the continental tectosphere and earth’s long-wavelength

Ž .Lithos 48 1999 135–152

The continental tectosphere and Earth’s long-wavelengthgravity field

Steven S. Shapiro 1, Bradford H. Hager ), Thomas H. JordanDepartment of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 19 October 1998; received in revised form 8 February 1999; accepted 15 February 1999

Abstract

To estimate the average density contrast associated with the continental tectosphere, we separately project the degree2–36 non-hydrostatic geoid and free-air gravity anomalies onto several tectonic regionalizations. Because both theregionalizations and the geoid have distinctly red spectra, we do not use conventional statistical analysis, which is based onthe assumption of white spectra. Rather, we utilize a Monte Carlo approach that incorporates the spectral properties of thesefields. These simulations reveal that the undulations of Earth’s geoid correlate with surface tectonics no better than theywould were it randomly oriented with respect to the surface. However, our simulations indicate that free-air gravityanomalies correlate with surface tectonics better than almost 98% of our trials in which the free-air gravity anomalies wererandomly oriented with respect to Earth’s surface. The average geoid anomaly and free-air gravity anomaly over platformsand shields are significant at slightly better than the one-standard-deviation level: y11"8 m and y4"3 mgal,

Ž .respectively. After removing from the geoid estimated contributions associated with 1 a simple model of the continentalŽ . Ž . Ž .crust and oceanic lithosphere, 2 the lower mantle, 3 subducted slabs, and 4 remnant glacial isostatic disequilibrium, we

estimate a platform and shield signal of y8"4 m. We conclude that there is little contribution of platforms and shields tothe gravity field, consistent with their keels having small density contrasts. Using this estimate of the platform and shieldsignal, and previous estimates of upper-mantle shear-wave travel-time perturbations, we find that the average value of

Ž .Eln rrElnn within the 140–440 km depth range is 0.04"0.02. A continental tectosphere with an isopycnic equal-densitysŽ .structure Eln rrElnn s0 enforced by compositional variations is consistent with this result at the 2.0s level. Withouts

compositional buoyancy, the continental tectosphere would have an average Eln rrElnn f0.25, exceeding our estimate bys

10s . q 1999 Published by Elsevier Science B.V. All rights reserved.

Keywords: Continental tectosphere; Earth; Long-wavelength gravity field; Geoid anomaly; Gravity anomaly

1. Introduction

ŽMotivated by seismological evidence e.g., Sipkin.and Jordan, 1975 and the lack of a strong correla-

) Corresponding author.1 Present address: Department of Physics, Guilford College,

Greensboro, NC 27410, USA

tion between continents and the long-wavelengthŽ . Ž .geoid e.g., Kaula, 1967 , Jordan 1975 proposed

Ž . Žthat continents are 1 characterized by thick ;400. Ž .km thermal boundary layers TBLs which translate

Ž .coherently during lateral plate motions, 2 stabilizedagainst small-scale convective disruption by gradi-ents in density due to compositional variations, andŽ .3 not observable in the long-wavelength gravity

0024-4937r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0024-4937 99 00027-4

( )S.S. Shapiro et al.rLithos 48 1999 135–152136

field. The simple plate cooling model, which enjoysmuch success in describing the structure of oceanicTBLs, cannot be extended to explain thicker conti-

Ž . Ž .nental TBLs Jordan, 1978 . Instead, Jordan 1978postulated that the thick continental TBL, continentaltectosphere, was formed early in Earth’s history byadvective thickening and has been stabilized againstconvective disruption by the compositional buoyancyprovided by a depletion of basaltic constituents. The

Ž . Ž .isopycnic equal-density hypothesis Jordan, 1988predicts that the compositional and thermal effectson density cancel at every depth between the base ofthe mechanical boundary layer and the base of theTBL. Such a structure would be neutrally buoyantwith respect to neighboring oceanic mantle, andwould not be visible in the long-wavelength gravityfield.

There has been much discussion during the past 2decades about the relations among the Earth’s long-wavelength gravity field, surface tectonics, and man-tle convection. For example, there is an obviousassociation of long-wavelength geoid highs with sub-

Žduction zones Kaula, 1972; Chase, 1979; Crough.and Jurdy, 1980; Hager, 1984 and with the distribu-

Žtion of hotspots Chase, 1979; Crough and Jurdy,.1980; Richards and Hager, 1988 . Most of the power

in the longest wavelength geoid can be explained interms of lower-mantle structure imaged by seismic

Žtomography e.g., Hager et al., 1985; Hager and.Clayton, 1989; Forte et al., 1993a . This lower man-

tle seismic structure has been linked to tectonicprocesses, in particular, to the history of subductionŽ .e.g., Richards and Engebretson, 1992 . Althoughthere is general agreement among geodynamiciststhat most of the geoid can be explained in terms offeatures such as subducted slabs and lower mantlestructure, there is significant quantitative disagree-

Žment among the predictions of various models e.g.,.Panasyuk, 1998 . Thus, it is not possible to estimate

with high confidence the ‘‘residual geoid’’ not ex-plained by lower mantle structure.

The contribution to the geoid of upper-mantlestructures, including variations in the thickness of thecrust and lithosphere, is a question whose answer isstill disputed. Assuming that plates approach anasymptotic thickness of approximately 120 km aftercooling about 80 My, the geoid would be expected tobe higher by roughly 10 m over continents and over

midoceanic ridges than over old ocean basins due tothe density dipole associated with isostatic compen-

Žsation Haxby and Turcotte, 1978; Parsons and.Richter, 1980; Hager, 1983 . At intermediate to short

wavelengths, the expected changes in the geoid overŽthese features are observed e.g., Haxby and Tur-.cotte, 1978; Doin et al., 1996 , but the isolation of

the geoid signatures of these features at long wave-lengths is problematic. Using broad spatial averages

Ž .over selected areas, Turcotte and McAdoo 1979concluded that there is no systematic difference inthe geoid signal between oceanic and continental

Ž .regions. But, Souriau and Souriau 1983 demon-strated that there is a significant correlation between

Ž .the geoid spherical harmonic degrees ls3–12 andŽ .the tectonic regionalization of Okal 1977 . From

Ž .degree-by-degree correlations ls2–20 , RichardsŽ .and Hager 1988 observed a weak association be-

tween geoid lows and shields. On the other hand,Ž .Forte et al. 1995 reported that the degree 2–8 geoid

Ž .correlates significantly 99% confidence with anocean–continent function.

