A new isostatic model of the lithosphere and gravity field

M. K. Kaban, P. Schwintzer, Ch. Reigber

GeoForschungsZentrum Potsdam, Dept. 1, Telegrafenberg A 17, 14473 Potsdam, Germanye-mail: kaban@gfz-potsdam.de;Tel.: +49-331-288-1172; Fax: +49-331-288-1169

Received: 28 August 2003 / Accepted: 16 June 2004 / Published online: 12 November 2004

Abstract. Based on the analysis of various factorscontrolling isostatic gravity anomalies and geoid undu-lations, it is concluded that it is essential to model thelithospheric density structure as accurately as possible.Otherwise, if computed in the ‘classical’ way (i.e. basedon the surface topography and the simple Airy com-pensation scheme), isostatic anomalies mostly reflectdifferences of the real lithosphere structure from thesimplified compensation model, and not necessarily thedeviations from isostatic equilibrium. Starting withglobal gravity, topography and crustal density models,isostatic gravity anomalies and geoid undulations havebeen determined. The initial crust and upper-mantledensity structure has been corrected in a least squaresadjustment using gravity. To model the long-wavelength(>2000 km) features in the gravity field, the isostaticcondition (i.e. equal mass for all columns above thecompensation level) is applied in the adjustment touncover the signals from the deep-Earth interior,including dynamic deformations of the Earth’s surface.The isostatic gravity anomalies and geoid undulations,rather than the observed fields, then represent the signalsfrom mantle convection and deep density inhomogene-ities including remnants of subducted slabs. The long-wavelength non-isostatic (i.e. the dynamic) topographywas estimated to range from –0.4 to 0.5 km. For shorterwavelengths (<2000 km), the isostatic condition is notapplied in the adjustment in order to obtain the non-isostatic topography due to regional deviations fromclassical Airy isostasy. The maximum deviations fromAiry isostasy ()1.5 to 1 km) occur at currently activeplate boundaries. As another result, a new global modelof the lithosphere density distribution is generated. Themost pronounced negative density anomalies in theupper mantle are found near large plume provinces,such as Iceland and East Africa, and in the vicinity ofthe mid-ocean ridge axes. Positive density anomalies inthe upper mantle under the continents are not correlatedwith the cold and thick lithosphere of cratons, indicatinga compensation mechanism due to thermal and compo-sitional density.

Key words: Isostasy – Lithosphere – Isostatic gravity –Isostatic geoid – Dynamic topography

1 Introduction

The concept of isostasy plays an important role ingeodesy and geophysics. Even early studies of theEarth’s interior started from this concept. The firstevidence for crustal roots, which compensate thetopographic masses, was obtained in the middle of the19th century as a result of gravity surveys carried out byPratt in India (Pratt 1858), long before seismic investi-gations started. The deflection of the pendulum from theradial direction was only one-third as large as expectedfrom calculations of the mass attraction of the Hima-layas and Tibet. Thus, something within the Earth’sinterior must counteract the topographic gravity effect.

Also, the isostatic reduction of the gravity field isconsidered as one of the most important reductionsapplied for various geological and geophysical purposes.Isostatic anomalies of the gravity field are computed bysubtracting the effect of the isostatically compensatedcrust/lithosphere (i.e. the topographic mass surplus andthe compensating mass deficiency) from the observedfield, either free-air gravity anomalies or geoid undula-tions. As the topographic masses are always closer to thepoint of observation than the compensating roots, theircombined effect on observed gravity anomalies or geoidheights is non-zero, even in the case of a complete iso-static balance. Thus, the isostatic anomalies are residualanomalies after subtraction of what is known or as-sumed about the density distribution within the litho-sphere.

Traditionally, they were computed using the surfacetopography and its compensation following the idealizedAiry or Pratt schemes (see e.g. Heiskanen and Vening-Meinesz 1958; Heiskanen and Moritz 1967). The iso-static anomalies computed in these ways were usuallyregarded as a measure of deviations from the isostaticCorrespondence to: M. K. Kaban

Journal of Geodesy (2004) 78: 368–385DOI 10.1007/s00190-004-0401-6

equilibrium of the Earth’s crust, showing the areaswhere local topographic load is supported by regionalbending of the elastic crust (Vening-Meinesz 1940).However, the reality turns out to be much more com-plex: the isostatic anomalies also reflect all heterogene-ities within the Earth, even if isostatically compensated,which are not accounted for in the simplified Airy andPratt isostatic models. Evans and Crompton (1946)showed, for example, that the isostatic anomalies mightcontain a significant effect of sedimentary layers due toneglecting their density anomalies. They wrote on page244: ‘‘In regions of thick sediments, corrections shouldprecede the application of data to hypotheses of isostaticequilibrium’’.

At that time, the knowledge about the spatial extentand density distribution of sedimentary basins was ra-ther poor, and the available computation techniques didnot allow evaluation of the gravity effect of complexbodies using a dense data discretization. Today, thereexist various crustal models, which vary from very de-tailed ones for some well-studied regions to generalizedglobal models (see e.g. Mooney et al. 1998, Bassin et al.2000). Thus, it is now possible to include these data inthe isostatic model of the lithosphere while computingthe isostatic anomalies. The first global map of suchanomalies, including the information on the crustalstructure, was computed by Kaban et al. (1999). Thecrustal models, especially when generalized over thewhole Earth, contain significant uncertainties, whichobviously produce artificial effects in isostatic anoma-lies. Therefore, it will be investigated in the followingwhether it is worthwhile to consider the a priori crustalstructure in the computation of isostatic anomalies.

Next, a new global model of isostatic gravity anom-alies and geoid undulations will be derived based on theup-to-date data sets. The previous global isostatic modelof the lithosphere (Kaban et al. 1999) was based on the5� · 5� crustal model of Mooney et al. (1998) and ageneralized isostatic compensation concept (not dis-criminating between tectonic structures) determinedfrom an admittance (cross-spectral) analysis betweengravity and topography after the initial reduction of thea priori given crustal and upper mantle density struc-ture. Newly available data and insights provide themotivation to re-compute a global isostatic model withnew approaches. First, the CHAMP mission yields animproved long-wavelength geoid model (Reigber et al.2002). Second, new crustal data, especially for NorthAmerica and Eurasia, have been compiled and includedinto the global crustal model that is now based on a2� · 2� grid. Third, the generalized isostatic concept willbe re-thought to account for peculiarities and specificcompensation mechanisms in individual tectonic struc-tures, and for the spatially heterogeneous quality of thecrustal data.

2 Isostatic gravity anomalies: what do they mean?

The starting point is the isostatic hypothesis, whichmeans a perfect equilibrium of masses from the top of

topography (load) down to the isostatic compensationlevel H that is assumed to be at the base of thelithosphere (i.e. including the crust and upper mantle)

ZT

�H

DqðtÞ R� tR

� �2

dt ¼ 0 ð1Þ

where DqðtÞ is the density anomaly (including topogra-phy and water) relative to a horizontally homogeneousreference model with zero density above the geoid, H isthe topographic height above the geoid (zero for marineareas), t is the depth below the geoid, and R is the radiusof a spherical Earth. The bottom level of isostaticcompensation (T) is here taken at the base of thelithosphere and not at the base of the crust, becausethere exist large subcrustal mantle density variationsthat take part in the isostatic compensation (Kabanet al. 1999; Panasyuk and Hager 2000).

The ‘mechanical’ definition of the lithosphere isadopted assuming that the lithospheric part of the uppermantle is preserved for a relatively long geological timeand does not participate in mantle convection. The ref-erence model, to which the density anomalies DqðtÞ arereferred, corresponds to the generalized model of old(180-Ma) oceanic lithosphere (Kaban et al. 1999). Itincludes a 6.4-km depth of water with a density of1030 kg/m3 over ocean areas, a 7.2-km thick crust with aconstant average density of 2850 kg/m3, and an uppermantle with a density distribution corresponding to thecooling plate model for the oceanic lithosphere with anage of 180 Ma. The thickness of the reference conti-nental crust, which is in balance with the oceanic refer-ence lithosphere, must be equal to 39.64 km, assumingzero topography. The continental upper mantle is thentaken to have the same reference density distribution asthe oceanic upper mantle.

