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Earth and Planetary Science Letters 295 (2010) 127–138

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Earth and Planetary Science Letters

j ourna l homepage: www.e lsev ie r.com/ locate /eps l

The influence of dynamically supported topography on estimates of TeDan McKenzie ⁎Department of Earth Sciences, Bullard Labs, Madingley Road, Cambridge CB3 0EZ, United Kingdom

⁎ Tel.: +44 1223 337191.E-mail address: [email protected].

0012-821X/$ – see front matter © 2010 Published by Edoi:10.1016/j.epsl.2010.03.033

a b s t r a c t
a r t i c l e i n f o
Article history:Received 27 September 2009Received in revised form 9 February 2010Accepted 19 March 2010Available online 7 April 2010

Editor: R.D. van der Hilst

Keywords:elastic thicknesspostglacial upliftmantle convection

The effective elastic thickness, Te, of continental lithosphere can be estimated from the relationship betweengravity and topography in the spectral domain. Two methods have been used, one of which depends on thecoherence between Bouguer gravity anomalies and topography, whereas the other uses the transfer function,commonly known as the admittance, between the free air gravity and topography. The two methods giveestimates of Te which differ by as much as an order of magnitude in those continental regions wherevariations in elevation are small. This problem has led to much controversy. An important concern is theextent to which estimates of Te are affected by dynamically maintained gravity and topography, arising frommantle convection and postglacial recovery. Unlike elastically supported anomalies, these processes cangenerate gravity and topographic long wavelength (N500 km) anomalies. If such anomalies are modelled asbeing supported by elastic forces, the resulting values of Te are overestimated, often by a large amount.

lsevier B.V.

© 2010 Published by Elsevier B.V.

1. Introduction

The thickness of the elastic layer Te, which supports topographicvariations for millions of years and transmits stresses across plates,can be estimated in a number of ways. One method that has beenwidely used is the relationship between gravity and topography (orbathymetry) in the spectral domain. In oceanic regions this approachgives values that depend on the age of the oceanic plate, and whichincrease from about 5 km on spreading ridges to 40 km beneath oldoceanic lithosphere. Watts (2001) showed that most estimates of Tefrom oceanic regions require the oceanic lithosphere to behaveelastically to the depths of the 300°–600 °C isotherms, calculated usinga model of the oceanic lithosphere with constant thermal conductivity.When the temperature dependence of the thermal conductivity is takeninto account, this temperature range decreases to 200°–500 °C. Theelastic thickness is everywhere less than the seismogenic thickness, Ts,which closely follows the 600 °C isotherm (Jackson et al., 2008) whenthe thermal conductivity is temperature dependent. The differencebetween Te and Ts is to be expected, because the strain accumulation andrelease involved in earthquake faulting occurs on a timescale of a few kawhereas the elastic stresses that support topography must be main-tained for 10–100 Ma. Oceanic bathymetry changes from beingelastically supported at wavelengths less then ∼500 km to beingdynamically supported by mantle convection at wavelengthsgreater than ∼1000 km. In oceanic regions the ratio of free air gravityto topography, the admittance Z, is approximately 30 mgal/km(1 mgal≡10 μms−2, 1 mgal/km≡10−8 s−2) at long wavelengths

(Crosby et al., 2006; Crosby and McKenzie, 2009) and is alsoapproximately constant. This framework for understanding the rheologyof the oceanic lithosphere is now generally accepted and is consistentwith the expected behaviour of mantle materials.

On the continents the situation is much less satisfactory and hasgenerated a long-running controversy. The most widely used ap-proach is that of Forsyth (1985), who argued that the observedbehaviour of Z(k), where k=2π/λ is the wavenumber and λ thewavelength, results in part from the topography produced by internalloads. He showed that the coherence between the topography andBouguer gravity could be used to estimate Te, and that this approachwas little affected by such loads. Forsyth's method has been widelyused, and gives estimates of Te which often exceed 100 km in shieldregions (see Kirby and Swain, 2009). Such values greatly exceed theseismogenic thickness, and require material as hot as 800–1000 °C tobe able to sustain elastic stresses for ∼100 Ma (Jackson et al., 2008).Therefore such large values of Te cannot be explained using the simpleideas that successfully account for the behaviour of the oceans.

This situation lead McKenzie and Fairhead (1997) and McKenzie(2003) to examine the assumptions that underlie Forsyth's (1985)method in more detail. They argued that the coherence methodprovides only an upper bound on Te when the correlation betweentopography and free air gravity is small. Internal loadingmust initiallydeflect the plate and hence generate topography. But erosion acts toremove topography and to fill depressions, and so produces a flatlandscape. However, it has little effect on the gravity anomaliesproduced by subsurface density contrasts. Erosion therefore generatessurface loads that are coherent with the internal loads, and their ratiois given by the condition that there must be no resulting topography(McKenzie, 2003). If such loads are present and the Earth's gravityfield could suddenly be set to zero, the resulting elastic rebound

