng, 2002).ial times also has some draw-mes and, to a less degree, BCDFost mantle structures (Song and000). The AB and DF branches areremantle boundary (CMB),whereproved to exist. Secondly, becauses and stations, the data coverageections. A subset of the rays sam-ate at certain ray angles, e.g., raysin Alaska and Canada, which may

Fig. 4. Comparison of observed PKpredictions. The mantle model is thMIT (Li et al., submitted for publicatithemantlemodel areABDF andBCangles < 40) with distances lessdata are a subset of the data used iexcluded to avoid contamination froplotted at the entry and exit pointsPositive andnegative residuals gene(b) Same as (a), but for the predictthe same scale as (a). (c) Observed rcorrelate with the predictions veryThe regression has a slope of 1.28.iased, particularly when the total

phy of the mantle has increasedt review by Romanowicz, 2003).P differential times and predic-hic model is encouraging, whichthe data and predictions in cer-eng et al., 2007). Fig. 4 comparesfferential times with the predic-omographic model from Li et al.antle predictions are calculated

based on 1D rays in the AK135 mof 2134 measurements) is ABDFequatorial paths with distance lesat antipodal distance (>170) arefrom the inner core anisotropy aCDDF data set is excluded becasimilar throughout themantle tharesolved by the current generatioThe observations and predictionscorrelation coefcient of 0.65 (Fidata with respect to the predictioodel. The subset of data (a totaland BCDF differential times fors than 170. Polar paths and dataexcluded to reduce the inuencend possible IMIC structure. Theuse the CD and DF paths are sot anymantle inuence cannot ben of mantle tomographic models.correlate rather well with cross-g. 4). The linear regression of thens yields a scaling factor of 1.28,


i.e., themodel under-predicts the data by about 28%on average. Thevariance reduction of these equatorial data is about 43%, after cor-recting for themantlemodelwith the scaling factor.We thus correctall of our PKP differential times using the MIT mantle model andthe scaling factor (1.28) and regard the corrected data set as the rawdata for the inversion of the inner core structure. This correction byno means removes all the mantle inuence. However, because ofthe good correlation of the data and the predictions, we believe thecorrection reduces signicantly the inuence of large-scale anoma-lies, such as the Central Pacic and the African Super plumes andthe circum Pacic high-velocity anomalies.

To reduce the inuence of uneven sampling of the data, weuse summary paths for the inversion. We sort all the paths intogroups. For each group, all the events are within a certain dis-tance from each other and all the stations are also within a certaindistance from each other. We then select the path which has themedian of all the residuals as the summary path of the group. Theselection strategy and algorithm are similar to that described inLiang et al. (2004), which is modied to apply not only to earth-quakesbut also to stations. Thegehas been used previously with reinner core (e.g., Shearer, 1994) orthe ray direction in the inner corchoose 200 kmas the distance ranstations, respectively. The summaistics of the distribution of the rawdata redundancy. We obtain a totWhile the coverage is quite unifopaths, the polar-path coverage is sof 200400 km below the ICB isthe central part of the inner core

3. Inversion method

In this section, we describedincluding 3D ray tracing, paramelation. We also show a syntheticand resolution of the method.

3.1. 3D ray tracing

Previous body-wave studies oftracing or waveform modeling usdepth changes of anisotropy are ucan deviate greatly from those bmodel (Ouzounis and Creager, 20samples is from SSI events to statwaveform modeling suggests a rthe top of the inner core to largeabout 200250 km (Song and H2002). An inversion based on 1D r50150km with smaller anisotro2001).

Here we modify a pseudo-beproposed by Koketsu and Sekinthrough 3D isotropic media in sphas been applied to regional cruWidiyantoro et al., 2000). A Cartproposed earlier by Um and Thuinclude discontinuities by Zhao etshown in theowchart in Fig. 6. Othe source to the center of the Earto the station. The initial path indiscontinuity points (intersecting

ore see coveragekm. Tt samp. The s

a continuity point (e.g., the midpoint)points needed to perturb the ray seg-The method iteratively perturbs each

inuity and in the continuous mediumminimum. For the continuous points,ich essentially perturb each point inocal velocity gradient. Our implemen-me in calculating thevelocity gradientg. 6). A large volume is included in cal-t initially to nd the global minimum,ing the ray is used eventually to obtaindient that is sampled by the ray. Theimum travel time between the sourcecontinuity points, we use a bisectionthe discontinuities so that Snells Law

mentation is very stable, reliable, and00 rays for which we have differentialts using both the PBR method and an1D model (AK135). The travel-time

thods is generally within 0.05 s withrgest path difference is within 24kmneral approachof summary raysspect to the turning point in thewith respect to the distance ande (Su and Dziewonski, 1995). Wege for the grouping of events andry paths preserve the character-data while greatly reducing the

al of 1673 summary rays (Fig. 5).rm for both equatorial and polartill quite sparse. The depth rangebest covered and the coverage ofis particularly poor.