Were there a significant ocean–continent signal,the continental tectosphere might have a substantialdensity anomaly associated with it, and might there-fore be expected to play an active role in the large-scale structure of mantle convection. For example,

Ž . Ž .Forte et al. 1993b and Pari and Peltier 1996 , intheir preferred models, assumed linear relationshipsbetween seismic velocity anomalies and densityanomalies. They proposed dynamic models of thelong-wavelength geoid in which the high velocityroots beneath continents are cold, dense down-wellings in the convecting mantle. Such down-wellings would depress the surface of continents

Ž .dynamically by about 2 km Forte et al., 1993b . Thelack of significant temporal variation in continentalfreeboard over geologic time would require that theseconvecting downwellings be extremely long-lived

Žand translate coherently with the continents e.g.,.Gurnis, 1993 . On the other hand, Hager and Richards

Ž . Ž .1989 and Forte et al. 1993b found the best fits oftheir dynamic models to the geoid by assuming anunusually small global proportionality between seis-mic velocity anomalies and density anomalies in the

Ž .upper mantle. Forte et al. 1995 showed that theycould improve their fit to the geoid if they allowedsubcontinental regions to have a different proportion-

( )S.S. Shapiro et al.rLithos 48 1999 135–152 137

Table 1Ž .GTR1 Jordan, 1981

Region Definition FractionalŽ .area %

Oceans 61Ž .A Young oceans 0–25 My 13

B Intermediate-age oceans 35Ž .25–100 My

Ž .C Old oceans )100 My 13Continents 39Q Phanerozoic orogenic zones 22P Phanerozoic platforms 10S Precambrian shields 7

and platforms

ality constant between velocity and density anoma-lies beneath continents than beneath oceans.

To quantify the association of surface tectonicsand Earth’s gravity field, we investigate the signifi-cance of the association between the six-region global

Ž . Žtectonic regionalization GTR1 Jordan, 1981 Table. Ž1, Fig. 1 and the geoid, EGM96 Lemoine et al.,

.1996 , referred to the hydrostatic figure of EarthŽ . Ž .Nakiboglu, 1982 Fig. 2a . Although we use GTR1Žand coarser regionalizations created by combining

.some of these regions for the bulk of this study, wealso compare our results with those obtained using

Ž .the tectonic regionalizations of Mauk 1977 andŽ .Okal 1977 , as well as the ocean-continent function.

Because the geoid spectrum is red, with the root-Ž .mean-square rms value of a coefficient of degree l

decreasing roughly as ly2 , and because the longestwavelengths are likely dominated by the effects of

Ždensity contrasts in the lower mantle Hager et al.,.1985 , we also investigate the relationship between

GTR1 and free-air gravity anomalies. The gravityfield at spherical harmonic degree l is proportionalto ly1, so the gravity anomalies are expected to havecorrespondingly smaller long-wavelength variationsthan the geoid does.

We calculate regional averages of the geoid andthe gravity field and estimate their uncertainties.Further, we try to refine the estimate of the contribu-tion of the continental tectosphere to the geoid by

Fig. 1. Tectonic regionalization, GTR1 displayed using a Hammer equal-area projection. See Table 1 for a description of each region.


subtracting other contributions from the geoid esti-mates. By combining the upper-mantle shear-wavetravel-time anomalies associated with platforms and

Ž .shields Shapiro, 1995 and the results from thisstudy, we estimate, with uncertainties, the average ofEln rrElnn within the depth range 140–440 km, ands

compare our estimate with the isopycnic hypothesisŽ .of Jordan 1988 .

2. Tectonic regionalization and inversion

GTR1 and regionalizations published by OkalŽ . Ž . Ž . Ž .1977 Table 2 and Mauk 1977 Table 3 containsix, seven, and 20 regions, respectively. Both GTR1

Ž .and the regionalization of Mauk 1977 are definedon a grid of 58=58 cells, whereas the model of OkalŽ .1977 is defined using 158=158 and 108=158

Ž .cells. The regionalization of Mauk 1977 allows foras many as 10 regions to be represented in a givencell, while the other regionalizations are defined withonly one region per cell. In GTR1, the three oceanic

Ž .regions including marginal basins are defined byequal increments in the square root of crustal age:

Ž . Ž . Ž .0–25 My A , 25–100 My B , and )100 My Cand the continental regions are classified by theirgeneralized tectonic behavior during the Phanero-

Ž .zoic: Phanerozoic orogenic zones Q , PhanerozoicŽ .platforms P , and Precambrian shields and platforms

Ž . Ž .S . Like GTR1, the oceanic regions of Mauk 1977are based largely on crustal age. However, the conti-

Ž .nental regions of Mauk 1977 are classified by agerather than by their tectonic behavior. The morecomplex parameterization associated with the region-

Ž .alization of Mauk 1977 does not offer us anysignificant advantage over GTR1; as we showthrough representative projections, the platform andshield signatures from the regionalization of MaukŽ .1977 and from GTR1 are consistent with eachother and only significant at slightly better than theone-standard-deviation level. The regionalization of

Table 2Ž .Okal 1977


Ž .D Ocean 0–30 My 12.0Ž .C Ocean 30–80 My 30.1Ž .B Ocean 80–135 My 12.3Ž .A Ocean )135 My 2.5

T Trenches and marginal seas 10.9M Phanerozoic mountains 11.6S Shields 20.4

Ž .Okal 1977 is limited in the accuracy of its designa-Ž .tion of regions. For example, Okal 1977 labels the

entire continent of Antarctica a shield, whereas aŽ .significant fraction f1r3 is orogenic in nature.