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128 D. McKenzie / Earth and Planetary Science Letters 295 (2010) 127–138

would generate topography. McKenzie and Fairhead (1997) calledgravity anomalies that had no topographic expression ‘noise’. Forsythexplicitly excluded internal loads that correlate with surface loadsfrom his model, and his expressions will therefore not give the correctvalue of Te if such loads are important. Crosby (2007) carried out avariety of numerical experiments which showed that the value of Teestimated using the coherence method is indeed larger than the truevalue in the presence of such noise, and that this difference increasesas the mean free air coherence, γ2

FA

――, decreases. He used Forsyth's

(1985) method to obtain estimates of Te, Teest., for various valuesof γ2

FA

――. When Te=15 km he showed that his results could be

described by

Test:e = 13:09−12:73ln γ2

FA

――

: ð1Þ

If the same relationship applies to other values of Te then

Test:e

Ttruee

= 0:9127−0:8487ln γ2FA

――

ð2Þ

can be used to correct Teest., obtained from the Bouguer coherence, forthe effect of loads with no topographic expression. Since γ2

FA

――is often

as small as 0.05 in shield areas with subdued topography, Crosby'sexpression suggests that Te

est./Tetrue can be at least as large as 3.5.McKenzie and Fairhead (1997) and McKenzie (2003) argued thatthere is no evidence that Te exceeds Ts, even though Te cannot alwaysbe estimated from Z(k) when the coherence between free air gravityand topography is small.

These conclusions have not been generally accepted (Watts, 2001;Perez-Gussinye and Watts, 2005; Kirby and Swain, 2009; Perez-Gussinye et al., 2009). Almost all work on this problem in the last tenyears has used Forsyth's coherence method and has obtained largevalues of Te in shield regions. Kirby and Swain (2009) have recentlycarried out a thorough and detailed study of this problem using datafromNorth America, shown here as Fig. 1, and both the coherence and

Fig. 1. Estimates of Te from Kirby and Swain (2009) using (a) the coherence method of Forsythe same at all wavelengths.

the admittance methods. In the continental interior the coherencemethod gives values of Te of 60 km or more, whereas the admittancevalues from the same regions are 30–40 km. However, north of about50°N both methods give values of 100 km or more. At a depth of100 km the temperature of the Canadian Shield is about 740 °C(Kopylova et al., 1998; Jackson et al., 2008). The difference inbehaviour between the southern and northern regions is surprisingand unexpected. The lithospheric thickness derived from surfacewave tomography (Priestley and McKenzie, 2006) is similar in bothregions, and therefore temperature differences are unlikely to beresponsible for this difference in behaviour. Neither McKenzie andFairhead (1997) nor McKenzie (2003) studied northern Canada. Onepurpose of this paper is to do so, in order to understand therelationship between topography and gravity in the interior regions ofCanada where the topography is subdued.

2. Data processing and modelling

The free air gravity data and topography at the points where it wasdetermined were obtained from the Geological Survey of Canada'sweb site, gdr.nrcan.gc.ca/gravity. The Bouguer anomalies Δgb for theUSA (Hittelman et al., 1994) were converted to free air anomalies Δgfusing

Δgf = Δgb + 0:11194t ð3Þ

where t is the topography in metres. The topography of the USA,Fennoscandia and Africa (V12.1) was obtained from Sandwell's website, topex.ucsd.edu/cgi-bin/get_data.cgi. Free air gravity data forFennoscandia was provided by GETECH (Fairhead, personal commu-nication). That for Africa is from Pavlis et al. (2008, EGM2008 earth-info.nga.mil/GandG). The spherical harmonic coefficients obtainedfrom GRACE data are from www-app2.gfz-potsdam.de/pb1/op/grace/results. Theywere used to generate gravity on a 1°×1° grid in the passband 4000≥λ≥500 km. Comparison of Δge from EGM2008 with the

th (1985), and (b) the admittance method with the fraction of internal to total loading

129D. McKenzie / Earth and Planetary Science Letters 295 (2010) 127–138

Canadian free air gravity data Δgc shows that—Δge =

—Δgc = 0:94 when

λ=152 km, where the bars denote averages, and decreases to 0.81when λ=103 km. Therefore no values of admittance and coherencewith λb150 km from EGM2008 were used to estimate Te.

Estimates of admittance and coherence were obtained using themultitaper method (Thomson, 1982) in the x and y directions, usingfour prolate spheriodal wave functions with frequency bandwidthsNW=3. The estimates were obtained by windowing and Fouriertransforming all the data within the boxes shown, and therefore theestimates obtained at all wavelengths apply to the same region.Where the surface was below sea level the water load was convertedto an equivalent thickness of rock using a density of 2.67 Mg m–3. Theadmittance was assumed to be isotropic, but was not assumed to bereal. In all cases yet examined the phase of the admittance, which isthe same as the phase of the coherency used by Kirby and Swain(2009), is not significantly different from 0° or 180°. Estimates of thecross spectra of gravity and topography were binned in annuli in kspace, with radii at constant intervals Δk. The binning only usedvalues of the cross spectra with kx≥0. Error estimates for themagnitude and phase of the admittance were obtained using theexpressions given in the caption to Fig. 4. Windowing and Fouriertransforming converts N independent values of gravity and topogra-phy into 8N complex values of the amplitude in the spectral domain.These complex amplitudes cannot therefore all be independent, asMcKenzie and Fairhead (1997) and McKenzie (2003) assumed. Thenumber of independent estimates in each annulus was estimated byreducing the number of values by a factor of NW2, and is not in generalan integer. The numbers for the three longest wavelengths are givenin the relevant figure captions. The number for the longestwavelength annulus is generally too small to provide accurateestimates of the admittance and coherence.