Equation (1) describes the ‘local’ type of isostasy andincorporates the Airy and Pratt models as special cases.Although it is just an ideal simplification of the realEarth’s structure, this classical definition is thought tobe more appropriate to describe the term ‘isostasy’ thanlater innovations implying elastic or visco-elastic reac-tion of the lithosphere to external (surface) and/orinternal loads. Regional compensation models accord-ing to Vening-Meinesz (1940) are closer to reality butrequire a comprehensive model of the lithosphere as astarting point, including the determination of its rheol-ogy and geological history. On the other hand, theclassical isostatic scheme may be taken as an initialapproximation to compute the isostatic reduction whenthere is not enough information about the Earth’sinterior to construct a realistic model of the lithosphere.In this case, the isostatic anomalies may be used toinvestigate perturbations from the initial approximation.

The difference of the integral in Eq. (1) from zerodirectly points to disturbances of the isostatic equilib-rium. This difference defines the non-isostatic residualtopography, and thus that part of the observed topog-raphy that is not compensated or over-compensated bythe lithosphere structure. If this structure were known in

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sufficient detail, the computation of the residual topog-raphy would close the isostatic analysis and the com-putation of isostatic anomalies would bestraightforward. However, this knowledge is not avail-able to date. Instead, a simplified model of the densityinhomogeneities (usually just the surface topography) isapplied in the classical isostatic condition [Eq. (1)] andthe residual isostatic anomalies are computed. Withthese simplifications, the isostatic anomalies of thegravity field (either gravity or geoid) generally reflect thefollowing sources.

1. Density heterogeneities in the upper crust (internalload), which are unknown and not considered in theisostatic model, e.g. sedimentary layers or deviationsof the upper crust density from the nominal value.

2. Density heterogeneities in the lower crust/uppermantle that differ from the adopted compensationscheme (e.g. Airy).

3. Deep density heterogeneities located below the levelof the isostatic compensation.

4. disturbances of the isostatic equilibrium which canalso be attributed to: (a) local disturbances due to therigidity of the lithosphere (regional compensation oflocal load); (b) deep dynamic effects (dynamic sup-port of topography by mantle convection); and (c)visco-elastic response to time-varying loads (e.g.postglacial rebound).

5. Errors in the initial data sets used.

Cases 1 to 4 are important when trying to resolvevarious geological and geophysical signals. These arebetter represented in the isostatic anomalies than in thefree-air or Bouguer anomalies. For example, the UnitedStates Geological Survey (USGS) makes extensive use ofisostatic anomalies to model the density and spatial ex-tent of sedimentary basins (see e.g. Morin et al. 1999).Thereby, it is assumed that this source dominates theisostatic anomalies. Actually all mass sources are alwayspresent in the isostatic anomalies, but their impact dif-fers depending on the initial data used for the isostaticreduction and the adopted isostatic model. Thus, theproblem of mass-source separation is crucial. Differentkinds of high- and low-pass filters are often used toseparate deep and near-surface sources. However, thehorizontal scale of upper-crustal heterogeneities (e.g.large sedimentary basins) may reach 1000 km, and thusbe comparable with the signatures of mantle convection.Therefore, it is important to include these heterogene-ities, as far as they are known, in the isostatic reductionfrom the very beginning, as was previously suggested byEvans and Crompton (1946).

Large amplitudes in small-scale isostatic gravityanomalies are often considered as clear evidence ofisostatic disturbances (regional compensation) due tolithospheric rigidity (case 4a). However, the minimumwavelength of the isostatic anomalies cannot be tooshort. These anomalies represent the effect of artificiallyfilling up (e.g. the Airy assumption) compensatingmasses, which are placed after the topographic reductionat some depth (Z), usually at the base of the crust, the

Mohorovicic (Moho) discontinuity. Thus, due to theincreased distance from the topography to the Moho,the minimum wavelength of the isostatic anomalyshould be approximately 2pZ » 200–250 km. This im-plies that a 1� · 1� equi-angular grid is sufficient torepresent isostatic gravity anomalies.

This statement is illustrated in Fig. 1. The very nar-row gravity signal due to an uncompensated topo-graphic feature (which is falsely assumed ascompensated by a variation in crustal thickness) trans-forms into a wider-spread isostatic gravity anomaly. Inthe considered case, a very strong isostatic disturbance isassumed: a non-compensated ridge, which is 1 km highand 40 km wide. However, the isostatic gravity anom-alies barely reach 30 mGal. In the following analysis, itwill be shown that the effect of compensated crustaldensity inhomogeneities that are not included in theisostatic model may be of comparable amount or evenmuch larger. Specifically, the isostatic anomalies com-puted with a simplified model should not be taken as adirect measure of isostatic disturbances.

The maximum wavelength of isostatic disturbancesdue to lithospheric rigidity is related to the value of theeffective elastic plate thickness (Te). The amount ofisostatic compensation (C) of any load at a specifiedwavelength (L) can be estimated from the elastic plateequation (see e.g. Turcotte and Schubert 1982) as

C ¼ Dqg=ðk4Dþ DqgÞ ð2Þ

where D is the flexural rigidity of the lithosphere(D ¼ ET 3

e =½12ð1� m2Þ�, with E being Young’s modulusand m being Poisson’s ratio), Dq is the average densitydifference between the load and upper mantle, k ¼ 2p/Lis the wave number, and g is gravitational acceleration.Assuming that the maximum value of the effectiveelastic thickness of the continental lithosphere is�70 km (see e.g. Djomani et al. 1999), it follows fromEq. (2) that the amount of direct isostatic compensationversus regional bending at 2000 km wavelength isC ¼ �0.95. This wavelength corresponds to a degree-20 spherical harmonic spectral expansion. Logically, thiswavelength boundary is found from empirical analyses

Fig. 1. Isostatic gravity anomalies obtained from an uncompensatedtopographic feature. The compensation is (falsely) assumed to start at40 km depth when computing the isostatic reduction

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of isostatic anomaly spectra (Artemjev et al. 1994;Kaban et al. 1999) as the boundary dividing differentgeophysical sources. It is therefore assumed here thattopographic or internal loads exceeding 2000 km wave-length (degree 20) are balanced within the lithosphere ifnot of dynamic origin, whereas for smaller-scale featuresthe simple isostatic condition may be violated.

3 Influence of crustal structure on isostatic anomalies

For geophysical interpretation, an approximation of thereal compensation situation is important when comput-ing isostatic anomalies. For example, the depth of theMoho discontinuity often differs over wide areas fromthat predicted by the Airy scheme. Woollard (1959) hasalready proposed that not taking into account the realMoho depth leads to largely biased isostatic anomalies.An incorrect Moho depth in the isostatic reductionmeans that an incorrect density profile is applied at thecrust/mantle transition. A similar effect arises from nottaking the density distribution within the crust intoaccount, i.e. the density within sedimentary layers andthe underlying consolidated crust. The reliability ofcrustal data, as well as the propagation of crustal dataerrors into the isostatic anomalies, is discussed in thesequel.

In the analyses, the surface gravity anomaly of anylayer within the Earth’s crust and upper mantle is cal-culated using a direct three-dimentional (3-D) algorithmfor a spherical Earth, taking into account changes ofdensity in the horizontal and vertical directions and theaverage elevation. Input parameters are gridded ingeographical coordinates (globally). The same algorithmas in Artemjev and Kaban (1994) is used, which is basedon the formulas of Strakhov et al. (1989). For regionalcalculations, a spatially denser grid (100 · 150, as in thefollowing example) is used for the inner zone (radius outto 222 km), and the same 1� · 1� grid as for the globalcalculations is used for the remaining area of the globe.The accuracy of the direct (forward modelling) gravitycalculations is estimated to be within 1 mGal.

3.1 Sedimentary cover

The importance of sedimentary layers for isostaticmodelling has been understood for a long time (Evansand Crompton 1946). Over recent decades, sedimentarybasins have been one of the main targets of geologistsand geophysicists. As a result, a huge amount ofsedimentary thickness and density data are now avail-able globally, such as on the ‘Tectonic map of theWorld’ (Exxon 1985). Only for the polar areas andGreenland is there a lack of information.