Ž . ŽOkal 1977 also classifies some islands e.g., Ice-.land and Great Britain as shields. Misidentifications

such as these might have a significant effect onresults from associated data projections.

In general, a tectonic regionalization containing Ndistinct regions can be described by N functions,

Ž .R ns1, N , each having unit value over its regionn

and zero elsewhere. By combining regions, we canconstruct other, coarser regionalizations. For exam-

Ž .ple, by consolidating young oceans A , intermedi-Ž . Ž .ate-age oceans B , and old oceans C of GTR1,

into one region, and Q, P, and S, into another region,Ž .we can create a two-component ocean–continent

Ž .tectonic regionalization ABC, QPS . For much ofthis analysis, we combine regions P and S into one

Ž .region PS .For any such regionalization, we expand each Rn

in spherical harmonics, omitting degrees zero andone from our analysis because geoid anomalies arereferred to the center of mass and any rearrangementof mass from internal forces cannot change an ob-ject’s center of mass. With coefficient Rlm represent-n

Ž .ing the l,m harmonic of region n, and coefficientlm Ž .d representing the l,m harmonic of the observed

Ž . Ž . Ž . Ž .Fig. 2. a Geoid, ls2–36 EGM96; Lemoine et al., 1996 , referred to the hydrostatic figure of Earth Nakiboglu, 1982 ; b Projection ofŽ . Ž . Ž . Ž .a onto A, B, C, Q, PS , and c Residual: a–b . All plots are displayed using a Hammer equal-area projection with coastlines drawn inwhite. Negative contour lines are dashed and the zero contour line is thick. The contour interval is 10 m.


Table 3Ž .Mauk 1977


Oceans 61.5Ž .1 Anomaly 0–5 0–10 My 4.0Ž .2 Anomaly 5–6 10–20 My 10.4Ž .3 Anomaly 6–13 20–38 My 6.9Ž .4 Anomaly 13–25 38–63 My 10.2

5 Late Cretaceous sea floor 21.1Ž .63–100 My

6 Early Cretaceous sea floor 5.4Ž .100–140 My

7 Sea floor older than 140 My 3.5

Continents 38.48 Island arcs 1.49 Shelf sediments 7.1

10 Intermontane basin fill 0.711 Mesozoic volcanics 0.412 Cenozoic volcanics 1.413 Cenozoic folding 1.814 Mesozoic orogeny 2.715 Post-Precambrian undeformed 9.516 Late Paleozoic orogeny 1.917 Early Paleozoic orogeny 1.818 Precambrian undeformed 1.519 Proterozoic shield 6.220 Archaean shield 2.0

Ž .or model geoid or gravity field, we use a least-squares approach to solve:

Rlmg sdlm 1Ž .n n

Ž .summation convention implied here and below forthe regional averages, g . We include the additionaln

constraint:

A g s0 2Ž .n n

where A represents the surface area spanned byn

region n. This constraint ensures that g have a zeronŽ .weighted average, as, by definition, do the geoid-

Ž .height and free-air gravity anomalies. Theweighted-least-squares solution can be written:

y1T Tw xgs R WR R Wd 3Ž .where the values Rlm and A are the elements of then n

matrix R, W is a weight matrix constructed from thecovariance matrix associated with d, g are then

elements of the vector g , and dlm and zero consti-tute the vector d.

We next consider the effect of errors in d on ouranalysis. Although we have available the covariancematrix for EGM96, this weight matrix is not theappropriate one for our analysis. As discussed previ-ously, most of the power in the long wavelengthparts of the geoid is the result not of surface tecton-ics, but of deep internal processes. Unfortunately, thecontribution of these deep processes cannot be deter-mined to anywhere near the accuracy of the observedgravity field, so the covariance matrix will beswamped by the contributions of the errors due toneglecting important dynamic processes. Quantita-tive estimation of the errors associated with esti-mates of the contributions of these deep processes

Ž Ž .has rarely been attempted Panasyuk 1998 is an.exception . Here, we simply assume the identity

matrix as our default weight matrix. For this matrix,the relatiÕe error in the harmonic expansion of the

2 Žgeoid increases as l or as l for the gravity anoma-.lies . This behavior is qualitatively consistent with

the result that dynamic models of the geoid do betterat fitting the longest wavelength components andprogressively worse at fitting shorter wavelengthcomponents, for example, because the effects oflateral variations in viscosity become more important

Žat shorter wavelengths e.g., Richards and Hager,.1989 . The sole exception to the identity weight

matrix is our application of a large weight, 1000, tothe surface-area constraint. Results from our inver-sions are insensitive to the value of this weight, solong as it is not less than ten times the weight

Ž .associated with the data in our case unity nor soŽ 6 .large )10 times the data weight that the inver-

sion becomes numerically unstable.