The standard models used to interpret the relationship betweengravity and topography (Forsyth, 1985; McKenzie, 2003) are static

Fig. 2. (a) The subaerial admittance Zc resulting from convection in a half space, driven by a toverlain by a lithosphere of thickness 120 km containing an elastic layer of thickness Te (Eq(c) The surface topography and gravity anomalies used to obtain Zc in (a). (d) and (e) Timeelastic layer of thickness Te (Eq. (A18)) for two values of mantle viscosity. The numbers ne

models, and apply surface and internal loads to an elastic layer ofthickness Te overlying a fluid half space. The theoretical admittancevalues below were obtained using McKenzie's (2003) expressions fora two layer crust (see Appendix A). Internal loading, if required, wasplaced on the boundary between the upper and lower crust. Theelastic thickness Te was obtained by minimising H

H =1N

∑N

n=1

Z on−Z c

n

ΔZ on

� �2" #1=2

; 500≥ λ≥ 150km ð4Þ

where Zno is the observed admittance with standard errorΔZno and Znc is

the calculated admittance. The values of H for the five boxes discussedbelow range from 0.13 to 0.58, and with an average of 0.34. Thesevalues suggest that the ad hoc method used here to estimate ΔZno mayoverestimate its value.

The static elastic model used to calculate Zf is clearly inadequate ifthe load is time dependent. An obvious extension is to allow the fluidhalf space to consist of a viscous fluid, which has the added advantageof allowing the effects of mantle convection to be included. Therelevant expressions are given in the Appendix A. They are plotted inFig. 2, and are used to explore the effects of postglacial uplift andmantle convection on estimates of Te. Eq. (A25) shows that themagnitude of the surface deformation resulting from a constant rateof loading or unloading ζ

.0 is τζ

.0, where τ is the time constant for

recovery from an initial displacement of the surface. Deglaciationproduces themost rapid change in lithospheric load, corresponding tothe removal of ∼3 km of ice in ∼10 ka. In comparison, the rate ofloading produced by erosion and volcanism are at least two ordersof magnitude slower. Though rainfall produces as rapid variations asdoes deglaciation, the total change in load is much smaller. Mantleconvection affects estimates of admittance because it supportscoherent long wavelength variations in gravity and topography.

emperature variation T=T0 cos kx imposed on the upper surface, when the half space is. (A18)). The numbers next to the curves show the relevant value of Te in km. (b) andconstants in ka for postglacial recovery of a submarine viscous half space overlain by anxt to the curves show the relevant values of Te in km.


Fig. 2a–c shows the behaviour of the admittance, topography and freeair gravity anomaly when they result from convection beneath elasticlayers of various thicknesses, the temperature variation maintainingthe convection is 100 °C, and the lithospheric thickness is 120 km. Atwavelengths less than about 500 km the resulting gravity andtopographic anomalies are small and are likely to be dominated bystatic lithospheric loads. At longer wavelengths the admittance is

Fig. 3. (a) North American lithospheric thickness (Priestley and McKenzie, 2006). The heavydimensions of box 1 are 3500 km×4500 km and of box 2 are 4000 km×5300 km. (b) Themaximum, and today (Lambeck et al., 2006 and personal communication). (c) Gravity anompass filtered to pass wavelengths between 4000 and 500 km. All maps are in Lambert equa

approximately constant, with a value of about 50 mgal/km. Thoughthe admittance changes sign with decreasing wavelength, both thetopographic and gravity anomalies from convection are small wherethis occurs, and it is unlikely that this behaviour will be observable.

Fig. 2d and e shows the behaviour of the time constant τ for twovalues of mantle viscosity for submarine postglacial recovery, becauseCanada and Fennoscandia were covered with sea water during the

lines show the outlines of the boxes used to calculate the quantities shown in Fig. 4. Thechange in ice thickness between 20 ka before present, close to the time of the glacialalies, calculated from potential coefficients determined from GRACE observations, bandl area projection, centred on 50°N, −100°E.


time most of the uplift took place. As Lambeck (1993) and Lambecket al. (1996) have emphasised, and as is shown in these figures,postglacial uplift observations allow a tradeoff between η and Te, withthe viscosity increasing with Te. The admittance of postglacialdeformation is given by Eq. (A26), and is little different from that ofuncompensated topography.