in detail our inversion process,terization, and inversion formu-test to demonstrate the validity

the inner core have relied on raying 1D models. If the lateral andp to several percent, the ray pathsased on a 1D isotropic reference01). One of the most anomalousions in Canada and Alaska, whereapid transition from isotropy atanisotropy of about 78% belowelmberger, 1998; Song and Xu,ay tracing places the boundary atpy below (Ouzounis and Creager,

nding ray tracing (PBR) methode (1998) for seismic ray tracingherical coordinates. This methodst and mantle tomography (e.g.,esian version of the method wasrber (1987) and was extended toal. (1992). Our basic algorithm isur initial ray is a straight line fromth and another straight line backcludes these end points, all thepoints with the discontinuities

Fig. 5. Path coverage of the inner cdata (a) and polar data (b). While thand polar paths, the polar-path covof the summary rays for every 200rays are plotted separately. The besfor both equatorial and polar pathscore is still poor along polar paths.

in the reference model), andbetween every two adjacentment in the continuous part.point of the ray at a discontso that the total travel time iswe use the PBR method, whthe direction normal to the ltation includes aexible scheneeded to perturb the ray (Ficulating the velocity gradienbut a small volume surroundthe precise local velocity granal ray is the one with minand the station. For the dismethod to nd the points atis satised.

Tests show that our implefast. We have traced over 40PKP travel-time measuremenexact shooting method for adifference from the two mea few up to 0.07 s and the lagments of summary rays from equatorialerage is quite uniform for both equatorialis still quite sparse. (c) Depth distributionhe number of total, equatorial, and polarling region is 200400km below the ICBampling for the central part of the inner


Fig. 6. Flow chart of our 3D pseudo-bendis discontinuity point (where there is a vel(where there is not a velocity discontinuioriginal perturbation length, Lmin is minimThe perturbation length is a critical paramvelocity gradient in the vicinity of the poinwe also include an enhancement factor (Uvary tomake the computationmore stablesegment length (S) between the two poinand Lmin is set to S/8. In step 4 or 9, the imof the travel time of the whole path (orprecision from each iteration is about 0.01

(Fig. 7). Even with a model of stroover thedepth rangeof 200300 kdifference is still less than 0.08 swithin 40km (Fig. 8).

The importanceof3Dray tracinthe PBR ray tracing and a hypothincrease over a 200 km depth tra150 turns deeper by 162km (lefthan what is expected from a 1Dlocations are shallow enough so tdeeper depth. For a ray sampling tbecause the velocity increase occthis distance. Fig. 9 b compares tlowermost mantle and the corethe hypothetical inner core. The dthe CMB is 280km for the hypothdo not recalculate the mantle cor

The bending method is valid foinner core is anisotropic. Fortunacore does not changemuch (withi

ethodTravel.03 s aexactl the, indichaning ray tracing (PBR) algorithm, where DPocity discontinuity), CP is continuity pointty), L is current perturbation length, L0 isum perturbation length, and p is 0.001%.eter used in the PBR method to estimatet to be perturbed. In our implementation,

Fig. 7. Ray tracing using the PBR mmodel is a 1D model (AK135). (a)The maximum difference is about 0two methods. Solid line is from anfrom the bending method. In generamaximum difference is about 24kmthe right ray. (c) Maximum ray anglemand Thurber, 1987), which is allowed to. In step 1, L and L0 are set to one half of thets adjacent to the point to be perturbed,provement is measured as a percentageabout 11001200 s for PKP(DF)), i.e. thes.

ng velocity gradient (5% increasembelow the ICB), the travel-timeand the largest path difference is

g isdemonstrated inFig. 9ausingetical model with a 5% velocitynsition. The ray path to distancet side) or 339km (top or bottom)ray. The velocity changes at thesehat the shortest time path is at ahe right side, there is no differenturs too deep to affect the ray forhe 3D ray and the 1D ray in thethat sample the top structure ofifference of the 3D and 1D rays atetical model. In our inversion, werections using the 3D rays.r 3D but isotropic media, yet the

tely, the ray direction in the innern 10) evenwith a strong velocity

in the inner core increases with depth, thchange much. The maximum change in ththat when tracing rays in 3D anisotropicstructure for a given ray as approximation

gradient in the inner core (Figs. 7structure of the inner core can be3D heterogeneous (but isotropic)3D anisotropy is known from prev3D isotropic model can be constrtracing can be carried out. This aimportant keys to our success in ithe inner core.

3.2. Parameterization

Our task is to invert the P wacore. This requires not only the sptional coverage of seismic rays forof theparameterization is guidededge, and tests on resolution andparameterization (for anisotropiradial direction, we assume uniforadius of 300 km. The assumptionconditions approaching the centeand the data coverage at antipodaand an exact shooting method. The test-time differences between two methods.t 150 . (b) Bottoming depth of rays fromshooting method, and the dashed line istwo lines match each other very well. Thecating the pseudo-bending code can ndge in the inner core. Although the velocitye ray direction in the inner core doesnt

is model is about 1.1 at 160 , suggestinginner core, we can use 3D heterogeneity.

and 8). Thus, the 3D anisotropicapproximated to the rst order asstructure for a given ray. Once theious iteration, the correspondingucted for that ray and the 3D raypproximation is one of the mostnverting for the 3D anisotropy of

ve anisotropy in the whole inneratial coverage but also the direc-the whole inner core. Our choice

by thedata coverage, prior knowl-stability. Fig. 10 shows our modelc coefcients, see below). In therm anisotropy for an IMIC with ais driven by the fact that physicalr of the Earth collapse to a pointl distance is sparse.