3. Statistical analysis procedure

Because neither the geoid nor the regionalizationhave white spectra, we do not use common statisticalestimates of uncertainties. In fact, their spectra arequite red, implying that uncertainties in parameterestimates based on the assumption of white spectrawill be substantially smaller than the actual uncer-tainties. Through the use of Monte Carlo techniques,we incorporate the spectral properties of these fieldsin our estimates of parameter uncertainties. For each


Ž .of 10,000 trials, we 1 randomly select an Eulerangle triple from a parent distribution in which allorientations are equally probable and then, in accord

with the selected triple, rigidly rotate the sphere onŽ lm lm lm .which the data residuals d sd yR g are de-res n n

fined, with respect to the sphere on which the surface

Ž . Ž . Ž . Ž . Ž . Ž .Fig. 3. Non-hydrostatic geoid EGM96, ls2–36 : Histograms of parameter values a g , b g , c g , d g , e g obtained fromA B C Q PS

projections onto the tectonic sphere of the correlated data combined with 10,000 random orientations of the data residual sphere,l̃m Ž .characterized by d see text . Gaussian distributions, determined by the standard deviation, mean, and area of each histogram, are

Ž . lmsuperposed. f Histogram of variance reduction resulting from 10,000 random rotations of the data sphere, characterized by d , withrespect to the tectonic sphere. The shaded and unshaded arrows indicate the variance reductions associated with the actual orientation andthe maximum variance reduction, respectively.


Ž . Ž .tectonics are defined ‘‘tectonic sphere’’ , 2 com-l̃m Žbine the rotated data residuals d ‘‘; ’’ denotesres

.rotated with the correlated data to produce pseudo

l̃m l̃m lm l̃mŽ . Ž .data, d sd qR g , and 3 project d ontores n nŽ .A, B, C, Q, PS . The resulting histograms of param-

Ž .eter values e.g., g , g , g , . . . approximateA B C

Ž . Ž . Ž . Ž . Ž . Ž .Fig. 4. Free-air gravity ls2–36 : Histograms of parameter values a g , b g , c g , d g , e g obtained from projections ontoA B C Q PSl̃m Žthe tectonic sphere of the correlated data combined with 10,000 random orientations of the data residual sphere, characterized by d see

. Ž .text . Gaussian distributions, determined by the standard deviation, mean, and area of each histogram, are superposed. f Histogram ofvariance reduction resulting from 10,000 random rotations of the data sphere, characterized by dlm, with respect to the tectonic sphere. Theshaded and unshaded arrows indicate the variance reductions associated with the actual orientation and the maximum variance reduction,respectively.


Gaussian distributions and, because the correlatedsignal is added to the rotated data residual beforeprojecting the composite, the resulting histograms ofparameter values are centered approximately on theparameter values corresponding to the actual orienta-tion of the ‘‘data sphere’’ with respect to the tectonic

Ž .sphere Figs. 3 and 4 . We take these latter para-meter values as our parameter estimates and thestandard deviations of these approximately Gaussiandistributions as the parameter uncertainties. Alterna-

Ž .tively, we could assign random white noise valuesto each coefficient describing the data-residual spherewhile constraining its power spectrum to be un-changed through a degree-by-degree scaling. His-tograms resulting from this approach yield very simi-lar distributions and virtually the same values for theparameter estimates and their standard errorsŽ .Shapiro, 1995 . If one relaxes the constraint byrequiring only that the total power remains un-changed, then the resulting histogram distributionsare narrower than the corresponding ones for whichthe spectra were scaled degree-by-degree. Thesesmaller values for the standard errors in the parame-

Ž .ter estimates likely coincide Shapiro, 1995 in thelimit of large numbers of trials with those deter-mined from the elements of the variance vector

2 �w T x-14 2 Žz'x diag R WR , where x , is the post-post post. 2fit x per degree of freedom.As a criterion for the success of the model in

fitting the data, we use the percent fractional differ-ence in the prefit and postfit x 2. This percent vari-ance reduction associated with each projection, i.e.,

w Ž 2 2 .xinversion, is thus defined by 100 1y x rx .post pre

From the results of the random rotations of the datasphere with respect to the tectonic sphere, we esti-mate significance levels in the variance reductionassociated with each projection. Specifically, we as-sociate the fraction of trials that yield lower variancereductions than the actual orientation with the confi-dence level of the variance reduction.

4. Projections

Table 4 shows the regional averages and theircorresponding statistical standard errors obtained byseparately projecting the geoid and the free-air grav-

Ž .ity anomalies onto A, B, C, Q, PS . Fig. 3a–eFig.4a–e graphically display the 10,000 parameter esti-mates obtained from the Monte Carlo simulationsthat lead to the uncertainties given in Table 4. With

Ž . Ž .the geoid, only regions C and PS have averageswhich are larger than their standard errors. However,the significance of these averages is only slightlyabove the one-standard-deviation level. For example,

Ž .with 95% 2s confidence, the geoid signature asso-ciated with platforms and shields is in the rangey27 to q6 m, a rather broad range which does noteven significantly constrain the sign of this signal.

Ž .The projection of the geoid onto A, B, C, Q, PS isshown in Fig. 2b and further demonstrates that verylittle of the long-wavelength non-hydrostatic geoidcan be explained simply in terms of surface tecton-ics. The magnitude of the geoid signal that is uncor-

Ž . Ž .related with A, B, C, Q, PS Fig. 2c is essentiallythe same as that of the geoid anomalies themselves,given by EGM96. Using the free-air gravity yields asomewhat different result: four regions have aver-

Ž .ages larger than their standard errors Fig. 4a–e .The significance of three of these averages is at orbelow the 1.5s level and the significance of the

Ž .fourth, g , is at the 2.5s level Table 4 .Q

With 95% confidence, the free-air gravity signa-ture associated with platforms and shields is in therange y10 to q1.4 mgal. Like with the geoid, thisrange is rather large and does not significantly con-strain the sign of this signal. However, unlike thegeoid projection, which explains less of the variancethan about two-thirds of the random orientations of

Ž .the data sphere Fig. 3f , the free-air gravity projec-tion explains more of the variance than about 98% ofprojections corresponding with random orientations

Table 4Ž .EGM96 ls2–36 : Regional averages and statistical standard errors from projections of the geoid and of perturbations to the free-air

Ž .gravity onto A, B, C, Q, PS

g g g g gA B C Q PS

Ž .Geoid m 0"12 1.8"5.6 17"16 y4.8"11 y10.5"8.3Ž .Gravity mgal 4.0"3.4 y1.7"1.7 y4.8"3.9 6.4"2.5 y4.3"2.9


Ž .of the data sphere Fig. 4f . However, this reductionin variance is only about 6% and does not producean impressive fit. Interestingly, Monte Carlo simula-tions using degrees 2–12 yield confidence levels ofless than 30%, suggesting that the association be-tween free-air gravity anomalies and surface tecton-ics is stronger in the higher frequencies.