Kirby and Swain (2009) point out that Eq. (A31) and Fig. A1(b) ofMcKenzie (2003) are in error, and give the correct expressions. Noother parts of McKenzie (2003) are affected. They also point out anerror in Crosby (2007), which is a typographic error (Crosby,personal communication): the correct expression was used in thecalculations.

Fig. 4. Estimates of themodulus (Zr2+Zi2)1/2, (a) and (f), of the complex admittance Z=Zr+ iZi b

the cross spectra bΔgh⋆N/bhh⋆Nwhere b N denotes averaging over a semicircular annulus of wcomplex conjugate, plotted as functions of the averagewavenumber in each annulus. (b) and (gestimate of the number of statistically independent valueswithin each annulus, using γ2

c = Bγ2

B−transforms, and NW is the frequency bandwidth. The one sigma error estimates in phase,σϕ, anσ2 = 1

B1γ2c−1

� �(Munk and Cartwright, 1966). The standard deviation of the coherence was o

functionofTewithminimumvaluesof 0.18 in (c) and0.58 in (h). Thenumber of independent valand in (f) is 1.8, 12.4, 23.1. (d) and (e) show the Bouguer coherence from the two boxes with

3. North America

The locations and values of the free air gravity anomalies andtopography were projected using a Lambert equal area projectioncentred on 50°N, 100°E, gridded with a grid spacing of 8 km, andFourier transformed. Two boxes, shown in Fig. 3a, were used to obtainthe quantities shown in Fig. 4. Their boundaries were chosen to bemostly within the region of thick lithosphere shown in Fig. 3a, and tocontain subdued topography. No part of either box includes thetopography of thewestern cordillera. The location of the northern box,2, was chosen to include the region in which Kirby and Swain (2009)obtained values of Te of more than 100 km (Fig. 1), whereas the

etween gravity and topography for the boxes in Fig. 3a, obtained from annular averages ofavenumbers k=(kx2+k2y)1/2 between k−Δk/2 and k+Δk/2 with kx≥0, and ⋆ denotes the) Estimates of the coherence, γc

2 obtained from γ2 = bΔgh⋆N2bhh⋆N bΔgΔg⋆N after correction for B, an

−11 where B=N/NW2,N is the number of values in each annulus fromall windowed Fourierd admittance, σz shown in the figure, were obtained from σϕ=1.96√σ2, σz=Zσϕ wherebtained from Simons et al.'s (2003) expression (B17). (c) and (h) show the misfit H as aues ink space used to obtain the three longestwavelength estimates in (a) is8.9, 39.1, 65.8;curves calculated form Forsyth's expressions and F2=0.3.


southern box, 1, contains a region where the coherence, but not theadmittance, method gave large values of Te. Both boxes are sufficientlylarge for spectral leakage to be unimportant (Crosby, 2007).

Fig. 4 shows the admittance, the coherence and the misfit as afunction of Te for the two boxes. The results for box 1 show that thefree air coherence remains low at the longest wavelengths. There is noindication of dynamic support: the observed admittance tends to zeroat long wavelengths. The best fitting model has an elastic thickness of19 km (Fig. 4c) when the internal load is a fraction of 0.3 of the totalload. The Bouguer coherence in Fig. 4d is best fit with Te=67 km if theinternal load fraction is the same. The free air coherence is about 0.05at a wavelength of 400 km. Crosby's empirical expression (Eq. (2))then gives an expected value for Teest. of 66 km. These values agree ingeneral with those of Kirby and Swain (2009) shown in Fig. 1 usingthe two approaches.

Unlike box 1, the behaviour of Z as k→0 for box 2 in Fig. 4f is notthat expected for flexural support, because it has constant value of

Fig. 5. (a) Topography of northern Europe, projected using an oblique Mercator projectiocalculate the quantities in Fig. 6. The dimensions of both boxes are 960 km×2200 km. (b) Lit(Priestley and McKenzie, 2006). (c) Estimated ice thickness 20 ka before present (Lambeck

about 64 mgal/km for wavelengths greater than 500 km. Furthermorethese values of admittance are associated with values of coherencethat exceed 0.7, whereas the values of coherence at short wavelengthsare about 0.1, or similar to those of box 1. The behaviour of theBouguer coherence in Fig. 4e shows none of these complications, andthe best fitting value of Te is in general agreement with the values inFig. 1a from Kirby and Swain (2009). The behaviour of the free airadmittance suggests that the longwavelength topography and gravityin the box 2 is not elastically supported. Fig. 4f shows the fit of aflexural model to values of Z for 150bλb500 km using the approachdescribed by McKenzie (2003), which produces estimates of both Teand the fraction of the load produced by internal density variations.The fit for Te=29 km is satisfactory, and is best when the internalload is zero.