To calculate the gravitational effect of the sedimen-tary basins, it is necessary to also know their densitystructure. Borehole well logs normally provide datadown to a depth of 3 km. These data show complexvariations in density with depth, including strong den-sity contrasts (see e.g. Beyer et al. 1985), even within a

single basin. Due to pronounced local lateral variations,it is generally not possible to assign these density con-trasts to more regular boundaries. A reasonable ap-proach is to construct a smooth density–depthrelationship based on averaged borehole data and onwell-determined density–compaction relations. Manyauthors (e.g. Jachens and Moring 1990; Artemjev andKaban 1994; Langenheim and Jachens 1996; Kaban andMooney 2001) have successfully applied this approach.Then, the isostatic residual anomalies will show all localdeflections from the generalized model.

A simple numerical experiment is now given todemonstrate the impact of errors in the sedimentarymass-density distribution on the isostatic anomalies. Thebottom of a typical medium- to large-sized sedimentarybasin is shown in Fig. 2. The assumed ‘real’ density-depth relationship within this basin is shown in Fig. 2c

Fig. 2. a schematic cross-section used to estimate the effect of errorscommitted in modelling a sedimentary basin’s density distribution onisostatic gravity anomalies. The dashed line shows the position of aMoho exactly compensating the sedimentary basin with a constant(2430 kg/m3) density; the solid line corresponds to a Mohocompensating sediments with a density distribution according to therelationship shown in c. The densities for the crust (upper and lower)and the upper mantle are explicitly given. b 1 Gravity profilerepresenting the effect of sediments with variable density (c) and theircompensation at the Moho (‘real’ case observation). This profilecorresponds to isostatic gravity anomalies when the effect ofsediments is completely ignored. 2 Isostatic gravity anomaliesobtained after removing the effect of the sediments assuming aconstant density of 2430 kg/m3 and their compensation at the Mohofrom the ‘observations’: equal to the modelling error, i.e. thedifference from the ‘real’ case isostatic gravity anomalies (beingzero). 3 Gravity effect of sediments alone without compensatingmasses, adopting a variable density (c). 4 Gravity effect of sedimentsalone without compensating masses adopting a constant density of2430 kg/m3.

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(solid line). This relation was found to be valid foroffshore basins on the western slope of North America(Kaban and Mooney 2001). The Moho (Fig. 2a, bot-tom, solid line) is the one that would provide a fullisostatic compensation of the sedimentary basin, takinginto account the ‘real’ density structure. The solidcurved line in Fig. 2b is then the observed gravity signalat the surface (no topography involved in this example).The same curve then would represent the isostaticgravity anomalies when completely ignoring the effect ofsediments. Note that the horizontal line (0 mGal) in theupper part of Fig. 2 represents the ‘true’ isostaticanomaly, i.e. after having applied the isostatic reductionto observed gravity taking into account the ‘real’ densitydistribution and ‘real’ Moho depths.

The dashed line in Fig. 2b shows the isostaticanomalies which we would obtain if the sedimentary

gravity effect was removed assuming a constant averagemass density of 2430 kg/m3 and its compensation at theMoho (dashed line in Fig. 2a) from observed gravity.The shape of the resulting anomaly is changed remark-ably. The positive peaks bounding the basin almostvanish and the new curve exactly reflects the densitydistribution within the sedimentary basin: denser centralpart and less dense outer sections. The maximumamplitude of the anomaly is reduced, and the wave-length of the anomalies becomes shorter. Thus, theresidual effect can be more easily separated from thedeep mass sources.

In reality, some data usually exist on the densitydistribution within a specific sedimentary basin. At least,it is always possible to use well-determined density–compaction relations together with information aboutthe formation age in order to improve the result illus-

Fig. 3. Southern part of the East Europeanplatform bounded from the south by the Cauca-sus, Black and Caspian seas. a Thickness ofsediments (km); b depth to Moho from seismicdeterminations (km); c isostatic anomalies cal-culated from topography compensated accordingto the Airy scheme, in which the ‘normal’ depthof Moho corresponds to the average depth in theregion according to b; d the same as in c, butincluding the correction for the sedimentarybasins shown in a; e in addition to d, the positionof Moho is taken from b rather than from theAiry scheme (Kaban 2001). The additionalcompensating masses not explained by Mohovariations are distributed within the consolidatedcrust (1/3) and upper mantle (2/3) following anadmittance analysis (Artemjev et al. 1994); f inaddition to e, density variations within theconsolidated crust as derived from seismicvelocities are taken into account. The remainingisostatic balance is provided by introducingdensity variations in the upper mantle (Kaban2002)

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trated in Fig. 2. Small-scale features, such as low-densitysalt domes or high-density magmatic intrusions, couldproduce rather significant anomalies, but these anoma-lies may be left to local studies. Thus, it follows thattaking into account the sedimentary correction will im-prove the power of computing isostatic gravity anoma-lies in order to uncover isostatic disturbances of thelithosphere, even when the density structure of sedi-ments is not known in detail.

To illustrate the importance of various correctionsfor practical computations of the isostatic anomalies,alternative isostatic reductions for the southern part ofthe East European platform are performed. This areahas been extensively studied for several decades, andboth the thickness and the density structure of sedimentsare very well known. Figure 3a shows the depths tobasement of the sediments. Approximately 50 density–depth relationships characterize the different parts ofthis area (Gordin and Kaban 1995).

In Fig. 3c the isostatic gravity anomalies calculatedfrom topography and compensation according to theAiry scheme (assuming an average Moho depth corre-sponding to seismic determinations) are given (no sedi-mentary correction). The same anomalies, but includingthe correction for the sedimentary basins, are shown inFig. 3d. The pattern looks remarkably different. Forexample the Indolo–Kuban basin (the western part ofthe pre-Caucasus foredeep) is hardly visible in the iso-static anomalies, while the Tersko–Kaspyski basin (theeastern part of the pre-Caucasus foredeep) still reveals agravity signal. This indicates that the isostatic state ofthe Great Caucasus and adjoining plate may differremarkably, which is important to know for geodynamicinterpretations (Kaban 2002).

3.2 Moho discontinuity

Many geophysical data indicate that the Moho discon-tinuity plays an important role in the isostatic compen-sation of continental near-surface structures. Since themiddle of the 19th century, the Airy hypothesis hasdominated the other isostatic model assumptions. How-ever, it is also known that significant deviations of thereal Moho from its approximation according to the Airyscheme exist (Artemjev et al. 1994). These deviationsmay be of two types.

First, the average depth of Moho (or ‘normal’thickness of the crust) varies on a large scale without acorresponding change in the average topographic load.For example, under the East European platform, theMoho is 10–15 km deeper than under western Europe.Figure 4 illustrates the effect in the isostatic reductionwhen the compensating boundary varies from 35 to45 km depth. The difference between the two levels ofcompensation amounts to 20 mGal, which is not negli-gible when judging the isostatic balance.

Second, the Moho undulations may not follow thetopography even if the crustal density is corrected forsedimentary layers. For example, Moho undulationsunder western Siberia with almost no topography have

amplitudes of up to 15 km (Artemjev et al. 1994). Theeffect of such variations is demonstrated in Fig. 3, againfor the area of the East European platform. The isostaticanomalies shown in Fig. 3e are computed including thesedimentary correction and based on the Moho depthsdetermined from seismic methods (Fig. 3b, Kaban2002). Additional compensation that is not described bythe crustal thickness variations is distributed within themiddle/lower crust (1/3) and the upper mantle (2/3).These proportions are obtained from admittance anal-ysis (Artemjev et al. 1994). Many patterns in Fig. 3e aredifferent to those in Fig. 3d (constant Moho). In par-ticular, the deep minimum in the central/southern partof the Caspian Sea, which has been the subject of manygeodynamic studies (see e.g. Gushtenko et al. 1993), isreduced significantly. Thus, taking into account the realMoho position instead of an Airy-derived Moho depthassumption is really important both for detailed mod-elling of the crust and for the study of the isostatic stateof the lithosphere.