Fig. 8. Same as Fig. 7 but for a model withThe increase starts at 200km below the ICto AK135. Note that the maximum travemaximum ray angle change in the innerthe strong gradient region. Even in suchapproximation for a given ray in anisotrop

We represent the rest of the inseven knots for each parameter (i150km). In the longitudinal directer using linear interpolation wit220 (or 140W), and 340 (or 20

one knot in the center of the QEHsent possible hemispherical pattour limited spatial and directionadence is included in this study,suggested latitudinal dependenceand Romanowicz, 1999). Thus ournates but two-dimensional in sph

3.3. Inversion formulation

The anisotropy of the inner co1997):

v

v= + cos2 + sin2 2,

where is the angle of the seism90 ), is the velocity perturbperturbation at polar direction (tand contributes to the anisotropparameters and are related tC11)/2C11 and = (4C44 + 2C13

Each anisotropy coefcient, ear interpolation in longitudina

model (AK135) and a 3D inner core model3D model is based on AK135 and includes anocity centered at (200km, 0, 0). The velocitym transition that starts at the depth of 100km00km in the right side of this cross section. (a)e 150 that sample four representative regions.n AK135 model and the 3D model, respectively.er mantle and the core for the 1D and 3D raysre of the 3D ray from the 1D ray is about 280km.a velocity increase inside the inner core.B to 5% at 300km below the ICB, relative

l-time difference is about 0.08 s, and thecore is about 8 when the ray samplesa model, the isotropic 3D heterogeneityic structure is still possible to rst order.

ner core using cubic splines with.e., with a radial spacing of abouttion, we represent each parame-h three knots at longitudes 100,W), respectively. This choice putsand two points in QWH, to repre-erns discussed above. Because ofl coverage, no latitudinal depen-although previous studies have

Fig. 9. Comparison of rays in a 1Dfrom the ray bending method. Theoffset inner sphere of increased velincreases from 0% to 5% over a 200kbelow the ICB in the left side and 5The rays in this gure are at distancThe dashed and solid lines are rays i(b) Display of ray paths in the lowshown at the top of (a). The departuat the CMB in this hypothetical case(Romanowicz et al., 1996; Durekmodel is 3D in Cartesian coordi-erical coordinates.

re can be expressed as (e.g., Song,

(1)

ic ray from the spin axis (i.e. =ation at equatorial plane, is thehe amplitude of the anisotropy),y in the intermediate angles. Theo elastic constants by: = (C33 C11 C33)/8C11., , or is represented by lin-

l direction and by cubic splines

Fig. 10. Schematic diagram of our model parameterization. This is a view fromthe North Pole. The outer circle is the ICB; the numbers labeled at the outer circleare longitudes. Our model includes an innermost inner core of uniform anisotropy(inner circle) with radius of 300km. At larger radii, each anisotropy coefcient(, , or ) is represented in radial direction by cubic splines with seven equallyspaced knots (knot spacing of about 150km) and in longitudinal direction by linearinterpolation of three equally spaced knots at longitudes 100 , 220 (or 140W),and 340 (or 20W), respectively.


(Michelini and Mcevilly, 1991) in radial direction as discussedabove. The knots have equal distance from each other (Fig. 10). Sup-pose the linear bases are Li(x), cubic spline bases are i(x), where idenotes the ith basis and x is location variable, then the anisotropycoefcient anywhere in the inner core can be calculated below

y(,, r) =

aijLi()j(r), (2)

where y is , , or ; , , and r are colatitude, longitude, and radius,respectively; and aij is the knot coefcient to be tted at the gridi (longitudinal) and j (radial). Once all knot coefcients (aij) areknown, the anisotropy coefcients can be calculated, so can theanisotropy.

The inversion problem can be expressed as

t =

taij

aij, (3)

where t is the PKP(DF) travel-time residual, thus t is thecorresponding PKP differential travel-time residual; and aij isperturbation to knot coefcient aij .

Since

t =

dlv

, (4)

t =

v

vdt, (5)

we have

taij

= (t)aij

=

( vv )aij

dt = f

Li()j(r) dt, (6)

where f is dened as

f ={

1 for cos2 for sin2 2 for

(7)

Thus from Eq. (3), we have a linear system:

di = Gijmj, (8)where di is differential travel-time residual for the ith ray, mj is thejth model parameter (perturbation to knot coefcient), and Gij isthe kernel matrix dened by Eq. (6).

Fig. 11. Synthetic test of the inversion. The input model includes depth and lateral variations in the anisotropic coefcients that resemble some main features of the realinversion. The data are synthesized using to thparameters (gray lines) and inversion resu nd 340of interpolated input parameters ( (left) iewedfor inversion results. (b) and (c) use the sathe ray bending code, then a Gaussian noise with 0.1 s standard deviation is addedlts (black lines) as a function of radius) at the three longitudinal knots 100 , 220 , a, (center) and (right)) as a function of radius and longitude for the inner core, vme gray scale for each parameter (bottom).e synthetic data. (a) Comparison of input from left to right, respectively. (b) Mapsfrom the north pole. (c) Same as (b), but


Fig. 12. Inversion results of the real data i tion of radius from the Earths center to the ICB.(b) Maps (polar views) of interpolated val vely.