There are two dynamical processes that could support thetopography and gravity in Canada and hence account for the behaviourof Z(k) in Fig. 4f: convection and postglacial uplift. Both effects are

n with axis 15°N, −110°E. The heavy lines mark the boundaries of the boxes used tohospheric thickness for the same region as (a), obtained from surface wave tomographyet al., 2006 and personal communication).


believed to be present (Simons and Hager, 1997; Tamisiea et al., 2007),and the value of Z(k)depends on their relative importance. Probably thebest estimate of the fraction of the present gravity field resulting fromthe glaciation comes from GRACE data, which allows dΔg/dt to beobtained. Tamisiea et al. (2007) determined dΔg/dt over North Americaand argued that a fraction β of between 0.25 and 0.45 of the free airgravity anomaly resulted from glaciation. This estimate is similar to thatobtained by Simons and Hager (1997). Tamisiea et al. were able toseparate convective support from postglacial depression because theformer is constant over timescales of ka. If the contributions fromconvection and from postglacial recovery are assumed to have acoherence of 1 at wavelengths N500 km, the admittance produced byboth dynamical effects is easily calculated from the expressions in theAppendix A. If the total gravity anomaly is Δg, then the postglacialcontribution is βΔg with topography βΔg/Zp, and the convectivecontribution is (1−β)Δg with topography (1−β)Δg/Zc, where Zp andZc are respectively the admittances of postglacial and convective

Fig. 6. As for Fig. 4, for the two boxes in Fig. 5a. The number of independent values in k spac(d) is 1.8, 16, 24.9. The minimum value of H in (c) is 0.51 and in (h) is 0.13. The postglacia

support. If both contributions to the gravity field have the same sign,as Fig. 2a suggests, the effective admittance Z is

Z�

=Δgh

=βZp

+1−βZc

" #−1

: ð5Þ

Tamisiea et al.'s estimates give average values of 53 and 63 mgal/kmfor λ≥500 km and Te=30 km. Though the theory used to obtain thesevalues is undoubtedly too simple, and it is unlikely that the coherencebetween the two processes will be 1, the general agreement with theobserved mean value of Z of 64±5 mgal/km is satisfactory.

The last issue concerns whether the time constant τ for postglacialrecovery is sufficiently long to account for the magnitude of thepresent depression. Fig. 2d and e show τ(k) for two values of mantleviscosity. The greatest thickness of ice was present about 20 ka ago,when it was about 3 km over much of northern Canada (Fig. 3b),

e used to obtain the three longest wavelength estimates in (a) is 1.8, 12.4, 21.3, and inl admittance in (a) is calculated from Eq. (A26).


corresponding to an equivalent rock thickness of about 1 km. The icesheetmust have been largely compensated, and thereforewould leavea depressionwhose gravity anomalywould have been about 140 mgal,or 100 mgal if the depression was filled with water. It is now about15 mgal (Fig. 3c), of which about half results from glacial depression.These values suggest that between two and three time constants haveelapsed since the ice melted, corresponding to τ∼7−10 ka. Theseestimates are compatiblewith Fig. 4 if Te=40 kmand η=5×1020 Pa s.

4. Fennoscandia

Another region that underwent extensiveglaciation is Fennoscandia.Perez-Gussinye and Watts (2005) have estimated Te for this regionusing Bouguer coherence, and obtained values in excess of 100 km. LikeCanada, Fennoscandia is now undergoing postglacial uplift. The

Fig. 7. (a) Smoothed topography, (b) band pass filtered gravity, (c) residual topography, andin (a) are A Ahaggar, Ca Cameroons, Co Congo, D Darfur, E Ethiopia, Ha Harare, Hu Huambo, Kdimensions 3400 km×2500 km is used to obtain Fig. 8.

resulting dynamical support of topography and gravity could thereforecause Te to beoverestimated. To examine this possibility, the gravity andtopography of the two boxes in Fig. 5a were Fourier transformed usingmultitapers. The regions were chosen to have subdued topography(Fig. 5a) and to be underlain by thick lithosphere (Fig. 5b). The NWbox,3, covers a region where the ice sheet was thick, whereas the SE box, 4,was largely ice-free20 kaago (Fig. 5c). Fig. 6 shows themagnitudeof theadmittance, the coherence and the variation of the misfit with Te. Forbox 3 the best fitting value of Te is greater than 100 km, in agreementwith the results of Perez-Gussinye andWatts (2005). But the behaviourof the coherence suggests thatmore than one process is involved.Whenthe wavelength is less than 500 km the coherence is below 0.3, but atlonger wavelengths is exceeds 0.4. Zp for postglacial uplift is shown inFig. 6a, and agrees well with the observed values for λ≥500 km. Box 3was chosen to include the regionwherepostglacial recovery is known to

(d) lithospheric thickness (Priestley andMcKenzie, 2006) for Africa. The labelled regionsKenya, L Lesotho, T Tibesti, WeWestern Rift, WiWindhoek. The data in box 5 in (d) with


be important. The three longest wavelength annuli in which thecoherence is large have well determined values of Z of more than110 mgal/km. There is therefore no evidence that either the topographyor gravity is affected by convection. The same is true for themeasurements of dΔg/dt from GRACE (Steffen et al., 2008), which canalso be described by a simple viscoelasticmodelwithout any convectivesupport.