Ž . ŽUsing the regionalization of Mauk 1977 the full20-region tectonic sphere as well as some representa-

.tive groupings of these regions leads to resultssimilar to those obtained from GTR1. In no case dowe find a significant signal that can be linked withthe continental tectosphere. Combining the regions

Ž .of Mauk 1977 into three groups based on crustalŽ w x w x w x.age regions 1–7 , 8–14, 16–17 , 15, 18–20 ,

yields regional averages which are roughly the sameŽmagnitude as their corresponding uncertainties Ta-

.ble 5 and a variance reduction of about 6%. AnotherŽw x w x wcontinental grouping 1–7 , 8–10, 12–13 , 11, 14–

Table 5Ž .EGM96 ls2–36 : Regional averages and statistical standard

errors from projections of the geoid onto several regionalizationsŽ . Žw x w x wbased on Mauk 1977 . Group 1: 1–7 , 8–14, 16–17 , 15,

x. Žw x w x w x w x.18–20 ; Group 2: 1–7 , 8–10, 12–13 , 11, 14–17 , 18–20 ;Group 3: 20 separate regions

Region Group 1 Group 2 Group 3Ž . Ž . Ž .g m g m g m

Oceans1 9"6 9"6 y13"142 9"6 9"6 10"143 9"6 9"6 9"84 9"6 9"6 5"75 9"6 9"6 4"106 9"6 9"6 15"147 9"6 9"6 40"25

Continents8 y22"16 y9"15 127"469 y22"16 y9"15 y41"19

10 y22"16 y9"15 62"4411 y22"16 y19"14 y10"5712 y22"16 y9"15 27"4113 y22"16 y9"15 6"3714 y22"16 y19"14 y34"2215 y7"9 y19"14 4"1416 y22"16 y19"14 y61"3317 y22"16 y19"14 y60"2618 y7"9 y12"13 37"2819 y7"9 y12"13 y27"1520 y7"9 y12"13 y23"23

x w x.17 , 18–20 based instead on a combination of ageŽ .and tectonic behavior, yields similar insignificant

Ž .results Table 5 , and even produces a slightly smallervariance reduction than the previous model, whichwas based on one fewer parameter. On the otherhand, when one uses the full 20-region tectonicsphere, the variance reduction associated with theprojection of the data sphere is about 20%. Thisresult by itself is not particularly surprising since onewould expect the variance reduction to increase withthe number of model parameters. However, usingthis regionalization, less than 10% of our MonteCarlo simulations result in a greater reduction invariance. While this result does not allow us to rejecta strong association between the geoid and the sur-

Ž .face tectonics defined by Mauk 1977 , the largeŽ .relative uncertainties and even differences in sign

Ž .associated with old continents Table 5 suggest thatthis association is indeed weak. In addition, there are

Ž .only three regions 8, 9, and 17 that have averagevalues that differ from zero by more than 2s .

Although there is substantial uncertainty in thepredictions of models of the contribution of otherprocesses to the long-wavelength geoid, perhaps wecould better isolate the tectosphere’s contribution by

Ž .subtracting from the observed non-hydrostatic geoidŽ .the effects of previously modeled components: 1 a

simplified representation of the upper 120 km basedon the oceanic plate cooling model and a uniform

Ž . Ž35-km-thick continental crust ls2–20 Hager,. Ž . Ž . Ž1983 ; 2 the lower mantle ls2–4 Hager and

. Ž . Ž . ŽClayton, 1989 ; 3 slabs ls2–9 Hager and Clay-. Ž .ton, 1989 ; and 4 remnant glacial isostatic disequi-

Ž . Ž .librium ls2–36 Simons and Hager, 1997 . Sepa-rately projecting each of these four contributions to

Ž .the model geoid onto A, B, C, Q, PS yields theresults given in Tables 6 and 7. Our resulting modelŽ . Ž .residual geoid, TECT-1 Fig. 5 , provides an esti-mate of the contributions to the geoid of the uppermantle structure below 120 km depth, excluding

Žsubducted slabs. For TECT-1, g f -8"4 m TablePS.6 .

ŽThe projections of TECT-1 separately onto A, B,. Ž . Ž .C, Q, PS , ABC, QPS , and ABCQ, PS lead to

reductions in variance that are listed in Table 7.From the percent of random trials that yield smallervariance reduction than that of the actual orientationŽ .confidence level , it is clear that the geoid signal


Table 6Ž .Regional averages and statistical standard errors from projections onto A, B, C, Q, PS , corresponding to contributions to the geoid from

five model geoids — each representing a separate contribution to the geoid. The bottom two represent projections of TECT-1, separately,Ž . Ž .onto ABC, QPS and ABCQ, PS

Ž . Ž . Ž . Ž . Ž .Geoid contributors g m g m g m g m g mA B C Q PS

Upper 120 km 4.3"1.0 y3.1"0.5 y6.1"1.0 3.1"0.9 3.5"0.8Lower Mantle y5"32 20"19 35"34 y76"48 34"35Slabs y11"9 y5"4 y1"11 21"10 y7"6Post-Glacial Rebound 1"0.5 1"0.3 0.7"0.5 y0.2"0.3 y3"0.5TECT-1 3"5 2"3 4"6 y1"4 y8"4