Our inversion also includes wconstraints on model parametersdata is not uniformthere are mpolar paths. We divide the whoangle into 10 bins and obtain tNi, i = 1, . . . , 9. We then weighWj =

N1/Nj , where j = int(/1

inversion more stable.We also impose constraints on

) at the ICB: we set them to zestudies which suggest weak anisas discussion above. The constrainlowing. First, we have a small amoWen, 2001) sampling the topmosnear the ICB then can be easily cotances. Secondly, our model parahas limited resolution. Even if wedata, the solution for the ICB wbelow.

We use the LSQR algorithm (Pafor the anisotropy coefcients iter(AK135). After each iteration, we uthe anisotropy coefcients pertuiteration. The rays are retraced usresiduals of the data with respectto invert for new anisotropy coefthe 6th iteration results as the nalready stable at 4th iteration.

3.4. Synthetic test

Fig. 11 shows the result of oneusing the samedata set, parameteThe input model (gray lines) incl

hat resemble our nal model (see dis-data are calculated using the bendingwith standard deviation of 0.1 s is

ifferential travel timesmeasurementslue. The inversion (black lines) repro-ell. The discrepancy lies mainly nearnsition. The inversion tends to pro-shallower depth. This is in part duebutlingn this study. (a) The coefcients are plotted at the three longitudinal knots as a funcues as a function of radius and longitude. From left to right are , , and , respecti

eighting on the data and a prior. The directional coverage of ourany more equatorial paths thanle data set according to the rayhe number of data in each bin:each datum with ray angle by0) + 1. The weighting makes the

the anisotropy parameters ( andro. This is motivated by previousotropy at the topmost inner corets are needed because of the fol-unt of CDDF data (fromNiu andt inner core. The weak anisotropyntaminated by data at larger dis-meterization using cubic splinesput more weight on the CDDF

ill be distorted by the structure

ige and Saunders, 1982) to invertatively.We start with a 1Dmodel

depth and lateral variations tcussion below). The syntheticcode, then a Gaussian noiseadded. The precision of our dis likely smaller than this vaduce the input model quite wthe boundaries of sharp traduce a gradual increase atto our spline interpolation,avoided without dense sampdirections.pdate the inner core model withrbations obtained from the lasting the new model, and the newto the new model are then usedcients perturbations. We choose

al model, although the results are

synthetic test that we performedrization, and inversionprocedure.udes some important features of

Fig. 13. Distributions of raw residuals (a(dashed), the residuals after the mantlecorrecting for the inner core model and tis the number of observations in a slidingand the sliding step of 0.1 s.it is a problem that cannot beof the inner core along differentfter correcting for the 1D model AK135)correction (grey), and the residuals afterhe mantle model (dark). The vertical axiswindow with the window width of 0.5 s


Fig. 14. Comparison of observed (grey crosses) ABDF and BCDF residuals and predictions (black circles) for our inner core anisotropy model. The observed residuals havebeen corrected for the mantle model (see text). The data and the predictions are divided into different distance ranges and plotted as a function of the ray angle for QEH (leftpanels) and QWH (right panels) hemisphere. It can also reproduce some of the complexities of the data, e.g., the data scatter in the QWH at distance 145153 , 156160 , or165173 and ray angle 6080 .

4. Results

Our inversion results of the anisotropy coefcients are shown inFig. 12. Fig. 13 shows the distribution of thewhole data set (not justthe summary rays) before and after correcting for the inner coreanisotropy. The distribution for the raw residuals (dashed curve)is asymmetrical and bi-modal with the tail of positive anomalies(greater than 2 s) coming from the polar paths. The distributionafter the mantle correction (grey solid curve) is also asymmetricaland bi-modal. It reduces the residuals for equatorial paths (resid-uals from about 2 to 2 s), but does not change much the residuals

of the polar paths. The curve after the inner core inversion wasmuch more symmetrical. It reduces further the residuals of theequatorial paths, but the major improvement is on the polar paths.The sidelobe between 3 and 3.5 s in the raw residuals is mostlyfrom polar paths originated from SSI earthquakes. The residualsare not completely corrected by our smoothed model and thusshows up in the sidelobe between 1 and 1.5 s after the mantle andinner core corrections.

The variances of the raw residuals, the residuals after mantlecorrection, and the residuals after both mantle and inner corecorrections are 2.12, 1.93, and 0.71 s2, respectively. Thus the vari-

Fig. 15. Averaged anisotropy coefcientsbelow the ICB) while the anisotropy in thin the two hemispheres as a function of radius. Note the anisotropy in QEH (left) remaie QWH (right) becomes strong at much shallower depth (around 200km).ns weak to a great depth (around 700km


Fig. 16. t ray a

ance reduction from the mantlereduction from the inner coremodcorrected data. The combined coinner core reduce the raw data vthe equatorial paths, the mantle cthe raw data by 36.4%; the innerof the mantle-corrected data verythe mantle correction reduces thinner core model reduces the vdata by 47.1%.