In contrast to Canada, it is not possible to estimate Te from Z(k).The misfit H in Fig. 6c can only be used for this purpose if it isdetermined from those values of Z that result from elastic support.Unfortunately the value of Z expected from postglacial recovery issimilar to that of uncompensated topography, so it is not clear whichvalues of Z(k) in Fig. 6a result from dynamic support. The onlyindication of where the change from dynamic to elastic support mayoccur is the change in coherence in Fig. 6b that takes place at awavelength of about 400 km. If this is caused by a change fromdynamic to elastic support with decreasing wavelength, then it isconsistent with a value of Te of ∼30 km. As expected, the results frombox 4 do not show any glacial effects. The coherence is low at allwavelengths and Z(k) is poorly determined. The misfit as a function ofTe has a poorly resolved minimum. Taken together, the plots in Fig. 6show that it is not even possible to obtain a bound on the value of Tein either of these two boxes from gravity and topography alone.Though Perez-Gussinye and Watts's map (their Fig. 1) show values ofTe of 70–256 km for the region within boxes 3 and 4, and values of120–215 km with error bars in their Fig. 2, in their supplementaryinformation they state that their RMS curves for this region had nominima, in agreement with the behaviour in Fig. 6d and e. Thisbehaviour occurs because the admittance of any postglacial depres-sion, of about 130 mgal/km, is similar to that of uncompensatedtopography, 112 mgal/km. Therefore calculation of the Bougueranomaly removes the gravity anomaly of the postglacial depression.

Fig. 8. As for Fig. 3, calculated from the EGM2008 gravity field and topography. The number of(a) is 1.8, 12.4, 23.1. The minimum value of H in (c) is 0.28.

5. Africa

Most of Africa is dominated by basins and swells which have longbeen believed to be the surface expression of mantle convection(McKenzie and Weiss, 1975; Burke, 1996). Perez-Gussinye et al.(2009) have used Forsyth's method to estimate Te from the EGM2008gravity field, and have obtained values in excess of 70 km over muchof the continent. Since such values are at least twice the maximumobserved seismogenic thickness of 35 km (Foster and Jackson, 1998;Brazier et al., 2005), it is of interest to discover whether they resultfrom convective support. The gravity field is now well determinedfrom GRACE orbits for wavelengths greater than 500 km. If this part ofthe gravity field is convectively supported it should have anadmittance of ∼50 mgal/km, or similar that Downey and Gurnis(2009) found for the Congo Basin. To test whether this is generally thecase for Africa, the gravity field from GRACE in Fig. 7b was divided by50 and subtracted from the filtered topography shown in Fig. 7c togive the residual topography shown in Fig. 7c (McKenzie, 1994). Thisoperation removes most of the topographic features in Fig. 7a, whichare therefore likely to be convectively supported. However, theEthiopian Plateau is not removed by this operation, probably becauseit is partly supported by thickened crust.

The gravity coverage of Africa is less good than is that of Canadaand Fennoscandia. Where it is poor, the admittance at wavelengthsshorter than 200 km, where no compensation occurs, is often as smallas 60–70 mgal/km, corresponding to a density of topography of only1400–1600 kg m−3. Box 5 in Fig. 7d was chosen to exclude suchregions, to be sufficiently large to give reliable estimates of admittance(Crosby, 2007), and to include the area where lower crustal earth-quakes occur to depths of 35 km (Foster and Jackson, 1998; Brazieret al., 2005). Box 5 is also largely underlain by thick lithosphere(Fig. 7d). The resulting free air admittance in Fig. 8a is best fit by an

independent values in k space used to obtain the three longest wavelength estimates in

Fig. 9. Comparison of the admittance from Figs. 4a and 8a with that for Atla Regio,Venus, (McKenzie and Nimmo, 1997), showing the physical processes which controlthe admittance in different wavenumber bands.


elastic layer of thickness 26 km, and shows evidence of convectiveeffects at wavelengths greater than 500 km. In contrast, the Bouguercoherence in Fig. 8d gives an estimate of 65 km, in broad agreementwith Perez-Gussinye et al. (2009). The estimate of Te from the free airadmittance is less than the seismogenic thickness of 35 km, whereasthat from the Bouguer coherence is not, probably because it is affectedby dynamic support. Because such support has an admittance of about50 mgal/km, it generates a large signal when the Bouguer gravity iscalculated using Eq. (3), and hence generates a large value of theBouguer coherence at long wavelengths. It is this signal that producesthe large values of Te.

6. Discussion

Most of the regions discussed above show clear evidence ofdynamic support, perhaps most obviously in the results from GRACE(Tamisiea et al., 2007; Steffen et al., 2008), which show that postglacialuplift is causing the gravity field to change with time. As is also wellknown, the surface of both regions is still undergoing uplift. Thereforeneither region is at present isostatically compensated, even at longwavelengths. If this lack of compensation is attributed to elasticsupport, large (N100 km) values of Te are required. But it is clearlyincorrect to use an elastic model when it is generally agreed that thelack of compensation is time dependent, and arises from viscous, notelastic, forces. It is only possible to estimate Te for glaciated regionsfrom gravity and topography alone if the contribution of glacialdeformation to both is now small. This condition is often satisfied inregions of rough topography, or where the topography and gravitylargely result from convective support (which has a lower value of Z).