On the other hand, uncertainties in Moho depthsmay introduce a significant error into the computedisostatic anomalies, especially in continental areas thatare poorly studied. Figure 5 illustrates this error source.The distance between nearest seismic determinations ofMoho depths is assumed to be about 500 km. Thisassumption is equivalent to an interpolation of a 5� · 5�grid into a denser one. Solid and dashed lines in thelower part of Fig. 5 represent two alternative depths ofthe interpolated Moho discontinuity. The gravity effectof this difference reaches 70 mGal, as shown in the up-per part of Fig. 5. When constructing the isostatic modelof the lithosphere the deviation from the isostatic con-dition, Eq. (1), is compensated by additional masses inthe crust and/or upper mantle. If these masses arepositioned in the upper mantle, holding the crustalstructure fixed, the uncertainty in the isostatic anomalyreaches 17 mGal, thus about four times smaller than the

Fig. 4. The effect of crustal thickness on the Airy isostatic gravityreduction: simplified model of a mountain ridge (bottom) compen-sated according to the Airy scheme but with different ‘normal’thickness of the crust (T1 ¼ 35 and T2 ¼ 45 km). The densitydifference between crust and mantle is in both cases equal to500 kg/m3. The upper line shows the differences in the isostatic gravityreduction calculated with T1 and T2

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effect of the difference in Moho depths itself. Thisuncertainty is even smaller (less than 10 mGal) if theadditional masses are distributed within the consoli-dated crust and upper mantle. Thus, the error in theinitial Moho position may, to a large extent, be com-pensated by adjusting the crust/mantle densities.

3.3 Density variations within the consolidated crust

Almost the only information available on densityvariations within the consolidated crust, excluding itsuppermost part, consists of seismic velocities. However,the reliability of this information is worse than that ofthe sediment and Moho depths discussed above. First ofall, the velocity-to-density conversion may introduce asignificant error (Christensen and Mooney 1995). It isalso likely that the lateral resolution of the velocitymodel often is insufficient to map the crustal structure,since the model is usually determined from refractionwaves only (see e.g. Egorkin 1998). The gravitationaleffect of the total error in the estimated density mayexceed 50 mGal for thick continental crust (Kaban andSchwintzer 2001), which is not acceptable when com-puting isostatic anomalies.

In the following example, the effect of the crustaldensity uncertainty on the isostatic anomaly is simu-lated. An error of 50 kg/m3 is assigned to the consoli-dated crust in one 5� · 5� compartment. Its gravityeffect is about 70 mGal (Fig. 6). The compensatingmasses that provide isostatic compensation of the lith-osphere are determined according to Eq. (1) and dis-

tributed in the upper mantle. The remaining error in theisostatic anomaly then is about 20 mGal over the wholecell. This example confirms that the consolidated crustdensity uncertainties are most critical for the computa-tion of isostatic anomalies.

This situation is also illustrated in Fig. 3f, where theisostatic anomalies obtained for the East Europeanplatform are shown, taking into account all availablecrustal data (Kaban 2002) including density variationswithin the consolidated crust converted from seismicvelocities. Complete isostatic mass balance is providedin this example by estimating density variations in theupper mantle compensating the assumed mass distribu-tion within the consolidated crust. The amplitudes of theresulting isostatic anomalies are about two times largerthan those computed without introducing a priori den-sity heterogeneities in the consolidated crust (Fig. 3e).Based on this analysis, it is concluded that the correctionfor density variations within the consolidated crustintroduces large errors and should not be taken as beingreliable at the present stage in isostatic gravity fieldmodelling. A way to improve these data is investigatedin the following section.

4 Evaluation of a global isostatic model of the lithosphereand of the isostatic gravity field

Following the principles established above, a globalisostatic model of the crust and upper mantle is nowderived from the given gravity, topography and crustaldata in order to compute the global isostatic gravity fieldand the global non-isostatic residual topography.

4.1 Initial gravity model

The gravity field model taken here is a sphericalharmonic expansion of the gravitational potential up

Fig. 5. The effect of Moho uncertainties (e.g. due to interpolation) onisostatic gravity anomaly computation. The distance between thenearest seismic profiles is assumed to be 500 km. The same error maybe introduced by an interpolation of a 5 · 5� grid (like in the modelCRUST5.1) into a denser one. Bottom: ‘real’ Moho (solid line) anderroneous Moho (dashed line); Top: gravity effect of real versuserroneous Moho (dash–dotted line), and error in isostatic reductionwhen compensating masses are placed partly within the crust andpartly within the upper mantle (dashed line) and totally in the uppermantle (solid line)

Fig. 6. The effect of consolidated crust density uncertainties on theisostatic gravity anomalies: density of consolidated crust from 3 to38 km depth is increased within a 500 · 500 km block by 50 kg/m3

(error assumption), thus leading to a change in forward-computedgravity (dashed line). The solid line shows the error in the isostaticgravity anomalies after having distributed additional compensatingmasses within a 100-km-thick layer below the Moho

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to degree and order 180, representing a spatial resolu-tion (half-wavelength) of k/2 ¼ 1�. It is based on theEIGEN-1S CHAMP-derived satellite-only globalgeopotential model (Reigber et al. 2002), yielding thelong-wavelength part up to about degree and order 30(k/2 ¼ 6�) with an internally estimated accuracy of10 cm and 1.0 mGal in terms of geoid heights andgravity anomalies, respectively. The intermediate wave-lengths up to degree and order 120 then are determinedby combining EIGEN-IS in a rigorous least-squares(LS) adjustment with land and marine gravity anomalies(Lemoine et al. 1998), ending up with an accuracy ofabout 25 cm and 2 mGal, respectively, at a resolution ofk/2 ¼ 1.5�. The coefficients up to degree and order 180are taken from the EGM96 model (Lemoine et al. 1998).The full model is considered to be accurate at the 30 cm(geoid) and 4 mGal (gravity anomaly) level, with ahigher accuracy over the oceans and a lower accuracyover the polar areas which lack complete surface datacoverage.

Gravity disturbances (Heiskanen and Moritz 1967, p.88), rather than gravity anomalies, are computed fromthe spherical harmonic coefficients so as to account forthe masses between the reference ellipsoid and geoid (cf.Chapman and Bodine 1979). However, the term ‘gravityanomaly’ is used throughout the text for convenience(cf. Hackney and Featherstone 2003). The geoid undu-lations are evaluated from the spherical harmonic coef-ficients at the surface of the ellipsoid, not taking intoaccount the difference between height anomalies andgeoid undulations (Heiskanen and Moritz 1967, p. 325ff)nor the effect on the geoid of the downward continua-tion of gravity from the surface (Sjoberg 1998).

4.2 Initial crustal data

High-resolution crustal data, such as used in theexample above for the East European platform, arenot available for the entire Earth. However, for globalstudies to uncover deep and dynamic effects from theEarth’s mantle, which are superimposed on the surfacewith lithospheric signals, the crustal structure would be

sufficiently resolved with wavelengths down to 1000 km.More detailed information is therefore needed to studythe isostatic state of the lithosphere.

The most recent 2� · 2� gridded global compilationof crustal layers’ thickness and density data, CRUST 2.0(Bassin et al. 2000), which is an improved version of the5� · 5� model CRUST5.1 (Mooney et al. 1998), is used.However, in specific regions, it is replaced by refineddata sets: over central and northern Eurasia by the datafrom Kaban (2001), and over North America by thedata from Kaban and Mooney (2001) and Mooney andChulik (2002). Both data sets have a horizontal resolu-tion of 1� · 1�, except for northern Canada. The geo-graphic distribution of Moho depths, representing themost up-to-date knowledge of this quantity, is shown inFig. 7. This Moho depth model is greatly improved withrespect to the older CRUST5.1 model. CRUST5.1 dif-fers from the refined Moho depth model by up to 10 km,especially under continents and their margins, even on alarge scale. The quality of the crustal model in definingMoho depths and crustal densities varies geographically.A detailed error discussion of the crustal model’suncertainties and their propagation to gravity is given inKaban and Schwintzer (2001). The effect of the crustalknowledge uncertainties on the computed isostaticanomalies is discussed in Sect. 3. The spatially hetero-geneous quality of crustal data is one reason to base thefollowing calculations mainly on spatial discretizationrather than on spectral analysis.

In addition, data on the age of the sea floor and thecooling lithosphere model are used to estimate the age-dependent density distribution in the oceanic uppermantle (Kaban et al. 1999). Unlike the situation underthe continents, this governs the isostatic compensationwithin the standard oceanic lithosphere.