Fig. 14 shows adirect comparisat larger distances (>145). The pthe observations at different distThe model can even reproduce sdata, such as the scatters at theranges in QWH near ray angle the important features of the inve

4.1. Hemispherical and depth varia

One of the most prominentwavelength, hemispherical pattecore. The longitudinal variation i(Fig. 12). The anisotropy structureand 340 longitudes) are quite dieastern knot (100 longitude). Yettwowestern knots are quite consi

he din thted bericainueoutionsAveraged P-wave velocity of inner core for QEH (gray) and QWH (black) at differen

correction is 9.2%. The varianceel is 63.1% relative to themantle-rrections for the mantle and theariance by 66.5%. Separately, fororrection reduces the variance ofcore model reduces the variancelittle (4.4%). For the polar paths,e variance very little (5.4%); the

a large-scale robust pattern. Tspheres can be seen directlysection, which is well predicin Fig. 14. Note the hemisphhave referred to and we contcal. The QEH is in fact only abexact division of the two regat hand.ariance of the mantle-corrected

onof the predictions and the dataredictions agree quite well with

ances and different hemispheres.ome of the ne structures in the145153 and 156160 distance= 60. Below we discuss some ofrsion results.

tion

features of our model is a longrn for the upper half of the inners quite obvious in the model plots at the two western knots (220

fferent from the structure at the, the anisotropy structures at thestent with each other, suggesting

To examine the hemisphere paious comparisons in Figs. 15 andin the isotropic term (), i.e., thdirection. The velocity in the equthan that of QWH from the IC(Fig. 16). The structure is generalies by Tanaka and Hamaguchi (The largest difference is near theby about 0.5%. The value is slighWen (2001) because of model smcussion below). The depth extentkm) is shallower than 500km esturning point depth of the ray(1997).

A more prominent featureanisotropy. The average anisotrothan 1% from ICB all the way dowAt the center of QEH or 100 longzero down to that depth (Fig. 12ngle .

istinction between the twohemi-e data as we discussed in the datay our anisotropy model as shownl pattern, which previous studiesto refer to here, is not symmetri-one-third of the hemisphere. Thecannot be resolved with the datattern more closely, we show var-16. One important difference ise velocity perturbation at EWatorial direction of QEH is fasterB down to about 200300 kmly consistent with previous stud-1997) and Niu and Wen (2001).ICB with the velocity differencetly smaller than that of Niu andoothing in our inversion (see dis-in our model (topmost 200300timated approximately using thepath in Tanaka and Hamaguchi

is hemispherical variation inpy amplitude () in QEH is lessn to about 700km depth (Fig. 15).itude, the anisotropy is virtuallya). On the other hand, the aver-


Fig. 17. Comparison of the anisotropy vs.core. At a shallow depth (300km), the QWin the QEH. But when the depth increase tin radius), the amplitude and shape of thhemispheres. The IMIC in our model hasin the upper inner core, with the minimuboth equatorial and polar directions.

age amplitude in QWH is about(Fig. 15) except in the topmost innhemispheric variation is consisteand Hamaguchi, 1997; Creager, 1The depth of weak anisotropy indeeper than 400km in Garcia anuncertainties of 400700km in C

The transition from isotropyinner core is rapid in the QWH0 at ICB to about 3% on averageBecause of model smoothing, theven sharper on average and m(see further discussion below). Othe Central America sampled byto Alaska and Canada, where thtion has been reported (Song aXu, 2002). In QEH, however, thesmall anisotropy amplitude at dtion would be expected from sarange.

4.2. Inner inner core (IIC)

An important result fromour inthe formof anisotropy at a radiushalf of the inner core radius. The dpaper (Sun and Song, submitted fanisotropy form is generally consIshii andDziewonski (2002). Howshallower depth, which is more cCormier and Stroujkova (2005) aWe refer to the inner part as the inpart as outer inner core (OIC).

us froand Qd das

risonitudecussseengestionion,e ofisp

pe at300kfrothe celativdenhere(OICf aboalueshe twchansumprs atlow.ray angle at different depths of the innerH has much larger anisotropy than that

o 800km below the ICB (or about 420kme anisotropy become similar in the twoa different shape of anisotropy than thatm velocity at about 40and maximums at

24% for the same depth rangeer core. The general pattern of thent with previous studies (Tanaka999; Garcia and Souriau, 2000).QEH in our model (700km) is

d Souriau (2000) but within thereager (1999).to anisotropy in the topmost

. The amplitude increases fromat 200km below ICB (Fig. 15).

e actual transition is likely to beuch sharper in certain localitiesne example is the region underthe path from SSI earthquakese evidence for seismic triplica-nd Helmberger, 1998; Song andincrease is slow because of theepth. Thus no seismic triplica-mples of this region and depth

Fig. 18. Ratio k as a function of radiaged values for QEH (grey solid line)at the two western knots (dotted an

Fig. 17 shows a compadepths. The anisotropy ampland with depth as we discloser examination, we canfunction of ray direction chain our model parameterizaequatorial and polar directray angle = 40. The shap(radius 420km) in either hem(Fig. 17b). However, the sha(Fig. 17a). The velocity atchanges little for ray angles