A different problem arises in mapping lateral variations in Te. Suchvariations are to be expected, because the rheology of the crust andmantle is principally controlled by temperature. But such lateralvariations are difficult to resolve because of the convective support oflongwavelength gravity and topography. As Fig. 4f shows, it is difficultto separate convective and elastic support. Very large boxes arerequired, because so few points in k space contribute to each annuluswhen |k| is small. Especially if the topography is subdued, the resultinguncertainty in Z is large. The number of independent values in the firstthree annuli in Figs. 4a, f, 6a, d and 8a is listed in the figure captions.Elastic and convective support can only be separated by their values ofZ, and therefore the resolution of lateral variations will be poor,especially in shield regions where the coherence is often low. Similarproblems do not affect maps of the seismogenic thickness Ts, which istherefore likely to provide a better method of mapping lateralvariations in rheology.

7. Conclusions

Fig. 9 shows the main results of this study, and also shows theimportance of presenting plots of Z(k). At long wavelengths(λ≥500 km) continental admittance is often controlled by dynami-cally supported gravity and topography. The support comes eitherfrom convection, or from postglacial adjustment, or from both. Thesetwo effects can be separated by using observations of dΔg/dt fromGRACE. They also have different values of Z, with postglacial effectshaving a value of ∼140 mgal/km whereas convective support givesZc∼50 mgal/km. At wavelengths shorter than about 500 km thebehaviour of Z is generally controlled by elastic forces in the crust andlithosphere. When estimating Te it is important to determine both thevalue of Z and its error, and to avoid using values of Z that result fromdynamical support. If dynamically supported values are included, thevalue of Te may be overestimated by an indeterminate amount, sinceZ→constant as k→0 for such support. The models used in this paperare simple, and are chosen principally because they have analyticsolutions which illustrate the physical processes involved. Moredetailed models can easily be used, but there seems little purpose in

doing so until there is more agreement between different authors. Ifaccurate estimates of Te are to be obtained, it is essential to resolve thedecrease in Z that must occur at long wavelengths when thetopography and gravity are elastically supported. It is easier todetermine Z(k) accurately in oceanic than in continental regions,because continental values of the coherence between free air gravityand topography are strongly reduced by subaerial erosion. For thesame reason the value of Te for Atla Regio on Venus, where there is noerosion (Fig. 9c, from McKenzie and Nimmo, 1997), is also welldetermined. When the coherence is large the value of Te determinedusing Forsyth's coherencemethod is similar to that obtained using theadmittance. But in regions where the coherence is small thecoherence method only provides an upper bound on Te, which canbe much larger than the true value. In shields the topography is oftenalmost flat, and has low coherence with the free air gravity. Underthese conditions no reliable estimates of Te can be obtained from the


relationship between gravity and topography. The only availablemethod then depends on recognising gravity anomalies that resultfrom the imposition of line loads by thrusting, and then fitting profilestaken at right angles to these loads to estimate Te. This methodestimates the value of Te when the anomalies were produced, ratherthan the present value (McKenzie, 2003). There is no evidence that Teexceeds the seismogenic thickness in continental or oceanic regions.

Acknowledgements

I would like to thank A. Crosby, C. Swain, D. Fairhead, J. Jackson, K.Lambeck, F. Nimmo, F. Simons, M. Tamisiea and three anonymousreferees for their help, and the Royal Society for support.

Appendix A

The presence of an elastic layer at the surface of a viscous halfspace affects the surface deformation associated with convection andthe response time τ of any initial deflection of the surface. Simpleexpressions for both these effects can be obtained by solving theequation for Stokes flow in a half space

0 = η∇2v−ρgz−∇P ðA1Þ

where η is the dynamic viscosity, v=(u, 0, w) the velocity and ρ thedensity of the fluid, g the acceleration due to gravity, z a unit vector inthe +z direction, taken to be upwards, and P is the fluid pressure. Theorigin is taken to be at the surface of the half space. The normal stressσzz at the surface z=0 is

σ zzðz = 0Þ = −Pðz = 0Þ + 2η∂w∂z

� �z=0

: ðA2Þ

The equation for the elastic deflection ζ of a thin sheet whensubjected to a load L is

D∇4ζ + gðρm−ρwÞζ = L ðA3Þ

where ρm is the density of the mantle, and ρw that of water when thetopography is below sea level, and of air when it is subaerial, and

D =ET 3

e

12ð1−σ 2Þ

where E is Young's modulus and σ is Poisson's ratio. The values usedthroughout this paper are E=9.5×1010 Pa and σ=0.29. σzz inEq. (A2) is the stress exerted on the surface of the fluid by the plateabove, whereas L in (A3) is the stress exerted by the fluid on the plate.Therefore L=−σzz by Newton's third law.