4.3 Construction of a global isostatic modelof the lithosphere

Studies of lithospheric structures indicate that the typeof isostatic compensation differs substantially, eitherbetween tectonic units or in wavelengths. For example,

Fig. 7. Global map of the Moho depths (in km) belowsea level based on CRUST 2.0 (Bassin et al. 2000)supplemented by more detailed data over central andnorthern Eurasia (Kaban 2001) and over North America(Kaban and Mooney 2001; Mooney and Chulik 2002)

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structures such as the Urals, the Great Valley in thewestern US, or the shields of the East Europeanplatform, reveal an unusually high density of the solidcrust and a deep Moho, whereas isostatic compensationof the Basin and Ranges in the western US is providedbelow the Moho by an anomalously light upper mantle(see e.g. Kaban and Mooney 2001). Isostasy in northernEurasia, on the other hand, is a composite of large- andsmall-scale components, which are compensated indifferent ways (Kaban 2001). This means that ageneralized statistical approach like by admittance orcross-spectral analysis between gravity and topography(see e.g. McKenzie and Fairhead 1997) is applicableonly as a first approximation.

Instead, the initial density model of the crust is usedhere as a basis for the isostatic model of the lithosphere,and then fitted to observed gravity and topography byan LS adjustment. As was shown above, the consoli-dated crust densities obtained from seismic velocitiesintroduce large errors in the resulting isostatic anoma-lies. On the other hand, they represent some objectiveinformation and trends, in particular on a large scale. Tomake use of this information in the solution, the LSadjustment is performed using the given density valuesfor the layers in the consolidated crust as weighted apriori information. The idea behind this method is tominimize the isostatic gravity anomalies by modifyingthe initial lithosphere density structure, while keepingthe strong isostatic condition [Eq. (1)]. Clearly, thisprocedure is non-unique and the resulting isostaticanomalies depend greatly on the parameters used tostabilize the inversion.

The isostatic model of the lithosphere is finally con-structed in the following way. In a weighted LS adjust-ment that fits the solved-for densities in the crust andupper mantle to the residual gravity anomalies, taken asobservations, the quadratic form v2 of the weightedresiduals to be minimized reads

v2 ¼X

i

pðgiÞ cosui Dgires �

Xj

Xk

fðdqkj Þ

!2

ð3Þ

where u is the latitude of a 1� · 1� surface block area i,Dgi

res (observations) is the residual block mean gravitywith weight p(g)i obtained after removing (a) the effectof the initial crustal model including topography andwater, and (b) the gravity effect of the cooling oceanlithosphere over ocean areas. In Eq. (3), dqk

j (solved-forparameters) are the corrections to the initial densitymodel at block j in the depth range of layer k, and f isthe function which converts density anomalies intogravity at the surface. The density distribution must fitto the isostatic condition [Eq. (1)], where the discretedensity anomaly distribution Dqc

j ¼ DqcjðinitialÞ þ dqk

j isused for each block j over depths c with corrections dqk

jfor k layers.

Taking the initial density distribution as weighted apriori information to stabilize the solution because ofthe non-uniqueness of the above-posed problem, thefollowing quadratic form q is added to the right hand-

side of Eq. (3), i.e. to the function to be minimized in theLS adjustment:

q ¼ aX

j

Xk

pðqjÞ Dqcj � Dqc

jðinitialÞ� �2

¼ apðqÞX

j

Xk

ðdqkj Þ

2 ð4Þ

With this additional condition [Eq. (4)] we control,depending on the choice for the stabilization factor a,the amplitude of the solved-for parameters, i.e. thedensity anomalies within the crust and upper mantlewith respect to their initial values with weight p(qj),which are taken from the initial lithosphere model. Theweight p(gi) in Eq. (3) is equal to 1 and p(qj) in Eq. (4)equal to 20952, with Dgi

restaken in mGal and Dqcj (initial)

in g/cm3. The value of 2095 corresponds approximatelyto the ratio of gravity variations to density variations forthe selected units and a 50-km, thick layer.

The adjustment is performed in the spectral domainin terms of spherical harmonic coefficients where thesummation of all parameters over block indices i or j isreplaced by the summation over degree and order of thespherical harmonic coefficients. In the spectral domain,the function f represents the gravity kernel as a functionof depth and degree as given, for example, in Corrieu etal. (1995). In the system represented by Eqs. (1) and (3)we try to estimate k unknowns (density perturbationsdqk

j for k layers) from one observation Dgires plus one

isostatic condition equation, which is a non-uniqueproblem for k > 2. The uniqueness and stability of thesolution are attained by adding the right-hand side ofEq. (4) to Eq. (3) to be minimized in the LS adjustmentunder the condition of Eq. (1). However, we have todefine the number and boundaries of all layers in whichdensity is solved for. Since the number has to be small,the final density model should be considered as a gen-eralized one.

4.4 Relationship between the stabilization factor andadjustment results

Here, density anomalies for three layers, representingthe main structural subdivisions of the crust and uppermantle, are determined. The first layer goes down to25 km depth (below the geoid) and contains the upperpart of the continental crust and over ocean areas thewhole crust, including the uppermost mantle. Upper-most crust and sediments are not discriminated here,presuming that the relative density properties aredescribed sufficiently well by the crustal model for thisglobal study. The second layer ranges from 25 to 50 kmdepth, and comprises the lower part of thick continentalcrust and parts of the upper mantle elsewhere. The thirdlayer continues down to 150 km depth in the upper-mantle. Cratonic roots of the continents may reach300 km depth (see e.g. Grand et al. 1997), but densityvariations below 120 km are supposed to be muchsmaller in amplitude than those above (Forte and Perry

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2000, Kaban et al. 2003) and therefore no additionallayer is introduced.

The third layer also includes some part of the sub-lithospheric mantle, in particular under the oceans, butthe effect of large-scale upper mantle density variations,whether from below or from inside the lithosphere, onthe vertical displacement of a lithospheric block is sim-ilar. This is confirmed by numerical calculations that usetopography kernels (Green’s functions) to model uppermantle density inhomogeneities. For long-wavelengthdensity anomalies at depths of less than 150 km, thetopography kernels are close to unity, which is equiva-lent to the isostatic condition (see e.g. Corrieu et al.1995). After the solution by matrix inversion, the ad-justed density anomaly distribution for the lithosphere isobtained by adding the initial density distribution, witha higher vertical resolution than just three layers, to theassociated corrections dqk

j .The stabilization factor a determines the range of

possible amplitudes of density corrections within thethree layers. The density corrections are closest to theirinitial values for large values of a and, because the iso-static condition has to be fulfilled, of equal size for aninfinite a. While decreasing the damping factor, theisostatic gravity anomalies are minimized through anincreasing deviation of the adjusted density perturba-tions from their initial values. This dependence isinvestigated to find the optimum value a that givesrealistic results.

The adjustment is applied to the global gravity andcrustal data (described earlier) up to spherical harmonicdegree 180. Maximum density variations as a function ofthe stabilization factor are shown in Fig. 8. The valuesare taken out from northern Eurasia and North Amer-ica, which have the most reliable initial crustal data(recall that CRUST2.0 was supplemented by detaileddata in these regions). The maximum density perturba-tions occur mostly within the upper layer, thus sup-porting the assumption that the initial densities for theconsolidated crust are the most uncertain. Whiledecreasing a from 8 to 2, the maximum density anomalyvaries only insignificantly, and the amplitude remains atabout 100 kg/m3. For a < 2, the amplitudes start toincrease rapidly, indicating that a ¼ 2 is the optimumchoice for providing solution stability and the maximumadmissible freedom for the solution range.

The amplitudes per degree of the spherical harmonicexpansion of the resulting isostatic gravity anomaliesand of the initial free-air anomalies are shown in Fig. 9.The solutions obtained with a ‡ 2 reveal similar powerspectra from degrees 2 to 15, thus proving that thesolution for the isostatic gravity/geoid behaves stablywithin this long-wavelength band. For higher degrees,the relative differences in the degree amplitudes start toincrease for different a. It may be concluded that thelongest-wavelength components of the isostatic anoma-lies, which are assumed to be due to deep density het-erogeneities and their dynamic effect on the surfacetopography, are reliably determined by this analysis,while the shorter-wavelength components are highlysensitive to the choice of a. From the considerations

given above related to maximum density variations as afunction of a, the value a ¼ 2 is taken in the sequel toderive the isostatic gravity anomalies and geoid undu-lations.