Thechange in the shapeofby the large amplitude of rthe ratio k= / , which wasas the reciprocal of the ratioabout 0.3 in the outer part(IIC) on average at a radius oabout 150km (Fig. 18). The vas the averaged values for tconsistently, suggesting theIt is not an artifact of our as300 km) as the change occuexplore this issue further be

4.3. Voigt velocityversion is the apparent change ofof about 600km, slightly less thanetails are presented in a separateor publication). The change of theistent with the IMIC proposed byever, the change occurs in amuchonsistent with recent studies bynd Cao and Romanowicz (2007).ner inner core (IIC), and the outer

We also calculate the Voigt avwhich is equivalent to the averagetions of a single crystal (CreagerVoigt average is + (/3) + (8/variation in anisotropy betweenVoigt averages are similar (withiThe perturbations of the averagewestern hemispheres are 0.027 krelative to AK135. This numbers aresults (0.035km/s and 0.044km/We t the Voigt average of the worderpolynomial and the result ism our anisotropy model. Plotted are aver-WH (dark solid line) as well as the values

hed lines). The ratio k is dened as / .

of the anisotropy at differentvaries strongly with hemisphereed above. Furthermore, with athe shape of the anisotropy as awith depth. The IMIC embeddedhas greater velocities at bothand its minimum is at aroundthe anisotropy at depth 800 kmhere is similar to that of the IMICdepth 300km is quite differentm depth in either hemispherem 0 to 40.urves in thedeeperpart is causede to . The shape is controlled byed earlier by Garcia (2002b) (but). The ratio changes sharply from) to about 0.8 in the inner partut 600km over a depth range ofat the two western knots as wello hemispheres show the trendge is quite robust in our model.tion of an IMIC (with a radius ofthe radius twice as large. We will

erage of the inner core (Fig. 19),d velocity along all the ray direc-, 1999). The perturbation of the15) . Surprisingly, despite largeQEH and QWH in the OIC, their

n 0.5%) at all depths (Fig. 19a).d Voigt velocities for eastern andm/s and 0.041 km/s, respectively,re comparable to Creagers (1999)s, respectively, relative to AK135).hole inner core using a second

:vVoigt = 0.0948r2n 0.1874rn +


Fig. 19. Voigt averages derived from our anisotropy model. (a) Perturbations relative to AK135. The difference between the two hemispheres is small at all depths, comparedto the large difference in anisotropy. (b) Absolute values as a function of radius. The Voigt velocity for the whole inner core (averaged over all longitudes) (dark solid line) isa t to the 2nd order polynomial. Shown also are Voigt averages for QEH (dotted line) and QWH (dashed line), respectively, and the reference model AK135 (grey solid line).

11.3471, where vVoigt is the Voigt average in the inner core, and rn isthe normalized radius (rn = r/ric) and ric=1217.5 (from AK135). TheVoigt average for the inner core is about 0.150.74% greater than the1D referencemodel AK135 (Fig. 19b). This is not surprising becausethe 1D model is derived from the travel-time data that are dom-inated by equatorial paths. The seismically derived Voigt averageprovides a new reference for mineral physics studies of the innercore It is a more appropriate reference than a 1D model such asAK135 or PREM because of the strong but complex anisotropy ofthe inner core.

5. Conclusion and discussion

Wepresent a self-consistentmodel of the anisotropyof the innercore using PKP differential travel times, 3D ray tracing, and splineparameterization. The large-scale features of our results t the dataquite well and our synthetic tests suggest they are robust and canbe readily resolved by our data and inversion (Fig. 11). These fea-tures are generally consistent with most of the previous studies,although there are some important differences in details, such asthe depth extent of the hemispherical differences and the radius

Fig. 20. Synthetic test similar to Fig. 11 bu(grey) and inversion results (dark) of the at using 9 knots in the radial direction. No IMIC of uniform anisotropy is assumed in the mnisotropy coefcients at the three longitudinal knots. (Bottom) map views of the inversiodel parameterization. (Top) Input modelon results.


Fig. 21. Same as Fig. 12 but for an inversion with 9 knots in the radial direction without imposing an IMIC constraint. The knot spacing is similar to the previous inversionwith seven knots and a uniform IMIC (about 150km).

of the IIC. The major features of the model are as follows. (1) Theinversion results show strong hemispherical and depth variation.The isotropic velocity in the topmost inner core (200300km orso) is greater in QEH than in QWH. The anisotropy amplitude ()is small (less than 1% on average) in QEH to 600700km belowthe ICB. The anisotropy is virtually zero down to that depth atthe center of QEH (or 100 longitude). In QWH, the anisotropyincreases at much shallower depth (about 100200km below ICB)to about 34%, then remains strong (24%) throughout the inner

core. (2) The OIC and IIC possess different forms of anisotropy. Theanisotropy form changes abruptly at the radius of about 600km,slightly less than half of the inner core radius. This is the resultof increased amplitude relative to in the IIC. (3) The abso-lute value of and is anti-correlated, which results in a nearlyuniform isotropic velocity (Voigt average) throughout the innercore. Below we discuss the robustness of our model and varioustests on model parameterization, mantle heterogeneity, and 3Dray tracing. We will also discussion the limitation, in particular,

Fig. 22. Polar views of k ratio as a functiabout half radius of the inner core at all locomputed.on of radius and longitude from inversions with (left) and without (right) an IMIC, respengitudes, producing a distinct IIC sphere. In the white area in the eastern hemisphere of tctively. In both cases, k value changes athe left panel, < 0.3% and the ratio is not


Fig. 23. Comparison of inversion with (dark) and without (grey) mantle correction. Plotted are anisotropy coefcients as a function of radius for the three longitudinal knots.The two inversions follow the same procedures in the selection of summary rays, model parameterization, and inversion formulation.

notable discrepancy with the data, for our model, which is highlysmoothed.