Ž .TECT-1r ABC, QPS 2.5"2 2.5"2 2.5"2 y4"3 y4"3Ž .TECT-1r ABCQ, PS 1.6"0.8 1.6"0.8 1.6"0.8 1.6"0.8 y8"4

represented by TECT-1 is, among these choices, bestrepresented by the two-region regionalization:Ž .ABCQ, PS . Although the projection of TECT-1

Ž .onto ABCQ, PS results in a variance reduction ofonly about 3%, this value exceeds those obtainedfrom almost 95% of the projections associated withrandom rotations of the data sphere. This result isconsistent with the roughly 2s result associatedwith the platform and shield signal represented in

Ž .TECT-1 Table 6 , but contrasts markedly with theŽ .results for the five-region grouping A, B, C, Q, PS ,

where the actual orientation of the data sphere ex-plains more of the variance than only 54% of therandom orientations. This apparent discrepancy arisesbecause random orientations of the other tectonicregions can ‘‘lock on’’ to regional features in thegeoid such as those associated with subduction zones,providing a better fit to the synthetic geoids globally,but not in regions spanned by the projection of PS.

Ž .At these wavelengths ls2–36 , if there were nocontribution from density contrasts at depths greaterthan 120 km, the geoid anomaly associated withisostatically compensated platforms and shieldswould be about q10 m, referenced to old oceanbasins, or 0 m, referenced to ocean crust of zero age

Ž .or to young continental crust e.g., Hager, 1983 .Our estimate of the geoid anomaly associated withold ocean basins, from the TECT-1 projections, is4"6 m, for oceans 0–25 Ma is 3"5 m, and foryoung continents is y1"4 m. Depending onwhether we take old oceans, young oceans, or youngcontinents as the reference value, our estimate of thesignal due to the tectosphere alone, correcting for theeffects of the crust, would be y22 m, y11 m, ory7 m. Because the old oceanic regions may stillhave some residual effect of subduction included intheir estimate, and because the area-weighted aver-age of young oceans and young continents is close to

Table 7Variance reductions and the corresponding confidence levels associated with the projection onto different groups of tectonic regions of fivemodel geoids — each representing a separate contribution to the geoid. Confidence level represents the percent of random trials that yield asmaller reduction in variance than that of the actual orientation of each geoid contributor

Ž . Ž .Geoid Contributor Projection Variance reduction % Confidence %

Upper 120 km A, B, C, Q, PS 80 100Lower Mantle A, B, C, Q, PS 37 88Slabs A, B, C, Q, PS 13 77Post-Glacial Rebound A, B, C, Q, PS 19 100TECT-1 A, B, C, Q, PS 4 54

TECT-1 ABC, QPS 2 78TECT-1 ABCQ, PS 3 94


zero, we retain the estimate of y8 m as the signaldue to the continental tectosphere.

5. Estimate of ≥ lnrrrrrr≥ lnns

The isostatic geoid height anomaly, dN, associ-ated with static density anomalies can be calculated

Žfor each lateral location from e.g., Haxby and Tur-.cotte, 1978 :

y2p GdNs D r z zd z 4Ž . Ž .H

g

where G is the universal gravitational constant, g isŽ .the acceleration due to gravity, and D r z is the

anomalous density at depth z. The integration ex-tends from the surface to the assumed depth ofcompensation. Assuming that Eln rrElnn is constants

within a specified depth interval, we may write thescaling there between fractional perturbations in den-sity and shear-wave velocity as:

D r E ln r Dtfy 5Ž .ž /ž /r E lnn ts

where r is obtained, for example, from the radialŽearth model PREM Dziewonski and Anderson,

.1981 , and the fractional perturbations in shear wavevelocity Dn rn are equal to the negative of thes s

fractional travel-time perturbations Dtrt , for smallperturbations. We base the subsequent calculation ona depth of compensation of 440 km. Below thisdepth, we assume that there is no platform and shieldcontribution to the geoid, as there is no significantdistinction at such depths between the shear-wavesignal beneath platforms and shields and the global

Ž .average Shapiro, 1995 .Ž .Using S12_WM13 Su et al., 1994 , we calcu-

lated regional averages of one-way shear-wavetravel-time perturbations for 100-km-thick layers be-tween 140 and 440 km depth. We then approximatethe integral of the depth-dependent density anomaly

Table 8Ž . Ž .S12_WM13 Su et al., 1994 ls1–12 : Platform and shield

averages and uncertainties corresponding to one-way S-waveŽ .travel-time anomalies Shapiro, 1995

Ž . Ž . Ž .Depth interval km Dt rt %PS

140–240 y2.3"0.2240–340 y1.6"0.2340–440 y1.0"0.2

as the sum of the anomalies for these layers. UsingŽ . Žthe travel-time perturbations Shapiro, 1995 Table

.8 and dN 'g fy7.7"3.9 m, we find that forPS PS

platforms and shields,the average value ofŽEln rrElnn is about 0.041"0.021. This estimate ofs

standard error is based only on that of dN . ThePS

uncertainties associated with the regionally averagedtravel-time perturbations have a much smaller effecton the value of Eln rrElnn than the uncertaintys

.associated with the geoid and are therefore ignored.