4.5 Isostatic anomalies of the gravity field and geoid

Figure 10 shows the observed gravity anomaly field(Fig. 10a), the lithospheric gravity field (Fig. 10b)obtained by foreward computation of gravity from theadjusted density distribution (a ¼ 2) and, by subtractingthe lithospheric effect from the observed field, theisostatic gravity anomaly field (Fig. 10c). The litho-spheric contribution to observed gravity is substantial

Fig. 8. Maximum density perturbations with respect the initial crustalmodel (northern Eurasia and North America only) after isostaticadjustment as a function of the stabilization factor a

Fig. 9. Signal degree amplitudes of the isostatic gravity anomaliesobtained with different stabilization factors a together with thespectrum of the initial (observed) free air gravity anomalies

377

(Fig. 10b): its amplitude exceeds ±200 mGal. The mostsignificant variations are found in active tectonic areas:Tibet and the Himalayas, the Alpine–Mediterraneanfold belt, the western US and the Andes. This result is ingeneral agreement with the classical theory, according towhich elevated areas, even if isostatically compensated,are characterized by gravity and geoid heights, however,the amplitude of these heights depends on the actualcrustal structure. Over the area of Tibet and theHimalayas, the regional gravity maximum almost com-pletely vanishes after the isostatic gravity reduction andthe isostatic gravity now bridges the Siberian and Indianminima. A similar reduction in amplitude takes place forthe Andes and the western US. In general, the long-wavelength isostatic anomalies exhibit much less corre-lation with the lithosphere patterns than the observedgravity. This confirms the necessity of using the isostaticfield for global modelling of mantle dynamics, ratherthan the observed field.

The isostatic reduction in terms of geoid undulationsis shown in Fig. 11. All general tendencies are similar to

the gravity anomaly field reduction (Fig. 10), althoughthe relative contribution of the lithosphere to the ob-served geoid is less, because the long wavelengths in thegeoid contain more power. Nevertheless, in the activetectonic areas, the isostatic geoid reduction ranges from)18 to +43 m. The maximum value is again reached inTibet, while the large negative values mostly extend over‘old’ ocean areas with a deep ocean floor. The differencein the isostatic reduction of 20 m between oceanic ridgesand the old ocean purely reflects the isostatic balance ofthe oceanic lithosphere. The unification of the Siberianand Indian local minima into a global minimum is evenmore pronounced in the isostatic geoid than in the iso-static gravity anomalies.

5 Dynamic versus isostatic residual topography

The residual topography is the integral in Eq. (1)computed with the initial lithospheric density model(corresponding to the residual gravity anomalies). In

Fig. 10. Isostatic reduction of free-air gravity anomalies (mGal). aFree-air gravity anomalies (from the spherical harmonic globalgeopotential model to degree and order 180); b the gravity effect ofthe isostatically compensated lithospheric model; c isostatic gravityanomalies (a minus b)

Fig. 11. Isostatic reduction of the geoid (m). a Geoid (from thespherical harmonic global geopotential model to degree and order180); b the geoid effect of the isostatically compensated lithosphericmodel; c isostatic geoid (a minus b)

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order to separate the surface topography (qtop ¼ Dqtop)and internal density inhomogeneities, Eq. (1) is slightlymodified as follows:

Hres ¼1

�qðqtopÞHobs þ

1

�q

ZT

0

DqðtÞ R� tR

� �2

dt ð5Þ

where qtop is the the average block density (including theeffects of ice and sediments) of the topography (Hobs), �qis the average density of the residual topography, set to2670 kg/m3 to convert the residual mass into theresidual topographic height Hres, and the other param-eters are defined as in Eq. (1).

The calculated residual topography, low-pass filteredto a spectral resolution up to degree and order 20(k/2 ¼ 1000 km), is shown in Fig. 12a. It varies between)1.0 and +2.5 km; the zero level corresponds to an old(180 Ma) ocean reference column. The long-wavelengthresidual topography arises (principally) from two sour-ces. First, it depends on the density distribution in thelithospheric part of the upper mantle: highs are sup-ported by low-density lithospheric roots, while lows arebalanced by high-density anchors. The second source of

the residual topography is normal stress at the base ofthe lithosphere due to mantle flow. This part of theresidual topography is defined as ‘dynamic’ topography.Large-scale upper-mantle density variations just belowthe lithosphere still contribute to isostatic compensationof topography, similarly to long-wavelength lithosphericdensity variations. Therefore, the dynamic effect isconsidered here to arise only from heterogeneities below300 km, which is the assumed maximum depth ofcontinental roots.

5.1 Long-wavelength residual topography

One of the main problems of present-day geodynamicsis to estimate the dynamic contribution of deepheterogeneities to the residual topography, i.e. thelong-wavelength non-isostatic component. The for-ward-computation method implies the construction ofa complete global dynamic model of the Earth, includ-ing the density and viscosity distribution. Due to thelarge uncertainties in the knowledge of these parameters,this kind of study shows largely varying results.Different authors give amplitudes of the dynamictopography ranging from 0.5 km (see e.g. Hager and

Richards 1989; Cadek and Fleitout 2003) to 1 km (seee.g. Steinberger et al. 2001) and 2 km (see e.g. Pari andPeltier 2000). In this paper, a simple empirical approachis tried to estimate this effect.

In order to separate in the computed residualtopography of the dynamic part (Fig. 12a) from that ofthe isostatic part, a correlation analysis between theresidual topography and the isostatic gravity anomalies(Fig. 10c) was performed. Both fields were developed interms of spherical harmonics and truncated after degreeand order 20, the wavelength boundary previouslyfound for deep-mantle signals. The idea is that thegravity signatures due to the dynamic topography (i.e.that part of the topography generated and supported bymantle flow below the lithosphere) are only reduced inamplitude by the isostatic reduction, but otherwisepreserved. As the origin of long-wavelength dynamictopography is probably in the deeper mantle, the dif-ferential gravity signal between topography and deepmantle should be larger than that between topographyand lower crust/upper mantle, where (falsely) compen-sating masses were placed in the adjustment whenapplying the isostatic condition [Eq. (1)]. On the otherhand, if the isostatic condition is justified, then theresidual topography is completely compensated withinthe crust and upper mantle in the adjustment and nogravitational signal should be left in the isostatic gravityanomalies.

The residual topography reflects a mixture of dy-namic and isostatic residual topography, and the iso-static gravity anomalies also contain modelling errors.Therefore, a simple correlation analysis between bothfields should reveal the broad features of the dynamictopography. Since the contribution of the dynamictopography to the isostatic anomalies and residualtopography depends on wavelength, their relationship is

Fig. 12. Residual topography (km) with a resolution correspondingto a spherical harmonic expansion up to degree and order 20. Theresidual topography was computed by subtracting—from thetopography—the compensating masses within the crust (initial densitydistribution) and over ocean areas, the additional compensatingmasses within the lithosphere (density distribution according to thecooling oceanic plate model as a function of ocean floor age). Thezero level corresponds to 180-Ma-old oceanic lithosphere (thereference model). a Overall residual topography (initially unexplainedmass excess and deficiency, respectively); b part of the residualtopography that is correlated with the long-wavelength isostaticgravity anomalies (dynamic part)

379

investigated in the spectral domain for different spheri-cal harmonic degrees. The admittance between theseparameters as a function of the harmonic degree isshown in Fig. 13. It was found that the admittance be-tween residual topography and isostatic anomalies isdifferent from zero up to degree 10, and varies within thedetermination errors around the average value of0.65 · 10)2 km/mGal, while for the higher degrees thecorrelation is practically lost. Thus, this single admit-tance coefficient is used to scale the isostatic anomaliesup to degree 10. The result, the correlated part of theresidual topography, is regarded to represent the dy-namic topography, and the remaining (uncorrelated)part is considered to reflect the isostatic part ofthe residual topography, which is due to an initiallyincomplete knowledge of the crustal/upper-mantlestructure.