In our parameterization,we articially imposed an IMIC (Fig. 10)because of limited data at near antipodal distances along differentdirections. We now explore the inuence of such a constrainton our inversion results. In our test, we parameterize the radialdirection using cubic splines with nine equally spaced knots fromthe ICB to the center of the Earth, but do not impose an IMIC ofuniform anisotropy. The knot spacing is identical (about 150km).The longitudinal parameterization remains the same. With thenew parameterization, we conducted a synthetic test (Fig. 20)for the same input model as before (Fig. 11) and a real inversionusing the same data and procedure (Fig. 21). The results of thereal inversion (Fig. 21) are very similar to the previous results(Fig. 12). The main difference is in the innermost region of 300 kmradius, where the anisotropy diminishes approaching the centerof the Earth. The synthetic test (Fig. 20) shows the same features,suggesting it is likely caused by the poor data sampling of theinnermost region. The test provided a motivation for our initialassumption of an IMIC in our mod

Our conclusion of the existenof anisotropy is further supporteshows a comparison of the k valueThe results are strikingly similar

similar depth (radius of about 600 km). In either case, the sharpchange occurs consistently at almost the same radius at differentlongitudes, producing a distinct sphere of the IIC with a larger kvalue in amplitude. The distinction in the anisotropy form betweenthe OIC and the IIC is robust in our data.

Our model has been corrected for mantle heterogeneity usinga mantle model as discussed above. However, one may questionthe accuracy of the current mantle models, particularly in the low-ermost mantle. The inuence of lowermost mantle structure onPKPdifferential times arewell-noted (Song andHelmberger, 1993b,1997; Breger et al., 1999, 2000; Luo et al., 2001). As such, differentialPKPmeasurements have been used in the tomographicmapping ofthe lowermost mantle (Karason and van der Hilst, 2001; Tkalcic etal., 2002; Sun et al., 2007).

The recentMITmodel (Li et al., submitted forpublication),whichwebased ourmantle corrections on, has also included PKP absoluteanddifferential travel times. It achieves signicantlybettervariancereduction than the corresponding earlier generation model by vander Hilst et al. (1997).

of mrectire inersiond w

Fig. 24. Comparison of the 1st (grey) andthus the display is a comparison of a lineael parameterization.ce of an IIC with a different formd by this new inversion. Fig. 22(k=/ ) from the two inversions.. The sharp changes start at the

To explore the inuenceextreme case: no mantle corWe follow the same proceduparameterization, and in invthe inversion results with a6th last iteration (dark) of the inversion for our inner core anisotropy model. Our initiar inversion (1st iteration) and a non-linear inversion (last iteration).antle heterogeneity, we test anon is made before the inversion.selecting summary rays, inmodeln formulation. Fig. 23 comparesithout mantle corrections. Thel model is a 1D reference model (AK135),


Fig. 25. Comparison of the PKiKPPKP(D erallydifference, but there is signicant discrep y fromthe inner core beneath central America, w of mo

major features are very similar,ences, particularly in mid-inner55.4%, with respect to the raw dapolar data is similar (53.4%); the vrial data is small (8.8%). Consideridifferential times of up to 4 s (or(Sun et al., 2007), the agreementremarkable. Fundamentally, the athe following two key observationthemantle and the inner core struity of thedata to the inner core is dcoverage of the inner core fromdierogeneity is averaged out whendependence. Second, the polar-pthan the equatorial ones. The asyresiduals (Fig. 13) shows that thehard to be explained by any mant

A major effort of this study istation of the PBR 3D ray tracingand the non-linear iterative invesion results change if we use oinversion. Fig. 24 compares the retions of our inversion. The 1st iteof the 1D reference model (AK1anisotropy patterns are in fact vbility of the inversion. The greatanisotropychangesmost rapidly, n600km in radius to the IMIC (at rprising because the change of rayto be the greatest where the velocear inversion overestimates absoby 29.9% and 15.0%, respectivelyinversion.