6. Discussion

Ž .None of the projections based on 1 the non-hy-Ž . Ž .drostatic geoid, 2 free-air gravity anomalies, or 3

our model geoid, TECT-1, yields a platform andshield signal that is significant at a level exceedingabout 2.0s. Our conclusion is in accord with that

Ž .reached by Doin et al. 1996 using a geologicregionalization based on the tectonic map of Sclater

Ž .et al. 1980 . They estimated that shields have ageoid difference from midoceanic ridges of betweeny10 m and 0 m; their corresponding estimate forplatforms, which they keep as a separate region, isy4 m to 1 m, while they found essentially nodifference in geoid for tectonically active continentalareas and ridges. They were unable to estimate for-mal errors because of the previously discussed rednature of the spectra, but these values represent theirsubjective estimates of confidence intervals. Al-though there are many differences in detail between

Ž . Ž . Ž . Ž . Ž . Ž .Fig. 5. a TECT-1, ls2–36; b Projection of a onto A, B, C, Q, PS , and c Residual: a–b . All plots are displayed using a Hammerequal-area projection with coastlines drawn in white. Negative contour lines are dashed and the zero contour line is thick. The contourinterval is 10 m.


their study and ours, their estimates fall within ouruncertainties, and their conclusion that the tecto-sphere is compositionally distinct is consistent withours.

These observations differ substantially from theŽ .highly significant 99% confidence correlation, re-

Ž .ported by Forte et al. 1995 , between an ocean–con-tinent function and the non-hydrostatic long-wave-

Ž .length ls2–8 geoid. However, a correlation coef-Ž .ficient r between different fields defined on a

Žsphere is only meaningful subject to tests of signifi-. Ž .cance for fields with significantly non-white spec-

tra if correlation coefficients are determined sepa-rately for each spherical harmonic degree of interestŽ .Eckhardt, 1984 . Given the appropriate number ofdegrees of freedom associated with the correlation,one can nonetheless estimate the confidence levelcorresponding to the assumption that the true correla-tion is zero. Therefore, we estimate the effectivenumber of degrees of freedom in the analysis of

Ž .Forte et al. 1995 and, using this value, estimate theprobability that the correlation which they obtainedis significantly different from zero.

Under the conditions outlined above, we can esti-mate the effective number of degrees of freedomusing Student’s t distribution. For uncorrelated fields,

w Ž 2 .x1r2the quantity tsr nr 1yr can be describedby Student’s t distribution with n degrees of free-

Ž .dom e.g., Cramer, 1946; see also O’Connell, 1971 .We create 10,000 degree-eight fields, each with thesame spectral properties as the non-hydrostatic geoid,by randomly selecting coefficients from a uniformdistribution and then scaling them degree-by-degreeso that the power spectrum of each ‘‘synthetic’’ fieldmatches that of the geoid. From these synthetic fieldsand an ocean-continent function derived from GTR1,we generate a collection of 10,000 correlation coeffi-

Ž .cients Fig. 6a . We then estimate n by minimizingthe x 2 in the fit of Student’s distribution to this set

Ž .of correlation coefficients Fig. 6b . Fig. 6c,d,e

demonstrate the sensitivity of the fits to the value ofn . As shown, values of n which differ from the

Ž .estimated value ns30 by even 5 degrees of free-dom, noticeably degrade the fit.

The correlation coefficient corresponding to theŽ . Ž .geoid and ABC, QPS ls2–8 is y0.18. How-

ever, using the geoid and an ocean-continent func-Ž .tion ls2–8 derived from the 58=58 tectonic re-

Ž .gionalization of Mauk 1977 , we obtained the sameŽ . Ž .value y0.28 as Forte et al. 1995 . With the

Ž .regionalization of Mauk 1977 , simulations likethose described above yield 31 as the estimate of theeffective number of degrees of freedom. The signifi-cance levels of the correlations associated with the

Ž .GTR1 and Mauk 1977 ocean–continent functionsare, respectively, about 85% and 95%. The dominantdegree-two term in the geoid governs this correlationand highlights a difficulty associated with attachingsignificance to the correlations between such fields.For example, if one considers only degrees ls3–8,the significance levels of the correlations associated

Ž .with the GTR1 and Mauk 1977 ocean–continentfunctions reduce to about 55% and 60%, respec-tively, and hence indicate insignificant correlations.

Our conclusion also differs substantially from thatŽ .of Souriau and Souriau 1983 who, using a Monte

Carlo scheme based on random rotations of the datasphere with respect to the tectonic sphere, found that

Ž .the non-hydrostatic geoid ls3–12 correlates sig-Ž .nificantly at the 95% confidence level with the

Ž .surface tectonics defined by Okal 1977 . The closegeoid-tectonic association obtained by Souriau and

Ž .Souriau 1983 is partially related to the fact that theŽ .regionalization of Okal 1977 includes subduction

zones; the association between the geoid and thisregionalization is a result of the strong geoid-slab

Ž .correlation e.g., Hager, 1984 . Unlike our study,Ž .Souriau and Souriau 1983 perform their projections

in the spatial rather than in the spherical harmonicdomain. After reproducing their results, we repeated

Ž . Ž .Fig. 6. a Histogram of correlations r between an ocean–continent function derived from GTR1 and 10,000 synthetic degree-eight fieldseach with the same spectral properties as the non-hydrostatic geoid. The shaded and unshaded arrows indicate the variance reductions

Ž . 2associated with the actual geoid and the maximum variance reduction, respectively. b x , calculated from the fit of Student’s t distributionwŽ Ž 2 .x1r2 Ž .with the set of t’s calculated from tsr nr 1yr , plotted as a function of the number of degrees of freedom n . The minimum

2 Ž .value of x corresponds with ns30. Histogram of values of t with Student’s t distribution with n degrees of freedom superposed: cŽ . Ž .ns25, d ns30, and e ns35.


their suite of projections in the spherical-harmonicdomain. We found that the correlation between the

Ž .long-wavelength geoid ls3-12 and the regional-Ž .ization of Okal 1977 is significant at about the 98%

confidence level, slightly higher than the result ofŽ .Souriau and Souriau 1983 of about 95% from a

spatial-domain analysis. However, when we substi-Žtute a slab-residual model geoid Hager and Clayton,

.1989 for the geoid, we find that the confidence levelreduces to about 50%, indicating that the signal

Ž .observed by Souriau and Souriau 1983 is largelydue to the correlation between slabs and the regional-ization.