The geographic distribution of the resulting dynamictopography is given in Fig. 12b. The dynamic topogra-phy creates, together with the direct effect of deep den-sity heterogeneities, the long-wavelength signal in theisostatic gravity anomalies and geoid heights. The ap-proach used does not necessarily pick up the overalldynamic topography, since—depending on the viscositydistribution in the Earth’s interior—the joint gravitycontribution of these two factors may cancel out, or thedensity inhomogeneity may dominate. However, since aknowledge of the dynamic topography is essential to anunderstanding of Earth dynamics, these first-order esti-mations are helpful when no other direct data areavailable. It was found that the dynamic topographyvaries between )0.5 (down-welling) and +0.4 km (up-welling). This order-of-magnitude estimate is in accor-dance with Hager and Richards (1989) and Cadek andFleitout (2003). Even if we assume that an equal amountof dynamic topography is hidden in the non-correlatedpart of the dynamic topography, the total amplitudewould increase only by �40%. Thus, it can be concludedthat the larger part of the residual topography withamplitudes of about 2 km is attributed to the initiallyunresolved isostatic topography.

5.2 Local non-isostatic topography

Small-scale isostatic gravity anomalies should reflectlocal disturbances of isostasy due to crustal rigidity(regional compensation) if the crustal density distribu-tion were perfectly known and modelled, i.e. therecognition of isostatic disturbances depends on theexistence and quality of the initial crustal data. Thus,because of the poor knowledge of the crustal densitydistribution over most parts of the globe, the computedlocal isostatic anomalies generally cannot be used as areliable criterion for the isostatic balance of tectonicstructures. Instead, observed gravity is used here toimprove the initial density model by LS adjustment. Theresidual topography (non-isostatic topography) is thencomputed according to Eq. (5), and taken as a measurefor the isostatic balance of the lithosphere. For thispurpose, the adjustment using Eqs. (3) and (4) is againapplied but omitting the isostatic condition of Eq. (1),because it is supposed that—on a small scale—theisostatic condition does not fit reality. Contrary to thelong-wavelength considerations given above, observedgravity is allowed to completely determine the litho-spheric density structure, but this depends upon thechoice for the stabilization factor a.

The appropriate factor a for constraining theadjustment is chosen according to the following criteria:maximum value of the adjusted density anomaly withrespect to the initial model, RMS (root mean square) ofthe gravity residuals after adjustment, and RMS of thenon-isostatic topography. Numerical values of theseparameters as a function of the stabilization factor areshown in Fig. 14. The RMS of the gravity residualsgradually decreases with decreasing a and equal 2 mGalat a ¼ 2. This value corresponds to the accuracy levelover the well-determined areas in the gravity field model,as described earlier. The maximum value of the adjusteddensity anomaly with respect to the initial model is ofcourse smaller than in the previous case where the iso-static condition [Eq. (1)] has been applied. It can be seenfrom Fig. 14 that the maximum density variation onlydecreases slowly for a ‡ 2. The same is true for the RMSof the non-isostatic topography, depending much less onthe choice of a than the RMS of the gravity residuals.The resulting non-isostatic topography can therefore beconsidered as a reliable measure for the isostatic balanceof the lithosphere. It is again concluded that a ¼ 2 is themost appropriate choice for the final solution.

This procedure clearly works better for estimatinglocal isostatic disturbances than interpreting isostaticgravity anomalies computed in the classical Airy way. Itis also easy to separate the local non-isostatic topogra-phy from the dynamic topography and effects of deepinhomogeneities, which are only present in the long-wavelength part of the spectrum. The full spectrum ofthe non-isostatic topography after inversion (a ¼ 2) isshown in Fig. 15. It differs significantly from the spec-trum shown in Fig. 9, found when applying the isostaticcondition [Eq. (1)]. A sharp minimum appears in Fig. 15at harmonic degree 20. Again, in accordance with thereasoning formulated above, the choice of degree 20 as

Fig. 13. The relationship (admittance in km/mGal) between residualtopography and isostatic gravity anomalies in Fig. 10 as a function ofspherical harmonic degree. The error estimates are based on a 95%confidence level. The dashed line shows the regression coefficient(0.65 · 10)2 km/mGal) obtained for all spherical harmonic coeffi-cients within the interval from degree 2 to 10

380

the boundary between long-wavelength dynamictopography (taken from the adjustment including theisostatic condition) and deep-mantle effects on one sideof the spectrum, and non-isostatic topography on theother, is confirmed. It is concluded that the non-isostatictopography derived here for degrees greater than 20characterizes the isostatic balance of the lithosphere.

The geographical distribution of the non-isostatictopography (degrees 21 to 180) is shown in Fig. 16. TheRMS over the entire Earth is about 0.2 km. Ignoring theuncertain polar regions, the maximum amplitudes()1.5 km, +1 km) occur along active ocean-to-conti-nent and intra-continental plate boundaries. Withinstable continental areas, non-isostatic topography varieswithin ±0.3 km and is supported by lithospheric stres-ses. The non-isostatic topography within stable oceanicregions is substantially smaller in amplitude, which justconfirms the belief that the whole oceanic lithosphere isweaker than the continental one.

The large variations of the non-isostatic topographyin near-polar areas are likely to be due to errors in theinitial gravity data, since such variations are not ob-served over Africa and South America, where the crustaldata are also unreliable. From this, it is concluded thatthe resulting non-isostatic topography can be reliablydetermined from accurate gravity data even if there are

large uncertainties in the initial crustal data. Forth-coming improvements in high-resolution gravity fieldmodels from the GRACE and GOCE satellite missions,as well as additional surface gravity data, will probablyresolve the polar cap problem.

5.3 Density model of the lithosphere

Based on the above adjustment results, a compositedensity model of the lithosphere—as one of the mainoutcomes of this study—is constructed by merging theresults related to some specified wavelength intervals.For the long wavelengths (up to degree 20), the model istaken from the adjustment including the isostaticcondition [Eq. (1)], and for the shorter wavelengthsfrom the adjustment excluding the isostatic condition.The transition has been smoothed a posteriori inbetween degrees 20 and 25. Figure 17a shows theaveraged initial density anomalies within the consoli-dated crust down to the Moho relative to the referencevalue of 2850 kg/m3. Figure 17b gives the density

Fig. 14. Top: estimation of non-isostatic, short-wavelength topogra-phy and residual gravity: root mean square (RMS) of residual gravity(solid line) and non-isostatic topography (northern Eurasia and NorthAmerica only) as a function at the stabilization factor a applied in theLS adjustment without the isostatic condition; bottom: the same forthe maximum density perturbation

Fig. 15. Signal degree amplitudes of the LS adjusted non-isostatictopography (stabilization factor a ¼ 2)

Fig. 16. Geographical distribution of non-isostatic topography (inkm) derived from the spherical harmonic representation from degrees21 to 180

381

anomalies within the consolidated crust obtained fromthe adjustments. The Moho position was not correctedand the obtained density model therefore includes Mohodepth uncertainties. This effect, which is not separablefrom pure density anomalies, is clearly visible in thoseareas lacking seismic data, especially near the bound-aries of Africa, Australia and South America. In theseismically well-studied areas of North America andEurasia, the changes made to the initial crustal modellook more reasonable. For example, the adjusted densityanomalies of the consolidated crust under westernEurope and the East European platform fit much betterthan the initial crustal data to the contours of the maintectonic units known from geological maps (e.g. the

Rhein graben, Black Sea, South Caspian depression,Forecaspian basin, Ukrainian shield).

Figure 17c shows the average density anomalieswithin the 100-km-thick sub-crustal layer below theMoho. The initial model has been computed followingthe cooling ocean lithosphere model under the oceansand adopting a constant value under continents. Themost significant negative density anomalies (down to–0.15 g/cm3) are found under East Africa and the NorthAtlantic in the vicinity of Iceland. Mid-ocean ridges arecharacterized by smaller, yet still significant, negativeanomalies (down to –0.1 g/cm3). Similar negativeanomalies are found in the south-western Pacific, wes-tern US and some parts of south-east Asia. The densityof the upper mantle under continental areas is on aver-age close to the density of the undisturbed old ocean.The results for the continents agree well with the resultsobtained earlier by Kaban et al. (2003).