Despite the major robust featudata and parameterization and tundoubtedly a smoothed versionis particularly true for any sharpsmall-scale variations. It is clear fra sharp boundary will be smooth(sharper than in our model) incluinner core (from isotropy to st

e inngeundto aathSSI echant 20oodetrucnly toet islitudd in

ampld americaicterepaat lars buhemmerotropdel shalloith aamion,rger,fromdel.s hasva, 2e to tmpleF) residuals with model predictions at distances 130143 . The predictions can genancy for those polar paths sampling QWH at larger distances. These data are mainlhere large anisotropy exists at shallow depth. The discrepancy is likely the artifact

although there are some differ-core. The variance reduction ista. The variance reduction for theariance reduction for the equato-ng some strong anomalies in PKP2 s) from mantle heterogeneitybetween the two inversions are

greement canbeunderstood froms. First, sensitivities of the data toctures are different. The sensitiv-irectiondependent.With enoughfferent directions, themantle het-we try to extract the directionalath anomalies are much greatermmetrical distribution of the rawpolar-path anomalies will be veryle structure.the development and implemen-(for an anisotropic inner core)

rsion scheme. How do our inver-nly 1D ray tracing and a linearsults of the 1st and the 6th itera-ration is based on the ray tracing35). The overall results and theery similar, suggesting the sta-est differences occur where theear200 kmdepthor so, and fromadius of 300km). This is not sur-paths by 3D structure is expectedity changesmost rapidly. The lin-lute values of and in the IMIC, compared with the non-linear

res, our model, with the limitedhe use of the spline functions, isof the inner core structure. This

between OIC and IIC (changbetween QEH and QWH (cha

For the topmost inner corethe transition from isotropythan in the model. Undernewaveform triplication fromand Alaska suggests that theup to 8% at the depth of abou1998; SongandXu,2002).Agmodelts ordoesnottne sdata set, which is sensitive o(Fig. 25). Because the data smeasurements and the ampals, they can be overwhelme(BCDF and ABDF times) smuch greater in number ancan reproduce the hemisphresiduals, although the predHowever, an important disc = 6070) sampling QWHdata have no large anomaliepositive residuals. Most of tAlaska sampling theCentralAals come fromthestronganisThis is likely an artifact ofmoden jump in anisotropy to a swould be more consistent w(greater change in anisotropyat a greater depth in this regmodeling (Song and Helmbe

The change in the k valueeven sharper than in our modistortion at greater distance2004; Cormier and Stroujkoare likely to be very sensitivextra care to detect. For exaboundaries in the inner core oromour synthetic test (Fig. 11) thated out. Possible sharp transitionsde the boundary in the topmostrong anisotropy), the boundary

400 to 300 km in the equatoriasmall waveform triplication, and600400km around = 35 madefocusing effect. Polar directionhemisphere across the OIC/IIC treproduced the pattern of hemisphericalstations in Canada and Alaska, sampling

del smoothing.

the k value), and the boundariesin the anisotropy strength).er parts of thewestern inner core,nisotropy is likely even sharperthe central America, evidence ofarthquakes to stations in Canadage from isotropy to anisotropy of0250km (Song and Helmberger,xampleofhowwell the inversiontureof the inner core is theCDDFthe top 100km of the inner coresmall in terms of the number ofe of the differential time residu-the inversion by other data setsing deeper structures, which areplitude. Nevertheless, the modell pattern in the differential timed difference is smaller (Fig. 25).ncy is the polar data (ray angleger distances (>137), where thet the predictions have signicantare from stations in Canada andica. Thepredictedpositive residu-y inQWHeven in the top100 km.moothing,which spreads a sud-wer depth. Thus the CDDF datamodel with a sharper transition

plitude over smaller depth range)as evidenced from the waveform1998; Song and Xu, 2002).the OIC to the IIC could also be

No strong evidence of waveformbeen found so far (Sun and Song,

005). Possible waveform changeshe ray directions, which requires, the velocity increase from radii

l direction ( = 0) may cause athe velocity decrease from radiiy in fact cause a shadow zones, particularly along the easternransition, are probably the best


candidate for detecting waveform changes because the velocitychange is the biggest.

Our anisotropy model will be useful in mapping the alignmentof the iron crystals in the inner core, which is responsible for theanisotropy, and in understanding the physical mechanisms of thepreferred alignment, a still elusive goal two decades after the dis-covery of the inner core anisotropy. The fact that the OIC has largelateral variation in anisotropy and yet small, if any, lateral varia-tion in Voigt average and in the k value suggests that the OIC islikely composed of the same type of iron crystals. Thus the seismicanisotropymodel can be used tomap directly the crystal alignmentas we presented in a separate paper (Sun and Song, submitted forpublication). The different formof anisotropy in the IICmay suggesteither a different type of crystal alignment of a different iron phase.The mapping of the inner core texture, however, critically dependson our knowledge of the inner core iron phase(s) and its elasticproperties. Increasingly, the progress in this eld requires multi-disciplinary effort, in seismology, as well as in mineral physics,geodynamics, and geomagnetism studies.

Acknowledgements

Part of the data we used areKen Creager, Fenglin Niu, Barbaand Satoru Tanaka. Most of ourLDEO and Caltech WWSSN lmORFEUS, GEOFON, NARS data cenwere provided by China Earthquthank the insightful reviews fromreviewer, which improve our mis supported by NSF EAR-0409Knowledge Innovation Program(KZCX3-SW-131).

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Tomographic inversion for three-dimensional anisotropy of Earth's inner coreIntroductionDataInversion method3D ray tracingParameterizationInversion formulationSynthetic test

ResultsHemispherical and depth variationInner inner core (IIC)Voigt velocity

Conclusion and discussionAcknowledgementsReferences

Xinlei Sun - Tomographic Inversion for Three-dimensional Anisotrophy of Earth's Inner Core

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Transcript of Xinlei Sun - Tomographic Inversion for Three-dimensional Anisotrophy of Earth's Inner Core