If a surface temperature of T0 cos kx, where T0 is a constant, isimposed at z=0, then the steady state temperature in the half space is

Tðx; zÞ = T0 expðkzÞcoskx ðA4Þ

if heat advection is neglected. The equation of state is

ρðTÞ = ρmð1−αTÞ ðA5Þ

where α is the thermal expansion coefficient. In two dimensions thesolution to Eq. (A1) with u=w=0 on z=0 can be written in terms ofa stream function ψ, where

v = −∂ψ∂z ;0;

∂ψ∂x

� �ðA6Þ

ψ =ρmgαT08ηk

z2expðkzÞsinkx ðA7Þ

and

σ zzðz = 0Þ = −3ρmgαT04k

cos kx ðA8Þ

Substitution into Eq. (A3) gives the contribution ζa that thevariations of temperature in the asthenosphere, assumed to occupythe region zb0, make to the topography

ζa =3ρmαT0

4kðρm−ρwÞð1 + FÞ cos kx ðA9Þ

where

F =Dk4

gðρm−ρwÞ: ðA10Þ

The resulting gravity anomaly Δga on z=0 is (McKenzie, 1977)

Δga = 2πG ðρm−ρwÞζa−ρmαT02k

cos kx� �

: ðA11Þ

Substitution for ζa gives

Δga =2πGρmαT0ð1−2FÞ

4kð1 + FÞ cos kx ðA12Þ

where G is the gravitational constant, Eq. (A11) shows that Δgabecomes negative as k increases, because the second term in thesquare brackets→0 as 1/k, whereas ζa→0 as 1/k4.

If lithosphere of thickness a overlies the convecting asthenosphereand the temperature on z=a is fixed at 0 °C, Eq. (A11) becomes

Δga = 2πG ðρm−ρwÞζa−ρmαT0expð−kaÞ

2kcos kx

� �ðA13Þ

when Δga is measured on the surface. The steady state temperaturewithin the lithosphere is

Tðx; zÞ = T0sinh kða−zÞsinh ka

cos kx ðA14Þ

The resulting gravity anomaly Δgl on z=a can be obtained byrequiring the gravitational potential U and ∂U/∂z to be continuous onz=(0, a)

Δgl = −2πGρmαT0expð−kaÞ2k

ka expðkaÞsinh ka

−1� �

cos kx ðA15Þ

The change in temperature within the lithosphere producestopographic uplift ζT at the surface z=a, where

ζT =αT0ðcosh ka−1Þ3kð1 + FÞsinh ka

cos kx ðA16Þ

associated with a gravity anomaly ΔgT, where

ΔgT = 2πGðρm−ρwÞζT ðA17Þ

The admittance Zc resulting from convection therefore is

Zc =Δga + Δgl + ΔgT

ζa + ζ TðA18Þ

and is plotted in Fig. 2a for various values of Te, with T0=100 °C,a=120 km, α=4×10−5 °C−1 and ρm=3300 kg m−3.


Postglacial recovery is driven by surface deformation, and not byvariations in mantle density. If ρ=ρm=constant and u=0 on z=0,the stream function is given by

ψ = Að1−kzÞexpðkzÞsin kx ðA19Þ

where A is a function of time alone, and

vðz = 0Þ = dζdt

= Ak cos kx ðA20Þ

σ zzðz = 0Þ = 2η Ak2 cos kx = −L: ðA21Þ

Substitution of Eq. (A21) into Eq. (A3) gives A, and hence Eq. (A20)becomes

dζdt

= − ζτ

ðA22Þ

where

τ =2ηk

gðρm−ρwÞð1 + FÞ ðA23Þ

in agreement with Cathles (1975 p52). The solution to Eq. (A22) isζ(t)=ζ0exp(–t/τ) where ζ0 is the initial surface deformation. Ifinstead the load due to the surface deformation is initially zero andaccumulates at a constant rate ζ0, then Eq. (A22) becomes

dζdt

= − ζτ

+ ζ 0 ðA24Þ

whose solution is

ζðtÞ = τζ 0½1−expð−t = τÞ� ðA25Þ

and the maximum value of ζ(t) is τζ0. For loads like rainfall t≪τ andζ(t)=ζ0t.

The admittance for subaerial postglacial uplift Zp is easily obtained,because glacial depression deflects the surface, the Moho and anycrustal layers by the same amount. It is therefore independent of Te.For the two layer crust used by McKenzie (2003)

Δg = Zph = 2πG½ρu + ðρl−ρuÞexpð−ktuÞ + ðρm−ρlÞexpð−ktcÞ�hðA26Þ

whereρu(=2400 kg m−3),ρl(=2900 kg m−3) andρm(=3300 kg m−3)are the densities of the upper crust, lower crust, andmantle respectively,tu(=15 km) is the thickness of the upper crust and tc(=35 km) that ofthe whole crust. Zp is plotted in Fig. 6a. At long wavelengths Eq. (A26)shows that Zp→2πGρm.

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