6 Discussion and conclusions

Isostatic gravity anomalies and geoid undulations reflectsignals of different origins that complicate their geo-physical interpretation. Therefore, it is important tomodel the isostatic compensation scheme, i.e. the densitydistribution within the lithosphere including the positionof the main discontinuities, as closely as possible toreality. Otherwise, when computing isostatic anomaliesin the classical way based on the surface topography andthe simple Airy scheme, the isostatic gravity field to alarge extent reflects differences of the real lithosphericstructure from the simplified compensation model.Thus, it is crucial to take into account all available dataabout the lithospheric density structure, in particularsedimentary basins and the depth of the Moho, in orderto obtain a reasonable computation of the isostaticreduction.

The reduction of isostatically balanced lithosphericsignals in both gravity and geoid, leading to the isostaticgravity anomalies and geoid undulations, is a pre-requisite when looking for signals originating from thedeep-Earth interior. Even for the geoid, where thelong-wavelength contribution dominates, the isostaticreduction reaches absolute values of up to 40 m. As awhole, and after the reduction, the isostatic gravityanomalies and geoid undulations do not correlate wellwith topographic and bathymetric features, unlike theobserved gravity fields.

Looking at the isostatic gravity anomalies and geoidundulations, a lot of geophysical phenomena can beidentified. At the largest scale (spherical harmonic de-gree <6), the minima over Siberia down to the Indianocean and over North America, and the maxima overthe North Atlantic, southern Africa and over the south-western Pacific, all appear to dominate. These large-scale structures are presumably a sign of global mantleconvection and/or deep density inhomogeneities.

The medium-wavelength isostatic anomalies relate toglobal tectonic features, e.g. plate convergence zonesand cratons. To emphasize these features, we remove

Fig. 17. Mass-density anomalies in the consolidated crust and uppermantle (in g/cm3). a Initial crustal density model (vertically averagedover consolidated crust relative to the reference density value of2.85 g/cm3): CRUST2.0 supplemented over North America andcentral and northern Eurasia by more detailed data (Kaban 2001,Kaban and Mooney 2001); b vertically averaged density anomalieswithin the consolidated crust after having adjusted the lithosphericdensity; c density anomalies in the upper mantle (averaged over a 100-km-thick layer below the Moho)

382

both long- and short-wavelength components from thetotal field in Fig. 10c. The isostatic anomalies in Fig. 18are limited to wavelengths in between spherical har-monic degrees 6 and 20 (7000 and 2000 km). Thedecomposition of long- and medium-wavelength fea-tures is not optimal in the spatial domain, because thespectrum (Fig. 9) is continuous over degrees 2 to 20, butthe main features become visible. Despite the reductionin magnitude of the free-air gravity anomalies overmountainous areas near active plate boundaries, theisostatic anomalies reveal significant broad linear max-ima that are associated with the Alpine–Mediterranean–Himalayan fold belt and active Pacific boundaries,including the western South America plate boundary.However, the locations of these positive anomalies differremarkably from those in the observed gravity field.Instead of the isolated Tibet maximum, the new linearmaximum bounds the Himalayas from the south. In themap of the isostatic gravity anomalies (Fig. 10c), thisstructure is covered by the large-scale minimum, butafter removing the longest-wavelength component, it isjoined with the maximum related to the Alpine–Medi-terranean plate convergence zone. For the Andes, afterelimination of the influence of the lithosphere, the broadlinear positive anomaly is spread to the east of the mainridge.

The underlying sources of these anomalies are notquite clear. One possible explanation is that theseanomalies are an image of remnants of subducted slabsin the upper mantle. However, the anomalies do notalways follow this hypothesis. For instance, at the Pa-cific/American boundary, these maxima are—as onewould expect—shifted towards the direction of thesubducting slabs (both present and past). However, thesituation is different in the area of the India–Eurasiacontinent-to-continent plate collision zone, where theisostatic maximum is located over the Indian plate, thusin front of the subduction zone. A second possibleexplanation is that these anomalies reflect a dynamicinstability in the upper mantle, which could also corre-spond to small-scale convection cells related to thesubducting slabs. This analysis provides valuable newconstraints for further geodynamic modelling to clarifythe origin of these anomalies.

Pronounced negative anomalies in Fig. 18 are some-times related to the cold, deep roots of the conti-nents—cratons such as the Canadian and Baltic shields,as well as the Australian cratons. The Baltic- andCanadian-shield anomalies also correlate with the mainuncompensated post-glacial deformations. However, theanomalies in the gravity field are about a factor of two,or more, larger for the Baltic (Prilepin et al. 2002) andthe Canadian shields (Vermeersen et al. 2003) than arethose predicted by glacial isostatic adjustment simula-tions. Thus, we have to search for additional processes,or data errors, contributing to these anomalies. Recentresults based on numerical forward modelling of mantleconvection show that continents, especially their colddeep roots, interact with mantle flow (Trubitsyn 2000).For instance, the cold, high-viscosity roots could ‘at-tract’ (on a geological time scale) down-going currents inthe mantle. The downward flow then may provide anexplanation for the additional depression of the Balticand Canadian shields and the Western Australian cra-ton.

The third length scale in the isostatic gravity anom-alies concerns features with spatial wavelengths smallerthan 2000 km (spherical harmonic degrees >20). Thisboundary shows as a sharp change in the computedspectra of isostatic gravity and non-isostatic topogra-phy. The short-wavelength component contains un-modelled density variations within the crust, localisostatic disturbances and uncertainties in the initial datasets. The use of short-wavelength isostatic gravityanomalies for the investigation of regional disturbancesof the isostatic condition proves to be somewhat useless,until the density structure of the crust is known ingreater detail.

Contrary to the short-wavelength isostatic gravityanomalies, stable results were obtained for the short-wavelength non-isostatic topography. This non-isostatictopography is estimated after having adjusted the crus-tal/upper-mantle density distribution using observedgravity. The residual, i.e. non-isostatic, topography isless sensitive to the redistribution of masses within thelithosphere than gravity and, therefore, is more ade-quate to estimate short-wavelength isostatic distur-bances originating from a rigid support of external andinternal load within the lithosphere. Logically, themaximum variations of this parameter ()1.5 to +1 km)are found along the active ocean-to-continent and intra-continental plate boundaries. The short-wavelengthnon-isostatic topography is generally correlated with thedistribution of effective elastic plate thickness of thelithosphere. Specifically, larger amplitudes, compared tooceanic areas, are observed over continental areas, evenif tectonically stable.

The amplitude of the long-wavelength non-isostatic(dynamic) topography that is supposed to be generatedby mantle dynamics from below 300 km was estimatedto range from )0.4 to +0.5 km. This result is at thelower limit of what was estimated from direct modellingof mantle convection and its effect on the Earth’s sur-face. The identification of dynamic topography is crucialwhen modelling deep-Earth processes. The gravitational

Fig. 18. Medium-wavelength isostatic gravity anomalies (in mGal)from spherical harmonic degrees 6 to 20. Shaded areas correspond topositive values (>10 mGal); negative isolines are shown by dashedlines

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effect of the dynamic topography is still present in theisostatic gravity anomalies and geoid undulations be-cause of the application of the isostatic condition in thereduction process.

The density distribution within the consolidatedcrust and the upper mantle obtained from the isostaticadjustment with the initial density model from seismicobservations as a priori information is one of theprincipal outcomes of this study. The crustal densitydistribution corresponds well to known geologicalprovinces. The upper-mantle density heterogeneitiesparticipating in the isostatic compensation are sub-stantial and exceed ±100 kg/m3. Since these anomaliesextend to a significant depth, the cumulative effect ofthe lithosphere is significant even at long wavelengths.The most pronounced negative anomalies are foundnear large-plume provinces such as Iceland and EastAfrica, and also in the vicinity of the mid-ocean ridges.Positive density anomalies in the upper mantle occur incontinental lithosphere, but not within cratons that areassociated with cold, thick lithosphere. This resultconfirms—in general—the isopycnic (equal-density)hypothesis of Jordan (1978) regarding density varia-tions of thermal and chemical origin, and is in agree-ment with the conclusions made on the basis of adetailed gravity modelling of the continental litho-sphere (Kaban et al. 2003).

Acknowledgements. Thoughtful reviews by one anonymous re-viewer, L.E. Sjoberg and B.L.A. Vermeersen improved the textgreatly.

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