Ž .The isopycnic hypothesis Jordan, 1988 predictsa value of zero for Eln rrElnn . This value is withins

2.0s of our estimate and indicates that at this levelof significance, the isopycnic hypothesis is consistentwith the average geoid anomaly associated with plat-forms and shields. We can also estimate the value ofEln rrElnn by considering only thermal effects ons

density:

Eln r 1rr drrdTŽ . Ž .f 6Ž .

Elnn 1rn dn rdTŽ . Ž .s s s

Using a coefficient of volume expansion of 3=10y5

y1 Ž .K , we make two estimates: 1 Eln rrElnn f0.23,s

using dn rdTfy0.6 m sy1 Ky1 from McNutt andsŽ .Judge 1990 and an average upper-mantle shear

y1 Ž .velocity of n f4.5 km s , and 2 Eln rrElnn fs sŽ . y4 y10.27, using Elnn rET fy1.1=10 K froms

Ž .Nataf and Ricard 1996 . The average of these esti-mates is inconsistent at about the 10s level with thevalue of Eln rrElnn that we estimate for the conti-s

nental tectosphere. Hence, our analysis indicates thata simple conversion of shear-wave velocity to den-sity via temperature dependence is inappropriate forthe continental tectosphere and that one must con-sider compositional effects.

Our conclusion could not differ more completelyŽ . Ž .from that of Pari and Peltier 1996 henceforth PP ,

who claim that they can rule out the hypothesis thatneutrally buoyant, compositionally distinct materialexists beneath ‘‘cratons.’’ Based on a match to the

Ž .peak amplitude of a severely truncated ls2–8Žfree-air gravity anomaly at one location Hudson

. Ž .Bay , they argue that 0.21- Eln rrElnn -0.26,s

consistent with the thermal estimate above, and in-

consistent at the 8–10s level with our estimate.However, there are several easily identifiable differ-ences between their approach and ours. Most impor-tantly, we use a geologic regionalization to definecratons. PP define ‘‘cratons’’ as any region, beneatheither continents or margins, that has high inferreddensities at 30 km depth in heterogeneity model

Ž .S.F1.KrWM13 Forte et al., 1994 . This definitionof ‘‘craton’’ is inappropriate for testing the composi-

Ž .tion of tectosphere for many reasons, including: 130 km depth beneath continents is generally withinthe lower crust, not within the proposed isopycnic

Ž .region of the continental tectosphere; 2 modelS.F1.KrWM13 is a heterogeneity model based on aweighted fit both to the gravity field and to theseismic data, assuming that density and velocityanomalies are proportional through assumed depth-

Ž .dependent values of Eln rrElnn which vary be-sŽ .tween 0.21 and 0.34 Karato, 1993 . In regions

where the seismic coverage is not good, this assump-tion introduces a strong gravitational bias into modelS.F1.KrWM13, making the use of this model in the

Ž .inversion for Eln rrElnn an example of circularsŽ .logic; 3 The use of this hybrid model fails to

identify the South African craton, a region with thickŽ . Ž .tectosphere e.g., Su et al., 1994 , as a craton. 4 PP

emphasize the value of the fit at Hudson Bay, whileour study weights all regions of the globe equally.We also note that the estimate of the amount of thispeak free-air gravity anomaly attributable to mantlestructure is suspect due to contamination from post-

Ž .glacial rebound Simons and Hager, 1997 . In sum-mary, given their approach, and their non-geologicdefinition of cratons, it is not surprising that PP find

Ž .a different value for Eln rrElnn than we do. Theirs

value applies to the mantle beneath regions of in-ferred high-density lower-crust in a model deter-mined from a joint inversion of gravity and seismic

Ž .data. Our value of Eln rrElnn applies to cratonss

defined by geological processes.In summary, to obtain realistic estimates of the

significance of correlations between data fields de-fined on a sphere requires that one consider thespectra of the data fields so that the number ofdegrees of freedom can be determined appropriately.Our analysis demonstrates that the relationship be-tween the long-wavelength geoid and the ocean–con-tinent function is tenuous. The large difference in


correlation that we obtain with different ocean–con-tinent functions further illustrates its insignificance.From error estimates that account for the redness inthe geoid, gravity field, and tectonic regionalizationspectra, we conclude that neither the geoid nor thefree-air gravity has a platform and shield signal that

Ž .differs significantly 2s from zero. AdditionallyŽ .see Shapiro, 1995 , by considering regionally aver-aged shear-wave travel-time anomalies together withour model of the continental tectosphere’s contribu-tion to the geoid, we find that Eln rrElnn is abouts

0.04"0.02. Although this estimate is consistent atthe 2.0s level with the isopycnic hypothesis of

Ž .Jordan 1988 , the slightly positive estimate suggeststhat the decreased density associated with composi-tional buoyancy does not completely balance theincreased density associated with low temperatures.We also note that convection calculations addressingthe stability and dynamics of the continental tecto-sphere indicate that Eln rrElnn is likely to varys

Žsomewhat with depth Shapiro et al., 1999; see also.Forte et al., 1995 . Thus, our estimate is a weighted

average of a quantity that may vary with position.

Acknowledgements

We thank P. Puster and G. Masters for computercode and T.A. Herring, P. Puster, W.L. Rodi, and M.Simons for helpful discussions. Richard J. O’Connellprovided a useful review. Figs. 1, 2 and 5 werecreated using the Generic Mapping Tools softwareŽ .Wessel and Smith, 1991 . We performed many ofthe calculations using the Guilford College ScientificComputation and Visualization Facility which wascreated with funds from a grant from the National

Ž .Science Foundation CDA-9601603 . This work wasalso supported by National Science Foundation grantEAR-9506427.

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